- Email: [email protected]
The Strict Bounded Real Lemma for Linear Time-Varying Systems
Journal of Mathematical Analysis and Applications 244, 120᎐132 Ž2000. doi:10.1006rjmaa.1999.6693, available online at http:rrwww.idealibrary.com on
The Strict Bounded Real Lemma for Linear Time-Varying Systems1 Wanyi Chen College of Mathematics, Nankai Uni¨ ersity, Tianjin 300071, People’s Republic of China
and Fengsheng Tu2 Department of Computer and System Sciences, Nankai Uni¨ ersity, Tianjin 300071, People’s Republic of China Submitted by Harold L. Stalford Received December 6, 1995
In this paper, we present a strict bounded real lemma for linear time-varying systems in the infinite-horizon case. Using some operator methods, we show that the strict bounded realness for the related IrO operators is equivalent to the solvability of a semidefinite or definite Riccati equation. We also apply this result to the problem of disturbance attenuation and H⬁ -optimization. All our results include current ones in the literature for linear time-invariant systems. 䊚 2000 Academic Press
Key Words: time-varying systems; IrO operator; Riccati equation; H⬁ -norm.
1. INTRODUCTION As it is well known, the bounded real lemma is a classical result in linear system theory; see w3x. More recently, it has been included in the famous H⬁ control theory and has been featured as a powerful tool in the problem of disturbance attenuation and H ⬁-optimization. For linear time-invariant systems, its infinite dimensional version has been obtained by Curtain w3x for systems generated by C0-semigroups, and its finite dimensional version was obtained by Doyle et al. w4x and Khargonekar et al. w7x. For linear 1 2
This work is supported by the National Natural Science Foundation of China. E-mail: [email protected]. 120
0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
121
time-varying systems, Tadmor w11x utilized the classical maximum principle and gave a characterization in terms of solvability of an indefinite Riccati equation. Our purpose in this paper is to generalize these results for finite dimensional linear time-varying systems for the infinite-horizon case. By some operator techniques, we present more necessary and sufficient Riccati equation type conditions. From this, we also obtain a result for disturbance attenuation in the sense of H⬁ norms. In this paper, we will study linear time-varying systems in the form
⌺0
¡˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . q E Ž t . uŽ t . ¢x Ž 0. s 0, t G 0,
s~ z Ž t . s C Ž t . x Ž t . q D Ž t . w Ž t . q F Ž t . u Ž t .
where x Ž t . g R n, uŽ t . g R m , w Ž t . g R q, and z Ž t . g R p are, respectively, the state, the control input, the disturbance input Žor the exogenous input., and the controlled output. Let Rqs w0, q⬁., AŽ⭈. g L⬁n= n Ž Rq. , B Ž⭈. g L⬁n= q Ž Rq. , C Ž⭈. g L⬁p= n Ž Rq. , DŽ⭈. g L⬁p= q Ž Rq. , EŽ⭈. g L⬁n= m Ž Rq. , and F Ž⭈. g L⬁p= m Ž Rq. . Throughout the paper, 5 ⭈ 5 will represent the Euclidean norm for vectors and the induced norm for matrices. Setting uŽ t . ' 0, we obtain the unforced form of ⌺ 0 ⌺1
¡˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . ¢x Ž 0. s 0.
s~ z Ž t . s C Ž t . x Ž t . q D Ž t . w Ž t .
From Ravi et al. w9x, we know that when the homogeneous system ˙ xŽ t . s AŽ t . x Ž t . is exponentially stable; in brief, AŽ⭈. is stable. Then ⌺ 1 generates a bounded input᎐output operator Tz w : Lq2 Ž Rq. ª L2p Ž Rq. , Tz w Ž w Ž ⭈ . . Ž t . s
t
H0 C Ž t . ⌽ Ž t , s . B Ž s . w Ž s . ds q D Ž t . w Ž t . ,
where w Ž⭈. g Lq2 Ž Rq. , and ⌽ Ž t, s . is the state transition matrix of the homogeneous system
˙x Ž t . s A Ž t . x Ž t . . One of our purposes in this paper is to give some characterizations for the estimation of the norm of Tz w . In ⌺ 0 , if all the states are available for feedback, we can design either static or dynamic state feedback controllers. First, we consider the case of static state feedback uŽ t . s K Ž t . x Ž t . ,
Ž 1.1.
122
CHEN AND TU
where K Ž⭈. g L⬁m= n. Let TzKw denote the input᎐output operator for the closed-loop system of ⌺ 0 with Ž1.1., and define
␥s s inf 5 TzKw 5 : K Ž ⭈ . g L⬁m= n Ž Rq . , A Ž ⭈ . q E Ž ⭈ . K Ž ⭈ . is stable 4 . Ž 1.2. Alternatively, for the dynamic state feedback ⌺d s
½
¨˙Ž t . s G Ž t . ¨ Ž t . q H Ž t . x Ž t .
uŽ t . s LŽ t . ¨ Ž t . q M Ž t . x Ž t . ,
where GŽ⭈. g L⬁r=r Ž Rq. , H Ž⭈. g L⬁r=n Ž Rq. , LŽ⭈. g L⬁m= r Ž Rq. , and M Ž⭈. g L⬁m= n Ž Rq. . The closed-loop system of ⌺ 0 with ⌺ d is in the form
¡X˙Ž t . s
⌺2 s
~
AŽ t . q E Ž t . M Ž t . HŽ t.
zŽ t. s CŽ t. q FŽ t. MŽ t.
¢
EŽ t . LŽ t . BŽ t. XŽ t. q wŽ t. GŽ t . 0 F Ž t . LŽ t . X Ž t . q D Ž t . w Ž t . ,
X Ž 0 . s 0.
Similar to Khargonekar et al. w8x, we define
½
D s Ž G, H , L, M . g L⬁r=r Ž Rq . = L⬁r=n Ž Rq . = L⬁m= r Ž Rq . =L⬁m= n Ž Rq . : AŽ ⭈. q E Ž ⭈. M Ž ⭈. H Ž ⭈.
E Ž ⭈. L Ž ⭈. G Ž ⭈.
is stable
5
and let Tzdw be the IrO operator of ⌺ 2 . Moreover, we define
␥d s inf 5 Tzdw 5 : Ž G, H , L, M . g D 4
Ž 1.3.
It is easy to see that for any natural number r,
␥s s inf 5 Tzdw 5 : Ž G, H , L, M . g D l 0 4 = 0 4 = 0 4 = L⬁m= n Ž Rq . 4 G ␥d We will show that ␥s s ␥d , which generalizes the result of Khargonekar et al. to the linear time-varying case.
