An improved stabilization method for linear time-delay ... - IEEE Xplore
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002
An Improved Stabilization Method for Linear Time-Delay Systems Emilia Fridman and Uri Shaked
1931
2 Rn2n , means that P is symmetric and positive definite. The space of vector functions that are square integrable over [0 1) is denoted by L2 .
P
II. A NEW STABILIZATION METHOD Abstract—In this note, we combine a new approach for linear timedelay systems based on a descriptor representation with a recent result on bounding of cross products of vectors. A delay-dependent criterion for determining the stability of systems with time-varying delays is obtained. This criterion is used to derive an efficient stabilizing state-feedback design method for systems with parameter uncertainty, of either the polytopic or the norm-bounded types. Index Terms—Delay-dependent stability, linear matrix inequality (LMI), stabilization, time-delay systems, time-varying delay.
I. INTRODUCTION The problem of reducing the conservatism entailed in applying finite-dimensional techniques to asses the stability of linear systems with time delay has attracted much attention in the past few years [1]–[6]. All these techniques provide sufficient conditions only for the asymptotic stability of these systems and they entail a considerable conservatism which stems from two main sources. The first cause for conservatism is the model transformation used to describe the system which makes it more amenable for analysis [7], [8] and the second reason for conservatism is the bounding method used to derive the bounds on weighted cross products of the state and its delayed version while trying to secure a negative value to the derivative of the corresponding Lyapunov–Krasovskii functional. The search for the most appropriate model transformation has led to four main approaches [9]–[11]. The most recent one [9], the one that is based on a descriptor representation of the system, which is equivalent to the original system, minimizes the overdesign that stems from the model transformation source of conservatism [11]. The conservatism that stems from the bounding of the cross terms has also been significantly reduced in the past few years. An important result for improving the standard bounding technique of, e.g., [2], has been proposed in [12]. Indeed, combining the later with the descriptor model transformation lead in [10] and [11] to an efficient delay-dependent stability criterion that was also used in synthesis for stabilization and optimal performance. Only recently, an improvement of the bounding technique has been proposed [13]. The latter generalizes the one in [12] and the resulting criteria that are obtained in [13] are, therefore, more efficient than those found in [12]. It is the purpose of this note to combine the bounding method of [13] with the descriptor model transformation of [9] and [11] in order to derive a most efficient stability criterion for systems with time-varying delays. This criterion is then applied to solve the problem of robust stabilizing the system in presence of either norm-bounded or polytopic uncertainties by means of state-feedback control. The resulting criterion is applied to an example taken from [13], and its superiority to the results of the latter is demonstrated. Notation: Throughout this note, the superscript T stands for matrix transposition, Rn denotes the n-dimensional Euclidean space, Rn2m is the set of all n 2 m real matrices, and the notation P > 0, for
We consider the following linear system with time-varying delays: 2
_( ) =
x t
( 0 i (t)) + Bu(t);
( ) = (t); t 2 [0h; 0]
Ai x t i=0
x t
(1) where x(t) 2 Rn is the system state, u(t) 2 Rq is the control input, 0 0, Ai and B are constant n 2 n matrices, is a continuously differentiable initial function, and h is an upper-bound on the timedelays i , i = 1; 2. For simplicity only, we took two delays 1 and 2 . The results of this section can be easily applied to the case of multiple delays 1 ; . . . ; m . The matrices of the system are not exactly known. Denoting
= [ A0
A1
A2
B
]
we assume that
=
N
fj j ; j =1
0 fj 1;
for some
N
fj j =1
=1
(2)
where the N vertices of the polytope are described by
j = [ A0(j )
(j )
A1
(j )
A2
]
B (j ) :
In Section III, we extend our results to the case where the uncertainty in the system parameters obeys the norm-bounded model [17]. As in [11], we consider two different cases for time-varying delays i (t) are differentiable functions, satisfying for all t 0:
0 i (t) hi ; ()
_ ( ) di < 1;
i t
i
= 1; 2:
(3)
0, 0 i (t)
i t are continuous functions, satisfying for all t hi , i ; .
