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IBBB TRANSACTIONSON
AUTOMATIC CONTROL,VOL AC-25, NO.
5, OCTOBER 1980
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181 E. I. Jury and 8. W. Lee, A u f o m . Remote Conrr, voL 26, p. 943. 1965.
191 A. G. J. MacFarbe and I. Postlethwaite, T h e generalizedNyquist stability
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and R K. Miller, "Qualitativeanalysis of large scale dynamical
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m
.I
__
Synthesis of State Feedback Control Laws with a
Specified Gain and Phase Margin
PER MOLANDER AND JAN C . WILLEMS
I.
-C.7
8.e
3.2
e.4
8.6
2.8
1.8
:.?
:.A
1.6
(b)
Fig. 2. Theorem 5 applied 10 Example 2. With 4- 0, we obtain
min(-0.105,-0.11).Thisgiveskc9.
-l / k <
RsFWBNCEs
111 A R Bergen and M. A. Sapiro, T h e parabola test for absolute stability," IEEE
Tram.Amomnt. C a r . , voL AC-12, 1967.
121 Y . ( 3 0 and K. S. Narendra. "An 0ff-axi.s circle criterion for the stability of
feedback systems with a monotonic nonlinearity," IEEE Tram. Awomm. Con@.,
AC-I2 pp. 413-416, 1968.
[3] P. A Cook, 'Circle criteria for stability in Hilbert spaa," SIAM 1. C m . , voL 13,
pp. 593-610. 1975.
141 -,
"Conditions for the absence of Limit cycles," IEEE
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INTRODUCTION
In aseminalpaper, Kalman [l] demonstrated that linearquadratic
regulators satisfy a certain frequency domain inequality which yields a
degree of robustness for these control laws. In terms of classical control
concepts, Kalman's inequalityimplies that thesecontrollers possess a
60" phase margin, infinite gain margin, and 50 percent gain reduction
tolerance. In terms of feedback stability concepts [2], [3], it implies that
l
iremain stable if an arbitrary nonlinear system
the closed-loop systemw
contained in the sector (1/2,00) is introduced in the loop. This result
holds under the sole assumption that the integrand in the performance
criterion is the usual sum of a positive definite term in the control and a
nonnegative definite term in the state.
In the last few years, there has been renewed interest in this sort of
results (see,e.g., [4]-[6D, motivated principally by robustness questions.
Indeed, where classical control synthesis techniques were very directly
concerned with robustness considerations, it appeared that modem control theory had somewhat neglected to formulate robustness questions as
such. In particular, the question has recently been raised whether one
can design a state feedback control law such that the closed-loop system
has a specified gain and phase margin. In this paper it will be shown how
this can be achieved.
Trm. A u r o m . Conrr,
0 0 1 8 - 9 2 8 6 / 8 0 / 1 ~ 2 8 $ ~ . 7 50 1980 IEEE
IEER TRANSACTIONS ON AUTOMATIC CONTROL, VOL.
929
AC-25,NO. 5, OCTOBER 1980
with A any symmetric m a t h with eigenvalues 1 G X, <g. It is said to
have p h e muwh + (06 < T ) if it is globally asymptoticallystable for
all control laws 0with A any unitary matrix (A*A =A A* = I ) possessing eigenvalues h,=exp(j.&) with I+,J<g. It is said to have gain
reduction rolerance p in percent if it is globally asymptotically stable for
all control laws (7) with A any symmetric matrix witheigenvalues
1-p/100<Xi<1.(p>O,andusuallyp<100,butp>100isalsopossible
and corresponds to gain reversal.) These are straightforward generalizations of the classical single-inputconcepts.
From (4) the following proposition follows (the proof is given in the
Appendix).
Propasition I : Assume that (2) has robustness sector (Kl, K2).Then it
t
+
'
K,o
has
gain margin g= K2
+
Fig. 1. An illustration of the seclor condition.
II.
phase margin with cos( 9) =
gainreduction tolerancep=(l-Kl).lOO.
