PASSIVITY-BASED ANALYSIS AND CONTROL OF
NONLINEAR SYSTEMS
a dissertation
submitted to the department of mechanical engineering
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Thomas E. Pare, Jr.
November 2000
c 2000 by Thomas E. Pare, Jr.
Copyright All Rights Reserved.
ii
I certify that I have read this dissertation and that in my opinion it is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Jonathan P. How
(Principal Adviser)
I certify that I have read this dissertation and that in my opinion it is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Stephen P. Boyd
I certify that I have read this dissertation and that in my opinion it is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Gene F. Franklin
I certify that I have read this dissertation and that in my opinion it is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
J. Christian Gerdes
Approved for the University Committee on Graduate Studies:
iii
iv
Abstract
A new set of tools for the stability analysis and robust control design for nonlinear
systems is introduced in this dissertation. The tools have a wide range of applicability, covering systems with common types of hysteresis, as well as systems with
memoryless forms such as saturation, or slope-restricted nonlinearities (e.g., parameter uncertainty or gain variation), and can be used to guarantee stability for systems
with both nonlinearities and additional norm-bounded uncertainty. These robust stability tests are developed using a combination of passivity and dissipation theories,
and are presented in both graphical (Nyquist) and numerical form using linear matrix
inequalities (LMIs). The LMI formulation yields tests that are eciently solved with
existing software packages, and allows the extension to the case of multiple nonlinearities. In particular, an asymptotic stability test is developed using LMIs for systems
with multiple hysteresis nonlinearities. The invariant set for such systems is shown
in general to be a polytopic region of the state space.
This analysis framework is extended to include LMI-based algorithms for the design of reduced order, output feedback controllers. Four new synthesis routines are
presented, each of which optimize an H1 performance metric. First, the basic full
order H1 and robust H1 design algorithms are reformulated to produce controllers
with an explicit constraint on controller order. The robust synthesis algorithm yields
controllers that give optimal closed loop L2 -gain performance for systems having norm
bounded uncertainties by performing a sequence of convex optimizations over LMI
constraints. Order reduction is accomplished by treating the controller order as part
of a multi-objective optimization. These two basic routines are then extended to solve
v
for controllers that are robust to sector bounded, memoryless and hysteresis nonlinearities. Control design for systems with sector bounded memoryless nonlinearities
with a stability guarantee based on the Popov criteria is referred to as Popov/H1 control design, and is widely known to be a nonconvex problem due to the bilinear form
of the corresponding matrix inequality constraints (i.e., BMIs). The new algorithm
presented here solves this BMI problem while yielding controllers that have order
lower than the plant, and is demonstrated to have improved convergence compared
to xed order algorithms that solve the same problem. This reduced order technique
is then adapted to produce H1 compensation for systems with hysteresis by building
upon the new robust stability criteria, and used to synthesize locally optimal controllers for systems with input saturation. Numerical examples are used throughout
the thesis to illustrate the utility of the new analysis and design algorithms. With
these developments, this research aims to broaden the application of absolute stability
and to extend robust control design to include these important classes of nonlinear
systems.
vi
Acknowledgements
My lasting memory of graduate school will be the people of Stanford I've come to
know. The professors, sta and students together combined to make the University
a truly unique learning atmosphere. Teaching, one realizes after enough years sitting
in a classroom, is the art of giving in its purest form. No more so than at Stanford,
where each class was an experience of new concepts and thought processes from an
experience of People along the way who contributed in this work:
Research advisor Jon How. Dedicated, for giving me the fundamental under-
standing of robust control theory and broadening my knowledge of system engineering. While his focus is primarily on the development of hardware projects,
he allowed me to pursue a mostly theoretical research. For this I am most
grateful, since this served to best complement my previous background and
work experience in hardware systems.
Excellent professors who gave clarity to and inspired learning the more dicult,
more abstract sciences: Stephen Boyd, Gene Franklin, Brad Parkinson, and
Bernie Roth.
Fellow classmates and collaborators: Bijan Sayir-Rodssari, Yuji Takahara, Arash
Hassibi.
Research group: David Banjerdpongchai, Heidi Schubert, Bruce Woodley, SungYung Lim, Hong Song Bae, Andrew Robertson
Support from Hughes Aircraft Fellowship program. In particular, Ronald Cubalchini, Dennis Pollet, Bernie Skehan, and Keith Yokomoto. Look forward to
vii
working again with in the future.
All those at Stanford, whom I haven't mentioned, who made the doctoral experience a memorable one.
Family who kept my perspective grounded.
Finally, to my wife, Dee Dee, who, to be sure, is the reason I try stu like this
thesis.
viii
List of Figures
1.1 The Lur'e-Postnikov system for absolute stability analysis. . . . . . .
1.2 Robust control design set-up for systems with hysteresis. . . . . . . .
2.1
2.2
2.3
2.4
Basic operator mapping , with y(t) = (x(t)). . . .
Denition of sector and incrementally sector bounded
A sector transformation. . . . . . . . . . . . . . . . .
System diagram for passivity analysis. . . . . . . . .
3.1
3.2
3.3
3.4
3.5
3.6
Typical passive hysteresis input-output relationship . . . . . . . . . .
General hysteresis operator (hysteron). . . . . . . . . . . . . . . . . .
Sector Transformation ~ 2 sector[0; 1) . . . . . . . . . . . . . . . . .
Block diagram relation of passive operator ~ , with (t) = ~()(t). . .
Transformation of hysteretic relay into passive operator. . . . . . . .
Block diagram model of backlash nonlinearity is transformed into passive operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Converting a backlash to a passive operator using generalized multiplier.
Loop transformation for absolute stability analysis. . . . . . . . . . .
Nyquist test for existence of stability multiplier. . . . . . . . . . . . .
Nominal system with nonlinearity and additional performance channel.
Example system with uncertainty and hysteresis nonlinearity. . . . . .
Nyquist plot for nominal system with superimposed uncertainty ellipses
Positive real Nyquist plot of system transformed with stability multiplier.
Typical initial condition response for system indicating robust stability.
Initial condition response with = ;2 indicates near instability. . . .
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
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3.16
3.17
3.18
3.19
Nyquist plot with uncertainty gain increased. . . . . . . . . . . . . . .
Input-output trajectories of hsyteresis near instability. . . . . . . . . .
Limit cycle indicating onset of system instability . . . . . . . . . . . .
Nyquist plot demonstrating intersection with nonlinearity describing
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.20 Hysteresis input/output with sustained oscillation. . . . . . . . . . . .
61
61
62
Sector transformation results in ~ as passive operator. . . . . . . . .
Smooth approximation for discontinuous nonlinearity . . . . . . . . .
Backlash, or play nonlinearity detail . . . . . . . . . . . . . . . . . . .
Typical Preisach hysteresis characteristic. . . . . . . . . . . . . . . . .
Nonlinear system and loop transformation. . . . . . . . . . . . . . . .
Multivariable version of Popov indirect control system . . . . . . . . .
Graphical criteria for determining invariant stationary set . . . . . . .
Block diagram depiction of main stability proof derivation. . . . . . .
Stability analysis using multipliers and transformation to Popov indirect control form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of state convergence to stationary set for system with hysteretic relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alternate view of state trajectories convergence of relay system . . . .
State convergence to polytopic stability region for multiple backlash
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alternate view of stable convergence of trajectories of multiple backlash
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
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87
93
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96
98
5.1 Block diagram framework for H1 control design . . . . . . . . . . . . 103
5.2 Robust H1 control design framework . . . . . . . . . . . . . . . . . . 107
5.3 Maximum singular value curves corresponding to various reduced order
designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4 Typical performance/controller order trade o using benchmark problem117
5.5 Root locus for systems with controllers of varying order. . . . . . . . 118
x
5.6 Robust H1 and Popov/H1 performance trade o curves using benchmark uncertainty problem . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Performance comparison of new reduced order H1 design technique to
controllers reduced by balanced real order reduction . . . . . . . . . .
5.8 Popov/H1 algorithm insensitivity to initial conditions . . . . . . . .
5.9 Alternative compensation designs achieved using design knob . . .
5.10 Convergence comparison between xed and reduced order design algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Well conditioned solution matrices lead to good convergence for reduced order algorithm . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 S=KS mixed sensitivity synthesis set up. . . . . . . . . . . . . . . . .
5.13 Robustness test for -design requires containment in unit disk. . . . .
5.14 Nyquist stability test for new control requires less conservative avoidance of restricted region. . . . . . . . . . . . . . . . . . . . . . . . . .
5.15 Loop shaping performance of , reduced and xed order controllers .
5.16 Comparison of closed loop performance with iteration of reduced and
xed order algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.17 Disturbance rejection comparison of and multiplier control design for
system with hysteresis: plant output y(t) . . . . . . . . . . . . . . . .
5.18 Disturbance rejection comparison of and multiplier control design for
system with hysteresis: control signal u(t) . . . . . . . . . . . . . . .
5.19 Hysteresis input-output: reduced order controller. . . . . . . . . . . .
5.20 Hysteresis input-output: ;controller. . . . . . . . . . . . . . . . . .
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Saturation locally sector bounded as parameterized by r.
Control system with saturation nonlinearity. . . . . . . .
Transformed system with deadzone nonlinearity. . . . . .
Deadzone nonlinearity and saturation parameter r. . . .
Inverted pendulum with disturbance. . . . . . . . . . . .
Performance and disturbance level dependence on r. . . .
Disturbance rejection vs. L2-gain performance trade-o. .
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146
152
153
160
161
162
B.1 Benchmark three mass used for algorithm comparisons. . . . . . . . . 177
D.1 Popov system with initial condition response as input. . . . . . . . . 186
xii
List of Tables
4.1 Comparison of new stability theorem to previous work on multiple
slope-restricted nonlinearities . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
Algorithm for reduced order H1 synthesis . . . . . . . .
Synthesis algorithm for reduced order, robust H1 control
Popov/H1 control synthesis . . . . . . . . . . . . . . . .
H1/hysteresis control synthesis . . . . . . . . . . . . . .
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115
119
122
123
xiv
Notation and Symbols
R
The set of real numbers
C
The set of complex numbers
R+
The set of nonnegative numbers
jR
The imaginary axis
Rm,
The set of real m-vectors
Rmn
The set of real m n real matrices
XT
The transpose of matrix X
X
The complex conjugate transpose matrix of X 2 Cmn .
Im
The identity matrix of dimension m m, or the identity operator. The subscript is omitted when m can be determined from
context.
0mn
An m n zero matrix.
X ;1
The matrix inverse, or the inverse of the linear operator, i.e.,
XX ;1 = I:
diag(A1; : : : ; Am )
Block diagonal matrix with A1; : : : ; Am on the diagonal.
(X ); (X )
maximum, minimum singular value of the matrix X:
(X )
The condition number of the matrix X .
xv
Tr X
The trace of a matrix X 2 Rmm .
X?
h
The orthogonal
complement of matrix X , i.e., XX? = 0, and
i
X X? is of minimum rank.
X > 0 (X 0)
The symmetric matrix X is positive denite (semidenite), that
is X = X T and yT Xy > 0 (yT Xy 0) for all y 2 Rn.
X > Y (X Y )
The symmetric X; Y 2 Rnn satisfy X ; Y > 0 (X ; Y 0).
E; M
Stationary set, invariant set
Co(x1 ; : : : ; xm)
Convex hull of elements x1 ; : : : ; xm
Ln2
Time domain square integrable signals in Rn.
Ln1
Time domain absolutely integrable signals in Rn.
Ln1
Time domain bounded signals in Rn.
H2
The subset of L2(j R) with functions analytic in Re(s) > 0.
H1
The set of L1(j R) functions analytic in Re(s) > 0.
"
A B
C D
#
Shorthand for state space realization C (sI ; A)B + D.
hx; yi
The inner product of x and y; e.g., if x; y 2 Rn+, then hx; yit =
Rt
T
0 x( ) y ( ) d . (Note that a subscript may be used when necessary to specify domain.)
xy
The convolution of x and y:
jj
Absolute value, or modulus of a number.
kxkp
The p-norm of a vector; typically p = 1, 2, or 1.
kX kp;i
The induced p-norm of an operator
xvi
0(a) (_ )
Derivitive of a function with respect to argument, i.e. 0(a) =
d
da (derivative with respect to time).
=s
Equivalent state space representation
=
Equal by denition
equivalent
2
Belongs to, or is a member of.
Subset
T
Intersection
9
exists
8
for all
!
an input to output mapping
)
implies, or it follows that
end of proof
LMI (BMI)
Linear matrix inequality (bilinear matrix inequality)
LTI
Linear time invariant
SISO
Single input, single output
SPR
Strictly positive real
xvii
xviii
Contents
Abstract
v
Acknowledgements
vii
List of Figures
viii
List of Tables
ix
Notation and Symbols
xiii
1 Introduction
1.1 Motivation and approach . . . . . . . . . . . . . . . . . . . . .
1.2 Research objectives . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Previous and related work . . . . . . . . . . . . . . . . . . . .
1.3.1 Robust Control Theory . . . . . . . . . . . . . . . . . .
1.3.2 Control Design for Systems with Hysteresis . . . . . . .
1.3.3 Linear Matrix Inequalities for Control System Design .
1.4 Research Contributions . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Absolute Stability Analysis for Hysteresis . . . . . . . .
1.4.2 Robust H1 Control Design . . . . . . . . . . . . . . .
1.4.3 Reduced Order Control Design . . . . . . . . . . . . .
1.4.4 Control Design for Systems with Saturating Actuators
1.5 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . .
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1
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2 Preliminaries
2.1 Linear and nonlinear operators
2.2 Linear Matrix Inequalities . . .
2.2.1 System analysis . . . . .
2.2.2 Control design . . . . . .
2.3 Signals and system norms . . .
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3 Input-Output Stability
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3.1 Introduction . . . . . . . . . . . . . . . . . . .
3.2 Hysteresis Denitions . . . . . . . . . . . . . .
3.3 Hysteresis as a Passive Operator . . . . . . . .
3.3.1 Sector Transformation . . . . . . . . .
3.3.2 Hysteresis Integral Properties . . . . .
3.3.3 Examples of Passive Hysteresis . . . .
3.4 Robust Stability Analysis . . . . . . . . . . .
3.4.1 Loop Transformation . . . . . . . . . .
3.4.2 Robust Stability . . . . . . . . . . . . .
3.4.3 Robust Performance . . . . . . . . . .
3.5 Numerical Example . . . . . . . . . . . . . . .
3.5.1 How conservative is this stability test?
3.6 Conclusions . . . . . . . . . . . . . . . . . . .
4 Multiple Hysteresis Nonlinearities
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4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Approach Overview . . . . . . . . . . . . . .
4.2 Nonlinearities and Sector Transformations . . . . .
4.2.1 Memoryless, Slope Restricted . . . . . . . .
4.2.2 Hysteresis . . . . . . . . . . . . . . . . . . .
4.3 System Description and Loop Transformation . . .
4.4 Stationary Sets and Stability Denitions . . . . . .
4.4.1 Stationary Set for Memoryless Nonlinearity
4.4.2 Stationary Sets for Hysteresis Nonlinearities
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23
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4.5
4.6
4.7
4.8
4.4.3 Denitions of Stability . . . . . . . . . . . . . . . . . . . . .
Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Passivity and Frequency Domain Interpretations . . . . . . . . . . .
Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.1 Computing the Maximum Allowed Slope for Nonlinearities .
4.7.2 Asymptotic Stability with Single Hysteretic Relay . . . . . .
4.7.3 Asymptotic Stability with Multiple Backlash Nonlinearities .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Reduced Order Control Design
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Synthesis Problem Statements . . . . . . . . . . . . . .
5.2.1 H1 Control . . . . . . . . . . . . . . . . . . . .
5.2.2 Robust H1 Control . . . . . . . . . . . . . . . .
5.2.3 Popov/H1 Control . . . . . . . . . . . . . . . .
5.2.4 H1 Control for Systems with Hysteresis . . . .
5.3 Algorithm Descriptions . . . . . . . . . . . . . . . . . .
5.3.1 Reduced order H1 design . . . . . . . . . . . .
5.3.2 Reduced Order Robust H1 control . . . . . . .
5.3.3 Reduced order Popov/H1 control . . . . . . . .
5.3.4 Reduced order H1/Hysteresis control . . . . . .
5.4 Numerical Examples . . . . . . . . . . . . . . . . . . .
5.4.1 Reduced order H1 . . . . . . . . . . . . . . . .
5.4.2 Popov/H1 convergence properties . . . . . . .
5.4.3 Robust loop shaping for system with hysteresis
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
6 Control Design for Systems with Saturation
6.1
6.2
6.3
6.4
Introduction . . . . . . . . . . . .
Problems of Local Control Design
The Design Approach . . . . . . .
System Model . . . . . . . . . . .
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6.5 Design Algorithms . . . . . . . . .
6.5.1 Stability Region (SR) . . . .
6.5.2 Disturbance rejection (DR)
6.5.3 Local L2-Gain (EG) . . . .
6.5.4 Controller Reconstruction .
6.5.5 Optimization Algorithms . .
6.6 L2-Gain Control Example . . . . .
6.7 Conclusions . . . . . . . . . . . . .
7 Conclusion
7.1 Summary of Main Results . .
7.1.1 Stability Analysis . . .
7.1.2 Robust Control Design
7.2 Future Research Directions . .
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153
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171
A Energy Storage and Dissipation Functions
175
B Benchmark Uncertainty Problem
177
C Controller Reconstruction
179
C.1 Popov/H1 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
C.2 Hysteresis/H1 Control . . . . . . . . . . . . . . . . . . . . . . . . . . 180
C.3 Region of Convergence Design . . . . . . . . . . . . . . . . . . . . . . 181
C.4 Local Disturbance Rejection Design . . . . . . . . . . . . . . . . . . . 182
C.5 Local L2-Gain Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
D Convergence Limit Proof
185
xxii
Chapter 1
Introduction
This thesis introduces a new set of tools for the stability analysis and compensation
design for nonlinear systems. In particular, for systems with hysteresis, the analytical tools provide numerical and graphical techniques to predict asymptotic stability.
These stability tests are formulated using linear matrix inequalities which allow the
extension to analyze systems with multiple nonlinearities, and to perform robust stability tests for systems with both hysteresis and bounded uncertainty. Moreover, the
analysis framework is developed to include the capability to synthesize controllers that
will guarantee stability while optimizing an H1 performance metric. This new design
approach for systems with hysteresis is a general technique that produces reduced order controllers, and can be readily adapted to other types of nonlinear systems. The
versatility of the framework is demonstrated with three new local control design algorithms for unstable plants with actuator saturation. The new routines synthesize
compensation for systems with saturating actuators that optimize disturbance rejection or H1 performance over a limited, or local, region of the system state space.
For its general and practical application, the results of this research provide valuable
new tools for the study of nonlinear control systems.
1
CHAPTER 1. INTRODUCTION
2
1.1 Motivation and approach
Stability and control of nonlinear systems has long been an important eld of engineering study, simply because most physical systems have some form of nonlinearity.
Hysteresis is such an example that occurs commonly in engineering practice, and
takes many forms. For an introduction to the notion of hysteresis, Webster's Ninth
New Collegiate Dictionary gives a rather technical denition:
hys-ter-e-sis:
n [NL, fr. Gk hysteresis shortcoming, fr. hysterein to be
late, fall short, fr. hysteros later] (1801): a retardation of
the eect when the forces acting upon a body are changed
(as if from viscosity or internal friction); esp: a lagging
in the values of resulting magnetization in a magnetic
material (as iron) due to a changing magnetizing force.
|hys-ter-et-ic adj
Basically, hysteresis represents the history dependence of physical systems, and captures the property of a system to remember its past inputs. If you push on something,
it will yield. When you release, does it spring back completely? If it does not, it is
exhibiting hysteresis, in some broad sense. The term is most commonly applied, as
Webster suggests, to magnetic materials: when the external eld generated by a write
head is removed, the magnetic prole on the hard drive does not return to its original
conguration. Of course this is by design, otherwise the data record would disappear!
The most common mechanical analogy of hysteresis is the stress-strain behavior of a
material undergoing plastic deformation.
Mathematically, hysteresis typically refers to the input-output relation between
two time-dependent quantities that can not be expressed as a single-valued function.
Instead, the relationship usually takes the form of loops that are traversed either
in a clockwise or counter-clockwise direction. Hysteresis loops are often associated
with some form of energy exchange, or energy loss. The pressure-volume relation of
a refrigeration system is an example in which the hysteresis loop (of the p-v curve)
equals the work done in the cycle. The work done is transferred to the surrounding
1.1. MOTIVATION AND APPROACH
3
environment in the form of heat [CS81], and of course, the direction of the loops is
strictly maintained, or else the system would violate the second law of thermodynamics. This unidirectionality is a fundamental property of a thermodynamic cycle,
such a process is said to be irreversible. The second law of thermodynamics governs
electro-magnetic systems as well. When the magnetic eld in a force actuator is reversed by changing the current applied to its windings, the magnetic eld remains.
Because of this, it takes an additional supply of current just to reverse the eld (in
essence, the magnetic eld can be thought of as having \memory"). On a microscopic
level, the coupling between the electric and magnetic elds creates eddy-currents in
the iron structure that completes the magnetic circuit. As the eld is varied, this current dissipates away in the form of heat. The overall eect results in hysteresis loops
in the electric-magnetic eld relation, and again, the area of these loops is directly
proportional to the energy lost to the device [May91, BS96].
Certainly, there are types of hysteresis that occur in engineering practice that
have little connection to thermodynamics. Imperfections in a gear train, for instance,
caused by manufacturing tolerances can often lead to backlash or the play nonlinearity
found in many mechanical drive systems. In this case, the area of the hysteresis loop
in the input-output graph does not have an exact energy interpretation. The same
can be said for a hysteretic relay that is part an electronic circuit. Nevertheless,
whether due to a thermodynamic law or some other physical constraint, these devices
are guaranteed to have certain input-output characteristics: e.g., an input-output
relation characterized by circulation that maintains a strict directional sense. While
easy enough to characterize, these nonlinearities can result in complex dynamics for
the systems in which they occur. A single hysteretic relay in a relatively simple
circuit, for example, can exhibit limit cycles or even chaotic behavior [NE86, Jin96].
Such phenomena is often undesirable in a control system, so that when hysteresis is
introduced into a system as part of a switching circuit, electromagnetic actuator, or
certain types of friction, a priori knowledge of the nonlinear eects is critical.
A primary goal of this research is to provide analytical tools to predict stability
for systems that have nonlinearities, and guarantee system operation free of limit
4
CHAPTER 1. INTRODUCTION
cycles or chaos. The stability tests developed here are cast as a set of linear matrix inequalities (LMIs). The feasibility of the LMIs is readily solved using widely
available software packages. The analysis problem is further extended to treat hysteretic systems that contain additional norm bounded uncertainties (either dynamic
or parametric), and posed as a convex semi-denite program. This allows the analyst
to ask basic questions such as, \How much gain variation can the nonlinear system
tolerate before going unstable?" Answering this question is referred to as robust stability analysis, and forms the basis for the robust control design problem. Thus, the
analysis for hysteretic systems is further extended in this dissertation to the synthesis
of robust controllers using an LMI framework. The essential element behind the stability criteria is based on the idea that the area contained by the characteristic loops
represents energy exchange, or energy lost to the hysteretic element. This concept
is utilized in the analysis by employing a particular transformation that converts the
nonlinearity into a passive operator. A passive operator is simply a system that has
a bounded (nite) amount of energy that can be extracted from it. In the electromagnetic case, the energy analogy is explicit; whereas, in general, the corresponding
energy terms are considered in a general sense, as is typical in Lyapunov stability
analysis. The overall approach proceeds by rst dening a class of hystereses for
which the passive transformation holds. Then, a combination of passivity, Lyapunov
and Popov stability theories are used to show that the nonlinear system must be
stable, and the state of the linear subsystem must converge to an equilibrium point of
the system. These asymptotic stability criteria are expressed as LMIs, which allows
the direct extension to robust control synthesis. Stability analysis approached in this
way in which conditions are derived for an entire class of nonlinearities is referred
to as absolute stability theory [AG64, Lef65, DV75, Kha96]. A brief background of
absolute stability approach is given below in order to place this research in the proper
historical context. This is followed by a discussion of previous work done in robust
control design, and in particular, concentrating on approaches that utilize an LMI
formulation.
1.2. RESEARCH OBJECTIVES
5
1.2 Research objectives
The main objective of this thesis is to provide a new set of robust control design
and stability analysis tools for nonlinear systems. The tools have a wide range of
applicability, covering systems with common types of hysteresis, as well as systems
with memoryless forms such as saturation, or slope-restricted (e.g., gain uncertainty
or variation) nonlinearities. In particular, the analysis tools will guarantee stability for uncertain systems with hysteresis and for systems having multiple hysteresis
or memoryless, slope-restricted nonlinearities. Stability tests presented are in both
graphical (Nyquist) and numerical form, using linear matrix inequalities (LMIs) that
are readily solved with existing software packages. This analysis framework is extended to include an LMI-based synthesis technique which produces output feedback
controllers for these nonlinear systems. Examples include controllers that optimize
an H1 performance metric while guaranteeing stability for systems with hysteresis
nonlinearity, and stabilizing controllers that achieve local L2 -gain performance for
systems with saturating actuation. The LMI synthesis technique is very ecient for
the user, requiring only a state space description of the system and various parameters that characterize the nonlinearity (maximum slopes, etc.), and will produce
controllers that are both xed and reduced order. With these developments, this
research aims to broaden the application of absolute stability and to extend robust
control design to include these important classes of nonlinear systems.
1.3 Previous and related work
The qualitative behavior of nonlinear systems having dynamics that can be modeled
as the feedback interconnection of linear and nonlinear subsystems G(s) and F (), as
depicted in Fig. 1.1, can be studied in a framework referred to as absolute stability
theory [Vid93]. The original analysis, often attributed to Lur'e and Postnikov [LP44,
Lur57], was motivated by the need to understand the eect of nonlinearities on control
systems due to elements such as imperfect actuators (e.g., d.c. motors, etc.) or sensors
that have gain or amplication that can vary over time. Within this framework
CHAPTER 1. INTRODUCTION
6
x
e
-
G(s)
y
F
Figure 1.1: Absolute stability framework for stability analysis: the Lur'e-Postnikov
system.
these nonlinearities are most commonly modeled as gain bounded, or sector bounded
uncertainties, and the stability tests are developed to guarantee the state x of the
linear system converges to the origin asymptotically, i.e., x(t) ! 0; as t ! 1:
Analysis of these systems is accomplished by extending the direct method of Lyapunov
[Lya92, Zub64] by augmenting the Lyapunov energy function with an integral of
the nonlinearity. In this way the Lur'e form of the Lyapunov function captures
the generalized total energy of the system, and stability is subsequently ensured by
guaranteeing that this energy, and thus the state, decreases asymptotically to zero.
The general solution to Lur'e problem requires the solution of a set of nonlinear
equations and methods available at the time limited practical application of the analysis to second and third order systems. A breakthrough for this problem came with
the introduction of frequency domain criteria in stability analysis by Popov [Pop61].
Incorporating Fourier integrals into the analysis removed any restriction on the order
of the system and led to his well known graphical test for stability in a modied
Nyquist plane. Popov developed tests for several dierent forms of nonlinear systems
(direct and indirect control, etc.) and these results are well documented in the early
monographs [AG64, Lef65, Cor73, NT73].
The new Popov criteria stirred a great deal of interest and progress in the eld beginning in the early 1960s. For example, equivalent algebraic conditions in the form
of frequency domain matrix inequalities were soon after developed by Yakubovich
1.3. PREVIOUS AND RELATED WORK
7
Yak:1964, Yak:1967a,Bar:1996; and similarly, Kalman showed that the Popov results correspond to the solvability of the original Lur'e equations [Kal63]. Popov
and Yakubovich provided further extensions to the case of systems with multiple
nonlinearities [Pop64, GY65]. Of course, the work of the latter three researchers is
captured by the celebrated Kalman-Yakubovich-Popov lemma, or the KYP lemma;
various forms of the lemma appear throughout the controls literature. One important feature of the KYP lemma is that it relates the internal (state) stability of a
nonlinear system to the input-output properties of its subsystems. In particular, a
linear subsystem satises the KYP lemma if and only if it is a passive, or positive
system. Popov referred to such systems as hyperstable, and generalized his original
stability theory to apply to the interconnection of two hyperstable blocks, neither of
which needs to be linear [Pop64, Pop73].
At the same time hyperstability theory was being developed in eastern Europe,
system analysis based on operator theory and functional analysis was gaining popularity in the West. Researchers such as Zames, Sandberg, and Willems approached
the stability question by viewing systems as operators which map signals from one
vector space into another [Zam63], with the most powerful results obtained when
the system is dened to operate on a Hilbert space. Sandberg introduced the idea
of an extended Hilbert space to show that feedback connections of operators could
yield stable mappings on the Hilbert space even though the subsystems themselves
were unstable, or unbounded operators [San64b, San64a]. This work of Sandberg
and Zames [Zam66a, Zam66b] ultimately resulted in the small gain theorem, which
in turn was used to prove the circle criterion [San65, Zam66b]. Zames formulated
the use of loop transforms to produce operators that satised either loop gain or
positivity conditions [Zam66a], which is equivalent to Popov's hyperstability theory. These methods formed the basis for what is known today as passivity theory.
He also introduced the use of RC and RL-type multipliers in order to strengthen
the small gain theorem, and showed that this method itself could lead directly
to the Popov criterion [Zam66a]. Multiplier methods were later generalized for
systems with memoryless nonlinearities having certain sector and slope restrictions
[BW65, O'S66, ZF68]; and subsequently, Cho and Narendra [CN68] found that the
8
CHAPTER 1. INTRODUCTION
existence of such multipliers could be established with an o-axis circle test in the
Nyquist plane. The functional analytic approach, including multipliers, passivity,
small gain theory and the relation to the Popov criterion, was later documented in
the monographs [Wil71a, Hol72, NT73, DV75].
Shortly after the work of Popov, a unifying framework incorporating passivity
and small gain concepts, referred to as dissipation theory, was developed by Willems
[Wil72b], and subsequently extended by Moylan and Hill [HM76, HM80b]. The key
idea behind this approach is that combining subsystems that absorb (or dissipate)
more energy than they produce (or supply) results in stable systems. Within this
framework, supplied energy is measured using an inner product of input and output
signals, and the subsystems are considered as operators on a Hilbert space which, in
addition, also supply, dissipate and store energy. Naturally, as one might expect, the
results using dissipation are quite similar to those using absolute and input-output
stability techniques. An advantage of this framework is that relatively complicated
systems can be described in terms of the scalar quantities that measure the energy
stored, dissipated or supplied by each of its subsystems. This allows the combination of a variety of conditions to be tested, such as worst case L2-gain across
one input-output channel, while maintaining passivity across another. The combination of diverse conditions into one analytical test has gained some recent interest in
the integral quadratic (IQC) framework [MR97, Jon98]. The IQC framework comes
equipped with a set of frequency domain multipliers that allow the user to construct
stability tests based on small gain, passivity, and the Popov criteria simultaneously,
thus combining the essential features of multiplier and dissipation theories.
Yakubovich rst applied absolute stability to systems with hysteresis nonlinearities [Yak67, BY79], using a combination of frequency domain criteria and Lyapunov
methods to derive sucient conditions for stability. In this work, Yakubovich introduced a variation of the Lur'e-Postnikov Lyapunov function which, because the
integral of the nonlinearity term is path-dependent, must take into account the circulation direction of the hysteresis loops. This approach is unique in that it utilized
a general set of properties including circulation direction and slope bounds, to dene
a class of hysteresis nonlinearities. The resulting test is a variation of the Popov
1.3. PREVIOUS AND RELATED WORK
9
test for memoryless nonlinearities and applies to a wide range of hysteresis, including
the Preisach type [May91, BS96] and backlash cases. By contrast, Lecoq and Hopkin [LH72] developed an analysis limited to a particular electromagnetic nonlinearity
called the Chua-Stromsmoe hysteresis. For this particular model, Lecoq and Hopkin
developed a positive real stability test using passivity theory that was equivalent to
the circle criteria. They further showed that when the time derivatives of the inputoutput relation of the hysteresis maintains a certain circulation direction, then the test
can be generalized to the Popov criteria. In general, however, checking the circulation
of the input-output time derivatives must be done by testing each Chua-Stromsmoe
model on a case-by-case basis, which makes the Popov test inconvenient in practice.
Safonov and Karimlou [SK83, SK84] later removed this inconvenience by showing that
a sequence of loop transformations will decompose the Chua-Stromsmoe model into
a pair of sector-bounded, memoryless nonlinearities and therefore justify the Popov
test for this particular hysteresis. The analysis by Safonov and Karimlou, however, is
limited since the constants used to parameterize the model are only valid at a particular excitation frequency. The results, therefore, are not valid on the general extended
L2 space. In more recent work, Gorbet et al.[GMW97, GM98] concentrates on the
Preisach hysteresis in connection with work involving shape memory alloys. By showing that dierentiating the output of the hysteretic relay results in a passive operator,
the Preisach model itself is ultimately passive because the relay is the model's basic
building block [May85, May91]. Although the analysis in Ref. [GMW97] does take
into account the circulation of the relay, the resulting stability test does not include
the multiplier introduced by Barabanov and Yakubovich [BY79] and is therefore not
as general. Finally, an IQC methodology was recently applied to systems with hysteresis. Rantzer and Megretski [RM96] develop frequency domain multipliers to test
stability for systems containing hysteretic relays; while Jonsson [Jon98] extends the
approach to study backlash systems that have uncertain elements.
While providing a benchmark for the later work on hysteresis, the approach by
Yakubovich involves a unique combination of Lyapunov and frequency domain inequalities that does not extend gracefully to treat more general cases, such as systems
with multiple hysteresis nonlinearities. This thesis extends the results of the previous
10
CHAPTER 1. INTRODUCTION
researchers with an analysis framework that is developed in a unique, and consistent
mathematical way. The most common forms of hysteresis (relays, backlash, etc.)
are shown to belong to a general class of nonlinearities that is dened by a set of
input-output properties; the nonlinear elements in this class are then demonstrated
to be passive operators under a particular loop transformation. Applying passivity
theory on the transformed system directly results in a stability test equivalent to that
of Barabanov and Yakubovich. More importantly, the distinct passivity approach
developed here allows several signicant extensions. First, the stability criterion is
generalized for hysteresis systems that have additional uncertain elements [PH98b],
and to the case of multiple hysteresis nonlinearities [PHH99a, PHH00]. In the latter
case, the notion of asymptotic stability to stationary sets is developed along with a
simple technique to identify the stability sets for several common types of hysteresis.