=12
Note that in the past, the Razumikhin’s approach was the only one that was to cope with Case I) of fastly varying delays. The Krasovskii approach for this case was introduced recently in [11]. We seek a control law
( ) = Kx(t)
u t
(4)
that will asymptotically stabilize the system. A. Stability Issue In this section, we consider B descriptor form [9]
_ ( ) =y(t)
= 0. Representing (1) in an equivalent
x t
0 = 0 y(t) +
(5a) 2
Ai i=0
( )0
2
x t
Ai i=1
t
0
t (t)
()
y s ds (5b)
or Manuscript received October 15, 2001; revised February 8, 2002 and June 28, 2002. Recommended by Associate Editor L. Pandolfi. This work was supported by the C&M Maus Chair at Tel Aviv University. The authors are with the Department of Electrical Engineering-Systems, Tel-Aviv University, Tel-Aviv 69978, Israel. Digital Object Identifier 10.1109/TAC.2002.804462
_ ( ) = x_ (0t) 2 0 I 0 = 2 Ai 0I x(t) 0 Ai i=1
Ex t
0018-9286/02$17.00 © 2002 IEEE
i=0
t
0
t (t)
()
y s ds (5c)
1932
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002
with x (t) = colfx(t); y(t)g, E = diagfI; 0g, the following Lyapunov–Krasovskii functional is applied:
V (t) = xT (t)EP x(t) + V2 + V3
and, hence, differentiating the first term of (6) with respect to t gives
d xT (t)EP x(t) dt
(6)
=
V2
=
V3
=
0
P1 P2
P3
2
0 i=1 0h 2 t
P1 > 0 EP
; t
+
t
=1 t0 (t)
i
= PTE 0
and
3
y T (s)Ri y (s)dsd
xT ( )Si x( )d :
(7a-e)
Y
9 =P T
0
Zi1
3
Zi2 ; Zi3
I 0 I 0I + A0 0I
A0
T
x
i (t)
= =
(t)EP x(t) = x (t)P1 x(t)
t
0
t t
0
t t
0
t t
0
91 < 0
t h
[ y (s) x (t) ] T
T
=1
i
Ri
YiT
y T (s)Ri y (s)ds + 2 y T (s)Ri y (s)ds + 2
t
0
t t
0
t
0N
Y
0 NT Z
a b (13)
N = Ni =
Yi 0 [ 0 ATi ] P Zi
Ri
3
and
t
(1 0 di )xT (t 0 i )Si x(t 0 i ) +
0 P T A0i
R
PT
0
T (t)01 (t)
where the first equation shown at the bottom of the next page holds, and where (t) = colfx(t); y (t); x(t 0 1 ); x(t 0 2 )g. Since 01 = 0 the LMIs in (8) lead to V_ < 0, while V 0 and, thus, (1) with B = 0 is asymptotically stable [5], [15]. Choosing in Lemma 1 Si ! 0 and Yi = [ 0 ATi ]P T we obtain the following result for the case B. Corollary 1: Under Case II), (1), with B = 0, is asymptotically stable if there exist n 2 n matrices 0 < P1 ; P2 ; P3 ; Zi1 ; Zi2 ; Zi3 and Ri > 0, i = 1; 2 that satisfy the following LMIs:
(9a-c)
2
Y
T
0
dV (t) dt
P
xT (t)0x(t) 0
T
, R = Ri , Z = Zi , Ai Y = Yi , a = y (s) and b = x(t), we obtain, for i = 1; 2, (14) found at the bottom of the page. Substituting the latter and (12) into (11), we obtain that
T
dV (t) dt
a b
we apply the latter on the expression we have previously obtained for
Proof: Note that T
Y Z
i . From (13), taking
i = 1; 2
2 Si 0 2 + hi Zi + i=1 2 i=1 0 hi Ri i=1 2 Y 2 Y T i i + + : 0 0 i=1 i=1
R YT
where
Zi
= [Yi1 Yi2 ] Zi =
2 T I 0 Ai 20 + P i=0 Ai 0I I 0I i=0 2 Si 0 + i=1 2 0 hi Ri i=1 t 0 1 i (t)= 0 2 xT (t)P T y (s)ds: (12) A i t0 Since, by [13], for any a 2 Rn , b 2 R2n , N 2 R2n2n , R 2 Rn2n , 2 Rn22n , Z 2 R2n22n , the following holds:
02bT N a
0 9 PT 0 Y1T P T A02 0 Y2T A1 0 = 3 0S1 (1 0 d1 ) 0 3 3 0S2 (1 0 d2 ) 0, i = 1; 2 that satisfy the following linear matrix inequalities (LMIs):
Ri
x_ (t)
Substituting (5) into (10), we obtain (11), as shown at the bottom of the page, where
where
P
= 2xT (t)P1 x_ (t) = 2xT (t)P T
0
t h
[ 0 ATi ]P T Zi
y T ( )Ri y ( )d
0;
i = 1; 2
0 i
(11)
y (s) ds x(t)
y T (s)(Yi 0 [ 0 ATi ] P ) x(t)ds +
t
0
t
x(t)T Zi x(t)ds
x_ T (s)(Yi 0 [ 0 ATi ] P ) x(t)ds + i x(t)T Zi x(t)
y T (s)Ri y (s)ds + 2xT (t)(Yi 0 [ 0 ATi ] P ) x(t) 0 2xT (t 0 i )(Yi 0 [ 0 ATi ] P ) x(t) + hi x(t)T Zi x(t):
(14)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002
where
Zi =
Zi1 Zi2 3 Zi3 ;
91 =P
T
2
0
i=0
Ai
0 + hi Zi + 0 i=1 2
In the sequel, it will be important to determine the conditions for achieving H1 norm of (1) less than 1, where u is the input vector and the controlled output is given by
i = 1; 2 I 0I +
2
0
i=0
2
i=1
Ai
z (t) = Lx(t) + L1 x(t 0 1 ) + L2 x(t 0 2 ):
T
I 0I
P
hi Ri :
Remark 2: A question may arise as to whether the standard Lyapunov criterion can be restored when letting h go to 0. Taking Ri = I , Zi = 01=2 , ! 1, Yi = [ 0 ATi ]P and 0 < Si ! 0 we obtain
