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Quadratic stabilization with an H ∞ performance bound of uncertain linear-time-varying systems
SYSTIUI S {m CONTROL U[TTI[RS ELSEVIER
Systems & Control Letters 28 (1996) 103-114
Quadratic stabilization with an )ff performance bound of uncertain linear-time-varying systems F. A m a t o * , A . P i r o n t i Dipartimento di Injormatica e Sistemistica, Universith degli Studi di Napoli "Federico H", via Claudio 21, 80125 Napoli, Italy
Received 10 July 1995; revised 14 December 1995
Abstract
In this paper we deal with linear-time-varying systems subject to uncertain parameters in the so-called one block, norm bounded form. We consider the problem of designing a controller which guarantees the quadratic stabilization of the closed-loop system while keeping the norm of the operator mapping the exogenus input to the controlled output under a prespecified level. Necessary and sufficient conditions for the existence of such a controller are given. Keywords: Linear systems; Time-varying systems; Uncertain systems; Quadratic stabilization; ,;¢~ optimal control
I. Introduction
It is well known that an unforced linear, time-invariant system subject to uncertainties is said to be quadratically stable if there exists a time-invariant quadratic Lyapunov function whose derivative along the trajectories of the system is negative definite for all values of the uncertain parameters in their bounding set (see [2]). In the same way a forced, uncertain system is said to be quadratically stabilizable via linear control if there exists a linear controller which makes the overall closed-loop system quadratically stable. In [13] to cope simultaneously with quadratic stability and robust performance the concept of quadratic stabilization with an ~ norm bound constraint has also been introduced. Necessary and sufficient conditions for quadratic stability, quadratic stabilizability and quadratic stabilizability with an ~ norm bound are known for systems subject to uncertainties in the so-called norm bounded, structured, one block form (see the tutorial paper [4] and the bibliography therein). All the involved conditions are expressed in terms o f suitable parametric algebraic Riccati equations as shown in [6, 7, 13]. Recently in [ 12, 5, 1] the situation in which the uncertainties are still in norm bounded, structured, one block form, but the matrices of the nominal system are time-varying, has been discussed. In this case the definition o f quadratic stability is naturally extended using time-varying quadratic Lyapunov functions. Necessary and sufficient conditions for quadratic stability and stabilizability have been found in [1]; these conditions are expressed in terms of parametric Riccati differential equations or in terms of a modified version of the Small Gain Theorem. Starting from the work done in [ 1], in this note we consider a particular ~,ug~ control problem for an uncertain, linear, time-varying system subject to norm bounded uncertainties. For this class of systems we consider * Corresponding author. Tel.: +39 81 7683513; fax: +39 81 7683186; e-mail: [email protected]. 0167-6911/96/$12.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PII S0167-691 1(96)0001 8-7
104
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the problem of finding a linear controller which yields quadratic stability for the closed-loop system while keeping the norm of the operator mapping the exogenus input to the controlled output under a prespecified level. Our main contribution (Section 3) consists of necessary and sufficient conditions for the existence of such a controller both in the state and the output feedback case. This paper can be seen as the extension of [13] to the time-varying setting•
2. Notation and preliminaries We denote by , ~ t o (~c~lo) the space of the uniformly bounded, piecewise continuous (piecewise continuously differentiable), real matrix-valued functions defined on [to, +c~) and by 5~t2o the space of the real vector-valued functions which are square integrable on ( t o , + ~ ) . Given an exponentially stable linear time-varying system .i(t) = A ( t ) x ( t ) + B ( t ) u ( t ) ,
t E [t0, +cx~),
(la) (lb)
y(t) = C(t)x(t),
it uniquely defines the linear operator F('~ • $ 2,o ~_~ ~,,,, 2
u(.) ~ y(.) = _y, U ° u(.) = C(.) ft~+~ @(.,s)B(s)u(s)ds,
(2)
where q~ denotes the state transition matrix of (1). The Euclidean vector norm and the corresponding induced matrix norm are denoted by I' [, the usual norm in LP2t0 by I]" II, while [[Fyt~ll denotes the operator norm induced by the norm in £,et20 Given R : [ t 0 , + ~ ) H ~nx,, we write R > 0 (~>0) to mean that R is positive definite (semidefinite), i.e. that there exists ~ > 0 such that for all v E MI := {v E ~": Iv[ = 1} and for all t E [ t o , + ~ ) vTR(t)v>~
(vTR(t)v>~O).
Given two matrix-valued functions of the same dimensions R and S, the notation R > S (R>>-S) means that R - S is positive definite (semidefinite). Finally, the symbols R < (~<)0, and R < (~<)S have obvious meaning. In the sequel we shall use the following technical result.
Lemma 1. Let us consider system (1). Then the followin9 statements are equivalent (i) [l~f~[I < 7; (ii) the Riccati differential equation P(t) + AT(t)P(t) + P(t)A(t) + 7-2p(t)B(t)BT(t)P(t) + c T ( t ) C ( t ) + el = O
(3)
admits, f o r some constant e > O, a positive-definite solution P E ~cg~,; (iii) there exists a positive-definite P E ~cg)o such that [~ + ATp + PA + 7-2pBBTp + c T c < O.
Proof. See the appendix. Now let us consider the following linear, unforced, time-varying system subject to time-varying uncertainties a?(t) = [A(t)+ AA(t)]x(t), where x(t) E ~ , A E ~ 0 block form AA(t) = Fl(t)A(t)Ej(t),
t>~O,
(4)
is exponentially stable and the uncertain part is in the so-called structured, one (5)
[( Amato, A. PirontilSystems & Control Letters 28 (1996) 103-114
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with F1 and E1 belonging to ~Cgo and A is an unknown matrix-valued function belonging to the class
:= {A : [ 0 , ÷ ~ ) ~
N*×JlA is Lebesgue measurable, AT(t)A(t)<~I Vt>~0}.
(6)
We shall refer to system (4) as an uncertain, linear, time-varying system, with uncertainties in norm bounded, structured, one-block form. Such a system is said to be quadratically stable if there exist a to ~>0 and a quadratic, time-varying Lyapunov function V(t,x) = xTp(t)x whose derivative along the solutions of (4) is negative definite for all A C @ (see [12, 1]). Using properties of quadratic forms we can state the following equivalent definition. Definition 1. System (4) is said to be quadratically stable if there exists a t0~>0 and a positive-definite ~ t o such that for all A E P C~
[~ + (A + FI AE1 )Tp + P(A + F1AEI ) < O. If system (4) is quadratically stable, directly from Definition 1 follows that the same system is exponentially stable for all A E 9 . In [1] it is shown that quadratic stability of system (4) also guarantees global uniform exponential stability of the closed-loop system
~(t)=A(t)x(t)+Ffft)~(t),
(7a)
2(t)=El(t)x(t),
(7b)
~(t)=A(~(t),t),
(7c)
where A is any continuous (possibly nonlinear) mapping such that IA(~,t)l ~<12[, for all ~ E ~J and for all t~>0. In the same paper necessary and sufficient conditions for quadratic stability of system (4) are given. We now introduce the concepts of "quadratic stability and stabilizability with an ~ norm bound y", extending to the time-varying setting the definitions given in [13], where the time-invariant case has been considered. Consider the uncertain system ,f(t) = [A(t) + AA(t)]x(t) + Bl(t)w(t) + [B2(t) + AB2(t)]u(t),
(8a)
z(t) = Cl(t)x(t) ÷ Dlz(t)u(t),
(8b)
y(t) = [C2(t) + dC2(t)]x(t) + D21(t)w(t) q- [D22(t) + AD22(t)]u(t),
(8c)
where
AC2(t)
ADz2(t)
= \F2(t)
A(t)(El(t) E2(t)),
(9)
u(t) C ~m and w(t) E •P are the control and the exogenus input, respectively, y(t) E ~l and z(t) E ~q are the measured and the controlled output, and Ba,C1,B2,D12, C2,D2bD22,F2,E2 belong to ~Cgo. Definition 2. Given to >/0 and V > 0, system (8) is said to be quadratically stable with an o ~ 7 in [to, +cxD) if there exists a positive-definite P c ~cg~0 such that for all A E
t 5 + [A +F1AEj]TP +P[A +F1AEI] + 7-2pB1BTp + cTICI < O.
norm bound
(10)
Note that, letting u = 0, Eqs. (8a) and (8b) define a family of input-output operators Tt°(A) depending on the realization of the uncertain matrix A. The next lemma justifies Definition 2.
