CONTROL SYSTEM DESIGN
USING H. OPTIMIZATION
by
Robert Playter
B.S.A.E., Ohio State University
(1986)
SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND
ASTRONAUTICS IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTSINSTITUTE OF TECHNOLOGY
June 7, 1988
(
Robert Playter
1988
The author hereby grants to M.I.T. and the C.S. Draper Laboratory, Inc. permission to reproduce and to distribute copies of this thesis document in whole
or in part.
Signature of Author
Department of Alronautics and/Astronautics
I
/ June 7, 1988
Certifiedby
John it. Dowdle, Ph.D.
Technical Staff
The Charles Stark Draper Laboratory, Inc.
K>
Cprtifpe hv
YJVVI
JI
-------------
Wallace ri. Vander Velde
PrQfeslor of Aeronautics and Astronautics
Accepted by-
Professor Harold Y. Wachman
-.
hairman, Departmental Graduate Committee
i
- ,
I
A·---
--
'
I·
SMMSASETS Immm
OFTE4OLOOY
SEP 07 1988
Arn'
A nna
__S~II
WItHRkWNr
18
T
"
1
£- we
-i
MA-T~~
CONTROL SYSTEM DESIGN
USING Hoo,OPTIMIZATION
by
Robert Playter
submitted to the Department of Aeronautics and
Astronautics in partial fulfillmentof the
requirements for the degree of Master of Science
ABSTRACT
The practical application of Ho, optimization to control system design is investigated. Two solutions of the Ho, optimal control problem are presented within
a general framework for control system synthesis. The synthesis approach is
amenable to optimal control system design using the H 2 norm (LQG optimal
control) or the Ho, norm. The most recent solution to the Ho optimal control
problem is significantly easier to apply than the previous solution, making the
new solution a viable alternative to established multivariable control system
design methodologies.
Weighting functions augmented to the open-loop system provide the necessary design freedom to achieve system requirements with the Ho procedure.
In this thesis, system performance is achieved through singular value response
shaping of the closed-loop transfer functions. Singular value inequalities are presented that provide tight bounds for the response of closed-loop transfer func-
tions in terms of the H,, norm and the singular value response of the weighting
functions.
H, optimization is included in a control system design procedure for largescale interconnected systems. In a design example, the procedure is applied to
an interconnected, multivariable system.
Technical Supervisor:
John R. Dowdle, Ph.D.
Title:
Technical Staff
The Charles Stark Draper Laboratory, Inc.
Thesis Supervisor:
Wallace E. Vander Velde
Title:
Professor of Aeronautics and Astronautics
ii
ACKNOWLEDGEMENT
I would like to thank the many people who have helped with the
research of this thesis and, more generally, provided me with inspiration and encouragement during my education thus far.
Dr. John Dowdle provided me with the opportunity to study
at Draper Laboratory. His interest in Ho optimal control theory
provided the impetus to begin my research on this topic. John's
expertise, interest, and patience were invaluable to the successful
completion of this project.
I thank my parents, for their ever present support, their counsel
and their friendship.
I would like to thank a few special friends, and you know who
you are, for providing me with the intellectual challenge and support of our unique friendship. It has been, and I hope will continue
to be, an inspiration to me to keep learning, and striving for the
betterment of myself and those about me.
Finally, I would like to thank Eftihea. Her strength, encouragement and love, have been instrumental in my education in every
respect.
I hereby assign my copyright of this thesis to The Charles Stark
Draper Laboratory, Inc., Cambridge, Massachusetts.
Robert Playter
Permission is hereby granted by The Charles Stark Draper Laboratory, Inc. to the Massachusetts Institute of Technology to reproduce any or all of this thesis.
iii
Contents
Abstract
Acknowledgement
Contents
List of Figures
Notation and Abbreviations
1 Introduction
1.1
1.2
1
Motivation ...............
Background ...............
1.3 Organization ..............
2
3
3
2 Mathematical Necessities
5
2.1
2.2
Function Spaces.
............
Riccati Equations and Factorizations
3 Summary of the Solution Approach
3.1
Motivation
.............
.................
3.2 Problem Statement .........
3.3 Youla's Parametrization ......
. . . . . . . . . . . . . . . . .
3.4
3.5
.................
.................
The Generalized Distance Problem
Best Approximation ........
3.6 Summary ..............
5
8
.
14
.16
18
.19
.20
. . . . . . . . . . . . . . . . .
. ...
. .22
. . . . ........
4 Problem Statement
22
24
4.1 Internal Stability ..........
4.2 Youla's Parametrization ......
25
4.3
4.4
28
31
27
Linear Fractional Transformations: Several Faces of One Problem
H, Optimization as a Generalized Distance Problem .......
4.5 Summary ...............................
iv
34
5 Optimal Solutions
The H2 Problem
. .
The Ho Problem . . .
. .
5.3 7--Iteration .
5.4 A Simpler Solution . .
. .
5.5 Summary .....
5.1
5.2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...................
...................
...................
...................
...................
6 Control System Design Using H-Infinity
6.1
6.2
6.3
6.4
System Definition.......................
Frequency Response Shaping . .
Weight Selection .........
. .
Examples .............
7 Design Example
A Pointing and Tracking Model
Performance Bounding ......
7.2.1 Kalman Filter Bounding
7.2.2 Wiener-Hopf Bounding.
7.3 Decentralized Control Design . .
7.3.1 Stabilized Platform . . . .
7.3.2 Gimbal.
7.3.3 Fast Steering Mirror . . .
7.3.4 Performance
7.4 Ho, Design.
7.1
7.2
7.4.1 Plant Definition .....
7.5
7.4.2 Weight Selection .....
Design Examples .........
.
Optimization
. . . . . . . . . . . .- .
..................
..................
..................
..............
..................
..................
..................
..................
..................
..................
..................
..................
..................
8 Conclusions and Suggestions for Further Research
v
35
36
37
40
44
48
49
50
52
55
56
76
77
82
82
82
84
86
86
86
91
93
93
95
98
117
List of Figures
3.1
Solution Methodology
........................
14
16
17
3.2 Control System Structure ......................
3.3 Standard Control Configuration ...................
4.1 Standard Control Configuration ...................
4.4
4.5 Ft(J,Q)................................
Modified System for Stability Analysis ...............
24
25
26
26
29
5.1
Scaled System Model ...........
46
4.2
Modified System
...........................
4.3 'Subsystem of Ft(P, K)........................
6.1 Control System Structure
6.2(a-e)
Example 1 Plots .
6.3(a-e)
Example 2 Plots .
6.4(a-e)
Example 3 Plots .
6.5(a-e)
Example 4 Plots .
6.6(a-e)
Example 5 Plots .
.....
. . . . . . . . . . . . . . . . .
50
59-61.................
62-64.................
65-67.................
68-70.................
71-73.................
7.1 Pointing and Tracking Model ..........
. . . . . . . . .
7.2 Block Diagram: Pointing and Tracking Model . . . . . . . . .
7.3 Loop Gains: Wiener-Hopf Controller .....
. . . . . . . . .
7.4 Stabilized Platform Loop ............
. . . . . . . . .
7.5 Stabilized Platform Loop Gain and Phase . . . . . . . . . . . .
7.6 Gimbal Loop ...................
. . . . . . . . .
7.7 Gimbal Loop Gain and Phase ..........
. . . . . . . . .
7.8 Fast Steering Mirror Loop ............
. . . . . . . .
7.9 Fast Steering Mirror Loop Gain and Phase . . . . . . . . . . .
7.10 Hoo Transfer Functions.
. . . . . . . . .
. . . . . . . . .
7.11 Weighted Transfer Function ..
.........
7.12 Example 5-Control Loop Gains .........
. . . . . . . . .
7.13(a-e) Design 1 Plots ...............
. . . . . . . . .
7.14(a-e) Design 2 Plots ...............
. . . . . . . . .
vi
. 77
. 79
. 85
. 87
. 88
. 89
. 90
. 91
. 92
. 95
. 96
. 97
. 99-101
. 102-104
7.15(a-e)
Design 3 Plots ........................
7.16(a-e) Final Design Plots ...................
7.17(a-d)
H 2 Design Plots
...................
vii
106-108
....
.....
111-113
115-116
NOTATION AND ABBREVIATIONS
R
the real numbers
C
the complex numbers
X
complex linear space
Re(f)
the real part of the complex function f
Rp
proper, real-rational functions
R nx m
real matrix of dimension n x m
A-1
inverse of the matrix A
At
the generalized inverse of the matrix A
A1
the orthogonal complement of the matrix A
AT
transpose of the matrix A
A*
Hermitian conjugate of the matrix A
A> 0
the matrix A is positive definite
A
the matrix A is positive semi-definite
O0
Ai(A)
the it h eigenvalue of the matrix A
oi(A)
the ith singular value of the matrix A
a(A)
the maximum singular value of the matrix A
r(A)
the minimum singular value of the matrix A
G'(a)
:= G T ( -s)
jw
point on the imaginary axis (w
VIII
viii
R)
the Banach space of essentially bounded
matrix valued functions, F(s), with norm
IIFIIoo= e
sup,, [F(jw)]
Ho:
the subspace of functions F(s) in Loo which
are analytic and bounded in Re a > 0
RH,o
real rational functions in Ho
L2
the Hilbert space of vector valued functions with norm
IIfll2= [o
<
f*(jw)f(jw)dw]1
and inner product
< f,g >= fo f (jw)g(jw)dw
H2
the subspace of functions f(s) in L 2
which are analytic and bounded in Re
>0
H,2
the orthogonal commplement of H 2 in L 2
PH,
the orthogonal projection from L 2 to H 2
HP-
the orthogonal projection from L 2 to H#
11112
L2 / H 2 norm
IlGlloo
Lo / Hoo norm
IIMII
the operator norm of M
MG
the multiplicative operator generated by G E Lo
MG]H2
the Laurent operator, MG, restricted to the
space of H 2 functions
HG
the Hankel operator generated by G E Loo
Ft(P, K) the linear fractional transformation of P and K
Ric[-]
the solution to the Riccati equation with associated Hamiltonian, [.]
ix
A
B ]
D + C(sI - A)-'B (shorthand notation for transfer functions)
rcf
right coprime factorization
Icf
left coprime factorization
SISO
single-input/single-output
MIMO
multiple-input/multiple-output
LQG
Linear Quadratic Gaussian
FDLTI
finite dimensional, linear, time-invariant
GDP
generalized distance problem
sup
supremum
x
Chapter
1
Introduction
Fundamental in the design of an optimal control system is the
selection of the performance measure. This measure is generally
a mathematical norm that quantifies the response of the closedloop system. Whether or not the optimal system has desirable
properties depends upon the selected norm and how pertinent it
is to design goals. The H, norm is the maximum singular value
over all frequency of a stable matrix transfer function. This norm
provides a measure of system response that arises naturally in a
broad class of input-output situations. Closed-loop systems that
are Hoooptimal will thus have application to a range of important
situations including the design of optimal feedback systems and robustly stable systems. Another asset of Ho optimal control design
is the flexibility of the framework used to define the solution approach. This framework is very general and encompasses a variety
of feedback problems and optimization methodologies making the
approach a valuable and versatile tool. Finally, the H~ optimal
control methodology offers the capability to perform explicit multivariable frequency response shaping, a useful characteristic. For
these reasons H,, optimal control theory is potentially an important addition to multivariable control system design methodologies.
1
1.1
Motivation
In his seminal papers [35,36],Zames pursued the idea of H~, norm
optimization of a weighted sensitivity function in order to avoid
the restrictions of schemes using quadratic norms. Optimization
with respect to quadratic norms, such as Wiener-Hopf and Linear Quadratic Gaussian (LQG) optimal control, hinge upon the
description of a particular power spectrum of the disturbance signals. Performance of systems designed with this criteria tends
to be sensitive to modest changes in the signal power spectrum
and plant model [13,26]. The Ho norm arises when optimization
over a broader class of input signals is considered, thus making
H, optimal control systems more robust to input uncertainty.
The maximum singular value has been recognized as a sufficient, although conservative, measure of stability in the presence
of plant uncertainty [23]. The conservatism arises because of the
inability to easily incorporate any known structure of the uncertainty. However, H, optimization, combined with the structured
singular value, denoted
[15], provides necessary and sufficient
conditions for robust stability and performance in the presence of
structured uncertainty [5]. The names i--synthesis [5]and Causality Recovery Method [24] have been coined for this current topic
of research.
As mentioned previously, the framework used to solve H, control
problems is very flexible and can be applied to a broad class
of problems. It also illustrates the inherent similarities between
several different optimal control methodologies. The solution approach first parametrizes all finite dimensional, linear, time invariant (FDLTI) compensators which stabilize a given plant, also required to have these characteristics, using Youla's parametrization.
This leads to a parametrization of all stable closed-loop systems
from which the optimization problem is derived. Several other optimal control design methodologies, Weiner-Hopf or H 2 , which
minimizes the RMS response, and L 1 [6,7] which minimizes the
maximum response, employ this general approach. The optimal
solutions to H2 and H, control problems are also strikingly similar.
They each require the solution of two remarkably similar Riccati
equations; H 2 requires a single solution of two Riccati equations
2
while Hoosolves the Riccati equations as part of an iteration to
find the optimum norm.
Finally, H,o optimization offers a capability for explicit loop
shaping, and thus systematic design of closed-loop performance.
Weighting functions, which act as design parameters, become, within
limitations, frequency response shaping functions for selected closedloop [25] and open-loop [28] transfer functions.
1.2
Background
The H/ optimization approach to control design was introduced by
Zames in [35] and developed for single-input/single-output systems
by Zames and Francis in [36] where the impetus was control design
in the face of disturbances and plant uncertainty. Multivariable extensions of the theory have depended upon complex mathematical
tools including Nevanlinna-Pick interpolation theory [9], the operator theory of Sarason [29] and Adamjan, Arov, and Krein [1,2],
and the geometric theory of Ball and Helton [4]. Developinga theory for robust stabilization with structured uncertainty has been
the impetus for the Ho optimization work of Doyle and Chu who
have led the practical extensions of the theory to state-space methods [5,13,14,15]. These solutions are dependent upon the theory
of Hankel operators and the state-space algorithms of [20]. A recent development by Doyle and Glover has led to a solution of the
H.o optimal control problem [16]that obviates a great deal of the
cumbersome mathematics previously necessary. It requires only
the solution of two Riccati equations in an iterative procedure.
1.3
Organization
This thesis presents two solutions of the Hoooptimal control problem. Application of the theory to control system design is motivated by a disturbance rejection problem. However, the flexibility
to address different control problems is emphasized. The theory
is applied to a multivariable example where solutions have been
obtained using both the solution of [5] and of [16]. Design exam-
ples focus on applications of the theory to shape singular value
3
responses of closed-loop transfer functions.
The organization of this thesis is as follows: Chapter 2 presents
mathematical necessities for the solution of the Hoooptimal control problem. The factorization approach of [5]is very lengthy and
complex; therefore, Chapter 3 summarizes the definition of the op-
timization problem and the solution approach. Chapter 4 describes
the problem statement in detail. In this chapter, successive transformations reduce the optimization problem to a more tractable
form, which is then solved in Chapter 5.
A new, significantly
simpler solution [16] of the H,o optimal control problem is then
presented at the end of Chapter 5. Many of the complex factorizations required for the approach of [5] are not necessary for this
simpler approach, although the latter solution is presented within
the same framework as the solution of [5], discussed in Chapters 3,
4 and 5. The general application of H/e optimization is discussed
in Chapter 6 where the focus is on frequency response shaping of
closed-loop systems. A design example for a multivariable sys-
tem is included in Chapter 7, where Ho, optimization is included
within a methodology for designing compensators for large-scale
interconnected systems. Finally, Chapter 8 presents conclusions
and suggestions for further research.
4
Chapter 2
Mathematical Necessities
Much of the challenge of learning H, optimization techniques is in
mastering the notation and the mathematical manipulations which
are used to render the problem soluble. This chapter introduces
the mathematical concepts required for understanding and solving the H, optimal control problem. First, the function spaces
that provide the foundation of the optimization problem are defined. Next, matrix coprime factorizations, which are essential to
this solution, are presented. Finally, solutions to Riccati equations
which yield state-space formulas for the desired factorizations are
discussed. The notation and properties presented in this chapter
will be used throughout this thesis.
2.1
Function Spaces
The H, paradigm is dependent upon the theory of Hilbert and
Banach spaces for the proof of some of its most important elements.
Two illustrations of this fact are,
* The parametrization of all stabilizing compensators for a given
plant is given in terms of an unknown function Q. As long
as Q is stable, i.e. belongs to to the Banach space H,, the
closed-loop system is stable.
* The optimization of the H, norm of a transfer function is
posed as the orthogonal projection of a function from the
5
Hilbert space H2 to its complementary space H2L. The in-
duced norm of the operator that performs this projection is
the norm to be minimized.
The concepts of norm, completeness, and inner product are addressed first and then used to define the desired function spaces. It
is assumed that the reader is familiar with the concept of a linear
space. The remainder of this section is largely taken from [18]. For
a more detailed discussion of the subjects presented in this section
see [11,18].