+ =
0
I 0 + A0 0I A0 2 2 Yi Yi
+
0
i=1
P2T (
2
i=0
For P3 = I; becomes
P1 (
i=1
Ai ) + (
3
I 0I
B. State-Feedback Stabilization The results of Lemma 1 can also be used to verify the stability of the closed loop obtained by applying (4) to (1) (with B 6= 0) if we replace A0 in (8a) by A0 + BK and verify that the resulting inequality is feasible over the polytope defined in (2) by solving the LMI simultaneously for all the N vertices, applying the same P1 , Pi , Si , Yi1 , Yi2 , and Ri , i = 1,2. The problem with (8a) is that it is linear in its variables, only when the state-feedback gain K is given. In order to find K , consider the inverse of P . It is obvious from the requirement of 0 < P1 , and the fact that in (8)0(P3 + P3T ) must be negative definite, that P is nonsingular. Defining
0
2
i=0
ATi )P2 P1 0 P2T + ( 3
Ai ) + (
2
i=0
2
1
ATi )P3
:
i=0 PT
0P 0
2
i=0
P
T
! 0 and P = P 2
T
3
> 0 the requirement that 9 < 0
ATi )P1 < 0;
P1 > 0:
(15)
Q1 0 Q2 Q3 1 =diagfQ; I g
It follows from (15) that if the system with h = 0 is asymptotically stable, then there exists P1 > 0 that solves (15) and, thus, (8a),(b) possess a solution for small enough h > 0. The latter can be readily used to verify the stability of (1) over the uncertainty polytope (2) [1]:
=
N j =1
j ; for some 0 fj fj
1;
N j =1
P 01 =Q =
fj = 1
Zi =
[Yi1
j = [ A(0j ) A(1j ) A(2j ) ]
01 =
i=1
hi Zi +
I
0
(Yi 0 [ 0 ATi ] P ) + YiT
3 3
9 PT
3 3 3 3
0
B
0Iq 3 3 3
Zi1 Zi2 = QT Zi Q and R i = Ri01 , i = 1,2 and choosing ZiT2 Zi3 Yi2 ] = "i AiT [P2 P3 ] , where "i 2 Rn2n is a diagonal
0 P T A0i
0Y T + P T A0 0S (1 0 d ) 3
[I 0]
1
1
1
PT
0
A1
0YT 1
0 0S1 (1 0 d1)
3 3
(18b)
matrix, we obtain, similarly to [14], the following. Theorem 1: The control law of (4) asymptotically stabilizes (1) for all the delays that belong to Case I) and for all the system parameters that reside in the uncertainty polytope, if for some diagonal matrices
by solving the LMI simultaneously for all the N vertices, applying the same P1 , Pi , Si , Yi1 , Yi2 , and Ri , i = 1,2.
2
(18a)
we multiply (8a) by 1T and 1, on the left and on the right, respectively, and (8b), on the left and on the right, by diagfRi01; QT g and diagfRi01; Qg, respectively . Applying Schur formula to the emerging quadratic term in Q, denoting Si = Si01 ,
where the N vertices of the polytope are described by
0 +
(16)
Similarly to the derivation of the bounded real lemma (BRL) in [11], we obtain the following. Lemma 2: Under Case I) the H1 norm of (1) and (16) is less than one if there exist n 2 n matrices 0 < P1 ; P2 ; P3 ; Si ; Yi1 ; Yi2 ; Zi1 ; Zi2 ; Zi3 and Ri , i = 1,2 that satisfy (17), as shown at the bottom of the page, and (8b), where 9 is given by (9c). Proof: Adding the term z T (t)z (t) 0 w(t)T w(t) to dV (t)=dt in (11) and substituting for z (t) from (16), the result follows from the arguments used to derive Lemma 1 where the last column and row blocks in (17) are obtained by applying the standard Schur’s formula [1].
0
Remark 1: It follows from (8a) that the diagonal elements 0Si (1 0 di ), i = 1,2 are negative and, thus, Si > 0, since by assumption di < 1.
9 =P T
1933
PT
0
A2
1
0Y T + P T A0 2
2
0 0S2 (1 0 d2)
0 Y T [ LT ] 2
0 0 0S2 (1 0 d2 ) 0
0
L1T L2T 0Ir
0; Zij i+1 n2n , i = 1,2, j = 1,2,3 and Y q2n that satisfy the LMIs shown in (19) at the bottom of the page, where
"1 ; "2
R
2R
2
2R
4(j) = Q3(j) 0 Q2(j)T + Q1 (A(0j)T + +
2
2
i=1
hi Z
(j ) i2
i=1
Corollary 2: In Case I), the control law of (21) asymptotically stabilizes (1) independently of the delay lengths, for all the system parameters that reside in the uncertainty polytope, if there exist:Q1 > 0, S1 , S2 , Q2(j ) , Q3(j ) 2 Rn2n and Yi 2 Rq2n , i = 0; 1; 2 that satisfy the equations shown at the bottom of the page where
"i Ai(j )T )
+ Y T B (j)T ; j
4g(j) = Q3(j) 0 Q2(j)T + Q1 A0(j)T + Y0T B (j)T ;
= 1; 2; . . . ; N:
K0
= Y Q01 1 :
(j )
we replace Ai
2
(j )
in (19) by Ai
Q2(j ) + Q2(j )T
h1 Q2(j )T h1 Q3(j )T
0 0 0 0
+
3 3 3 3 3 3 3 3
+ B (j) Ki and obtain the following.