F. Amato. A. Pironti/Systems & 6bntrol Letters 28 (1996) 103-114
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Lemma 2. If system (8) is' quadratically stable with an ~ ically stable and I[ T_~,(A)[I < 7 fi )r all A C ~.
norm bound 7 in [t0,+cxz) then it & quadrat-
Proof. Since y-2pBIB~P + C(CI is positive semidefinite, from (10) follows that for all A E
P + [A + F1AEI ]Tp + P[A + FI AE~] < O,
(11 )
and hence the quadratic stability of system (8). Moreover, applying Lemma 1, Eq. (10) guarantees that II ~°~.(A)II < Y- [] Definition 3. Given to ~>0 and 7 > 0, system (8) is said to be quadratically stabilizable with an ~ norm bound y in [ t 0 , + ~ ) via linear state (output)feedback control if there exists a linear, finite-dimensional controller u = K ~ x (u -~ KuyY ) such that the overall closed-loop system is quadratically stable with an ,g,c~ norm bound y.
3. Main results In this section we provide necessary and sufficient conditions for quadratic stability and stabilizability with an o ~ norm bound 7- First we need the following preliminary lemma. Lemma 3. Given to >10 and 7 > 0, consider the operator Tj~(#) defined by the ,system
Yc(t) = A(t)x(t) + (y-IBl(t) #(t)Fl(t))~(t),
(12a)
£(t) = ( C,(t) ) x(t). \#-'(t)El(t)
(12b)
The following statements are equivalent: (i) there exists a positive-definite scalar Junction # E J)°~to such that
IIT~(#)ll < l; (ii) there exists a positive-definite scalar Junction It E , ~ t o such that the Riccati differential equation
P(t) + AT(t)P(t) + P(t)A(t) + 7-2p(t)Bl(t)B~(t)P(t) + C~(t)CT(t) +#-2(t)E~(t)El(t) + #2(t)P(t)Ffft)F[(t)P(t) + ,:1 = O
(13)
admits, Jor some constant ~: > 0, a positive-definite solution P C ' ~ ] 0 ; (iii) system (8) is' quadratically stable with an ~ norm bound 7 in [ t 0 , + ~ ) . ProoL See the appendix. Remark 1. The idea underlying Lemma 3 is the introduction of the scaling function #. It is important to note that, in this way, we are able to relate the quadratic stability with an jug~ norm bound y of system (8) to a small gain condition for the fictitious system (12). In the time-invariant case, that is when A, BI, etc., are constant matrices, Lemma 3 states that system (8) is quadratically stable with an Jcg~ norm bound 7 if and only if there exists a constant # > 0 such that the Riccati algebraic equation
ATp + PA + y-2PB1B~P + CXlC1 + #-2E~E1 + Ft2pF1F~P + e l = 0
(14)
admits, for some e, > O, a positive-definite solution P. In the same case a necessary and sufficient condition for quadratic stability (without the J g ~ norm bound constraint) is the existence of a constant # > 0 such
F. Amato, A. PirontilSystems & Control Letters 28 (1996) 103-114
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that the equation
ATp + PA + p-2EXlE1 + p2pF1F[P + ~I = O
(15)
admits a positive-definite solution P. The change of variable/5 = pZp in (15) shows that for quadratic stability it is necessary and sufficient to solve Eq. (15) for p = 1, avoiding in this way the search for the parameter p. One could ask if a similar property holds for Eq. (14). Unfortunately, the answer to this question is in the negative; indeed, the above change of variable does not work in this case. To convince ourselves definitely about this point, note that, according to L e m m a 3(i), a necessary and sufficient condition for quadratic stability with an , ~ norm bound 7 is the existence of p > 0 such that ,[
to r~o(p)ll =
(
7-'C'(sI-A)-IB'
pCI(sI-A)-'F')
p-17-1El(sI - A ) - l B l
~
El(sI -A)-tFj
< 1.
(16)
Now from (16) follows that, in general, II rz:;(1)ll # II r~°(~)ll for p # 1. It is simple to build examples for which [[ T ~ ( p ) [ I < 1~< HT~°(1)I[ for some p # 1; the conclusion is that, in the time-invariant case, the introduction of a constant scaling factor p in the condition for quadratic stability with the .Zf ~ norm bound is necessary. Now we will show, by a simple example, that in the time-varying case it may be necessary that such a scalar factor be a function of time (as stated in Lemma 3). E x a m p l e 1. Consider the scalar system 2(t) = ( - 2 +
fl(t)f(t)el(t))x(t) + w(t),
(17a)
z(t) = x(t),
(17b)
with fl(t)=l,
el(t)=0,
tE[2nT,(2n+l)T[, n = 0 , 1 .... ,
(18a)
fl(t)=0,
el(t)=4,
tE[(2n+l)T,2(n+l)T[,
(18b)
n=0,1 .....
where T = 1; furthermore, assume that 7 = 1 and to = 0. Since
IIr°,,ll = I I ( s + 2 ) - ' l l ~
=
~1 < 1,
(19)
from Lemma 1 follows the existence of a positive-definite p E ~cgl which satisfies
p+2ap+7-Zb2p 2 + c ~ = p - 4 p +
p2 + 1 < 0.
(20)
Since Aa(t) = fl(t)6(t)el(t) is identically zero we can conclude that condition (10) in Definition 2 is satisfied and therefore the system (17) is quadratically stable with an . ~ norm bound 7 = 1 in [ 0 , + c c ) . The following result, is proved in the appendix. Fact 1.
Let us define, according to Lemma 3, the operator T°~(p) associated to system
2(t) = - 2 x ( t ) + (1
pfl(t))~(t),
(21a)
-~(t) =
x(t);
(21b)
p-tel(t)
then inf,,>0 II ~°~(p)ll > 1. Fact 1 shows that, although system (17) is quadratically stable with an ~ norm bound 7 = 1, there does not exist a positive constant scaling factor which satisfies condition (i) in Lemma 3. On the other
1:( Amato, A. Pironti/Systems & Control Letters 28 (1996) 103-114
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hand, the same lemma guarantees the existence of a time-varying scaling factor p which satisfies condition (i) (or equivalently (ii)). Indeed since (19) holds, from Lemma 1 follows that there exists a positive-definite p E ~cg~ which satisfies (20). Now, following continuity arguments, one can find a sequence of sufficiently small positive numbers #n and a sequence of sufficiently big positive numbers /7n such that for all integer n p-4p+p2+l+p_~p
2 <0,
D-4p+p2+l+fi22
< O,
tE[2nT,(2n+l)T[, tE[(2n+l)T,2(n+l)T[.
(22) (23)
Obviously, the piecewise constant function {-P-n if t E [ 2 n T , ( 2 n + l ) T [ , #(t):=
fin
if t c [ ( 2 n + l ) T , 2 ( n + l ) T [ ,
(24)
is such that [ ~ - 4 p + p2 + l +t~ -2 +I~2p 2 < 0.