In addition, it is shown that a simple modication to the loop transformation allows
a multiplier analysis for systems with backlash, with the same generality as that developed by Zames and Falb [ZF68] for memoryless, slope restricted nonlinearities.
In each case, the stability tests are developed as a set of linear matrix inequalities,
and can thus be easily computed with existing numerical software. Lastly, the LMI
framework is used to design controllers for systems that have hysteresis using the
robust control set up depicted in Fig. 1.2. A controller, K (s), designed to optimize a
performance metric while guaranteeing closed loop stability for the nonlinear system
is referred to as an optimal, robust controller. The synthesis technique developed
in this thesis is a high level and ecient means for computing robust controllers for
these nonlinear systems. An overview of the robust control eld given next serves as
background and motivates the need for the new design method as an alternative to
existing nonrobust synthesis techniques.
1.3.1 Robust Control Theory
The early developments in stability theory formed the foundations of the system analysis and design framework referred to today as robust control [Zho98, ZDG96, SP96,
1.3. PREVIOUS AND RELATED WORK
11
p
q
G(s)
w
u
z
y
K (s)
Figure 1.2: Robust control design set-up for systems with hysteresis.
DD95]. Whereas the Lur'e system involves an isolated nonlinearity, the robust control problem typically addresses the stability and control design for systems with an
isolated uncertainty. The framework is often attributed to Zames, who introduced the
use of small gain concepts to analyze systems having uncertain elements [Zam66b],
and then later extended the results to robust control synthesis [Zam81]. In the typical
robust control setting, depicted in Fig. 1.2, the uncertainty is assumed to be unity
norm bounded, and stability is subsequently guaranteed provided the linear system
(with input-output p ! q) has H1 norm less than one. Robust synthesis is then
aimed at producing a controller K (s) to satisfy the stability constraint while possibly
optimizing additional performance metrics. The need for this type of analysis and
synthesis was highlighted by Doyle [Doy78], who showed that standard LQG control could lead to designs with innitesimally small tolerance to system uncertainty.
This approach, however, can lead to overly conservative tests, particularly when the
number of uncertainties grows, or when the uncertainty has a known structure such
as block diagonal or real parametric. To address this concern, the H1 theory was
combined with the multiplier techniques of the previous decade to result in the multivariable stability margin, Km, by Safonov [Saf82] and the structured singular value,
12
CHAPTER 1. INTRODUCTION
, by Doyle [Doy82, PD93]. These techniques employ frequency dependent multipliers that exploit the known mathematical structure of the system uncertainties and
thereby reduce conservativeness of the standard H1 theory. Sharper stability predictions were later developed for systems with mixed (real and complex) uncertainties,
known as mixed- or mixed-Km analysis [FTD91, You93, LCGS95]. While dicult
to calculate exactly, the mixed- test is accomplished by computing the corresponding upper and lower bounds, or by using techniques based on the Popov criterion or
dissipation theory developed by How and Hall [HH93b, HH93a]. The Popov analysis was further rened for the particular case when the nonlinearities are monotonic
or odd monotonic [How93, HHHB92], which applies to systems which, for example,
contain parametric uncertainty. An extension of this idea involves combining both
stability and performance into the same analytical framework. Feron and Balakrishnan [Fer94, Bal94, Bal97] used an LMI formulation to include an H2 metric along
with dynamic or parametric uncertainty in order to develop analytical tests for robust performance. Banjerdpongchai and How [BH96] and Yang et al.[YLH96] later
extended these results to design controllers that provide guaranteed closed loop H2
performance robust to parametric uncertainty.
A common element in all the stability tests discussed is that the analyses are
improved when more information about the uncertainty or nonlinearity is taken into
account. Usually such information is captured as either a norm bounded, sector
bounded, or passive property. Taking these properties into account can relax the
stability constraints, and allow for less conservative stability predictions. Naturally,
this benet is an essential feature in optimal robust control synthesis, where less
restrictive stability constraints permit controllers that can achieve better performance.
This has been the trend in robust control for the past fteen years where, since the
basic foundations of H1 were established [Dor87, DGKF89], researchers have sought
reliable analytical and numerical means to design better performing robust controllers.
The =Km-synthesis framework by Doyle and Safonov [BDG+93, CS94] has proven to
be an eective design technique for systems with complex uncertainty. This technique
utilizes multipliers that exploit the diagonal structure of the uncertainty. The design
is accomplished in an iterative fashion by rst solving for an upper bound of the
1.3. PREVIOUS AND RELATED WORK
13
structured singular value, ; over a set of distinct frequency values, and computing
the stability multipliers D(s) by curve tting the diagonal scalings to the points.
The controller K (s) for the particular interation is obtained by solving the standard
H1 problem for the system augmented with the multiplier. This method, so called
the D{K iteration, is common in practice today, but it is limited by the curve tting
procedure which can be time consuming if a dense grid of frequency points is used
and lead to large controllers when a high order curve t is required to calculate
D(s): This synthesis framework was further extended by Young [You93] to include
real complex and real uncertainties, known as a mixed- constraint. This mixed-
synthesis involves a D; G{K iteration in which the G component takes into account
the known phase (i.e., 0 or 180 degrees) of the real uncertainty, but also requires a
curve tting in the frequency domain to produce the stability multipliers.
Safonov and Chiang [SC93] eectively eliminated the curve tting requirement by
parameterizing the multipliers with a nite set of basis functions and replacing the
D; G step in the iteration with a positivity constraint on a set of xed order, diagonal
scaling matrices. An important feature of this approach is that the resulting analysis
is convex and nite dimensional. Subsequently, the mixed- synthesis problem was
recast by Goh et al.[GLTS94] as an optimization over bilinear matrix inequalities

(BMIs). While in general a dicult problem to solve directly [TO95],
the solution
to the BMI problem in practice is also obtained by using a two part optimization.
The rst step, called the analysis, requires xing the controller and optimizing over
the multipliers. This is then followed by the synthesis phase, in which the multipliers
are xed and the optimal controller is computed. The distinguishing feature of this
approach is that both the analysis and synthesis steps are optimizations over LMIs
and are thus convex programs. Convexity enables the application of very ecient
interior point algorithms and greatly simplies the design procedure by eliminating
the curve tting required using the -synthesis, and has even been shown to yield
lower bounds with no increase in controller order [GLTS94]. Perhaps the most
important advantage of a BMI approach is the exibility it provides by allowing the
engineer to solve a wide range of problems. El Ghaoui and Folcher, for example,
use a BMI synthesis approach to maximize regions of convergence in state space for
14
CHAPTER 1. INTRODUCTION
uncertain systems [EF96], while Banjerdpongchai and How, as mentioned above, use
a BMI approach to solve for parametric robust H2 controllers. As an extension of
these and other works, this thesis will show how to apply a BMI synthesis to produce
optimal controllers that are robust to hysteresis nonlinearities. This is a signicant
advancement over existing control techniques for hysteresis systems that are typically
limited to inverse and constructive design methods, and which are discussed below.
1.3.2 Control Design for Systems with Hysteresis
A natural extension of the analysis is the capability to design controllers that stabilize a particular nonlinear system. In the robust control framework, control design,
or synthesis, is based on explicit use of the stability criteria so that the resulting
compensation guarantees closed loop stability. This has been done for a variety of
nonlinear systems.
For hysteresis nonlinearities, several dierent synthesis approaches that have been
pursued. The inverse approach is a technique whereby an inverse of the hysteresis
model is used to cancel the nonlinearity of the system. Most often the nonlinearity
of concern appears in a control actuator, and the hysteretic eects are nullied by
incorporating the inverse model at the output of the controller. With the nonlinearity eectively canceled, the control design is completed by treating the modied
system as linear. Of course, design in this way can be complicated when the hysteresis involved is hard to model accurately, as is the case for electro-magnetic types
of hysteresis, and can fail (lead to instabilities) if the characteristics of the nonlinearity change over time. In order to alleviate this potential problem, the inverse
control approach was recently extended with algorithms which estimate the model
parameters on-line, so that the controller can adapt to changing plant conditions and
improve operational performance [TK96, TT98, MT99]. This technique is referred
to as adaptive inverse control. However, the inversion of a relatively simple hysteresis such as backlash requires a fair eort from the designer [TK96, chp. 2], and
indeed, inverting a more complicated form such as the Preisach model may not even
be possible [HW95, DHW96]. In any case, neither the standard nor adaptive inverse
1.3. PREVIOUS AND RELATED WORK
15
control schemes can guarantee closed loop stability or system performance. Control
design techniques are available which can address this shortcoming by drawing upon
the prior developments in absolute stability theory and robust control. For example,
the analytical work by Gorbet on the passivity of the Preisach model was incorporated into a design method that enabled the design of PID controllers [GW98] for
systems actuated using shape memory alloys. Other passivity based approaches, such
as the constructive techniques that employ backstepping or forwarding algorithms,
have the potential to apply to systems with hysteresis [SJK97]. To date, such applications have not been made, most likely because hysteresis nonlinearities do not t
the standard denition of a passive operator (i.e., memoryless, sector-bounded). The
passivity results of this research, in fact, will allow constructive methods to be applied
to systems with hysteresis. In any event, techniques such as backstepping are limited
to systems that satisfy a strict feedback or feedforward structure (see [JLK99] for a
recent exception), and assume full state feedback information is available. Indeed,
the latter condition is, in general, not true in most practical applications. This full
state feedback assumption was also used recently to derive bang-bang controllers for
systems with hysteretic actuators [Oss97]. The synthesis technique developed in this
thesis avoids this assumption, and considers the more general case of output feedback control utilizing an LMI formulation, along the same lines as that developed by
Banjerdpongchai and How for systems with parametric uncertainty.
1.3.3 Linear Matrix Inequalities for Control System Design
The popularity of LMIs as a framework to analyze the stability of uncertain systems
and to design linear robust controllers has grown rapidly over the last ve years.
Linear matrix inequalities allow the user the freedom to express diverse concepts
such as Lyapunov stability, dissipation theory, passivity and energy gain all in a
single compact notational form [BEFB94], most often as feasibility or optimization
problems. The availability of software [GNLC95, WB96] that can eciently solve for
the resulting problems has led to widespread application. Recently LMI's have found
extensive use in H1 multi-objective control design [Gah96, GA94, Iwa93, SIG97].
16
CHAPTER 1. INTRODUCTION
Most of this work involves full-order design, whereby the controllers designed have
the same order as the plant. However, it is commonly reported in these works that
the order of controller is tied directly to the rank of a certain positive matrix that
forms part of the closed loop stability guarantee. Because the rank of a matrix
is not a convex constraint, optimal control synthesis aimed specically at producing
reduced order designs has been achieved with only limited success due to the numerical
complexity introduced by the nonconvex condition [GI94, BG96]. Researchers have
since noted that replacing the rank of the matrix with its trace often leads to good
low order stabilizing controllers [Mes99, GB99].
In this thesis, the Trace() function is treated as a convex (actually linear) relaxation for the matrix rank to develop new robust control algorithms for systems
with nonlinearities and uncertainties. In particular, the Trace() is used to include
controller order as an explicit component of several new multi-objective design algorithms which allow the user to trade o closed loop performance against controller
size. First, the basic full order H1 and robust H1 design algorithms are reformulated
to produce controllers with an explicit constraint on controller order. The robust synthesis algorithm yields controllers that give optimal closed loop L2-gain performance
for systems having norm bounded uncertainties by performing a sequence of convex
optimizations over LMI constraints. These two basic routines are then extended to
solve for controllers that are robust to sector bounded, memoryless and hysteresis nonlinearities. Control design for systems with sector bounded memoryless nonlinearities
with a stability guarantee based on the Popov criteria is referred to as Popov/H1
control design, and is widely known to be a nonconvex problem due to the bilinear
form of the corresponding matrix inequality constraints (i.e., BMIs). The new solution to this BMI problem presented here is a reduced order alternative to recent
xed order approaches [Ban97, BH97a]. A main benet of the reduced order BMI
synthesis over existing techniques is in improved numerical reliability. A common step
in all LMI-based output feedback control design algorithms is the controller reconstruction, which requires a matrix inversion. In general the associated matrix can be
poorly conditioned and, in extreme cases, completely prevent a design solution. The
approach presented in this thesis systematically isolates and removes the subspace
1.4. RESEARCH CONTRIBUTIONS
17
that has small aect on the overall design, resulting in matrices that are reduced in
size and readily inverted. Another advantage, of course, it that the controllers are
of reduced order which is advantageous when real-time computational resources are
limited.
1.4 Research Contributions
The main results of this thesis are the development of stability analysis and control
synthesis techniques for nonlinear and uncertain systems. In doing so, this research
leverages many of the advances made over the last decade in the eld of absolute
stability and robust control theory. While much of this literature has focused on systems with uncertainties that can be modeled as either memoryless (e.g., parametric)
or norm bounded and linear time invariant (LTI), more complicated eects such as
hysteresis have been neglected. Thus, many of the analysis and synthesis techniques
available today that utilize software packages that solve convex programs via interior
point algorithms were not applicable to this class of systems. In addition, existing
synthesis approaches for these systems can lead to xed order formulations that are
dicult to solve numerically. This thesis bridges this gap by developing new analytical techniques for systems with hysteresis and saturation nonlinearities. Further,
a new synthesis framework is presented that produces reduced order controllers by
solving a set of well conditioned LMI problems. The main contributions are further
described below.
1.4.1 Absolute Stability Analysis for Hysteresis
This thesis introduces a new framework, based rmly in absolute stability theory, for
the study of hysteretic systems. By employing a unique approach in which hysteresis
nonlinearities are transformed into passive operators, a passivity based analysis is
developed for the treatment of this important class of nonlinear systems. The new
framework provides both graphical and numerical means to test for stability. For
systems with a single nonlinearity the graphical test is a simple variation of the
18
CHAPTER 1. INTRODUCTION
familiar Popov test, which is carried out using a modied Nyquist plot. A simple
frequency domain graph of the transformed linear subsystem gives a guarantee of
stability provided the curve avoids a certain restricted region in the Nyquist plane.
Equivalently, the test can be performed numerically by solving for the feasibility of
a set of linear matrix inequalities, which are a function of the state space matrices
representing the subsystem. The LMI test is a convenient alternative since there are a
wide range of software packages available that can eciently solve the LMI feasibility
problem. Moreover, as shown in this thesis, the new stability framework allows for
numerical extensions of the simple analysis to tackle several signicant engineering
problems.
The robust analysis of uncertain systems with hysteresis is solved for using
dissipation theory, and then cast as a convex programming problem over a set
of linear matrix inequalities. Under the typical assumption of norm bounded
uncertainties, solution of the convex optimization problem enables the analyst to
assess the level of uncertainty that is tolerable while still being able to guarantee
system stability.
New absolute stability criteria for systems with multiple hysteresis nonlineari-
ties are given in this thesis. This new result extends the passivity based solution
for the scalar case by augmenting feasibility LMI set with an additional residue
matrix inequality that must be satised. For systems satisfying the stability
criteria, the system state is guaranteed to converge asymptotically to a stationary set rather than to the origin, which is characteristic of systems containing
multi-valued nonlinearities. Because it is often important to not only determine
stability but also to predict the asymptotic behavior of the state, mathematical
descriptions of the (asymptotic) stationary sets corresponding to typical types
of hysteresis (relay, backlash, etc.) are provided in detail.
For the backlash hysteresis, the stability result is further extended to a multiplier analysis of the same form and generality as that developed by Zames for
monotonic, memoryless nonlinearities. Thus, the framework incorporates an
even broader class of systems for a very common type of nonlinearity.
1.4. RESEARCH CONTRIBUTIONS
19
Connections to related work are made throughout this thesis. In particular, the basic
analysis stability result for the scalar system essentially provides a passivity interpretation of the work by Yakubovich. This is important since it demonstrates how a
new perspective on earlier work can lead to signicant advances in the theory. In this
case, the passivity conditions expressed in terms of linear matrix inequalities resulted
in the multivariable extension, and allowed the development of a new robust control
design capability, as is outlined below. As another example, because slope restricted
memoryless nonlinearities can be thought of as a special case of hysteresis, the new
stability criteria developed for multiple hystereses can apply to the memoryless case
as well. In so doing, the new results serve to generalize recently published results
[HK95, PBK98] by providing less restrictive criteria, and thus broadening the class
of systems that can be studied.
1.4.2 Robust H1 Control Design
A new procedure for the design of robust controllers for systems with hysteresis is
introduced in this thesis. Developed as a direct extension of stability analysis, this
synthesis method utilizes an LMI framework to produce controllers that are guaranteed to stabilize the nonlinear system while optimizing an H1 performance metric.
In contrast to previous design procedures that assume full state information, this new
approach requires the less restrictive, and more realistic case of output feedback for
control. That is, the new technique requires only partial state information, and that
the typical assumptions of controllability and observability conditions hold. Similarly, while existing passivity-based constructive techniques, such as backstepping or
forwarding, are only applicable for systems with a particular structure (e.g., upper
triangular), and are limited in practice to relatively low order systems, this LMI synthesis is a high level state space solution that allows control design for systems of any
order or structure with the same relative ease. Thus, as a general design tool for systems with hysteresis, this new synthesis technique represents an important extension
of robust control theory to include this important class of nonlinear systems.
20
CHAPTER 1. INTRODUCTION
1.4.3 Reduced Order Control Design
While there has been much work done in recent years on optimal robust control synthesis, most frameworks produce compensators that are full order. To a large extent,
reliable algorithms that produce reduced order controllers, that is controllers that
have smaller dimension than the plant, are still needed in the engineering community. Direct reduced order design involves a nonconvex constraint that corresponds to
the rank of a certain matrix in the LMI formulation. This inherent nonconvexity has
been a primary challenge for control designers. To address the problem, this thesis
introduces three new LMI-based algorithms for producing reduced order robust H1
controllers. In each case, the optimization is accomplished by utilizing a Trace()
objective as a convex relaxation for the rank constraint. One algorithm simply optimizes the H1 performance subject to an explicit constraint on the order. This basic
routine is then extended to synthesis robust H1 and Popov/H1 compensators by
using a two part objective involving the Trace() and the closed loop performance.
Using the combined objective, the designer is then free to select the relative weighting on the two parts of the objective cost in order to trade-o performance versus
controller order. In this way, this algorithm provides a valuable tool that allows control designers to perform a multi-objective design analysis in the practical situations
when performance is critical and the order of the controller must be reduced because
of either real-time control hardware limitations or the excessive order of the plant.
Popov/H1 controllers are designed to optimize an H1 performance metric while
guaranteeing stability based on the Popov criteria. Technically, the solution for this
controller is bilinear in the multiplier and controller parameters, and hence is a BMI,
as opposed to an LMI problem. While this problem has been previously solved
[BH97b], the new reduced order synthesis is potentially more reliable since the new
algorithm systematically eliminates poorly conditioned subspaces of matrices which
could otherwise lead to numerical instability in the reconstruction of the controller.
A simple numerical example is used to illustrate a case in which a xed order algorithm fails due to poor conditioning while the new reduced order algorithm converges
reliably, from a wide range of initial conditions.
1.4. RESEARCH CONTRIBUTIONS
21
1.4.4 Control Design for Systems with Saturating Actuators
Lastly, this research oers new techniques for control design for unstable systems with
actuators that are subject to saturation. For this common situation, standard techniques are fundamentally limited and the saturation can often lead to sensitive closed
loop systems that are driven to instability by relatively small external disturbances.
Compensator designs in this case are considered local, since for a system with actuator
saturation stabilization can only be guaranteed for a limited region of the state space.
In this thesis three new design algorithms are presented which enable the designer to
optimize performance of a system with limited actuation. In all cases, the stability
analyses are based on the Popov stability criterion and, in keeping within the context
of the thesis, the synthesis routines are given in terms of LMI/BMI algorithms that
oer a systematic way to achieve three critical performance metrics through output
feedback for this important class of nonlinear system.
By utilizing a performance metric proportional to the volume in state space for
which stability can be guaranteed, the rst algorithm will produce compensation
that maximize the regions of attraction for the closed loop system. In this case
the controller is designed to guarantee stability while allowing for the largest
range of initial conditions.
The second routine is specically aimed at reducing closed loop sensitivity to
external disturbances. The resulting controllers optimize disturbance rejection
capability by maximizing the allowed energy of external disturbances that are
applied to the plant.
Building on the disturbance rejection objective, the last algorithm oers the
designer optimal L2 -gain across a performance channel for disturbances of a
specied energy level. It is shown in this research that the optimal disturbance
rejection and L2-gain performance are competing objectives, and in practice,
it may be desired to trade o between the two metrics in order to accomplish
the nal design. In this way, the two algorithms together provide a unique
advantage for the design of saturating controllers.
22
CHAPTER 1. INTRODUCTION
Some of the contributions described above have either appeared in preliminary
form or have been accepted for publication in the controls literature. The scalar
robust stability analysis can be found in Ref. [PH98b], and the extension to the case
of multiple hysteresis nonlinearities published in [PHH99a], with a more thorough
treatment to appear in [PHH00]. Control design for hysteresis systems was introduced
in [PH98a]. The reduced order control algorithms were rst documented in [PH99a],
while the results on local control design for systems with limited actuation can be
found in [PHHB98, PHH99b]. Additional publications on robust control design that
have appeared can be found in [PH99b, FPPH98].
1.5 Thesis organization
The mathematical preliminaries are followed with the robust stability analysis in
Chapter 3; various examples of common hysteresis and numerical examples are used
to demonstrate the utility of the new results. Stability criteria for multiple nonlinearities are developed in Chapter 4 along with the concepts and denitions for the
associated stationary sets. Included there is related analysis for sector bounded, memoryless nonlinearities, which in a sense, can be treated as special cases of hysteresis.
Using parallel developments for the two nonlinearities, the new analysis is shown to
generalize recent results for this class of systems through the use of numerical examples. Chapter 5 describes the reduced order control algorithms for the basic H1,
Popov/H1 , and for systems with hysteresis; while Chapter 6 describes local control
design for systems with saturating actuators. A summary of the results along with
ideas for future study are then provided in Chapter 7.
Chapter 2
Preliminaries
This chapter provides some denitions of nonlinear operators, and basic concepts from
stability theory and linear matrix inequalities that will be referred to later. Most of
the notation is standard, however, hysteresis nonlinearities are only described here in
connection to passive operators and details of specic forms delayed until Chapters 3
and 4.
2.1 Linear and nonlinear operators
An operator can be considered as simply a mapping between an input and an
output, each dened in an appropriate vector space, is written as : X ! Y , as
depicted in Figure 2.1. The notation y(t) = (x(t)) will be used to specify the output
value of at a particular time t, while, more generally, y = x will denote the
output signal in the given vector space. For example, if : X ! Y , then we write
y = x 2 L2.
A multiple input, multiple output (MIMIO) linear system G : Rm ! Rl , maps m
real inputs to l outputs. When G(s) = C (sI ; A);1 B + D, the state space realization
will be denoted as
"
#
A
B
G =s
:
(2.1)
C D
23
CHAPTER 2. PRELIMINARIES
24
x
y
Figure 2.1: Basic operator mapping , with y(t) = (x(t)).
An operator, , with y = (x), is L2-stable if for some 0; > 0
hx; xiT + kxT k; 8T 0;
where the subscript T on the signal is used to denote truncation:
(
xT =
x(t) 0 t T
0 else.
The minimum for which the inequality holds is the L2-gain of .
A passive operator , with y = (x), satises
hx; yiT ;; 8T 0;
(2.2)
for some 0, where the inner product is dened as
hx; yiT =
Z
0
T
xT y dt:
Also, an operator is called strictly passive if for some > 0; 0
hx; yiT ; + hx; xiT
(2.3)
is satised 8T 0 [DV75, p. 173].
A sector bounded nonlinearity is one for which the characteristic remains contained
in the input-output plane sector dened by the half-planes
h
;k1 1
i
"
#
"
#
h
i
x
0; and ;k2 1 x 0:
y
y
A nonlinearity is said to lie in sector [k1; k2], or simply 2 sect[k1 ; k2] if it satises the half plane conditions. Sector bounded nonlinearities are commonly depicted
2.1. LINEAR AND NONLINEAR OPERATORS
y
(x)
k2
25
y
k1
k1
x
(a)
(x)
k2
x
(b)
Fig. 2.2: (a) Sector bounded, and (b) incrementally sector bounded nonlinearities.
graphically, with the input-output characteristic lying between the lines y = k1 x and
y = k2x, as shown in Figure 2.2a. Similarly, if at any point the local slope 0(x)
is bounded between k1 and k2, the nonlinearity is said to be incrementally sector
bounded. This condition is expressed 0(x) 2 sector[k1 ; k2]. Incrementally sector
bounded is a stronger condition than sector bounded. This fact is utilized later in
Chapter 4. The operator is memoryless if the output y(t) = (x(t)) depends only on
the input x at time instant t, and not on the time history of x(t); t 2 [0; t].
A nonlinearity 2 sector[k1; k2] can be characterized by its center c, and radius
r, as
c = 21 (k1 + k2 )
(2.4a)
r = 12 (k2 ; k1);
(2.4b)
assuming k2 > k1: Using these denitions, we can dene an transformation that will
convert a nonlinearity with center and radius (c; r) to one with arbitrary (~c; r~). A
transformation that accomplishes this is depicted in Figure 2.3, and is parameterized
by the pair (a; b):
a = r~=r
b = c~r=r~ ; c;
(2.5a)
(2.5b)
CHAPTER 2. PRELIMINARIES
26
x
y
a
y~
b
Figure 2.3: A sector transformation.
and results in a new operator ~ , with y~ = ~(x), given by:
y~ = a((x) + bx):
(2.6)
For example, a gain bounded memoryless uncertainty 2 sector[;1; 1] can be converted to ~ 2 sector[0; 1] by selecting a = 1=2 and b = 1: This transformation is used
in Chapter 5 as part of the control synthesis algorithm for systems with parametric
uncertainties. Memoryless operators that are sector bounded [0; 1] are passive, by
the denitions given above, since their input-output pairs are positive. That is,
2 sect[0; 1] ) (x)x > 0 ) is passive,
with the constant = 0:
A hysteresis is a functional mapping with memory operating on a particular input
vector space. Hysteresis, in general, is not passive by the denitions given above since
as depicted in Figure 3.2, for example, it may violate the sector conditions. In the
sequel a hysteresis will be expressed as : L2e ! L2e such that for y = (x) we have
y(t) = (x([0; t]); y0):
For simplicity of notation, we will often drop the dependence on initial conditions,
and write y(t) = (x)(t).
2.2. LINEAR MATRIX INEQUALITIES
27
2.2 Linear Matrix Inequalities
A linear matrix inequality (LMI) is a matrix inequality of the form
F (x) = F0 +
m
X
i=1
xiFi > 0
(2.7)
where x 2 Rm is a vector of free parameters and Fi = FiT 2 Rmm ; i = 1; : : : ; m
are constant matrices particular to the given problem. Basic optimization problems
expressed as LMIs are solvable using a wide range of software packages [GNLC95,
WB96]. The use of LMIs has become widespread in system analysis and control
because conditions for Lyapunov stability conditions, passivity, dissipation and small
gain can be concisely written in LMI form and numerically solved. Below are some
concepts from stability theory expressed as LMIs that are used in this thesis.
2.2.1 System analysis
Lyapunov Stability
A basic LMI appearing in stability analysis is due to Lyapunov. For a linear system
with a state space dynamic representation
x_ = Ax; x(0) = x0
will have asymptotic stability, x(t) ! 0, if and only if there exists a matrix P =
P T > 0 such that
AT P + PA < 0:
(2.8)
It is straightforward to show that the inequality (2.8) can be expressed in the basic
form (2.7), with the entries of P comprising the free parameters in (2.8), and the
matrices Fi; i = 0; : : : ; m dened by the system matrix A: For example, with the
matrices in the Lyapunov inequality (2.8) given as
"
#
"
#
1 0:5
p p
A=
; and P = 1 2 ;
3 7
p2 p3
CHAPTER 2. PRELIMINARIES
28
u1
e1
y1
H1
-
y2
e2
H2
+
+
u2
Figure 2.4: System diagram for passivity analysis.
then the equivalent expression for (2.7) becomes F (p) = F0 +
"
#
"
P3
#
i=1 pi Fi
< 0; with
"
#
2 0:5
6 8
0 3
F0 = 0; F1 =
; F2 =
; and F3 =
:
0:5 0
8 1
3 14
Small Gain
In addition to internal stability expressed by (2.8), system input-output properties
are easily expressed in LMI form. The system (2.1) is said to be non-expansive, and
satises the small gain condition:
kGk1 (2.9)
if and only if it satises the dissipation inequality [TW91]:
d V (x(t)) ;2uT u ; yT y
(2.10)
dt
where the storage function V (x(t)) = xT Px with P = P T > 0. This equivalently
results in the LMI
"
#
AT P + PA + C T C PB + C T D
0:
B T P + DT C
DT D ; ;2 I
(2.11)
Passivity
A fundamental result from absolute stability theory, passivity is used throughout this
thesis to develop new stability criteria for nonlinear systems. The theorem refers to
2.2. LINEAR MATRIX INEQUALITIES
29
the Lur'e type system depicted in Figure 2.4, and described by the equations:
y1 = H1e1 = H1 (u1 ; y2)
y2 = H2e2 = H2 (u2 + y1):
(2.12a)
(2.12b)
Given below is one particular passivity theorem; for other versions see [NA89],[Vid93],
or [VDS00]. This form appears in the stability analysis in Chapters 3 and 4.
Theorem 2.2.1 (Passivity) If in Eqn. (2.12), u2(t) 0, H1 is passive, H2 is
strictly passive, and u1 2 L2 , then y1 2 L2.
Proof: Refer to [DV75]. In the sequel, the subsystem H2 is a stable LTI system G(s)
(2.1), and its passivity tested with the feasibility of an LMI analogous to that for
the small gain test. The system is strictly passive if and only if G(s) satises the
dissipation inequality [TW91]:
d V (x(t)) uT y ; uT u;
(2.13)
dt
where the storage function is V (x(t)) = xT Px with P = P T > 0, and some > 0.
This leads directly to the LMI
"
#
AT P + PA PB ; C T
0:
B T P ; C I ; (D + DT )
(2.14)
Of course, for a given system, the passivity and small gain inequalities, (2.14) and
(2.11), respectively, can be expressed in the standard form (2.7).
2.2.2 Control design
The following two results are often used for compensation design using linear matrix
inequalities. The Elimination Lemma is used to remove the controller parameters
from the LMI expressions for closed loop stability, while the Completion Lemma is
employed in the controller reconstruction. Both are used for reduced order control
design in Chapter 5.
CHAPTER 2. PRELIMINARIES
30
Lemma 2.2.2 (Elimination) Let G 2 Rnn, U 2 Rnp and V 2 Rnq . We dene
U? to be an orthogonal complement of U . Similarly, V? is the orthogonal complement
of V . There exists a matrix X 2 Rpq such that
if and only if
G + V X T U T + UXV T < 0;
(2.15)
V?T GV? < 0; U?T GU? < 0:
(2.16)
For detailed proof see, for example [BGFB94, pages 32{33].
Lemma 2.2.3 (Completion) Let P and Q 2 Rmm be positive denite matrices.
There exists a positive denite matrix P~ 2 R2m2m such that the upper left m m
block of P~ is P , and that of P~ ;1 is Q if and only if
"
#
P I
0:
I Q
(2.17)
See [PZPB91] for proof.
For each pair of matrices P and Q that satisfy (2.17), the set of matrices P~
satisfying the conditions of the Completion Lemma is parameterized by
"
I 0
P~ =
0 MT
#"
P
I
I (P ; Q;1 );1
#"
I 0
0 M
#
(2.18)
where M 2 Rmm is an arbitrary invertible matrix. Then,
"
I 0
Q~ = P~ ;1 =
0 NT
#"
P
I
I (Q ; P ;1);1
#"
I 0
0 N
#
(2.19)
where N = (I ; QP )M ;1 .
2.3 Signals and system norms
Throughout the dissertation various forms of system stability will refer to stable
mappings between vector spaces. For example, as previously mentioned, the system
2.3. SIGNALS AND SYSTEM NORMS
31
P depicted in Fig. 2.1 is L2-stable if it maps inputs x 2 L2 to outputs y 2 L2. A
signal x : R+ ! Rn belongs to the vector space L2 if it has nite 2-norm, dened as:
kxk2 =
n
1X
Z
0
i=1
!1=2
xi (t)2 dt
=
n
X
i=1
kxi k22
!1=2
:
(2.20)
Other useful norms appearing in the sequel are the 1-norm, given as
kxk1 =
n
1X
Z
!
kxik1;
(2.21)
kxk1 = 1max
kx k = sup 1max
jx (t)j:
in i 1
in i
(2.22)
0
i=1
jxi(t)j dt =
n
X
i=1
and the 1-norm for a vector, dened by
t0
32
Chapter 3
Input-Output Stability
There has been extensive work done in recent years on the analysis and synthesis of
systems having memoryless, sector bounded nonlinearities and uncertainties. In this
chapter a fundamentally dierent approach is taken to develop tests of the stability
of systems with hysteresis nonlinearities which, in general, have memory and are not
sector bounded. Using an operator perspective, and considering a hysteresis that
obeys a strict circulation direction, a transformation is developed which converts
a hysteresis nonlinearity into a passive operator. In the passivity framework, this
transformation leads directly to a stability multiplier of the same form investigated
in earlier work by Yakubovich. The main stability theorem then provides a simple
Nyquist test (for a SISO system) or a linear matrix inequality (LMI) which is extended
to include a provision for a robust performance test. A simple numerical example
it then used to illustrate the benet of the multiplier introduced for this class of
nonlinearities.