2
i=1
hi Zi(1j )
4(j)
0Q j 0 Q j T + i 3 3 3 3 3 3 3
h2 Q2(j )T h2 Q3(j )T
0 0 0 0 0
1
Q2 + Q2T
3 3 3 3 3
( ) 3
2
=1
= Y0 Q101
Ki
= Yi Si01 ;
i = 1; 2:
hi Zi(3j )
0
0
Q1
Q1
A(1j ) (In 0 "1 )S1
A(2j ) (In 0 "2 )S2
0 0 0 0S1
0 0 0 0 0S2
0(1 0 d )S 3 3 3 3 3 3 1
0
1
0(1 0 d )S 3 3 3 3 3 2
2
3 3 3 3
0, Q2 ,
(20)
The previous result represents a delay-dependent sufficient condition for the controller of (4) to guarantee, for Case I), stability over the entire uncertainty polytope. The corresponding delay-independent result is i = In and obtained, still for Case I), by substituting "i = 0, R Zi = 0 in (19) and taking the limit where tends to infinity. The i ! last two row and column blocks of (19a) will disappear due to R 1. Considering, still in the delay-independent case, the more general control law
u(t) =
= 1; 2; . . . ; N:
The state-feedback gains are then given by
The state-feedback gain is then given by
K
j
( ) 3
( ) 1
( )
1
1
1
( ) 2
( )
2
2
1
2
2
1
2
; Zij ; 2 Rn2n , i ,2, j ,2,3 and Y 2 Rq2n that satisfy the LMIs, shown in the equation at the bottom of the next page. where
The state-feedback gain is then given by (20). III. STABILIZATION OF SYSTEMS WITH NORM-BOUNDED UNCERTAINTIES
1( )
The results of Section II were derived for the case where the unknown parameters of (1) lie in a given polytope. An alternative way of dealing with uncertain systems is to assume that the deviation of the system parameters from their nominal values is norm bounded [17]. In our case, consider the system
()
2
()
_ ( ) = (Ai + H 1(t)Ei )x(t 0 i (t)) i + (B + H 1(t)E )u(t) (24) x(s) = (s)s 0 where x(t) and u(t) are defined in Section II and the time delays are = ^ defined in (3). The matrices Ai , i = 0,1,2, B , H and Ei , i = 0; . . . ; 3 are constant matrices of appropriate dimensions. The matrix 1(t) is a 1( ) time-varying matrix of uncertain parameters satisfying ^ 0 0 0 1T (t)1(t) I 8 t: (25) =1 =1 We consider also, for a given positive scalar "^, the following augmented system: 4g = Q 0QT +Q (AT + "i AiT )+ hi Zi +Y T BT : (28a-c) i i 0 _(t) = Ai (t 0 i (t)) + Bu(t) + "^ Hw(t) The state-feedback gain is then given by i 0 K = Y Q : (29) (s) = (s) s 0 Remark 4: The delay-independent version in Case I) is obtained by "^Ei (t 0 i ) + "^E u(t) (26a,b) z (t) =^ "E (t) + solving (28a) and (28c), where "i = 0, Zi = 0 and where the eighth x t
=0
3
3
2
2
1
2
2
=1
=1
2
0
1
=0
1
1
2
0
3
i=1
with the performance index
( )=
1
J w
0
(zT z 0 wT w)d
and the ninth row and column blocks are omitted. In Case II), the corresponding delay-dependent result is obtained by solving the LMIs of Theorem 2 for "i I, i ; , where in (28a) the fourth, fifth, sixth, and seventh row and column blocks are deleted. Remark 5: The results of Theorems 1 and 2 apply the tuning parameters "1 and "2 . The question arises how to find the optimal combination of these parameters. One way to address the tuning issue is to choose for a cost function the parameter tmin that is obtained while solving the feasibility problem using Matlab’s LMI toolbox [18]. This scalar parameter is positive in cases where the combination of the tuning parameters is one that does not allow a feasible solution to the set of LMIs considered. Applying a numerical optimization algorithm, such as the program fminsearch in the optimization toolbox of Matlab
=
(27)
where w 2 L2 is an exogenous signal. It has been explicitly proved in [17], in the case without delays, that the existence of a solution to the Riccati equations or LMIs that are obtained when solving the H1 state-feedback control problem for the augmented system (26) with the index (27), without delays, guarantees the stability of (24), under the same feedback law, for all t that satisfy (25). The proof follows, in fact, from the small gain theorem [16] which can also be applied to our case of retarded systems. The
1( )
(j )
Q2
+Q j T + ( ) 2
3 3 3
2
i=1
j
4^ j
( )
(j )T
( )
hi Zi1
h1 Q2
0Q j 0 Q j T + i 3 3 ( ) 3
( ) 3
2
=1
j
( )
hi Zi3
(j )T
h1 Q3
0h R 3 1
1
=12
(j )T
h2 Q2
(j )T
h2 Q3
0
0h R 2
2
An Improved Stabilization Method for Linear Time-Delay Systems Emilia Fridman and Uri Shaked
1931
2 Rn2n , means that P is symmetric and positive definite. The space of vector functions that are square integrable over [0 1) is denoted by L2 .