(25)
Using again Lemma 1 the existence of a positive-definite p satisfying the Riccati equation in condition (ii) of Lemma 3 follows. The following theorems give necessary and sufficient conditions for quadratic stabilizability of system (8) with an ,X~ norm bound 7. First we consider the state feedback case. Theorem 1. Given to >~0 and 7 > O, consider the system 2(t) = A(t)x(t) + (7-1Bt(t) #(t)Fl(t))r?(t) + Bz(t)u(t),
(26a)
(Cl(t) ~ ~ D12(t) ) 5(t) = \ t ~ - l ( t ) E x ( t ) J x(t) + \ # -l(t)E2(t) u(t).
(26b)
Let Z~°(t~) be the closed-loop operator obtained by applyin9 a linear, finite-dimensional state feedback control in the Jorm u = K~xx; then system (8) /s quadratically stabilizable with an J f ~ norm bound 7 in [to, +oc) via linear state feedback control if and only if there exist a positive-definite scalar function ] < 1. In this case K is a /~ C ~cgt0 and a memoryless operator Kux = K E ~cgto such that ]]Z'e~(/~)t to compensator which quadratically stabilizes system (8) with an ~ norm bound 7. Proof. Let p(t) = J(t)p(t) + L(t)x(t),
(27a)
u(t) = M(t)p(t) + K(t)x(t),
(27b)
a state space representation of the compensator K~. The feedback connection of system (8) and (27) gives the closed-loop system 2c(t) -- [Ac(t) + AAc(t)]Xc(t) + Bic(t)w(t),
(28a)
z(t) = C~c(t)Xc(t),
(28b)
F. Amato, A. PirontiISystems & Control Letters 28 (1996) 103-114
109
where Xc = (x r pV)7, and Ac(t) = ( A ( t ) +B2(t)K(t)
L(t)
B2(tlM(t)~ J(t) J '
AAc(t) = Fc(t)A(t)Ec(t),
Fc(t) = ( F ~ t ) ) , Ec(t) = (El(t) + E2(t)g(t) E2(t)M(t)),
C~c(t) = (Cl(t) + Dlz(t)K(t) Dlz(t)m(t)). From Lemma 3 it follows that system (28) is quadratically stable with an Jut~ norm bound ~' in [to, + e c ) if and only if there exists a positive-definite scalar function /~ E ~cgt 0 such that the input-output operator associated to the system
ic(t) = Ac(t)Xc(t) + (7-1Blc(t) #(t)Fc(t))~(t), (C'c(t) 2(t) =
)xc(t )
(29a) (29b)
~-~(t)Ec(t)
has norm less than unity. Now, considering that system (29) is also a state space representation of the feedback connection of system (26) and Ku~, we can conclude that system (8) is quadratically stable with an ~ norm bound 7 if and only if there exists a finite-dimensional system K~ such that IIZe~(#)ll to < 1. This is an ~¢go~ control problem with state feedback for which we can search, without loss of generality, for memoryless controllers (see [11]). [] The next theorem focuses on the output feedback case. Theorem 2. Given to >~0 and 7 > O, consider the system
2(t) = A(t)x(t) + (7-1Bl(t) It(t)Fl(t))~(t) + B2(t)u(t), Z(t)= (Cl(t))x(t)+
IL-~(t)E1 (t)
(Dl:(t))u(t), I~-I(t)E2(t)
y(t) = C2(t)x(t) + (7-1D21(t) lt(t)F2(t))v~(t) + D22(t)u(t).
(30a) (30b) (30c)
to Let Se~,(# ) be the closed-loop opertor obtained by applying a linear, finite-dimensional output feedback control in the form u = K u y Y ; then system (8) is quadratically stabilizable with an ~ norm bound 7 in [to, + o c ) via linear output feedback control if and only if there exist a positive-definite scalar function p E ~ t o and a K,y such that IlSz'-~-(~)[I < 1. In this case a minimal realization of K,y (see [3]) is a compensator which quadratically stabilizes system (8) with an , ~ norm bound 7.
Proof. Let
p(t) = J(t)p(t) + L(t)y(t),
(31a)
u(t) = M(t)p(t),
(31b)
a state space representation of the compensator Kuy. The feedback connection of system (8) and (31) gives the closed-loop system 2c(t) = [Ac(t) + AAc(t)]Xc(t) + Blc(t)w(t),
(32a)
110
IY Amato, A. Pironti/Svstems & Control Letters 28 (1996) 103-114 z(t) = Cic(t)Xc(t),
(32b)
where xc = (x v pT)T, and
/
A(t)
Ac(t) = !
\ L(t)O2(t)
B2(t)M(t) J(t) + L(t)D22(t)M(t) J '
AAc(t) = Fc(t)d(t)Ec(t), Fc(,) = ( F , ( t ) k L(t)F2(t) ) ' Ec(t) = (Ej(t) E2(t)M(t)),
( Bl(t) ) B i t ( t ) = \ L(t)Dzl(t)
'
Clc(t) = ( G ( t ) Dlz(t)M(t)). Then the proof continues as in Theorem 1. Note that in (31) we focus on strictly proper compensators; this does not cause any loss of generality because, as shown in [8, 1 1], in an , ~ control problem the existence of a proper controller guarantees the existence of a strictly proper one. []
Remark 2. It is important to note that Theorems 1 and 2 transforms the quadratic stabilizability with an Jt ° ~ norm bound 7 problem into an equivalent ~ , ~ problem, which can be solved via existing techniques (see U1,8]).
4. Conclusions In this paper we have considered linear systems subject to uncertainties in norm bounded, structured, one block form. The novelty consists in the fact that the nominal system and the uncertainties scaling matrices are time varying. Necessary and sufficient conditions for quadratic stability and quadratic stabilizability with the simultaneous satisfaction of an , ~ performance bound, between the exogenus input and the controlled output, have been provided in terms of equivalent conditions involving the Small Gain Theorem and the solvability of a certain Riccati differential equation.
Appendix
Proof of L e m m a 1. The equivalence between proposition (i) and (ii) is stated in the main theorem of [10]. to (i) =~ (iii) If ]]Fvull < ~ then the Riccati differential equation (3) admits a positive-definite solution P E , ~ ? ~ for some r, > 0. In this case we have that, in [to, + ~ ] , [~ + ATp + PA + 7 - 2 p B B T p + CTC < O. (iii) =¢- (i) Let
S(t) = -([~(t) + AZ(t)P(t) + P(t)A(t) + 7-2p(t)B(t)BT(t)P(t) + cT(t)C(t) + eI), where e, > 0 is such that S > 0. In this way the matrix-valued function P satisfies the Riccati differential equation
[~(t) + AY(t)P(t) + P(t)A(t) + ~-2p(t)B(t)BT(t)P(t) + c T ( t ) C ( t ) + sl/2(t)SI/2(t) + el = O.
E Amato. A. Pironti/Systerns & Control Letters 28 (1996) 103-114
111
^t0 defined by the system Due to the equivalence between proposition (i) and (ii), the linear operator F~u, :f(t) = A ( t ) x ( t ) + B(t)u(t),
t E [to, +cxD)
(c(t) "~ s12(t) ) x(t) has norm less than 7. Hence, for all u E L~a2, we have to < z'. [] that Ilry,,ll
IlYll < v,llull, and, since IlY[I~< IlYlI, we can conclude
In order to prove Lemma 3 we need the following result which has been proven in [l]; for the selfcontainedness of the paper we repeat the proof here. Fact 2. Given X, Y , Z belonging to . ~ t o and such that, .for some ?) > 0, X>~0,
Y < 0,
Z>~0,
(A.1)
( J Y ( t ) v ) 2 -4vXX(t)vvXZ(t)v>>-6
Vv E ~1, Vt E [ t 0 , + ~ ) ,
(A.2)
then there exists a positive-definite scalar function # E "@%o satL~fyin,q #2X + ItY + Z < O. Proof. Let
a(v,t) := vTX(t)v;
b(v,t) := vTy(t)v;
c(v,t) := vTZ(t)v.