PROBL~MFORMULATION
Consider the b
e
a
r time-invariant finitedimensional system
(1)
i=Ax+Bu
with XER" and u € R m . It will be assumed throughout that ( A , B is
) a
controllable pair, although this is sometimes irrelevant and s t a b h b i l i t y
oftensuffices.Assume that the desired input is givenby the linear
time-invariant control law
and that it is neceSSary to choose L such that the closed-loop systemis
stable when the actual input is a perturbation of (2). Specifically,(2) wil
be called arobust control law with robustness sector ( K , , K 2 )(- 00 < K , <
1 < K2 < 00) if the closed-loop system is globally asymptotically stable
when the actual input is given by
u=
-f(LTx, r )
(3)
withf: W""R+BPm any nonlinear function satisfying
0
Consequently, in order to design a feedback law (2) with gain margin g
and phase margin 9 it suffices to take for robustness sector (Kl, K 2 )
with
and
K 2 =g.
(2)
u= -LTx
K,K2+1
4 +K2
Notice that if cos(+)6 1/g, then X , G 0 G 1 < K 2 and hence A will have
to be asymptotically stable in order for this design to be possible.
The circle criterion inequalities will be used to design control laws
with a given robustness sector.Of course,there may be other techniques
which yield a prescribed gain and phase margin., but such results do not
seem to be part of the linearquadratic ideas on which, after all, also the
nonlinear feedback stability results (Propositions 2and 3) are based. The
main facts from stability theory to be used are given in the following
section.
111.
PRELIMINARlEs
Let { A , BC, ) denote the system
or, equivalently,
Z: i = A x + B u ,
y=CTx.
TheminimalsystemXissaidtobeinridefhesector[a,~](-m<a<~<
/?<03)if A is asymptotically stable and if, with x(O)=O, and for all
u ( * )EL2(0,a),the responsey(.)satisfies
where
I - 1 denotes the Euclidean
norm on R'". The sector condition is
illustrated in Fig. 1for the singleinput case.
Moreover, the feedback system
i = A x + yB=uL+TGxo, ,
z=CTx
(5)
with u(.) determined fromy(-) by a nonlinear input/output system with
nonanticipatingsystem function F: y ( * ) + - u ( * ) satisfying
(Y(.)-au(,).Y(.)--BU(.)IL2(0,m)~0.
It is said to be outside the sector [a,/?]
(O<a<D< 03)if A+/?-'BCT is
asymptotically stable and
(Y(.)-au(.),r(.)-BU(.)ILI(O,m,)>O.
(If /?- 00,the first inequality should read
<r(.)-au(.),
should be h(0,co) input/output stable as a system from u(*) to .z(*).
When K2= co, (4) and (6) should be interpreted as holding for some
K 2 i 00,and a similar modification is required when KI= - 00.
The robustness requirements also allowfor an interpretation in terms
of gain and phase margin and gain reduction tolerance. To be precise,
the control law (2) has a gain margin g (g > 1) if the closed-loop systemis
globally asymptoticallystable for all control laws
u= - A L ~
(7)
U(.))L*(O,OD)>O,
with a similar modification for a= - 00.)
The following results follow
from standard calculations [2], [3] and the
Kalman-Yacubovich-Popv l e m m a or its extensions [q-[9].
Prqwsition 2; Let Z = { A , B , C ) be minimal and G(s)=CT(sZA)-'B be its transfer function. Then the following conditionsare equivalent.
i) Z is inside the sector [a,81.
ii)A is asymptotically stable and for all w ,
Re<G(jw)-a,G(jw)-/3)
iiii There exists a P=PT>O such that
<O.
930
IEEB TRANSACTIONS ON AUTOMATIC CONTRO& VOL.
+
A ~ PPA -
AC-25, NO. 5, OCIDBER 1980
(PB+D)(PB+D)T+DDT+Q=O
4K,K2
(K,+K2),
or, equivalently, if a/?#O,
ATP+PA+CCT-(PB-TC
)iB(
-
P B - - a+B
c )2
Go.