P
II. A NEW STABILIZATION METHOD Abstract—In this note, we combine a new approach for linear timedelay systems based on a descriptor representation with a recent result on bounding of cross products of vectors. A delay-dependent criterion for determining the stability of systems with time-varying delays is obtained. This criterion is used to derive an efficient stabilizing state-feedback design method for systems with parameter uncertainty, of either the polytopic or the norm-bounded types. Index Terms—Delay-dependent stability, linear matrix inequality (LMI), stabilization, time-delay systems, time-varying delay.
I. INTRODUCTION The problem of reducing the conservatism entailed in applying finite-dimensional techniques to asses the stability of linear systems with time delay has attracted much attention in the past few years [1]–[6]. All these techniques provide sufficient conditions only for the asymptotic stability of these systems and they entail a considerable conservatism which stems from two main sources. The first cause for conservatism is the model transformation used to describe the system which makes it more amenable for analysis [7], [8] and the second reason for conservatism is the bounding method used to derive the bounds on weighted cross products of the state and its delayed version while trying to secure a negative value to the derivative of the corresponding Lyapunov–Krasovskii functional. The search for the most appropriate model transformation has led to four main approaches [9]–[11]. The most recent one [9], the one that is based on a descriptor representation of the system, which is equivalent to the original system, minimizes the overdesign that stems from the model transformation source of conservatism [11]. The conservatism that stems from the bounding of the cross terms has also been significantly reduced in the past few years. An important result for improving the standard bounding technique of, e.g., [2], has been proposed in [12]. Indeed, combining the later with the descriptor model transformation lead in [10] and [11] to an efficient delay-dependent stability criterion that was also used in synthesis for stabilization and optimal performance. Only recently, an improvement of the bounding technique has been proposed [13]. The latter generalizes the one in [12] and the resulting criteria that are obtained in [13] are, therefore, more efficient than those found in [12]. It is the purpose of this note to combine the bounding method of [13] with the descriptor model transformation of [9] and [11] in order to derive a most efficient stability criterion for systems with time-varying delays. This criterion is then applied to solve the problem of robust stabilizing the system in presence of either norm-bounded or polytopic uncertainties by means of state-feedback control. The resulting criterion is applied to an example taken from [13], and its superiority to the results of the latter is demonstrated. Notation: Throughout this note, the superscript T stands for matrix transposition, Rn denotes the n-dimensional Euclidean space, Rn2m is the set of all n 2 m real matrices, and the notation P > 0, for
We consider the following linear system with time-varying delays: 2
_( ) =
x t
( 0 i (t)) + Bu(t);
( ) = (t); t 2 [0h; 0]
Ai x t i=0
x t
(1) where x(t) 2 Rn is the system state, u(t) 2 Rq is the control input, 0 0, Ai and B are constant n 2 n matrices, is a continuously differentiable initial function, and h is an upper-bound on the timedelays i , i = 1; 2. For simplicity only, we took two delays 1 and 2 . The results of this section can be easily applied to the case of multiple delays 1 ; . . . ; m . The matrices of the system are not exactly known. Denoting
= [ A0
A1
A2
B
]
we assume that
=
N
fj j ; j =1
0 fj 1;
for some
N
fj j =1
=1
(2)
where the N vertices of the polytope are described by
j = [ A0(j )
(j )
A1
(j )
A2
]
B (j ) :
In Section III, we extend our results to the case where the uncertainty in the system parameters obeys the norm-bounded model [17]. As in [11], we consider two different cases for time-varying delays i (t) are differentiable functions, satisfying for all t 0:
0 i (t) hi ; ()
_ ( ) di < 1;
i t
i
= 1; 2:
(3)
0, 0 i (t)
i t are continuous functions, satisfying for all t hi , i ; .