(A.3)
Since X and Z are uniformly bounded, there exists 7 > 0 such that
Ix(t)l ~<'/,
Iz(t)l ~<7,
Vt E [t0,+o~);
then set gg()~,v, t) := ,~2a(v,t) + 2b(v,t) + c(v,t) + e,
(A.4)
where 0 < ~: < 6/47. From (A.1) there exists ~ > 0 such that
a(v,t)>lO,
b(v,t)<. - ct,
c(v,t)>~O
V(v,t) E ~1 × [t0,+oc).
(A.5)
Now observe that Eq. (A.2) implies
A(v,t) := bZ(v,t) -4a(v,t)[c(v,t)+e]>Jfi - 4 7 e > 0
V(v,t) E ;~1 × [ t 0 , + ~ ) ,
(A.6)
from which follows that the second-order equation in the variable )~, ,q(2, v, t) = 0, admits two real distinct solutions 21(v,t) and 22(v,t), with 21(v,t) < 22(v,t), for all (v,t) E ~1 × [t0,+oo). Moreover, by applying Cartesius rule of sign of solutions and recalling (A.5), we can conclude that
0 < )~l(v,t) < )~2(v,t)
V(v,t) E ,~l × [t0,+oc).
(A.7)
We claim that ).l(V,t)
< 2 < 22(v,t) 4:~ ,q()~,v,t) < 0
V(v,t) E ~1 × [t0,+OO).
(A.8)
Indeed, for fixed v and t, the graph of y(2) = `q(2, v, t) in the plane (2, y) is either a concave parabola (when a(v, t) ¢ 0) with vertex ( - b ( v , t)/2a(v, t), - ( b 2 ( v , t ) - 4a(v, t)(c(v, t) + ~,))/4a(v, t)) and intersections in 21 (v, t) and )~2(v,t) on the 2-axis, or a straight line (when a(v, t) = 0) with negative slope and intersection in 21 (v, t) on the 2-axis (in this latter case 22(v,t) = + ~ ) . In both cases (A.8) is proved.
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Now from Lemma 3.2 in [9], for a fixed t there exists ,~ such that 9(2, v, t) < 0 for all v E ~ l . By virtue of (A.8) we can conclude that ~, is such that
),l(V,t) < ,~ < 22(v,t)
Vv E .~1,
(A.9)
or equivalently 0 < A l ( t ) : = maxAl(v,t) < ,~ < rain 2 2 ( v , t ) = : A~(t). vG,~)
(A. 10)
vE~l
Now observe that, since X, Y and Z are piecewise continuous, it is possible to find a sequence of intervals {J,.} such that U,~l J i = [t0,+oc) and a(v,.), b(v,.) and c(v,.) are continuous on each interval .h. From this follows that ),l(v, .) is continuous on each of these intervals and in turn, being ~1 a compact set, A1 is continuous. We can conclude that the function AI is piecewise continuous on [to, +cx~). Now pick any positive-definite scalar function /~ E ,~'~gt0 such that for all t i> to A1 (t) ~
(A. 11 )
Note that # can be chosen (i) uniformly bounded because Ai is, being Al(t)~<2(y + a)/~ for all t E [t0,+oc); (ii) piecewise continuous because A I is; (iii) positive definite because A2 is, being A2(t)~>a/2y for all t E [to, +oo). From (A.8), (A.11) and (A.4) it follows that
#2(t)a(v,t) + #(t)b(v,t) + c(v,t)<~ - e < 0 and hence the proof of the lemma.
V(v,t) E ~ i x [t0,+cx~),
(A.12)
[]
P r o o f of L e m m a 3. The equivalence between (i) and (ii) follows directly from Lemma 1. (ii) =¢, (iii) Suppose that, for some s, > 0 and some definite positive scalar function # E ~cgto, there exists a positive-definite solution of Eq. (13). From
(p-~ AEI - pFT p ) T ( # - ' AEI - #FT p)>JO
(A.13)
it follows that
ET ATFTp @ PF1AEI <<,p2pF1FTp + 11-2ETA T AEI <~ t~EpF1FTp + t~-2ETE1,
(A.14)
therefore we have that for all A E
/6 + [A + F1AE~]T P + P[A + F~AE~] + cT c~ + 7 EpB~BT P < 0. (iii) =:~ (ii) Suppose there exists a positive-definite P E ; ~ o
(A.15)
satisfying the matrix inequality
/6 + [A + FIAE1]T P + P[A + F1AEI] + cT cI + 7-zPB1BT p < 0
(A. 16)
for all A E ~ . Now let us define L : = / 6 + ATp + PA + cTc~ + ~' 2pB~BTp < 0.
(A.17)
From (A.16) it follows that for all A E ~ , for all v E Ml, for all t E [t0,+oo) and for some ~ > 0
2vT P(t)Fl(t)A(t)El(t)v < --vT L(t)v -- 7;
(A.18)
on the other hand, since the set ~ is symmetric around zero, A E ~ implies - A E ~ , hence from (A.18) follows
--2vT p(t)F1 (t)A(t)E1 (t)v < --vT L(t)v -- ~.
(A. 19)
F. Amato, A. PirontilSystems & Control Letters 28 (1996) 103-114
113
Eqs. (A.18) and (A.19) imply that for all v E M1, for all t E [t0,+c~) and for some c( > 0
(vT L(t)v) 2 >14max{(vTp(t)Fl(t)A(t)El(t)v) 2} + ~' = 4vTp(t)F1 (t)FT(t)e(t)vvTE~(t)El (t)v + ~',
(A.20)
where the last equality is proved in L e m m a 3.1 in [9]. From (A.20), using Fact 2, it follows that there exists a positive-definite function # E ~c~t o such that
#4pFIF~P + #2L + E~El < 0.
(A.21)
Now dividing by #2(0 we obtain
[~ + ATp + PA + cTc1 + ?-2pBIB~P + #2PF1FTp + #-2ETE, < 0.
(A.22)
From (A.22) and using L e m m a 1 the thesis follows.
Proof of Fact 1. First note that [I ~°,(#)t[ :=
]1Te°~(#)wl[ >~II T~(U)W o • I[ ,
sup
wcz0~_(0~
Ilwll
(1.23)
[[w*ll
where w* :=
{(4
--
0
for t E [0, 1), for t~> 1.
(A.24)
Since
IIw*ll
:=
w*T(t)w*(t)dt
= 1
(1.25)
from (A.23) it follows that
II T°(#)ll >/ql T°~(#)w* II•
(1.26)
Now a simple but cumbersome calculation shows that
II T°,(#)w* II =
[0.0710(1 + #)2 + 0.3664(1 + #-1)211/2 := g(#).
(A.27)
The minimum of g is attained at # = 1.728 and is equal to 1.2007. This last observation, together with (A.26), proves our claim.
References [1] F. Amato, A. Pironti and S. Scala, Necessary and sufficient conditions for quadratic stability and stabilizability of uncertain linear time-varying systems, IEEE Trans. Automat. Control AC-41 (1996) 125-128. [2] B,R. Barmish, Stabilization of uncertain systems via linear control, IEEE Trans. Automat. Control AC-28 (1983) 848-850. [3] R,W. Brockett, Finite Dimensional Linear Systems (Wiley, New York, 1970). [4] M. Corless, Robust stability analysis and controller design with quadratic Lyapunov functions, in: A.S.I. Zinober, ed., Variable Structure and Lyapunov Control, Lectures Notes in Control and Information Sciences, Vol. 193 (1993) 181-203. [5] D. Hinrichsen, A. Ilchmann and A.J. Pritchard, Robustness of stability of time-varying linear systems, J. Differential Equations 82 (1989) 219-250. [6] D. Hinrichsen and A.J. Pritchard, Stability radius for structured perturbations and the algebraic Riccati equation, Systems Control Lett. 7 (1986) 1-10.