Let X denote the linear space over the complex numbers, C. A
norm on X is a function,
xIIll,which maps X to the field of real
numbers, R, and has the following four properties:
1. 1111
2.
0,
x11
= 0 iff x= 0,
3. Ilcxll = IclIxlJ, c E C,
4. Ixa + yll < 1l~l + IIYll.i
Completeness of a linear space is defined with the aid of a norm: a
sequence {xkh} in X is said to convergeto its limit, x, also in X, if
the real numbers {IIxk- x11}converge to zero as k - oo. If all the
sequences in X which tend towards convergence do so in X, then
X is complete.
A Banach space is a normed linear space which is complete.
There are many norms defined for a linear space populated
by complex transfer function matrices. We wish to minimize the
Ho norm, (jw), the maximum over frequency of the maximum
singular value of a complex transfer function matrix. Associated
with this norm are the following two Banach spaces:
Definition 2.1
Lo : The Banach space of essentially bounded (bounded ev-
erywhere ezcept possibly at a finite number of singularities)
matriz valued functions, F(s), with norm
IIFIloo
= ess sup
&[F(jw)].
w
6
Definition 2.2
H, : The subspaceof functions F(s) in L,, which are analytic
and bounded in Re s > 0.
We will be working primarily with the real, rational functions in
Hoo. These matrix functions compose the subspace RH, C H,.
The definition of a Hilbert space includes the additional concept
of an inner product. An inner product on the complex linear space,
X, is a function denoted < , y > which maps the space X x X to
the set of complex numbers C having the following properties
1. < x, x > is real and nonnegative,
2. < ,x >= Oiff =0,
3. the mapping of y to < ,y > is linear,
4. < x,y >*= < y, >,
where * denotes complex conjugate transpose.
The inner product on X may be used to define a norm,
IIxI =<
X,X >/2.
A Hilbert space is a normed linear space which has an inner
product and is complete.
The Hilbert spaces which we will use are now defined.
Definition 2.3
L2 : The Hilbert space of vector valued functions with norm
IIlf112 [f, f*(jW)f (jw)dw] ' < a)
and inner product
< f,g >= f
f*(jw)g(jw)dw.
Definition 2.4
H2 : The subspace of functions f(s) in L2 which are analytic
and bounded in Re s > 0.
7
The H, norm is related to the H2 norm through an important
and useful property. The maximum singular value is a measure of
the frequency dependent gain of a transfer function matrix that is
multiplied by an H2 vector. This is an induced norm of the matrix
[18]. The supremum over frequency of this gain is the H, norm of
the matrix. More precisely, if F E H,, then
IlFloo= sup{llFgll 2 :g E H 2 , 11912<
1}.
The inner product on a Hilbert space, X, also defines the concept of orthogonality. Two vectors x,y in X are orthogonal if
< z, y > = 0. If S is a subset of X then S is the set of all vectors
orthogonal to the vectors in S. If S' is closed then it is called an
orthogonal complement to S.
2.2 Riccati Equations and Factorizations
The solution of the H, optimal control problem depends heavily on
transfer function matrix factorizations. The appeal of the present
solution is largely due to the fact that it can be solved using statespace formulas. The relationships between solutions of Riccati
equations and the desired factorizations provide this state-space
capability.
The solution of certain Riccati equations are used to obtain
coprime factorizations, necessary for the parametrization of all
stabilizing compensators (Chapter 4).
Special forms of coprime
factorizations, inner-outer and outer-inner factorizations, are also
used to yield the generalizeddistance problem (Chapter 4). Finally,
spectral factorizations are necessary to solve the best approzimation problem (Chapter 5), the last step in the H,. optimal control
problem. The theorems of this section establish the relationships
between Riccati equations and the required factorizations.
Consider the concept of coprime factorization. It is a fact [14,32]
that every proper, real-rational matrix, G E Rp ( denotes the set
of proper, real-rational matrices), has a right-coprime factorization
(rcj). Denote this factorization as G = NM-1.
By duality a
left-coprime factorization (Icf) also exists such that G = M-lN.
Definitions of right and left coprimeness follow [14,16]:
8
Definition 2.5 (Right Coprimeness)
Two matrices, N, M E RHo. with the same number of columns,
m, are right coprime if there ezist X, Y E RHo. such that
XM + YN = Im.
(2.1)
Definition 2.6 (Left Coprimeness)
Two matrices, N,
E RH,, with the same number of rows,
p, are left coprime if there eist X, Y E RH.o such that
MX + NY = Ip.
(2.2)
The above definitions are equivalent to saying that the combined
matrices
N
and
have a left, and respectively, a right inverse in RH..
The utility
of these coprime factorizations is their capability to express every
proper, real-rational matrix as a function of stable, or Hoo, factors.
Equations 2.1 and 2.2 are called Bezout or Diophantine identities.
The solution of certain Riccati equations will be employed to compute the coprime factors of Eqs. 2.1 and 2.2.
In the following, denote the transfer function matrix
G(s)= D + C(sI -A)-B
by
A B
Consider the Riccati equation
ETX +XE- XWX +Q
(2.3)
where E,W,Q E Rnn,W = WT > 0, and Q = QT with the
associated Hamiltonian matrix
-ET
AH =-Q
9
Denote the necessarily symmetric solution of the Riccati equation
as
X = Ric(A).The following theorems hold [14,16].
Theorem 2.1
Let W = GGT. Then the stabilizability of (E,G) and the
requirement that AH has no purely imaginary eigenvalues are
both necessary and sufficient for the existence of a unique
solution to Eq. 2.3 which stabilizes (E - WX).
Theorem 2.2
Let A, B, P, S, and R be matrices of compatible dimensions
such that P = pT, R = RT > 0 with (A, B) stabilizable and
(A, P) detectable. Then
1. The parahermitian rational matrix
(s)-[BT(_8I
- A T )-'
I] [ PT S [ (sI - A)-'B ]
S
R
I
satisfies
r(jw) >
w.
2. For
(a) E = A - BR-ST
(b) W = BR-1BT and
(c) Q = P - SR - 'S T
there exists a unique, real, symmetric X satisfying Eq 2.3
and (E - WX) is stable.
3. There exists an M E Rp such that M - 1 E RHO and
r(s) = M*(s)RM(s).
One such M is
where
F= ( +BX)
whereF = -R-l(ST + BTX).
10
Note that 3 above provides a mechanism for performing spectral
factorizations.
Next, a characterization of inner transfer functions will lead to
inner-outer factorizations via the solution of a Riccati equation.
Let G
[-C-D-].
Then G is inner if G*G = I and co-inner
if GG* = I. (Note that G does not need to be square.) This
definition means that the singular value response of an inner or
co-inner function is constant and equal to 1. An all-pass func-
tion is a constant matrix times an inner (or co-inner) function.
If G E RX
m,
(p > m) is inner then G. E R(P
-m )
is called a
complementary inner factor if [G Gi] is square and inner. The following result characterizes inner functions in terms of the solution
of a Lyapunov equation.
Theorem 2.3 (Inner Functions)
Let G =
X
C[
]
e Rpxm . Suppose there exists a symmetric
RnXn satisfying
1. ATX + XA + CTC = Oand
2. BTX + DTC = O
Then G is inner if
3. DTD =I
The next theorem characterizes coprime factorizations with the
solution of a Riccati equation.
Theorem 2.4 (Coprime Factorization)
C
Suppose that G =
- , where (A, B) is stabilizable and
(A, C) is detectable. A state feedback gain, F, stabilizing
the matrix (A + BF), yields a right coprime factorization,
G = NM - 1 where
A BF BZ
[
F
and Z is any nonsingular matrix.
11
Z
(2.4)
(The notation used above means that M and N have identical
'A' and 'B' matrices but different 'C' and 'D' matrices.) A state
feedback gain, F, can be found by defining the well recognized
terms of a linear quadratic regulator for Eq. 2.3:
E=A
T
W = BR-1B
Q = CCT
then
F = -R-1BTX
By duality, an output injection gain, H, which stabilizes (A+ HC),
yields a left coprime factorization G = M-1N where
[ ]
[M
[A+HC H B+HD
]
ZC
Z
(2.5)
ZD
(2.5)
When the requirements for an inner function are combined with
a realization for coprime factorizations, a formula for an innerouter factorization results. A function M(s) E RH, is outer if
M - 1 exists and is stable and proper. The next two theorems apply
to stable systems. Denote by R /2 the square root matrix such
that (R1/2 )TR1/2 = R (or R1/2(R1/2 )T = R) and let DI be any
orthogonal complement of D such that [DR - 1/2 Di] is square and
orthogonal.
Theorem 2.5 (Inner-Outer Factorization)
[
Assume thatG=
E RH PX m (p
D
m) is
a minimal
realization of G. Then there exists a right-coprime factorization G = NM - ', with N inner and M outer if and only
if G*G > 0 on the imaginary axis, including at infinity. A
particular realization is
[
A + BF
N
BR-1/ 2
R - 1/2
F
C + DF DR- 1/ 2
where
12
(2.6)
=
R = DTD > O
F = -R-l(BTX + DTC)
and
X
(ARic- BR-
1
DTC)
-BR-1BT
CT(I - DR-' DT)C -(A - BR-' DTC)T
Theorem 2.6 (Complementary Inner Factor)
If p > m in Theorem 2.5, then there exists a complementary
inner factor N1 such that [N N±] is square and inner. A
particular realization is
N1 = A + BF -XtCTDI
C + DF
Di
where t denotes a generalized matrix inverse.
An outer-inner factorization is dual to the inner-outer factorization
and may be determined by applying the above result to GT(s)
resulting in G = /1--N1
Finally, an Icf with an inner denominator will be needed to
perform spectral factorizations.
Theorem 2.7
There exists an Icf, G
M-N,
such that M is inner if and
only if G has no poles on the jw axis. A particular realization
is
A±HC
H
C
I B
D
__
where
H = -YCT
and
Y = Ric
-A
-A
With the above theory serving as background we next turn attention to the Hoooptimal control problem.
13
Chapter 3
Summary of the Solution
Approach
This chapter motivates the H~ optimal control problem and summarizes the solution. The methodology consecutively reduces the
complexity of the problem until it reaches a soluble form (see Fig-
ure 3.1). There are several steps in the problem reduction, and
the mechanics at each step are complex. First, a general model is
defined which can address a broad class of systems. The optimiza-
tion problem is then stated as the minimization of the Ho norm
of a weighted closed-loop transfer function subject to closed-loop
stability. The optimal compensator is denoted K,. Next, the compensators which stabilize the system are parametrized in terms
of the stable matrix Q using the so-called Youla parametrization.
This yields an equivalent but simpler representation of the closed-
loop system. Employing a factorization approach, this optimization problem is transformed to a generalized distance problem.
Finally, the generalized distance problem is reduced to a special
form, the best approximation problem, which is solved iteratively
using 7y-iteration. Once the iteration procedure has converged,
the stabilizing compensator that achieves the minimum norm can
be computed. The details of these steps are found in subsequent
chapters.
(Note: Most of the functions we deal with are frequency dependent, however the explicit dependency on s will often be suppressed
for convenience).
14
Origi
Prob.
Equiv
Prob
Generalized
Distance
Problem
Appro
Pr
-Iteration
Figure 3.1: Solution Methodology
15
3.1
Motivation
The seminal work [35,36]of Hoooptimal control addressed the minimization of the H norm, or maximum singular value over all
frequency, of a weighted sensitivity function.
In Figure 3.2 this
corresponds to minimizing the Hoonorm of the closed-loop transfer function from q to e. The equation for this is written,
e = Wy(S)S(S)Wq(S)
qo
where the sensitivity transfer function is defined as
S(s) = (I + Gp(s)K(s))- '
This problem statement allows the designer to shape the sensitiv-
Figure 3.2: Control System Structure
ity response through the choice of the weighting functions Wq and
Wy. However, it does not allow a direct method for considering
disturbance rejection or control limitations.
An alternative problem definition is to consider the transfer
from the external signals v
=
[ ° ] to the weighted performance
do
16
variables e =
e ]. This equation is written
e,
WSWq
WySGpWd
V.
WuKSWq W,,KSGpWd.
(3.1)
Observe that with Wd and W, zero the minimization of this transfer function includes the original problem as a special case. A
block diagram of such a system is given in Figure 3.3. This fig-
ure represents the general configuration of plant and controller
common to the study of Ho optimal control. All linear, lumped,
time-invariant systems of interest can be expressed in this form.
The plant augmented with weighting functions is denoted by P(s)
and the compensator is K(s). For the systems to be considered in
this thesis, the measurements, y, and the controls, u, are defined
by the plant while the exogenous signal, v, consists of disturbances.
The performance variable, e, defines the key variables for system
optimization and evaluation.
Figure 3.3: Standard Control Configuration
In [35,36] Zames argues that minimizing the H,~ norm of the
closed-loop transfer function of Figure 3.3 is preferable to mini-
mizing a quadratic norm as in LQG and Wiener-Hopf optimal
control. The H2 norm, a quadratic norm defined in Chapter 2, of
the closed-loop system arises in two practical optimization problems [5]:
17
1. when the input signal, v, is stochastic with a fixed power
spectral density and it is desired to minimize the variance of
e, and
2. when v is a fixed signal, such as the impulse response of a stable transfer function, and it is desired to minimize the L2 norm
of e.
These norms depend upon precise descriptions of the input signals;
thus systems that are H2optimal tend not to be robust to modest
changes in the input signal.
The Hoonorm, as a measure of system response, applies to a
broader class of physical circumstances, making it useful in design
for performance and for robust stability. Three cases where the
Hoonorm of the closed-loop system is a useful measure of optimality are [5]:
1. if the input is any L 2 function and it is desired to minimize
the worst case L2 norm of the response, e,
2. if the input is a sinusoid and it is desired to minimize the
worst case amplitude of the response, e,
3. if robust stability is required for a nominally stable closed-loop
system with plant perturbations.
In the first two cases, the set of input signals is very general and
Ho optimization minimizes the worst possible response. The last
case is important in the design of robustly stable systems.
3.2 Problem Statement
Figure 3.3 corresponds to a partitioning of the plant transfer function by input and output dimensions as
P21(s
) P 2 ()
]
(3.2)
where the input-output relationships are
e(s)
=
P1 1(s)v(s) + P1 2(s)u(s)
(3.3)
y(s) -
P21(s)v(s) + P22(s)u(s)
(3.4)
18
with
Pij(s) = Di + C,(I - A)-'Bi.
(3.5)
The control signal is generated via feedback compensation as
u(s) = K(s)y(s).
(3.6)
The following notation will be adopted to represent the Pij in a
convenient state-space format
P(s)=
A
B1
B2
C
D
.D 12
C2
D2 1
D2 2
(3.7)
When Eqs. 3.6 and 3.4 are used in Eq. 3.3, the closed-loop
transfer function from v to e can be written as a linear fractional
transformation,
e = (P11 + P12K(I - P2 2K)-P 2 1) v.
(3.8)
or
e = Ft(P,K) v
The goal of the Ho optimal control problem is to design a compensator which minimizes the Hoonorm of this transfer function
subject to internal stability of the closed-loop system:
Ko(s) = argK(,)EnR
min IFt(P, K)[oo.
In what follows this optimization problem will be restated in several equivalent forms until it ultimately reaches a form that can be
solved iteratively.
3.3
Youla's Parametrization
The first simplification of this problem identifies the form of the
compensator to within one unknown dynamic transfer function, Q.
This is the Youla parametrization of all stabilizing compensators
[10,31]:
K(s) = (U + MQ)(V + NQ) - 1
where U,V, M, N, and Q E RH,.
19
(3.9)
Youla's result states that any FDLTI compensator for an FDLTI
P(s) can be represented in the form of Eq. 3.9. The matrices
U, V, M, and N are coprime factors (Eqs. 2.1 and 2.2) of, respec-
tively, some arbitrary stabilizing compensator and the plant, P(s).
Then as Q ranges over RH,,
all stabilizing compensators are re-
alized. (In subsequent chapters a realization of K as a function of
Q will be found using observer and state feed-back theory. Equation 3.9 can be rewritten as another linear fractional transformation which will be useful with the simple solution of section 5.4:
K = Ft(J,Q) = J + J12Q(I - J22 Q)1J2
(3.10)
An important advantage of Youla's parametrization is that the
closed-loop transfer function can be cast as a linear function of
Q. When Eq. 3.9 is used in Eq. 3.8 the following linear fractional
transformation results:
FI(T, Q) = Ft(P, K) = T 11 + T12 QT21
(3.11)
Now the objective is to find an optimal Q which minimizes the
H, norm of FI(T, Q):
Qo (s) = argQERH. min ijFt(T, Q)j,.
3.4
The Generalized Distance Problem
Further simplification of the problem utilizes the properties of
inner-outer factorizations. The Tij of Eq. 3.11 are functions of the
coprime factors U, V, M, and N. Coprime factors are not unique
(see Eq. 2.4), so they can be selected such that T12 and T2 l have
inner-outer and outer-inner factorizations given by
T12 = N 12z
12
(3.12)
T21 = 0 21 N 21 ,
(3.13)
and
where N12 and N2 l are inner and co-inner respectively and 812 and
021 are outer functions; in fact, the outer functions are constant.