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
123
2. MAIN RESULTS THEOREM 2.1 ŽThe Strict Bounded Real Lemma.. For system ⌺ 1 , if DŽ t . ' 0, then the following conditions are mutually equi¨ alent: Ž1. The homogeneous system ˙ x Ž t . s AŽ t . x Ž t . is exponentially stable and 5 Tz w 5 - 1. Ž2. There exists a bounded and continuous function QŽ⭈.: Rqª R n=n such that QŽ⭈. G 0, ᭙ t g Rq, and following the Riccati equation
˙Ž t . q Q Ž t . AŽ t . q A⬘ Ž t . Q Ž t . q Q Ž t . B Ž t . B⬘ Ž t . Q Ž t . q C⬘ Ž t . C Ž t . Q s0 holds for almost all t g Rq. Moreo¨ er, the homogeneous system ˙ xŽ t . s w A q xŽ . Ž . BB⬘Q t x t is exponentially stable. Ž3. There exist a Lyapuno¨ matrix function P Ž⭈.: Rqª R n=n and a positi¨ e constant ␣ 0 such that the following Riccati inequality holds: P˙Ž t . q P Ž t . A Ž t . q A⬘ Ž t . P Ž t . q P Ž t . B Ž t . B⬘ Ž t . P Ž t . a.e., t g Rq.
q C⬘ Ž t . C Ž t . F y␣ 0 In ,
Here the Lyapuno¨ matrix function refers to bounded uniformly positi¨ edefinite function on Rq Ž see Rotea and Khargonekar w10x., and In denotes the n = n identity. x Ž t . s AŽ t . x Ž t . is exponentially stable and 5 Tz w 5 LEMMA 2.2. In ⌺ 1 , if ˙ 1, then 5 DŽ⭈.5 ⬁ - 1. LEMMA 2.3. In ⌺ 1 , AŽ⭈. is stable, and the induced IrO operator has norm 5 Tz w 5 - 1 if and only if the following conditions are true: Ž1. Ž2.
5 DŽ⭈.5 ⬁ - 1, and A1Ž⭈. is stable. The induced IrO operator by the system ⌺ r is strictly contracti¨ e ⌺r s
½
˙x Ž t . s A1 Ž t . x Ž t . q B1 Ž t . w Ž t . z Ž t . s C1 Ž t . w Ž t . , x Ž 0 . s 0;
here A1 Ž t . s A q B Ž Iq y D⬘D . B1 Ž t . s B Ž Iq y D⬘D . C1 Ž t . s
Ž Ip y DD⬘ .
1 2
y 12
y1
Ž t.
C Ž t. .
C Ž t.
124
CHEN AND TU
From this lemma, we obtain the following result which gives a characterization for the IrO operator of ⌺1 and has potential usage in disturbance attenuation and H ⬁-optimization. THEOREM 2.4. equi¨ alent:
For system ⌺ 1 , the following conditions are mutually
Ž1. The homogeneous system ˙ x Ž t . s AŽ t . x Ž t . is exponentially stable and 5 Tz w 5 - 1. Ž2. 5 DŽ⭈.5 ⬁ - 1. There exists a bounded and continuous function QŽ⭈.: Rqª R n=n such that QŽ t . G 0, ᭙ t g Rq, and the Riccati equation
˙Ž t . q C⬘ Ž t . C Ž t . q Q Ž t . AŽ t . q A⬘ Ž t . Q Ž t . Q q Ž QB q C⬘D . Ž t . Ž Iq y D⬘D .
y1
Ž t . Ž B⬘Q q D⬘C . Ž t . s 0
holds for almost all t g Rq. Moreo¨ er, w A q B Ž Iq y D⬘D .y1 Ž B⬘Q q D⬘C .xŽ⭈. is stable. Ž3. 5 DŽ⭈.5 ⬁ - 1. There exist a Lyapuno¨ matrix function P Ž⭈.: Rqª n= n R and a positi¨ e constant ␣ 1 such that following Riccati inequality holds for almost all t g Rq: P˙Ž t . q C⬘ Ž t . C Ž t . q P Ž t . A Ž t . q A⬘ Ž t . P Ž t . q P Ž t . B Ž t . q C⬘ Ž t . D Ž t .
Iq y D⬘ Ž t . D Ž t .
y1
= B⬘ Ž t . P Ž t . q D⬘ Ž t . C Ž t . F y␣ 1 In . From this theorem, we can obtain the following result on disturbance attenuation, which shows that dynamic state feedback offers no advantage over static state feedback in the minimization of the H⬁ norm of the IrO operator of the closed-loop system. Also, this result is a natural generalization of Khargonekar et al. w8x to the time-varying case. THEOREM 2.5. Consider the dynamical system ⌺ 0 and the state feedback controllers Ž1.1. and ⌺ d . Define ␥s and ␥d as in Ž1.2. and Ž1.3., respecti¨ ely; then ␥s s ␥d . 3. PROOFS Proof of Theorem 2.1. Ž1. « Ž2. This can be finished by a similar method used in the proof of w9, Theorem 3.1x. Thus, we omit it.
125
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
Ž2. « Ž1. First, we show that ˙ x Ž t . s AŽ t . x Ž t . is exponentially stable. Let ⌽ Ž t, s . denote its state transition matrix, 0 F s F t - q⬁, and let ⌿ Ž t, s . denote the state transition matrix generated by Ž A q BB⬘Q .Ž⭈.; then ⌽Ž t, s. s ⌿Ž t, s. y
t
Hs ⌿ Ž t , . B Ž . B⬘ Ž . Q Ž . ⌽ Ž , s . d .
Thus for any x g R n, d² ⌽ Ž t , s . x, Q Ž t . ⌽ Ž t , s . x : dt s ² A Ž t . ⌽ Ž t , s . x, Q Ž t . ⌽ Ž t , s . x : q ² ⌽ Ž t , s . x, Q Ž t . A Ž t . ⌽ Ž t , s . x :
˙Ž t . ⌽ Ž t , s . x : q² ⌽ Ž t , s . x, Q s y5 B⬘ Ž t . Q Ž t . ⌽ Ž t , s . x 5 2 y 5 C Ž t . ⌽ Ž t , s . x 5 2 . For any t 1 ) s G 0, integrating from s to t 1 and noting that ⌽ Ž s, s . s In , we obtain t1
Hs 5 B⬘ Ž . Q Ž . ⌽ Ž , s . x 5
d F ² Ž Q Ž s . x, x . F  5 x 5 2 .
2
Here  is chosen by the assumption that QŽ⭈. is bounded on Rq; hence, there exists a  ) 0 such that QŽ s . F  In , for all s g Rq. However, the system ˙ x Ž t . s Ž A q BB⬘Q .Ž t . x Ž t . is exponentially stable, so by Brocket w1x, there exist positive constants m 0 , m1 , and m 2 such that q⬁
Hs
5 ⌿ Ž t , s . 5 2 dt F m 0 ,
q⬁
Hs
5 ⌿ Ž t , s . 5 dt F m1 ,
t
H0 5 ⌿ Ž t , . 5 d F m
2,
᭙s G 0, ᭙s G 0, ᭙ t G 0.