3.1 Introduction
Hysteresis is a property of a wide range of physical systems and devices, such as
electro-magnetic elds, mechanical ball bearings, and electronic relay circuits. The
term hysteresis typically refers to the input-output relation between two time dependent quantities that can not be expressed as a single-valued function. Instead, the
33
CHAPTER 3. INPUT-OUTPUT STABILITY
34
Passive hysteresis (Preisach−type)
1
0.5
y (output)
Major loop
0
Minor loop
−0.5
−1
−3
−2
−1
0
x (input)
1
2
3
Fig. 3.1: Typical passive hysteresis input-output relationship
relationship usually takes the form of loops that are traversed either in a clockwise or
counter-clockwise direction. A hysteresis with counter-clockwise loops is sometimes
referred to as a passive hysteresis [HM68, p. 366]. A particular example of a passive
hysteresis is depicted in Figure 3.1. This nonlinerity could represent the relationship
between the electric and magnetic elds of an electro-magnetic actuator, and is used
later in x3.5 to illustrate the robust stability analysis developed in this chapter.
The area enclosed by the loops is often thought of as representing energy loss into
the hysteretic element [May91, p. 44]. Because this phenomenon is so prevalent, it
is important to be able to predict its eect on systems in which it occurs. Early
stability formulations for linear systems with hysteretic nonlinearities were done by
Yakubovich [Yak67, BY79], who used a Lyapunov approach to arrive at a test for
stability, similar to the Popov criterion, involving a multiplier of the form ( + s)=s.
3.1. INTRODUCTION
35
More recently, Jonsson suggested using this multiplier for stability analysis of systems
with hysteresis in an integral quadratic constraint (IQC) setting [Jon98], with the rst
numerical results illustrating the utility of the multiplier appearing in [PH98b], which
served as a preliminary version of this chapter. This particular multiplier was also
discussed in [Kap96] for use in determining stability of systems with sector and slope
restricted nonlinearities. Other recent work [GMW97] considered the time-derivative
of the output of the nonlinear element as the basis for guaranteed stability of systems
with Preisach-type hysteresis. Stability of systems with frictional hysteresis using
passivity concepts was also investigated in [dWOA93], where a certain product of
variables in the nonlinear model was shown to represent a passive operator.
A hysteresis function is said to have memory since its output at any given time
depends on the entire history of the input signal, and its input-output relation is often
not sector-bounded because of the characteristic loops, as depicted in Figure 3.1.
Because of these two properties, much of the work done in recent years in robust
analysis, with the exception of work cited, is not applicable to systems with hysteresis
nonlinearities. For instance, parameter uncertainty is typically considered in the class
of sector-bounded, memoryless perturbations (see [Ban97] and references therein, for
examples).
This chapter investigates the robustness analysis of systems which have nonlinearities that are described by a passive hysteresis. The approach diers from previous
work by taking a distinct operator perspective. In particular it is shown that, under
the proper transformation, the passive hysteresis becomes a passive operator. This
transformation establishes the mathematical connection between the passive hysteresis as an energy absorbing element and the property of a passive operator as having
bounded extractible energy. The connection is proven using only the basic integral
properties of the hysteresis (e.g., counterclockwise circulation). This transformation
then allows the problem to be cast into a passivity framework where a form of the
Passivity Theorem [DV75] is used to guarantee the L2-stability of systems containing these nonlinearities. The main stability theorem leads to a simple graphical test
in the Nyquist plane (for the SISO case) or to a particular linear matrix inequality
(LMI) which can be readily extended to systems with multiple hystereses. While the
36
CHAPTER 3. INPUT-OUTPUT STABILITY
work in [GMW97] also took a passivity approach to study the stability of systems
with the Preisach nonlinearity, the test developed here is less conservative because
the analysis captures more information about the behavior of the nonlinearity. The
result is then extended to include a test for robust performance by using dissipation
theory [Wil72b]. This again is expressed as an LMI, which is readily solved using
existing optimization codes. Lastly, an example is given which illustrates the utility
of the stability theorem.
The chapter is organized as follows. Section 3.2 denes passive and active hystereses; section 3.3 details the properties of the passive hysteresis, and the transformation
which will make the hysteresis a passive operator. In section 3.4, the transformation
and the Passivity Theorem are used to develop the robust stability tests. Section 3.5
presents a simple numerical example to illustrate the benets of the analysis herein.
3.2 Hysteresis Denitions
Passive hysteresis P : L2e ! L2e, having the input/output property with loops
that are traversed counter-clockwise. As a result, for bounded input x([t1 ; t2]), for
some A 0,
Z x(t2 )
;
y dx ;A:
x(t1 )
Similarly, an active hysteresis, A has loops that are strictly clockwise, and thus
will have path integral with the lower bound
Z x(t2 )
x(t1 )
y dx ;A:
Note that this convention of active and passive hystereses as tied to the sense of
the loop circulation is taken from the early reference [HM68, p. 351], but has by
no means become standard. However, as shown in the following section, it is the
counter-clockwise circulation of the passive hysteresis that allows it to be converted
to a passive operator by using a particular transformation.
3.3. HYSTERESIS AS A PASSIVE OPERATOR
37
3.3 Hysteresis as a Passive Operator
(x)
m3
m2
m1
x
Fig. 3.2: General hysteresis operator.
Hysteresis is a property of a wide range of physical systems and devices, such
as electro-magnetic elds, mechanical stress-strain elements, and electronic relay circuits. In general, the memory and looping characteristics can be quite complicated,
and adequate models of these eects often require the composition of many basic hysteretic elements, called hysterons [KP89]. A typical hysteron, with counter-clockwise
input-output circulation is depicted in Fig. 3.2. In order simplify the development, a
set of properties is given which will limit the analysis to a particular class of hysteresis. These properties naturally characterize the basic hysteron in Fig. 3.2, and the
class dened is general enough to include many models that occur in practice, such
as the hysteretic relay, backlash, and Preisach hysteresis [BS96]. In the next sections,
a sector transform and the properties of particular scalar hystereses are detailed, and
the class of hysteresis nonlinearities is dened using these properties.
As discussed in the introduction, the area enclosed by complete cycles in the graph
of an input/output relation for a hysteresis is often thought of as representing some
measure of energy exchange. For the case of a passive hysteresis, the area enclosed
by the loops has a negative value, and thus results in the area constraint given in
the denition above. Physically this means that energy is owing into the hysteretic
element, and so a passive hysteresis is said to essentially absorb energy over the long
term. Similarly, the denition of a passive operator expresses a bound on energy
CHAPTER 3. INPUT-OUTPUT STABILITY
38
exchange. For a passive operator the constraint is on the time integral of the product
of the input and output variable. When the input and output variables are power
conjugates, the integral is thought of as specifying a constraint on the total energy
that can be extracted from a system across the input and output junction. Since the
concepts of a passive hysteresis and a passive operator are so closely related, it is
natural to expect a mathematical link between the two. This section provides such a
link.
Below a simple sector transformation commonly used in stability analysis is dened which converts a nonlinearity with nite sector width to one with innite width.
This transformation is then used with a set of integral properties to dene a class of
hysteresis nonlinearities; elements of this class are then shown to be passive under a
transformation which incorporates a modied Popov multiplier. Examples of some
common hystereses which belong to this class are then given.
3.3.1 Sector Transformation
Using the approach of [ZF68, NT73], we note that a nonlinearity with local slope
conned to a nite sector can be converted to a nonlinearity with innite sector
width. The transformation requires a positive feedback around the nonlinearity, as
depicted in Figure 3.3.
y
+
(y)
1=
Fig. 3.3: Sector Transformation ~ 2 sector[0; 1)
Lemma 3.3.1 (Finite/Innite Sector Transform) A slope restricted function :
R ! R with (y)=y 2 sector[0; ) under positive feedback with gain 1=, as
3.3. HYSTERESIS AS A PASSIVE OPERATOR
39
depicted in Figure 3.3, is is converted to a nonlinearity ~ : R ! R with the innite
slope bounds satisfying ~i()= 2 sector[0; 1).
Proof: See [NT73, pp. 108{109].
The eect of this transformation on the hysteron in Fig. 3.2, for example, is to alter
the segment slopes so that the steepest leg, with slope , becomes vertical, and the
legs m1 , m2 , and m3 increase in slope as well. Note, however, that the circulation
direction under this transformation remains unchanged. A simple consequence of the
slope restricted sector condition is that the time derivitives of the relation are always
in phase. That is,
_ (t)~_ (t) 0;
(3.1)
which means, simply stated, that the input-output pair of signals always increase or
decrease together in time.
3.3.2 Hysteresis Integral Properties
Prop. 1 Non-local memory. Unlike memoryless nonlinearities, hysteresis output at any
given time is a function of the entire history of the input, and the initial condition of the output, 0. Considering a general hysteresis as the mapping between
continuous signals, : (R; C (0; t)) ! C (0; t), an output signal w(t) can be expressed as
w(t) = (0; x([0; t))
= [x; 0 ](t):
(3.2)
(3.3)
To simplify notation, we will drop the explicit dependence on 0 below.
Prop. 2 Causality, time invariance and rate independence. The hystereses considered
are causal and time-invariant operators, as given by the standard denitions
Note that the sector is half-open, and essentially does not include innity. More precisely, the
transformation should have positive feedback of 1=( ; ), where 0 < : This is the approach
taken in Ref. [ZF68], and likewise, we assume this adjustment is included in the sector transform,
but for simplicity this will not be expressed explicitly.
CHAPTER 3. INPUT-OUTPUT STABILITY
40
[DV75]. They are also rate-independent, which essentially means that the inputoutput relation, as depicted on a graph such as Fig. 3.2, is unchanged for an
arbitrary time scaling of the input function, such as changes in the frequency
of cycling.
Prop. 3 Counterclockwise circulation. Closed loops that occur on the input-output characteristic are strictly counterclockwise. That is, a periodic input x(t), with
period T > 0, will result in a closed curve relation
Z
T
0
x(s)[x]0 (s)ds
=
=
Z t+T
t
I
x(s)w0(s) ds
w(t+T )
w(t)
x(s) dw(s) 0;
(3.4a)
(3.4b)
with equality achieved, for the backlash example, when x(t) remains in the
backlash deadzone. The value of the integral (3.4), when the path is closed, is
equal to the area enclosed by the hysteresis loop. For partial, unclosed loops,
the integral represents the area between the path traversed and the hysteresis
output axis (i.e., -axis in Fig. 3.2).
Prop. 4 Positive Path Integral. Let be the intersection of the output -axis and the
hysteresis characteristic curvesy . For any input-output path
; = f(x(t); w(t)) j t 2 [0; T ]g ;
R
originating in , the path integral ; x dw is non-negative. That is, if x(t); t 2
[0; T ] with x(0) = 0 generates the path ;, joining points p 2 and some
arbitrary b, we have
Z
T
0
x(s)[x]0 (s) ds =
Z
T
0
x(s)w0(s) ds =
Z
x dw 0:
;!
p
(3.5)
b
Similarly, now let ; denote the path joining any two points on the hysteresis
graph, and note that this path may involve many complete cycles, as in (3.4)
y For the unit relay, this set consists of two points: = f(0; 1); (0; ;1)g, for the backlash and
Preisach models, is the corresponding line segment on the -axis.
3.3. HYSTERESIS AS A PASSIVE OPERATOR
41
above. Let ;ab denote the shortest path joining the two points a and b, not
containing any complete cycles. Assuming ; results from input x(t); t 2 [0; T ]
and taking a third point p 2 , we have that
Z
0
T
x[x]0 (t) dt
=
Z
T
Z0
x(t)w0 (t) dt
x(t) dw(t)
;Z
ab
= ;
x(t)x(t) dw(t) +
;!
p
a
; (x(0); 0)
where
(3.6a)
(x(0); 0) =
Z
;!
p
(3.6b)
Z
x(t) dw(t)
;!
x(t) dw(t) 0:
p
b
(3.6c)
(3.7)
a
The rst inequality (3.6b) holds from the circulation condition (3.4), while the
second inequality (3.6c), and the positivity of is a result of (3.5).
Prop. 5 Finally, we require that the above Properties 3 and 4 hold when the nonlinearity
is sector transformed in accordance with Lemma 3.3.1. In essence it is required
that, under this transformation, the new hysteresis maintains the circulation
and positivity properties, and satises the slope condition: ~0() 2 [0; 1].
Remarks: While this collection of integral properties may seem restrictive, they can
be used to form a quite general set of hysteresis nonlinearities that includes many
common forms, including the relay circuit, backlash and Preisach hysteresis. The
properties are used below to form a class of nonlinearities, and in fact, these types of
hysteresis are shown in x3.3.3 to belong to the class. Note that properties (3.3.2.3{
3.3.2.5) express lower bounds on these integral expressions, which will be instrumental
in satisfying the desired passivity.
CHAPTER 3. INPUT-OUTPUT STABILITY
42
Class of Hysteresis Nonlinearities
Using the set of properties for hysteresis nonlinearities above, the following class, or
set, is now readily dened. We dene h, a hysteresis class as:
8
9
>
>
is dierentiable a.e. in R
>
>
<
=
h = > : R ! R 0 (y)=y < :
(3.8)
>
>
>
:
has Properties 3.3.2.1{3.3.2.5 ;
The set, or class, h consists of hysteresis nonlinearities that are locally slope bounded
(wherever the nonlinearity is dierentiable) and conforming to the properties detailed
in the previous section, such as counterclockwise rotation, positive area integrals, etc.
Lemma 3.3.2 (Passive Operator, Scalar case) Consider the hysteresis nonlinearities, : (R; C (0; t)) ! C (0; t), in the class dened (3.8). Then the input-output
relation of the new operator ~ h dened with (t) as the input to ~ h and output
(t) = dtd h(), the time derivative of sector transformed hysteresis h() (as depicted in Figure 3.4) is passive for all 0.
Proof: For all T 0, the sequence of inequalities hold:
Z
T
0
Z
T
( + d ) d ~() dt
dt dt
0
Z T
dtd ~() dt
0
dt =
= = Z
T
Z0
Z;
;
(
(3.9a)
(3.9b)
w0(t) dt
(3.9c)
(t) dw(t)
(3.9d)
(t) dw(t)
ab
= ;
Z
(t) dw(t) +
;!
p
(3.9e)
((0); w(0)) = (y(0); 0) = p
Z
b
(t) dw(t) 0;
;!
p
(t) dw(t)
;!
a
; ((0); w(0))
where
)
Z
a
(3.9f)
(3.9g)
(3.10)
3.3. HYSTERESIS AS A PASSIVE OPERATOR
1
+s
y
+
43
(y)
s
1
Fig. 3.4: Block diagram relation of passive operator ~ , with (t) = ~()(t).
according to properties 3.3.2.3{3.3.2.5 of the class. Hence, the input-output relation
is passive, by the denition given in [DV75, p. 173].
The rst inequality (3.9a) above is a result of the slope restricted property, given
by (3.1), while (3.9d) is the path integral equivalent to the integral over time (3.9c).
Inequality (3.9e) is a consequence of positive area of enclosed paths caused by counterclockwise circulation, per property 3.3.2.3. Finally, positivity of is a result of
the positive path integral relationships described by property 3.3.2.4, and the given
assumption 0.
Notes:
The denition of a passive operator typically uses a lower bound = 0 (see
[GMW97] or [Vid93, p 352] for examples), probably because it is most commonly
used in conjunction with sector-bounded nonlinearities. Here we require the
more general denition ( 0) essentially because we are considering with
nonlinearities that have memory and are not sector bounded. The constant in (3.10) is sometimes given the interpretation of the maximum energy that can
be extracted (available energy) from the nonlinear operator with a given set of
initial conditions [Wil72b].
While the list of properties given above may appear overly restrictive, many
common hysteresis have these properties. Application of the properties and
Lemma 3.3.2 to common types of hysteresis are provided below.
CHAPTER 3. INPUT-OUTPUT STABILITY
44
σ
1
τ+s
x
y
s
ξ
σ
Hysteretic
Relay
1
τ+s
x
ξ
b: Tranformed Hysteretic relay
a: Hysteretic relay with transformation
Figure 3.5: (a.) Hysteretic relay under transformation and (b.) Transformed relay
with dierentiated input-output relationship
3.3.3 Examples of Passive Hysteresis
Here we give three examples of the usefulness of the transformation given above in
converting a passive hysteresis into a passive operator.
Hysteretic Relay
The most simple example of a hysteresis with the properties above is the hysteretic
relay common in electronic switches, shown in Figure 3.5. It has only two (stable)
output states, and transitions from the low state to the high state only when the
input is increasing, and similarly only transitions from the high to the low when the
input is decreasing. Clearly, the maximum slope = 1, and as a result the feedback
channel in the transformation (Figure 3.4) is shut o, resulting in the operator shown
in Figure 3.5a. Absorbing the dierentiator into the operator results in a nonlinearity
characterized by impulse-like functions, as depicted in Figure 3.5b. The signicance
is that this operator is now sector bounded [0; 1) and is thus a passive operator.
A similar result was found in [GMW97], using integral properties of the relay in
conjunction with the Preisach hysteresis model.
The inclusion of the Popov-like multiplier 1=( + s) at the input maintains the
passivity of the net operator ~ : ! since, referring to Figure 3.5b, we have that
Z
T
0
dt =
Z
Z
T
0
0
T
(x + x_ )dt
xdt
_
3.3. HYSTERESIS AS A PASSIVE OPERATOR
=
Z
x(T )
x(0)
45
dx 0:
The Backlash Nonlinearity Model (Netushil) [Net73]
The backlash, or play-type operator (see Figure 3.6a) is another example of a hysteresis
belonging to the class (3.8) which can be transformed into a passive operator as
detailed by Lemma 3.3.2. Using the mathematical representation [Net73, pp.475{476]
shown in Figure 3.6b, it is readily seen that under the transformation, the backlash
becomes a memoryless, sector bounded operator as depicted in Figure 3.6c. Then, of
course, having the Popov multiplier at the input to the sector bounded nonlinearity
maintains the passivity of the input-output relation, by the same reasoning given for
the hysteretic relayz.
Passivity using Generalized Multipliers
The analysis for the backlash nonlinearity can be extended to include a generalized
multiplier of the form originally formulated by Zames and Falb [ZF68] for the analysis
of systems with memoryless, slope-restricted nonlinearities. The benet of the more
general form is that it provides additional degrees of freedom that can ultimately yield
less conservative stability analyses for the corresponding class of nonlinear systems.
The new multiplier for backlash systems has the form 1+s1;z(s) , where the additional
term z(s) is taken to be an LTI system with impulse response properties:
kz(t)k1 1
z(t) 0
z_ (t) 0:
(3.11a)
(3.11b)
(3.11c)
To show passivity of the transformed backlash depicted in Fig. 3.7, we use explicit
characteristic equations instead of the Netushil block diagram model. For this we
describe input-output behavior of a backlash (Fig. 3.6a) by two modes of operation,
z For an alternate proof that the backlash satises the conditions of Lemma 3.3.2 not involving
the block diagram model, see Appendix A.
CHAPTER 3. INPUT-OUTPUT STABILITY
46
x
D
-
y
1
s
-D
y
µ
1
µ
-D
D
(b.) Netushil backlash model
x
σ
(a.) Backlash nonlinearity characteristic
1
τ+s
ξ
-D
D
(c.) Transformed backlash model
Figure 3.6: The transformation, as detailed in Lemma 3.3.2, converts backlash into
passive operator: (a.) Backlash nonlinearity; (b.) Block diagram representation of backlash and (c.) Transformed (passive) operator.
as either tracking or in the deadzone, for which we dene:
(
y_ > 0; y = (x ; D) or
(3.12)
y_ < 0; y = (x + D);
Deadzone: y_ = 0 jy ; xj D;
where 2D is the deadzone width and is the slope of the tracking region, as indicated
in Fig. 3.6. Using the relations (3.12), it is straightforward to show
Tracking: y_ = x_
= Dj j = Djx_ j
(3.13)
Using these relations we can now assert the passivity of the transformed backlash in
Fig. 3.7 with the following lemma.
Lemma 3.3.3 If is a backlash with input-output characterized by (3.12), then ~
with = ~( ) as depicted in Fig. 3.7, with 0, and z(t) characterized by (3.11),
is a passive operator.
3.3. HYSTERESIS AS A PASSIVE OPERATOR
1
;z(s)+1+s
x
+
47
y
s
1
Figure 3.7: Converting a backlash to a passive operator using generalized multiplier.
Proof: Since, from Fig. 3.7,
then
=
1
;z(s) + 1 + s
(t) = ; (t) z(t) + + _ (t);
and the input-output pair satises
h; iT = ;h z; iT + h ; iT + h_ ; _iT
;h z; iT + h ; iT
(3.14)
(3.15)
(3.16a)
(3.16b)
with the inequality due to the slope restricted property of the nonlinearity. Now using
the backlash model (3.12)
h; iT ;h z; iT + h ; iT
= h(x ; 1 y) z; iT + h(x ; 1 y); iT
= ;Dhz; iT + DkT k1
;Dkzk1 kT k1 + DkT k1
= D(1 ; kzk1 )kT k1 0:
(3.17a)
(3.17b)
(3.17c)
(3.17d)
(3.17e)
The transformed backlash operator is therefore passive by the denition (2.2). Note
that in simplifying from (3.17b) to (3.17c) either positive or negative tracking conditions can be used, either of which leads to the nal result.
48
CHAPTER 3. INPUT-OUTPUT STABILITY
The Preisach Hysteresis Model [May91]
The Preisach model also satises the properties 3.3.2.1{3.3.2.5 of the hysteresis set h.
The model depicted in Figure 3.1 is used extensively to model the memory associated
ferro-magnetic materials. This operator is essentially constructed with a bank of the
hysteretic relay elements shown above with varying input switch points and output
weightings [May91, BS96]. As such, the basic property of the relay, (i.e., that the
total output increases only with increasing input, etc.) is maintained. Therefore,
under the transformation dened, the Preisach hysteresis will behave as a passive
operator. The exception here is that, with a large enough bank of relay elements, the
input-output characteristic is smooth and the eective slope is no longer innite. For
example, the hysteresis shown in Figure 3.1 was constructed with = 1.
3.4 Robust Stability Analysis
Employing Lemma 3.3.2 and the transformation depicted in Fig. 3.4, passivity arguments can be used to analyze the stability of systems containing passive hystereses.
Here using standard passivity techniques, we will apply a loop transformation [Vid93,
pp. 224{5] to incorporate the transformation and then derive constraints on the linear
portion of the system to guarantee robust stability. These constraints will be in the
form of a linear matrix inequality (LMI), which is then extended to include bounds
on a separate performance channel.
3.4.1 Loop Transformation
We assume that the total system to be analyzed has a nonlinearity, , that appears in
a feedback conguration with a linear system G(s), as depicted in Figure 3.8a, below.
Introducing the transformation, as shown in Figure 3.8b, results in the system with
modied feedforward and feedback elements, ~ and G~ (s). Of course, if the original
was a passive hysteresis with the properties 3.3.2.1{3.3.2.5 given above, then ~
is a passive operator, according to Lemma 3.3.2. An essential feature of the loop
transformation in Figure (3.8b) is that the input-output property of the net system
3.4. ROBUST STABILITY ANALYSIS
x
u
-
49
y
G
a. Linear system G(s), with nonlinearity .
u
+s
u
e
-
1
+s
x
+
1
q
+s
+
y
s
y_
~
G
1
s
1
~
G
1
s
y
p
b. Transformed system G~ (s), with passive operator ~ .
Figure 3.8: a. Block diagram of original system; and b. transformed system with
G~ (s) and passive operator ~ .
CHAPTER 3. INPUT-OUTPUT STABILITY
50
from u to y is unchanged. Therefore, any stability conclusions we make regarding the
transformed system will be applicable to the original system. We note here that if
the original system G : e 7! y has a state space representation
"
#
A B
G(s) =s
;
C D
then a representation for the transformed system G~ : p 7! q is given as
2
3
B~p
A~
5;
G~ (s) =s 4
C~q1 + C~q2 D~ qp
where
and
"
A 0
A~ =
C 0
h
C~q1 = C 0
i
#
"
(3.18)
#
B
B~p =
;
D + 1=
h
C~q2 = 0 1
i
D~ qp = [D + 1=] :
(3.19)
(3.20)
3.4.2 Robust Stability
Here we use a form of the Passivity Theorem to derive the positive-real constraint
on the linear portion of the system to guarantee the L2 -stability of the closed loop
system.
Theorem 3.4.1 The feedback relation Rfb : u 7! y_ shown in Figure 3.8 is L2 -stable
if, for some > 0, we have the three conditions:
1. The feedback element 2 h , i.e., is a passive hysteresis with properties
3.3.2.1{3.3.2.5,
2. The linear system G~ : p 7! q is dissipative with respect to the supply rate:
r(p; q) = pT q ; pT p;
and
(3.21)
3.4. ROBUST STABILITY ANALYSIS
51
3. u_ 2 L2 .
As a result, we will have that y(t) ! yss, where the steady state value jyssj < 1.
Proof: First, note that by condition (3) we have that if u 2 L2 , then u 2 L2, where
u = u + u_ . Then by condition (1) we have that if 2 h , then by Lemma 3.3.2, ~ ,
is passive; i.e.,
he; ~ eiT ;;
for some > 0, where, from Figure 3.8, e = u ;q. Next, let V (x) be a storage function
[Wil72b] for the system G~ . Then by condition (2) we have that, with reference to
Figure 3.8b,
dV dt
=
) V (x(T )) ; V (x(0)) =
qT p ; pT p
pT (u ; e) ; pT p
hu; piT ; hp; eiT ; hp; piT
hu; piT ; h~ e; eiT ; hp; piT
kuT k2kpT k2 + ; kpT k22
) kpT k22 ; kuT k2 kpT k2 V (x(0)) + :
Completing the square on the left hand side yields, after some simplication
1=2
1
V
(
x
(0))
+
1
2
+ 42 kuk2
kpT k2 2 kuk2 +
h
i
1
1
=
2
) kpT k2 kuk2 + ((V (x(0)) + )) ;
and, hence, the feedback relation Rfb : u 7! p is L2 -stable. It follows then since
p = y_ 2 L2, as t ! 1 we have y_ ! 0. We conclude then that y(t) ! yss, where
steady state yss is bounded.
Notes:
1. Convergence to a bounded steady state value is consistent with the ndings in
[Yak67], where the set of steady state values is dened as the intersection of
CHAPTER 3. INPUT-OUTPUT STABILITY
52
the line y = G(0)e and graph of the hysteresis characteristic (see also [Jon98]).
The idea of stability to stationary sets is developed in greater detail in the next
chapter, for the case of multiple nonlinearities.
2. Dissipation with respect to the supply rate (3.21) is equivalent to the strict
passivity of the linear system, as discussed in x2.2.1. Thus, we have that G~
is strictly passive, and ~ is passive in order to guarantee stability{a common
condition for the Passivity Theorem [Vid93, p. 350].
3. The last inequality clearly shows the L2-gain of Rfb : u 7! p is 1= and the
\bias" is due to terms involving the initial stored energy in the linear system G~
and the hysteresis, ~ .
4. Having u; u_ 2 L2 means that u belongs to a Sobolev vector space [NS82, p. 281].
LMI Test for Robust Stability
We note here that the dissipation condition (3.21) can be expressed as a set of linear
matrix inequalities. By letting x be the state of system G~ : p 7! q and the storage
function V (x) = 21 xT Px where P = P T > 0, then we have
~ + B~pu)
pT q ; pT p ; xT P x_ = pT ((C~q1 + C~q2)x + D~ qpp) ; pT p ; xT P (Ax
"
#T "
#"
#
x
;
A~T P ; P A~ (C~q1 + C~q2 )T ; P B~p
x
1
= 2
;
p
p
()T12
D~ qp + D~ qpT ; 2I
where ()T12 simply denotes the transpose of the (1; 2)-entry of the matrix. Assuming
that conditions 1 and 3 of Theorem 3.4.2 hold, an equivalent test of stability for the
closed loop system is the feasibility of the set of matrix inequalities:
"
> 0; > 0; P > 0
#
;A~T P ; P A~
(C~q1 + C~q2 )T ; P B~p
0;
(C~q1 + C~q2 ) ; B~pT P D~ qp + D~ qpT ; 2I
(3.22)
where the last inequality in (3.22) enforces the dissipation constraint, and is equivalent
to the strict passivity inequality (2.14).
3.4. ROBUST STABILITY ANALYSIS
Imag
53
G(j!)
G(1)
G(0)
;1
Real
Restricted
region
Stability boundary
Figure 3.9: Nyquist test for existence of > 0
Graphical Test for Stability
An equivalent condition for the strict passivity of an LTI system, H (s) is that H (j!)+
H (j!) > 0; 8! 2 R ([Vid93, p. 223]), which is equivalent to H (s) being strictly
positive real. For the SISO system G~ qp = +s s (G(s) + 1=), where the original system
G 2 RH1, it is straightforward to show that we can test for the existence of a 0
that will satisfy the strict passivity of G~ qp using the Nyquist plot of the original
system. In particular, we have then that 9 0 for which G~ qp is positive real if the
graph of G(j!); ! 0 does not enter the portion of third quadrant of the Nyquist
plane to the left of the point (;1=; 0). This graphical test is depicted in Figure 3.9.
(See [Kap96] for further discussion in using this graphical test in conjunction with
sector bounded, slope restricted nonlinearities.)
However, there are some obvious cases for G 2= RH1 that will require = 0 in
order to have G~ qp strictly passivex . For example, consider G(s) with a single pole at
zero, and assume we can expand G(s) so that
G(s) = Rs0 + Gr (s)
x The set of rational transfer functions F (s) such that sup
kF (s)k < 1 is the space RH1
(see [CGL97], for example).
Re(s)
0
CHAPTER 3. INPUT-OUTPUT STABILITY
54
w
z
G(s)
p
q
Figure 3.10: G(s) with performance channel and nonlinearity.
where R0 > 0 and for the reduced system Gr we have jGr (0)j < 1. Then for strict
passivity of G~ we require that 8! 0,
+ 1)(G(j!) + 1=)g =
Ref( j!
; !2 R0 + RefGr (j!)g + ! ImfGr (j!)g + 1= > 0:
Since here jImfGr (j!)gj and jRefGr (j!)gj are bounded as ! ! 0, the term ; !2 R0
will dominate in the limit. In fact, by construction, we have
jImfGr (j!)gj ! 0
as ! ! 0. Therefore, we must have = 0 in order to satisfy the passivity requirement.
In that case the multiplier reduces to identity, and we are left with
RefGr (j!)g > ; 1 8! 0;
which is simply a positive real test, for guaranteed stability.
3.4.3 Robust Performance
We can extend the analysis techniques for robust stability to include performance as
well. Consider the system G(s) with a performance channel from w to z, as shown in
Figure 3.10.
We can simultaneously satisfy a norm bound constraint on the performance channel while guaranteeing L2-stability by simply augmenting the supply rate (3.21) with
3.5. NUMERICAL EXAMPLE
55
a term corresponding to the system performance (see [BGFB94, p. 124] for a similar
example). With the supply rate then given as
r(p; q; w; z) = 2 wT w ; zT z + pT q ; pT p;
(3.23)
we can then bound the L2 performance by solving the optimization problem,
minimize 2
subject to: ; ; > 0
M 0; P > 0
where
2
(3.24)
3
;A~T P ; P A~ ; C~zT C~z C~qT1 + C~qT2 ; P B~p ; C~zT D~ zp ;P B~w ; C~zT D~ zw 7
6
M = 64
()T12
(D~ qp + D~ qpT ) ; 2I ; D~ zpT D~ zp D~ qw ; D~ zpT D~ zw 75 :
T D
~ zw
()T13
()T23
2 ; D~ zw
The robust stability test given in (3.24) is used as the basis for control design later
in x5.2.4. Next, a simple example is used to illustrate its use for stability analysis.
3.5 Numerical Example
Here we will use the analysis given above to test the robust stability of a system
with a passive hysteresis and plant output multiplicative uncertainty. Consider the
nominal system, G0 (s) shown in Figure 3.11, with passive hysteresis, , in the feedback channel, and a norm bounded uncertainty, . Using standard practice we let
2 where = f 2 RH1 : kk 1g{. Our approach is to minimize the upper
bound on , which is the L2-gain from w to z. If this upper bound is less than unity,
then we can conclude that the system will be stable 8 2 . We will then examine
the conservativeness of the upper bound by considering particular that exceed the
norm bound.
Consider the case with G0(s) and the weighting function, W (s), given as
2
G0(s) = 7:5 s3s +;2s02:2+s +2s0+:11 ;
W (s) = 0:97 s +s 10 :
{ For examples and further detail, see [ZDG96, Chpt. 9] or [SP96, Chpts. 7{8].
CHAPTER 3. INPUT-OUTPUT STABILITY
56
z
p
r=0
-
w
W (s)
G0(s)
q
Figure 3.11: Nominal plant G0 (s), with uncertainty , and hysteresis .
and the nonlinearity is a Preisach type with = 1, as shown in Figure 3.1. A
corresponding state space representation for G(s) is given as:
2
2
3
A Bp Bw 7
s 6
6
G(s) = 4 Cq Dqp Dqw 75 =
Cz Dzp Dzw
6
6
6
6
6
6
6
6
6
6
4
;2 ;2 ;1
1
0
0
7:5
7:5
0
0
0
0
1
0
0
0
0 ;10
;1:5 0:75 ;9:7
;1:5 0:75 0
3
1 0 7
0 0 77
7
0 0 77
:
0 1 77
7
0 0:97 75
0 0
Augmenting the system with the stability multiplier, using the system representation
(3.18{3.20), gives:
2
A~
6
G~ (s) =s 64 C~q1 + C~q2
C~z
B~p
D~ qp
D~ zp
B~w
D~ qw
D~ zw
3
7
7
5
3.5. NUMERICAL EXAMPLE
2
=
6
6
6
6
6
6
6
6
6
6
6 h
6
6
6
6
6
4
;2 ;2 ;1
0 0
1 0
0
0 0
0 1
0
0 0
0 0
0 ;10 0
7:5 ;1:5 0:75 ;9:7 0 i
7:5 ;h1:5 0:75 ;9:7i 0
+ 0 0 0 0 1
7:5 ;1:5 0:75 0 0
57
1
0
0
0
1
0
0
0
1
0:97
1
0:97
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
:
With the augmented system, solving the optimization problem (3.24) (using [WB96],
for example) yields the minimized upper bound, opt = 0:99, which indicates robust
stability over the set of dynamic uncertainty . The corresponding stability (multiplier) parameters are = 1:129, = 2:097, and = 1:99e;9. This stability condition
is consistent with the Nyquist plot, and uncertainty ellipses, of the nominal plant
G0, shown in Figure 3.12, avoiding the restricted region of the lower left quadrant.