(Recall from Chapter 2 that an inner function is a function G(s)
20
such that G*G = I and an outer function is a function F(s) such
that F-l(s) is stable and proper.) Additionally, the complementary inner factors NL and N
can be found such that
[N12 N]
and
N21
are square and inner. The utility of these factorizations is derived
from the fact that multiplication by an inner function preserves
the H2 and Hoonorms [14,16]and thus the norm of the closed-loop
transfer function can be rewritten as
IFt(T,Q) 11 =
]F(T, Q) [N.*
When the factorizations of T12 and T21 are applied to F(T, Q)
significant cancellations occur and the norm can be written as
IFe(T, Q)1l00
[ R11-Q
R 12
(3.14)
where Q is a simple function of Q. The minimization of this norm
is a generalizeddistance problem (GDP). The name is derived from
the notion of minimizing the 'distance', in an Lo, norm sense, from
R, which is an anti-stable matrix (all unstable poles), to Q which is
a stable matrix. Note that here the Lo norm is appropriate rather
than the Hoonorm because the matrix is no longer stable. For
the case in which R is a real-rational matrix an optimum solution
is known to exist [14,16]. However, the proof of existence does
not yield computable formulas. Thus the final step in solving the
problem is to reduce the generalized distance problem to a best (or
Hankel) approximation problem.
3.5 Best Approximation
The generalized distance problem is reduced to solving a series
of best approximation problems [1,2,20,29]using matrix dilation
21
theory in a process called --iteration. The optimization problem
is written as
min
~(o)¢/RHo
1
R 1 1 -Q
R12
R21
R22 ]|'
The iterative procedure is initialized by using an estimate of the
optimal norm, 7. Then applying the properties of matrix dilation
[8,14] to Eq. 3.15 the following 'one-block' problem results:
f(y)
min
0(a)ERH 00
IR - 0lloo
(3.16)
where R is a function of the Rij in Eq. 3.15 and -y. The initializing
y is less than yo if and only if f(y)
< 1. If y is selected to be
exactly equal to the optimum norm then the one-block problem
will yield f(7y) = 1. Thus yo can be found from the zero crossing
of the function f ()- 1.
is updated, using bisection for example,
according to the value of f ().
Equation 3.16 is called a 'best approximation' problem because
we are approximating an antistable matrix, R, by a stable matrix,
Q, while minimizing the L, norm of the difference. Clearly this
is a special 'one block' case of the generalized distance problem.
The approximation is equivalent to a projection using the Hankel
operator. Therefore, f(y) is equal to the Hankel norm of R. Once
the optimum cost is known, a Q, which achieves the L
norm, can
be determined using state-space methods [20]. Then Q,, Qo and
Ko are computed by inverting the necessary transformations. The
resulting frequency response of FI(P, K) has constant maximum
singular value response equal to 7y.
3.6
Summary
This chapter has described the solution to the H, optimal control
problem at a very high level as the minimization of the H, norm
of FI(P,K)over
all stabilizing compensators in Rp. The Youla
parametrization, the generalized distance problem, 7--iteration,
and best approximation problems were presented as key steps toward obtaining an Ho optimal compensator. In the chapters which
22
follow, details of the computational procedures required in carrying
out these steps are presented.
23
Chapter 4
Problem Statement
This chapter contains the detailed statement of the H~, optimal
control problem. Requirements for internal stability and 'optimal'
performance are set forth. The optimization problem itself can be
represented in several distinct forms; equivalent representations
will be employed which sequentially reduce the complexity of the
problem until a soluble form is reached. The major results of this
chapter are quickly previewed.
The configuration of plant and compensator that will be used is
shown again in Figure 4.1. The goal of H~ optimization is to find
Figure 4.1: Standard Control Configuration
a real, proper, stabilizing compensator to optimize the Ho. norm of
24
the closed-loop transfer function, Ft(P, K). Youla's parametrization yields a linear fractional transformation,
FI(J, Q), for all lin-
ear, time invariant, finite dimensional compensators which stabilize the given system. Then a linear fractional transformation for
the closed-loop transfer function that is linear in Q, FI(T, Q), is
derived. Special forms of coprime factorizations, namely inner-
outer and outer-inner factorizations, then allow the optimization
problem to be cast as a generalized distance problem. State-space
formulas for the required steps are provided.
4.1 Internal Stability
To aid in the definition of internal stability, introduce two fictitious
signals wl and w 2 as in Figure 4.2. To insure that the transfer func-
Figure 4.2: Modified System
tion from v, wl, and w2 to u exists and is proper, (I-P 22 (oo)K(oo))
must be invertible [5,14]. This condition for well posedness is as-
sumed in the remainder of this thesis. If the transfer functions
from wl, w 2, and v to e, u and y are stable then the system is
considered internally stable. Alternatively, let v be set to zero in
Figure 4.1. The system is considered internally stable if the state
vectors of P(s) and K(s) tend to zero from every initial condition.
(This is the usual definition of asymptotic stability.)
It is a fact [5,14] that the system of Figure 4.1 is internally stable
if and only if the system of Figure 4.3 is internally stable. This
becomes intuitive when v is set to zero as previously described;
25
in this case the only feedback loop involves P22 and K. In the
following, the system of Figure 4.3 will be used to characterize
closed-loop stability for the entire system.
Figure 4.3: Subsystem of Ft(P, K)
Consider the system of Figure 4.4. The input-output relationships for this system are given by
[
-P22 I
[:e2= [:
(4.1)
This system is internally stable if and only if the followingtransfer
function matrix is stable:
I -K 1_ I + K(I - P22K)-1P22
K(I- P22 K)-1
-P2 2
=
(I - P 22K)-P 2
(I - P22K)-1
(4.2)
Therefore, the closed-loop system of Figure 4.1 is internally stable
if and only if the matrix of Eq. 4.2 is an element of RH.
Youla's
parametrization provides an equation for the compensator that
insures that this requirement is satisfied.
Figure 4.4: Modified System for Stability Analysis
26
4.2
Youla's Parametrization
Youla's parametrization is presented here in a general form; it requires knowledge of the plant and some arbitrary stabilizing com-
pensator. The result is an equation, with one free transfer matrix, Q RH,, that defines all compensators which stabilize the
closed-loop system. In the next section a particular form of Youla's
parametrization is defined using gains derived from observer and
state feedback theory.
Recall from Chapter 2 that every proper, real-rational matrix
has a right and left coprime factorization [14]. Begin with right
and left coprime factorizations of P22 and Ka, some stabilizing
compensator:
P22 = NM -1 = -1 N
(4.3)
(4.4)
Ka = UV-1 = fV-1U
Substituting these expressions into the left hand side of Eq. 4.2
results in
o] -= ]
-0
(4.5)
For the system of Figure 4.1 to be internally stable, both of the
block matrices on the right hand side of Eq. 4.5 must be in RH,.
The right-most matrix of Eq. 4.5 is clearly stable by virtue of the
fact that coprime factors are stable.
When the factorizations of Eq. 4.3 and 4.4 simultaneously satisfy Bezout identities (Eqs. 2.1 and 2.2) the following equation,
often called the double bezout identity, holds:
[U M
f [
N V
I]
(4.6)
or
V
-t
e= N V
From this expression it can be immediately deduced that the inverted matrix on the right hand side of Eq. 4.5 is stable by the
inverse defined in Eq. 4.6, again because the coprime factors are
27
stable. Therefore, any Ka and P22 combination whose coprime
factorizations satisfy Eq. 4.6 yield an internally stable system.
If instead of using the coprime factorizations of Eq. 4.4, we use
Uo=-U+MQ
V =V+NQ
UO=U+QM
VzV + N
then the double bezout identity of Equation 4.6 is still satisfied for
any Q C RH,,. By using the above characterization of the coprime
factors of K, the parametrization of all stabilizing compensators
is given by the following right and left coprime factorizations
K = (U + MQ)(V + NQ)- 1 = (X/+ QN)-(U + QMT) (4.7)
which is the desired result.
4.3
Linear Fractional Transformations: Several
Faces of One Problem
The previous section utilized coprime factorizations to show that
all stabilizing compensators can be parametrized by one simple
formula, Eq. 4.7. This section uses that form to show that the
compensator and the closed-looptransfer function can both be represented by linear fractional transformations. Realizations of these
equations will be given using the state-space formulas of Chapter
2.
Recall that the closed-loop transfer function is given by the
linear fractional transformation,
Ft(P, K) = P11 + P 12K(I - P2 2K)-P 2,1 .
(4.8)
Youla's parametrization led to an expression for all stabilizing com-
pensators, which is itself a linear fractional transformation. It can
be shown [14] that Eq. 4.7 can be written as
K = UV-' + V-'Q(I + V-1 NQ)-lV - 1
28
or
K = FI(J, Q) = J11+ J12Q(I - J22Q)-J 2 1
(4.9)
where
J =
UV-
(4.10)
-V"'N
The linear fractional transformation for the compensator is represented pictorially in Figure 4.5. A stabilizing compensator may be
Figure 4.5: Ft(J, Q)
found by computing the coprime factors of Eq. 4.10. Begin with
the realization of P22 ,
P
=
A
P22=[C2
22
D22
A stabilizing state feedback gain, F, and output injection gain, H,
yield right and left coprime factorizations via Eqs. 2.4 and 2.5:
P22 = NM- 1 = M-1N
+]
[ A + B2F
C2 + D22F
A + HC 2
piB
V-]
C2
29
IH
I
B2
(4.11)
D22
B 2 + HD
D 22
22
(4.12)
The factors of Eq. 4.10 are derived from the model based com-
pensator associated with F and H, which is realized by the state
equations
= A +B2u + H(C2 + D22 u- y)
u = F.
This compensator has the state-space representation
A + B2F+HC2 + HD22F -H
K.
F
0]
To identify an rcf and an Icf for this compensator define
A = A+B 2F+HC2 + HD22F
b = -H
C= F
1D =
0
tF = C2 + D 22F
t = -(B 2 +HD 22)
Then, using Eqs. 2.4 and 2.5, the coprime factorizations of the
model based compensator are identified in terms of the state-space
parameters
of P 2 2:
A±B2F -H
C2 + D22F
I
F
(4.13)
0
(4.14)
[V~] = [ A + HC 2 -(B 2 + HD 2 ) -H
With these factorizations available, a realization of J, and thus
of all stabilizing compensators, is immediate. When Eqs. 4.11,
4.13, and 4.14 are applied to Eq. 4.10, the following representation
of J results:
J=
A + B 2F + HC2 + HD22 F -H
F
0
-(C2 + D22F)
I
30
B2 + HD22
I
-D22
We can go one step further with the approach of linear fractional transformations. Using Eqs. 4.3, 4.6 and 4.7, the following
expression for the closed-loop transfer is obtained:
FI(P,K) = (P11 + P12UIUMP
2 1) + (P12M)Q(MP21).
This is itself a linear fractional transformation. The importance of
this particularly simple form for the closed-loop transfer function
is that it is linear in Q:
Ft (P,K) = FL(T,Q) = T 1 + T12 QT2 1
where
T=[P1
+ P12 UMP21 P12M
MP21
°
Realizations of the Tij can be found from the realizations of
the Pj and the coprime factors of P22 and K0 [5]. After much
simplification, the Tij can be represented as
Tn =
A + B2 F
0
-HC 2
-HD21
A + HC 2 B + HD21
C1 + D12 F
T12 =
C1
CA[
(4.15)
Dll
z
D+ 2 12F
(4.16)
and
T2 1 = [
4.4
A + HC2 B1 +HD2 1
C2
D2 1
(4.17)
Ho Optimization as a Generalized Distance
Problem
In the previous section a particular realization of Youla's parametrization of all stabilizing compensators was used to derive statespace formulas for all stabilizing compensators, F(J, Q), and all
stable closed-loop systems, Ft(T,Q).
A state feedback gain, F,
and an output injection gain, H, for the subsystem P22 were used
to obtain these realizations. In this section further specifications
31
on the derivation of F and H yield forms for T12 and T21 of
Ft(T, Q) that are particularly useful. These are an inner-outer factorization of T12 and an outer-inner factorization of T21 . They allow
the optimization of the transfer function, FI(T, Q), to be stated in
terms of a generalized distance problem.
The inner-outer factorization of T12 is given by T12 = N 12Z 17The complementary inner function, N±, is also found such that
T12 = [N 1 2 N]
[
]
where
[N12 N]
is square and inner. The outer function Z 1 is a constant matrix.
Similarly, the outer-inner factorization of T21and the complementary inner function, N1 are found such that
1 = [Z1
0o]
[21
where
iN21
is square and inner and Z 2 is a constant. The utility of these fac-
torizations lies in the fact that multiplication by an inner function
is norm preserving [14,16]. Thus
F(T,Q)11.
=
[
12 FI(T,Q) [2 1N]
Using the inner-outer and outer-inner factorizations of the Tij
the argument simplifies to the form of the generalized distance
problem,
IIFt(T, Q)l
=
R 11 -Q
R122
I
(4.18)
where
R1
= N; 2 T11N2
R12 = N*2TllN
32
(4.19)
(4.20)
R2
= N;T 11N*1
(4.21)
R22 = NjT 11Ni
(4.22)
and
Q = -ZlQZ_ -.
(4.23)
The convenience of the constant outer functions is apparent when
Q needs to be computed from a realization of Q. For the case of
constant outer functions, the required inversions are very simple.
State-space realizations of the required factorizations result from
Theorems 2.5 and 2.6.
If T12 is an n x r matrix and n > r then the complementary
inner factor, NL, may also be found. Applying Theorems 2.5 and
2.6 gives the following form for an inner-outer factorization of T12
with complementary inner factor:
[N12 N]
Tl 2 = [N12 N]
[
Z1 = (DTD
12 )
[ A+B 2 F
B2Z1
(4.24)
1/ 2
(4.25)
XC
(4.26)
D±DT = I - D12ZlZ[ D2
(4.27)
F = -Z 1Z[ (BX + DT2C)
(4.28)
with
F is given by
and
X
=
Ric [ A- B 2Z1ZTDTC
-CTDIDTC 1
-B 2 Z 1ZTBT
-(A - B2Z 1ZTDC 1 )T
(4.29)
Note that D12 must be full rank so that Zl is non-singular.
Duality principles may be applied to obtain the outer-inner factorization of T21 E RXm, p < m. If p < m the complementary
inner factor, N 1 can also be found:
= [Z
]
33
[N
]
(4.30)
(4.31)
Z2 = (D21D21
A+HC
N 2i1
2
Z2C2
N
L
-DLBTYt
with
bTb
B 1 + HD 2 1
Z2D21
Dl
(4.32)
I
= I - DZT:Z 2D 2 1 -
(4.33)
+ B,1D:)Z
212~lu Z 2
(4.34)
H is given by
H = -(YC
2
and
DT ZT Z 2C 2)T
Y= Ric (A --BB 1DTDB
BT
1
-C2Z2 Z2C2
-(A - BDTZTZ2C)
'
(4.35)
Similarly, D21 must be full rank so that Z2 is non-singular.
4.5
Summary
Using the results of this chapter, all FDLTI stabilizing compensators are parametrized by a linear fractional transformation. In
addition, the closed-looptransfer function is parametrized by a single linear fractional transformation that is linear in Q. This latter
expression allows the optimization problem to be cast into a generalized distance problem. State-space formulas for the computation
of all necessary transformations have been provided.
The solution of the generalized distance problem is the subject
of the next chapter.
34
Chapter 5
Optimal Solutions
Youla's parametrization and the factorization approach of Chapter 4 have allowed the optimization problem to be cast as a gen-
eralized distance problem. This chapter is devoted to the solution
of this problem.
All the transformations of the problem to this point are actually independent of whether the H2 norm or the Ho, norm is chosen as the measure of optimality for the GDP since both norms
are invariant to transformations by inner functions. Therefore, the
framework of this procedure can be used to solve either optimiza-
tion problem. Considering this generalization, the GDP is stated
as the minimization of
1FI(T,Q)11a
-
R 11 -
R21
R12
R22(5.1)
where a equal to 2 or oo refers to the H2 or Ho, optimization problem. The H2 optimization problem is equivalent to solving a Linear
Quadratic Gaussian (LQG) problem with, potentially, frequency
dependent weighting functions [3,21]. Its solution is particularly
simple at this point and is equivalent to solving a projection problem. There is also an indication that this form of the H2 solution
may obviate difficulties associated with the LQG/LTR (Linear
Quadratic Gaussian with Loop Transfer Recovery) method, such
as requiring infinite gains to adequately recover the target loop
[30]. The Ho, solution being addressed here is more complex and
involves the solution of a related best approximation problem.
35
In this chapter, the existence of a solution to the GDP will be
discussed briefly. An abstract operator point of view is employed
which utilizes the 'geometry' of the problem. This operator perspective leads to a solution for a very special case of GDP, the
'one-block' problem. More general cases, two and four-block problems, that result from Hoooptimization must be solved by itera-
tively reducing the problem to a one-block problem in a process
called 7--iteration.
This chapter concludes with a brief presentation of the latest
solution to the H, optimization problem [16], which drastically
simplifies the solution procedure. This is the solution that was
most easily applied to the design example of this thesis (see Chapters 6 and 7).