Hence, q⬁
žH s
5 ⌽ Ž t , s . x 5 dt
F
2
q⬁
žH s
1 2
/
5 ⌿ Ž t , s . x 5 dt 2
q m1 m2 ⭈ 5 B Ž ⭈ . 5 ⬁ 1 2
1 2
1 2
/ q⬁
žH s
5 B⬘ Ž t . Q Ž t . ⌽ Ž t , s . x 5 2 dt
F Ž m 0 q m1 m2 5 B Ž ⭈ . 5 ⬁  . 5 x 5 . 1 2
1 2
1 2
1 2
/
126
CHEN AND TU
Therefore from Gurov and Tadmor w5x, the homogeneous system ˙ xŽ t . s AŽ t . x Ž t . is exponentially stable. Second, we will show that 5 Tz w 5 - 1. For any T ) 0, consider the related system in w0, T x, T
H0 Ž z⬘z y w⬘w . dt T
½
s
H0
z⬘z y w⬘w q
s
H0 x⬘
T
d Ž x⬘Qx . dt
5
dt y x⬘ Ž T . Q Ž T . x Ž T .
˙ q QA q A⬘Q q QBB⬘Q q C⬘C x dt Q Ž 3.1.
T
H0 w w y B⬘Qx x ⬘ w w y B⬘Qx x dt y x⬘ Ž T . Q Ž T . x Ž T .
y
T
H0 w w y B⬘Qx x ⬘ w w y B⬘Qx x dt y x⬘ Ž T . Q Ž T . x Ž T .
sy
T
H0 5 w y w*5
Fy
2
dt,
where w*Ž t . s yB⬘Ž t . QŽ t . x Ž t ., t g Rq. Letting T ª q⬁ in the previous inequality, we have q⬁
H0
q⬁
Ž z⬘z y w⬘w . dt F yH
Ž w y w* . ⬘ Ž w y w* . dt.
0
Suppose L is the IrO operator of the system
˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . Ž w y w* . Ž t . s y Ž B⬘Q . Ž t . x Ž t . q w Ž t . ,
x Ž 0. s 0
which maps w to w y w*. Since Ly1 exists as Ly1 s
½
˙x Ž t . s Ž A q BB⬘Q . Ž t . x Ž t . q B Ž t . Ž w y w* . Ž t . w Ž t . s Ž B⬘Q . Ž t . x Ž t . q Ž w y w* . Ž t . , x Ž 0. s 0
and Ž A q BB⬘Q .Ž⭈. is stable, Ly1 is also a bounded operator on Lq2 Ž Rq. . Also, w s Ly1 Ž w y w*., q⬁
žH
0
1 2
w⬘w dt
/
F 5 Ly1 5
q⬁
žH
0
1 2
Ž w y w* . ⬘ Ž w y w* . dt .
/
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
127
Combining this with Ž3.1., we have q⬁
H0
q⬁
Ž z⬘z y w⬘w . dt F yH
Ž w y w* . ⬘ Ž w y w* . dt
0
Fy
1
q⬁
y1 5 2
5L
H0
w⬘w dt.
Hence q⬁
H0
z⬘z dt F 1 y
ž
5 Tz w 5 F 1 y
ž
1 5 L5
/
2
q⬁
H0
w⬘w dt
1 2
1 5 Ly1 5 2
- 1.
/
Ž3. « Ž1. Note that z Ž t . s C Ž t . x Ž t . and C Ž⭈. is essentially bounded on Rq; therefore, there exists a positive constant , such that C⬘C F In . Thus for any T ) 0, T
T
T
H0 z⬘z dt s H0 x⬘C⬘Cx dt F H0 x⬘ x dt.
Ž 3.2.
Also under the given assumption, P Ž⭈.: Rqª R n=n is a Lyapunov matrix function for ˙ x Ž t . s AŽ t . x Ž t .; hence, it is exponentially stable. For T ) 0, by the given Riccati equation and Ž3.2. used, we have T
H0 Ž z⬘z y w⬘w . dt s yx⬘ Ž T . P Ž T . x Ž T . T
q
H0
s
ž
z⬘z y w⬘w q
d Ž x⬘Px . dt
/
dt
T
H0 x⬘ P˙ q A⬘P q PA q PBB⬘P q C⬘C 4 x dt y x⬘ Ž T . P Ž T . x Ž T . y
F y␣ 0
␣0
T
T
H0 w w y B⬘Px x ⬘ w w y B⬘Px x dt T
H0 x⬘ x dt F y H0 z⬘z dt.
Let T ª q⬁, we obtain q⬁
H0
z⬘z dt F
␣0
T
H w⬘w dt. q 0
128
CHEN AND TU
Therefore, 5 Tz w 5 F
(
␣0 q
- 1.
Ž1. « Ž3. First, we show that there exists a 0 ) 0 such that the reduced IrO operator Tz 1 w by the system ⌺ 0
¡˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . CŽ t. xŽ t. s~ ¢z Ž t . s x Ž t . , x Ž 0. s 0 1
0
has the norm 5 Tz 1 w 5 - 1. In fact, since 5 Tz w 5 - 1, there exists a g Ž0, 1. such that 5 Tz w 5 - - 1. Since AŽ⭈. is stable, the IrO operator of the system Hy w s
½
˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . y Ž t . s x Ž t . , x Ž 0. s 0
is also bounded. q⬁
H0
zX1 z1 dt s 5 Tz w w 5 2 q 02 5 Hy w w 5 2 F 2 5 w 5 2 q 02 5 Hy w 5 2 5 w 5 2 s 2 q 02 5 Hy w 5 2 5 w 5 2 .
ž
/
Choose 0 - 0 -
'1 y 2 5 Hy w 5 q 1
;
then
2 q 02 5 Hy w 5 2 - 1 and 5 Tz w 5 F 1
'
2
q 02 5 Hy w 5 2 .
For ⌺ 0 , by Ž2., there exists a bounded and continuous function QŽ⭈.: Rqª R n=n such that QŽ⭈. G 0, ᭙ t g Rq, and the Riccati equation
˙Ž t . q Q Ž t . A Ž t . q A⬘ Ž t . Q Ž t . q Q Ž t . B Ž t . B⬘ Ž t . Q Ž t . Q q C⬘ Ž t . C Ž t . q 02 In s 0
Ž 3.3.
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
129
holds for almost all t g Rq. For i s 1, 2, . . . , let ⌬ i s t g Rq: QŽ t . F 1i In4 . We will show that there exists an i such that M Ž ⌬ i . s 0 Žhere the measure is the usual Lebesgue measure on real line.. Otherwise, choose a natural number i sufficiently large, such that 2 5 AŽ ⭈. 5 ⬁ i
q
5 B Ž ⭈ . 5 ⬁2 i2
-
02 2
;
then ᭙ x g R n, t g ⌬ i , x / 0, <² Q Ž t . x, A Ž t . x : q ² A Ž t . x, Q Ž t . x : q ² Q Ž t . B Ž t . B⬘ Ž t . Q Ž t . x, x :< F
ž
2 5 AŽ ⭈. 5 ⬁
q
i
5 B Ž ⭈ . 5 ⬁2 i2
/
5 x52 -
02 2
5 x52.