~ B~p; C~q1 + C~q2 ; D~ qp),
As expected, the Nyquist plot of the augmented system, (A;
shown in Figure 3.13, is strictly positive real, and a typical initial condition response
for the case = ;1, Figure 3.14, indicates that the system is robustly stable.
3.5.1 How conservative is this stability test?
Note that without the stability multiplier, ( = 0), the stability test reduces to that
considered in [Jon98] and in this case the analysis would fail for this system with
= 1. Thus, for this system we have established a less conservative test than that in
[Jon98].
However, this test is still conservative to a certain degree. Consider two additional
cases for 2= . For = ;2, the initial condition response, Figure 3.15, indicates
stability which is consistent with the Nyquist plot, Figure 3.16, which clearly passes
the stability test. At this level, we see that while the system is \active" in the
sense that the Nyquist plot extends into the third quadrant, the passive hysteresis
is able to absorb energy at a rate fast enough to maintain stability. This is seen
graphically by the input/output graph, Figure 3.17, which displays the hysteresis
CHAPTER 3. INPUT-OUTPUT STABILITY
58
Nyquist for Gqp(s)
5
4
3
imag
2
1
0
−1
−2
Stability boundary
−3
−2
−1
0
1
2
3
4
5
real
Figure 3.12: Nyquist plot of G0(s) with uncertainty ellipses.
Nyquist of Gqp(s) with multiplier (positive real)
10
5
0
imag
−5
−10
−15
−20
−25
−30
0
5
real
10
15
Figure 3.13: Nyquist plot of G~ qp, = 1:129, is positive real.
3.5. NUMERICAL EXAMPLE
59
Initial condition response, stable case (Delta=−1)
0.5
0
−0.5
q
−1
−1.5
−2
−2.5
0
5
10
15
20
25
time
30
35
40
45
50
Figure 3.14: Typical stable initial condition response for sytem with = ;1.
loops decreasing in area. Increasing the perturbation to = ;3, however, does cause
instability, as indicated by a sustained 1:3 Hz oscillation depicted in Figure 3.18. At
this perturbation level, the Nyquist plot extends well into the third quadrant, and
intersects the describing function for the nonlinearity, as seen in Figure 3.19. The
intersection of the Nyquist and describing function plots does predict a sustained
oscillation at this frequency, (see [Coo94, p. 66])k. As the Nyquist plot indicates that
the system at this level of perturbation is very active, we expect that at the limit
cycle, the passive hysteresis is absorbing energy at a rate equal to the rate at which
the system is producing it. The input/output diagram, Figure 3.20 shows the steady
state orbit of the nonlinearity, and the area enclosed by the graph is a measure of the
absorption rate.
In this example, the describing function analysis is less conservative than the
Nyquist test since, as shown in Figure 3.19, the stability region boundary completely
contains the describing function graph. It should be noted however, that, unlike
k Note that there are two points of intersection, but at 1:3 Hz the intersection occurs with an
angle and direction that means sustained oscillation is more likely [Coo94, p. 66].
CHAPTER 3. INPUT-OUTPUT STABILITY
60
Initial condition response, stable case (Delta=−2)
8
6
4
q
2
0
−2
−4
−6
0
5
10
15
20
25
time
30
35
40
45
50
Figure 3.15: Initial condition response with = ;2 indicates near instability.
the multiplier analysis, the describing function test does not necessarily guarantee
system stability. That is, because the describing function is only a frequency domain
approximation of the nonlinearity (typically rst or second order), the closed loop
system may be unstable even though no intersection with the Nyquist plot occurs.
In addition, the passivity-based analysis provides a direct extension to robust control
design. Robust control design for systems with hysteresis based on this multiplier
stability analysis is developed in x5.3.4.
3.6 Conclusions
In this chapter we have investigated the stability of systems with hysteresis nonlinearities. By restricting our attention to hysteresis that has strictly counter-clockwise
circulation we motivated the use a a particular transformation which converts this
nonlinearity into a passive operator. In particular, for the backlash case, a straightforward extension of the approach results in a stability multiplier with the same
generality as that developed by Zames for memoryless nonlinearities. The transformation is subsequently used in a passivity framework to develop a stability theorem
3.6. CONCLUSIONS
61
Perturbed plant, (Delta=−2), and describing function
5
4
3
imag
2
Nominal
Perturb.
1
0
−1
−2
−3
Describing function
−4
−3
−2
−1
0
1
2
3
4
5
real
Figure 3.16: Nyquist plot with = ;2 still avoids stability boundary.
Input−Output response of nonlinearity, (Delta=−2)
1.5
1
0.5
(q) 0
−0.5
−1
−1.5
−6
−4
−2
0
q
2
4
6
8
Figure 3.17: Input/output graph with = ;2 shows decreasing hysteresis loops.
CHAPTER 3. INPUT-OUTPUT STABILITY
62
Initial condition response, sustained oscillation (Delta=−3)
8
6
4
2
q
0
−2
−4
−6
0
5
10
15
20
25
30
time
35
40
45
50
Figure 3.18: At = ;3, system becomes unstable, with onset of limit cycle.
Perturbed plant, (Delta=−3), and describing function
5
4
3
imag
2
Nominal
Perturb.
1
0
−1
−2
−3
−4
−3
−2
−1
0
1
2
real
3
4
5
6
Figure 3.19: Nyquist plot with = ;3 crosses over boundary, intersects describing
function.
3.6. CONCLUSIONS
63
Input−Output response of nonlinearity, (Delta=−3)
1.5
1
0.5
(q) 0
−0.5
−1
−1.5
−6
−4
−2
0
q
2
4
6
8
Figure 3.20: Limit cycle results in constant loop in hysteresis input/output trajectory with uncertainty = ;3.
for systems having nonlinearities with the prescribed characteristics. This stability
test, for the SISO case, is easily veried by a simple graphical test in the Nyquist
plane, and is readily computed by solving an LMI. The LMI framework allows for
a straightforward extension of the test to include robust performance. A simple numerical example is also presented to illustrate the utility of this particular form of
the multiplier in testing for the robust stability of a linear system with a hysteresis
nonlinearity and a multiplicative uncertainty in the plant output.
The results presented here for scalar systems are extended in the following chapter
to treat the case of multiple hysteresis nonlinearities. In Chapter 5, the stability
analysis is used to develop algorithms for robust H1 control design for systems with
hysteresis.
64
Chapter 4
Multiple Hysteresis Nonlinearities
Absolute stability criteria for systems with multiple hysteresis nonlinearities are given
in this chapter. It is shown that the stability guarantee is achieved with a simple two
part test on the linear subsystem. If the linear subsystem satises a particular linear
matrix inequality and a simple residue condition, then, as is proven, the nonlinear
system will be asymptotically stable. The main stability theorem is developed using
a combination of passivity, Lyapunov, and Popov stability theories to show that
the state describing the linear system dynamics must converge to an equilibrium
position of the nonlinear closed loop system. The invariant sets that contain all
such possible equilibrium points are described in detail for several common types of
hystereses. The class of nonlinearities covered by the analysis is very general and
includes multiple slope-restricted memoryless nonlinearities as a special case. Simple
numerical examples are used to demonstrate the eectiveness of the new analysis
in comparison to other recent results, and graphically illustrate state asymptotic
stability.
4.1 Introduction
The Popov stability criteria [Pop61] has long been the standard analytical tool for
systems having memoryless, sector bounded nonlinearities. Details of Popov's analytical approach can be found in the standard texts by Desoer, Vidyasagar and Khalil
65
66
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
[DV75, Vid93, Kha96]. When nonlinearities, in addition to being sector bounded, are
also monotonic and slope restricted, Zames and Falb [ZF68] proved that the Popov
analysis can be further sharpened by employing a more general type of multiplier, often called the Zames-Falb multiplier. Subsequently, Cho and Narendra [CN68] found
that the existence of such multipliers could be established with an o-axis circle test
in the Nyquist plane. While this early work was limited to a scalar nonlinearity,
an extension by Safonov [Saf84] considered multiple nonlinearities and established
criteria through loop shifting and diagonal frequency dependent matrix multipliers,
as is now common in the =Km-analysis approach, introduced by Doyle and Safonov [Doy82, Saf82]. An alternate approach for the slope restricted case pursued by
Singh [Sin84] and Rasvan [Ras88] utilized a multiplier rst introduced by Yakubovich
[Yak65] for systems with dierentiable nonlinearities. Although not as general as
the Zames-Falb multiplier, the simple form of the Yakubovich multiplier makes it
a valuable complement to the Popov analysis. More recently, Haddad and Kapila
[HK95], and Park [PBK98] have attempted to generalize the results in [Sin84, Ras88]
to the case of multiple slope restricted nonlinearities. The resulting criteria oered,
however, restrict the value of the linear system transfer matrix, G(s), in a variety of
ways. In both papers, for instance, the systems are restricted to be strictly proper
(i.e., the feedthrough term D = 0). Also, in [HK95], the value of the system matrix
at s = 0, G(0) must be either nonsingular or identically zero, while in [PBK98] the
stability guarantee requires that G(0) = G(0)T > 0.
In this chapter the analysis for multiple nonlinearities is generalized in several
ways. First the extension to non-strictly proper systems, where D 6= 0, is provided,
and the positivity requirement relaxed to G(0) = G(0)T > ;M ;1 , where M > 0 is the
diagonal matrix of the maximum slopes occurring in the vector of nonlinearities. More
importantly, it is shown that the same analysis that applies to the slope restricted
case is valid for a class of multiple hysteresis nonlinearities as well. This is a rather
signicant generalization since hysteresis is not sector bounded and has memory, and
thus is functionally very dierent than a memoryless, slope restricted nonlinearity.
These results eectively generalize the early scalar hysteresis work by Yakubovich
and Barabanov [Yak67, BY79] and more recent analysis given in [PH98b, PH98a], to
4.1. INTRODUCTION
67
the case of multiple hysteresis nonlinearities.
Using an approach similar to Park [PBK98], a linear matrix inequality is developed
which, if feasible in a set of free matrix variables, proves the asymptotic stability of
the system. For the slope restricted nonlinearity, asymptotic stability means the state
converges to the origin, which is assumed to be the unique equilibrium point of the
nonlinear system. Since a typical hysteresis is in general multivalued, convergence is
not to a single point, but rather to a stationary set, dened by the intersection of
the nonlinearity and the DC value of the system matrix. Several of these sets are
explicitly dened for some commonly occurring types of hystereses. In contrast to the
previous work of Haddad, Kapila [HK95] and Park [PBK98], the Lyapunov function
is a function of the system state, and not its time derivative. This dierence results
in a more straightforward conclusion of asymptotic stability.
4.1.1 Approach Overview
The original general form of Popov's stability criterion [Pop61, Hah67] requires the
linear portion of the system to be stable and strictly proper. However, the general
form does allow for a single pure integrator in the system. This is sometimes referred
to as the indirect form or the indirect control form of Popov's criteria (see texts [AG64,
NT73, Vid93] for scalar versions), and it commonly has associated with it a three term
Lyapunov function. In this chapter, this form is extended to the vector case using, as a
guide, the procedure of Narendra and Taylor [NT73, p. 100] for the single nonlinearity,
which we summarize in three simple steps. First, a loop transformation is applied
that changes the slope sector bounds, dierentiates the output of the nonlinearity, and
results in an integrator state in the transformed linear subsystem, G~ (s). Provided the
original linear subsystem G(s) is stable, G~ (s) is then cast in Popov's indirect control
form. A three part Lyapunov functional, V (t), is then formed that is quadratic
in the state of G~ (s) and includes a particular integral of the nonlinearity. When the
nonlinearity is a hysteresis having memory, the value of the integral is path dependent;
while in the memoryless case, it is not. Lastly, the requirement that V_ 0 is enforced
by the existence of a certain LMI, and subsequently, this condition is used to conclude
68
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
asymptotic stability of certain stationary sets.
The outline of the chapter is as follows. First, the next section details the properties of the nonlinearities, and in particular, limit the hysteresis class to multi-valued
functions having an input-output relationship with characteristic loops that circulate
in a strict direction. Following that, in x4.3, the nonlinear system is dened and the
loop transform used for the analysis is given. The stationary, or equilibrium sets, for
the various nonlinear systems are in general polytopic regions of state space, and are
detailed in x4.4. This leads directly to the main stability theorem, which is proved
in x4.5. Frequency domain and passivity interpretations of the Lyapunov result are
discussed in x4.6. Simple numerical examples are then presented in x4.7 which conrm the benets of the new approach with respect to prior stability criteria and give
a graphical illustration of the asymptotic stability to the stationary sets.
4.2 Nonlinearities and Sector Transformations
4.2.1 Memoryless, Slope Restricted
Following the denition given by Haddad and Kapila [HK95], we dene the class of
nonlinearities as
8
>
>
>
>
<
9
(y) = [1 (y1); : : : ; m(ym)]T >
>
>
>
=
m
is
dierentiable
a.e.
2
R
m
m
=> :R !R
(4.1)
0 < i ; i = 1; : : : ; m
>
0
>
>
i
>
>
>
>
:
;
(0) = 0
The set consists of m decoupled scalar nonlinearities, with each scalar component
locally slope sector bounded obeying the slope restriction:
a
b
(4.2)
0 i(yi a) ; bi(yi ) < i;
yi ; yi
for any yia; yib 2 R. This sector property is sometimes denoted as 0i 2 sector[0; i),
or given the discrete representation [NT73]:
i(yi)=yi 2 sector[0; i):
(4.3)
4.2. NONLINEARITIES AND SECTOR TRANSFORMATIONS
69
The slope restriction (4.3) on a function is a stronger than the standard sector bound
condition on a function. This idea is formalized with the following proposition.
Proposition 4.2.1 (Sector Bound Property) A function i : R ! R satisfying the
conditions (0) = 0 and (4.3) is necessarily sector bounded, with the same bounds.
That is, i 2 sector[0; i).
Proof: Simply set yib = 0 in (4.2) and multiply through by (yia)2 to get the relation:
0 i(yia)yia < i(yia)2;
and thus i 2 sector[0; i), which is the standard sector bound condition on i.
2
Recall from the previous chapter, that a nonlinearity with local slope conned to
a nite sector can be converted to a nonlinearity with semi-innite sector width using
a transformation involving a positive feedback around the nonlinearity, as depicted
in Figure 3.3. Also, as a consequence of Lemma 3.3.1, the transformed nonlinearity
slope condition:
0 ~0i(i ) < 1;
(4.4)
is equivalent to the sector condition between the time derivatives of the input-output
pair:
0 ~_i_i < 1:
(4.5)
To develop the Popov-based analysis for the vector case, it is convenient to employ
a simplied version of the transformation used earlier for the scalar case, shown in
Figure 3.4. Here the sector transform is applied to each scalar component of and a
new operator dened by dierentiating the vector output, as depicted in Figure 4.1,
where M = diag(1; : : : ; m) > 0 is the diagonal matrix of maximum slopes occurring
in . Note that this transform does not include the multiplier, and that, as might be
expected from the results of x3.3.2, the input-output relation from to , as dened
in Figure 4.1, is passive. This is detailed by the following lemma.
Lemma 4.2.2 (Passive Operator, Memoryless, Slope Restricted Case) Consider a
slope restricted nonlinearity ~ : Rm ! Rm with decoupled scalar components satisfying 0 ~0i( ) < 1. Then the input-output relation dened with (t) as the input to
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
70
~
y
+
sI
M ;1
Figure 4.1: Sector transformation results in ~ as passive operator.
~ and output (t) = dtd ~ (), the time derivative of ~ () (as depicted in Figure 4.1)
is passive.
Proof: For all T 0 we have
Z
T
0
T dt
=
=
=
=
m Z
X
T
i=0 0
m Z T
X
i=0 0
m Z T
X
i i dt
(4.6a)
i dtd ~i(i) dt
(4.6b)
i ~0i(i)_i dt
(4.6c)
i=0 0
m Z (T )
X
i
i~0i(i ) di(t)
i=0 ( (0)
Z (0)
m
X
(4.6d)
i
=
i=0
;
i
0
i ~0i(i ) di(t) +
; ((0))
where
((0)) =
m Z
X
i=0 0
(0)
i
Z
(T )
i
0
i~0i(i ) di(t)
i~0i(i ) di(t) 0;
)
(4.6e)
(4.6f)
(4.7)
since each scalar kernel, ki(i) = i~0i(i ), is a memoryless, sector bounded function,
with ki 2 sector[0; 1): Therefore, the input-output relation is passive, by the denition given in reference [DV75, p. 173].
4.2. NONLINEARITIES AND SECTOR TRANSFORMATIONS
71
(x)
(x)
1
1
;D
D
-1
a.) Ideal, discontinuous relay.
x
;D
D
x
-1
b.) Smooth, analytical approximation.
Figure 4.2: Discontinuous relay replaced with smooth approximation for stability
analysis.
Having dened the passive transformation for the memoryless class of slope restricted
nonlinearities, we turn now to the hysteresis case.
4.2.2 Hysteresis
The same approach used in the previous section for memoryless nonlinearities is applied in this section to convert a vector hysteresis into a passive operator. The result
is a natural extension to the scalar case presented in x3.3.2, and is useful in practice
since a particular system of interest may include several hysteretic eects, of possibly dierent types. Analytical techniques with generality sucient for this case are
developed here using a passivity-based LMI formulation analagous to that given for
the single nonlinearity in the previous chapter. First, a class of multiple, decoupled
nonlinearities is dened using the scalar hysteresis properties (3.3.2.1{5). Recall that
limiting consideration to this particular class simplies the analysis, but the results
still apply to many nonlinearities that occur in practice, such as the hysteretic relay, backlash, and Preisach hysteresis [May91], which are depicted in Figures 4.2, 4.3
and 4.4, respectively . The transformation used for the memoryless set (Figure 4.1)
While counter-clockwise circulation is an assumed property of the class, it is possible to include
clockwise behavior by employing a coordinate transformation that eectively reverses the circulation,
72
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
(x)
;r
r
x
Figure 4.3: Backlash with deadzone width = 2r.
will be used again to convert members of this hysteresis class into passive operators.
As before, proof of passivity is then detailed in the corresponding lemma. By utilizing a common transformation, the subsequent stability criteria developed in x4.5
is unied, applying to both the multiple memoryless and hysteresis nonlinearities.
To simplify the following analysis, continuous approximations for certain hysteretic
forms will be assumed, as discussed next.
Smooth approximations for discontinuities
Nonlinearities with discontinuities, such as the relay depicted in Fig. 4.2, can present
diculties for the stability analysis because the transform used (shown in Fig. 4.1)
involves the time derivitive of the nonlinear output. As such, the transformed nonlinearity will result in an unbounded operator, mapping continuous input signals, with
bounded velocities, to an output signal with innite rate of change. Naturally, this
would violate the sector bound (4.5) established for the memoryless case. In order to
use the same analytical approach for hysteresis with discontinuities, certain smooth
approximations must be assumed. Smooth approximations for relay-type nonlinearities, as depicted in Fig. 4.2b, will be assumed for the subsequent analysis, so that the
as discussed in [HM68, p. 366].
4.2. NONLINEARITIES AND SECTOR TRANSFORMATIONS
(x)
SATURATION
ZONE
+
Minor Loop
73
Major Loop
1st Order
Transition
x;
x+
x
A
;
Figure 4.4: Typical Preisach hysteresis characteristic.
local slope,
d
dx
= 0(x), satises the bound:
0 0(x) < 1;
(4.8)
where the upper bound is stricty. An important consequence of this condition is
that the approximation maps continuous input signals into continuous outputs. This
allows us to establish a local Lipschitz condition for the nonlinearity. That is, if the
input x : R ! C0[0; t], where x(t2 ) is a suciently small (local) perturbation of x(t1 )
on an interval:
jx(t1 ) ; x(t)j < ; 8t 2 [t1 ; t2]
(4.9)
then the local Lipschitz condition:
j(x(t1 )) ; (x(t2 ))j 0jx(t1) ; x(t2 )j
(4.10)
will hold, where 0 0(x) 0 < 1. In this case 0 = , the maximum slope
appearing in the nonlinearity. This property will ensure that the transformed operator
y See [Vis88] for a similar approximation for the hysteretic relay.
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
74
is a bounded operator on the space of continuous signals, C0[0; t]z.
Multiple Hysteresis Nonlinearities
Using the given properties of the scalar hysteresis nonlinearities, dening the class
for the vector case is straightforward. We dene h, the multiple hysteresis class as:
8
>
>
>
>
<
9
>
(y) = [1 (y1); : : : ; m(ym)]T
>
>
>
=
obeys
local
Lipschitz
property
(4.10)
i
m
m
h = > : R ! R
: (4.11)
>
0 0i < i; i = 1; : : : ; m
>
>
>
>
>
>
:
;
i has Properties 3.3.2.1{3.3.2.5
The set h consists of m decoupled scalar nonlinearities, with each scalar component
locally slope bounded (wherever the nonlinearity is dierentiable) and conforming to
the properties detailed previously in x3.3.2.
Lemma 4.2.3 (Passive Operator, Hysteresis case) Consider a vector hysteresis nonlinearity h : Rm ! Rm in the class dened (4.11). Then the input-output relation
of the sector transformed operator ~ h dened with (t) as the input to ~ h and output
(t) = dtd ~ h (), the time derivative of ~ h() (as depicted in Figure 4.1) is passive.
Proof: For all T 0
Z
T
0
T dt
=
=
=
m Z
X
T
i dtd ~i(i ) dt
(4.12a)
iwi0 (t) dt
(4.12b)
i (t) dwi(t)
(4.12c)
i=0 0
m Z T
X
i=0 0
m Z
X
i=0 ;
m Z
X
i
i (t) dwi(t)
i=0 (;
Z
m
X
abi
=
i=0
;
; !
; ((0))
pi
ai
z See [BS96, p. 24], for a similar discussion.
i (t) dwi(t) +
(4.12d)
)
Z
; !
pi
bi
i(t) dwi(t)
(4.12e)
(4.12f)
4.3. SYSTEM DESCRIPTION AND LOOP TRANSFORMATION
0
e
0
-
p
G(s)
y
-
e
s;1 I
G(s)
y
75
G~ (s)
+
M ;1
a.) Original nonlinear system.
p
sI
y
+
M ;1
~
b.) Loop transformed system.
Fig. 4.5: Nonlinear system and loop transformation.
where
((0); w(0)) = (y(0); 0) =
m Z
X
i=0 ; !
pi
i (t) dwi(t) 0;
(4.13)
ai
according to properties ( 3.3.2.4{3.3.2.5) of the class. Hence, the input-output relation
is passive, by the denition given in [DV75, p. 173].
Note, that the proof is structured in a way analogous to the memoryless case. Instead
of positive (sector bounded, path independent) line integrals, the corresponding steps
here involve positive path integrals.
4.3 System Description and Loop Transformation
As in the standard absolute stability analysis framework, it is assumed that the
nonlinearity can be isolated from the linear dynamics and placed into a feedback
76
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
path, as is shown in Fig. 4.5a. Assuming the linear dynamics G(s) has a minimal
state space representation (A; B; C; D), with A Hurwitz, the nonlinear (Lur'e) system
is described as
x_ = Ax + Be
(4.14)
y = Cx + De
pi(t) = i(yi(t)); i = 1; : : : ; m;
where p(t) 2 Rm and 2 or 2 h , belonging to either the multiple memoryless
(4.1) or hysteresis classes (4.11), as dened above. In order to convert the nonlinearity
into a passive operator, in accordance with Lemma 4.2.1 and 4.2.2 we introduce the
loop transform, as described in Fig. 4.1, to give the equivalent system shown in
Fig. 4.5b. Note that ~ is now passive, and that the transformed linear system:
G~ (s) = (G(s) + M ;1 )(s;1I );
(4.15)
has the state space representation:
2 "
6
G~ =s 664
h
A 0
C 0
0 I
#
i
"
B
D + M ;1
0
# 3
7
7
7
5
:
(4.16)
By the Hurwitz assumption, we have that A is invertible, and thus by introducing
the similarity transform:
"
#
I 0
T=
;
CA;1 I
the augmented system G~ (s) can be decomposed into its stable and constant dynamic
components as:
G~ (s) = G~ r (s) + s;1R;
(4.17)
where R = G(0) + M ;1 with G(0) = ;CA;1 B + D; and the stable component G~ r is
reduced by the integrator states and has the state space description:
"
#
A B
G~ r =
:
CA;1 0
s
(4.18)
4.3. SYSTEM DESCRIPTION AND LOOP TRANSFORMATION
u
0
G~ r (s)
-
_
1I
s
77
y
+
R
0
sI
M (t) M
Figure 4.6: Popov indirect control form.
With the linear dynamics decomposed in this way, the nonlinear, closed loop
system can then be expressed in the vector version of Popov's indirect control form
(see [Vid93, p. 231], for example), as is depicted in Fig. 4.6. The dynamics of the
original Lur'e system (4.14) corresponding now to the Popov form are equivalently
given as:
x_ = Ax + Bu = Ax ; B
_ = ;; (0) = ;M (0)
(4.19)
= CA;1 x + R
= _ M (t)
Proper initialization of the integral state , as shown in Fig. 4.6, leads to the identities:
(t) = ;M (t)
_ (t) = ; = ;_ M (t):
(4.20a)
(4.20b)
The stable (equilibrium) conditions for the hysteresis case diers than that which
results from memoryless, slope-restricted nonlinearities because the hysteresis is multivalued. As a result, while the equilibrium point for the memoryless nonlinear system
is unique (e.g., the origin), convergence for the hysteresis system is to an invariant
78
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
set, which may consist of an innite number of points. The next section provides
explicit descriptions of these stability sets.
4.4 Stationary Sets and Stability Denitions
Stability theory is often used to determine whether or not an autonomous system will
achieve some sort of steady state condition. Generally speaking, in steady state, the
system state may be at an equilibrium point (at rest with x_ = 0), or in a limit cycle.
In either case, the state x(t) belongs to an invariant set [Hah63, Vid93]. The largest
invariant set M Rn, for a particular system, is the union of all equilibrium points
and the sets containing all possible limit cycles. The equilibrium, or stationary, set
E M, for the nonlinear system (4.14) is dened as:
n
o
E = x 2 Rn such that (4.22) is satised ;
(4.21)
where (4.22) is the set of algebraic conditions:
yss = [;CA;1 B + D]ess = G(0)ess
ess = ;(yss)
xss = ;A;1 Bess:
(4.22a)
(4.22b)
(4.22c)
Naturally, E is unique to each system (4.14) and, in particular, depends on the type
of nonlinearity present. Various stationary sets are given below.
4.4.1 Stationary Set for Memoryless Nonlinearity
For the slope-restricted nonlinearity, we assume there exists a unique equilibrium
point x = 0, for the closed loop system (4.14). That is, Em is a singleton:
Em = f0g :
(4.23)
This result is consistent with the sector bounded property of the class , and the
assumption G(0) > ;M ;1 . Geometrically, this condition means that the graph of
i-th nonlinearity i(yi ) and the line i = ;yi=Gii (0) intersect only once, at the origin.
4.4. STATIONARY SETS AND STABILITY DEFINITIONS
i
=
;1
Gii(0) yi
79
i
i
i
yi
Fig. 4.7: Graphical criteria for determining stationary set E.
This intersection is necessarily non-unique in the hysteresis case, and as a result, Eh
is comprised of nite regions in state space. These sets are dened below for various
special cases.
4.4.2 Stationary Sets for Hysteresis Nonlinearities
The stationary sets for multiple hysteresis can be dened with a simple extension of
the graphical technique for the scalar case originally detailed by [BY79]x. To proceed,
consider a generic Preisach nonlinearity, and note that conditions (4.22a{b) together
can be depicted graphically, as shown in Fig. 4.7, as the intersection of the line
i = ;yi =Gii(0) and the graph of the hysteresis. This intersection denes the range
of outputs for each nonlinearity i 2 [i; i ] which must be satised simultaneously
for each i; i = 1; : : : ; m: Then letting each i vary over the allowed range maps out
x A similar denition for (4.21) is given in Ref. [Jon98].
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
80
the invariant set E, according to the condition (4.22c) x = ;A;1 Be, where e = ;:
Note that if Gii(0) = 0, then the corresponding limits i; i are simply the extreme
values of intersection of the hysteresis with the {axis. The stationary sets for the
relay, backlash, and Preisach hysteresis nonlinearities are given next.
Hysteretic Relay
For a system with a bank of m unit relays, as shown in Fig. 4.2, the stationary set is
given by:
(
Erelay =
x 2 Rn
x = ;A;1 Be
e 2 Rm ; ei 2 f;1; 1g; i = 1; : : : ; m
)
(4.24)
Erelay consists of 2m discrete points in Rn. Each point is essentially the steady state
solution of the open loop system G(s) in response to a particular constant input
vector e consisting of elements ei = +1; or ; 1:
Backlash and Preisach Nonlinearities
The equilibrium sets for these two types of nonlinearities are dened in the same way,
since both operators admit outputs that range continuously over a prescribed interval.
Once the output limits are dened, the stationary set is completely determined.
(
Ebacklash; EPreisach =
x 2 Rn
x = ;A;1 Be
e 2 Rm ; ei 2 [i; i]; i = 1; : : : ; m
)
(4.25)
Note that these sets are polytopic regions, and are equivalently dened as the convex
hull of the corresponding set of limiting vectors:
Ebacklash; EPreisach = Co fvi ; vi; : : : ; vm ; vm g ;
where vi; vi 2 Rn, with
vi = ;A;1 Bz; where z j =
and vi dened similarly.
(
i; j = i
0; else,
(4.26)
4.5. STABILITY THEOREM
81
The denitions for the stationary sets E provide a clear idea of the position of
x 2 Rn should the system achieve the equilibrium condition dened by x_ = 0: Before
providing the stability criteria that guarantees the system is indeed stable, we give
precise denitions of what it means for a system to be stable with respect to an
invariant set.
4.4.3 Denitions of Stability
Using standard notation (as by [Hah63], for example), dene the trajectory of motion
for an initial condition x(0) = x0 of some arbitrary system as q(x0 ; t). For an invariant
set M of the system, the distance to the set from any arbitrary point is given by:
dist(x; M) = inf kx ; yk; y 2 M;
with dist(x; M) = 0 for x 2 M. A closed invariant set M is called stable, if for every
> 0 a number > 0 can be found such that for all t > 0,
dist(q(x0 ; t); M) < provided
If in addition,
dist(x0 ; M) < :
dist(q(x0 ; t); M) ! 0; as t ! 0;
then M is said to be asymptotically stable.
4.5 Stability Theorem
This section provides a Lyapunov-based asymptotic stability theory for the systems
with either slope-restricted (memoryless) or hysteresis nonlinearities. The Lyapunov
function used refers to the transformed system dened in x4.3 and includes the integral
of the nonlinearity that is positive, as a result of the passive properties dened in x4.2.
Negativity of the Lyapunov derivative is enforced by a certain matrix inequality of
a form similar to that associated with the well-known KYP Lemma (see [BGFB94,
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
82
p. 120], for one treatment). The theorem then concludes asymptotic stability of
the origin in the case of the memoryless, slope-restricted nonlinearities, and for the
equilibrium sets given x4.4.2 in the hysteresis case by using the Lyapunov conditions
and employing basic analytical results.
Theorem 4.5.1 (Asymptotic Stability) If there exists constants P; N; , with
P 2 Rnn; P = P T > 0
2 Rmm ; = T > 0
N = diag(n1; : : : ; nm ); ni > 0; i = 1; : : : ; m
such that
"
(4.27)
#
;AT P ; PA C T N + A;T C T ; PB
0;
()T12
ND + DT N + 2NM ;1 ; (4.28)
and R = RT > 0; R = G(0) + M ;1 , then the closed loop system (4.19) is asymptotically stable. In this case, the Lyapunov functional:
V (x(t); (t); t) = x(t)T Px(t) + 2
Z
t
0
T ( ) ( ) d + (0 ; 0) + T (t)R(t) (4.29)
proves stability.
Proof: Choosing as (4.7) for the slope-restricted nonlinearity, or as (4.13) when
the nonlinearity is a multiple hysteresis{ , then V 0; and since P; R > 0, V ! 1
whenever (x; ) ! 1, so V is positive denite and hence, a valid Lyapunov candidate.
In order to assert V_ 0, rst note that matrix inequality (4.28) implies, for all
x 2 Rn ; u 2 Rm
xT (AT P + PA)x 2xT (C T N + A;T C T ; PB )u + uT M22 u;
(4.30)
where M22 is the (2; 2) entry of the LMI (4.28). Using this fact, and (4.20) we have
V_ (x; ) = xT (PA + AT P )x ; 2xT PB _ M + 2T _ M + 2T R_
;2xT (C T N + A;T C T )_ M + 2T _ M + 2TM R_ M + _ TM M22 _ M (4.31a)
{ In the particular case when the nonlinearity is of the multiple backlash type, = 0, as discussed
in x3.3.3.
4.5. STABILITY THEOREM
83
= ;2(_ + (D + M ;1 )_ M )T N _ M ; 2( + RM )T _ M + 2T _ M
+ 2TM R_ M + _ TM M22 _ M
= ;2_ T N _ M ; _ TM _ M
;_ TM _ M
;j_ M j2
0;
(4.31b)
(4.31c)
(4.31d)
where the rst inequality (4.31a) is due to the LMI condition, the second (4.31b)
a result of the time-derivative sector condition (4.5), and the last two (4.31c-4.31d)
follow from the constraint > 0 (4.27), and the assumption that is the minimum
eigenvalue of . Now since V is positive denite in x; and V_ 0, we conclude the
closed loop system is stable, or, simply that x and are bounded. To nd asymptotic
stability, rst note that,
V_ ;j_ M j2 ) _ M (t) ! 0 as t ! 1;
(4.32)
since V (t) is bounded below. Further, using (4.31c), we have
V (t) ; V (0) ;
which, can be rewritten as
Z
t
0
Z
0
t
j_ M j2 dt;
j_ M j2 dt 1 (V (0) ; V (t)) V (0)=;
(4.33)
(4.34)
which implies _ 2 L2, and as a result y(t) 2 L2 as well since G~ r is L2-stable (i.e.,
A Hurwitz). Using the system dynamics (4.19), the signal y and its derivative are
expressed as
y(t) =
y_ (t) =
CA;1
eAt x(0) +
Z
t
eA(t; ) Bu( ) d
0
Z t
CeAtx(0) ; C eA(t; ) B _ M ( ) d
0
; CA;1B _ M (t):
(4.35a)
(4.35b)
Assuming Lipschitz continuous nonlinearities so that _ M (t) exists (i.e., _ M (t) 2 L1),
we have that y_ 2 L1.k In this case, the two conditions y(t) 2 L2, y_ 2 L1 imply
k Recall continuous approximation to establish local Lipschitz condition (4.10), and see Remark 2
below for alternate treatment allowing for discontinuities.