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1~ Amat o, A. Pironti / Systems & Control Letters 28 (1996) 103-114
[7] P.P. Khargonekar, 1.R. Petersen and K. Zhou, Robust stabilization of uncertain linear systems: quadratic stabilizability and , ~ control theory, 1EEE Trans. Automat. Control AC-35 (1990) 356-361. [8] D.J.N. Limebeer, B.D.O. Anderson, P.P. Khargonekar and M. Green, A game theoretic approach to ,h~ control tbr time-varying systems, S I A M J. Control Optim. 30 (1992) 262--283. [9] I.R. Petersen, A stabilization algorithm lbr a class of uncertain linear systems, Systems Control Lett. 8 (1986) 351-357. [10] G. Tadmor, Input/output norms in general linear systems, Int. J. Control 51 (1990) 911 921. [11] G. Yadmor, Worst-case design in time domain: the maximum principle and the standard .¢g~ problem, M C S S 3 (1990) 301 324. [12] G. Tadmor, Uncertain feedback loops and robustness in general linear systems, A utomatica 27 (1991) 1039-1042. [13] L. Xie, M. Fu and C.E. de Souza, ,¢? ~ control and quadratic stabilization of systems with parameter uncertainty via output feedback, 1EEE Trans. Automat. 6bntrol AC-37 (1992) 1253 1256.
Systems & Control Letters 28 (1996) 103-114
Quadratic stabilization with an )ff performance bound of uncertain linear-time-varying systems F. A m a t o * , A . P i r o n t i Dipartimento di Injormatica e Sistemistica, Universith degli Studi di Napoli "Federico H", via Claudio 21, 80125 Napoli, Italy
Received 10 July 1995; revised 14 December 1995
Abstract
In this paper we deal with linear-time-varying systems subject to uncertain parameters in the so-called one block, norm bounded form. We consider the problem of designing a controller which guarantees the quadratic stabilization of the closed-loop system while keeping the norm of the operator mapping the exogenus input to the controlled output under a prespecified level. Necessary and sufficient conditions for the existence of such a controller are given. Keywords: Linear systems; Time-varying systems; Uncertain systems; Quadratic stabilization; ,;¢~ optimal control
I. Introduction
It is well known that an unforced linear, time-invariant system subject to uncertainties is said to be quadratically stable if there exists a time-invariant quadratic Lyapunov function whose derivative along the trajectories of the system is negative definite for all values of the uncertain parameters in their bounding set (see [2]). In the same way a forced, uncertain system is said to be quadratically stabilizable via linear control if there exists a linear controller which makes the overall closed-loop system quadratically stable. In [13] to cope simultaneously with quadratic stability and robust performance the concept of quadratic stabilization with an ~ norm bound constraint has also been introduced. Necessary and sufficient conditions for quadratic stability, quadratic stabilizability and quadratic stabilizability with an ~ norm bound are known for systems subject to uncertainties in the so-called norm bounded, structured, one block form (see the tutorial paper [4] and the bibliography therein). All the involved conditions are expressed in terms o f suitable parametric algebraic Riccati equations as shown in [6, 7, 13]. Recently in [ 12, 5, 1] the situation in which the uncertainties are still in norm bounded, structured, one block form, but the matrices of the nominal system are time-varying, has been discussed. In this case the definition o f quadratic stability is naturally extended using time-varying quadratic Lyapunov functions. Necessary and sufficient conditions for quadratic stability and stabilizability have been found in [1]; these conditions are expressed in terms of parametric Riccati differential equations or in terms of a modified version of the Small Gain Theorem. Starting from the work done in [ 1], in this note we consider a particular ~,ug~ control problem for an uncertain, linear, time-varying system subject to norm bounded uncertainties. For this class of systems we consider * Corresponding author. Tel.: +39 81 7683513; fax: +39 81 7683186; e-mail: [email protected]. 0167-6911/96/$12.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PII S0167-691 1(96)0001 8-7
104
1~ Amato, A. Pironti/Svstems & Control Letters 28 (1996) 103 114
the problem of finding a linear controller which yields quadratic stability for the closed-loop system while keeping the norm of the operator mapping the exogenus input to the controlled output under a prespecified level. Our main contribution (Section 3) consists of necessary and sufficient conditions for the existence of such a controller both in the state and the output feedback case. This paper can be seen as the extension of [13] to the time-varying setting•
2. Notation and preliminaries We denote by , ~ t o (~c~lo) the space of the uniformly bounded, piecewise continuous (piecewise continuously differentiable), real matrix-valued functions defined on [to, +c~) and by 5~t2o the space of the real vector-valued functions which are square integrable on ( t o , + ~ ) . Given an exponentially stable linear time-varying system .i(t) = A ( t ) x ( t ) + B ( t ) u ( t ) ,
t E [t0, +cx~),
(la) (lb)
y(t) = C(t)x(t),
it uniquely defines the linear operator F('~ • $ 2,o ~_~ ~,,,, 2
u(.) ~ y(.) = _y, U ° u(.) = C(.) ft~+~ @(.,s)B(s)u(s)ds,
(2)
where q~ denotes the state transition matrix of (1). The Euclidean vector norm and the corresponding induced matrix norm are denoted by I' [, the usual norm in LP2t0 by I]" II, while [[Fyt~ll denotes the operator norm induced by the norm in £,et20 Given R : [ t 0 , + ~ ) H ~nx,, we write R > 0 (~>0) to mean that R is positive definite (semidefinite), i.e. that there exists ~ > 0 such that for all v E MI := {v E ~": Iv[ = 1} and for all t E [ t o , + ~ ) vTR(t)v>~
(vTR(t)v>~O).
Given two matrix-valued functions of the same dimensions R and S, the notation R > S (R>>-S) means that R - S is positive definite (semidefinite). Finally, the symbols R < (~<)0, and R < (~<)S have obvious meaning. In the sequel we shall use the following technical result.
Lemma 1. Let us consider system (1). Then the followin9 statements are equivalent (i) [l~f~[I < 7; (ii) the Riccati differential equation P(t) + AT(t)P(t) + P(t)A(t) + 7-2p(t)B(t)BT(t)P(t) + c T ( t ) C ( t ) + el = O
(3)
admits, f o r some constant e > O, a positive-definite solution P E ~cg~,; (iii) there exists a positive-definite P E ~cg)o such that [~ + ATp + PA + 7-2pBBTp + c T c < O.
Proof. See the appendix. Now let us consider the following linear, unforced, time-varying system subject to time-varying uncertainties a?(t) = [A(t)+ AA(t)]x(t), where x(t) E ~ , A E ~ 0 block form AA(t) = Fl(t)A(t)Ej(t),
t>~O,
(4)
is exponentially stable and the uncertain part is in the so-called structured, one (5)
[( Amato, A. PirontilSystems & Control Letters 28 (1996) 103-114
105
with F1 and E1 belonging to ~Cgo and A is an unknown matrix-valued function belonging to the class
:= {A : [ 0 , ÷ ~ ) ~
N*×JlA is Lebesgue measurable, AT(t)A(t)<~I Vt>~0}.