=12
Note that in the past, the Razumikhin’s approach was the only one that was to cope with Case I) of fastly varying delays. The Krasovskii approach for this case was introduced recently in [11]. We seek a control law
( ) = Kx(t)
u t
(4)
that will asymptotically stabilize the system. A. Stability Issue In this section, we consider B descriptor form [9]
_ ( ) =y(t)
= 0. Representing (1) in an equivalent
x t
0 = 0 y(t) +
(5a) 2
Ai i=0
( )0
2
x t
Ai i=1
t
0
t (t)
()
y s ds (5b)
or Manuscript received October 15, 2001; revised February 8, 2002 and June 28, 2002. Recommended by Associate Editor L. Pandolfi. This work was supported by the C&M Maus Chair at Tel Aviv University. The authors are with the Department of Electrical Engineering-Systems, Tel-Aviv University, Tel-Aviv 69978, Israel. Digital Object Identifier 10.1109/TAC.2002.804462
_ ( ) = x_ (0t) 2 0 I 0 = 2 Ai 0I x(t) 0 Ai i=1
Ex t
0018-9286/02$17.00 © 2002 IEEE
i=0
t
0
t (t)
()
y s ds (5c)
1932
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002
with x (t) = colfx(t); y(t)g, E = diagfI; 0g, the following Lyapunov–Krasovskii functional is applied:
V (t) = xT (t)EP x(t) + V2 + V3
and, hence, differentiating the first term of (6) with respect to t gives
d xT (t)EP x(t) dt
(6)
=
V2
=
V3
=
0
P1 P2
P3
2
0 i=1 0h 2 t
P1 > 0 EP
; t
+
t
=1 t0 (t)
i
= PTE 0
and
3
y T (s)Ri y (s)dsd
xT ( )Si x( )d :
(7a-e)
Y
9 =P T
0
Zi1
3
Zi2 ; Zi3
I 0 I 0I + A0 0I
A0
T
x
i (t)
= =
(t)EP x(t) = x (t)P1 x(t)
t
0
t t
0
t t
0
t t
0
91 < 0
t h
[ y (s) x (t) ] T
T
=1
i
Ri
YiT
y T (s)Ri y (s)ds + 2 y T (s)Ri y (s)ds + 2
t
0
t t
0
t
0N
Y
0 NT Z
a b (13)
N = Ni =
Yi 0 [ 0 ATi ] P Zi
Ri
3
and
t
(1 0 di )xT (t 0 i )Si x(t 0 i ) +
0 P T A0i
R
PT
0
T (t)01 (t)
where the first equation shown at the bottom of the next page holds, and where (t) = colfx(t); y (t); x(t 0 1 ); x(t 0 2 )g. Since 01 = 0 the LMIs in (8) lead to V_ < 0, while V 0 and, thus, (1) with B = 0 is asymptotically stable [5], [15]. Choosing in Lemma 1 Si ! 0 and Yi = [ 0 ATi ]P T we obtain the following result for the case B. Corollary 1: Under Case II), (1), with B = 0, is asymptotically stable if there exist n 2 n matrices 0 < P1 ; P2 ; P3 ; Zi1 ; Zi2 ; Zi3 and Ri > 0, i = 1; 2 that satisfy the following LMIs:
(9a-c)
2
Y
T
0
dV (t) dt
P
xT (t)0x(t) 0
T
, R = Ri , Z = Zi , Ai Y = Yi , a = y (s) and b = x(t), we obtain, for i = 1; 2, (14) found at the bottom of the page. Substituting the latter and (12) into (11), we obtain that
T
dV (t) dt
a b
we apply the latter on the expression we have previously obtained for
Proof: Note that T
Y Z
i . From (13), taking
i = 1; 2
2 Si 0 2 + hi Zi + i=1 2 i=1 0 hi Ri i=1 2 Y 2 Y T i i + + : 0 0 i=1 i=1
R YT
where
Zi
= [Yi1 Yi2 ] Zi =
2 T I 0 Ai 20 + P i=0 Ai 0I I 0I i=0 2 Si 0 + i=1 2 0 hi Ri i=1 t 0 1 i (t)= 0 2 xT (t)P T y (s)ds: (12) A i t0 Since, by [13], for any a 2 Rn , b 2 R2n , N 2 R2n2n , R 2 Rn2n , 2 Rn22n , Z 2 R2n22n , the following holds:
02bT N a
0 9 PT 0 Y1T P T A02 0 Y2T A1 0 = 3 0S1 (1 0 d1 ) 0 3 3 0S2 (1 0 d2 ) 0, i = 1; 2 that satisfy the following linear matrix inequalities (LMIs):
Ri
x_ (t)
Substituting (5) into (10), we obtain (11), as shown at the bottom of the page, where
where
P
= 2xT (t)P1 x_ (t) = 2xT (t)P T
0
t h
[ 0 ATi ]P T Zi
y T ( )Ri y ( )d
0;
i = 1; 2
0 i
(11)
y (s) ds x(t)
y T (s)(Yi 0 [ 0 ATi ] P ) x(t)ds +
t
0
t
x(t)T Zi x(t)ds
x_ T (s)(Yi 0 [ 0 ATi ] P ) x(t)ds + i x(t)T Zi x(t)
y T (s)Ri y (s)ds + 2xT (t)(Yi 0 [ 0 ATi ] P ) x(t) 0 2xT (t 0 i )(Yi 0 [ 0 ATi ] P ) x(t) + hi x(t)T Zi x(t):
(14)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002
where
Zi =
Zi1 Zi2 3 Zi3 ;
91 =P
T
2
0
i=0
Ai
0 + hi Zi + 0 i=1 2
In the sequel, it will be important to determine the conditions for achieving H1 norm of (1) less than 1, where u is the input vector and the controlled output is given by
i = 1; 2 I 0I +
2
0
i=0
2
i=1
Ai
z (t) = Lx(t) + L1 x(t 0 1 ) + L2 x(t 0 2 ):
T
I 0I
P
hi Ri :
Remark 2: A question may arise as to whether the standard Lyapunov criterion can be restored when letting h go to 0. Taking Ri = I , Zi = 01=2 , ! 1, Yi = [ 0 ATi ]P and 0 < Si ! 0 we obtain