5.1
The H2 Problem
In order to demonstrate the utility of the generalized distance problem formulation, we first consider the H2 problem. The H2norm
of the closed-loop system may be represented by
IIF(T, Q)12 =
R -Q
R 21
R 12
R
22
2
The optimization of JIFt(T,Q)112is particularly simple because
R 1 1-Q
R12
R2 2
2
=-IR 11
2-QI+
2
R 21 R 22
2
and the optimal Q is merely the stable part of R 11 [16]. Since R 11
is completely unstable Qo is equal to the constant part of R 11:
Qo = Z 1D 12 D 1 1D
2 1Z 2.
The optimal Q is found by inverting the transformation of Eq. 4.23,
which results in
Qo = -D12DljD21The optimal H2 compensator can now be found by employing the
realization of F( J, Q) .
36
5.2
The HooProblem
We now consider utilizing the generalized distance problem formu-
lation to solve the Hoooptimal control problem. Recall that the
problem in its present form was obtained by factoring the linear
fractional transformation,
F(T, Q) = T11 + T 2QT21.
A sufficient condition for the existence of an optimal solution is
given below [17].
Theorem 5.1
The optimal Ho norm of FI(T, Q)is achieved if the matrices
T12 and T21 are full rank for all frequencies.
(These conditions usually hold for well-definedproblems.)
The optimization problem calls for minimizing the 'distance', in
an Loonorm sense, from a stable matrix Q E Ho to the antistable
matrix R E Lo,. (Note that the L,o norm is required because R is
unstable.) Let us examine the simplest case first.
If T12 and T21 are both square and full rank, then so are their
inner factors, N12 and N21, which implies the complementary inner
functions Nl and Nj do not exist. In this case R11 is the only
non-zero element of R (see Eqs. 4.19-4.22). The Hoooptimization
problem in this case is equivalent to the one-block problem given
by
(5.2)
IIF(P,K)IIo = IIR1 - Qll
and a Qo that achieves the optimum is sought. This is the socalled best appoximation problem, and its solution is central to
the solution of the general case. Therefore, we examine it in some
detail starting with the operator theoretic perspective.
H2 is a closed subspace of L 2 ; let Hjldenote its orthogonal com-
plement. The multiplicative operator associated with a matrix
R1 E Lx is referred to as the Laurent operator and, when operating on a function f E L2, is denoted by MR,:
MRf = Rlf.
37
The Laurent operator defined by a matrix, Q E Ho,, leaves the
H2 subspace of L2 invariant. That is if
f E H2
then
MQf = Qf E H2.
A useful fact is that the H norm of a matrix is equal to the
supremum over frequency of the induced norm of the associated
Laurent operator on H2,
IIlko = sup{llGfll12 = IIMGi : f
E
H2,
Ilfll2
11
where the induced operator norm of the Laurent operator is denoted by IIMG
J. Let PHldenote the orthogonal projection from
L2 to Hi'. The invariance of the Laurent operator induced by an
H,
matrix, Q, is equivalent to
PH.MQIH2 = 0.
Stated simply, the orthogonal projection onto Hof an H2function
is zero.
Using these properties, the minimum distance, in an Loonorm
sense, between an unstable matrix R 1 and some stable matrix Q
can be written as
dist(R1,Q)
= min IIR - QII: Q E H.
= min IIMR 1 - MQII: Q E H
> min IIPH(MR, - MQ)IH21 : Q e H.
=
IIPHLMR
1 IH211
(5.3)
(5.4)
(5.5)
(5.6)
where equality actually holds [17,32].
Theorem 5.2
The minimum distance in an L norm sense from a known
matrix R 1 E L, to a matrix Q E H, equals the norm of the
operator PHI MR IH2 from H2 to HI.
38
The minimum L distance of Theorem 5.2 is the minimum
Hoonorm of the closed-loop transfer function of Eq. 5.2 that we
seek. Denote this norm by y,,.
The operator, PHIMR1 IH2 , is the Hankel operator, HR, defined
herefore, y, for Eq. 5.2 is the Hankel norm
by the matrix R 1.
of R 1. The continuous time Hankel operator is often described,
by analogy to the convolution operator, as an anti-causal operator
[20].
A state-space formula for computing the Hankel norm is used to
be a minimal state-space realization
find 3y. Let D+ C(sI-A)-'B
of the antistable matrix R 1. The controllability and observability
gramians, LC and L,, associated with R1 are the unique solutions
of the Lyapunov equations
AL, + L,AT = BBT
ATLo + LoA = CT C .
The Hankel norm of R1 is the square-root of the maximum singular
value of the product LCL, [20].
The state-space formula for a realization of a Q that achieves
the minimum Lo norm of the GDP and thus the minimum Ho, norm
of FI(T, Q) results in a constant maximum singular value response
of the closed-loop system [20]. This realization requires a balanced
realization of R 1. A realization is balanced if the controllability
and observability gramians are diagonal and equal [20]. Now, let
the matrix R 1 have a minimal, balanced realization given by
with controllability and observability gramians given by
where a is the greatest diagonal element of W with multiplicity r.
Partition A, B and C conformally with W,
A:
All
A21
A[
A22
B,
]
B2
I
39
= [C1C2
and choose D such that
DB + C 1 = 0
/bbT = r21.
Set
B = -(a2I _-
2 )-1(B
2
+ C2 )
A = (-A 22 + B 2 BT)T
C = C2 S+bB.
Then a state-space realization of Q is
So, for the special one-block case, or best approximation prob-
lem, the H~ optimization problem is solved by computing the
Hankel norm of the antistable matrix, R 1, and solving for the QO
which achieves that norm. In this case the optimal norm is exactly
achievable. For the more general two-block or four-block case the
minimum Lo norm is also equivalent to an operator norm. However, the exact solution involves an infinite dimensional eigenvalue
problem for which there is no computable formula. The solution
approach then depends upon reducing the general case to the best
approximation problem using matrix dilations and then computing
nearly optimal solutions by a process called 7--iteration.
5.3
y-Iteration
Nearly optimal solutions for the generalized distance problem are
found by iterating on an estimate of the optimal L, norm, denoted by y, and reducing the problem to a best approximation
problem. This approach allows the solution to be approximated
to any degree of accuracy. The necessary relations presented here
are derived from the dilation of constant matrices [8,14].
First, consider the two-block case which results when T2l has
full column rank, thus yielding NL = 0 (see Eqs. 4.19-4.22). The
40
problem is to find Qo:
Q =
arg4ERH.
min
R2Q
||
An upper bound for the minimization is provided by the H/ norm
of any other stabilizing solution, such as the H2 solution which is
easily computable within the current framework (Section 5.1).
The initializing estimate of y, allows the problem to be reduced
to a best approximation problem as follows:
Theorem 5.3
If Q E Ha0 and 7 > 70 then
R -Q
R2
<
|oo-
if and only if
f() = Il(R1 -
)M' 11 <
where
M*M = (y2I - RR 2).
yl = IR2 1llooprovides a lower bound on yo.
M is called a spectral factor (see Theorem 2.2) of the para-Hermitian
matrix (721- RR 2) and M -1 E RHoo. According to the theorem
the desired optimal norm, 70, is less than the chosen y if and only
if the norm of the related best approximation problem, f(y), is
less than one. If y is selected to be exactly equal to the optimum
norm then the one-block problem will yield f(y,) = 1. Thus, 7o
can be found from the zero crossing of the function f(y) - 1.
is updated, using bisection for example, according to the value of
f(Y).
It is known that f(y) is continuous, strictly monotonically decreasing and convex in 7 for 7 < JIR2 ll00 [5]. Thus, theoretically,
convergence is assured. By repeated application of Theorem 5.3,
70 may be approximated to any accuracy. The Q which achieves
the optimum norm can then be found by applying the best approximation algorithm of the previous section.
A general description of the 7--iteration procedure for the twoblock problem follows:
41
-y-Iteration
for the Two-Block Case
1. Compute bounds for y.
2. Choose y to satisfy the bounds.
3. Find the spectral factor M = (y2 - RR2)
1/ 2.
4. Compute f(y) = IHRM-1 I|1
(a) if f(y) > 1, go to 2 and increase y.
(b) if f(y) < 1, go to 2 and decrease .
(c) if f(y) = 1, go to 5.
5. The value of y is the minimum achievable norm. Solve for the
product QoM- 1, which is the best approximation of R 1M - 1'.
6. The optimal solution is Q.
Because y can not be achieved exactly for the general problem,
f(7y) is never equal to one. Therefore, the iteration is considered
converged when f(-y) is 'close' to one.
The four-block problem is solved in an analogous fashion [5].
Theorem 5.4
If Q C Ho, and y > 7y, then
[
R21
<
R 22
7
if and only if
f() = IT-1{Fl(y- R, -1R
2)-
-Q*}T-loo_< 1
where
T
(I- LL*)1/ 2
T=
(I-
L = R12S-
L
S
S-R
-
S -
(721-
L)
~/
1
21
R22R22)1/2
(72 - R22R22)1/2
42
and
F(y - R, r-'R;,)
* = - 1 {R 11 + R12(
2
- R;,R22)-IRaR2}.
A lower bound on the optimum norm is given by
= max{II[R 21 R 2 2 111
R22 '
}
Again, an upper bound can be taken as the Ho norm of any other
stable closed-loop system. More refined upper and lower bounds
are found in [5,14]. A general description of the
cedure for the four-block problem follows:
--iteration
pro-
7-yIteration for the Four-Block Problem
1. Compute bounds for y,.
2. Choose 7 to satisfy the bounds.
3. Find the factors S, S.
4. Find the spectral factors T and T.
5. Compute f(y).
(a) if f(7) > 1, go to 2 and increase 7.
(b) if f(y) < 1, go to 2 and decrease y.
(c) if f( 7 ) = 1, go to 6.
6. The value of 7 is the minimum achievable norm. Solve for the
best approximation of
T- Fl(-'R, -a'R;
2 )Twhich is denoted by
-T-Q}o-.
7. The optimal solution is Q.
Again 7o is not actually achieved and f(y) is never exactly one.
43
5.4
A Simpler Solution
As of the writing of this thesis, the solution approach just presented is the one currently available in the literature. This solution
hinges upon the complex inner-outer and outer-inner factorizations
of Chapter 4 as well as the balanced realizations and spectral fac-
torizations required in this Chapter. They have been successfully
implemented, yet they are doubtlessly cumbersome.
A new solution exists due to Doyle and Glover [16,24] which
requires the solution of two Riccati equations in a y-iteration
procedure. For this approach, the plant is constrained to have a
model of the form
P(s)=
A
C1
B1
B2
0
.D12
C2
Da21
(5.7)
Note that here the terms Dll and D22 are required to be zero.
This implies that there will be no feedthrough terms from the
exogenous signal, v, to the performance variable, e, and from the
control signal, u, to the measurement, y.
Given that
y > %o,
the following equations parametrize all stabilizing compensators,
K, such that
llFt(P, K)11oo<.
The solution employs scaled versions of P and K denoted by M
and L such that the previous inequality is satisfied if
IIFI(M,L)IIoo < 1.
To obtain the scaled system, M, define the following:
Z1 = (DT2Dl2) - 1/ 2
Z2
=
(D21lD21/
and
ul = ZU
Y1
=
7YZ 2Y
V1
=
7V.
44
Then the scaled system is defined by
e]
=
[
M()]
]
where
A
M() =
B1
B2
C1 o
D12.
0
D 21
C2
and
A=A
B1 = B /,
B2 = B 2Z 1
C1
C1
C2 =
D1 2 =
7 Z 2C 2
Da2
Z 2D 21.
D 12 Z 1
=
Note that as a result of this scaling
Dt2Dl = I
The scaled system is shown in Figure 5.1. The scaling of u and y
can be absorbed into the compensator to establish a relationship
between the compensator for the scaled system, L, and that for
the unscaled system, K:
K = 7 Z1 LZ 2.
The set of all stabilizing compensators for the scaled system of
Figure 5.1 such that
IIFI(M, L)I.
<1
is given by the following linear fractional transformation (Eq. 4.9):
L = Jll + J12Q(I - J22Q)- 1J2
45
1
V1
V
e
Figure 5.1: Scale System Model
where Q is stable with
IlQlloo < 1
and
Ak
Bk
Gk
I .
J(s)= Ck 0
Hk I
0
The parameters of J are defined via
Ak = A-BkC 2 - +
2Ck
Bk = YC 2 + Bl21
YC (C - Dl2Ck)
Ck = (B2x + DTC0
1)(I -X)
Gk = YCT1 2 + B
Hk = (D21 B - 2)(I-
x
where
X- = Ric
-
(A-
-I('
-(B 2
D2 1T2)
- D D))
12 1 -(A -
-B B
1 T)
2D, 2C 1)
I
and
y = RiC
( - B,
21
- Dr)r
=-iB(,
21 :,)B
46
-C 2T
_ CCI)
2 2- DC)
1
-(
-(A-
1
16b2) ]
Finally, for a solution to exist both X and Y must be positive
definite and the maximum eigenvalue of YX must be less than 1.0
[24].
Q can be chosen to be zero, thus simplifying the equations sub-
stantially, resulting in
L(s) = J11(s)
and therefore
K(s) = ZlJll(8)Z2
where
J(=
[ A-k
O-]1
The process of obtaining an optimal solution is still iterative as
it was with y--iteration of the previous solution. In this case the
maximum eigenvalue (YX) plays the part of f(y). A lower bound
for y may be taken as zero and an upper bound can be computed
from the Hoonorm of any other stable closed-loop system, such as
that using an LQG compensator.
A general description of the iterative procedure for the new
solution follows:
a-Iteration
for the Simpler Solution
1. Compute bounds for 70.
2. Choose 7 to satisfy the bounds.
3. Scale the system as directed.
4. Solve for X and Y
5. Compute A(YX).
(a) if A(YX) > 1, go to 2 and increase 7.
(b) if A(YX) < 1, go to 2 and decrease 7.
(c) if (YX) = 1, go to 6.
6. The value of 7 is the minimum achievable norm. Solve for the
scaled compensator, L.
47
7. The optimal compensator is K = yZLZ 2.
The resulting compensator has an intuitively appealing structure in that it resembles the LQG compensator in many ways. For
Q = 0, a realization of L(s) is
x = At+Y 1 + B(y-C 2 .)
=
where
is an 'estimate' of the error signal e:
e=
5.5
Ck2
- D12 1 = Cli- D12u.
Summary
This chapter has completed the detailed solution of the Ho optimal
control problem. In the following two chapters, applications issues
associated with practical H,, optimal control system designs will
be addressed.
48
Chapter 6
Control System Design
Using H-Infinity
Optimization
Thus far, this thesis has focused on the theory of obtaining a solution to an Hoooptimal control problem once it has been defined.
However, the most difficult task in design is choosing a problem
with a numerically tenable solution that will result in a system with
desirable closed-loop properties. This chapter addresses configuring the plant, P(s), of the Hoooptimal control problem to allow
shaping of selected frequency responses.
Closed-loop properties, such as stability robustness, disturbance
attenuation, and control limitations, can be designed through the
shaping, i.e. specifying the maximum frequency response, of ap-
propriate closed-looptransfer functions of a given system. In what
follows, the system is first defined by the set of closed-loop transfer
functions that we wish to shape. Weighting functions augmented
to this system act as tools for shaping the closed-loop response.
Singular value inequalities are then presented which establish relationships between weight selection and the closed-loop singular
value response. The chapter concludes with design examples that
illustrate the effect of weight selection on closed-loop singular value
responses.
49
6.1
System Definition
In the original H, work of Zames [35,36] the weighted plant was de-
fined to be a weighted sensitivity function. However,this methodology can encompass much more. The block diagram in Figure 6.1
represents a closed-loop system which can be used to identify some
typical transfer functions of interest.
qO
Figure 6.1: Control System Structure
It is known that robust stability and disturbance attenuation, as
well as explicit shaping of the loop gain, can be achieved by shap-
ing the sensitivity (S) and complementary sensitivity (T) transfer
functions [13,27,28]. In Figure 6.1 these correspond to
S = (I + GpK)T = I-S
= (I + GpK)-GpK.
Control limitations as well as robust stability can be addressed by
shaping the transfer function from output disturbance to control,
KS [25]. These capabilities suggest that a valid design approach is
to achieve performance through the shaping of closed-looptransfer
functions.
50
To define the weighted plant, P(s), used in the H paradigm
(see Figure 3.3), one must select the four signals:
1. v, exogenousinput consisting of disturbances, commands, etc.,
2. e, weighted performance variable which may be arbitrary com-
binations of the state, output, control, error signals, etc.,
3. y, measurements which are defined by the particular system,
4. u, control signals which are also defined by the system.
The open-loop transfer functions between these signals compose
P(s).
When the loop is closed via the feedback compensation,
u = Ky, the closed-loop transfer function, F1 (P, K), is defined.
Clearly, this choice of signals will define the closed-loop transfer
functions and thus the performance to be optimized.