(6)
We shall refer to system (4) as an uncertain, linear, time-varying system, with uncertainties in norm bounded, structured, one-block form. Such a system is said to be quadratically stable if there exist a to ~>0 and a quadratic, time-varying Lyapunov function V(t,x) = xTp(t)x whose derivative along the solutions of (4) is negative definite for all A C @ (see [12, 1]). Using properties of quadratic forms we can state the following equivalent definition. Definition 1. System (4) is said to be quadratically stable if there exists a t0~>0 and a positive-definite ~ t o such that for all A E P C~
[~ + (A + FI AE1 )Tp + P(A + F1AEI ) < O. If system (4) is quadratically stable, directly from Definition 1 follows that the same system is exponentially stable for all A E 9 . In [1] it is shown that quadratic stability of system (4) also guarantees global uniform exponential stability of the closed-loop system
~(t)=A(t)x(t)+Ffft)~(t),
(7a)
2(t)=El(t)x(t),
(7b)
~(t)=A(~(t),t),
(7c)
where A is any continuous (possibly nonlinear) mapping such that IA(~,t)l ~<12[, for all ~ E ~J and for all t~>0. In the same paper necessary and sufficient conditions for quadratic stability of system (4) are given. We now introduce the concepts of "quadratic stability and stabilizability with an ~ norm bound y", extending to the time-varying setting the definitions given in [13], where the time-invariant case has been considered. Consider the uncertain system ,f(t) = [A(t) + AA(t)]x(t) + Bl(t)w(t) + [B2(t) + AB2(t)]u(t),
(8a)
z(t) = Cl(t)x(t) ÷ Dlz(t)u(t),
(8b)
y(t) = [C2(t) + dC2(t)]x(t) + D21(t)w(t) q- [D22(t) + AD22(t)]u(t),
(8c)
where
AC2(t)
ADz2(t)
= \F2(t)
A(t)(El(t) E2(t)),
(9)
u(t) C ~m and w(t) E •P are the control and the exogenus input, respectively, y(t) E ~l and z(t) E ~q are the measured and the controlled output, and Ba,C1,B2,D12, C2,D2bD22,F2,E2 belong to ~Cgo. Definition 2. Given to >/0 and V > 0, system (8) is said to be quadratically stable with an o ~ 7 in [to, +cxD) if there exists a positive-definite P c ~cg~0 such that for all A E
t 5 + [A +F1AEj]TP +P[A +F1AEI] + 7-2pB1BTp + cTICI < O.
norm bound
(10)
Note that, letting u = 0, Eqs. (8a) and (8b) define a family of input-output operators Tt°(A) depending on the realization of the uncertain matrix A. The next lemma justifies Definition 2.
F. Amato. A. Pironti/Systems & 6bntrol Letters 28 (1996) 103-114
106
Lemma 2. If system (8) is' quadratically stable with an ~ ically stable and I[ T_~,(A)[I < 7 fi )r all A C ~.
norm bound 7 in [t0,+cxz) then it & quadrat-
Proof. Since y-2pBIB~P + C(CI is positive semidefinite, from (10) follows that for all A E
P + [A + F1AEI ]Tp + P[A + FI AE~] < O,
(11 )
and hence the quadratic stability of system (8). Moreover, applying Lemma 1, Eq. (10) guarantees that II ~°~.(A)II < Y- [] Definition 3. Given to ~>0 and 7 > 0, system (8) is said to be quadratically stabilizable with an ~ norm bound y in [ t 0 , + ~ ) via linear state (output)feedback control if there exists a linear, finite-dimensional controller u = K ~ x (u -~ KuyY ) such that the overall closed-loop system is quadratically stable with an ,g,c~ norm bound y.
3. Main results In this section we provide necessary and sufficient conditions for quadratic stability and stabilizability with an o ~ norm bound 7- First we need the following preliminary lemma. Lemma 3. Given to >10 and 7 > 0, consider the operator Tj~(#) defined by the ,system
Yc(t) = A(t)x(t) + (y-IBl(t) #(t)Fl(t))~(t),
(12a)
£(t) = ( C,(t) ) x(t). \#-'(t)El(t)
(12b)
The following statements are equivalent: (i) there exists a positive-definite scalar Junction # E J)°~to such that
IIT~(#)ll < l; (ii) there exists a positive-definite scalar Junction It E , ~ t o such that the Riccati differential equation
P(t) + AT(t)P(t) + P(t)A(t) + 7-2p(t)Bl(t)B~(t)P(t) + C~(t)CT(t) +#-2(t)E~(t)El(t) + #2(t)P(t)Ffft)F[(t)P(t) + ,:1 = O
(13)
admits, Jor some constant ~: > 0, a positive-definite solution P C ' ~ ] 0 ; (iii) system (8) is' quadratically stable with an ~ norm bound 7 in [ t 0 , + ~ ) . ProoL See the appendix. Remark 1. The idea underlying Lemma 3 is the introduction of the scaling function #. It is important to note that, in this way, we are able to relate the quadratic stability with an jug~ norm bound y of system (8) to a small gain condition for the fictitious system (12). In the time-invariant case, that is when A, BI, etc., are constant matrices, Lemma 3 states that system (8) is quadratically stable with an Jcg~ norm bound 7 if and only if there exists a constant # > 0 such that the Riccati algebraic equation
ATp + PA + y-2PB1B~P + CXlC1 + #-2E~E1 + Ft2pF1F~P + e l = 0
(14)
admits, for some e, > O, a positive-definite solution P. In the same case a necessary and sufficient condition for quadratic stability (without the J g ~ norm bound constraint) is the existence of a constant # > 0 such
F. Amato, A. PirontilSystems & Control Letters 28 (1996) 103-114
107
that the equation
ATp + PA + p-2EXlE1 + p2pF1F[P + ~I = O
(15)
admits a positive-definite solution P. The change of variable/5 = pZp in (15) shows that for quadratic stability it is necessary and sufficient to solve Eq. (15) for p = 1, avoiding in this way the search for the parameter p. One could ask if a similar property holds for Eq. (14). Unfortunately, the answer to this question is in the negative; indeed, the above change of variable does not work in this case. To convince ourselves definitely about this point, note that, according to L e m m a 3(i), a necessary and sufficient condition for quadratic stability with an , ~ norm bound 7 is the existence of p > 0 such that ,[
to r~o(p)ll =
(
7-'C'(sI-A)-IB'
pCI(sI-A)-'F')
p-17-1El(sI - A ) - l B l
~
El(sI -A)-tFj
< 1.
(16)
Now from (16) follows that, in general, II rz:;(1)ll # II r~°(~)ll for p # 1. It is simple to build examples for which [[ T ~ ( p ) [ I < 1~< HT~°(1)I[ for some p # 1; the conclusion is that, in the time-invariant case, the introduction of a constant scaling factor p in the condition for quadratic stability with the .Zf ~ norm bound is necessary. Now we will show, by a simple example, that in the time-varying case it may be necessary that such a scalar factor be a function of time (as stated in Lemma 3). E x a m p l e 1. Consider the scalar system 2(t) = ( - 2 +
fl(t)f(t)el(t))x(t) + w(t),
(17a)
z(t) = x(t),
(17b)
with fl(t)=l,
el(t)=0,
tE[2nT,(2n+l)T[, n = 0 , 1 .... ,
(18a)
fl(t)=0,
el(t)=4,
tE[(2n+l)T,2(n+l)T[,
(18b)
n=0,1 .....
where T = 1; furthermore, assume that 7 = 1 and to = 0. Since
IIr°,,ll = I I ( s + 2 ) - ' l l ~
=
~1 < 1,
(19)
from Lemma 1 follows the existence of a positive-definite p E ~cgl which satisfies
p+2ap+7-Zb2p 2 + c ~ = p - 4 p +
p2 + 1 < 0.
(20)
Since Aa(t) = fl(t)6(t)el(t) is identically zero we can conclude that condition (10) in Definition 2 is satisfied and therefore the system (17) is quadratically stable with an . ~ norm bound 7 = 1 in [ 0 , + c c ) . The following result, is proved in the appendix. Fact 1.
Let us define, according to Lemma 3, the operator T°~(p) associated to system
2(t) = - 2 x ( t ) + (1
pfl(t))~(t),
(21a)
-~(t) =
x(t);
(21b)
p-tel(t)
then inf,,>0 II ~°~(p)ll > 1. Fact 1 shows that, although system (17) is quadratically stable with an ~ norm bound 7 = 1, there does not exist a positive constant scaling factor which satisfies condition (i) in Lemma 3. On the other
1:( Amato, A. Pironti/Systems & Control Letters 28 (1996) 103-114
108
hand, the same lemma guarantees the existence of a time-varying scaling factor p which satisfies condition (i) (or equivalently (ii)). Indeed since (19) holds, from Lemma 1 follows that there exists a positive-definite p E ~cg~ which satisfies (20). Now, following continuity arguments, one can find a sequence of sufficiently small positive numbers #n and a sequence of sufficiently big positive numbers /7n such that for all integer n p-4p+p2+l+p_~p
2 <0,
D-4p+p2+l+fi22
< O,
tE[2nT,(2n+l)T[, tE[(2n+l)T,2(n+l)T[.