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
84
that y(t) ! 0 as t ! 1 (see, for example, [NA89, Lemma 2.1.2]). The asymptotic
conditions y(t); _ M (t) ! 0 together require that the closed loop system must approach an equilibrium condition as t ! 0. To see this, rst note that the conditions
= ;_ M ! 0 and y ! 0 imply that all signals of the Popov system (4.19) contained
within the dashed region of the block diagram in Fig. 4.8a approach zero asymptotically. Secondly, ; = _ ! 0, together with the condition _ (t) 2 L1, established
in Appx. D, implies that limt!1 (t) exists. Further, recall that the initialization of
variable (0) = ;M (0) implies that (t) = ;M (t) 8t 0, as given by Eqn. (4.20).
Thus, in the limit, the zero signals can be eliminated and the system reduced to that
shown in Fig. 4.8b, where the signal equivalence mentioned above is indicated by the
dashed line. Reversing the sector transformation further simplies the diagram to
that in Fig. 4.8c, which corresponds to the equivalent algebraic conditions:
yss = G(0)uss
uss = ;(yss)
x = ;A;1 Buss;
(4.36a)
(4.36b)
(4.36c)
which are identical to the conditions (4.22) that describe the stationary set E. Therefore, in the hysteresis case, we conclude global asymptotic stability of the set E. In
the special case of the system with multiple slope-restricted nonlinearities, the set E
is simply the origin, as noted by Eqn. (4.23).
Remarks:
1. This proof utilizes a combination of Lyapunov and input-output stability theories. Of course, connections between Lyapunov and input-output stability
concepts have been well established [Wil71b, HM80a, BY89]. In this case, passivity conditions are used to establish Lyapunov stability arguments for slope
restricted/hysteresis nonlinear systems, all within the analytical framework of
Popov's indirect control form. An alternate approach could proceed using passivity (as is done in [PH98b]) or Popov's hyperstability theorem [Pop73], exclusively. However, the Lyapunov component included here enables the additional
4.5. STABILITY THEOREM
85
u
0
G~ r (s)
-
_
1I
s
y
+
R
0
_ M
M
sI
(a)
0
-
uss G(0)
R
-
M () M
(b)
yss
(yss)
(c)
Figure 4.8: The condition V_ 0 implies steady state condition on the Popov sys-
tem. The signals contained in the dashed region seen in (a) tend to zero
asymptotically. In the limit, this allows reduction to the system (b),
which is the transform equivalent to (c), that describes the steady state
equilibrium condition (4.36).
86
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
conclusion of asymptotic stability of the set E. Positive real and passivity interpretations of the analysis are further explored in the following section.
2. Note that the hysteresis set h (4.11) includes only smooth approximations for
discontinuous nonlinearities such as the hysteretic relay. Of course this was
done to maintain simplicity. A more exact treatment could be developed for
these discontinuous nonlinearities using Filippov [Fil88] state solutions, onesided Lyapunov derivatives as described by [Hah63, Cla83], and the generalized
version of LaSalles Invariance Principle [LaS76].
3. The condition R = RT > 0 is not overly restrictive. For instance, the odiagonal elements G(s) can often be arbitrarily scaled using diagonal scaling matrices. In this way the matrix G(0) can be made symmetric with the necessary
gain adjustments incorporated into the nonlinearity. The condition R = RT > 0
is less restrictive than the condition G(0) = 0 given by [HK95], and the criterion
G(0) = G(0)T > 0 required by [PBK98], whenever the nonlinearity has nite
maximum slope. The criteria in Ref. [PBK98] includes the additional constraint
that NG(0) = G(0)T N , which limits N to a scalar quantity in the case when
G(0) is a full matrix. This can further restrict the analysis, as is illustrated
with a simple example in x4.7.
4.6 Passivity and Frequency Domain Interpretations
The LMI (4.28) is recognized as a strict passivity condition on the linear system:
"
#
A
B
G~ ra =
(4.37)
;
1
NC + CA N (D + M ;1 )
which is an augmented version of the reduced system G~ r . Strict passivity of this
augmented system is a requirement for stability that could have been derived using
an equivalent analysis of the system in Fig. 4.5 that employs noncausal multipliers,
as is detailed in Ref. [Wil71a, Ch. 6]. A robust stability analysis using passivity and
4.6. PASSIVITY AND FREQUENCY DOMAIN INTERPRETATIONS
87
G~ a
0
;
s;1 I
G + M ;1
sI
M
Ns + I
(Ns + I );1
~
a.) Augmented, transformed nonlinear system.
G~ ra
0
;
s;1R
qr
G~ a
+
qI
+
~
b.) System in Popov Indirect form.
Figure 4.9: Equivalent passivity-based stability analysis can be accomplished by (a.)
including multipliers and (b.) transforming to Popov indirect control
form.
88
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
multipliers for the specic case of systems having a single hysteresis was recently done
in Ref. [PH98b]. To proceed, introduce the multiplier W (s) = Ns + I (with N as
dened in (4.27)) into the transformed system, as shown in Fig. 4.9a. In this case,
premultiplying the hysteresis M with W ;1 as shown results in a new nonlinearity ~
in the feedback path which is passive. This passivity condition is ascertained using the
steps in the proofs of Lemma 4.2.1 and 4.2.2 and using the additional time derivative
constraint (4.5). Introducing the multiplier similarly leads to the transformed linear
system:
G~ a (s) = W (s)(G(s) + M ;1 )(s;1I ) = (Ns + I )(G(s) + M ;1 )(s;1I ):
(4.38)
This decomposes, as was done in Eqs. (4.15-4.17), to
G~ a(s) = G~ ra (s) + s;1R;
(4.39)
where again R = G(0) + M ;1 , and Gra(s) is the augmented system (4.38) reduced
by the integrator states and has the state space representation (4.37). This leads
directly to the Popov indirect form, with a parallel combination of the augmented
system G~ ra and the constant dynamics s;1R, as depicted in Fig. 4.9b. In the passivity
framework, stability requires either the feedforward or feedback operator be strictly
passive. In this case, strict passivity is achieved by conditions on the reduced system
G~ ra and strict positivity of R, as detailed in the following corollary.
Corollary 4.6.1 (Strict Passivity) If there exists N = diag(n1 ; : : : ; nm ) 0; =
T > 0 such that the following two conditions:
1. R = RT ; R = G(0) + M ;1 > 0
2. The reduced system Gra , given by (4.37) is dissipative with respect
to the supply rate:
r(p; q) = pT q ; pT p;
(4.40)
are satised, then the system G~ a (s) is strictly passive. In this case, the closed loop
system (4.14) is asymptotically stable.
4.6. PASSIVITY AND FREQUENCY DOMAIN INTERPRETATIONS
89
Proof: Let : R+ ! Rm represent the integrator state with (0) = 0 , V : Rn ! R+
be a storage function for G~ a and qI ; qr be the outputs of the integrator and G~ ra,
respectively, as depicted in Fig. 4.9b. Then for any T 0 we have
Z
T
0
qT p dt
=
Z
Z
T
0
(qI + qr )T p dt
Z T
d
T p dt
=
(
t
)
dt
+
q
r
dt
0
0
1
2 (TT RT ; 0T R0 ) + V (xT ) ; V (x0 ) + hp; pi2T
; (0; x0 ) + kpT k22;
T
T (t)RT
where (0; x0) = 0T R0 =2 + V (x0) 0 and > 0 is the minimum eigenvalue of .
Thus, G~ a is strictly passive by the denition given in [DV75]. Then, since the loop
transformed system consists of a passive (transformed) nonlinearity in feedback with
a strictly passive linear system, we thus conclude the closed loop system will converge
asymptotically to the equilibrium conditions.
Corollary 4.6.1 essentially uses the idea that an operator consisting of the parallel
combination of passive (nonstrict) and a strictly passive operators is strictly passive. Condition 1 ensures the passivity (nonstrict) of the integrator component, while
condition 2 enforces the strict passivity of the reduced system G~ ra. The necessary
dissipation for the parallel system is ultimately guaranteed by the existence of some
> 0. Naturally, the scalar analogy for the positivity condition: R = RT > 0 on
the integrator term is the simple capacitor, which is passive provided the capacitance value is positive. The notion that a linear system can be strictly passive even
though it has zero eigenvalues is not intuitive, but similar results are available in the
literature, and usually involve decomposing the system into its stable and constant
dynamic components, as is done here for the indirect Popov criterion. In Ref. [AV73,
p. 216], for example, it is shown that systems with purely imaginary poles are positive real only if the associated residue matrices are nonnegative denite Hermitian.
A similar state space diagonalization is used to establish Lyapunov stability criteria in [BGFB94, pp. 20{22] for systems having eigenvalues with a zero real part.
Assuming the input/output relation across the capacitor terminals is current/voltage.
90
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
In essence, Thm. 4.5.1 is an extension of these ideas to a particular version of the
KYP Lemma, and in eect could be called the Indirect Control KYP Lemma, for the
historical reasons cited in x4.1.
Of course, as is well known, a linear system is strictly passive if and only if its
Hermitian form is strictly positive denite for all frequencies [DV75, p. 174]; that is,
a system H (s) is strictly passive if and only if, for some > 0,
H (j!) + H (j!) > I; 8! 0:
(4.41)
Hence, the stability question can be addressed by asking the equivalent question:
When is a square, linear system having zero eigenvalues strictly passive? Note that,
unlike the approach taken in [HK95, PBK98], we do not require the linear system to
be strictly positive real (SPR) [Kha96, Wen88], which is a stronger condition than
strict passivity. In fact, the transformed system G~ a in general can not be SPR since
the multiplier W (s) introduces a zero eigenvalue (see [Kha96, pp 404-405]); however,
it is clear that G~ a satisfying the conditions of Theorem 4.5.1 are strictly passive. This
follows since,
1 (R ; RT )
G~ a (j!) + G~ a (j!) = G~ ra (j!) + G~ ra(j!) + j!
(4.42a)
= G~ ra (j!) + G~ ra(j!)
(4.42b)
> (4.42c)
I;
(4.42d)
where is the minimum eigenvalue of . Therefore the strict passivity condition (4.41) is achieved. Here again, as in the Corollary 4.6.1, the role of symmetric R
is apparent, this time in the frequency domain.
4.7. NUMERICAL EXAMPLES
91
4.7 Numerical Examples
4.7.1 Computing the Maximum Allowed Slope for Nonlinearities
A common engineering problem that often arises is that of nding the maximum
sloped nonlinearity that a given system can tolerate before going unstable. This
problem was posed in [PBK98], and an LMI solution was suggested based on the
analysis given in that paper. The same problem in terms of the conditions of Theorem 4.5.1 is stated as:
(
max subject to:
(4.27), (4.28)
R = RT > 0
(4.43)
where M = Im. Solving (4.43) for the arbitrary 2 2 system G(s) given as
"
G1(s) =
s2 ;0:2s+0:1
s2 ;0:4s+0:75
s3 +2s2 +2s+1
s3 +32s2 +3s+1
0:1s2 +5s+0:75 0:15(s +s+0:75)
s3 +1:33s2 +2s+1 s3 +s2 +1:1s+1
#
;
(4.44)
by using the LMI solver [GNLC95], yields a maximum allowed slope value of =
0:940. By comparison, the equivalent problem using the stability criteria from [PBK98]
results in a maximum slope of 0:392, approximately a factor of 2 smaller. Obviously,
Theorem 4.5.1 is less conservative in guaranteeing stability for this system. The reason
for this is that while G(0) is symmetric, and thus satises the criteria in [PBK98], G(0)
is a full matrix. As a result of Park's additional constraint, NG(0) ; G(0)T N = 0, the
multiplier N must reduce to a scalar, positive number. By contrast, Theorem 4.5.1
poses no such condition on G(0), and allows N to remain a diagonal matrix with
two degrees of freedom, and is thus able to give less conservative stability guarantees.
This relative advantage is likely to increase as the number of nonlinearities increases
in the case of non-diagonal G(0). This follows since Theorem 4.5.1 will allow one
additional degree of freedom for each nonlinearity, while the criteria from [PBK98]
restricts the multiplier to a single scalar number (i.e., N = nIm) regardless of the
problem size.
92
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
[HK95] [PBK98] Theorem 4.5.1
G1 (s)
n=a
0:39
0:94
G2 (s)
n=a
n=a
0:99
Table 4.1: New analysis extends previous results for systems with multiple slope-
restricted nonlinearities. In each case, Thm. 4.5.1 allows larger nonlinearity slopes, and applies to a wider class of systems (note, n/a: not
applicable).
As a second example, consider
"
G2 (s) =
s;0:2
0:1s2 +1
s +2s2 +2s+1
s22+3s+1
0:1s2 +5s+1 0:2(s +s+0:75)
s2 +1:33s+1 s3 +s2 +1:1s+1
3
#
;
(4.45)
and note the state space version of this system has a non-zero feedthrough term,
D 6= 0, and the system matrix at s = 0:
"
;0:2 1:0
G2(0) =
#
1:0 0:15
has a negative eigenvalue. For either of these reasons, the recent results of [PBK98]
and [HK95] do not apply in this case. Within the context of absolute stability then,
it is fair to conclude the criteria in [PBK98, HK95] can guarantee stability only
for nonlinearities having zero maximum slope (i.e., only when nonlinearities are not
present). However, Theorem 4.5.1 does apply and guarantees stability for all nonlinearities in the classes described in x4.2 that have a maximum slope less than 0:996:
The corresponding stability multiplier is N = diag(25:327; 11:134):
The results of these two numerical examples are summarized in Table 4.1. Note
that for these examples the new criteria provides a less conservative analytical tool
than the previous results by predicting larger allowed slopes in the nonlinearities, and
applying to systems which otherwise could not be evaluated.
4.7. NUMERICAL EXAMPLES
93
4.7.2 Asymptotic Stability with Single Hysteretic Relay
As a simple example of an application of Theorem 4.5.1 for a system with a single
nonlinearity, consider a third order system:
2 0:01s + 0:25
(4.46)
G(s) = (ss ++1)(
s2 + s + 1)
that is attached in negative feedback with a hysteretic relay (Fig. 4.2). A simple
graphical check, as described in x4.4.2 shows that the line = ;y=G(0) intersects the
nonlinearity in two stable points, = 1, and does not intersect the discontinuous
portion of the characteristic. In this case the stationary set is well dened and,
according to denition (4.24), is simply two discrete points in state space:
8
>
>
<
2
0
E = > 0
>
:
2
6
6
4
39
>
>
7=
7
5>
>
;
:
(4.47)
To prove asymptotic stability of E, we solve the LMI (4.28) by approximating the
innite slope of the relay with the value = 1106. Using the LMI toolbox [GNLC95],
the stability parameters are found to be
2
3
5:0826 ;0:02149 0:16304 7
6
P = 64 ;0:02149 4:7911 ;0:02991 75 ;
0:16304 ;0:02991 3:038
N = 4:7078, and = 4:77 10;6, which proves the global asymptotic stability of
the set E. Note that in this case G(s) is not positive real, and thus an analysis of
this hysteretic relay system based on the circle criteria, such as the IQC technique
given by [RM96], would fail. However, it is straightforward to show that the graph of
G(j!); w 0 does not enter the third quadrant of the Nyquist plane and therefore
satises less restrictive stability criteria for systems with scalar hysteresis nonlinearities, as discussed in x3.4.2, and depicted in Fig. 3.9yy. Several simulations of the
nonlinear system conrm this result. The set is clearly visible in Fig. 4.10, as initial
yy See [PH98b] for further details.
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
94
State trajectories for system with hysteretic relay.
10
x3
5
0
-5
6
4
-10
-4
2
-2
0
-2
0
-4
2
4
x1
-6
x2
Figure 4.10: For system with hysteretic relay, analysis shows all solutions converge
to the stationary set E, which consists of two distinct points in state
space.
conditions at various locations in state space converge to either of the two discrete
points. The nonlinear behavior of the system is evident in Fig. 4.11, which shows
nonsmooth trajectories of the state that result at times when the relay switches. The
nonlinear switching is also the cause of the asymmetric pattern of the state trajectories, as seen in the x1 -x3 plane.
4.7. NUMERICAL EXAMPLES
95
Asymmetric trajectory pattern caused by relay hysteresis.
6
4
x2
2
0
-2
-4
-6
-4
-3
-2
-1
0
x1
1
2
3
4
Fig. 4.11: Alternate view depicts nonsmooth trajectories that result from relay
switching. In this view E appears as single point in state space.
4.7.3 Asymptotic Stability with Multiple Backlash Nonlinearities
Here we investigate the stability of the two-input, two-output system:
2
"
#
A B
G=
=
C D
s
6
6
6
6
6
6
6
4
;2
;1
3
;0:5 0:19365 0:41312 7
0
0
;0:41312 77
2
0
0
1
0
1:875 ;0:1875 0:09375
1
0:75
1
0
0
0
0
0
0
7
7
7
7
5
(4.48)
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
96
Multiple backlash nonlinearity results in polytopic stationary set.
0.6
0.4
0.2
x3
0
-0.2
-0.4
-0.6
-0.8
-0.2
-0.1
0.2
0.1
0
0
0.1
-0.1
0.2
x1
-0.2
x2
Fig. 4.12: The stationary set E for a multiple backlash nonlinearity is a rectangular
region in the x1 -x3 plane.
that is attached in feedback with two backlash nonlinearities, described in Fig. 4.3,
each having unit slope and deadband width (; D = 1). The system matrix at s = 0:
"
0:0363 0:3873
G(0) =
0:3873 0:20656
#
is symmetric, and has eigenvalues = ;0:275; 0:518, so that the criteria R = RT > 0,
where R = G(0)+ I is satised. Solving the LMI (4.28) yields the stability parameters
2
3
4:2914 ;1:921 ;3:7638 7
6
P = 64 ;1:921 7:6573 ;2:3389 75
;3:7638 ;2:3389 18:354
4.8. CONCLUSIONS
and
97
"
#
"
#
1:7292 0
0:75697 ;0:18976
N=
=
:
0 1:6253
;0:18976 0:69497
Positivity of these matrices proves global asymptotic stability for the set E, as per
Theorem 4.5.1. In this case, E is a polytopic region, as described for the backlash
nonlinearity by Eqn. (4.26), given by:
8
>
>
<
E = Co >
>
:
2
3
2
0:1712 7
0
6
0 75 ; 64 0
0
0:3737
6
6
4
3 9
>
>
7 =
7
5 >
>
;
:
(4.49)
The stationary set E (4.49) is shown dashed in Fig. 4.12. Simulation of the nonlinear
system with six dierent initial conditions conrms the stability of the set. All trajectories terminate in E, as shown in Fig. 4.12. The perspective looking down onto
the x2 -x3 plane, given in Fig. 4.13, conrms that the second component of the state
indeed converges to zero, since the various trajectories all end in the corresponding
segment of the x3 -axis.
4.8 Conclusions
This chapter establishes absolute stability criteria for systems with multiple hysteresis
and slope-restricted nonlinearities. Using Popov's indirect control form as an analytical framework, a Lyapunov stability proof is developed to guarantee stability for
these two classes of nonlinear systems. The analyses for the two dierent cases are
eectively unied by introducing a transformation that converts either nonlinearity
into a passive operator. In the hysteresis case, the Lyapunov function includes an
integral term that is dependent on the nonlinearity input-output path, while the corresponding Lyapunov term for the memoryless nonlinearity is not. As a result of the
new analysis, early work performed by Yakubovich for a scalar hysteresis is extended
to handle multiple nonlinearities, and recent work on multiple slope-restricted nonlinearities is further generalized. The stability guarantee is presented in terms of a
simple linear matrix inequality (LMI) in the given system matrices, and a certain
CHAPTER 4. MULTIPLE HYSTERESIS NONLINEARITIES
98
All solutions converge to x2 = 0.
0.5
0.4
0.3
0.2
x3
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
x2
0.1
0.2
0.3
0.4
0.5
Fig. 4.13: As viewed in the x2 -x3 plane, the set E appears as a segment of the x3 -axis
in which all trajectories terminate.
residue matrix condition that must be satised. Asymptotic stability is with respect
to a subset of state space that contains all equilibrium positions of the nonlinear
system. Descriptions of these stationary sets for several common hysteresis types are
given in detail. Simple numerical examples are then used to demonstrate the eectiveness of the new analysis in comparison to other recent results, and graphically
illustrate state asymptotic stability. By contrast to the previous work, the analysis
allows for non-strictly proper systems and, except for trivial cases such as a diagonal
system matrix, the stability multiplier is allowed to be more general and leads to less
conservative stability predictions.
Chapter 5
Reduced Order Control Design
While there has been much work done in recent years on robust control synthesis,
most frameworks are constrained to produce controllers that are full order, whereby
the controller and plant transfer functions have an equal number of poles. Reduced
order design algorithms are required in the cases when robust performance is critical
but the controller order must be constrained due to limitations of control hardware,
or excessive order of the plant. This chapter presents LMI-based algorithms that explicitly synthesize reduced order robust controllers. Order reduction is accomplished
by treating the controller order as part of a multi-objective optimization. As expected, the resulting controllers typically achieve degraded performance as the order
is further constrained. This trade-o is explored in this chapter with new algorithms
for the basic H1 and robust H1 cases. The robust synthesis algorithm yields controllers that give optimal closed loop L2-gain performance for systems having norm
bounded uncertainties by performing a sequence of convex optimizations over LMI
constraints. These basic routines are then extended to solve for controllers that are
robust to sector bounded, memoryless and hysteresis nonlinearities. Control design
for systems with sector bounded memoryless nonlinearities with a stability guarantee
based on the Popov criteria is referred to as Popov/H1 control design, and is widely
known to be a nonconvex problem due to the bilinear form of the corresponding matrix inequality constraints (i.e., BMIs). The new algorithm presented here solves this
BMI problem while yielding controllers that have order lower than the plant. This
99
100
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
new synthesis technique is then adapted to produce H1 controllers for systems with
hysteresis, using the stability criteria detailed in Chapter 3. The utility of the basic
H1, robust H1 and Popov BMI routines is demonstrated using a typical benchmark
system with uncertainty; while the hysteresis design algorithm is applied to synthesize
a reduced order loop shaping controller for a system with a Preisach nonlinearity.
5.1 Introduction
The popularity of linear matrix inequalities as a framework to analyze the stability
of uncertain systems and to design linear robust controllers has grown rapidly over
the last ve years. Linear matrix inequalities allow the user the freedom to express
diverse concepts such as Lyapunov stability, passivity and energy gain all in a single
compact notational form [BEFB94]. Recently LMI's have found extensive use in
H1 multi-objective control design [Gah96, GA94, Iwa93, SIG97]. Most of this work
involves full-order design in which the compensation has the same order as the plant.
However, it is commonly reported in these works that the order of controller is tied
directly to the rank of a certain positive matrix of the form:
"
R I
M=
I S
#
(5.1)
that occurs in the design process, where the dimensions of R and S correspond to
that of the plant and controller, respectively. Ultimately, minimum order control
design requires solving for positive matrices, R and S , that simultaneous minimize
the rank of M and satisfy the set of constraints that guarantee a preselected performance level. Unfortunately, while the performance constraint set is convex, the
minimum rank set is not, and therefore the joint problem is not suitable for current LMI software [GNLC95, WB96]. Several attempts have been made to overcome
this problem. In [GI94], for example, the eigendecomposition of the nonconvex constraint is used to produce an approximate subgradient for a descent direction in an
optimization routine. It is noted, however, that this technique is numerically cumbersome in practice, and the nondierentiability of the constraints often leads to
5.1. INTRODUCTION
101
convergence problems. A more recent approach utilizes an alternating projection
algorithm [GS94, BG96, GB98] which produces minimum order controllers by projecting the two matrices (R and S ) onto the convex and nonconvex constraint sets.
As reported in Ref. [GB98], the alternating projection scheme is often not necessary
in practice, and that minimizing the linear objective c = Trace(M ) often leads to the
minimum order stabilizing controller. The connection between the rank of the matrix
M (5.1) and its trace is explored in recent preliminary work by Mesbahi [Mes99]. Essentially, Ref. [Mes99] points out that positive matrices of the form (5.1) that satisfy
the closed loop stability constraints form a type of set referred to as a hyperlattice, and
minimum rank elements in this set will also have minimum trace (see also [MP97]).
In this chapter this observation about Trace(M ) is used as the basis to generate
reduced order controllers. In particular, the Trace() function is treated as a convex
relaxation of the Rank() objective to develop LMI based algorithms to synthesize
controllers that optimize an H1 performance objective subject to an order condition
on the controller. First, we incorporate the trace objective into a bisection algorithm
to perform a Pareto optimal investigation that trades o controller order for performance. We then extend the design objective to include a robustness constraint in
addition to performance. For this case the objective function has two parts, one part
pertaining to closed loop performance and the other corresponding to compensation
order. This approach is used to develop synthesis algorithms for systems with both
norm bounded uncertainties, and uncertainties that can be modeled as memoryless,
sector bounded nonlinearities (such as saturation, or linear parametric uncertainty).
In the latter case, the algorithm yields solutions that are referred to as reduced order Popov/H1 controllers, since the stability constraint used is based on the Popov
criterion. Using a three-mass benchmark problem, we show that this algorithm consistently produces reduced order controllers that give better performance than those
designed using the robust H1 algorithm. Finally, an algorithm that parallels the
Popov synthesis routine is developed based on the stability analysis from Chapter 3
that produces reduced order controllers for systems with hysteresis nonlinearities.
The utility of the new algorithm is demonstrated by solving an H1 loop shaping
problem for a system with a Preisach hysteresis.
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
102
The chapter is organized as follows. In x5.2, four reduced order synthesis problem
statements are dened. The dierent cases considered,
Reduced order H1,
Robust/H1,
Popov/H1, and
H1 for systems with hysteresis
are presented in a uniform manner, with all design parameters concisely detailed.
These problem denitions are then followed in x5.3 with the corresponding LMI/BMI
design algorithms which can be used to solve these problems. Numerical examples are
provided in x5.4 to compare the new reduced order design approach to a balanced realization/truncation technique using a typical benchmark problem, and demonstrate
H1 control design for a system with hysteresis.
5.2 Synthesis Problem Statements
The design for the controller K (s) that will achieve a closed loop L2-gain across
some performance channel using an LMI framework is well documented in the literature [Gah96, GA94, Iwa93, SIG97]. All presentations start with the Bounded Real
Lemma [BEFB94, p. 23], and use the Elimination and Completion Lemmas to convert the problem into an optimization over a set of convex constraints which is readily
solved using available LMI solvers [WB96, GNLC95]. Here we use those well established techniques to arrive at the equivalent convex optimization problem for reduced
order solutions. The typical problem, depicted in Fig. 5.1, is to design a proper, linear
controller:
"
#
s Ac Bc
K=
;
(5.2)
Cc Dc
5.2. SYNTHESIS PROBLEM STATEMENTS
w
103
z
G(s)
u
y
K (s)
Figure 5.1: Standard H1 control design framework.
that will achieve a closed loop L2-gain of from input disturbance w to performance
output z of the linear plant G(s), given as:
2
3
A Bw Bu 7
G = 4 Cz Dzw Dzu 75 :
Cy Dyw 0
s 6
6
5.2.1
H1 Control
(5.3)
By starting with the small gain LMI (2.11) expressed in terms of the closed loop
system matrices, and applying the Elimination Lemma 2.2.2, it can be shown ([Gah96,
GA94, Iwa93, SIG97]) that there exists a controller of the form (5.2) achieving the
H1 upper bound if and only if there exist positive denite matrices R and S such
that the conditions:
2
RCzT
3
Bw 7
Cz R
;I Dzw 75 U? < 0
?4
T ;I
BwT
Dzw
2
3
T S + SA SBw C T
A
z 7
6
T
T
T
6
V? 4 Bw S
;I Dzw 75 V? < 0;
Cz
Dzw ;I
6
UT 6
AR + RAT
(5.4a)
(5.4b)
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
104
are satised, with
2 "
3
#?
2 "
Bu
6
0 77
6
U? =
Dzu
5 ; and V? = 4
0
I
and the matrices R; S related by:
6
6
4
"
R
I
"
R
Rank
I
I
S
I
S
#
#
CyT
T
Dyw
#?
3
0
I
0
7
7
5
0
(5.5a)
n + nc:
(5.5b)
The matrix pair R; S that satises (5.4) is a convex set (this follows simply because
the set is linear in R; S ). Following the notation in Ref. [GB98] we refer to this convex
set simply as:
o
n
(5.6)
;convex = (R; S ) R; S 2 Sn; (5:4a{b) :
However, as noted, the set described by the rank condition (5.5) is non-convex, and
will be referred to by:
n
o
Zn = (R; S ) R; S 2 Sn ; (5:5a{b) :
(5.7)
c
Therefore, there exists a controller of the form (5.2) that achieves a given performance
level , if and only if there exists a pair of matrices R; S > 0 such that (R; S ) 2
;convex T Zn . The optimal control problem is now stated below.
c
H1 Control Problem
Find the matrix pair R; S that solves the optimization problem:
min : T
such that : R; S > 0; (R; S ) 2 ;convex Zn :
(5.8)
c
The pair R; S that solves (5.8) completely parameterizes the optimal H1 controller (5.2).
An algorithm which solves this reduced order H1 control problem (5.8) is detailed in
x5.3.1.
5.2. SYNTHESIS PROBLEM STATEMENTS
105
Controller Reconstruction
Given the matrix pair R; S , performance level and desired controller order nc, recovering the optimal controller requires a feasible solution of a linear matrix inequality.
The rst step in the procedure, often called controller reconstruction, requires rst
the decomposition of the quadratic stability matrix (corresponding to the controller
subspace):
Q = R ; S ;1;
(5.9)
using the singular value decomposition (svd, [HJ92]), as
W; ; W T = svd(Q):
(5.10)
Selecting the columns of W corresponding to the nc most signicant singular values,
Wr = [w1 ; : : : ; wn ]
(5.11)
c
and the reduced order matrix of singular values r = diag(1; : : : ; n ) then allows
the formation of the reduced order, stability matrix:
c
"
#
S Wr
S~ =
:
(5.12)
WrT r
Recovering the controller K then requires the solution of the LMI feasibility problem
~ V~ T + V~ K T U~ T < 0;
M + UK
(5.13)
where M is dened in terms of the plant matrices and performance level as:
2
3
T
T
~
~
~
A
S
+
SA
SB
C
t
w;t
t
z;t 7
6
T
T 7:
6
~
M = 4 Bw;tS
(5.14)
;I Dzw
5
Cz;t
Dzw ;I
The system matrices appearing above are simply those that result by applying the
Elimination Lemma 2.2.2 to the closed loop H1 LMI (2.11). The matrix elements
comprising (5.14) have the reduced order controller dimension nc, and are given as
"
A 0
At =
0 0n
#
c
"
#
"
#
T
B
T = Cz :
; Bw;t = w ; and Cz;t
0
0
(5.15)
106
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
The outer matrices in the reconstruction inequality (5.13) are
2
3
2
3
T
~SBt
Ct
~U = 664 0 775 ; and V~ T = 664 D2T;t 775 ;
D1;t
0
where similarly,
"
#
"
#
0 Bu
0 I
Bt =
; Ct =
;
I 0
Cy 0
and
"
#
h
i
0
D1;t = 0 Dzu ; D2;t =
:
Dyw
(5.16)
(5.17)
(5.18)
5.2.2 Robust H1 Control
Following the formulation for H1 control, we consider now the corresponding robust
control problem in which the plant has uncertainty , as is depicted in the standard
three-block system set-up shown in Fig. 5.2. As is common practice in control theory
[SP96], the optimal control design for the uncertain plant is achieved using either the
Bounded Real Lemma (small gain), or the Popov criteria. In that framework, we seek
a linear controller K (s) for a plant G that has norm-bounded uncertainty captured
by the -block, where 2 , with
= f 2 RH1 : kk1 < 1g :
(5.19)
The performance channel is then scaled by a constant that is used to determine
the achievable performance of the closed loop system. The scaled system is dened
as:
2
3
pB
A
Bp
Bu
w
6
7
pD
7
C
D
D
q
qp
qw
qu
s 6
7
G = 66 p
(5.20a)
p
p
7
C
D
D
D
4
z
zp
zw
zu 5
pD
Cy
Dyp
0
yw
2
3
A Bw Bu 7
6
D 7 ;
= 64 Cz Dzw
(5.20b)
zu 5
Cy Dyw
0
5.2. SYNTHESIS PROBLEM STATEMENTS
107
p
q
G(s)
w
z
u
y
K (s)
Figure 5.2: Robust H1 control design set-up includes additional channel for system
uncertainty ().
where we articially combine the uncertainty and scaled performance channels into
a single channel. Making the substitutions, Bw Bw , etc., in the set of inequalities (5.4), we dene a new convex set ;convex in terms of the scaled inequalities.
Thus, we say there exists a robust controller that stabilizes the system G for all uncertainties 2 if and only if there exists a pair of matrices R; S > 0 such that
T
(R; S ) 2 ;convex Zn , with the parameter xed at = 1. We can now state the
robust H1 control problem.
c
Robust H1 Control Problem
Find the matrix pair R; S that solves the convex optimization problem:
max : T
subject to : (R; S ) 2 ;convex Zn ; = 1:
(5.21)
c
The pair R; S that solves (5.21) parameterizes the controller that achieves optimal,
robust H1 performance. The optimal performance is 1=opt, where opt solves (5.21),
and is guaranteed for all 2 . The algorithm which produces robust H1 controllers, detailed in problem (5.21), is given in x5.3.2.
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
108
Controller Reconstruction
Recovering the robust H1 controller is accomplished by solving LMI feasibility problem,
~ V~ T + V~ K T U~ T < 0;
M + UK
(5.22)
where, after combing the uncertainty and performance channels, the matrices M ; U~
and V~ are identical to those described in x5.2.1.