˙Ž t . G 0, a.e. t g Rq. Also, C⬘Ž t .C Ž t . G 0, But QŽ t . G 0, ᭙ t g Rq; hence Q q Ž . for all t g R . From 3.3 , y
02 2
5 x 5 2 F y 02 5 x 5 2 ,
that is, an obvious contradiction. Therefore, there exists a natural number i 0 , such that mŽ ⌬ i 0 . s 0. So, QŽ t . G Ž1ri 0 . In , a.e. t g Rq, and QŽ⭈.: Rqª R n= n is a Lyapunov matrix function. Proof of Lemma 2.2. As 5 Tz w 5 - 1, there exist  1 ,  2 ) 0, such that 5 Tz w 5 -  1 -  2 - 1. For any fixed t 0 g Rq, by the selection theorem used Žsee w6x., there exists a measurable function wUt 0 Ž⭈.: w t 0 , t 0 q 1x ª R q, such that 5 wUt 0 Ž t .5 s 1, and 5 DŽ t . wUt 0 Ž t .5 s 5 DŽ t .5, a.e. t g w t 0 , t 0 q 1x. For i s 1, 2, . . . , set the L2 integrable function on Rq, wit 0
¡ ¢0
' Ž t . s~ i w
U t0
Ž t.
t0 F t F t0 q
1 i
otherwise.
Then ⬁
H0 5 w
t0 5 2 i
dt s 1, i s 1, 2, . . . ,
and when t ) t 0 , x it 0 Ž t . s s
t
H0 ⌽ Ž t , s . B Ž s . w
t0 i
Ž s . ds
Ht t q ⌽ Ž t , s . B Ž s . 'i w 0
0
1 i
U t0
Ž s . ds.
130
CHEN AND TU
When t F t 0 , x it 0 Ž t . s 0. Since AŽ⭈. is stable, there are two constants M1 , 1 ) 0, such that 5 ⌽ Ž t, s .5 F M1 ey 1Ž tys., for all 0 F s F t - q⬁. Thus when t ) t 0 , 5 x it 0 Ž t . 5 F
Ht t q M e 1 i
0
y 1Ž tys.
1
'i 5 B Ž ⭈. 5 ⬁ ds
0
F
M1
1
1 i
5 B Ž ⭈ . 5 ⬁ 'i e t 0 Ž e y 1 . ey 1 t .
Then q⬁
H0
5 x it 0 Ž t . 5 2 dt F
M12 5 B Ž ⭈ . 5 ⬁2 1 2 1
i
2
1 i
ž
e y1 1 i
/
.
Note that 1 i
2
1 i
ž
e y1 1 i
/
ª 0 Ž i ª q⬁ . ;
x it 0 4 converges to zero uniformly in the L2 norm Žindependent of the choice of t 0 g Rq. . However, z it 0 Ž t . s C Ž t . x it 0 Ž t . q DŽ t . wit 0 Ž t . and also both C Ž⭈. and DŽ⭈. are essentially bounded for Rq; lim 5 z it 0 Ž ⭈ . y D Ž ⭈ . wit 0 5 2 s lim 5 C Ž ⭈ . x it 0 5 2 s 0
iª⬁
iª⬁
uniformly for t 0 g Rq. Thus q⬁
H ² z iª⬁ 0 lim
t0 i ,
z it 0 : y ² Dwit 0 , Dwit 0 : 4 dt s 0
uniformly for all t g Rq. As 5 Tz w 5 -  1 -  2 - 1, q⬁
H0
² z it 0 , z it 0 : F  1
q⬁
H0
² wit 0 , wit 0 : dt s  1 .
Hence there exists a natural number i1 Žindependent of t 0 g Rq. such that when i G i1 , q⬁
H0
² Dwit 0 , Dwit 0 : dt F  2 .
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
131
So for any t 0 g Rq, t 0q1ri
Ht
'i 2 5 D Ž t . 5 2 dt F  2 ;
0
i.e., i
t 0q1ri
Ht
5 D Ž t . 5 2 dt F  2 , ᭙ t 0 G 0, i G i 0 .
Ž 3.4.
0
Set f Ž t. s
t
H0 5 D Ž s . 5
2
ds;
then f Ž t . is differentiable for almost all t g Rq, and f ⬘Ž t . s 5 DŽ t .5 2 , a.e. t g Rq. But from Ž3.4., f ⬘Ž t . F  2 ; hence 5 DŽ t .5 2 F  2 , a.e. t g Rq. Therefore, 5 DŽ⭈.5 ⬁ F  2 - 1. Lemma 2.3 and Theorems 2.4 and 2.5 can be proved by arguments similar to those used for the time-invariant systems, and the operator techniques applied behind are almost the same; see Chen and Tu w2x.
'
4. CONCLUSIONS This paper investigates the strict bounded real lemma for linear timevarying systems, which presents some characterizations based on Riccati equations. We also study its potential usage in disturbance attenuation and H ⬁-optimization. Our results are natural continuations to the current results for linear time-invariant systems.
REFERENCES 1. R. W. Brocket, ‘‘Finite Dimensional Linear Systems,’’ Wiley, New York, 1970. 2. W. Chen and F. Tu, Disturbance attenuation via state-feedback controllers, Systems Science and Mathematical Sciences, 12 Ž1999., 338᎐343. 3. R. F. Curtain, The strict bounded real lemma in infinite dimensions, Systems Control Lett., 20 Ž1993., 113᎐116. 4. J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, State-space solutions to standard H2 and H⬁ control problems, IEEE Trans. Automat. Control 34 Ž1989., 831᎐847. 5. I. R. Gurov and G. Tadmor, On robustness of H⬁ state feedback control to non-linear perturbations, Automatica 30 Ž1994., 499᎐502. 6. C. J. Himmelberg et al. Measurable multifunctions, selectors, and Filippov implicit functions lemma, J. Math. Anal. Appl. 25 Ž1969., 276᎐285.
132
CHEN AND TU
7. P. P. Khargonekar et al. Robust stabilization of uncertain linear systems: Quadratic stabilizability and H ⬁ control theory, IEEE Trans. Automat. Control 35 Ž1990., 356᎐361. 8. P. P. Khargonekar et al. H⬁ -optimal control with state-feedback, IEEE Trans. Automat. Control 33 Ž1988., 786᎐788. 9. R. Ravi et al., H ⬁ control of linear time-varying systems: A state-space approach, SIAM J. Control Optim. 29 Ž1991., 1394᎐1413. 10. M. A. Rotea and P. P. Khargonekar, Stabilizability of linear time-varying and uncertain linear systems, IEEE Trans. Automat. Control 33 Ž1988., 884᎐887. 11. G. Tadmor, Inputroutput norms in general linear systems, Internat. J. Control 51 Ž1990., 911᎐921.