(22) (23)
Obviously, the piecewise constant function {-P-n if t E [ 2 n T , ( 2 n + l ) T [ , #(t):=
fin
if t c [ ( 2 n + l ) T , 2 ( n + l ) T [ ,
(24)
is such that [ ~ - 4 p + p2 + l +t~ -2 +I~2p 2 < 0.
(25)
Using again Lemma 1 the existence of a positive-definite p satisfying the Riccati equation in condition (ii) of Lemma 3 follows. The following theorems give necessary and sufficient conditions for quadratic stabilizability of system (8) with an ,X~ norm bound 7. First we consider the state feedback case. Theorem 1. Given to >~0 and 7 > O, consider the system 2(t) = A(t)x(t) + (7-1Bt(t) #(t)Fl(t))r?(t) + Bz(t)u(t),
(26a)
(Cl(t) ~ ~ D12(t) ) 5(t) = \ t ~ - l ( t ) E x ( t ) J x(t) + \ # -l(t)E2(t) u(t).
(26b)
Let Z~°(t~) be the closed-loop operator obtained by applyin9 a linear, finite-dimensional state feedback control in the Jorm u = K~xx; then system (8) /s quadratically stabilizable with an J f ~ norm bound 7 in [to, +oc) via linear state feedback control if and only if there exist a positive-definite scalar function ] < 1. In this case K is a /~ C ~cgt0 and a memoryless operator Kux = K E ~cgto such that ]]Z'e~(/~)t to compensator which quadratically stabilizes system (8) with an ~ norm bound 7. Proof. Let p(t) = J(t)p(t) + L(t)x(t),
(27a)
u(t) = M(t)p(t) + K(t)x(t),
(27b)
a state space representation of the compensator K~. The feedback connection of system (8) and (27) gives the closed-loop system 2c(t) -- [Ac(t) + AAc(t)]Xc(t) + Bic(t)w(t),
(28a)
z(t) = C~c(t)Xc(t),
(28b)
F. Amato, A. PirontiISystems & Control Letters 28 (1996) 103-114
109
where Xc = (x r pV)7, and Ac(t) = ( A ( t ) +B2(t)K(t)
L(t)
B2(tlM(t)~ J(t) J '
AAc(t) = Fc(t)A(t)Ec(t),
Fc(t) = ( F ~ t ) ) , Ec(t) = (El(t) + E2(t)g(t) E2(t)M(t)),
C~c(t) = (Cl(t) + Dlz(t)K(t) Dlz(t)m(t)). From Lemma 3 it follows that system (28) is quadratically stable with an Jut~ norm bound ~' in [to, + e c ) if and only if there exists a positive-definite scalar function /~ E ~cgt 0 such that the input-output operator associated to the system
ic(t) = Ac(t)Xc(t) + (7-1Blc(t) #(t)Fc(t))~(t), (C'c(t) 2(t) =
)xc(t )
(29a) (29b)
~-~(t)Ec(t)
has norm less than unity. Now, considering that system (29) is also a state space representation of the feedback connection of system (26) and Ku~, we can conclude that system (8) is quadratically stable with an ~ norm bound 7 if and only if there exists a finite-dimensional system K~ such that IIZe~(#)ll to < 1. This is an ~¢go~ control problem with state feedback for which we can search, without loss of generality, for memoryless controllers (see [11]). [] The next theorem focuses on the output feedback case. Theorem 2. Given to >~0 and 7 > O, consider the system
2(t) = A(t)x(t) + (7-1Bl(t) It(t)Fl(t))~(t) + B2(t)u(t), Z(t)= (Cl(t))x(t)+
IL-~(t)E1 (t)
(Dl:(t))u(t), I~-I(t)E2(t)
y(t) = C2(t)x(t) + (7-1D21(t) lt(t)F2(t))v~(t) + D22(t)u(t).
(30a) (30b) (30c)
to Let Se~,(# ) be the closed-loop opertor obtained by applying a linear, finite-dimensional output feedback control in the form u = K u y Y ; then system (8) is quadratically stabilizable with an ~ norm bound 7 in [to, + o c ) via linear output feedback control if and only if there exist a positive-definite scalar function p E ~ t o and a K,y such that IlSz'-~-(~)[I < 1. In this case a minimal realization of K,y (see [3]) is a compensator which quadratically stabilizes system (8) with an , ~ norm bound 7.
Proof. Let
p(t) = J(t)p(t) + L(t)y(t),
(31a)
u(t) = M(t)p(t),
(31b)
a state space representation of the compensator Kuy. The feedback connection of system (8) and (31) gives the closed-loop system 2c(t) = [Ac(t) + AAc(t)]Xc(t) + Blc(t)w(t),
(32a)
110
IY Amato, A. Pironti/Svstems & Control Letters 28 (1996) 103-114 z(t) = Cic(t)Xc(t),
(32b)
where xc = (x v pT)T, and
/
A(t)
Ac(t) = !
\ L(t)O2(t)
B2(t)M(t) J(t) + L(t)D22(t)M(t) J '
AAc(t) = Fc(t)d(t)Ec(t), Fc(,) = ( F , ( t ) k L(t)F2(t) ) ' Ec(t) = (Ej(t) E2(t)M(t)),
( Bl(t) ) B i t ( t ) = \ L(t)Dzl(t)
'
Clc(t) = ( G ( t ) Dlz(t)M(t)). Then the proof continues as in Theorem 1. Note that in (31) we focus on strictly proper compensators; this does not cause any loss of generality because, as shown in [8, 1 1], in an , ~ control problem the existence of a proper controller guarantees the existence of a strictly proper one. []
Remark 2. It is important to note that Theorems 1 and 2 transforms the quadratic stabilizability with an Jt ° ~ norm bound 7 problem into an equivalent ~ , ~ problem, which can be solved via existing techniques (see U1,8]).
4. Conclusions In this paper we have considered linear systems subject to uncertainties in norm bounded, structured, one block form. The novelty consists in the fact that the nominal system and the uncertainties scaling matrices are time varying. Necessary and sufficient conditions for quadratic stability and quadratic stabilizability with the simultaneous satisfaction of an , ~ performance bound, between the exogenus input and the controlled output, have been provided in terms of equivalent conditions involving the Small Gain Theorem and the solvability of a certain Riccati differential equation.
Appendix
Proof of L e m m a 1. The equivalence between proposition (i) and (ii) is stated in the main theorem of [10]. to (i) =~ (iii) If ]]Fvull < ~ then the Riccati differential equation (3) admits a positive-definite solution P E , ~ ? ~ for some r, > 0. In this case we have that, in [to, + ~ ] , [~ + ATp + PA + 7 - 2 p B B T p + CTC < O. (iii) =¢- (i) Let
S(t) = -([~(t) + AZ(t)P(t) + P(t)A(t) + 7-2p(t)B(t)BT(t)P(t) + cT(t)C(t) + eI), where e, > 0 is such that S > 0. In this way the matrix-valued function P satisfies the Riccati differential equation
[~(t) + AY(t)P(t) + P(t)A(t) + ~-2p(t)B(t)BT(t)P(t) + c T ( t ) C ( t ) + sl/2(t)SI/2(t) + el = O.