5.2.3 Popov/H1 Control
The robust design approach in x5.2.2 is known to result in conservative controllers if
the plant uncertainty can be modeled as a sector-bounded, memoryless nonlinearity.
This occurs, for instance, when the plant G includes a constant real parameter with
a value only known to within a certain tolerance. A better approach in such cases
is to use the Popov stability criterion, which can be much less restrictive than the
small gain constraint [BH97b]. Robust performance for a system with uncertainty
2 sector[0; 1], using this criterion is guaranteed by the LMI [BEFB94, p. 122]:
2
6
6
4
A~T P + P A~ + C~zT C~z P B~p + A~T C~q + C~ T T P B~w
~ p + BpT C~ T ; 2T C~ B~w
()T12
CB
B~wT P
B~wT C~ T ; 2 I
3
7
7
5
0;
(5.23)
where ; T 2 Rn n are diagonal, non-negative and referred to as the stability multipliers. The system matrices in (5.23) are assumed to represent the closed loop
dynamics depicted in Fig. 5.2. Note here that unlike the robust H1 case, the uncertainty is treated separately as a passivity constraint and not combined with the performance channel. Thus, the scaling between the two channels is not needed here,
and instead, the LMI contains an additional row and column to satisfy the robustness
and performance conditions simultaneously. To employ a Popov analysis, the simple
sector transformation depicted in Fig. 2.3 can be used to convert a norm bounded
uncertainty, 2 sector[;1; 1], to one with a positive sector, 2 sector[0; 1]. In the
Using a = and b = 1, as discussed in x2.1.
q
1
2
q
5.2. SYNTHESIS PROBLEM STATEMENTS
109
sequel, it will be assumed that the transformation required to re-sector the nonlinearity has been applied, and the resulting open loop system for consideration has the
form:
2
3
A Bp Bw Bu
6
7
7
C
0
0
0
q
s 6
7:
GP = 66
(5.24)
7
4 Cz 0 Dzw Dzu 5
Cy 0 Dyw 0
Note that the matrices Dqp, Dqw , Dqu, Dzp, and Dyp are assumed zero to simplify the
application of the Popov stability analysis. The controller, initially assumed full-order
as dened by (5.2), can again be eliminated from the Popov/H1 constraint (5.23)
resulting in the equivalent pair inequalities that guarantee existence of the Popov/H1
controller:
2
AR + RAT Bp + RAT CqT + RCqT T
6
6
()T12
Cq Bp + BpT CqT ; 2T
T
U?66
Cz R
0
4
T
T
Bw
Bw CqT 2
AT S + SA SBp + AT CqT + CqT T
6
6
()T12
Cq Bp + BpT CqT ; 2T
T
V? 66
BwT S
BwT CqT 4
Cz
0
where the outer matrices are
22
3?
Bu 7
66
66
7
U? = 666 4 Cq Bu 5
Dzu
4
0
3
3
RCzT Bw
7
0 Cq Bw 77
U <0
;I Dzw 75 ?
T
Dzw
;I
3
SBw CzT
7
Cq Bw 0 77
T 7V? < 0;
;I Dzw
5
Dzw ;I
22
w
(5.25b)
3
3?
CT
7
66 y 7
66
0 77
0 75
6
7 ; V? = 6 4
7
6 DT
yw
5
4
In
0
(5.25a)
0
In
7
7
7
7
7
5
:
(5.26)
z
These constraints have the same form as those in (5.4). Note that these inequalities
are again linear in the pair R; S , and in the multiplier matrices ; T . However, the set
is not jointly linear in both pairs of variables since products of R with and T appear
in the (1; 2) entry of the center matrix of constraint (5.25a). Subsequently, Eqn. (5.25)
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
110
is considered a bilinear matrix inequality (BMI), and is thus not convex jointly in the
two sets of variables. The common approach (see [BH97b], and references therein)
is to consider convex subsets that result when xing either the pair R; S or the
multipliers ; T: Fixing ; T gives the set:
n
o
;p(; T ) = (R; S ) (5:25a{b) ;
(5.27)
and, similarly, holding R; S xed denes the convex set:
;p(R; S ) =
(
"
S
I
(; T ) (5:23); P =
I (S ; R;1 );1
# )
:
(5.28)
Note, that for ;p(R; S ) it is assumed that the feasible Popov/H1 controller corresponding to the given R; S is used to form the closed loop system matrices in the
Popov LMI (5.23). Details of this reconstruction are discussed below. For convenience, dene the set of diagonal m m non-negative matrices as Dm+ . Now the
quadruples that satisfy the sets above can be used to dene a Popov solution. Specifically, let ;Popov : Sn Sn Dm+ Dm+ be dened as:
;Popov =
(
(R; S ) 2 ;p(; T )
(R; S; ; T )
(; T ) 2 ;p(R; S )
)
:
(5.29)
We can now state the Popov/H1 control problem as an optimization problem similar
to the H1 and robust H1 cases. However, unlike these previous cases, the Popov/H1
is not a convex optimization problem since, as alluded to, the constraint set ;Popov is
not convex.
Popov/H1 Control Problem
There exists a controller that robustly stabilizes the system if there exists a quadruple
(R; S; ; T ) 2 ;Popov . The minimum that can be achieved for any such quadruple
is the optimal Popov/H1 solution. This leads to the following optimization problem:
min (
subject to:
(R; S; ; T ) 2 ;Popov
(R; S ) 2 Zn
c
(5.30)
5.2. SYNTHESIS PROBLEM STATEMENTS
111
An iterative algorithm which performs this minimization over the BMI constraints is
detailed in x5.3.3.
Controller Reconstruction
Having solved (5.30) the corresponding controller can be recovered by nding a K
satisfying
~ V~ T + V~ K T U~ T < 0:
M (; T ) + UK
(5.31)
The matrices comprising this LMI are detailed in Appendix C.1.
5.2.4
H1 Control for Systems with Hysteresis
The robust stability test (3.24) for systems with hysteresis given in terms of optimization over an LMI constraint containing multipliers and the gain term has the
same form as in the Popov LMI (5.23). Because of this, the control design problem
for systems with hysteresis can be stated in a manner analogous the Popov synthesis
case. Now, instead of a sector bounded, memoryless nonlinearity, it is assumed the
{block in the closed loop system, Fig. 5.2, represents a hysteresis that obeys the
properties dened in x3.3.2. We seek a controller K (s) that minimizes the L2{gain ,
while guaranteeing stability in the presence of the nonlinearity. For notational consistency with the previous sections, we rewrite the corresponding hysteresis inequality
from (3.24) as:
2
6
6
4
A~T P + P A~ + C~zT C~z P B~p + C~z D~ zp ; C~qT1 ; C~qT2T P B~w + C~zT D~ zw
()T12
D~ zpT D~ zp + 2I ; D~ qp ; D~ qpT D~ zpT D~ zw ; D~ qw
T D
~ zw ; 2 I
()T13
()T23
D~ zw
3
7
7
5
0;
(5.32)
where again ; T 2 Rn n are the diagonal, non-negative stability multipliers. Here
we assume the closed loop system matrices in (5.32) have been augmented with the
multiplier dynamics, as described by Eqns. (3.18{3.20). Assuming a full-order controller, as given by (5.2), the same approach used to formulate the stability constraints
for the previous algorithms results in the corresponding inequalities that guarantee
q
q
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
112
existence of the H1 controller for systems with hysteresis:
2
AR + RAT Bp ; RCqT1 ; RCqT2T ;RCzT
6
T
6
()T12
2I ; Dqp ; Dqp Dzp
T
U? 66
;Cz R
Dzp
;I
4
T T
BwT
Dqw
Dzw
2
AT S + SA SBp ; CqT1 ; CqT2 T SBw
6
6
()T12
2I ; Dqp ; Dqp Dqw
T
V? 66
T BwT S
Dqw
;I
4
;Cz
Dzp
Dzw
where the outer matrices are
2 2
Bu
6 6
6 6
U? = 666 4 Dqu
Dzu
4
0
3?
7
7
5
3
3
Bw
7
Dqw 77
U < 0
Dzw 75 ?
;I
3
;CzT
7
DzpT 77
V? < 0;
T 7
Dzw
5
;I
2 2
CyT
7
6 6
6 6
0 77
DypT
6
7 ; and V? = 6 4
T
7
6
Dyw
5
4
In
0
w
(5.33b)
3
3?
7
7
5
(5.33a)
0
In
7
7
7
7
7
5
:
(5.34)
z
These constraints have the same form as those in (5.25), and again these inequalities
are bilinear in the matrix pairs (R; S ) and (; T ), with products of R with and T
appearing in the (1; 2) term of (5.33a). Following the same approach taken for the
Popov case, we dene the convex subsets that result when xing either the pair R; S
or the multipliers ; T: In this case, xing ; T gives the set:
n
o
;h(; T ) = (R; S ) (5:33a{b) ;
(5.35)
and, similarly, holding R; S xed denes the convex set:
;h(R; S ) =
(
"
S
I
(; T ) (5:32); P =
I (S ; R;1);1
# )
:
(5.36)
Note that for ;h(R; S ) it is assumed that the stabilizing controller satisfying the
H1 performance bound is formed using the given R; S and used to construct the
closed loop system matrices in the hysteresis LMI (5.32). Again, in terms of the set
of diagonal m m non-negative matrices Dm+ , the quadruples that satisfy the sets
5.3. ALGORITHM DESCRIPTIONS
113
above can be used to dene an H1 controller solution for a system with hysteresis.
Specically, let ;hyst : Sn Sn Dm+ Dm+ be dened as:
n
o
;hyst = (R; S; ; T ) (R; S ) 2 ;h(; T ); (; T ) 2 ;h(R; S ) :
(5.37)
We can now state the hysteresis control problem as an optimization problem analogous
to the Popov/H1 problem. Note again that the hysteresis synthesis problem is not
a convex optimization problem for the same reasons stated for the Popov/H1 case.
H1/Hysteresis Control Problem
There exists a controller K (s) of the form (5.2) that robustly stabilizes the system
G(s) having a hysteresis nonlinearity, as shown in Fig. 5.2, if there exists a quadruple
(R; S; ; T ) 2 ;hyst. The minimum that can be achieved for any such quadruple is
the optimal H1/hysteresis solution. This leads to the following optimization problem:
min (
subject to:
(R; S; ; T ) 2 ;hyst
(R; S ) 2 Zn
(5.38)
c
An iterative algorithm which performs this minimization over the BMI constraints is
detailed in x5.3.4.
Controller Reconstruction
As was the case for the Popov controller, with the solution to (5.38) the corresponding controller can be recovered by following the reconstruction procedure detailed in
Appendix C.2.
5.3 Algorithm Descriptions
How can these control problems be practically solved? In particular, for a desired order of controller, what are the algorithms that can be used to numerically implement
solutions for these problems to yield the optimal solution? We answer these questions
114
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
with a series of algorithms in this section. Naturally, they range in complication from
easiest to hardest, with the Popov/H1 and hysteresis designs requiring the most sophistication. Note that in each case, the algorithms produce a pair of matrices (R; S )
that completely parameterize the reduced order solution. To obtain the controller K
from a pair (R; S ) that solves one of the optimization problems requires the solution
of a feasibility problem:
M (R; S ) + UKV T + V K T U T < 0
(5.39)
where U; V are the orthogonal complements of the corresponding matrices above, and
M (R; S ) is a constant matrix involving the open loop system matrices, the specied
performance level, the pair (R; S ) and when appropriate, the stability multipliers
(; T ). The matrix M is simply the portion of the matrix from inequalities (5.23,
5.32, etc.) not dependent on the controller term K ; simplied versions of M appear
as the center matrix in the expressions (5.25) and (5.33). These matrices are easily
derived by employing the Elimination, Completion Lemmas and Schur complements
in a now standard technique well documented in [Gah96]. Some of the details are
omitted above for brevity, and instead, the nal results are given in sections x5.2.1,
C.1, and C.2.
We illustrate the use of the reduced order H1, robust H1 and Popov/H1 algorithms using a benchmark three mass-spring system (see Appendix B for details).
The benchmark system is characterized by a rigid body mode and two exible modes,
one of which is non-colocated. By including a small amount of damping in between
the masses, the minimum order stabilizing controller is known in advance to be rst
order. This simply corresponds to a rst order lead network required to stabilize
the pole pair at the origin that models the system rigid body mode. We can expect
however, that any controller designed to be rst order will not perform very well due
to the presence of the lightly damped, non-colocated mode.
5.3.1 Reduced order H1 design
Solving for reduced H1 controllers is fairly simple, and we present one approach
that utilizes a simple bisection search and extends the algorithms for reduced order
5.3. ALGORITHM DESCRIPTIONS
115
1. Set upper, lower performance bounds, convergence tolerance: u; l ; tol
2. Set desired controller order: nc;des
3. Repeat f
(a) = 21 (U + L)
(b) Solve: min: Tr(R + S ); subject to: (R; S ) 2 ;convex
(c) Decompose matrix: [U T ; ; U ] = svd(R ; S ;1)
(d) nc = length(diag())
(e) if nc > nc;des, l = , else u = g
until: (u ; l ) < tol l
4. Reconstruct controller, solving (5.13).
Table 5.1: Algorithm for reduced order H1 synthesis
stabilizing compensation design in [GB98, Mes99]. Specically, a ve-step iteration
that nds the best xed-order H1 controller is provided in Table 5.1. This algorithm produces a controller of prescribed order that gives the best H1 performance.
It performs a bisection, or line search, in order to determine the best performance
possible for the given controller order. The upper and lower bounds on the search
are adjusted according to whether or not a feasible controller exists that achieves the
particular performance level : The order of the controller is equal to the number of
non-zero elements of the singular value decomposition (svd()) of the matrix R ; S ;1
(see [Gah96, GA94, Iwa93] for discussion). Of course, the accuracy of the search can
be adjusted using the tolerance level tol:
For the benchmark problem, the six controllers (ve reduced and one full order)
were generated using this algorithm. The corresponding maximum singular value
curves are depicted in Fig. 5.3, along with the bar graph which summarizes the
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
116
Performance curves for dierent order controllers.
2
10
open loop
nc=1
nc=2
nc=3
nc=4
nc=5,6
1
max
10
0
10
-1
10
-1
10
0
!
(rads/sec)
10
1
10
Figure 5.3: Reduced order designs for benchmark problem illustrate varying per-
formance levels; peaks in maximum singular value curves, (!), are
reduced as order of controller is allowed to increase. (Recall that
opt = maxw (w).)
performance as a function of controller order, shown in Fig. 5.4. Clearly, as expected,
achievable performance reduces with controller order. Controller orders of ve and
six both achieve the H1 limit of = 3:7, and have at max curve typical of a fullorder optimal controller. As the controllers are restricted to fourth order and lower,
peaks appear in the max curves since the reduced order controllers cannot completely
notch the exible body modes; lower order controllers result in higher peaks. This
relationship between controller order and performance is captured by the bar graph in
Fig. 5.4. In a sense, Fig. 5.4 can be considered a Pareto optimal curve which depicts
the trade-o between two competing design objectives.
5.3. ALGORITHM DESCRIPTIONS
117
120
100
opt
80
60
40
20
0
1
2
controller order nc
3
4
5
6
Figure 5.4: Bar graph depicts trade o between controller order and closed loop
performance; opt increases from unconstrained limit of = 3:7 with
nc = 6, to = 117 when controller reduced to a single state (nc = 1).
In addition to the performance curves, the root loci for the various systems are
shown in Fig. 5.5. The progression of control designs can clearly be seen in these
plots. Limited to rst order, the controller is a simple lead which draws the rigid
body poles just slightly into the left half plane. The controller imposes little authority
beyond this since any more would destabilize the non-colocated pole pair. A second
order controller immediately changes strategy and places a minimum phase notch
which stabilizes both the rigid body and non-colocated modes. The resulting eect
on performance is signicant, reducing the performance by nearly one-fourth, from
110 to less than 30: As the order of control increases, the notching that occurs becomes
more pronounced, with non-minimum phase zeros used for orders above nc = 4: Note
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
118
nc = 2
2
2
1
1
imag
imag
nc = 1
0
0
-1
-1
-2
-2
-0.1
real
nc = 3
0
0.1
-0.2
2
2
1
1
imag
imag
-0.2
0
-1
-2
-2
0
real
nc = 5
-0.4
2
2
1
1
imag
imag
-0.2
0
-1
-2
-2
-0.8
-0.6
-0.4
real
-0.2
0
0
0.1
-0.2
0
real
nc = 6
0
-1
-1
real
nc = 4
0
-1
-0.4
-0.1
-1
-0.8
-0.6
-0.4
real
-0.2
0
Fig. 5.5: Root locus for systems with controllers of varying order.
5.3. ALGORITHM DESCRIPTIONS
119
1. Set performance channel bounds, convergence tolerance: u; l ; and tol
2. Set desired relative weight: 2 [0; 1]
3. Repeat f
(a) = 21 (U + L)
min: + (1 ; )Tr(R + S )
subject to: (R; S ) 2 ;convex
(c) if > 1, u = , else l = g
until: (u ; l ) < tol l .
4. Reconstruct controller, solving (5.22).
(b) Solve:
Table 5.2: Synthesis algorithm for reduced order, robust H1 control
that the controllers of the fth and sixth (full order) controllers are virtually identical;
in fact, the only dierence between the two is a near perfect pole-zero cancellation,
and thus yield the same root loci and the same (at-line) performance curves.
5.3.2 Reduced Order Robust H1 control
The design for robust H 1 controllers requires only a slight modication to the previous algorithm. Here the bisection is on the performance channel weighting , and
not the control order explicitly. The objective function now consists of two parts: one
part aecting controller order, Trace(R + S ) and the other performance : The same
trade-o between order and performance is achieved by varying ; the relative weight
between the two cost components. The algorithm is described in Table 5.2. Note
that the iteration on continues until the scaled performance = 1; the optimal
performance that results is then opt = 1=.
Once again, using the benchmark problem, and now assuming that the spring
120
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
constant between the second and third masses is only known to within 10%, reduced
order controllers were produced by running the algorithm for a range of values.
The resulting performance curve for the swept values of is depicted in Fig. 5.6;
note that the controller orders corresponding to the performance levels are labeled
alongside the curve. As expected, as increases, more weight is put on in the cost
objective, and results in improved performance. Of course, the cost improvement
comes at the expense of controller order. Note that the order increases monotonically
with , along with the performance improvement. As ! 1, the tends toward the
unconstrained H1 limit value of 3:7: It is interesting that a rst order controller could
not be found with this algorithm. This is most probably due to the conservativeness
of the small gain stability guarantee. Essentially, this suggests that a 5% variation
in the spring constant is too much uncertainty to guarantee stability when using the
small gain criterion and limited to rst order controllers.
5.3.3 Reduced order Popov/H1 control
The Popov/H1 algorithm is slightly more complicated than the previous two cases.
The cost objective once again has two parts, but each step of the iteration involves
two optimization steps. The rst step optimizes over the controller matrices, while
the second step is with respect to the multipliers. The algorithm is summarized in
Table 5.3. This algorithm was executed for a range of and the performance and
corresponding controller orders are plotted in Fig. 5.6, along with the robust H1
results. There are several interesting aspects to this data. First, as we might expect,
the performance curve for the Popov controllers lies strictly below that for the robust
H1 designs. This follows since the Popov constraint is less conservative when the uncertainties are assumed to be sector bounded and memoryless. Also, the performance
of the controllers monotonically improves as the performance weight increases, just
as the previous case. However, the controller order does not strictly increase as performance improves. For example, several second order compensators are produced
with weighting values between 0:55 and 0:75 that give better performance than the
third and fourth order controllers that result from lower values of : It is dicult to
5.3. ALGORITHM DESCRIPTIONS
121
Robust H1 and Popov/H1 Control: Reduced Order comparison.
3
10
nc=2
Robust H∞
Popov H
∞
H∞ limit
nc=3
2
10
nc=3
3
1
3
nc=4
opt
3
4
nc=4
2
2
nc=4
2
nc=4
1
4
10
5
5
5
0
10
0
0.1
0.2
0.3
0.4
0.5
0.6
relative objective weight ( )
0.7
0.8
0.9
1
Figure 5.6: Performance/controller order trade o for robust and Popov design using benchmark problem with uncertainty. Note that because Popov
criteria is less restrictive, reduced order Popov controllers consistently
outperform robust H1 designs.
attribute this observation to anything other than the known nonconvexity of the sets
over which the optimization is performed.
Note also that a rst order controller is produced using the Popov algorithm that
has better performance than the non-robust H1 controller. This seems to be an
indication that while using the Trace() objective might be a good convex approach
to minimizing rank, it is still just an approximation. This is evident here since, even
though the Popov optimization has the additional stability constraint, the routine
was able to nd a lower minimum.
122
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
1. Initialize controller K1(s) and multipliers (; T )
2. Set desired relative weight: 2 [0; 1]
3. Repeat f
(a) Solve: min: + (1 ; )Trace(R + S ); subject to: (R; S ) 2 ;p(; T )
(b) Find feasible controller, Kk (s), by solving problem (5.31)
(c) Form closed loop system matrices
(d) Solve: min: + (1 ; )Trace(P ); subject to: (5:23) g
until: jk ; k;1j < tol k
Table 5.3: Popov/H1 control synthesis
5.3.4 Reduced order H1/Hysteresis control
Because the stability multiplier for hysteresis and Popov analyses are cast in the
same form, the resulting design problems given in (5.30) and (5.38) have the same
structure. As a result, the algorithm for robust hysteresis synthesis is very similar
to that given for the Popov control, consisting of a two part cost function, with each
iteration involving two optimization steps and a feasible controller reconstruction.
The algorithm is provided in Table 5.4. The utility of this algorithm is demonstrated
with a numerical example in x5.4.3.
5.4. NUMERICAL EXAMPLES
123
1. Initialize controller K1 (s) and multipliers (; T ) that satisfy inequality
(5.32)
2. Set desired relative weight: 2 [0; 1]
3. Repeat f
(a) Solve: min: + (1 ; )Trace(R + S ); subject to: (R; S ) 2 ;h(; T )
(b) Find feasible controller, Kk (s) following procedure in xC.2
(c) Form closed loop matrices
(d) Solve: min: + (1 ; )Trace(P ); subject to: (5:32) g
until: jk ; k;1j < tol k
Table 5.4: H1/hysteresis control synthesis
5.4 Numerical Examples
5.4.1 Reduced order H1
In order to assess the eectiveness of the new algorithms in generating optimal, reduced order controllers, we compare our results to another reduced order design approach. Depicted in Fig. 5.7 are the same results of the reduced order H1 design
algorithm along with the performance of several other reduced order sub-optimal controllers. These sub-optimal cases were designed by taking full-order LQG controllers,
designed using various control and performance weightings, and reducing the order
by means of a balanced real order reduction. Clearly, in all cases, controllers designed
using this reduced order technique could not approach the optimal performance levels
of the reduced order algorithm from x5.3.1. As the controllers are truncated, the performance degrades in each case, and the corresponding performance lines all lie above
the bar graph. For instance, reducing a full order controller with a performance of
124
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
= 25; can yield a second order controller with a = 47; however, the new algorithm
produces a second order controller with = 23, which is an improvement of about
50%: Pushing the performance of the full order LQG controllers in an attempt to
achieve better low order controllers leads to dramatic degradation in -levels. In fact,
no stabilizing rst order controller could be found using the balance real reduction
scheme.
We note that the search for good low order controllers using the balanced real,
order reduction technique was by no means exhaustive. Indeed, there may exist controllers that give better performance than those indicated by the bar graph. However,
nding the true optimal reduced order controllers is still an open (non-convex) design
problem; the algorithms presented in this chapter are an ecient, convex approach
toward that goal.
5.4.2 Popov/H1 convergence properties
As noted in previous sections, the Popov/H1 and robust hysteresis control design
problems are not convex, due to bilinear matrix inequality constraints. In general,
iterative solutions for BMI design, such as those described, are not guaranteed to
converge to a global optimum. In fact, in practice, if the algorithm converges at all,
the solution may be at a local minima, which could depend heavily on the initial condition. If the initial feasible controller is too far from optimal, the resulting controller
may not give acceptable performance. Of course, from the designer's point of view,
algorithms that do not converge reliably, require many iterations, or are sensitive to
initial conditions are of limited practical use. The reduced order approach for BMI
design presented in this chapter, however, is demonstrated to provide reliable convergence to near optimal solutions. For example, consider again the three-mass benchmark problem with 5% spring constant uncertainty. Setting = 0, the Popov/H1
algorithm was initialized with six dierent controllers, of third, fourth and fth order
having performance levels ranging from = 5 to = 85. As depicted in Fig. 5.8,
in each case, the algorithm converges within 5 iterations. All nal controllers were
5th order and, with the exception of the extreme case (3rd order initial controller), all
5.4. NUMERICAL EXAMPLES
125
150
100
opt
50
0
1
2
3
4
5
6
controller order
Figure 5.7: Comparing reduced order H1 performance to controllers reduced via
balanced real reduction, note that as controller orders are reduced, the
resulting performance curves are always above bar graph values.
controllers were within 0:25% of = 3:76: The nal case resulted in a performance
of = 3:85: Since for this system the unconstrained performance limit is 3:7, the
algorithm can be said to be robust to changing initial conditions.
Alternatively, the user may have an existing compensator, with a certain order
and performance capability, but may seek a design with either lower complexity, or
higher performance. Using the given controller as a starting condition and a choice
of depending on the objective, the Popov routine can be used for this investigation.
Choosing = 0:75 with an initial 4th order controller, the algorithm converges to
a new controller of order 2, as shown in Fig. 5.9. Of course, performance has been
sacriced with lower order ( increases from 11:9 to 20). If higher performance is
desired instead, setting = 0:95 or = 1:0 will provide the higher performing 5th
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
126
Convergence robustness to initial conditions.
2
10
nc= 3
nc= 4
nc= 4
nc= 4
nc= 4
1
10
nc= 5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
iteration
Figure 5.8: Popov/H1 algorithm insensitivity to starting conditions illustrated with
consistent convergence using a wide range of initial controllers.
order solutions. Note again that in each case, only 16 iterations were required for
each design.
The feature of the algorithm that allows reduced order solutions provides the
added benet of enhanced reliability. In fact, experience shows that forcing full order
solutions when it is not required for optimal performance can lead to ill conditioned
matrices and thus poor convergence. In particular, the algorithm, as described in
x5.3.3 calls for a singular value decomposition of the controller matrix R ; S ;1 prior
to taking its inverse for use in the controller reconstruction. Selecting the subspace
corresponding to most signicant singular values and eliminating the values near zero
tends to improve the condition of the inverse and maintain numerical reliability. Forcing a full order (6th order) solution for the simple benchmark problem, for example,
5.4. NUMERICAL EXAMPLES
127
Convergence to dierent order controllers
22
20
nc=2
nc=4
18
16
β = 0.75
β = 0.95
β = 1.00
nc=3
14
nc=4
12
10
8
6
nc=5
nc=5
4
nc=5
nc=5
2
0
2
4
6
8
10
12
14
16
18
iteration
Figure 5.9: Alternate conpensation designs of varying complexity and performance
can be investigated by simply varying the design parameter . Here,
starting with 4th order compensation, three other Popov/H1 designs
are investigated.
can lead to numerical problems as optimal performance is achieved with a 5th order
controller, and the eect of the additional state is nullied by a near perfect pole-zero
cancellationy. In this case design routines constrained to full order solutions may encounter problems. Using the full order Popov/H1 algorithm from Banjerdpongchai
[Ban97, p. 49], for example, can experience convergence diculties. Using an initial
controller with near optimal performance ( = 8:14) results in a erratic convergence,
oscillating between 7 and 40, as shown in Fig. 5.10, while the reduced order algorithm
converges steadily to a performance level of = 4:6. The poor convergence in this
case of the xed order algorithm is best explained by observing the condition number
y Recall discussion in x5.3.1.
128
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
(R ; S ;1), where z is dened as
(A) = max((AA)) :
min
(5.40)
Following the rst controller optimization, > 1015, which means that inverting R ;
S ;1 may be problematic. Indeed, the rst controller computed after reconstruction
with this matrix results in a large performance degradation, with increasing by a
factor of 12, to > 100, as depicted in Fig. 5.11. This large deviation introduces an
oscillation from which the algorithm can not recover. The reduced order algorithm,
however, experiences no such problems. The rapid convergence is accompanied by a
well conditioned controller matrix, with approximately 50 throughout the iterations.
5.4.3 Robust loop shaping for system with hysteresis
As a nal example, we consider the use of the robust synthesis method described in
x5.3.4 to design a controller for a system with hysteresis that optimizes an S=KS
mixed sensitivity cost objective [SP96, pages 369{375]. This controller will be compared to a full (xed) order solution designed using a similar BMI algorithm described
in [PH98a]. The set-up for the synthesis is shown in Figure 5.12, where the hysteresis
: q 7! p is taken to be passive, with maximum slope = 1. The plant transfer
function is given as
2
G1 (s) = 7:s5(3 +s 2;s20+:2s2s++0:11) ;
which has a pair of nonminimum phase zeros at 0:1 0:3j . The performance weights,
given as
s + 28 ; W (s) = 150s + 75 ;
W1 (s) = 350
2
500s + 1
s + 100
are used to frequency weight the loop transfer functions. The objective of the S=KS
mixed sensitivity optimization is to design a controller K (s) that minimizes the cost
objective
"
#
W1S = ;
W2KS 1
z Sometimes is called the spectral condition of a matrix [HJ91, p. 158].
5.4. NUMERICAL EXAMPLES
129
2
10
reduced order
fixed order
1
10
0
2
4
6
8
10
12
14
16
iteration
Figure 5.10: Reduced order formulation can provide more reliable convergence. As
depicted above, the xed order performance oscillates widely with
each iteration, while the reduced order algorithm converges steadily.
where the sensitivity function, S = (I + KG);1 , is a measure of the closed loop
disturbance rejection. By minimizing , we simultaneously minimize the peak gain in
the error to a disturbance w and limit the bandwidth of the controller. The reduced
order algorithm for hysteresis systems from x5.3.4 applies directly to this system.
Because for the open loop system, Gqp(0) = ;0:75, we have that Gqp(0) > ;1=
and since the system has no zero eigenvalues, we can use the parameterization of
the system given in Ref. [PH98a], and also design a xed order controller K with
dimension equal to that of the original plant with the augmented weights, that is,
Ak 2 R55. In order to obtain reduced order controller with comparable performance
to the xed order solution while eliminating any excess controller states, the relative
weighting was set at 0:95.
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
130
condition of reconstruction matrices
18
10
16
10
14
10
12
10
(R ; S ;1)
10
10
8
10
6
10
-1
full order P-Q
-1
reduced P-Q
4
10
2
10
0
10
0
2
4
6
8
10
12
14
16
18
iteration
Figure 5.11: Convergence failure of full order algorithm can be traced to poorly
conditioned controller matrix. Note that condition of corresponding
reduced order matrix is steady throughout each iteration.
To initialize the synthesis algorithms, a controller was designed that optimizes a
skewed-, or s metric [SP96, page 321].x This controller gives the best H1 performance and provides a stability guarantee for the system with a hysteresis nonlinearity
by maintaining a norm bound constraint on the robustness channel, so that for the
closed loop system jG~ qp(j!)j < 1, 8! 0: This controller serves as a good initial
condition for our synthesis algorithm since the norm bound means that the graph of
G~ qp(j!) will not enter the forbidden region of the Nyquist plane and thus existence
of an initial set of stability multipliers , , and is guaranteed. The design was
accomplished by computing the optimal H1 controller for the system G(s) scaled by
x Note, the variable is double booked here, but there should be no confusion when taken in
context.
5.4. NUMERICAL EXAMPLES
(q)(t)
131
q(t)
W1(s)
z1 (t)
p(t)
-
u(t)
y(t)
G1(s)
-
w(t)
K (s)
W2(s)
z2 (t)
Figure 5.12: S=KS mixed sensitivity synthesis set up.
the matrix
"
I 0
Km =
0 km I
#
and iterating the design on the scalar km until kG~ k1 < 1. Note here that Km is
partitioned in accordance with the uncertainty structure = diagf1 ; P g with 1 ,
and P as full, complex blocks, and by denition s = 1=km. For this particular
system the nal control design requires km = 0:01 and guarantees robust stability
since the gain through the robustness channel (simply a scalar here) is less than unity.
The resulting closed loop transfer function for the system with the -design is shown
in a Nyquist plot, Figure 5.13. Indeed for the nal design we have (G~ ) = 0:95 and
as shown in the plot, the graph of G~ qp(j!) stays within the unit circle. (See [FT92]
for more on the s performance metric.)
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
132
Nyquist of design
5
µ design
open loop
4
3
imag
2
1
0
-1
-2
-2
-1
0
1
real
2
3
4
5
Figure 5.13: Robustness test for -design requires containment in unit disk.
However, as is consistent with having km 1; the performance of the -design is
not very good. As shown in Figure 5.15, both the sensitivity S and the KS objective
exceed the desired loop shaping requirements. At low frequency, the sensitivity requirement is violated by factor of 10 while the KS criteria exceeds the desired gain by
nearly a factor of 25. This design, while not very appealing from a performance point
of view, does provide guaranteed stability. Containment in the unit circle implies that
the -design represents a valid initial condition for the multiplier synthesis algorithm.
Using the controller as the initial condition, the xed order synthesis algorithm ran
18 iterations before converging to a new controller, reaching the stopping criterion
set by tol = 0:0005. During the iteration, the performance improved by a factor of
50, with reducing from 0:75 down to a value of 0:0122 as indicated in Figure 5.16.
Similarly, the reduced order algorithm required 10 iterations to produce a 4th-order
controller with a = 0:0132, which is within 8:5% of the full order result. With either
controller, the S=KS performance objective is now satised. It is clearly evident in
5.4. NUMERICAL EXAMPLES
5
133
Nyquist of xed, reduced order designs
fixed order
reduced order
open loop
4
3
imag
2
1
0
-1
-2
-2
-1
0
1
real
2
3
4
5
Figure 5.14: Stability test for new control requires less conservative avoidance of
restricted region in the Nyquist plane. In this case, restricted region
is in the third quadrant to the left of the ;1 point on real axis.