Similarly, the following conditions are equivalent.
i) Z is outside the sector [ a , / ? ] .
ii) A + / ? - ’ B C T is asymptotically stable and for all w
Re(G(jw)-a,G(jw)-/l>>O.
iii) There exists a P = P T > O such that
has a rymmetric solution (such D and Q will exist iff A is arynptotically
stable).
2) Pick any ymmetric solution of this Riccati equation (which will
aomatical!y be > 0).
3) Take L = 2 / K
( , +K2)(PB+ D).
Proof: The propertiesof the Riccati equation follow fromthe results
in [9] and the claimed robustness from the fact that ( A , B , L ) satisfies
condition iii)of Proposition 2 with a = - 1 / K 2 and /?=- l / K , .
0
Remurk: The above procedure collapses if K , + K, =O. For a treatment of thiscase,
singular
see [12].
0
The case K , = ca is treated in the following theorem whichis proved in
a similar fashion.
Theorem 3: The following procedure yields a feedback law (2) with
robustness sector ( K , , 00).
Case I-K, > 0:
I ) Pickany(nXn)matrixQ=QT>Osuchthat(A,Q)isobseroable.
2) Solw the algebraic Riccnti equation
ATP+PA-2K,PBBTP+Q=0
or, equivalently, if aj?#O,
(
A = P + P A - C C ~ + PB+-c
-
)ai(
PB+-c
a;B
GO.
Obviousmodifications are necessary for a = - ca or /?=00.
0
The following proposition gives the basic stabilityresult in which
much of the work on feedback stability has culminated.
Proposition 3: Assume that ( A , B ,L ) i s minimal. Then the control
law (2) has robustness sector ( K , , K 2 ) if either
i) O < K , < K2 and ( A , B , L ) is outside the sector [ - l / K , , - I/K2J
or
i
i
) K , G O < K , and { A , BL, ) is inside the sector [ - 1 / K 2 , - 1 / K , ] . 0
In fact, xTpx with P as in the above proposition will be a Lyapunov
function for the control law (3).
Various conditions (conicity, etc.)equivalent to those of the above
theorems or specific cases in which the conditions hold may befound in
the literature [2],[3], [IO], [1 I ] .
lV. ROBU~INESS
SYNTHESIS
I : Asswne 0 < K , < K 2 < 00. Thenthe
Theorem
following procedure
yields a feedback law (2) with robustness sector ( K , , K 2 ) .
I ) Pick any ( n X m ) matrix D such that ( A , Dis) an obsercuble pair and
any ( n x n ) matrix Q = Q ~ > o .
2) Solw the algebraic Riccati equation
for its (unique) solution P = PT’>Q. (This solution automatical& exists
when K , > 0 and iff A is asynptotical& stable when K , =0).
3) Take L = 2 / ( K , + K 2 ) ( P B + D ) .
Proof: By Proposition 3, one must show that P is well-defined in 2
and that ( A , B15)
, satisfies condition iii) of Propnsition 2 with a = - l/Kl
and /?=- l / K 2 . Part 1 ) follows from the standard linear-quadratic
results since the constant term of the Riccati equation equals
for its (unique) solution P = P T >0 (this solution automatically exists when
K , >O and iff A is arymptoticd!y stable when K,=O inwhichcase this
equation becomes a L y q m equation).
3) Take L = PB.
Case 2-K, 6 0:
I ) Pick nny ( n x n ) matrix Q=Q’>Osuch that ( A , Q ) is obserwble
and the algebraic Riccati equation
ATP+PA-2K,PBBTP+Q=0
has a ymmetric solution (such a Q will exist #A
is arynptoticcz& stable).
2) Pick any rymmetric solution of this Riccati equation (which will
atuomatically be > 0).
3) Take L= PB.
V. DISCUSSION
1) The robustness problem considered here is not well-posed. If A is
asymptotically stable,L= 0 will give a robustness sector (- w,m) ,and
if A is not, then a robustness sector(c, 00) (c >0) is obtainable from the
linearquadratic theory of 1 by using 1/2e times the optimal gain for L.
Nevertheless, the theorems serveto yield techniques for generating other
control laws. In these theorems one may regard the choices of D and Q
as in some sense equivalent (albeit much less intuitivelyinterpretable) to
choosing the performance criterionof a linear-quadratic design.