The Strict Bounded Real Lemma for Linear Time-Varying Systems1 Wanyi Chen College of Mathematics, Nankai Uni¨ ersity, Tianjin 300071, People’s Republic of China
and Fengsheng Tu2 Department of Computer and System Sciences, Nankai Uni¨ ersity, Tianjin 300071, People’s Republic of China Submitted by Harold L. Stalford Received December 6, 1995
In this paper, we present a strict bounded real lemma for linear time-varying systems in the infinite-horizon case. Using some operator methods, we show that the strict bounded realness for the related IrO operators is equivalent to the solvability of a semidefinite or definite Riccati equation. We also apply this result to the problem of disturbance attenuation and H⬁ -optimization. All our results include current ones in the literature for linear time-invariant systems. 䊚 2000 Academic Press
Key Words: time-varying systems; IrO operator; Riccati equation; H⬁ -norm.
1. INTRODUCTION As it is well known, the bounded real lemma is a classical result in linear system theory; see w3x. More recently, it has been included in the famous H⬁ control theory and has been featured as a powerful tool in the problem of disturbance attenuation and H ⬁-optimization. For linear time-invariant systems, its infinite dimensional version has been obtained by Curtain w3x for systems generated by C0-semigroups, and its finite dimensional version was obtained by Doyle et al. w4x and Khargonekar et al. w7x. For linear 1 2
This work is supported by the National Natural Science Foundation of China. E-mail: [email protected]. 120
0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
121
time-varying systems, Tadmor w11x utilized the classical maximum principle and gave a characterization in terms of solvability of an indefinite Riccati equation. Our purpose in this paper is to generalize these results for finite dimensional linear time-varying systems for the infinite-horizon case. By some operator techniques, we present more necessary and sufficient Riccati equation type conditions. From this, we also obtain a result for disturbance attenuation in the sense of H⬁ norms. In this paper, we will study linear time-varying systems in the form
⌺0
¡˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . q E Ž t . uŽ t . ¢x Ž 0. s 0, t G 0,
s~ z Ž t . s C Ž t . x Ž t . q D Ž t . w Ž t . q F Ž t . u Ž t .
where x Ž t . g R n, uŽ t . g R m , w Ž t . g R q, and z Ž t . g R p are, respectively, the state, the control input, the disturbance input Žor the exogenous input., and the controlled output. Let Rqs w0, q⬁., AŽ⭈. g L⬁n= n Ž Rq. , B Ž⭈. g L⬁n= q Ž Rq. , C Ž⭈. g L⬁p= n Ž Rq. , DŽ⭈. g L⬁p= q Ž Rq. , EŽ⭈. g L⬁n= m Ž Rq. , and F Ž⭈. g L⬁p= m Ž Rq. . Throughout the paper, 5 ⭈ 5 will represent the Euclidean norm for vectors and the induced norm for matrices. Setting uŽ t . ' 0, we obtain the unforced form of ⌺ 0 ⌺1
¡˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . ¢x Ž 0. s 0.
s~ z Ž t . s C Ž t . x Ž t . q D Ž t . w Ž t .
From Ravi et al. w9x, we know that when the homogeneous system ˙ xŽ t . s AŽ t . x Ž t . is exponentially stable; in brief, AŽ⭈. is stable. Then ⌺ 1 generates a bounded input᎐output operator Tz w : Lq2 Ž Rq. ª L2p Ž Rq. , Tz w Ž w Ž ⭈ . . Ž t . s
t
H0 C Ž t . ⌽ Ž t , s . B Ž s . w Ž s . ds q D Ž t . w Ž t . ,
where w Ž⭈. g Lq2 Ž Rq. , and ⌽ Ž t, s . is the state transition matrix of the homogeneous system
˙x Ž t . s A Ž t . x Ž t . . One of our purposes in this paper is to give some characterizations for the estimation of the norm of Tz w . In ⌺ 0 , if all the states are available for feedback, we can design either static or dynamic state feedback controllers. First, we consider the case of static state feedback uŽ t . s K Ž t . x Ž t . ,
Ž 1.1.
122
CHEN AND TU
where K Ž⭈. g L⬁m= n. Let TzKw denote the input᎐output operator for the closed-loop system of ⌺ 0 with Ž1.1., and define
␥s s inf 5 TzKw 5 : K Ž ⭈ . g L⬁m= n Ž Rq . , A Ž ⭈ . q E Ž ⭈ . K Ž ⭈ . is stable 4 . Ž 1.2. Alternatively, for the dynamic state feedback ⌺d s
½
¨˙Ž t . s G Ž t . ¨ Ž t . q H Ž t . x Ž t .
uŽ t . s LŽ t . ¨ Ž t . q M Ž t . x Ž t . ,
where GŽ⭈. g L⬁r=r Ž Rq. , H Ž⭈. g L⬁r=n Ž Rq. , LŽ⭈. g L⬁m= r Ž Rq. , and M Ž⭈. g L⬁m= n Ž Rq. . The closed-loop system of ⌺ 0 with ⌺ d is in the form
¡X˙Ž t . s
⌺2 s
~
AŽ t . q E Ž t . M Ž t . HŽ t.
zŽ t. s CŽ t. q FŽ t. MŽ t.
¢
EŽ t . LŽ t . BŽ t. XŽ t. q wŽ t. GŽ t . 0 F Ž t . LŽ t . X Ž t . q D Ž t . w Ž t . ,
X Ž 0 . s 0.
Similar to Khargonekar et al. w8x, we define
½
D s Ž G, H , L, M . g L⬁r=r Ž Rq . = L⬁r=n Ž Rq . = L⬁m= r Ž Rq . =L⬁m= n Ž Rq . : AŽ ⭈. q E Ž ⭈. M Ž ⭈. H Ž ⭈.
E Ž ⭈. L Ž ⭈. G Ž ⭈.
is stable
5
and let Tzdw be the IrO operator of ⌺ 2 . Moreover, we define
␥d s inf 5 Tzdw 5 : Ž G, H , L, M . g D 4
Ž 1.3.
It is easy to see that for any natural number r,
␥s s inf 5 Tzdw 5 : Ž G, H , L, M . g D l 0 4 = 0 4 = 0 4 = L⬁m= n Ž Rq . 4 G ␥d We will show that ␥s s ␥d , which generalizes the result of Khargonekar et al. to the linear time-varying case.