Figure 5.15 that the desired loop shaping design goal was attained since the graphs
of the unweighted S and KS functions lie under the required shaping curves. The
robustness guarantee is indicated in Figure 5.14, where the graphs of the closed loop
transfer functions G~ qp do not enter into the restricted region of the third quadrant, to
the left of the point (;1; 0), thus satisfying the graphical test discussed in x3.4.2. It is
evident in comparing these Nyquist plots to the corresponding plot in Figure 5.13
that the robustness and performance requirements are competing. That is, in maintaining the graph inside the unit circle, the -controller had to sacrice a great deal
of performance. The new controllers, however, take advantage of the less restrictive
stability requirement and are thus able to meet the performance objectives.
Closed loop simulations comparing the responses of the system with {controller
and the reduced order design conrms the benets of the improved performance. For
the simulation the nonlinearity was chosen with unity maximum slope, with typical
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
134
Sensitivity Performance Comparison
0
z1 =w
10
µ design
Fixed order
reduced order
requirement
-1
10
-2
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
Controller Bandwidth Comparison
0
z2 =w
10
-2
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
freq (rad/sec)
Figure 5.15: Because of restrictive stability criteria of -controller, the corresponding performance does not satisfy loop shaping objective. With less
conservative analysis, the new design approach meets design requirements while providing closed loop stability guarantee.
input/output characterized in Figures 5.19 and 5.20. This is the Preisach nonlinearity
which is commonly used to model the hysteresis associated with electromagnetic
actuators [May91]. The disturbance w(t) 2 L2 is displayed in Figure 5.17 with
the time histories of the system response y(t) using the two dierent controllers.
As shown, the response y(t) for the system using the 4th -order design is smaller in
amplitude (after the initial transient) than that for the ;controller and dies out
after 50 seconds. The response y(t) decays at a much slower rate, and has amplitude
near unity after 70 seconds. The control response u(t) shows similar improvement.
The reduced order control response, as indicated in Figure 5.18 peaks at a value of
only one third of that for the ;controller. The improved response dies out after
5.5. CONCLUSIONS
135
Performance with iteration
0
10
Fixed Order
Variable Order
10
-1
4th Order
5th Order
-2
10
0
2
4
6
8
10
12
14
16
18
20
iteration
Figure 5.16: Closed loop performance converges using either reduced or xed
order algorithms, within 11 (reduced order) or 19 iterations (xed, full
order).
40 seconds while the ;control amplitude is greater than unity after 70 seconds.
Certainly, these results provide a strong conrmation as to the eectiveness of the
new technique for robust control with an L2;gain performance criteria.
5.5 Conclusions
Four dierent LMI-based algorithms for producing reduced order H1 controllers are
detailed in this chapter. All four procedures utilize the Trace() objective function as
a means to constrain the order of the controller. One algorithm simply optimizes the
H1 performance subject to an explicit constraint on the order. The other algorithms
produce reduced order robust controllers using a two part objective involving the
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
136
Closed loop output response, y(t)
6
multiplier design
µ design
disturbance
5
4
3
y(t)
2
1
0
-1
-2
-3
0
10
20
30
40
50
60
70
time (sec)
Figure 5.17: New multiplier compensation design shows better disturbance rejec-
tion, with faster settling and smaller output (y(t)) response than that
provided by -controller.
control response
4
multiplier design
µ design
3.5
3
2.5
2
u(t)
1.5
1
0.5
0
-0.5
-1
0
10
20
30
40
time (sec)
50
60
70
Figure 5.18: Improved disturbance rejection is also evident in response of control
signal u(t) to disturbance; note large energy in -control compared to
new controller.
5.5. CONCLUSIONS
137
Nonlinear input-output response (multipier design)
1.5
1
Φ(q)(t)
0.5
0
-0.5
-1
-1.5
-4
-3
-2
-1
0
1
2
3
q(t)
Figure 5.19: Hysteresis input-output: reduced order controller.
Nonlinear input-output response ( µ design)
1.5
1
Φ(q)(t)
0.5
0
-0.5
-1
-1.5
-2
-1.5
-1
-0.5
0
0.5
q(t)
1
1.5
2
2.5
3
Figure 5.20: Hysteresis input-output: ;controller.
138
CHAPTER 5. REDUCED ORDER CONTROL DESIGN
Trace() and the closed loop performance. Using the combined objective, the designer
is then free to select the relative weighting on the two parts of the objective cost in
order to trade-o performance versus controller order. In this way, these algorithms
provide valuable tools that allow control designers to perform multi-objective design
analyses in the practical situations when performance is critical and the order of the
controller must be reduced because of either real-time control hardware limitations
or the excessive order of the plant.
The Trace() is used here as a convex relaxation to the rank condition and enables
ecient design of reduced order controllers. Also, the reduced order BMI algorithm
for the design of Popov/H1 controllers is more reliable than previous xed order design techniques since the new algorithm systematically eliminates poorly conditioned
subspaces of matrices which could otherwise lead to numerical instability in the reconstruction of the controller. As a result, the new algorithm converges reliably and
quickly from a wide range of initial conditions, allowing the user to evaluate several
dierent controllers of varying complexity and performance levels.
While the Popov/H1 design is specically geared to produce controllers robust
to uncertainties that can be classied as sector bounded, memoryless nonlinearities,
the general BMI framework presented in this chapter is readily extended for reduced
order synthesis for systems with hysteresis by incorporating the stability analysis
from Chapter 3. This new algorithm oers signicant practical advantages over the
recently developed xed order procedure for hysteresis described in Ref. [PH98a]. In
addition to the numerical advantages described above, the new approach eliminates
the need to remove the constant eigenspace associated with the stability multiplier
introduced in the analysis, which can be cumbersome and in general not always possible, since it requires the open loop plant to have an invertible system matrix.{ The
benets of the new design technique are illustrated in solving a loop shaping control
problem for a non-minimum phase system. The new technique allows for improved
performance by both reducing tracking error and controller bandwidth compared to
that for a suitably designed ;controller. This improvement is achievable with the
{ When not invertible, the system matrix includes a zero eigenvalue. In practice, approximate
xed order solutions by replacing the constant eigenspace with a subspace characterized by a very
small and stable eigenvalue.
5.5. CONCLUSIONS
139
new synthesis procedure since the new method utilizes stability constraints which are
less restrictive than those of the ;synthesis, or small gain design procedures.
Indeed, the general BMI technique developed here for memoryless and hysteresis
nonlinearities has broader application. In the next chapter it is used in the practical
case of control synthesis for unstable systems that have saturating actuators.
140
Chapter 6
Control Design for Systems with
Saturation
A large number of problems occurring in the practice of control theory can be thought
of as feedback stabilization of systems using bounded controls. Indeed, the control
input to any realizable system is ultimately limited by physical constraints. In cases in
which a linear plant has eigenvalues with positive real part, the closed loop system is,
in general, not globally stable. Instead, stability analysis must be limited to consider
local, or semi-global, regions of convergence. In other words, initial conditions that
are too large can result in a saturation condition from which these systems cannot
recover. Similarly, an underdesigned system operating normally can be driven to
instability by external disturbances. Of course, these are problems that have faced
engineers for many years, and remain a topic of much current research. This chapter
details local control design approaches for systems with actuators that are subject
to saturation. Three design algorithms are presented which produce output feedback
controllers that either maximize regions of attraction, maximize disturbance rejection,
or optimize an L2-gain performance metric. In all cases, the stability analyses are
based on the Popov stability criterion and, using the same approach detailed in the
previous chapter for reduced order design, the controllers are formulated in terms of
LMIs that can be solved eciently as semidenite programs.
141
142 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
sat(q)
q
1
1
;1
1
rq
p
Figure 6.1: Saturation locally sector bounded as parameterized by r.
6.1 Introduction
The eects of saturation on system behavior has long been a focus of attention for
control system designers. Actuator saturation can lead to instability, or at the least,
result in degraded closed loop performance. Traditionally, control system design has
either ignored these eects altogether by intentionally over designing the actuation
system so as to avoid possible saturation, or by designing the best compensation possible and post-analyzing the resulting system to ensure acceptable performance, and
subsequently redesigning (e.g., reducing or increasing control bandwidth) to achieve
desired closed loop behavior. This chapter takes a fundamentally dierent approach
to saturation by basing the design process on the analysis which will guarantee certain
levels of performance are achieved by the resulting closed loop system. The analysis,
in keeping with the methodology of the previous chapters, is based on absolute stability theory, and employs sector transformations and properties of the nonlinearity to
derive expressions for stability and performance levels, which in turn are developed
into a synthesis algorithms.
Perhaps the most fundamental analysis of a system with saturation is a prediction
of regions of attraction. Early work applied absolute stability theory to address this
question by isolating the nonlinearity, and casting the problem in the Lur'e-Postnikov
framework, depicted in Fig. 1.1 [WM67, Wei68, DK71]. As is typical in these works,
a Lyapunov function is used to dene regions in the state space in which energy is
6.1. INTRODUCTION
143
guaranteed to decrease, while the nonlinearity is bounded by a prescribed amplitude.
With these conditions satised, it is then shown that initial conditions in these regions
will result in state trajectories that converge to the origin. When the input to the
nonlinearity remains within certain bounds, that is if jq(t)j < r; 8t 0, the saturation
can then be treated as memoryless nonlinearity, locally sector-bounded in [; 1], where
= 1 ; 1r . Using this observation, the early analysis was later extended to produce
better stability region estimates by utilizing the circle stability criteria [Kos83, GS83].
More recently, the local stability analysis for multiple nonlinearities using both the
circle and Popov criteria was formulated in an LMI setting [PTB97, HB98]. The
exibility provided by the LMI framework allowed for a generalization of the stability
guarantee to include performance as measured by external disturbance rejection and
local L2{gain. These latter results provide the analytical basis for the local control
design algorithms described in this chapter.
The practical importance of control design for systems using bounded input keeps
this an active area of research [TG97]. Various linear, piecewise linear and nonlinear
methods can be found throughout the literature. A Lyapunov approach has been used
to construct nonlinear approximations to linear, time-optimal control laws [SSDA94],
while Kamarov proposed a nonlinear technique based upon an inner estimation of
attainable ellipsoidal sets by solving a matrix dierential equation [Kom95]. Although these nonlinear techniques explore the limits of performance possible using
bounded control, their application is limited to relatively simple systems, and by the
common assumption of state feedback. Low-and-high gain control (LHG) is another
state feedback technique with a simple implementation, approaching the problem
by rst designing a low gain controller so as to avoid actuator saturation and thus
widening the region of attraction for the system. Because the low gain controller
is intentionally conservative, resulting in sluggish transient performance, the design
is then augmented with a high gain outer loop which can help speed up system response [SLT96]. Piecewise linear LQ control (PLC) [WB94] is a less conservative state
feedback technique in which the feedback gains are increased in a piecewise fashion
as the state approaches the origin. While providing improved response, PLC lacks
See also [Kha96, pp. 407{419] for a detailed scalar example.
144 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
robustness to large uncertainties and the ability to reject arbitrarily large bounded
disturbances. Recently, a state feedback design framework combining the LHG and
PLC techniques in order to achieve a desirable mix of the robustness and performance
oered by each was proposed in Ref. [LPBS97].
The PLC and LHG designs represent some of the latest attempts at achieving
closed loop performance with bounded control. In other recent work, state feedback
compensation for global stabilization with eigenvalue assignment and local L2{gain
performance has been achieved based on the solution to an algebraic Riccati equation
(ARE) [SARSD98, SARSIV97]. ARE-based approaches have also been applied to the
more general output feedback case by utilizing state observers [Lin97, Lin98], and to
local LQG design by solving coupled Riccati equations [TB97]. As with these later
approaches, this chapter introduces synthesis algorithms for output feedback control
aimed at specic local performance criteria. In particular, the new design methods
build on the recent local saturation analysis in which the authors formulate bounds
for regions of attraction, disturbance rejection and local L2{gain performance, all
based on the Popov stability criterion. The analysis generalizes to treat the multiple
nonlinearity case, and because the results are posed in terms of LMIs, the bounds are
readily computed by solving semidenite programs [PTB97, HB98].
Synthesis of local regions of attraction via state feedback based on the circle
criterion was introduced in [PTB97] and later extended to the case of output feedback [PHHB98]. In these works, the maximized stability region was dened in state
space by the ellipse,
EP = f x xT Px < 1; P = P T > 0 g;
(6.1)
and the optimal controllers guaranteed that any initial condition x0 2 EP resulted
in the stable trajectory x(t) ! 0: In [PHHB98], the controllers were dynamic, of
the same order as the given linear plant, and computed using an LMI approach. The
resulting stability regions are not invariant, but have a property referred to as pseudo{
invariance [EF96], which means that state trajectories originating in the region may
exit but will eventually return as the state converges.
This chapter presents the analogous local stability design method based on the
6.2. PROBLEMS OF LOCAL CONTROL DESIGN
145
less conservative Popov criteria, using the analysis from [HB98]y, and builds this basic
approach into two new design algorithms that yield dynamic compensation for closed
loop robustness or local L2{gain performance for systems with actuator saturation. In
the rst case, the controllers are designed to reduce the sensitivity of the closed loop
system to external disturbances. This is accomplished with designs that guarantee
stability in the presence of all external disturbances, w(t), that are bounded by a
given level of energy, max (i.e., kwk2 max ). The second algorithm solves for
controllers that minimize the L2{gain across a performance channel. In general,
the local performance and external disturbance rejection problems have competing
objectives. That is, controllers with the best L2{gains correspond to designs that will
only tolerate relatively small external disturbances. In cases where both robustness
and performance are critical, the analysis and design algorithms can be used in an
iterative fashion to arrive at an optimal combination of the two objectives. This
trade-o is illustrated with a simple numerical example at the end of the chapter.
6.2 Problems of Local Control Design
A typical control system that includes bounded, or saturating control inputs is depicted in the standard three block framework shown in Fig. 6.2. In the diagram sat()
is the unit saturation function, H is an LTI plant that is to be controlled and K is an
LTI controller that is to be designed. Note that the linear plant includes the block L,
which is some strictly proper very high bandwidth low pass system which is included
for technical reasons discussed later. Using standard robust control notation, the signal w is the disturbance input, and z is the performance variable. Systems operating
in saturation can exhibit nonlinear behavior such as local stability, nite disturbance
rejection, and performance degradation. These eects are analyzed and quantied in
[HB98] with bounds that are computable by solving a convex optimization over a set
of linear matrix inequalities. The analysis is continued here with the consideration of
three basic engineering design problems. To simplify the development, the case of a
y Applying Popov criteria leads to regions of attraction dened by a constraint with both quadratic
and nonquadratic components and thus generalizes the region dened in (6.1).
146 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
sat()
1
1
r
q
p
w
z
H
L
y
u
K
Figure 6.2: Control system with saturation nonlinearity. The system within inner
dashed line is original plant H augmentented with lowpass lter L, while
outer dashed region is closed loop system.
single saturation nonlinearity is considered, although the results are readily extended
to the MIMO case. Let Umax be the maximum available actuator level (subsequently
assumed normalized to 1) and let pmax bound the energy of input disturbances w
that can be rejected by the system starting with zero initial conditions:
max = supf j kwk22 ; x(0) = 0; tlim
x(t) = 0g;
!1
(6.2)
where x is the overall state of the linear plant (consisting of H and L) and the
controller K . Similarly, dene the L2-gain (energy gain from w to z) 22 of the closed
loop system, starting from a zero state as:
kzk2 :
22 =
sup
(6.3)
kwk2
w(t) 2 L2
x(0) = 0
6.2. PROBLEMS OF LOCAL CONTROL DESIGN
147
Finally, let D be a region in the state space of H of initial conditions from which the
closed loop system is guaranteed to be brought back to zero with the nite actuator
authority in the absence of disturbance:
D=
n
x 2 Rn
o
x(0) 2 D; w 0 ) x(t) ! 0, as t ! 1 :
(6.4)
Now consider following design problems, where we are given a set of design specications and asked to design a controller K that achieves one of the following objectives:
(SR) :
Given : H; Umax = 1; w 0
Find : K which maximizes D
(DR) :
Given : H; Umax = 1; 22 spec
Find : K which maximizes max
(EG) :
Given : H; Umax = 1; max spec
Find : K which minimizes 22
In the (SR) case, we are asked to nd a controller which maximizes, in some sense,
the stability region; in (DR) we seek a controller which optimizes the closed loop
disturbance rejection, while maintaining an L2-gain less than some value of spec ;
whereas in (EG) we require a controller which minimizes gain, while maintaining a
disturbance rejection bound of at least spec .
It might appear that we have created an articial dependence between disturbance
rejection and L2-gain in problems (DR) and (EG), since each of these is stated in
terms of both max and 22, while in the global denitions (6.2) and (6.3), the two
seem unrelated. In practice, however, solving for the optimal (EG) controller without
requiring max spec can lead to closed loop systems that are not suciently robust
to external disturbances. Of course, in the extreme case, an unconstrained (EG)
controller designed by ignoring the eects of saturation altogether will achieve the
highest performance, but can result in a closed loop system highly sensitive to external
disturbances. Similarly, controllers that provide good disturbance rejection do so by
148 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
maintaining stability despite large levels of actuator saturationz. This relationship
motivates the need for a systematic approach that enables the designer to strike a
balance between these two objectives. A natural choice for a design variable then
becomes the saturation parameter, r, depicted in Figs. 6.1 and 6.2, that can be used
to control the relative amount of nonlinear operation that a system will encounter
when acted on by external disturbances, or when starting from some initial condition.
The designer should expect better (EG) performance if operation is maintained \close
to linear" by setting the parameter r not much larger than 1x; while better disturbance
rejection (DR) could be achieved by allowing higher values of r.
This motivates the following local versions of the global metrics given in (6.2, 6.3,
and 6.4) [HB98] that have explicit dependence on r. First, consider the r-level local
stability region Dr as the maximum volume set of initial conditions in the state space
of H , for which the control never exceeds r, in the absence of disturbance, dened by,
"
Dr = x 2 Rn x(0) 2 Dr ; w 0 )
(
x(t) ! 0, as t ! 1
jq(t)j r; 8t 0
#
:
(6.5)
Similarly, the r-level local disturbance rejection is dened by the largest disturbance
that results in r-bounded control response:
r =
max
supf j kwk22 ; x(0) = 0; tlim
x(t) = 0; kqk1 rg:
!1
(6.6)
r is equivalent to computing the
Note that from (6.6), it follows that computing max
r-level local L2-to-L1-gain from w to q:
kqk1 p r ;
1r 2 =
sup
(6.7)
r
k
wk2
max
2
r
kwk2 max
x(0) = 0
z This relationship between the (DR) and (EG) designs is illustrated with a simple example in
x6.6
x Note that r < 1 means that the system operates linearly.
6.3. THE DESIGN APPROACH
and nally, the associated r-level local L2-gain from w to z is dened as:
kzk2 :
22r =
sup
kwk2
r
kwk22 max
x(0) = 0
149
(6.8)
It follows immediately from these denitions, that for a given closed loop system,
r , r and r are all monotonically increasing functions of r, which tend to their
max
12
22
global values for large r. Therefore, they will always produce less conservative bounds
than their global counterparts. Thus it seems reasonable to recast problems (SR),
r , r and r . Note that, in contrast to the global
(DR), and (EG) in terms of max
12
22
r , r , r , and r.
case, there is now a denite relationship among max
12 22
6.3 The Design Approach
The local denitions above make it possible to talk about the local stability and
performance of linear systems with saturation in a precise way. Unfortunately, at
r , and r for general linear systems with
this time, the exact computation of Dr , max
22
saturation such as Fig. 6.2 operating with r > 1 is still an open problem, and the same
goes for the synthesis problems above. Therefore, in our actual design procedures for
problems (SR), (DR), and (EG), we will make use of estimates of these objects.
r , a lower bound
Specically, we compute: D^r , an inner approximation to Dr ; ^max
r , (and hence an upper bound on r , ^r ); and ^r , an upper bound on
on max
12 12
22
r
L2-gain 22. These estimates are computed using LMI/BMI techniques, by applying
the Popov criterion to the r-level sector model of the system which will be described
in the following sections.
Consider the design of a controller for problem (EG). Strictly speaking, this problem is a mixed L2-gain and L2-to-L1-gain optimization problem. Such mixed norm
multiobjective problems are, in general, not very easy to solve, even in the linear
case. So one reasonable approach would be to simply start by trying a linear design,
ie, assuming that (EG) can be solved with a controller which does not saturate, i.e.,
150 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
r = 1. One can then solve the single objective controller synthesis problem of computing a K which minimizes the H1-norm of the closed loop system using standard
techniques. The closed loop system with K can then be post-analyzed to ensure that
1 the max
spec constraint is met. If this is the case, then the problem is solved.
1 , and
(Note that when the system operates linearly, it is possible to compute D1 , max
221 exactly.)
1 If the max
spec constraint is not satised, then the r = 1 saturation level
disturbance rejection bound is too small. At this point, one might wonder if it would
be possible to increase max by allowing the system to saturate slightly, while possibly
trading o some L2 performance. One can try increasing r to a value slightly greater
than 1, and redoing the synthesis, this time computing a controller K which minimizes
^22r using BMI synthesis. Equation (6.7) shows that if the relative increase in the
r
gain 1r 2 for the new controller is smaller than the relative increase in r, then max
will increase. Then the closed loop system can once again be post-analyzed: LMI
r and the constraints are checked (conservatively)
techniques are used to compute ^max
r spec . If not, then r can be increased once more and the process
by checking if ^max
of BMI synthesis and LMI post-analysis can be repeated, until either the LMI's used
in the computations become infeasible, or the specication is achieved.
Similar BMI-synthesis/LMI-postanalysis design methods can be proposed for (SR)
and (DR). Such methods, while seemingly crude, can often do a good job at tuning
an initial controller to satisfy some desired specications. A numerical example in
x3.5 is used to demonstrate this design approach.
6.4 System Model
We will now describe the models that we use for the components of Fig. 6.2. The
linear plant H (s) shown is given by the dynamics:
H
8
>
>
<
>
>
:
x_ P = APxP + BPw w + BPup
z = CPz xP + DPzw w + DPzup
y = CPy xP + DPyw w;
(6.9)
6.4. SYSTEM MODEL
151
where it is assumed the matrix A may have unstable eigenvalues, and that the system
is both observable and controllable. The control u is assumed to be ltered with the
high bandwidth, lowpass network L(s):
(
x_ L = AL xL + BLu
(6.10)
q = CL xL :
Without this lter, the control signal feedthrough to the nonlinearity would signicantly complicate the controller elimination in the synthesis in x6.5. The lter output
q is subject to saturation
p = sat(q):
(6.11)
We will consider the design and analysis of proper, linear controllers K of the form
L
(
x_ c = Acxc + Bcy
u = Ccxc + Dcy:
K
(6.12)
Following the analysis in Ref. [HB98], dene the deadzone nonlinearity dzn() as
sat() + dzn() = 1()
(6.13)
and apply the loop transformation
p = dzn(q) = ;(sat(q) ; 1(q))
(6.14)
which transforms the system in Fig. 6.2 to that in Fig. 6.3, which is the nominal
closed loop system (i.e., with no saturation) perturbed by the dzn() nonlinearity{.
The corresponding open loop plant G : (p; w; u) 7! (q; z; y), shown by the inner
dashed line in Fig. 6.3 is dened by
2
AP BPuCL ;BPu BPw
6
6 0
AL
0
0
6
s 6
G = 66 0
CL
0
0
6 C
4 Pz DPzu CL ;DPzu DPzw
CPy
0
0 DPyw
0
BL
0
0
0
3
7
7
7
7
7
7
7
5
(6.15)
{ See also [BB91, pp. 230{31] for another analysis of saturation employing this loop transformation
which converts sat() into a dzn() nonlinearity.
152 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
dzn()
1
q
p
w
H
;
L
z
u
y
K
Figure 6.3: Transformed system with deadzone nonlinearity. Transformation converts saturation into deadzone nonlinearity, and creates feedthrough
path from lowpass system L(s) to linear plant H (s).
2
A
6
6 C
= 66 q
4 Cz
Cy
Bp
Dqp
Dzp
Dyp
Bw
Dqw
Dzw
Dyw
3
Bu
7
Dqu 77
:
Dzu 75
Dyu
This model is used in the synthesis phase of the design procedure. Similarly, the
closed loop plant, shown by the outer dashed line in Fig. 6.3, G~ : (p; w) 7! (q; z) is
2
A11 BuCc
6
6 B C
Ac
G~ =s 66 c y
0
4 Cq
C13 0
Bp M14
0 BcDyw
0
0
Dzp Dzw
3
7
7
7
7
5
(6.16)
6.5. DESIGN ALGORITHMS
153
dzn(q)
q
1
;r ;1
;1
r
r q
1
p
Figure 6.4: Deadzone nonlinearity and saturation parameter r.
2
A~
6
= 64 C~q
C~z
B~p
D~ qp
D~ zp
3
B~w 7
D~ qw 75 ;
D~ zw
where A11 = A + BuDcCy , M14 = Bw + BuDcDyw , and C13 = Cz + Cq Dzp. This
model is used in the analysis phase of the design procedure. The level r relates to the
transformed system, Fig. 6.3, as the limit on the input q to the dzn nonlinearity. Now
if the input signal is limited such that jq(t)j r; 8t; then the dzn is guaranteed to
act as a sector bounded nonlinearity, in sector[0; r ], where r = (1 ; 1r ), as depicted
in Fig. 6.4. Normalizing the sector to [0; 1] then results in a multiplication of the
corresponding matrices, Bp; Dqp; Dzp; Dyp, of system (6.16) by a factor of r , and
similarly for the closed loop matrices dened in (6.17). In the sequel, these scaled
transformed matrices will be designated with superscript r (i.e., B~pr , etc.).
6.5 Design Algorithms
Because the dzn() function is a sector bounded, memoryless nonlinearity, the system,
Fig. 6.3, is in a form suitable for Popov analysis [BGFB94]. In this section we
present the analysis based on the Popov criterion to estimate the stability regions,
disturbance rejection capability, and local L2 -gain performance for the closed loop
system, and provide the corresponding (SR), (DR) and (EG) design algorithms. Note
that each case is based on an assumed r-level being maintained, so that the local sector
154 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
model described above is satised (depicted in Fig. 6.4). The (SR), (DR) and (EG)
analyses and design solutions presented in the following sections extend the LMI local
analysis and synthesis work in [PHHB98, HB98, BH97a], and are based on the Popov
Lyapunov function of the closed loop system state x:
~ +2
V (x) = xT Px
nq
X
i=1
i
Z C~q;i x
0
dzn() d;
(6.17)
with P~ = P~ T > 0, i > 0, and where C~q;i denotes the ith row of the system output
matrix, C~q . For brevity, the proofs are not included here; the interested reader can
refer to [HB98].
6.5.1 Stability Region (SR)
In this case, we set the disturbance w = 0, and consider the simple objective of
designing a controller K that maximizes the region of attraction for the system (6.17).
Adapting the general analysis presented in [HB98, PTB97] to the system (6.17) we
have the following theorem, which denes a region of attraction D^r for the closed
loop system in terms of a level set of the Lyapunov function (6.17). Here a level set
is simply the region in state space which satises the inequality
n
o
lev"V (x) = x 2 Rn V (x) " :
(6.18)
Theorem 6.5.1: An r-level region of attraction D^r is given by the invariant
set lev1V , where V is the Popov function obtained by solving the following convex
optimization problem in the variables P~ = P T 2 Rn~ n~ , = diag(1; : : : ; n ), and
T = diag(t1; : : : ; tn ) [HB98]:
x
x
q
q
min Tr
(P~ + C~qT #C~q )
"
r2 C~
such that: ~ iT ~q;i 0;
Cq;i P
P"~ > 0; > 0; T > 0;
A~T P~ + P~ A~ P~ B~pr + A~T C~qT + C~qT T
()T12
;2T
(6.19)
#
< 0;
6.5. DESIGN ALGORITHMS
155
for i = 1; : : : ; nq where the closed loop system matrices are dened in (6.17). The
corresponding control design is posed by the following corollary.
Corollary 6.5.1: A controller which maximizes the region of attraction D^r dened
by a set of matrices satisfying Thm. 6.5.1 is parameterized by the matrices R, S that
solve the following optimization problem:
min Tr
R + Tr CqT#Cq
"
ri2 Cq;i
such that:
0;
T
C
R
q;i
"
#
R I
> 0; > 0; T > 0;
I" S
#
T M12
AR
+
RA
U?T
U? < 0
T
M
;
2
T
12
"
#
T S + SA N12
A
V?T
V < 0;
N12T
;2T ?
(6.20)
for i = 1; : : : ; nq , where M12 = Bpr + RAT CqT + RCqT T , N12 = SBpr + AT CqT + CqT T ,
and
2
3
"
#
T
C
Bu
U? =
and V? = 4 ry T 5 ;
(6.21)
Cq Bu ?
Dyp
?
with U? being the orthogonal complement of U , and the open loop system matrices
dened in (6.16).
6.5.2 Disturbance rejection (DR)
The local disturbance rejection problem can be presented in a similar way. The
theorem below computes an upper bound on the level of disturbance energy that a
closed loop system can tolerate before instability for a given saturation level r. This
theorem is then followed with the corresponding corollary for control design in terms
of a semidenite program solution.
r ,
Theorem 6.5.2: For system (6.17), an r-level disturbance rejection bound, ^max
156 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
can be computed as
Z r
2
r
+ 2 '() d;
max = max
2[0;1] C~q P;1 C~qT
0
^ r
(6.22)
where for each 2 [0; 1], (P; ) is the optimal value of the following convex semidefinite program in the variables s1 ; s2 2 R, P~ = P~ T 2 Rn n ,
x
x
min (1
; )s1 + s
2
"
#
"
#
~
s1 Cq;i
s2 1
such that:
0;
0
T P~
C~q;i
1 P2~ > 0; > 0; s > 0;
3
~T P~ + P~ A~ M12
~ B~w
A
P
6
7
T
6
~
~
M
;
2
T
C
B
q w 7
12
4
5 0;
B~wT P~
B~wT C~qT ;I
(6.23)
for i = 1; : : : ; nq , where M12 = P~ B~pr + A~T C~qT + C~qT T .
Corollary 6.5.2: A controller which maximizes the disturbance rejection is parameterized by the matrices R, S that solve the following optimization problem:
min (1 ; )s1 + s2
"
#
"
#
s1 Cq;i
s2 1
such that:
0
;
0
T R
C
1
q;i
"
#
R I
> 0; > 0; s > 0;
I2 S
3
T M12
AR
+
RA
B
w
6
7
T
T
6
U? 4 M12
;2T Cq Bw 75 U? < 0
BwT
()T23 ;I 3
2
AT S + AA SBw N12
7
6
V?T 64 BwT S
;I BwT CqT 75 V? < 0;
N12T
()T23 ;2T
(6.24)
6.5. DESIGN ALGORITHMS
for i = 1; : : : ; nq ; where
2 "
6
U? = 64
Bu
Cq Bu
0
157
3
#
?
0
I
7
7
5
2 "
6
; V? = 64
CyT
T
Dyw
0
3
#
?
0
I
7
7
5
;
(6.25)
and N12 , M12 are dened as in (6.20). An extension of Theorem 6.5.2 and the corresponding design corollary to the case of multiple nonlinearities is possible using
the analytical extension given in [HB98]. We now consider the case with an output
performance variable z(t):
6.5.3 Local L2-Gain (EG)
Extending the disturbance rejection case to the local L2-gain design can be accomplished by considering the small gain dissipation inequality (2.10):
d V (x(t)) ;2wT w ; zT z;
(6.26)
dt
where the storage quantity V (x(t)) is taken to be the Lyapunov function (6.17). Of
course, the resulting controller will simply minimize the gain from disturbance w to
performance variable z. Closed loop stability will still be subject to the absolute
size of the disturbance (in an L2 sense), and can be checked using the disturbance
rejection (DR) test (6.23).
r , and x(0) = 0, an
Theorem 6.5.3: For system (6.17), whenever kwk2 ^max
r-level L2-gain bound bound, ^r , can be computed by solving the convex semidenite
program [BGFB94]:
min ^r2
such that: 2P~ > 0; > 0; T > 0; and
A~T P~ + P~ A~ + C~zT C~zT M12 + C~zT D~ zpr
P~ B~w + C~zT D~ zw
6
r )T D
r ; 2T C~q B~w + (D
r )T D
6
~ zp
~ zp
~ zw
()T12
(D~ zp
4
T C~z
T D
r
T D
~ zp
~ zw ; ^r2I
B~wT P~ + D~ zw
B~wT C~qT + D~ zw
D~ zw
3
7
7
5
0;
(6.27)
158 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
where M12 is dened as in (6.23).
Corollary 6.5.3: A controller K;r which minimizes the L2-gain bound is parameterized by the matrices R, S that solve the following optimization problem:
min "^r2
#
R I
such that:
> 0; > 0; T > 0; and
I2 S
Bw
AR + RAT M12
6
6
M12T
;2T Cq Bw
U?T 66
BwT
BwT CqT ;^r I
4
Cz R
Dzpr
Dzw
2
N13
AT S + SA SBw
6
T
6
;^r I BwT CqT V?T 66 BwTS
N13
Cq Bw ;2T
4
Dzpr
Cz
Dzw
3
RCzT
r )T 7
7
(Dzp
7 U? < 0
T
Dzw 75
;^r I 3
CzT
T 7
7
Dzw
7 V? < 0;
r
T
(Dzp) 75
;^r I
(6.28)
where M12 = Bpr + RAT CqT + RCqT T , N13 = SBpr + AT CqT + CqT T , and the outer
perpendicular matrices are
2"
#
3
2"
#
3
Bu
CyT
0
0 77
6
7
6
T
U? = 64 Cq Bu ? 75 and V? = 64 Dyw
(6.29)
? 5:
0
I
0
I
The identity matrices appearing above in (6.29) for U? and V? have dimension nw + nz
and np + nz , respectively.