2) If K , > 0,the robustness designis always possible and one is free in
choosing D and Q in the procedure of Theorems 1 and 3. In case
K , < O< K , , A must be asymptotically stable. In this case, the matrices
D and Q of Theorems 2 and 3 should be chosen such that the Riccati
equation has a solution. In fact, they should be such that for
i=Ax+Bu,
y=DTx
with x(O)=O and for all u ( - ) , one has
in Theorem 2, and
and the required controllability and observability conditions are satisfied. To see part 2). solve for D in the algorithm getting D = ( K , +
K 2 ) L / 2- PB. Inserting this into the R
i
a
t
i equation yields the inequality in iii) of Proposition 2, modulo
a
factor K 1 K 2 .
0
Theorem 2: Assume - 00 < K , < 0 < K 2 < ca. Then the foilow*ngprocedure yielak a feedback law (2) with robusmess sector ( K , , K 2 ) .
I ) Pick any ( n x m ) matrix D andm ( n x n ) matrix Q = Q T > O such
that ( A , Dis) obseroable and the algebraic Riccati equation
in Theorem 3.
3) Theorem 3 with K , = 1 / 2 yields Kalman’s original result. In fact,
the other cases of Theorem 3 with K,> O may also be interpreted from
this result since the procedure used actually corresponds to taking an
ordinary linearquadratic optimal design, thus obtaining the robustness
sector ( l / 2 , a),and then using 2 K , times the optimal gain to obtain the
robustness sector (K,,
03).
It follows from the circle criterion that the condition on the Nyquist
l o c u s of G(s)=LT(sZ-A)-'B pertaining to condition (4) is that it be
T'"'
contained in a circle through (- 1/K2,0) and (- l/Kl, 0), symmetric
with respect to the real
This implies that the Nyquist l o c u s intersects the unit circle at an angle at least degrees away fromthe negative
real
Inserting the coordinates (- cos(+),-sin(+)) into the equation
of the circle bounding the Nyquist locus yields
axis.
+
axis.
whence
cos( +) =
Fig. 2. Deduction of the phase-margin equation in the singlainput case.
An interesting special case of Theorem 3 is K, =0, in which situation
one obtains the robustness sector [O,m) by picking ( A , Q) observable
and solving
[I]
[2]
[3]
[4]
[SI
161
[7]
181
[9]
on i = A x + B u with C and M such that
[lo]
I u J 2 + 2 u = c = x + x ~ M x ~ y 2 1 u 1 2 ( l y l1)
g
[ 111
[I21
will yield a robustness sector (l/(l+lyl), l/(l-lyl)).
K,K2+l
K,+K2
The multivariable case may be reduced to the above situation by using
the fact that a unitary matrix is unitarily equivalent to a diagonal one. 0
ATP+PA= -Q.
L = PB then yields an infinite gain margin, a 90" phase margin, and 100
percent gain reduction tolerance. These are so-called Lyapunov designs.
It is possible to show that all the L's thus obtained are also linearquadratic optimal designs, but by considering themas such, it would not
be possible to guarantee * h i s robustness sector.
4) AU of the results of Theorems 1 to 3 admit an optimal control
interpretation. It is easily verified using the by now standard frequency
domain manipulations,which yieldthe results of [l] andits multivariable
extensions, that using
~
Similarly, if
R E KaIman, "when is a linea control system optimal?," Trans. ASME, J . Basic
Eng, voL 86, pp. 1-10, 1964.
G. Zames, "On the input-output stability of time-varying leedback systems" (Parts I
& It), IEEE
Automat. Contr., vol. AC-11. pp. 228-238,465-476, 1966.
J. C.Willems, The Am&sis of Feedback S y s r m . Cambridge, MA: M.I.T. Res.
1972.
M. G. Safonov and M. Athans, "Gain and phase margins for multiloop LQO
regulator%"IEEE Trans. Automm. Cuntr., vol. AG22, pp. 173-179, 1977.