The performance variable, z, of Figure 6.1 is generally defined
as a linear combination of the states, output, and the control:
z = D + Fy + Gu
(6.1)
Note that in the case of Figure 6.1, z is simply the output and zu
is the control. The weighted performance variables compose e as
in the following equation:
e-
1ey
eu
[ W,
WY
0
0]ZY
WI
zu
For the example to be presented, the exogenous signal, v, consists
of the zero-mean, white, gaussian process and measurement noise:
do
while Wq and Wd act as input weighting functions.
The transfer functions between e, y, v, and u define P(s):
[eu
=P(s)
do
tt
51
Let the constituent transfer functions have state-space representations given by
where a denotes the particular transfer function: p for plant, y
for output weighting, u for control weighting, q for measurement
noise weighting, and d for process noise weighting. A state-space
representation of P(s) is formed by augmenting the various transfer
functions, which for the plant of Figure 6.1 results in:
0
0
0
o
o
BpCd
Ad
0
0
0
O
AP
P(s) =
Bd
0
0
ByCp
0
Aq
ByCq
0
Ay
0
0
Bq
ByDq
0
0
O
O
0
0
Au
o
0
0
O
, Cp
O
o
o
Cu
0
Ca
0
0
_
_
_
BpDd -Bp,
o
Da
0
0
0
o
Bu
0
O
D,
0
0
0
where it is assumed that Dp and D, are zero. (This is because
the simpler Hoosolution requires that D
and D22 be zero, as
discussed in Section 6.3. The more complex solution presented in
Chapters 3, 4, and 5 does not impose this restriction on Dll and
D 2 2 .)
The choice of input and output signals in Figure 6.1 results in
a closed-loop system that can be written compactly as
e
I eJ
1
[ WYSWq
WySGpWd
WuKSWq WKSGpWd
[
(6.2)
qo
[ do
'
(
This is the 2 x 2 block-matrix transfer function from the disturbances qo and do to the weighted output, e, and the weighted
control, e. Minimizing the Hoonorm of this transfer function corresponds to a disturbance rejection problem.
6.2
Frequency Response Shaping
The desired shapes of the closed-looptransfer functions are achieved
through the selection of the weighting functions. Singular value re52
lationships involving block matrices are now given which will prove
useful in guiding the selection of the weighting functions.
Theorem 6.1 (Singular Value Inequalities: Two Block Case)
Consider the block matrix,
C
Denote the maximum singular value as a function of frequency
by &(.). The following inequalities then hold [5,28]:
&(Z) < max{e(A)
(C) < (Z).
Theorem 6.2 (Singular Value Inequalities: Four Block Case)
Consider the block matrix,
Z=A C B
D
The following inequalities hold [5]:
(Z) < maxl(A),
(B), (C),
(D) < (Z).
These inequalities provide bounds, accurate to a factor of 1/x/2
or 3 dB for the two block case and a factor of 1/2 or 6 dB in the
four block case, on the frequency response of the constituent blocks
of Z. The bounds apply to the block with the greatest maximum
singular value response; which block this refers to may change with
frequency.
The above singular value inequalities can be applied to an Ho. op-
timal closed-loop transfer function. Consider the (1,1) block of
Eq. 6.2. Assume that the weighting functions of Eq. 6.2 are selected
so that WySWq is dominant over a particular frequency range, f,.
The previous singular value inequalities then provide bounds on
the frequency response of the sensitivity, S.
53
To derive the appropriate relationships, the following inequalities involving the (1,1) block are used [11],
a(s) =
<
(W-' WSWq Wq 1)
(WY 1) (WVSW9) (W;
&(WySWq)
a(W) a(Wq)
)
or equivalently
(6.3)
(WySq)
&(S) < 0(Wy)
(Wq)
and
&(WY)&(S)(Wq)
>
(S)
&(Wyswq)
(WV) (Wq)
a(W)(W)
(6.4)
Eqs. 6.3 and 6.4 provide upper and lower bounds on the frequency
response of S.
An important property of the solution to the Hoooptimal control
problem that makes frequency response shaping reasonable is that
the maximum singular value response of the closed-loop system
is constant (and equal to 'o,) [20]. Since WySWq is the dominant
block over fl,, its frequency response is bounded by the inequality
of Theorem 6.2;
< (WsWq)
< o
Applying Eqs. 6.3 and 6.4 yields the following inequality for the
frequency range fl,:
Ifisa-< w(S)
<-lC~(WY)j(Wq)'
a
_
2&(Wy)&(Wg)
(6.5)
If instead we consider the two-block case that results when Wd is
zero, the resulting inequality is
70
< (S) <
<
V2&(Wy)&(Wq)
70
(6.6)
(Wy)O)(w,)
If Wy and Wq are chosen to have equal minimum and maximum
singular values then these relationships prescribe the shape of the
54
sensitivity to within 6 or 3 dB respectively. The shape of the sen-
sitivity transfer function is determined solely by the shape of the
weighting functions, Wy and Wq because y, is a constant. Similar
relationships can be written for any block that is dominant, or has
the maximum singular value response, over some frequency range.
6.3
Weight Selection
As with all analyses using singular values as a measure of system
response, the performance variables must be scaled so that they are
of comparable importance. This is accomplished with the weighting functions. The relative magnitudes and frequency content of
the controlled variable weights is an indication of the relative importance of each controlled variable. In general these weights are
selected to be large over the frequencies that the performance variables are to be kept small. It is also desirable to choose diagonal,
stable and minimum phase weights.
The previous inequalities provide additional guidelines for selecting the weights to achieve desired transfer function shapes.
This allows a great deal of flexibility in designing the closed-loop
frequency response. However, some restrictions do apply to the
achievable performance and to the definition of P(s).
In certain cases, the frequency response may be shaped almost
arbitrarily, for example, shaping the sensitivity of a stable, minimum phase system with no control constraints. However, as more
realistic constraints are applied to the problem the limitations and
implicit tradeoffs inherent in control system design make frequency
response shaping more difficult. Certainly, any control system design can not violate certain fundamental restrictions [13,19,23,27],
such as the limitations on sensitivity minimization imposed by nonminimum phase zeros and unstable poles, and the inherent tradeoff between the sensitivity and complementary sensitivity transfer
functions implied by the equation
T + S = I.
Weight selection and target loop shapes should be chosen in coor-
dination with these inherent limitations.
55
Additionally there are some basic conditions that P(s) must satisfy. Recall from Chapter 4 that the inner-outer and outer-inner
factorizations of T12 and T21 require certain rank conditions of D 12
and D21 . If D 12 E RnX then we must have n > r and the rank
of D 12 must equal r. This is so that Z1 exists and is invertible.
Dual conditions hold for D21. If D21 E RmX P then we must have
m < p and the rank of D21 must equal m. The simpler solution
of Section 5.4 also requires that these conditions be met. These
requirements can be interpreted as follows: the rank requirement
on D12 states that all controls must be weighted at all frequencies.
Similarly, from the conditions on D21 all measurements must have
some feedthrough term (noise) from v. The simpler solution also
requires that D11 and D22 be zero matrices; this implies that there
be no feedthrough terms from v to e or from u to y. The more general solution did not have this latter restriction, but it is thought
that this requirement is not an unrealistic one.
6.4
Examples
The following are examples of H optimal control systems that
are presented to illustrate
the effect of weight selection on the
closed-loop frequency response. In many instances the results are
intuitive, but as with any complex optimization problem there are
trade-offs that are not immediately apparent.
The design problem that provides the example is investigated in
detail in the following chapter. It is a disturbance rejection problem of the form indicated by Eq. 6.2 with an additional matrix,
A, that generates the performance variable, z, from a linear combination of the output. The weighted closed-loop system of the
example problem is given below:
ey
e]
WyASWq
[ WKSWq
WyASGWd
qO
WKSGWd jdo
(6.7)
(
)
Illustration comes in the form of singular value response plots
of important closed-loop transfer functions identified in Eq. 6.7. In
the examples, the input weights, Wd and Wq, are coloring filters for
the noise sources; therefore, they are not design parameters. The
56
explicit representation of the plant and input weighting functions
is not necessary for the subject of this chapter.
For the examples, the following plots are given:
1. Frequency Response of W,: A singular value response plot of
the output weighting function.
2. Frequency Response of Wu: A singular value response plot of
the control weighting function.
3. Block Frequency Responses: A singular value response plot
for each block of the closed-loop transfer function, Eq. 6.7.
4. Frequency Response of z:
A singular value response plot of
the disturbance-to-performance
variable transfer function,
[ASWq ASGWd].
5. Frequency Response of z: A singular value response plot of
the disturbance-to-performance
variable transfer function,
[KSWq KSG1Wd].
In the examples, the design goal is to make the response from
the disturbance to the controlled variable, zy, as small as possible
at low frequencies while keeping the response of the control, zu,
'reasonable'.
Example
1
In this example the output weighting function indicates an interest in disturbance attenuation for frequencies below 10 rad/s
(Figure 6.2(a)). The control weighting reflects increased penalties
on the use of control at frequencies greater than 1 rad/s (Fig-
ure 6.2(b)).
Recall that the maximum singular value response of the optimal
closed-loop system is constant; in this case 'y,, = 22.5. This means
that the frequency responses of the individual blocks of the closedloop system (Eq. 6.7) must complement each other in a way to
achieve a flat response for all frequency. Figure 6.2(c) shows how
the separate blocks of the weighted system combine to achieve
the flat closed-loop response. The upper bound of Theorem 6.2 is
57
obviously satisfied since the Ho norm of each block never exceeds
70.
Notice that as the response of the dominating (1,2) block begins
to drop off, the response of the (2,2) block and finally that of the
(2,1) block increase to the value of 7o. In each of these cases a
singular value relationship similar to Eq. 6.5 can be derived from
the dominant block to describe the response of z, or z, for the particular frequency range. For frequencies less than 6 rad/s the (1,2)
block dominates the response yielding the following inequality:
2aO <((ASGWd)) <<
or
22.5
22.5
1.125 = 210 < (ASGWd) < 10 = 2.25.
The response of zy from the measurement noise is small ((1,1)
block of Figure 6.2(c)) so the response of zy is dominated by the
(1,2) block. Therefore, the previous inequality accurately describes
the frequency response plot of z for the given frequency range
(Figure 6.2(d)). If the response to measurement noise is assumed
to be negligible then the problem is essentially a two-block case for
which the following inequality holds:
1.591=
22.5
'1
22.5
< (ASGWd) < -
,,F0
10
2.25.
These bounds are even closer to the actual response of z in Fig-
ure 6.2(d).
Even though the inequalities do not provide a bound on the
response of the control, z,, at low frequencies (because the (2,1) or
(2,2) block do not dominate), the response of z, is obviously influ-
enced by W, at low frequencies (note the similar break frequencies
(Figures 6.2(b) and 6.2(e))). This example nicely illustrates some
of the singular value relationships but it is a poor design since the
performance variable response to disturbances is greater than 1.0
at low frequencies (Figure 6.2(d)).
58
3
10
2
10
Ci)
oCL
CL
1
10
1
0
.2
10
\i
-1
10
0
-2
1
-1
101
o
0
Exampl
t'g
6.2(a):
2
10
10
1C
-Frequency Response of W
Figure 6.2(a): Example 1-Frequency Response of Wy
2
10
-I.- .___
. I ._-, . , , - -.1
10
a)
0
10
-1
.
co
10
i72
10
-2
-3
10
10
.o2
. 100
.c. . . . 10
Iio .
- 2
10
-
10 0
101
2
10
10 3
1 )4
Fre(llvm-lrcy rad ';
Figure 6.2(b): Example 1-Frequency Response of W,
59
2
1(
1
U)
1
co
aC
04
CU
;:I
Co
10
::I
10
I:;
. C.
Cd
Cd
10
10 10 -2
-
10
100
1
101
Frequency
1102
103
radcl/s
Figure 6.2(c): Example 1-Block Frequency Responses
10
a)
(
o
Ca
rC,
2o
.1
Er
M,
. 01
)
(t-td
-ocltlencv
Figure 6.2(d): Example 1-Frequency Response of z
60
104
10
i),
~or
2
10
U)
aV)
1,~~~~~-
a)
(13
I)
3
10
1
I,
O
0
1010 -2
10-1
100
101
Fi-ctl(enlc'V
10
103
(rad l '-)
Figure 6.2(e): Example 1-Frequency Response of z,
61
1
Example
2
This example illustrates how a change of weights in one frequency
regime can change the response over all frequencies. The high frequency content of W. is increased while the low frequency content
of W. (Figure 6.3(b)) and Wy (Figure 6.3(a)) remain identical to
those of Example 1. The H, norm increases to 7yo= 35.0.
The most noticeable change is the decrease in control response
at high frequencies (Figure 6.3(e)), a rather intuitive result. However, in order to achieve a constant a, the low frequency response of
the closed-loopsystem increased. This results in an increase of the
response of zy (Figure 6.3(d)) and, therefore, poorer performance.
3
1C
1C
CL
aQ.)
ao
1C
5
.n
tD
Uo
1C
10
lrc'qitlencyOrad,'s
Figure 6.3(a): Example 2-Frequency Response of W,
62
2
10
1l
10
v)
oC2.
0
0
-1
C-
10
-2
10
-3
~2 2
10
-2
I 10
o10
100
101
102
'' (q
I"
t (-n'V
(-'-' - l
03
3
11
lI
Figure 6.3(b): Example 2-Frequency Response of W,
2
O
0
a
C,
.CQ,
t)
Frequency rad/s
Figure 6.3(c): Example 2-Block Frequency Responses
63
a,
VI
0
o
A
a)
:I
M
-C
co
CI
.E
U)
1-
2
.
o-
10
1
01
102
103
1
Figure 6.3(d): Example 2-Frequency Response of zy
3
10
a,
2
V)
CO
10
en
a)
a)
CO
CO
CO
'x.,
10
j~~~~~~~~~~
\\
\\
C
On
0
lu
_
1
__
2
-1
100
101
102
103
104
Figure 6.3(e): Example 2-Frequency Response of zu
64
Example 3
In this example Wv is identical to that of Example 1 (Figure 6.4(a)).
The control weight is increased to reflect a higher weight on the
use of control at frequencies beyond 0.1 rad/s (Figure 6.4(b)). The
Hoonorm is 39.3.
The (2,2) block of the closed-loop system is now the dominant
block at low frequencies (Figure 6.4(c)). The control response
from the measurement noise is small in this frequency range so
the following singular value relationship describes the response of
z, (Figure 6.4(e)) to approximately 400 rad/s:
7o
<o(KSGWd)
2&<(WS)
<
<
The response of the control, z,, is effectively described by the
inverse of the frequency response of Wu scaled by y,. This results
in an increase in control response which improves the performance
of z (Figure 6.4(d)) at low frequency. Note, however, that the
response of z v is no longer dictated by the shape of Wv .
.
.
z
10
.
.
.
...
.
.
...
.
..
, . . ...............
, ,
2
W
cn
10
0a.
1
3
-1
10
-1
0 10 - 2
_.
."
..
"
.
1
.
10
.
10
01
o2
103
104
Fr-eqllncy v irad '
Figure 6.4(a): Example 3-Frequency Response of Wv
65
2
l_
10
.
·
a
__
_
X
__
_|lr
·
·
_·
·
_
sww
1
10
(n
W
0
0.
a
a,
L.
Q)
:j
0
10
-1
10
CO
hi
c
-2
Q,
10
-3
L
10 102
· · · 1
1
10-1
· 1·I
·
10°
·
·· J
o1
· · ·
I
·
lo2
· · · ·
103
·
· · ·
1 4
Ft'rc
( tlen-cv adl '.
Figure 6.4(b): Example 3-Frequency Response of W,
10
10
2
1
Ea
0a
10
P::
Q,
W
10
-1
d
(U
c4
10
-2
C:
.
10
-3
-4
10 2
10-1
100
101
1o2
Frequency rad/s
Figure 6.4(c): Example 3-Block Frequency Responses
66
·
10
.. .
.I I
. ..
I ..
.11 , ,
,
. . ..-
a)
o
cu
C0
1
(:
U)
:.
~~~~~~~~~~~\1
------
C3
.1
D
r)
ci L_LIILL·
10-2
1
1111
1
· 1·(
101-
100
10
2
cd
(ta>-td
Vi''ei-recv-
· ·
I
1,11
101
I
Li
10
3
·
i·
104
)
Figure 6.4(d): Example 3-Frequency Response of zy
3
10
a)
C)
10
C
C
r-
I
a:C
C
9._
En
a_,
Q,
10
CD
U)
M
1 210
1-1
F10
210
1
[s ?<r-'
I
IX
orlC'v
102
('cacl'.)
10
3
Figure 6.4(e): Example 3-Frequency Response of z,
67
104
Example 4
In this example, the low frequency output weighting is increased
by a factor of 10 from Example 1 (Figure 6.5(a)). However, beyond
10 rad/s the output weight is unchanged. The control weighting
(Figure 6.5(b)) is identical to that of Example 1 and the optimum
cost does not change dramatically; y, = 25.1.