6.5.4 Controller Reconstruction
To obtain the controller K from a pair (R; S ) that solves one of the optimization
problems requires the solution of a feasibility problem:
M;r + UKV T + V K T U T < 0
(6.30)
where U; V are complements of the corresponding matrices above in (6.21, 6.25, and
6.29). The matrix M;r is a constant matrix involving the open loop system matrices
6.6. L2-GAIN CONTROL EXAMPLE
159
and the specied performance and saturation levels. The form of the matrices M;r , U
and V are easily derived by isolating terms involving the controller K , which is a step
necessary prior to application of the Elimination Lemma (2.2.2). Optimization determines the matrices (R; S; ; T ) and parameters r; which then, through the use of the
Completion Lemma (2.2.3), completely dene the feasibility problem (6.30). Details
of this elimination/completion technique is well documented in [Gah96] and omitted
here for brevity. Instead, as was done for the reduced order design in Chapter 5, the
nal matrices necessary to solve (6.30) for the (SR), (DR) and (EG) problems are
detailed in appendix sections C.3, C.4 and C.5, respectively.
6.5.5 Optimization Algorithms
The optimization problems are bilinear in the multipliers (; T ) and the Lyapunov
variables (R; S ). As discussed in the previous chapter, this problem is well known to
be nonconvex, and typically solved by successive optimization over the two sets of variables. This technique is referred to as BMI synthesis [GLTS94, BH97a, PH99a], and
applicable to the local designs described above. The successive iteration algorithms
described in Chapter 5 are readily applied to the (SR), (DR) and (EG) problems and
are not detailed here. In particular, the reduced order Popov/H1 algorithm described
in x5.3.3 was used directly to solve the local L2-gain problem described below.
6.6 L2-Gain Control Example
Here we consider the linearized inverted pendulum problem, depicted in Fig. 6.5, with
control force f input and angle output, and dynamics =f = s2;1 k2 , with k = 0:1.
The disturbance is a force fd, entering the system in the same way as the control. The
performance z = [W1 W2 u]T is a weighted combination of angle and control eort,
with W1 = (0:1s +1)=(s +1) and W2 = W1;1, and the lowpass set at L = 1=(s=200+1).
Using a BMI synthesis algorithm based on Corollary 6.5.3, L2-gain controllers were
designed for this system using saturation levels ranging from r = 1:1 to r = 3:0, in
increments of 0:10. For each r value, the algorithm was initialized with a controller
160 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
w
m
y=
u
Figure 6.5: Inverted pendulum with disturbance.
designed by extending the Circle criterion method in [PHHB98] to optimize the given
performance. The BMI algorithm for this example typically converged to a minimum
after 13 iterations, and yielded a new controller, based on the Popov analysis, that
improved performance by about 20% over that based on the Circle criteria.
The performance, 22r , of the new controllers is shown for each r value in Fig. 6.6.
Performance degrades as r increases, with 22r increasing from 1:7 to 4:5 as r ranges
from 1:1 to 3:0: This trend should be expected since higher values of r correspond to
larger sector widths, which allows the possibility of more nonlinear control behavior.
However, widening the sector generally improves the disturbance rejection capability.
r , computed using Theorem 6.5.2,
The disturbance energy bound for this system, ^max
increases by 66% over the full range of r (see Fig. 6.6). So that while performance
does degrade, designs using higher values of r give the closed loop systems that
can tolerate larger disturbances. The tradeo between performance and disturbance
r
rejection is depicted with a graph of 22r vs. 1=^max
in Fig. 6.7. Systems with good
performance (low values of 22) have relatively poor disturbance rejection (high values
r ). Conversely, designs with improved disturbance rejection are necessarily
of 1=^max
penalized with degraded performance. Of course, if disturbance rejection is critical
6.7. CONCLUSIONS
161
4.5
4
3.5
22(r)
r ; r
^max
22
3
2.5
2
1.5
^max(r)
1
0.5
1
1.2
1.4
1.6
1.8
2
r
2.2
2.4
2.6
2.8
3
Figure 6.6: Performance and disturbance level dependence on r.
and performance is not an issue, then the controller should be designed directly using
a BMI algorithm based on Corollary 6.5.2.
6.7 Conclusions
This chapter detailed control synthesis for systems with actuators that are subject
to saturation. Optimal control designs for three dierent performance objectives are
considered, and each is formulated as a solution to an LMI/BMI optimization problem. The rst design method produces controllers that maximize region of attraction
for an unstable plant. In this case, the optimal controller guarantees the largest region in state space in which all initial conditions will result in stable trajectories. The
solution to the second optimization problem gives controllers that allow the closed
162 CHAPTER 6. CONTROL DESIGN FOR SYSTEMS WITH SATURATION
r = 3:0
4.5
4
22r
3.5
3
2.5
r = 1:1
2
1.5
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1=^r
max
Figure 6.7: Disturbance rejection vs. L2-gain performance trade-o.
loop system to reject external disturbances which have a maximum level of energy
content. The last design algorithm yields controllers that optimize the L2-gain across
a specied performance channel.
Each design algorithm is based on the Popov stability criterion, and employs a
sector model of the saturation nonlinearity. The BMI implementations are eciently
solved using available LMI software. For a simple numerical example the algorithms
typically converged after 13 iterations and produced Popov controllers that achieved
a 20% improvement in L2-gain performance compared to controllers designed based
on the Circle criteria.
One limitation in the L2-gain design case is that each closed loop system must be
post analyzed to determine the worst case disturbance levels the system can tolerate
while achieving the optimal disturbance attenuation. As a practical consideration, it
6.7. CONCLUSIONS
163
may be desired to trade o performance against disturbance energy levels in order to
accomplish the nal design. Control design using this approach is explored with the
numerical example.
Finally, the BMI solutions for each of the local control designs are presented in a
form consistent with the reduced order designs detailed in Chapter 5. This means that
the algorithms that solve these problems can be structured to provide reduced order
solutions as well. Of course order reduction is desirable when controller complexity
needs to be avoided; reduced order algorithms will allow the designer to trade o
the local performance against compensation size in the same way illustrated with the
numerical examples in x5.3.3.
164
Chapter 7
Conclusion
7.1 Summary of Main Results
This thesis introduced a new set of tools for the stability analysis and compensation
design for nonlinear systems. Particular nonlinear forms considered include hysteresis, memoryless slope-restricted nonlinearities, and saturation. Stability tests are
developed for these systems as both state space and frequency domain criteria. The
state space formulation can be solved numerically as a convex optimization problem over linear matrix inequalities (LMIs) using widely available software packages
[WB96, GNLC95]. For systems with a single nonlinearity, the frequency domain criterion reduces to a graphical form which is a simple variation of the familiar Popov
test. Stability is guaranteed provided the frequency response of the transformed linear
subsystem avoids a certain restricted region in the Nyquist plane. The LMI analysis
extends gracefully to provide companion robust control design techniques for a variety
of nonlinear systems.
The main results of the research in nonlinear system analysis and control design
are overviewed in the following sections.
165
CHAPTER 7. CONCLUSION
166
7.1.1 Stability Analysis
The stability results are valid for a class of hysteresis nonlinearities dened by a set
of properties (slope bounds, positivity of certain path integrals, circulation direction,
etc.), and includes the most commonly occurring types, such as backlash and relays.
A particular transformation is developed that converts these nonlinearities into a passive operators and stability is subsequently ensured by requiring strict passivity on
the transformed linear subsystem. This fundamental result provides a passivity interpretation of the earlier work by Yakubovich [Yak67, BY79], and, more importantly,
the new analysis framework permits a straightforward extension to treat both multiple hysteresis and memoryless nonlinearities. In the latter case, the new analysis
extends recent work [HK95, PBK98] with less restrictive criteria that applies to a
broader class of nonlinear systems. This benet is illustrated with several numerical
examples. In addition, linear matrix inequalities are combined with basic dissipation
theory [Wil72a, Wil72b] in order to develop a multiplier analysis and robust stability
test for uncertain nonlinear systems. These results and others are highlighted below.
Robust analysis of systems with norm bounded uncertainties and hysteresis is
solved for using dissipation theory, and then cast as a convex optimization over
a set of linear matrix inequalities. Solution of the convex optimization problem
enables the analyst to assess the level of uncertainty that is tolerable while still
being able to guarantee system stability. In turn, this numerical analysis is
used as the basis for algorithms that produce robust controllers for this class of
nonlinear systems.
Absolute stability criteria is developed to treat systems with multiple hystere-
sis nonlinearities. This new result extends the passivity based solution for the
scalar case by augmenting feasibility LMI set with an additional residue matrix
inequality that must be satised. For systems satisfying the stability criteria,
the system state is guaranteed to converge asymptotically to a stationary set
rather than to the origin, which is characteristic of systems containing multivalued nonlinearities. In addition, to predict the asymptotic behavior of the
7.1. SUMMARY OF MAIN RESULTS
167
state, mathematical descriptions of the (asymptotic) stationary sets corresponding to typical types of hysteresis (relay, backlash, etc.) are provided in detail.
For the backlash hysteresis, the stability result is further extended to a multiplier analysis of the same form and generality as that developed by Zames for
monotonic, memoryless nonlinearities. Thus, the framework incorporates an
even broader class of systems for a very common type of nonlinearity.
Hysteresis was chosen as the particular nonlinearity for study primarily because
much of the absolute stability results to date consider only simpler memoryless forms.
Accordingly, since hysteresis occurs commonly in engineering practice in a wide range
of forms, this thesis provides new analytical tools for an important class of nonlinear
systems. However, it is important to note that this does not limit the application
of the analysis to other system classes. Indeed, any memoryless, slope restricted
nonlinearity can be considered as a special case of a hysteresis, and can be treated as
a passive operator under the transformations dened in x3.3. Thus, all of the stability
criteria can be applied directly. In particular, the analysis can be used to study eects
such as saturation and linear parameter uncertainty as well as hysteresis.
Throughout the research, an emphasis was placed on utility and usability of the
results. In their nal form, analytical expressions for stability are expressed as LMIs,
which are readily solved for reasonably sized systems (i.e., 20{30 states) using moderate computational resources. Similarly, the stability criteria are easily formulated
at system level, requiring only the matrices of a state space representation for the
linear subsystem and properties of the nonlinearity, such as minimum and maximum
slope bounds.
7.1.2 Robust Control Design
The simplicity of LMI analysis framework leads to a straightforward formulation for
the control synthesis algorithms which, again, require only system level inputs by
the designer. Other passivity based, constructive techniques, such as backstepping
or forwarding [SJK97], require the designer to rst transform the linear subsystem
168
CHAPTER 7. CONCLUSION
into a special form and then to build up controllers one state at a time, maintaining
passivity at each stage in the process. These techniques can become cumbersome,
and are thus practically limited to relatively low order systems. Also, while the
constructive methods often assume full state information, the routines developed in
this thesis design for the more general output feedback case.
Building on the robust stability analysis, several new algorithms are introduced
which synthesize reduced order controllers that guarantee stability while optimizing
an H1 performance metric. In particular, a reduced order version of the standard
H1 design algorithm is formulated and then extended to include the robust H1,
Popov/H1 synthesis cases, as well as robust control design for systems with hysteresis. The new framework is a general technique that produces reduced order controllers,
and can be readily adapted to other types of nonlinear systems. The versatility of
the approach is demonstrated with three new local control design algorithms for unstable plants with actuator saturation. The new routines synthesize compensation
for systems with saturating actuators that optimize disturbance rejection or H1 performance over a limited, or local, region of the system state space. For its general
and practical application, the results of this research provide valuable new tools for
the study of nonlinear control systems. An overview of the reduced order design and
bounded control results is given below.
Reduced order control
While there has been much work done in recent years on optimal robust control
synthesis, most frameworks produce compensators that are full order, having as
many states as the original plant [DGKF89, ZDG96]. Iterative algorithms such as {
synthesis [ZDG96, SP96, Zho98] often yield controllers with order much higher than
the plant when accurate computation requires high order frequency domain curve
tting. Large controllers designed to give optimal performance, must be reduced in
size in order for practical, real time implementation. Reduction schemes can result in
lost optimality and, in severe cases, even closed loop instability, and must be checked
For example, the system matrix for backstepping must be lower triangular; and similarly, for
forwarding must be upper triangular.
7.1. SUMMARY OF MAIN RESULTS
169
through post-analysis. For an alternate approach, this research focused on developing
reliable algorithms that explicitly produce reduced order controllers { that is { controllers with smaller dimension than the plant. Direct reduced order design involves
a nonconvex constraint that corresponds to the rank of a certain matrix in the LMI
formulation. This inherent nonconvexity has been a primary challenge for control designers. To address the problem, this thesis introduces new LMI-based reduced order
algorithms for producing robust H1 controllers that are useful in situations where
performance is critical but real time computational resources are limited. In each
case, the optimization is accomplished by utilizing a Trace() objective as a convex
relaxation for the rank constraint. Features of the four new methods are described
below.
1. The H1 performance metric and Trace() objective are incorporated into a bisection routine to produce output feedback controllers that optimize the H1
performance subject to an explicit constraint on the order. This simple algorithm provides a simple extension to basic H1 design, allowing the designer to
explore the relative trade-o between controller order and closed loop performance.
2. This basic routine is then extended to synthesis robust H1 and Popov/H1 compensators by using a two part objective involving the Trace() and the closed
loop performance. Using the combined objective, the designer is then free to
select the relative weighting on the two parts of the objective cost in order to
trade-o performance versus controller order. In this way, this algorithm provides a valuable tool that allows control designers to perform a multi-objective
design analysis in the practical situations when performance is critical and the
order of the controller must be reduced because of either real-time control hardware limitations or the excessive order of the plant.
3. Popov/H1 controllers are designed to optimize an H1 performance metric
while guaranteeing stability based on the Popov criteria. The Popov criteria is
preferred when the uncertainty, or nonlinearity can be modeled as memoryless
and sector-bounded, such as the case with linear parameter uncertainty or time
CHAPTER 7. CONCLUSION
170
invariant gain variation. Technically, the solution for this controller is bilinear
in the multiplier and controller parameters, and hence is a BMI, as opposed to
an LMI problem. While this problem has been previously solved [BH97b], the
new reduced order synthesis is potentially more reliable since the new algorithm
systematically eliminates poorly conditioned subspaces of matrices which could
otherwise lead to numerical instability in the reconstruction of the controller.
4. Finally, a new procedure for the design of robust controllers for systems with
hysteresis is introduced. Utilizing the same two part objective as the previous
robust designs, this new algorithm produces reduced order, output feedback
controllers that optimize H1 performance. Similarly, while existing passivitybased constructive techniques, such as backstepping or forwarding, are only
applicable for systems with a particular structure (e.g., upper triangular), and
are limited in practice to relatively low order systems, this LMI synthesis is
a high level state space solution that allows control design for systems of any
order or structure with the same relative ease. Thus, as a general design tool for
systems with hysteresis, this new synthesis technique represents an important
extension of robust control theory for this important class of nonlinear systems.
Bounded control design
The nal chapter of this thesis addresses the common control problem of feedback
stabilization of systems using bounded controls. Indeed, this is an important problem since, as noted, control input to any realizable system is ultimately limited by
physical constraints. In cases in which the plant is open loop unstable, the closed
loop system with bounded control is in general not globally stable. Instead, stability analysis must seek local, or semi-global, regions of convergence. In other words,
initial conditions that are too large can result in a saturation condition from which
these systems cannot recover. Similarly, a poorly designed system operating normally
can be driven to instability by external disturbances. Of course, these problems have
faced engineers for many years, and remain a topic of much current research. This
research details local control design approaches for systems with actuators that are
7.1. SUMMARY OF MAIN RESULTS
171
subject to saturation. Three design algorithms are presented which produce output
feedback controllers that either maximize regions of attraction, maximize disturbance
rejection, or optimize an L2-gain performance metric. In all cases, the stability analyses are based on the Popov stability criterion and, using the same approach detailed
for the reduced order design, the controllers are formulated in terms of LMIs that are
eciently solved as semidenite programs. These three new design algorithms are
described below.
1. By utilizing a performance metric proportional to the volume in state space for
which stability can be guaranteed, the rst algorithm will produce compensation
that maximize the regions of attraction for the closed loop system. In this case
the controller is designed to guarantee stability while allowing for the largest
range of initial conditions.
2. The second routine is specically aimed at reducing closed loop sensitivity to
external disturbances. The resulting controllers optimize disturbance rejection
capability by maximizing the allowed energy of external disturbances that are
applied to the plant.
3. Building on the disturbance rejection objective, the last algorithm oers the
designer optimal L2 -gain across a performance channel for disturbances of a
specied energy level. It is shown in this research that the optimal disturbance
rejection and L2-gain performance are competing objectives, and in practice,
it may be desired to trade o between the two metrics in order to accomplish
the nal design. In this way, the two algorithms together provide a unique
advantage for the design of saturating controllers.
For each algorithm, the saturation parameter r can be used as a design parameter.
Selecting larger r values allows larger regions of attraction, and better disturbance
rejection, while reduced values of r yields controllers with better L2-gain performance.
172
CHAPTER 7. CONCLUSION
7.2 Future Research Directions
All good things must come to a close, but along the way several ideas for new research
directions presented themselves. A few of these are discussed below.
Local stability analysis/synthesis for hysteretic actuation
In the case of bounded control, the analysis is simplied by the fact that the saturation
operator is well behaved, and can be treated simply as a piecewise linear eect.
Analytical bounds on region of convergence and local performance for a given system
with actuator saturation have been derived using various forms of absolute stability
theory. However, a new level of complexity is added to the analysis when actuator
models include hysteresis, which introduce eects such as memory. A local analysis for
hysteresis would provide a valuable extension to the stability analysis given in Chpts. 3
and 4. Of course, this would require expanding the denitions of stationary sets
detailed in x4.4 to include oscillatory trajectories for hysteresis forms with a deadzone
that include the origin, such as a relay or backlash (see Figs. 3.6 and 4.2). Other forms
such as the Preisach model that have a continuous characteristic near the origin, but
saturate for large input may be analyzed in a piecewise fashion, as is done for the
memoryless case, to establish local stability regions. Developing synthesis algorithms
based on these new local analysis would provide further valuable engineering tools.
Local and reduced order H2 design
This thesis primarily focuses on compensation design that optimize L2{gain performance. These algorithms require that the external disturbances acting on a system
can be modeled a signals in an L2 vector space, which means the signals have bounded
energy, and thus asymptotically approach zero. Often, disturbances are persistent,
and are better characterized as being bounded in a mean square sense. In these cases
linear quadratic design (LQG) may be more appropriate, and optimal performance
achieved by minimizing the H2, and not the H1 system norm. Recent research eorts
have yielded LMI formulations for the H2 control design problem, but are limited to
the global case [YLY96, BH98], while others are able to achieve local xed order
7.2. FUTURE RESEARCH DIRECTIONS
173
control using a coupled Riccati equation approach [TB97]. A useful extension of the
H1 design presented in this thesis would be the extension to an LMI-based reduced
order, local H2 design. This would further unify these two important design formulations, as they are often considered jointly throughout the literature (see, for examples
[DGKF89, SP96, CGL97]).
Ecient alternatives to LMI design
Finally, it must be noted that while providing notationally concise expressions for
absolute stability, LMI formulations of control design problems can become computationally intensive when the order of the system grows above 30{40 states. Routines
that can improve computational eciency by taking advantage of sparse matrices are
currently under development that may alleviate the problem for higher order systems [VBA99, VB96]. Until that time, alternate representations for the analyses and
control designs given in this thesis { possibly reformulated as Riccati inequalities following the approach in [YLY96]{ that can lead to more ecient numerical solutions
may be benecial.
174
Appendix A
Energy Storage and Dissipation
Functions
Energy storage and dissipation for the Backlash hysteresis
Here we show the common backlash nonlinearity conforms to the properties 3.3.2.1{5
described in x3.3.2. In particular, we give a simple mathematical representation for
the nonlinearity, and then show that the positivity constraint (3.6) holds under the
sector transformation indicated by property 3.3.2.5.
The input-output behaviour of a backlash (Fig. 4.3) can be described by two
modes of operation, as either tracking or in the deadzone, for which we dene:
(
Tracking: w_ = y_
Deadzone: w_ = 0
y_ > 0; w = (y ; r) or
y_ < 0; w = (y + r);
jw ; yj r;
(A.1)
where 2r is the deadzone width and is the slope of the tracking region, as indicated
in Fig. 4.3. Applying the sector transformation, shown in Fig. 3.3, we have, when
tracking with positive velocity
_ = w_ = (y ; 1 w)w_ = ry_
(A.2)
and, similarly for negative tracking: w_ = ;ry_ . This quantity is then expressed for
175
176
APPENDIX A. ENERGY STORAGE AND DISSIPATION FUNCTIONS
all times as
(
rjy_ j when tracking;
(A.3)
0
in deadzone.
Dening the interval I = [0; T ], for some T 0, and Ttrk I encompassing all
the subintervals in I for which tracking occurs, the integral (3.6) for the backlash
becomes
Z T
Z
0
(t)w (t) dt = r
jy_ (t)j dt 0:
(A.4)
w_ =
0
t2T
trk
Thus, = 0, which means that the sector transformed nonlinearity has zero stored
(or available) energy. In this case, it can be shown that the transformation induces
a dissipation equality. In particular, the energy balance, as noted by Brokate and
Sprekels [BS96, p. 69], is given as
M0(t) ; U 0 (t) = jD0(t)j
(A.5)
where the terms from (A.2{A.3) are identied with: M0(t) = wy
_ as the mechanical
1
0
work rate; U (t) = ww_ , the rate of hysteresis potential [BS96] energy storage; and
D0(t) = ry_ as the energy dissipation into the hysteretic element. The transformation
(Fig. 4.1) strips the energy potential and leaves only the energy dissipation term in
the integrand in (A.4). Expressed this way, we can exactly account for all energy
components associated with the nonlinear operator. Explicit potential, work and
dissipation expressions for more complicated hysteresis operators, such as the Preisach
and Prandtl models, is discussed in [BS96]. While being very powerful analytical
tools, they are not pursued further herein.
Appendix B
Benchmark Uncertainty Problem
b1
m1
k1
b2
m2
k2
x2
x1 ; d
m3
x3 ; F
Fig. B.1: Benchmark three mass used for algorithm comparisons.
A typical mechanical benchmark problem for robust control design is depicted in
Fig. B.1; this problem was used to demonstrate the reduced order algorithms. For this
system it is assumed the third spring constant k3 has 10% uncertainty, and we would
like to control the system by applying a force F to the third mass, m3 using a position
measurement, x1 of the rst mass, while the rst mass is subject to a disturbance
force d. For this system, the performance variable was chosen as z = [z1 z2]T ; where
z1 = x1 + x2 corresponds to the average position of the rst two masses, and z2 = u
is the control force. The system parameters were chosen to be m1 = m2 = m3 = 1:0;
b1 = b2 = 0:015; and k1 = k2 = 1:0: Note that the addition of a small amount of
damping to the mass/spring system will mean that the minimum order controller for
this system is simply rst order, corresponding to a stabilization of the rigid body
177
178
APPENDIX B. BENCHMARK UNCERTAINTY PROBLEM
mode. This problem is used widely throughout the controls literature as a benchmark
plant to compare various robust control design methods (see [Ban97], for example and
references therein).
Appendix C
Controller Reconstruction
C.1 Popov/H1 Control
Given a solution to the BMI problem (5.30) consisting of the set of matrices (R; S; ; T ),
the corresponding optimal controller can be recovered by nding a K (s) (5.2) satisfying
~ V~ T + V~ K T U~ T < 0:
M (; T ) + UK
(C.1)
The matrices comprising this LMI are described here. First, given the matrix pair
R; S , and desired controller order nc, the quadratic stability matrix is computed
Q = R ; S ;1;
(C.2)
and decomposed using the singular value decomposition as
W; ; W T = svd(Q):
(C.3)
Using the same reduction procedure described in x5.2.1 for standard H1 case, the
columns of W corresponding to the nc most signicant singular values are selected,
Wr = [w1 ; : : : ; wn ]
(C.4)
and the reduced order matrix of singular values r = diag(1; : : : ; n ) then allows
the formation of the reduced order, stability matrix
"
#
S
W
r
S~ =
:
(C.5)
WrT r
c
c
179
APPENDIX C. CONTROLLER RECONSTRUCTION
180
Using S~ the matrix M (; T ) is dened in terms of the plant matrices and performance
level as:
2
3
T + CT T
T
~ Tt SB
~ p;t + ATt Cq;t
~ w;t
At S~ + SA
SB
Cz;t
q;t
6
7
T
T C T ; 2T C B
6
7
(
)
C
B
+
B
0
q;t
p;t
q;t
w;t
n
n
12
p;t
q;t
7:
M = 66
(C.6)
T S~
T CT T 7
Bw;t
Bw;t
;
I
D
4
5
n
q;t
zw
Cz;t
0n n
Dzw ;In
q
z
w
z
q
z
Again, the matrices appearing in (C.6) result by applying the Elimination Lemma 2.2.2
to the closed loop Popov/H1 LMI (5.23). The t-subscripted matrices have the reduced order controller dimension, and are given as
"
A 0
At =
0 0n
c
with
h
#
"
B
; Bp;t = p
0
#
i
"
#
B
Bw;t = w ;
0
h
i
Cq;t = Cq 0 ; and Cz;t = Cz 0 :
The outer matrices in the reconstruction inequality (C.1) are
2
2
3
~ t 3
SB
CtT
6
7
6
7
6 Cq;t Bt 7
6 0 7
7 ; and V
~ = 66
7;
U~ = 66
7
7
T
0
D
4
5
4 2;t 5
D1;t
0
where the additional matrices in (C.9) are
"
#
"
h
D1;t = 0 Dzu
i
(C.8)
(C.9)
#
0 Bu
0 I
Bt =
; Ct =
;
I 0
Cy 0
and
(C.7)
"
(C.10)
#
0
; D2;t =
:
Dyw
(C.11)
C.2 Hysteresis/H1 Control
Reconstruction of the optimal K for sytems with hysteresis again requires the solution
of a feasibility problem very similar to the Popov case in the previous section. In this
C.3. REGION OF CONVERGENCE DESIGN
181
case, starting with the hysteresis inequality (5.32) the corresponding matrix
2
~ Tt SB
~ p;t ; CqT1;t ; CqT2;tT
At S~ + SA
6
6
()T12
2I ; Dqp ; Dqp
M = 66
T S~
T Bw;t
;Dqw
4
;Cz;t
;Dzp
~ w;t
SB
;Dqw
;In
;Dzw
w
3
;Cz;tT
;Dzp0 777 :
T 7
;Dzw
5
;In
(C.12)
z
is found in the same manner by forming the S~ as in (C.5) and applying the Elimination
Lemma. Note that because the stability multiplier allows for a more general plant
having more non-zero throughput matrices (Dqw , etc.), there are no zero entries in
(C.12). The t-subscripted matrices are as dened above in (C.7-C.8), with the new
terms given by
h
i
h
i
Cq1;t = Cq1 0 ; and Cq2;t = Cq2 0 :
(C.13)
where Cq1 = Cq and Cq2 = A;1Cq , as discussed in x3.4. The outer matrices
2
U~ =
6
6
6
6
4
~ t
;SB
3
2
3
CtT
7
6
7
6 D4T;t 7
D3;t 77
; and V~ = 66 T 77 ;
0 75
4 D2;t 5
D1;t
0
(C.14)
where, similarly, the new terms here are
h
D3;t = 0 Dqu
i
"
#
0
; D4;t =
:
Dyp
(C.15)
C.3 Region of Convergence Design
The local control design for systems with saturation requires a similar reconstruction
to obtain the nal controller. First, solving for the optimal solution (R; S; ; T ) to
the region of convergence problem (6.20), the corresponding feasibility inequality
~ V~ T + V~ K T U~ T < 0
M (; T ) + UK
r
(C.16)
can be established. In this case, the matrix M is a function of the parameter r
used to sector bound the dzn() nonlinearity, as detailed in x6.4 and Fig. 6.4. Here
r
APPENDIX C. CONTROLLER RECONSTRUCTION
182
we have
"
#
T + CT T
~ T SB
~ p;t + ATt Cq;t
A S~ + SA
q;t
M (; T ) = t T t
;
()12
;2T
and the outer matrices are given as
"
#
"
#
T
~ t
SB
C
U~ =
; and V~ = t :
Cq;tBt
0
r
(C.17)
(C.18)
Again, the stability matrix S~ is formed as in (C.5), and the t-subscripted matrices are
the same as above in (C.7{C.10). However, care must be taken to use the matrices
augmented with the lowpass lter, and incorporating the sector parameter r as
dened by the system (6.16).
C.4 Local Disturbance Rejection Design
The corresponding feasibility LMI that must be solved using the optimal solution to
the disturbance rejection problem (6.24) again has the form (C.16). In this case the
matrix
2
3
T
T
T
T
~
~
~
~
At S + SAt SBp;t + At Cq;t + Cq;tT SBw;t 7
6
6
M (; T ) = 4 ()T12
(C.19)
;2T
Cq;tBw;t 75 ;
T S~
T CT Bw;t
Bw;t
;I
q;t
r
and the outer matrices are
2
3
2
3
T
~ t
SB
C
t
6
7
6
7
U~ = 64 Cq;tBt 75 ; and V~ = 64 0 75 :
0
D2;t
(C.20)
where again, all t-subsrcipted matrices are dened in xC.1 and the system matrices
are the augmented versions, as discussed in x6.4.
C.5. LOCAL L2-GAIN DESIGN
183
C.5 Local L2-Gain Design
For the local L2 optimal compensation, reconstruction requires solution of the feasibility LMI (C.16), where
2
T + CT T
~ Tt SB
~ p;t + ATt Cq;t
~ w;t
At S~ + SA
SB
q;t
6
6
()T12
;2T
Cq;tBw;t
M (; T; ) = 66
T
T
T
Bw;tS~
Bw;tCq;t
;In
4
Cz;t
Dzp
Dzw
r
w
T
Cz;t
DzpT
T
Dzw
;In
3
7
7
7
7
5
:
z
(C.21)
is now also a function of the performance metric : Similarly, the outer matrices for
the corresponding feasibility are given as:
2
2
3
~ t 3
SB
CtT
6
7
6
7
6 Cq;t Bt 7
6 0 7
7 ; and V
~ = 66
7
U~ = 66
T 7:
0 75
4
4 D2;t 5
0
0
(C.22)
Note that Eqns. (C.21) and (C.22) have the same form as (C.6) and (C.9), respectively.
This is expected since this problem is simply the local version of the Popov/H1 design
case. Here, due to restrictions and simplications imposed by the local system model,
various terms in the above matrices (C.21) and (C.22) must be set to zero. Also, the
various system matrices here are dened according to the design model described in
x6.4.
184
Appendix D
Convergence Limit Proof
In this appendix, the existence of limt!1 (t) is established. This is done rst by
showing rst that _ 2 L1, and then by employing the Lebesgue Monotone Convergence theorem. Recall rst, as a result of the Lyapunov proof, and the dierentiability
of the nonlinearity, that _ 2 L1. We note that the dynamics of the transformed system in Fig. 4.6 can be equivalently depicted as resulting from an external input signal
that is the initial condition response of the (open loop) linear subsystem,
u2(t) = yh(t) + R(0);
where yh = CA;1 eAt x(0); as is shown in Fig. D.1. The initial state of the linear
subsystem, G~ r (s) + 1s I , is then considered zero, and its output
y1(t) = yf (t) + R(t);
R
with yf (t) = CA;1 0t eA(t; ) Bu( ) d . We then have
_ = ; dtd M ((t))
= ;0M () d (t)
dt
= ;0M () d fy1 + u2g
dt
d fy + R + y + R(0)g ;
0
= ;M () dt
f
h
185
(D.1a)
(D.1b)
(D.1c)
(D.1d)
APPENDIX D. CONVERGENCE LIMIT PROOF
186
u
0
G~ r (s)
-
_
1I
s
yf
+
R
M (t) M
sI
yh + R(0)
Figure D.1: Popov system with initial condition response as input.
where 0M () is the diagonal matrix of local slopes occurring at the m scalar nonlinearities,
0M () = diag f0i(i); : : : ; 0m(m )g > 0:
(D.2)
Inserting the identities
y_f (t) = C
and
Z
0
t
eA(t; ) B _ ( ) d + CA;1 B _ (t);
R_ (t) = (;CA;1B + D + M ;1 )_ (t);
into (D.1d) results in
_ = ;0M (G + M ;1 )_ + CeAt x(0) ;
(D.3)
where G : e 7! y is original system operator (as depicted in Fig. 4.5a), M ;1 is the
diagonal matrix of maximum slopes and, for simplicity of notation, the dependence
on is dropped. Solving for _ gives
;1
_ (t) = ; I + 0M (G + M ;1 )
0M CeAtx(0);
(D.4)
where the inverse exists, since _ is bounded. That is, the input-output mapping
;1
G = I + 0M (G + M ;1 )
0M
(D.5)
BIBLIOGRAPHY
187
is L1 stable. Designating the peak gain as kGk1;i < 1, _ can be bounded pointwise
in time with c1 ; c2 > 0, as
j_ (t)j kGk1;ijCeAt x(0)j
c1 kGk1;i1e;c t
(D.6a)
(D.6b)
2
where
c1
e;c2t
max
sup jyk (t)j
k
t0
k = 1; : : : ; m; with y(t) = CeAt x(0) and 1 2 Rm is a vector of 1's. In particular,
;c2 is the real part of the \slowest" eigenvalue of A (assumed Hurwitz), and c1 is
chosen simply to ensure the exponential envelope bounds all elements of jyk (t)j; t 0.
Therefore, the following holds:
k_ k1 =
m Z
X
1
j_k (t)j dt
k=1 0 Z
1
m max
j_k (t)j dt
k 0
Z 1
mc1kGk1;i e;c2t dt
0
c
1
m c kGk1;i
2
(D.7a)
(D.7b)
(D.7c)
=
< 1:
(D.7d)
(D.7e)
R
Thus, _ 2 L1, which ensures the existence of limt!1 0t _ ( ) d y. Therefore, (t)
asymptotically becomes constant. Furthermore, in this case, convergence to that
constant is exponential.
Sometimes called induced L1 -norm of the operator.
y A straightforward way to see this is to write R _ ( ) d as a Lebesgue integral and perform the
0
R
R
R
standard
decomposition
into
positive/negative
sequences,
_ d = [0 ] _ + d ; [0 ] _ ; d. Since
[0
]
R
R
_ ; d are monotonically increasing and bounded, the limit is guaranteed by
_ + d and
both
t
;t
[0;t]
[0;t]
;t
;t
the Lebesgue Monotone Convergence theorem (see [Rud87, OD96], for example, or any real analysis
text).
188
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