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
123
2. MAIN RESULTS THEOREM 2.1 ŽThe Strict Bounded Real Lemma.. For system ⌺ 1 , if DŽ t . ' 0, then the following conditions are mutually equi¨ alent: Ž1. The homogeneous system ˙ x Ž t . s AŽ t . x Ž t . is exponentially stable and 5 Tz w 5 - 1. Ž2. There exists a bounded and continuous function QŽ⭈.: Rqª R n=n such that QŽ⭈. G 0, ᭙ t g Rq, and following the Riccati equation
˙Ž t . q Q Ž t . AŽ t . q A⬘ Ž t . Q Ž t . q Q Ž t . B Ž t . B⬘ Ž t . Q Ž t . q C⬘ Ž t . C Ž t . Q s0 holds for almost all t g Rq. Moreo¨ er, the homogeneous system ˙ xŽ t . s w A q xŽ . Ž . BB⬘Q t x t is exponentially stable. Ž3. There exist a Lyapuno¨ matrix function P Ž⭈.: Rqª R n=n and a positi¨ e constant ␣ 0 such that the following Riccati inequality holds: P˙Ž t . q P Ž t . A Ž t . q A⬘ Ž t . P Ž t . q P Ž t . B Ž t . B⬘ Ž t . P Ž t . a.e., t g Rq.
q C⬘ Ž t . C Ž t . F y␣ 0 In ,
Here the Lyapuno¨ matrix function refers to bounded uniformly positi¨ edefinite function on Rq Ž see Rotea and Khargonekar w10x., and In denotes the n = n identity. x Ž t . s AŽ t . x Ž t . is exponentially stable and 5 Tz w 5 LEMMA 2.2. In ⌺ 1 , if ˙ 1, then 5 DŽ⭈.5 ⬁ - 1. LEMMA 2.3. In ⌺ 1 , AŽ⭈. is stable, and the induced IrO operator has norm 5 Tz w 5 - 1 if and only if the following conditions are true: Ž1. Ž2.
5 DŽ⭈.5 ⬁ - 1, and A1Ž⭈. is stable. The induced IrO operator by the system ⌺ r is strictly contracti¨ e ⌺r s
½
˙x Ž t . s A1 Ž t . x Ž t . q B1 Ž t . w Ž t . z Ž t . s C1 Ž t . w Ž t . , x Ž 0 . s 0;
here A1 Ž t . s A q B Ž Iq y D⬘D . B1 Ž t . s B Ž Iq y D⬘D . C1 Ž t . s
Ž Ip y DD⬘ .
1 2
y 12
y1
Ž t.
C Ž t. .
C Ž t.
124
CHEN AND TU
From this lemma, we obtain the following result which gives a characterization for the IrO operator of ⌺1 and has potential usage in disturbance attenuation and H ⬁-optimization. THEOREM 2.4. equi¨ alent:
For system ⌺ 1 , the following conditions are mutually
Ž1. The homogeneous system ˙ x Ž t . s AŽ t . x Ž t . is exponentially stable and 5 Tz w 5 - 1. Ž2. 5 DŽ⭈.5 ⬁ - 1. There exists a bounded and continuous function QŽ⭈.: Rqª R n=n such that QŽ t . G 0, ᭙ t g Rq, and the Riccati equation
˙Ž t . q C⬘ Ž t . C Ž t . q Q Ž t . AŽ t . q A⬘ Ž t . Q Ž t . Q q Ž QB q C⬘D . Ž t . Ž Iq y D⬘D .
y1
Ž t . Ž B⬘Q q D⬘C . Ž t . s 0
holds for almost all t g Rq. Moreo¨ er, w A q B Ž Iq y D⬘D .y1 Ž B⬘Q q D⬘C .xŽ⭈. is stable. Ž3. 5 DŽ⭈.5 ⬁ - 1. There exist a Lyapuno¨ matrix function P Ž⭈.: Rqª n= n R and a positi¨ e constant ␣ 1 such that following Riccati inequality holds for almost all t g Rq: P˙Ž t . q C⬘ Ž t . C Ž t . q P Ž t . A Ž t . q A⬘ Ž t . P Ž t . q P Ž t . B Ž t . q C⬘ Ž t . D Ž t .
Iq y D⬘ Ž t . D Ž t .
y1
= B⬘ Ž t . P Ž t . q D⬘ Ž t . C Ž t . F y␣ 1 In . From this theorem, we can obtain the following result on disturbance attenuation, which shows that dynamic state feedback offers no advantage over static state feedback in the minimization of the H⬁ norm of the IrO operator of the closed-loop system. Also, this result is a natural generalization of Khargonekar et al. w8x to the time-varying case. THEOREM 2.5. Consider the dynamical system ⌺ 0 and the state feedback controllers Ž1.1. and ⌺ d . Define ␥s and ␥d as in Ž1.2. and Ž1.3., respecti¨ ely; then ␥s s ␥d . 3. PROOFS Proof of Theorem 2.1. Ž1. « Ž2. This can be finished by a similar method used in the proof of w9, Theorem 3.1x. Thus, we omit it.
125
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
Ž2. « Ž1. First, we show that ˙ x Ž t . s AŽ t . x Ž t . is exponentially stable. Let ⌽ Ž t, s . denote its state transition matrix, 0 F s F t - q⬁, and let ⌿ Ž t, s . denote the state transition matrix generated by Ž A q BB⬘Q .Ž⭈.; then ⌽Ž t, s. s ⌿Ž t, s. y
t
Hs ⌿ Ž t , . B Ž . B⬘ Ž . Q Ž . ⌽ Ž , s . d .
Thus for any x g R n, d² ⌽ Ž t , s . x, Q Ž t . ⌽ Ž t , s . x : dt s ² A Ž t . ⌽ Ž t , s . x, Q Ž t . ⌽ Ž t , s . x : q ² ⌽ Ž t , s . x, Q Ž t . A Ž t . ⌽ Ž t , s . x :
˙Ž t . ⌽ Ž t , s . x : q² ⌽ Ž t , s . x, Q s y5 B⬘ Ž t . Q Ž t . ⌽ Ž t , s . x 5 2 y 5 C Ž t . ⌽ Ž t , s . x 5 2 . For any t 1 ) s G 0, integrating from s to t 1 and noting that ⌽ Ž s, s . s In , we obtain t1
Hs 5 B⬘ Ž . Q Ž . ⌽ Ž , s . x 5
d F ² Ž Q Ž s . x, x . F  5 x 5 2 .
2
Here  is chosen by the assumption that QŽ⭈. is bounded on Rq; hence, there exists a  ) 0 such that QŽ s . F  In , for all s g Rq. However, the system ˙ x Ž t . s Ž A q BB⬘Q .Ž t . x Ž t . is exponentially stable, so by Brocket w1x, there exist positive constants m 0 , m1 , and m 2 such that q⬁
Hs
5 ⌿ Ž t , s . 5 2 dt F m 0 ,
q⬁
Hs
5 ⌿ Ž t , s . 5 dt F m1 ,
t
H0 5 ⌿ Ž t , . 5 d F m
2,
᭙s G 0, ᭙s G 0, ᭙ t G 0.