The response of zy decreased by a factor of 10 at low frequency
when compared to Example 1 in coordination with the increase
in W.. It can be seen from Figure 6.4(d) that the response of
zY follows the response of the inverse of Wy up to 6 rad/s, where
the (1,2) block loses dominance (Figure 6.5(c)). Interestingly, this
improved low frequency performance is achieved with less control
at low frequency than in Example 1, although the control energy
expended at high frequency increased (Figure 6.5(e)). In this case
the sharp increase in the response of z near 6 rad/s is not necessarily a result of restricted use of the control; the response of zy
is simply following the inverse of Wy 'too' well. An adjustment of
output weights and perhaps control weights are needed to flatten
the response.
a)
0
e0o;:a
10
1
a
a
It
m0
10
1=
U)
I
10 10-2
10-1
100
101
102
103
10 4
S
re l
Figr
' (aI e -Feen-Figure 6.5(a): Example 4-Frequency Response of Wy
68
2
10
10
0.
C
::1.
c
-
10
an
c7
-
10
-
10
0-2
10
-1
100
101
|I'OqCi
lncy
10 4
10 3
102
S.
r-act
Figure 6.5(b): Example 4-Frequency Response of Wu
2
to
CL
as
0)
ED
U2
10
10
-2
10-
1
10 °
10 1
lo 2
Frequency rad/s
10
3
104
Figure 6.5(c): Example 4-Block Frequency Responses
69
rl
o)n
vo5
C3_
cc'
CI
.
U
10 -
2
10-1
10
10
10
10
(radc 's)
Ploqliencv
Figure 6.5(d): Example 4-Frequency Response of zy
-,I ,.,
I
i
co
10
2
f-_.
L
c,.
1.
10
0
10 O0 -2
·.
.
.
1-1
. .
100
.
.
.
101
lo2
10
3
104
IF'-ocileoncv (radl "s)
Figure 6.5(e): Example 4-Frequency Response of zu
70
Example 5
W, and Wu are changed substantially; the gain and bandwidth
of Wy is increased (Figure 6.6(a)) while the control weight is de-
creased (Figure 6.6(b)). The H norm is 17.3.
In each of the previous examples the response of the (1,2) block
began to roll off at the break frequency of W,. In this example the
(1,2) block is dominant to approximately 40 rad/s (Figure 6.6(c)),
short of the 100 rad/s break frequency of W,. Thus, the response
of z (Figure 6.6(d)) follows the inverse of W, up to approximately
40 rad/s. The (1,2) block does not remain dominant up to the
100 rad/s break frequency of W, at least partially because of an
inherent characteristic of the plant. In this example, G of Eq. 6.7 is
essentially a double integrator with crossover near 20 rad/s. This
forces the (1,2) block to roll off near this frequency. In other words,
beyond 20 rad/s, attenuation of the disturbance begins to occur
naturally and thus does not need control action.
3
10
W
cn
10
o
0
10
usCU
c
ic
10
10 2
10
10
10
102
103
F'reqluency- r-d
Figure 6.6(a): Example 5-Frequency Response of W,
71
104
10
10
a)
0o
a)
Is
L..
75
)4
Ti'eq iex cv rcl s
Figure 6.6(b): Example 5-Frequency Response of W,
'2
10
--
1
- -- -
-
-
--.-
-
-
...
....
.
.
10
,
0
10
\
,
.
0
0)
'
\
.
-1
-
10
.
-T
10
10
\
\
-2
-3
(1.1) Block
(1,2) Block
- -
(2,1) Block
''
(2,2) Block
*.
*
.........
/
\
.
-4
102
10
10
10
10
Frequency rad/s
10
10
Figure 6.6(c): Example 5-Block Frequency Responses
72
10 ---T----·-·----5---T-r---r
C)
1
To
V)
r0
P.
c)
13
M
a)
.1
gt
.
\1
Oz
L-
nl
Us
10-2
II
10-1
100
103
102
101
Figure 6.6(d): Example 5-Frequency Response of zy
3
10
co
0
:
I I I I.
I. . I~
I~ . ~ . ,
2
I~
I ·
II
_ K-'I~~~~~~~~
10
0,
clu
An
a,
-T
IZ0
7Ct
0
10
10 10 -2
100
0-1
l
lo
1
lo
(rc:td
',c,~t~(-lac\'
2
10
3
'.-)
Figure 6.6(e): Example 5-Frequency Response of z.
73
4
10
6.5
Summary
The above examples illustrate some basic requirements and trends
for designing the weighting functions and thus the target closedloop responses.
*The block containing the transfer function to be explicitly
shaped, for example ASWd of the block WyASWd, must dominate the other blocks of the system in the desired frequency
range. This generally can be achieved through selection of the
weights. If dominance occurs, singular value inequalities can
be derived which bound the shape of the transfer function of
interest.
This is evident to some extent in each example. In Example 1 the
response of z is shaped by Wy because of the dominance of the
(1,2) block as shown by the singular value inequality derived for
that example. In Example 3, although unintentional, the response
of z explicitly follows the inverse of W, because of the dominance
of the (2,2) block.
*The constituent blocks of the closed-loop system must be designed so that the frequency response of at least one of the
blocks can attain the optimal cost at all frequencies without
degrading performance.
In Example 2 the response of zYhad to increase with a subsequent
degradation of performance when W, was increased so that the
closed-loop response could be flat. An increase in output weighting at low frequency may have achieved the same result without
sacrificing performance.
*Target shapes for transfer functions must be consistent with
the restrictions imposed by the plant itself.
In Example 5 the (1,2) block did dominate up to the roll off of W,
because of the nature of G. In this case the high frequency shape
of the (1,2) block was dictated more by the 1/8s2of G than the roll
off of W,.
In this chapter, the design of control systems using Ho optimization is oriented toward the goal of minimizing the response to
74
disturbance by shaping the closed-loop frequency responses. Sin-
gular value inequalities that can be used to bound the response
of certain transfer functions were presented and several design examples demonstrating the effect of weight selection on closed-loop
responses were shown. In the next chapter, a detailed control system design study which employs H, optimization is presented.
75
Chapter 7
Design Example
To investigate the efficacy of Hoooptimization, in this chapter the
H,o paradigm is used to design a control system for a generic pointing and tracking model. The approach used here for control system design is one applicable to large-scale interconnected systems
(LSIS) [12]. The model for this design example, although modest
in size, is representative of a complex interconnected system due
to the distinct subsystems which must be coordinated to achieve
performance requirements.
A brief description of the design approach for LSISs follows.
Once a model has been defined for the LSIS, a requirements flow-
down is performed. This is the preliminary determination of tolerable disturbance levels and performance requirements such that
the complete system can meet design specifications. Kalman filter
estimation and Wiener-Hopf optimal control theory provide the
necessary disturbance and performance bounds. The connectivity of the system is then analyzed to determine strongly connected
components from which a decentralized controller may be designed.
Total system performance and robustness properties are then evaluated. If requirements are not met, the entire process is repeated
until a satisfactory design is reached.
76
Figure 7.1: Pointing and Tracking Model
7.1
A Pointing and Tracking Model
The task of the pointing and tracking system is to accurately point
a laser beam at a distant target. System requirements are given in
terms of the root-mean-square (RMS) value of the pointing error,
control bandwidth limitations and maximum control deflections.
A sketch of the system is given in Figure 7.1. The major subsystems of the model are an optical tracker, a gyro-stabilized
reference platform (SP), a fast steering mirror (FSM) for disturbance attenuation, an alignment sensor (AS) to measure the laser
illumination angle, and a gimballed beam expander (from now on
referred to as the gimbal) upon which the entire mechanism pivots.
Torque disturbances enter the system through the gimbal and all
measurements have some noise component.
77
A block diagram of this system is given in Figure 7.2. The op-
tical tracker is a sampled data system operating at 5 Hz. It is
rigidly attached to the gimbal and measures the angle between the
target and gimbal, OTgt/Gim,as well as the reference angle between
the stabilized platform and gimbal, OSP/Gim. The alignment sensor
is a high bandwidth device that measures the angle between the
laser line-of-sight (LOS) and the stabilized platform, OLos/sP.
Its signals can be used to drive the fast steering mirror to compensate for high frequency disturbances. The magnification of the
beam expander is represented by the gain term on the input to the
alignment sensor (Figure 7.2). Also available are the gyro mea-
surement of stabilized platform rate, OsP,and the angle between
the fast steering mirror and the gimbal, OFSM/Gim. The primary
performance variable of this system, which should be kept small,
is the pointing error, the small angle between the LOS and the
distant target. A measurement of the pointing error is obtained
by combiningthe measurementsOLOS/ISP,OTgt/Gim, and OSP/Gim.
The motion of the gimbal, fast steering mirror, and stable platform is governed by the rotational dynamics of each subsystem.
Referring to the block diagram, their transfer functions are represented by simple double integrators; the inertia terms have been
absorbed into the control gain terms. The inputs are non-dimensional control torques of the gimbal, stabilized platform and fast
steering mirror. They have been normalized by the maximum possible torque capability of the respective inputs. The process noise,
d, entering the gimbal is also assumed to take the form of a non-
dimensional torque disturbance.
The inertia of the entire mechanism is dominated by the gimbal, which is neutrally stable. Spring and damping constants in
the state-space model reflect the interconnection of the gimbal,
stabilized platform and fast steering mirror. These terms result
in stable eigenvalues for the stabilized platform and fast steering
mirror relative to the gimbal. The stabilized platform and fast
steering mirror are both driven by push pull actuators which react
against the gimbal. An action-reaction
effect is present between
the stabilized platform and the gimbal because of the appreciable
relative inertias (20 to 1). This effect is not present in the inter-
78
nFSM/ Gim
OFSM/ Gim
l
nAS
eLOSSP
I
Gim
+
eTgt
+
nTracker-1
TTacker -2
Nim
TsP
+
nGy o
* S & H - Sample and Hold
Figure 7.2: Block Diagram: Pointing and Tracking Model
79
connection of the fast steering mirror and gimbal because of the
low relative inertia of the fast steering mirror.
The continuous time, state-space model of the system is given
below:
x = Ax+Bu+Ld
y = Cx+v
with
0
-30
A=
1
-5
0
0
0
0
0
0
0
0
30
5
0
0
1
0
30
0
0
5
1
0
0
0
0
0
0
0
0
0
0
O0
O
0
0 -30
0
0
0
0
O
-5
0
1.5
0.25
-1.5
0
0
0
-10
0 -10
0
10
0
-10
0
B-
-0.25
O
10 7
0
0
0
0
0
0
0
0
4 105
0
0
0
0
-2 10 4 4.103
0
0
0
0
0
0
-0
0
L=
0
0
O
0
0 '
O
O
O
O
O
O
50
0
0
0
0
0
00
80
10
0
0 -10
10
C=
0
0
-9
0
0
0 0 0 000
0
0
0
1
-1
0
0 1
0
0
0
0
0
0
0
1
1
0
0 0 0 0
-1
0
0
0
The performance variables compose the vector z. They are defined
as a linear combination of the state, output, and the control via
the following equation,
z = D + Fy +Gu
(7.1)
These are the primary variables for control design and performance
analysis. The precise definition is specific to the design, so explicit
representation is given in subsequent sections.
Six states of the state-space model correspond to physical variables. They are the angles and angular rates of the fast steering
mirror, stabilized reference platform, and the gimbal. The last two
states are included to model the nature of the sampled-data optical
tracker systems. Each optical tracker measurement is modeled using a first order Pad6 approximation of a sample and hold with a 5
Hz sampling rate. The per-sample measurement noise of the optical tracker, bandlimited by the sampling rate, is modeled by passing zero-mean, white gaussian noise through the low pass filters
resulting from the approximation of the sample and holds. This
requires that the optical tracker noise be included as a process noise
with appropriate gain terms contained in L. Thus, the columns
of L correspond to the gimbal torque disturbance and noise of the
OTgt/Gim and OSP/Gim measurements. This approach eliminates
two additional states otherwise needed to bandlimit the noise. Relative magnitudes of the input distribution terms reflect bandwidth
limits of the controls. The bandwidth limits of the FSM, SP, and
gimbal are assumed to be respectively 500, 100, and 10 Hz. The
five measurements are of OLOS/SP, OSP/Gim OTgt/Gim, Sp, and
OFSM/Gim.
Both continuous and discrete time forms of the model are used
in design and analysis. For discrete time analysis a 100 Hz sam-
pling rate is used. The nominal performance objective is to make
the RMS value of the pointing error less than 1 rad while not
81
exceeding the previous bandwidth limitations. In addition, the
maximum values of OFSM/Gim and OSP/Gim are constrained to be
less than 20 milliradians and 10 milliradians respectively.
7.2
Performance Bounding
7.2.1
Kalman Filter Bounding
A Kalman filter bounding analysis is performed to derive preliminary requirements on measurement and process noise for which
the system can meet the RMS pointing error specification. The
target is assumed to be distant, thus the performance specification
pertains to a stationary target. The Kalman filter estimation error
covariance of the pointing error is used to provide a lower bound
on performance since the pointing error can not be controlled any
better than it can be estimated.
Measurement and process noises are assumed to be zero-mean,
white and gaussian. Nominal levels of measurement noise covariance are estimated to start the bounding process. The final values
of tolerable process and measurement noise are selected to represent the noise environment for the rest of the design process.
The discrete time noise statistics determined for the model resulted in an RMS error of the performance variable of 0.33 prad.
The RMS error is very sensitive to the covariance levels of the gim-
bal disturbance, gyro measurement, and the optical tracker. The
discrete time noise covariance levels determined from this analysis
are
Gimbal Disturbance
1 10-10 non-dimensional
Optical Tracker
2. 10-"1
1.10 -12
Gyro
Alignment Sensor
FSM/Gim
7.2.2
2.10 - 1 3
1 - 10-
rad2
(rad/s) 2
rad2
rad2
Wiener-Hopf Bounding
The Kalman filter bounding procedure provides an idealistic bound
on performance since compensator dynamics are not addressed. A
82
more realistic bound is obtained by using the previously determined noise statistics in a well established optimal control design
methodology to assess a performance bound that includes compen[31] controller yields an opti-
sator dynamics. The Wiener-Hopf
mal, stable, closed-loop system with optimality measured by the
RMS value of the performance variable. If the performance objectives are solely in terms of RMS measures, then the Wiener-Hopf
controller is the 'best' controller. The Wiener--Hopf controller
is approached by an LQG controller [22] as the control weighting
goes to zero.
For the LQG controller the state weighting term is defined by
DTD where D is from the performance variable equation (Eq. 7.1).
In this case it represents the uncorrupted pointing error:
0
D=[10
0
0
-9
0
0
0 ].
The process noise covariance matrix, Y, and the measurement noise
covariance, 0, are given by
1 1010
E =
0
0
0
0
2.10-11
0
0
2 10 - 11
0
0
I .10
0
0
0
0
0
1 10 - 12
0
and
O=
2. 10 o
0
O
0
13
0
1 · 10 0
0
0
17
- 17
0
0
0
0
1 10
- 15
E and O are the values determined from the Kalman filter bounding. The small measurement noise for the optical tracker measurements in O are present because all measurements must have
a non-zero term in the covariance matrix. (Recall that the optical tracker noise is accounted for with process noise terms.) Note
that a distinction has been made between the measurement of the
pointing error and the actual pointing error. The actual pointing error is only a function of the physical states and the target
position, whereas the measurement of the pointing error involves
83
the sample and hold states of the optical tracker. For the static
case, the target position can be treated as a constant input, which
is chosen to be zero, and thus does not affect the measurement
equation.
The steady-state RMS value of the pointing error resulting from
this design is
cr = 0.772 prad.
The steady-state RMS values for the non-dimensional control inputs are
Gimbal
Stabilized Platform
Fast Steering Mirror
3.47. 10 - 8
9.59. 10-10
1.96 10-
These numbers are very small because the scale factors, effectively
the input distribution terms of the B matrix, are very large. The
loop gains of each control are found by breaking the loop at the
input of each integrator while the other loops are closed. The
resulting frequency response plots are shown in Figure 7.3. The
distinct crossover frequencies reflect the different capabilities of
each subsystem. In particular, the FSM clearly has the highest
bandwidth with a crossover frequency of 200 rad/s reflecting its
capability to reject high frequency disturbances. The SP crossover
is 7 rad/s and the Gimbal has a crossover at 1.5 rad/s.
7.3
Decentralized Control Design
The goal in using a decentralized approach on this system is to
identify simple subsystems for which control loops can be designed
simply and independently. This approach obviates many difficulties associated with control system design for large-scale intercon-
nected systems that may not be amenable to multivariable control
laws. For example, order of magnitude differences in bandwidths
can be handled more easily using a decentralized approach, as can
very large systems for which the number of states makes full state
feedback prohibitive. However, this approach may lack stability
guarantees and may neglect important coupling of the subsystems.
84
10
0.)
:co
a
t-
-
6o
-4
10
2
l
10- 1
100
0
10
l1
1022
Frequency rad/s
Figure 7.3: Loop Gains: Wiener-Hopf
°
103
104
Controller
For this problem, a control system is designed by applying classical control methods to subsystems that can serve as SISO plants.
The control loops are designed and closed separately. Each loop
must exhibit favorable gain and phase characteristics and the error signals must complement one another to eliminate the pointing
error.