E Amato. A. Pironti/Systerns & Control Letters 28 (1996) 103-114
111
^t0 defined by the system Due to the equivalence between proposition (i) and (ii), the linear operator F~u, :f(t) = A ( t ) x ( t ) + B(t)u(t),
t E [to, +cxD)
(c(t) "~ s12(t) ) x(t) has norm less than 7. Hence, for all u E L~a2, we have to < z'. [] that Ilry,,ll
IlYll < v,llull, and, since IlY[I~< IlYlI, we can conclude
In order to prove Lemma 3 we need the following result which has been proven in [l]; for the selfcontainedness of the paper we repeat the proof here. Fact 2. Given X, Y , Z belonging to . ~ t o and such that, .for some ?) > 0, X>~0,
Y < 0,
Z>~0,
(A.1)
( J Y ( t ) v ) 2 -4vXX(t)vvXZ(t)v>>-6
Vv E ~1, Vt E [ t 0 , + ~ ) ,
(A.2)
then there exists a positive-definite scalar function # E "@%o satL~fyin,q #2X + ItY + Z < O. Proof. Let
a(v,t) := vTX(t)v;
b(v,t) := vTy(t)v;
c(v,t) := vTZ(t)v.
(A.3)
Since X and Z are uniformly bounded, there exists 7 > 0 such that
Ix(t)l ~<'/,
Iz(t)l ~<7,
Vt E [t0,+o~);
then set gg()~,v, t) := ,~2a(v,t) + 2b(v,t) + c(v,t) + e,
(A.4)
where 0 < ~: < 6/47. From (A.1) there exists ~ > 0 such that
a(v,t)>lO,
b(v,t)<. - ct,
c(v,t)>~O
V(v,t) E ~1 × [t0,+oc).
(A.5)
Now observe that Eq. (A.2) implies
A(v,t) := bZ(v,t) -4a(v,t)[c(v,t)+e]>Jfi - 4 7 e > 0
V(v,t) E ;~1 × [ t 0 , + ~ ) ,
(A.6)
from which follows that the second-order equation in the variable )~, ,q(2, v, t) = 0, admits two real distinct solutions 21(v,t) and 22(v,t), with 21(v,t) < 22(v,t), for all (v,t) E ~1 × [t0,+oo). Moreover, by applying Cartesius rule of sign of solutions and recalling (A.5), we can conclude that
0 < )~l(v,t) < )~2(v,t)
V(v,t) E ,~l × [t0,+oc).
(A.7)
We claim that ).l(V,t)
< 2 < 22(v,t) 4:~ ,q()~,v,t) < 0
V(v,t) E ~1 × [t0,+OO).
(A.8)
Indeed, for fixed v and t, the graph of y(2) = `q(2, v, t) in the plane (2, y) is either a concave parabola (when a(v, t) ¢ 0) with vertex ( - b ( v , t)/2a(v, t), - ( b 2 ( v , t ) - 4a(v, t)(c(v, t) + ~,))/4a(v, t)) and intersections in 21 (v, t) and )~2(v,t) on the 2-axis, or a straight line (when a(v, t) = 0) with negative slope and intersection in 21 (v, t) on the 2-axis (in this latter case 22(v,t) = + ~ ) . In both cases (A.8) is proved.
F Amato, A. Pironti/Systems & Control Letters 28 (1996) 103-114
112
Now from Lemma 3.2 in [9], for a fixed t there exists ,~ such that 9(2, v, t) < 0 for all v E ~ l . By virtue of (A.8) we can conclude that ~, is such that
),l(V,t) < ,~ < 22(v,t)
Vv E .~1,
(A.9)
or equivalently 0 < A l ( t ) : = maxAl(v,t) < ,~ < rain 2 2 ( v , t ) = : A~(t). vG,~)
(A. 10)
vE~l
Now observe that, since X, Y and Z are piecewise continuous, it is possible to find a sequence of intervals {J,.} such that U,~l J i = [t0,+oc) and a(v,.), b(v,.) and c(v,.) are continuous on each interval .h. From this follows that ),l(v, .) is continuous on each of these intervals and in turn, being ~1 a compact set, A1 is continuous. We can conclude that the function AI is piecewise continuous on [to, +cx~). Now pick any positive-definite scalar function /~ E ,~'~gt0 such that for all t i> to A1 (t) ~
(A. 11 )
Note that # can be chosen (i) uniformly bounded because Ai is, being Al(t)~<2(y + a)/~ for all t E [t0,+oc); (ii) piecewise continuous because A I is; (iii) positive definite because A2 is, being A2(t)~>a/2y for all t E [to, +oo). From (A.8), (A.11) and (A.4) it follows that
#2(t)a(v,t) + #(t)b(v,t) + c(v,t)<~ - e < 0 and hence the proof of the lemma.
V(v,t) E ~ i x [t0,+cx~),
(A.12)
[]
P r o o f of L e m m a 3. The equivalence between (i) and (ii) follows directly from Lemma 1. (ii) =¢, (iii) Suppose that, for some s, > 0 and some definite positive scalar function # E ~cgto, there exists a positive-definite solution of Eq. (13). From
(p-~ AEI - pFT p ) T ( # - ' AEI - #FT p)>JO
(A.13)
it follows that
ET ATFTp @ PF1AEI <<,p2pF1FTp + 11-2ETA T AEI <~ t~EpF1FTp + t~-2ETE1,
(A.14)
therefore we have that for all A E
/6 + [A + F1AE~]T P + P[A + F~AE~] + cT c~ + 7 EpB~BT P < 0. (iii) =:~ (ii) Suppose there exists a positive-definite P E ; ~ o
(A.15)
satisfying the matrix inequality
/6 + [A + FIAE1]T P + P[A + F1AEI] + cT cI + 7-zPB1BT p < 0
(A. 16)
for all A E ~ . Now let us define L : = / 6 + ATp + PA + cTc~ + ~' 2pB~BTp < 0.
(A.17)
From (A.16) it follows that for all A E ~ , for all v E Ml, for all t E [t0,+oo) and for some ~ > 0
2vT P(t)Fl(t)A(t)El(t)v < --vT L(t)v -- 7;
(A.18)
on the other hand, since the set ~ is symmetric around zero, A E ~ implies - A E ~ , hence from (A.18) follows
--2vT p(t)F1 (t)A(t)E1 (t)v < --vT L(t)v -- ~.
(A. 19)
F. Amato, A. PirontilSystems & Control Letters 28 (1996) 103-114
113
Eqs. (A.18) and (A.19) imply that for all v E M1, for all t E [t0,+c~) and for some c( > 0
(vT L(t)v) 2 >14max{(vTp(t)Fl(t)A(t)El(t)v) 2} + ~' = 4vTp(t)F1 (t)FT(t)e(t)vvTE~(t)El (t)v + ~',
(A.20)
where the last equality is proved in L e m m a 3.1 in [9]. From (A.20), using Fact 2, it follows that there exists a positive-definite function # E ~c~t o such that
#4pFIF~P + #2L + E~El < 0.
(A.21)
Now dividing by #2(0 we obtain
[~ + ATp + PA + cTc1 + ?-2pBIB~P + #2PF1FTp + #-2ETE, < 0.
(A.22)
From (A.22) and using L e m m a 1 the thesis follows.
Proof of Fact 1. First note that [I ~°,(#)t[ :=
]1Te°~(#)wl[ >~II T~(U)W o • I[ ,
sup
wcz0~_(0~
Ilwll
(1.23)
[[w*ll
where w* :=
{(4
--
0
for t E [0, 1), for t~> 1.
(A.24)
Since
IIw*ll
:=
w*T(t)w*(t)dt
= 1
(1.25)
from (A.23) it follows that
II T°(#)ll >/ql T°~(#)w* II•
(1.26)
Now a simple but cumbersome calculation shows that
II T°,(#)w* II =
[0.0710(1 + #)2 + 0.3664(1 + #-1)211/2 := g(#).
(A.27)
The minimum of g is attained at # = 1.728 and is equal to 1.2007. This last observation, together with (A.26), proves our claim.
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