Hence, q⬁
žH s
5 ⌽ Ž t , s . x 5 dt
F
2
q⬁
žH s
1 2
/
5 ⌿ Ž t , s . x 5 dt 2
q m1 m2 ⭈ 5 B Ž ⭈ . 5 ⬁ 1 2
1 2
1 2
/ q⬁
žH s
5 B⬘ Ž t . Q Ž t . ⌽ Ž t , s . x 5 2 dt
F Ž m 0 q m1 m2 5 B Ž ⭈ . 5 ⬁  . 5 x 5 . 1 2
1 2
1 2
1 2
/
126
CHEN AND TU
Therefore from Gurov and Tadmor w5x, the homogeneous system ˙ xŽ t . s AŽ t . x Ž t . is exponentially stable. Second, we will show that 5 Tz w 5 - 1. For any T ) 0, consider the related system in w0, T x, T
H0 Ž z⬘z y w⬘w . dt T
½
s
H0
z⬘z y w⬘w q
s
H0 x⬘
T
d Ž x⬘Qx . dt
5
dt y x⬘ Ž T . Q Ž T . x Ž T .
˙ q QA q A⬘Q q QBB⬘Q q C⬘C x dt Q Ž 3.1.
T
H0 w w y B⬘Qx x ⬘ w w y B⬘Qx x dt y x⬘ Ž T . Q Ž T . x Ž T .
y
T
H0 w w y B⬘Qx x ⬘ w w y B⬘Qx x dt y x⬘ Ž T . Q Ž T . x Ž T .
sy
T
H0 5 w y w*5
Fy
2
dt,
where w*Ž t . s yB⬘Ž t . QŽ t . x Ž t ., t g Rq. Letting T ª q⬁ in the previous inequality, we have q⬁
H0
q⬁
Ž z⬘z y w⬘w . dt F yH
Ž w y w* . ⬘ Ž w y w* . dt.
0
Suppose L is the IrO operator of the system
˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . Ž w y w* . Ž t . s y Ž B⬘Q . Ž t . x Ž t . q w Ž t . ,
x Ž 0. s 0
which maps w to w y w*. Since Ly1 exists as Ly1 s
½
˙x Ž t . s Ž A q BB⬘Q . Ž t . x Ž t . q B Ž t . Ž w y w* . Ž t . w Ž t . s Ž B⬘Q . Ž t . x Ž t . q Ž w y w* . Ž t . , x Ž 0. s 0
and Ž A q BB⬘Q .Ž⭈. is stable, Ly1 is also a bounded operator on Lq2 Ž Rq. . Also, w s Ly1 Ž w y w*., q⬁
žH
0
1 2
w⬘w dt
/
F 5 Ly1 5
q⬁
žH
0
1 2
Ž w y w* . ⬘ Ž w y w* . dt .
/
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
127
Combining this with Ž3.1., we have q⬁
H0
q⬁
Ž z⬘z y w⬘w . dt F yH
Ž w y w* . ⬘ Ž w y w* . dt
0
Fy
1
q⬁
y1 5 2
5L
H0
w⬘w dt.
Hence q⬁
H0
z⬘z dt F 1 y
ž
5 Tz w 5 F 1 y
ž
1 5 L5
/
2
q⬁
H0
w⬘w dt
1 2
1 5 Ly1 5 2
- 1.
/
Ž3. « Ž1. Note that z Ž t . s C Ž t . x Ž t . and C Ž⭈. is essentially bounded on Rq; therefore, there exists a positive constant , such that C⬘C F In . Thus for any T ) 0, T
T
T
H0 z⬘z dt s H0 x⬘C⬘Cx dt F H0 x⬘ x dt.
Ž 3.2.
Also under the given assumption, P Ž⭈.: Rqª R n=n is a Lyapunov matrix function for ˙ x Ž t . s AŽ t . x Ž t .; hence, it is exponentially stable. For T ) 0, by the given Riccati equation and Ž3.2. used, we have T
H0 Ž z⬘z y w⬘w . dt s yx⬘ Ž T . P Ž T . x Ž T . T
q
H0
s
ž
z⬘z y w⬘w q
d Ž x⬘Px . dt
/
dt
T
H0 x⬘ P˙ q A⬘P q PA q PBB⬘P q C⬘C 4 x dt y x⬘ Ž T . P Ž T . x Ž T . y
F y␣ 0
␣0
T
T
H0 w w y B⬘Px x ⬘ w w y B⬘Px x dt T
H0 x⬘ x dt F y H0 z⬘z dt.
Let T ª q⬁, we obtain q⬁
H0
z⬘z dt F
␣0
T
H w⬘w dt. q 0
128
CHEN AND TU
Therefore, 5 Tz w 5 F
(
␣0 q
- 1.
Ž1. « Ž3. First, we show that there exists a 0 ) 0 such that the reduced IrO operator Tz 1 w by the system ⌺ 0
¡˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . CŽ t. xŽ t. s~ ¢z Ž t . s x Ž t . , x Ž 0. s 0 1
0
has the norm 5 Tz 1 w 5 - 1. In fact, since 5 Tz w 5 - 1, there exists a g Ž0, 1. such that 5 Tz w 5 - - 1. Since AŽ⭈. is stable, the IrO operator of the system Hy w s
½
˙x Ž t . s AŽ t . x Ž t . q B Ž t . w Ž t . y Ž t . s x Ž t . , x Ž 0. s 0
is also bounded. q⬁
H0
zX1 z1 dt s 5 Tz w w 5 2 q 02 5 Hy w w 5 2 F 2 5 w 5 2 q 02 5 Hy w 5 2 5 w 5 2 s 2 q 02 5 Hy w 5 2 5 w 5 2 .
ž
/
Choose 0 - 0 -
'1 y 2 5 Hy w 5 q 1
;
then
2 q 02 5 Hy w 5 2 - 1 and 5 Tz w 5 F 1
'
2
q 02 5 Hy w 5 2 .
For ⌺ 0 , by Ž2., there exists a bounded and continuous function QŽ⭈.: Rqª R n=n such that QŽ⭈. G 0, ᭙ t g Rq, and the Riccati equation
˙Ž t . q Q Ž t . A Ž t . q A⬘ Ž t . Q Ž t . q Q Ž t . B Ž t . B⬘ Ž t . Q Ž t . Q q C⬘ Ž t . C Ž t . q 02 In s 0
Ž 3.3.
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
129
holds for almost all t g Rq. For i s 1, 2, . . . , let ⌬ i s t g Rq: QŽ t . F 1i In4 . We will show that there exists an i such that M Ž ⌬ i . s 0 Žhere the measure is the usual Lebesgue measure on real line.. Otherwise, choose a natural number i sufficiently large, such that 2 5 AŽ ⭈. 5 ⬁ i
q
5 B Ž ⭈ . 5 ⬁2 i2
-
02 2
;
then ᭙ x g R n, t g ⌬ i , x / 0, <² Q Ž t . x, A Ž t . x : q ² A Ž t . x, Q Ž t . x : q ² Q Ž t . B Ž t . B⬘ Ž t . Q Ž t . x, x :< F
ž
2 5 AŽ ⭈. 5 ⬁
q
i
5 B Ž ⭈ . 5 ⬁2 i2
/
5 x52 -
02 2
5 x52.