The identifiable subsystems that need to be controlled are the
fast steering mirror, the stabilized platform, and the gimbal. Strong
coupling between the SP and the gimbal necessitates the inclusion
of both of their dynamics in the design of the respective SISO
control loops. Gimbal motion affects the FSM but not vice versa,
so the compensator for the FSM is designed with only the FSM
dynamics in consideration. Final analysis utilizes the complete
closed-loop system.
The three feedback loops are chosen to be:
1. OsP and OTgt/sP are fed back to the SP input to stabilize
the reference platform rate and to keep the platform oriented
toward the target.
2. OTgt/Gim is used to drive the gimbal input to reject distur85
bances and keep the gimbal oriented toward the target.
3. OLOSISP and OSP/Tgt are combined to form a total pointing
error which drives the fast steering mirror to eliminate the
pointing error.
The design goals for each loop are to achieve high gain at DC
and desirable phase and gain margins. Several configurations with
varying control bandwidths were tested and the following decentralized design was selected based upon the steady-state RMS performance.
7.3.1
Stabilized Platform
Stabilizing the reference platform rate is important because of the
need for a stable reference for the system measurements. The SP
bandwidth is selected to be high so as to provide a stable reference
for all frequencies that disturbance rejection is desired. Using the
gyro measurement for this loop makes intuitive sense. However,
stabilizing the rate does not guarantee position stability. Therefore, the stabilized platform loop uses Osp and OSP/Tgt to stabilize
rate and position (see Figure 7.4). This control loop is dominated
by the rate feedback. A plot of the gain and phase of the loop broken at the SP torque input is given in Figure 7.5. The crossover
frequency is 200 rad/s.
7.3.2
Gimbal
The primary purpose of the gimbal loop is to orient the beam expander in the direction of the target. This is a low bandwidth
requirement for the distant target case. OTgt/Gimis used as feedback to the gimbal input. A block diagram of the control loop is
shown in Figure 7.6 and a plot of gain and phase of the loop broken
at the gimbal torque input is given in Figure 7.7. Note that the
crossover frequency is 6 rad/s.
7.3.3
Fast Steering Mirror
The fine pointing of the system is achieved with the fast steering
mirror. In this case, a total pointing error is the appropriate signal
86
esP
A_
eTgt/SP
J
-I
I
394.8(a+10)
a+200
Rate Compensation
Figure 7.4: Stabilized Platform Loop
87
Position Compensation
5
10
4
10
3
10
2
10
10
i1
)
1
I
0
10
3 . .. . .. .
-1
10
-2
r-'
M
-
Is
..
10
1\
-3
10
1 00
101
2
10
Frequency rac,/'
3
10
104
-135
-140
-145
-150
-155
C,
-160
-165
-170
-175
-180
101
10
102
Frequency rad/s
103
Figure 7.5: Stabilized Platform Loop Gain and Phase
88
at/Gim
TsP
Figure 7.6: Gimbal Loop
89
A
10
3
10
C
C:
'(8
10
-e
10
-6
103
2
102
lol
101
loO0
100
101
F]equency racl /'
En or
-1
.
. ......
.
.
.
.
.............
.. .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.
-140.
IX
-160.
/
'x
,
-180
uw
-200
'\\
-220
-240
-260
-)QI
10 - 1
i
2~~~~ II . .. . II .
00
101
Frequency recl/s
102
Figure 7.7: Gimbal Loop Gain and Phase
90
.
II
10
3
-t
OLos/Tgt
OGim
OSP/Gim
Figure 7.8: Fast Steering Mirror Loop
for feedback so that accurate pointing and disturbance attenuation
can be achieved. The alignment sensor and tracker measurements
are combined to form a measurement of the pointing error. The
crossover frequency of this loop is chosen to be 80 rad/s. A block
diagram of the control loop is shown in Figure 7.8 and a plot of
gain and phase of the loop broken at the FSM torque input is given
in Figure 7.9.
7.3.4
Performance
The complete compensator has eight states. The steady-state RMS
value of the pointing error is
a = 2.829/trad
while the non-dimensional control signals have steady-state RMS
values given by
91
(Tgt/SP
3
10
2
10
10
x
0
10
-1
10
10
-2
L
10 -1
·
·
·
·
100
101
Frequenc}
,·
·
·
·
,
102
I-ac. / s
0.00
-20
-40
-60.
-80.
rJn
f
-100.
-120
-140.
-160.
-1
1
O.
10
100
101
Frequcrncy
102
- c3d,/ s
Figure 7.9: Fast Steering Mirror Loop Gain and Phase
92
9.051 · 10 - 7
Gimbal
Stabilized Platform 4.848 .10-8
Fast Steering Mirror 1.801 10-10
The present SISO design resulted from a trade study of control bandwidths. For this feedback structure, this design is close
to optimal. However, this system does not meet the performance
specifications. If the SISO decentralized design were to be pursued further, the measurement or process noise would have to be
decreased to meet requirements.
7.4
HooDesign
The previous classical and Wiener-Hopf designs are now used
to provide guidelines for the Ho design. Once again, the main
performance requirement is a limit on the RMS response of the
pointing error. If this is strictly adhered to then no compensator
can do better than the Wiener-Hopf controller. The Ho. optimal
control design methodology is investigated because of its utility in
shaping frequency responses and subsequent capability for designing good closed-loop properties (see Chapter 6). The objectives for
the Ho design are to minimize the singular value response from the
disturbances to the performance variables and to impose control
bandwidth limitations.
The Ho designs use continuous time mathematics because of
the availability of state-space algorithms. Therefore, noise covariances of the previous sections are scaled by the time step of the
discrete time analysis, 0.01 s, to obtain continuous time noise intensities. The RMS performance of identical discrete time and
continuous time systems, although comparable, will not be equal;
in general the continuous time system analysis will be slightly better.
7.4.1
Plant Definition
The pointing error,
is still the most important performance variable, but in some of the following examples additional
LOS/Tgt,
93
performance variables are defined. To avoid excessive off-axis point-
ing and subsequent saturation of the fast-steering mirror actuators,
OTgt/Gi,, must be kept small. To avoid saturation of the stabilized
platform actuators, OTgt/SP must also be kept small. Finally, to
emphasize the importance of a stable reference platform, 5p may
also be included as a performance variable. Note that these are the
same signals chosen for feedback in the classical design. The performance variables defined for the following designs do not always
include each of these variables. However, in each of the designs,
the performance variable definition is made clear. To design differ-
ent bandwidths for each of the actuators, all three control signals
are included as performance variables. (Recall that the H,o theory
requires at least constant weighting on each control.)
Analogous to Eq. 6.1, the performance variables are defined as
a linear combination of the output and the control, and can be
written as
[ zy ]
[A
I
u[]
For the case where all the previously mentioned performance variables are included, the equation for z is written explicitly as
1
eLOS/Tgt
Tgt/Gim
OSP/Tgt
°°
-1
10
LOS/GimS
1 0
0Tgt/Gim
OSPIGim
S/Gim
-1 0
0 0
(7.2)
A block diagram for this pointing and tracking model used to
illustrate the transfer functions of interest is shown in Figure 7.10.
G1 is the transfer from the control, u, to the output, y, whereas
G2 is the transfer from the process noise, d, to the output. q is
measurement noise. The closed-loop transfer function from the
disturbances to the performance variables is written as
Z] [KS KSG2
where S is the sensitivity transfer function:
S = (I-G1 K) - 1.
94
d
(7.3)
Figure 7.10: Hoo Transfer Functions
In Figure 7.10, q and d are measurement and process noises with
spectra known from the previous bounding analysis. We assume
that the disturbances are of unit variance, white and gaussian,
therefore the coloring filters Wq and Wd are used to produce the
appropriate spectra.
The remaining design variables are the performance variable
weighting functions, Wy and W,. A block diagram of the weighted
system is given in Figure 7.11. The closed-loop transfer function
to be optimized by the HO algorithm is given by Eq. 7.4.
ey
1
e, =
7.4.2
[ WyASWq
WASG
2
Wd
qo
1
(7.4)
Wu,KSWq W,KSG 2 Wd j[ do j
Weight Selection
The general principles discussed in Chapter 6.3 are applied here
to select the weighting functions. As mentioned previously, the
input weights act as coloring filters to produce the disturbance
spectra determined from the bounding analysis. Wq and Wd are
then simply constant diagonal matrices with the square root of the
95
Figure 7.11: Weighted Transfer Function
appropriate noise intensity on the diagonal:
1l0-12
Wd =
0
2
0
10-
0
2 10 -15
o
Wq =
0
1. 1
0O
- 19
0
o
13
1/2
0
2 . 10- 1 3
0
0
°
0
0
0
0
0
1l-,*i
O
O
O
0
0
0
,
1
10
0
-14
1/2
0
1 10 - 1 7
Recall that these are the discrete time noise covariances scaled for
a continuous time system.
The previous analyses, and intuition, suggest that the bandwidths of the gimbal, stabilized platform, and fast steering mirror
will be different. This will require control weights that differ in
each channel. An example of the control loop gains for a system
with identical control weights in each channel is shown Figure 7.12.
The poor roll-off characteristics of these loops clearly needs to be
improved. This plot is taken from the design of Example 5 of the
previous chapter which had a good frequency response for z,.
96
4
0
on
0
a.
:
at
,..
-
r
10
2
10-1
100
101
102
Frequency rad/s
103
104
Figure 7.12: Example 5-Control Loop Gains
Additionally, in some of the examples more than one perfor-
mance variable is included in z,. This also requires different weights
in each channel of Wy. The singular value relationships of the previous chapter lose some of their utility with such weighting. This
is because the a and a of the weighting functions are now different
and the bounds of Theorem 6.2 are subsequently less tight.
For the case where zy includes more than one variable, a singular value response plot of zy includes some linear combination of
all the elements of zv. This makes the response more difficult to
interpret. It is also more difficult to shape the response of a particular variable when it is an element of a vector of performance
variables. Despite these difficulties, the trends quantified by the
singular value relationships are still present and with an appropriate choice of weights, the designer has a great deal of control over
the system response.
97
Design Examples
7.5
In the following, several examples of Ho optimal control designs
are given.
With each example the following plots are shown:
1. Frequency Response of W,: A singular value response plot of
the output weighting functions.
2. Frequency Response of Wu: A singular value response plot of
the control weighting functions.
3. Block Frequency Responses: A singular value response plot
for each block of the closed-loop transfer function.
4. Frequency Response of OLOS/Tgt: A singular value response
plot of the pointing error.
5. Control Loop Gains: A frequency response plot of the control
loops.
Note that there is a distinction between the response of z, and
OLOS/Tgt if Zy has more than one variable.
Design
1
The performance variable, z, is chosen to be the pointing error.
Wy of Figure 7.13(a) indicates that good pointing is desired to a
bandwidth of 100 rad/s. W, of Figure 7.13(b) has different weights
for the gimbal, SP, and FSM to reflect the desire for distinct bandwidths for each subsystem. The gimbal weight thus has the lowest
bandwidth, the FSM weight has the next highest, and the SP
weight has the highest bandwidth. The optimal cost is 19.0.
The (1,2) block of the closed-loop transfer function is dominant
for low frequencies (Figure 7.13(c)) and thus the pointing error is
shaped according to W,-' in this frequency range (Figure 7.13(d)).
Note that as long as z is comprised of only the pointing error,
singular value inequalities similar to those of Chapter 6 can still
be derived to bound the response of the pointing error. This design
meets the RMS pointing error requirement but the gimbal gain is
considered undesirable because it does not roll off quickly after
98
crossover (Figure 7.13(e)). Also, the FSM gain is unacceptably
small at low frequencies. However, the loop gains are more distinct
and rolling off slightly better than those of Figure 7.12. As in the
examples of the previous chapter, the response due to measurement
noise is generally small when compared to that of the process noise
(Figure 7.13(c)).
3
10
(D
10
o
a
1
10
r-
co_=
oz
1
10
10
2
10-1
100
101
102
F' eclllency rad s
10
104
Figure 7.13(a): Design -Frequency Response of %W
99
2
10
Key
Olmbl Weight
Fur Steering Mirror Weiht
1
10
Stabilised Pltform
-
Weight
-
-
CL
o
0
10
/
:t'
-1
I
y~~~~~
10
C-
CO
._n
CO'
10
//
/
/
2
/
-3
-2
10
10 -
10 3
10
o2
10
1 0
Frequency racl/s
10 4
Figure 7.13(b): Design 1-Frequency Response of W,.
2
10
"I
. --" '
-C·-R-C-r---T--·----
i
1-
1
10
I
-
r.
.
4)
C/)
CO
0
0
10
/
-1
10
.
\/
4)
: CO
,
\/
io~~~~~~
. . I . . ......
\
\
./
\\
CO
co
W
d::
t~D
C
10
-2
V,
Key
-3
(1,1)
(1,2)
(2,1)
(2,2)
10
-4
10
.
1,0-2
\
Block
Blocd
Block
BlocI
.
·. It
·
o-1
· · ·
I
100
r
· · rl
101
Frequency
Figure 7.13(c):
I
I··1
102
·
L1
I
l
.
...
3
10
rad/cls
Design 1-Block Frequency Responses
100
110
10
a)
Drn
g;
0o
1
Ww
co
D
-)
.1
0
. _
ffi
0o-2
o10-1
100
ol
10 3
lo2
I'-cclrlcunec (racl
104
.s)
Figure 7.13(d): Design 1-Frequency Response of eLOS/Tgt
4
10
3
10
2
D
10
Us
c:
o0
1
0
a)
:
10
(
0
-2
-3
10
10
-4
n
10-Z
10
1O-I 101 100
1
i
101
n
1n
2
2
3 In
Frequency iradl/s
Figure 7.13(e): Design 1-Control Loop Gains
101
S
4
Design 2
In this example, the high frequency weighting on the gimbal is
increased (Figure 7.14(b)) to encourage the gimbal control gain
to roll off at high frequencies. zy is again the pointing error with
the same weighting function as in Design 1 (Figure 7.14(a)). The
optimal cost increased slightly to 20.5.
Dominance of the (1,2) block (Figure 7.14(c)) still insures the
shaping of z for low frequencies (Figure 7.14(d)) .
The new
weights not only force the gimbal control gain to roll off faster but
it decreases the gain over all frequencies (Figure 7.13(e)). In ad-
dition the SP gain decreased slightly and the FSM gain increased,
an unexpected, but desirable result.
3
10
2
10
o
V)
10
r
,.
U29
10
0
-1
10 10-2
101
100
101
Pi'ociierli"
102
ricnd ,-
Figure 7.14(a): Design 2-Frequency Response of Wy
102
2
10
10
C/
o
0a;
0)
10
c
LS
-1
10
(.5
-2
10
-3
10
4
9
1
Frequency racl/s;
Figure 7.14(b): Design 2-Frequency Response of W±
2
sj
w
10
j
j
f
j
j
f
v
r
s
v
l
v
|
j
g
S
s
7
s
.
1
s
1
l
.
.
%
1
.
f
S
.
l
.
- - - - - - - - - - - - - \
10
/'\
0)
'\\ \ \
/
/ \ \
10
CV)
0)
::n
t
.
-1
\
\
10
\
t.,
-2
10
\
Key
-3
10
(1,1) Bloclk
(1,2) Block
(2,1) nolck
(2,2) lock
-4
10
......
'
1(0
-
2
. ., .. ...
- - - ...
.
10o-
\
\
.
100
l
'"'
4
101
Frequency
'
'
s
102
s
a
10
-
3
ra(l/s
Figure 7.14(c): Design 2-Block Frequency Responses
103
.
10
10
a)
1
eo
V)
0
r-
:>
1
cc.
=3
I
z
cc
'''
n1
ui
10-2
10-1
10 0
10
102
10 3
Figure 7.14(d): Design 2-Frequency Response of
10 4
OLOS/Tgt
,
0,
0.
a)
CD
-5
tt
r._3
10
i 2
10_1
100
10
102
Frequency ]-ad/s
Figure 7.14(e): Design 2-Control Loop Gains
104
Design 3
As previously discussed it was thought that a very high bandwidth on the SP is desirable. For this reason, efforts were made to
increase the stabilized platform bandwidth through the selection
of the control weights. This approach proved to be unsuccessful.
However,the inclusion of OsP as a controlled variable has a marked
affect on the gain of the SP. In this example, the pointing error
and stabilized platform rate are both included in z.
The band-
width of the sP weight is purposefully chosen to be far beyond
gimbal capabilities (Figure 7.15(a)) so that the SP will be forced
to respond to disturbances. The optimal H~ norm for this system
is 21.4.
The most noticeable difference is the increase in gain of the
SP, especially near crossover (Figure 7.15(e)).
Notice also that
the gimbal bandwidth increased even though the gimbal gain now
decreases more quickly at high frequency. With further efforts the
bandwidth of the SP was pushed to 180 rad/s. However,these high
SP bandwidth designs, although closed-loop stable, as guaranteed
by the theory, resulted in unstable compensators, an undesirable
occurence if there is any plant uncertainty. Therefore, the high SP
control bandwidth requirement was dropped.