+ =
0
I 0 + A0 0I A0 2 2 Yi Yi
+
0
i=1
P2T (
2
i=0
For P3 = I; becomes
P1 (
i=1
Ai ) + (
3
I 0I
B. State-Feedback Stabilization The results of Lemma 1 can also be used to verify the stability of the closed loop obtained by applying (4) to (1) (with B 6= 0) if we replace A0 in (8a) by A0 + BK and verify that the resulting inequality is feasible over the polytope defined in (2) by solving the LMI simultaneously for all the N vertices, applying the same P1 , Pi , Si , Yi1 , Yi2 , and Ri , i = 1,2. The problem with (8a) is that it is linear in its variables, only when the state-feedback gain K is given. In order to find K , consider the inverse of P . It is obvious from the requirement of 0 < P1 , and the fact that in (8)0(P3 + P3T ) must be negative definite, that P is nonsingular. Defining
0
2
i=0
ATi )P2 P1 0 P2T + ( 3
Ai ) + (
2
i=0
2
1
ATi )P3
:
i=0 PT
0P 0
2
i=0
P
T
! 0 and P = P 2
T
3
> 0 the requirement that 9 < 0
ATi )P1 < 0;
P1 > 0:
(15)
Q1 0 Q2 Q3 1 =diagfQ; I g
It follows from (15) that if the system with h = 0 is asymptotically stable, then there exists P1 > 0 that solves (15) and, thus, (8a),(b) possess a solution for small enough h > 0. The latter can be readily used to verify the stability of (1) over the uncertainty polytope (2) [1]:
=
N j =1
j ; for some 0 fj fj
1;
N j =1
P 01 =Q =
fj = 1
Zi =
[Yi1
j = [ A(0j ) A(1j ) A(2j ) ]
01 =
i=1
hi Zi +
I
0
(Yi 0 [ 0 ATi ] P ) + YiT
3 3
9 PT
3 3 3 3
0
B
0Iq 3 3 3
Zi1 Zi2 = QT Zi Q and R i = Ri01 , i = 1,2 and choosing ZiT2 Zi3 Yi2 ] = "i AiT [P2 P3 ] , where "i 2 Rn2n is a diagonal
0 P T A0i
0Y T + P T A0 0S (1 0 d ) 3
[I 0]
1
1
1
PT
0
A1
0YT 1
0 0S1 (1 0 d1)
3 3
(18b)
matrix, we obtain, similarly to [14], the following. Theorem 1: The control law of (4) asymptotically stabilizes (1) for all the delays that belong to Case I) and for all the system parameters that reside in the uncertainty polytope, if for some diagonal matrices
by solving the LMI simultaneously for all the N vertices, applying the same P1 , Pi , Si , Yi1 , Yi2 , and Ri , i = 1,2.
2
(18a)
we multiply (8a) by 1T and 1, on the left and on the right, respectively, and (8b), on the left and on the right, by diagfRi01; QT g and diagfRi01; Qg, respectively . Applying Schur formula to the emerging quadratic term in Q, denoting Si = Si01 ,
where the N vertices of the polytope are described by
0 +
(16)
Similarly to the derivation of the bounded real lemma (BRL) in [11], we obtain the following. Lemma 2: Under Case I) the H1 norm of (1) and (16) is less than one if there exist n 2 n matrices 0 < P1 ; P2 ; P3 ; Si ; Yi1 ; Yi2 ; Zi1 ; Zi2 ; Zi3 and Ri , i = 1,2 that satisfy (17), as shown at the bottom of the page, and (8b), where 9 is given by (9c). Proof: Adding the term z T (t)z (t) 0 w(t)T w(t) to dV (t)=dt in (11) and substituting for z (t) from (16), the result follows from the arguments used to derive Lemma 1 where the last column and row blocks in (17) are obtained by applying the standard Schur’s formula [1].
0
Remark 1: It follows from (8a) that the diagonal elements 0Si (1 0 di ), i = 1,2 are negative and, thus, Si > 0, since by assumption di < 1.
9 =P T
1933
PT
0
A2
1
0Y T + P T A0 2
2
0 0S2 (1 0 d2)
0 Y T [ LT ] 2
0 0 0S2 (1 0 d2 ) 0
0
L1T L2T 0Ir
0; Zij i+1 n2n , i = 1,2, j = 1,2,3 and Y q2n that satisfy the LMIs shown in (19) at the bottom of the page, where
"1 ; "2
R
2R
2
2R
4(j) = Q3(j) 0 Q2(j)T + Q1 (A(0j)T + +
2
2
i=1
hi Z
(j ) i2
i=1
Corollary 2: In Case I), the control law of (21) asymptotically stabilizes (1) independently of the delay lengths, for all the system parameters that reside in the uncertainty polytope, if there exist:Q1 > 0, S1 , S2 , Q2(j ) , Q3(j ) 2 Rn2n and Yi 2 Rq2n , i = 0; 1; 2 that satisfy the equations shown at the bottom of the page where
"i Ai(j )T )
+ Y T B (j)T ; j
4g(j) = Q3(j) 0 Q2(j)T + Q1 A0(j)T + Y0T B (j)T ;
= 1; 2; . . . ; N:
K0
= Y Q01 1 :
(j )
we replace Ai
2
(j )
in (19) by Ai
Q2(j ) + Q2(j )T
h1 Q2(j )T h1 Q3(j )T
0 0 0 0
+
3 3 3 3 3 3 3 3
+ B (j) Ki and obtain the following.