˙Ž t . G 0, a.e. t g Rq. Also, C⬘Ž t .C Ž t . G 0, But QŽ t . G 0, ᭙ t g Rq; hence Q q Ž . for all t g R . From 3.3 , y
02 2
5 x 5 2 F y 02 5 x 5 2 ,
that is, an obvious contradiction. Therefore, there exists a natural number i 0 , such that mŽ ⌬ i 0 . s 0. So, QŽ t . G Ž1ri 0 . In , a.e. t g Rq, and QŽ⭈.: Rqª R n= n is a Lyapunov matrix function. Proof of Lemma 2.2. As 5 Tz w 5 - 1, there exist  1 ,  2 ) 0, such that 5 Tz w 5 -  1 -  2 - 1. For any fixed t 0 g Rq, by the selection theorem used Žsee w6x., there exists a measurable function wUt 0 Ž⭈.: w t 0 , t 0 q 1x ª R q, such that 5 wUt 0 Ž t .5 s 1, and 5 DŽ t . wUt 0 Ž t .5 s 5 DŽ t .5, a.e. t g w t 0 , t 0 q 1x. For i s 1, 2, . . . , set the L2 integrable function on Rq, wit 0
¡ ¢0
' Ž t . s~ i w
U t0
Ž t.
t0 F t F t0 q
1 i
otherwise.
Then ⬁
H0 5 w
t0 5 2 i
dt s 1, i s 1, 2, . . . ,
and when t ) t 0 , x it 0 Ž t . s s
t
H0 ⌽ Ž t , s . B Ž s . w
t0 i
Ž s . ds
Ht t q ⌽ Ž t , s . B Ž s . 'i w 0
0
1 i
U t0
Ž s . ds.
130
CHEN AND TU
When t F t 0 , x it 0 Ž t . s 0. Since AŽ⭈. is stable, there are two constants M1 , 1 ) 0, such that 5 ⌽ Ž t, s .5 F M1 ey 1Ž tys., for all 0 F s F t - q⬁. Thus when t ) t 0 , 5 x it 0 Ž t . 5 F
Ht t q M e 1 i
0
y 1Ž tys.
1
'i 5 B Ž ⭈. 5 ⬁ ds
0
F
M1
1
1 i
5 B Ž ⭈ . 5 ⬁ 'i e t 0 Ž e y 1 . ey 1 t .
Then q⬁
H0
5 x it 0 Ž t . 5 2 dt F
M12 5 B Ž ⭈ . 5 ⬁2 1 2 1
i
2
1 i
ž
e y1 1 i
/
.
Note that 1 i
2
1 i
ž
e y1 1 i
/
ª 0 Ž i ª q⬁ . ;
x it 0 4 converges to zero uniformly in the L2 norm Žindependent of the choice of t 0 g Rq. . However, z it 0 Ž t . s C Ž t . x it 0 Ž t . q DŽ t . wit 0 Ž t . and also both C Ž⭈. and DŽ⭈. are essentially bounded for Rq; lim 5 z it 0 Ž ⭈ . y D Ž ⭈ . wit 0 5 2 s lim 5 C Ž ⭈ . x it 0 5 2 s 0
iª⬁
iª⬁
uniformly for t 0 g Rq. Thus q⬁
H ² z iª⬁ 0 lim
t0 i ,
z it 0 : y ² Dwit 0 , Dwit 0 : 4 dt s 0
uniformly for all t g Rq. As 5 Tz w 5 -  1 -  2 - 1, q⬁
H0
² z it 0 , z it 0 : F  1
q⬁
H0
² wit 0 , wit 0 : dt s  1 .
Hence there exists a natural number i1 Žindependent of t 0 g Rq. such that when i G i1 , q⬁
H0
² Dwit 0 , Dwit 0 : dt F  2 .
BOUNDED REAL LEMMA FOR LINEAR SYSTEMS
131
So for any t 0 g Rq, t 0q1ri
Ht
'i 2 5 D Ž t . 5 2 dt F  2 ;
0
i.e., i
t 0q1ri
Ht
5 D Ž t . 5 2 dt F  2 , ᭙ t 0 G 0, i G i 0 .
Ž 3.4.
0
Set f Ž t. s
t
H0 5 D Ž s . 5
2
ds;
then f Ž t . is differentiable for almost all t g Rq, and f ⬘Ž t . s 5 DŽ t .5 2 , a.e. t g Rq. But from Ž3.4., f ⬘Ž t . F  2 ; hence 5 DŽ t .5 2 F  2 , a.e. t g Rq. Therefore, 5 DŽ⭈.5 ⬁ F  2 - 1. Lemma 2.3 and Theorems 2.4 and 2.5 can be proved by arguments similar to those used for the time-invariant systems, and the operator techniques applied behind are almost the same; see Chen and Tu w2x.
'
4. CONCLUSIONS This paper investigates the strict bounded real lemma for linear timevarying systems, which presents some characterizations based on Riccati equations. We also study its potential usage in disturbance attenuation and H ⬁-optimization. Our results are natural continuations to the current results for linear time-invariant systems.
REFERENCES 1. R. W. Brocket, ‘‘Finite Dimensional Linear Systems,’’ Wiley, New York, 1970. 2. W. Chen and F. Tu, Disturbance attenuation via state-feedback controllers, Systems Science and Mathematical Sciences, 12 Ž1999., 338᎐343. 3. R. F. Curtain, The strict bounded real lemma in infinite dimensions, Systems Control Lett., 20 Ž1993., 113᎐116. 4. J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, State-space solutions to standard H2 and H⬁ control problems, IEEE Trans. Automat. Control 34 Ž1989., 831᎐847. 5. I. R. Gurov and G. Tadmor, On robustness of H⬁ state feedback control to non-linear perturbations, Automatica 30 Ž1994., 499᎐502. 6. C. J. Himmelberg et al. Measurable multifunctions, selectors, and Filippov implicit functions lemma, J. Math. Anal. Appl. 25 Ž1969., 276᎐285.
132
CHEN AND TU
7. P. P. Khargonekar et al. Robust stabilization of uncertain linear systems: Quadratic stabilizability and H ⬁ control theory, IEEE Trans. Automat. Control 35 Ž1990., 356᎐361. 8. P. P. Khargonekar et al. H⬁ -optimal control with state-feedback, IEEE Trans. Automat. Control 33 Ž1988., 786᎐788. 9. R. Ravi et al., H ⬁ control of linear time-varying systems: A state-space approach, SIAM J. Control Optim. 29 Ž1991., 1394᎐1413. 10. M. A. Rotea and P. P. Khargonekar, Stabilizability of linear time-varying and uncertain linear systems, IEEE Trans. Automat. Control 33 Ž1988., 884᎐887. 11. G. Tadmor, Inputroutput norms in general linear systems, Internat. J. Control 51 Ž1990., 911᎐921.