P. K. Won& G. Stein, and M. Athans, ''Structural reliability and robustness
properties 01 optimal linearquadratic multivariable regulators,'' Prqrinrs, Helrinki
IFAC Congr., 1978, pp. 1797-1805
P. Molander,"Stabilisation 01 uncertain systems." Dep. Automat. Contr.. Lund
Inst. Technol., Lund, Sweden, CODEN: LUTFD2/(TFRT-l020)/1-111/(1979).
1979.
V.M.Popov, Hyperstabili@ of Contrd System. New York:Springer, 1973.
P. Faurre, M. Clerget, and F. Germain, @iratetus Rntionne& Posirifs. Paris:
ked, 1979.
J. C. Willems, "Leastsquares stationary optimal control and the algebraic Riccati
equation," IEEE Trans. Automar. Cuntr.. voL AC-16, pp. 621-634. 1971.
C. A. DeJoer and M. Vidyasagar, Feedbrrck Systems: Iq~ut-outpulProperla.
New Yo& Academic. 1975.
M. Vidyasagar, Nodineur Systems A m b s i s . Englewood Cliffs, N J Rentice-Hall,
1978.
B. Friedland, "Limiting forms 01 optimal stochastic Linear regulators," in Prm. Joint
Autorrmr. Conrr. Conf.., 1970, pp. 212-220.
TJWLS.
lul2+2u=c=x+x=Mx<y2lu(2()yl> I),
and if the optimal control problem has a solution, which is then not
guaranteed, one obtains a robustness sector (1/( 1- Iy I), 1/( 1+ Iy I)). By
multiplying the optimal gain by an appropriate factor, it is possible to
obtain an arbitrary preassigned robustness sector. Theorems 1 to 3 are
easily interpreted in this vein. The performance criterion is
+
and the gain used is (Kl K2)/2 K, K2 times the optimal gain.
2uTCTx in thecost
The idea of includingacross-productterm
functional deserves in our opinion some attention in the linearquadratic
theory, where often only the case (uTRu+xTMx) with R = R T > 0 and
M = M T > O is treated. This extra flexibility makesit possible to generate
optimal control laws that may be superior from the sensitivity or accuracy point of view. For a treatment of such optimal control problems,
Connections Between Finite-Gain and Asymptotic
Stability
DAVID J. HILL AND PETER J. MOYLAN
Abstrm-The relationship W e e n inpnt-outpnt and Lyapumv stabiiity properties for nonlinear system is studied. Wd-knom d e f i i for
the i
nputproperties of finite-gain and passivity, even with qnite
reasonable minimality assumptions m a state-spme representation,do not
necessarily imply any form of stability for the state. Attention is given to
the precise versiopls of input-output and olk3efvabii p
r
o
p
e
r
e
t
is whi&
guarantee asymptotic stability. Particular emphasii is given to the possibility of multiple equilibria for the d
y
n
a
m
d
i system.
I.
see [9].
5) The theory is easily generalized to the case that Kl and K2 become
arbitrary diagonal matrices. This admits robustness design with unequal
gain and phase margin requirements in the different loops.
APPENDIX
Proof of Proposition 1: Consider first the single-input case. The gainmargin and gain reduction tolerance assertions are immediate from the
definitions. To deduce the phase-margin result, consider Fig. 2, whick
hasbeendrawnforthecaseKl<O<Kz.
~ODUCXION
For causal linear time-invariant systems, there are well-known strong
of definitions of stability [ 11. In particular,
equivalences among a variety
& finite-gainstabilityimplies, under minimalityassumptions,global
Manuscript reseived April 12. 1979; revised September 10, 1979 and March 27, 1980.
Paper ~ecammendedby M. Vidyasagar, chairman 01 theStability,Nonlinear,
and
Distributed Systems Committee.
D. H
ill is withtheDepartment of Electrical EngineeringandComputer Sce
i nces,
University of California,Berkeley, CA 94720.
P. MOYh is with the Department of Electrical Engineering, University 01 Newcastle,
Newcastle, N.S.W., Australia
0018-9286/80/1000-0931$00.75
01980 IEEE