105
3
10
2
10
oQ)
VC)
Cd
():n
10
0
10
-1
10 10o2
103
102
101
10
lo-1
104
TI'leqtuencv I'ac ,"
Figure 7.15(a): Design 3-Frequency Response of Wy
2
10
I
1
___
1
_1_·
·
_
j___
T_
__
· _1·
Key
Weight
Glmbl
1
10
Ft
Steering Mirror Weight
-
Stabilired Platform Weight
-
-
-
a)
0
0)
10
I
Q)
-1
..
C,
10
7
I
C:
10
10
i
-2
-3
.
1 0 -2
'·
~
' *11I
10 -l
s11
I I I11
100
r
·
l
lo 2
Frecquencyrad/s
10 3
Figure 7.15(b): Design 3-Frequency Response of W
106
L
-
10 4
· r_·
_ _1_·
._···
10
·
1
10
1
rT·
.- .'\
!
·
1
· 11·
*'..
10
O
0
a,
at,
-1
10
h)
S.
I.
-2
(d
Cbl
10
*Ke'
-3
\\
(1,1) Bltk
10
-
- (2,1) Blo%.
-4
10
·..
-2
10C
-
..........
Blok
(2,2)
Bloek
.(2,2)
,
...
10-1
!··
.................
_
2
_
103
1o
101
100
Frequency rad/s
Figure 7.15(c): Design 3-Block Frequency Responses
10
1
0v,
6>
1
t
10 -2
1'- v
eo
o
r
lo
2
Plto<lencv (racl ')
)
Figure 7.15(d): Design 3-Frequency Response of
107
104
eLOS/Tgt
10
3
10
2
cu
10
o
0
1
V)
Q)
10
::
10
0
L.
10
%c
10
-2
10
-3
-4
10
1 0
10-
-1
loU
101
Frequency
1o2
103
rad/s
Figure 7.15(e): Design 3-Control Loop Gains
108
104
Final Design
For the final design, the controlled variables of Eq. 7.2 are used for
zY. The selection of the weights for each variable is described as
follows (Figure 7.16(a)):
* OLOS/Tgt:This is the most important performance variable.
The magnitude and bandwidth of the corresponding weighting
function reflect its importance and the bandwidth over which
disturbance rejection is desired. The weights on the other
variables are chosen so as to not overwhelm the importance
of this variable.
*
Tgt/Gi,,m: This variable is included to insure that the gimbal remains oriented toward the target to prevent controller
saturation. This is a low bandwidth requirement since the
target is essentially static. Excessive weighting of this vari-
able forces the gimbal response to dominate the performance,
so the weighting function is kept small.
* Osp: The stabilized platform must provide a good reference
for all frequencies at which disturbance rejection is desired.
Therefore, the SP weighting function is selected to have the
same break frequency as the pointing error weight. The weight
is kept small enough to insure that the compensator remains
stable.
* OSP/Tgt: This variable is included to keep the SP oriented
toward the target.
The
control
weights
are identical
to
those
of Design
2
(Figure 7.16(b)) and the optimal Hoonorm is 21.4.
The (1,2) block is dominant at low frequencies (Figure 7.16(c)).
Therefore, the response of the pointing error is reasonably shaped
by its weighting function (Figure 7.16(d)) since the other weights
were selected so as to not dominate its response. Note however,
that a precise singular value inequality to bound this response
would be difficult to derive. The control loop gains are acceptable
(Figure 7.16(e)). All the loops have high gain at low frequency and
roll off well at high frequency. The control bandwidths indicate
109
that all three subsystems are contributing to disturbance rejection
without an overwhelmingly large response from any one subsystem.
The continuous time, steady-state RMS values of several important
variables are given below:
the pointing error,
a = 0.741urad
the non-dimensional control inputs
- °0
Gimbal
2.21 .10
Stabilized Platform
1.45 10-
Fast Steering Mirror 1.93.
9
10- s
relative angles,
OFSM/Gim 1.77. 10OSP/Gim
1.75 .10
- 5
This design exceeds the performance requirements for this problem. The RMS values of the control are noticeably different than
those of the Wiener-Hopf and classical designs. The H, design
requires significantly less gimbal motion and places more of the
burden of control on the FSM than in either of the previous two
designs. The RMS values of the relative angles, OFSM/Gim and
OSP/Gim, are included to illustrate that these steady-state values
do not exceed the limits of 20 and 10 milliradians imposed as a
constraint.
110
10
w
10
0
rn
c;
w
,
Cd
Ico
C10
Ed
tm
.g
10
LU
10 -2
10-1
100
10
102
Frequency rad/s
10 3
lo
Figure 7.16(a): Final Design-Frequency Response of W,
2
10
10
oen
Qa
10
11
L:.
10
Co
On
10
4
Frequency rad/s
Figure 7.16(b): Final Design-Frequency Response of Wu
111
2
10
-7.
~..?'-....
1
10
,)
..
0
·
. /,
*
\
/
-
·
0
10
/
a)
e,
\
10
-1
3co
-
L>
f.
10
-2
\.I.,
\
,
/'
\
\
/
.
\
I=
V)
Key
\
-3
10
(1,1) Bloc
(1,2) lock
\
(1,1) Block
(2,2) Block .
-4
10 10-2
10-2
-.-.,,
1
l101Fr
o
l
10e
10
Frequency
102
ra cl /s
Figure 7.16(c): Final Design-Block
,U
.
AU
-·
· · ·
· -·--
Frequency Responses
t,
C)
1
0
0o
V)
C)_
a_
1
M0
._
n1
_
0 -2
10-
___
101
10
o
· · )
10o2
10 o
}V' ,(VIIl
o cv
(Irncd
41·1
3
10
103
·1
-
-
)
Figure 7.16(d): Final Design-Frequency Response of ®LOS/Tgt
112
'4
1
1
6D
I
C
to
10
2-
I
v)
10
1
in
110 -2
10
1010-11
10 0
101
10o2
Frequency rad/s
10 3
Figure 7.16(e): Final Design-Control Loop Gains
113
Comparison to an H2 Solution
It is interesting to compare this Ho. solution to the H2solution for
the same model and weighting functions. Recall from Section 5.1
that the two problems can be solved using the same framework.
The following are the results from the H2 solution: The H**norm
is
&(jw) = 36.7
While the H, norm of the optimal Ho solution is 21.4 and the
closed-loop frequency response is flat, the H., norm of the H2 solution is greater, as expected, but rolls off at high frequency as can be
seen from Figure 7.17(a). This closed-loop response is illustrated
more clearly in the block response plot of Figure 7.17(b). The
shape of the responses of the constituent blocks is very nearly the
same as the responses for the Ho. solution (Figure 7.16(c)) except
for the roll-off at high frequency. The RMS value of the pointing
error is
a = 1.33prad
A plot of the response of the pointing error is shown in Fig-
ure 7.17(c). The fact that the H,o solution has a better RMS performance than the H2 solution may be surprising at first since the
H2 solution optimizes RMS performance. However, the optimality
in this case refers to the RMS performance of the entire weighted,
closed-loop transfer function; the RMS value of e is less for the
H2 solution than for the Ho. solution:
ae' =
a,=
187.3
2.0701
104
A plot of the control gains for the H2 solution (Figure 7.17(d) shows
that they are nearly identical to those for the H,. solution.
114
Ij. 4Lb
75
·
n_
55.070
cr
o
C
)
36.713
o-A:
18.357
0 000 Lo-3
1o- 2
10-1
G:
Figure
7
100
(rps)
ol
lo 2
.17(a): H 2 Closed-Loop Frequency Response
2
10
10
0
V)
(o
C
-:j
aL.
-5
AD
C:o
10
-1
10
-2
10
._
On
-3
10
-4
10 lO
MU
10 3
IV
I_
10aiu-
Frequency
10
rac/s
Figure 7.17(b): H 2 Block Frequency Response
115
I
10
1 7r
·
I ·
_ _7
oa)
rn
C
C:
C)
1
-
\
1
i
I
_
N1
. U
10 -2
j··
l
1
0
10
10
10
Fr equen c
Figure 7.17(c):
102
-adc 'S
10 3
H 2 Frequency Responses of OLOS/Tgt
10
a)
0
C,
a)
CC,
(,
0Q)
co
.v
-4
10
r
.Lu
IU
I
10 4
IU
-
Frequency
10
o
racld/s
Figure 7.17(d): H2 Control Loop Gains
116
Chapter 8
Conclusions and
Suggestions for Further
Research
This thesis has presented the basic theory of H, optimal control,
described the capabilities of the approach, and demonstrated the
utility on a practical design example. Although H, optimal control
is computationally intensive, the new solution of Doyle and Glover
[16] drastically simplifies the required calculations and appears to
be applicable to a broad class of plants.
The methodology demonstrated here, including the use of the
Youla parametrization and the subsequent norm minimization of
the weighted closed-loop transfer function, is a valuable and flexible tool for the synthesis of control systems. The approach eas-
ily handles the H, and H 2 optimization problems within a single
framework, making comparison of the design methodologies a simple procedure.
The capability to perform explicit loop shaping was demonstrated on a multivariable example. Weighting functions may be
used to shape closed-looptransfer functions to within tight bounds
provided by simple singular value inequalities. Important closed-
loop performance measures can be incorporated into the design
procedure with such shaping. This approach may be somewhat
limited by the necessity of having simple weighting functions when
117
several performance variables are defined, since weighting functions
with equal minimum and maximum singular values are needed to
provide tight bounds for singular value response. To extend the
capabilities of multivariable loop shaping, the effect of weighting
functions on closed-loop response needs to be examined more care-
fully. The implicit tradeoffs that occur with optimization of the
weighted system are not always intuitive and may easily lead to undesirable performance. Foreknowledgeof these tradeoffs and their
effect on singular value response may lead to a more systematic
method for choosing weights to achieve performance.
Given the similar approaches to the design of H2and Hoooptimal
systems using the present framework, an interesting topic is the
effect of the different norms on the properties of the closed-loop
system. In the final design of Chapter 7, the Hooand H2 solutions
were compared. Even though the RMS performance of the complete system was superior for the H2 design, as guaranteed by the
theory, the Hoooptimal system had better RMS performance of the
pointing error than the H2 optimal system. Clearly, the weighting
functions affect optimization of each norm differently and a good
weighting function for an Ho design is not necessarily a good one
for an H2 design.
Also, determining if the Hoooptimal system does actually have
better robustness properties than an H2 optimal system, simply because of the characteristics of each norm, is an important question
that needs further research. This last question applies to the design of robust controllers, using Ho optimization. The Ho norm,
used in conjunction with the structured singular value, can provide
a necessary and sufficient condition for robust stability. This is po-
tentially, a very important contribution of the theory of Hoooptimization that warrants continued research.
118
Bibliography
[1] V.M. Adamjan, D.Z. Arov, M.G. Krein, Analytic properties of
Schmidt pairs for a Hankel operator and the generalized SchurTakagi problem, Math USSR Sbornik, 15 no. 1, pp. 31-73, 1971.
[2] -
, Infinite block Hankel matrices and related eztension prob-
lems, AMS Transl., 111, pp. 133-156, 1978.
[3] B.D.O. Anderson and J.B. Moore, Relations Between Frequency-
Dependent Control and State Weighting in LQG Problems, Optimal Control Applications and Methods, vol. 8, pp. 109-127, 1987.
[4] J.A. Ball and J.W. Helton, A Beurling-Laz theorem for the Lie
group U(m,n) which contains most classical interpolation theory,
J. Operator Theory, 9, pp. 107-142, 1983.
[5] Cheng-Chih Chu, H Optimization and Robust Multivariable
Control, Ph.D. dissertation, Dept. of Electrical Engineering, University of Minnesota, Minneapolis, MN, 1985.
[6] M.A. Dahleh and J.B. Pearson, 1' optimal feedback controllers
discrete-time systems, Proceedings ACC, Seattle, WA, June 1986.
[7]
, I 1 optimal feedbackcontrollersfor MIMO discrete-time systems, IEEE Trans. Automat. Contr., vol. AC-32,no.6, April 1987.
[8] C. Davis, W.M. Kahan and H.F. Weinberger, Norm-preserving
dilations and their applications to optmal error bounds, SIAM J.
Numer. Anal., 19, 1982, pp. 445-469.
[9] P. Delsarte, Y. Genin and Y. Kamp, The Nevanlinna-Pickproblem
for matriz--valued
functions, SIAM. J. Appl. Math., 36, pp. 47-
61, 1979.
[10] C.A. Desoer, R.W. Liu, J. Murray, R. Sacks, Feedback system
design: The fractional representation approach to analysis and
synthesis, IEEE Trans. Automat. Contr., vol. 25, pp. 399-412,
June 1980.
119
[11] C.A. Desoer and M. Vidyasagar, Feedback Systems: InputOutput Properties, Academic Press Inc., New York, 1975.
[12] J.R. Dowdle, A Control System Design Methodologyfor LargeScale Interconnected Systems, CSDL-R-1946,
Draper Laboratory, Inc., Cambridge, Feb. 1987.
Charles
Stark
[13] J.C. Doyleand G. Stein, MultivariableFeedbackDesign: Concepts
for a Classical/ModernSynthesis, IEEE Trans. Automat. Contr.,
vol. AC-26, pp. 4-16, Feb. 1981.
[14] J.C. Doyle, Matriz Interpolation Theory and Optimal Control,
Ph.D. dissertation, University of California, Berkeley, 1984.
[15] -
, Analysis of FeedbackSystems with Structured Uncertainties,
IEE Proc., vol. 129, Pt. D, no. 6, November 1982.
[16] J.C. Doyle and K. Glover, A New Approach to H-Infinity
Opti-
mal Control, paper to appear American Control Conference, Atlanta, Georgia, June 15-17, 1988.
[17] B.A. Francis and J.C. Doyle, Linear Control Theory with an
H, Optimality Criterion, SIAM J. Control and Optimization, vol
25, no. 4, pp. 815-844, July 1987.
[18] B.A. Francis, A Course in Hoo Control Theory, Lecture Notes in
Control and Information Sciences, Springer-Verlag, New York.
[19] J.S. Freudenberg and D.P. Loose, Right Half Plane Poles and
Zeros and Design radeoffs in FeedbackSystems, IEEE Trans.
Automat. Contr., vol. AC-30, no. 6, pp. 555-565, June 1985.
[20] K. Glover, All optimal Hankel-norm approzimations of linear multivariable systems and their L °° error bounds, Int. J. Control, 39,
no. 6, pp. 1115-1193, 1984.
[21] N.K. Gupta, Frequency-ShapedCost Functionals: Etension of
the Linear-Quadratic-GaussianDesign Methods, J. Guidance and
Control, vol. 3, no. 6, pp. 529-535, Nov-Dec 1980.
[22] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems,
Wiley-Interscience, 1972.
[23] N.A. Lehtomaki, N.R. Sandell and M. Athans. Robustness Re-
sults in Linear-Quadratic Gaussian Based Multivariable Control
Designs, IEEE Trans. Automat. Contr., vol. AC-26, pp. 75-92,
February 1981.
120
[24] D.A. Milich, A Methodology for the Synthesis of Robust Feedback
Systems, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 1988.
[25] S.D. Postlethwaite and O'Young, and D.W. Gu, Stable-H User's
Guide, OUEL 1687/87, Univ. of Oxford, Dept. of Eng. Sci.
[26] M.G. Safonov and M. Athans, Gain and Phase Margin of Multiloop LQG Regulators, IEEE Trans. Automat. Contr., vol. AC-22,
pp. 173-178, April 1977.
[27] M.G. Safonov, A. Laub and G. Hartmann, Feedback Properties of
Multivariable Systems: The Role and Use of the Return Difference
Matriz, IEEE Trans. Automat. Contr., vol. AC-26, no. 1, pp. 4765, February
1981.
[28] M.G. Safonov and R. Y. Chiang, CACSD Using the State-Space
Loo Theory-A Design Ezample, IEEE Trans. Automat. Contr.
vol. 33, pp. 477-479, May 1988.
[29] D.E. Sarason Generalized interpolation in H °° , Trans. Amer.
Math. Soc. 127, pp. 179-203, May 1967.
[30] H.-H. Yeh, S.S. Banda, P.J. Lynch and T.E. McQuade, Loop-
TransferRecoveryvia Hardy-SpaceOptimization, Journal of Guidance, Dynamics, and Control, vol. 11, pp. 86-90, Jan.-Feb. 1988.
[31] D.C. Youla, H.A. Jabr and J.J. Bongiorno,Jr., Modern WeinerHopf design of optimal controllers,Part II, IEEE Trans. Automat.
Contr., vol. 21, pp. 319-338, June 1976.
[32] M. Vidyasagar, Control System Synthesis: A Factorization Approach, MIT Press, 1985.
[33] M. Vidyasagar, Input-Output Stability of Large-Scale Interconnected Systems, Lecture Notes in Control and Information Sciences, Springer-Verlag, 1981.
[34] G. Zames, Optimal Sensitivity and Feedback: Weighted Semi-
norms, Approzimate Inverses, and Plant Invariant Schemes,
Proc. Allerton Conf., 1979.
[35] -, Feedbackand Optimal Sensitivity: Model Reference Transformations, MultiplicativeSeminorms, and Approzimate Inverses,
IEEE Trans. Automat. Contr., vol. AC-26, no. 2, pp. 301-320,
April 1981.
121