2
i=1
hi Zi(1j )
4(j)
0Q j 0 Q j T + i 3 3 3 3 3 3 3
h2 Q2(j )T h2 Q3(j )T
0 0 0 0 0
1
Q2 + Q2T
3 3 3 3 3
( ) 3
2
=1
= Y0 Q101
Ki
= Yi Si01 ;
i = 1; 2:
hi Zi(3j )
0
0
Q1
Q1
A(1j ) (In 0 "1 )S1
A(2j ) (In 0 "2 )S2
0 0 0 0S1
0 0 0 0 0S2
0(1 0 d )S 3 3 3 3 3 3 1
0
1
0(1 0 d )S 3 3 3 3 3 2
2
3 3 3 3
0, Q2 ,
(20)
The previous result represents a delay-dependent sufficient condition for the controller of (4) to guarantee, for Case I), stability over the entire uncertainty polytope. The corresponding delay-independent result is i = In and obtained, still for Case I), by substituting "i = 0, R Zi = 0 in (19) and taking the limit where tends to infinity. The i ! last two row and column blocks of (19a) will disappear due to R 1. Considering, still in the delay-independent case, the more general control law
u(t) =
= 1; 2; . . . ; N:
The state-feedback gains are then given by
The state-feedback gain is then given by
K
j
( ) 3
( ) 1
( )
1
1
1
( ) 2
( )
2
2
1
2
2
1
2
; Zij ; 2 Rn2n , i ,2, j ,2,3 and Y 2 Rq2n that satisfy the LMIs, shown in the equation at the bottom of the next page. where
The state-feedback gain is then given by (20). III. STABILIZATION OF SYSTEMS WITH NORM-BOUNDED UNCERTAINTIES
1( )
The results of Section II were derived for the case where the unknown parameters of (1) lie in a given polytope. An alternative way of dealing with uncertain systems is to assume that the deviation of the system parameters from their nominal values is norm bounded [17]. In our case, consider the system
()
2
()
_ ( ) = (Ai + H 1(t)Ei )x(t 0 i (t)) i + (B + H 1(t)E )u(t) (24) x(s) = (s)s 0 where x(t) and u(t) are defined in Section II and the time delays are = ^ defined in (3). The matrices Ai , i = 0,1,2, B , H and Ei , i = 0; . . . ; 3 are constant matrices of appropriate dimensions. The matrix 1(t) is a 1( ) time-varying matrix of uncertain parameters satisfying ^ 0 0 0 1T (t)1(t) I 8 t: (25) =1 =1 We consider also, for a given positive scalar "^, the following augmented system: 4g = Q 0QT +Q (AT + "i AiT )+ hi Zi +Y T BT : (28a-c) i i 0 _(t) = Ai (t 0 i (t)) + Bu(t) + "^ Hw(t) The state-feedback gain is then given by i 0 K = Y Q : (29) (s) = (s) s 0 Remark 4: The delay-independent version in Case I) is obtained by "^Ei (t 0 i ) + "^E u(t) (26a,b) z (t) =^ "E (t) + solving (28a) and (28c), where "i = 0, Zi = 0 and where the eighth x t
=0
3
3
2
2
1
2
2
=1
=1
2
0
1
=0
1
1
2
0
3
i=1
with the performance index
( )=
1
J w
0
(zT z 0 wT w)d
and the ninth row and column blocks are omitted. In Case II), the corresponding delay-dependent result is obtained by solving the LMIs of Theorem 2 for "i I, i ; , where in (28a) the fourth, fifth, sixth, and seventh row and column blocks are deleted. Remark 5: The results of Theorems 1 and 2 apply the tuning parameters "1 and "2 . The question arises how to find the optimal combination of these parameters. One way to address the tuning issue is to choose for a cost function the parameter tmin that is obtained while solving the feasibility problem using Matlab’s LMI toolbox [18]. This scalar parameter is positive in cases where the combination of the tuning parameters is one that does not allow a feasible solution to the set of LMIs considered. Applying a numerical optimization algorithm, such as the program fminsearch in the optimization toolbox of Matlab
=
(27)
where w 2 L2 is an exogenous signal. It has been explicitly proved in [17], in the case without delays, that the existence of a solution to the Riccati equations or LMIs that are obtained when solving the H1 state-feedback control problem for the augmented system (26) with the index (27), without delays, guarantees the stability of (24), under the same feedback law, for all t that satisfy (25). The proof follows, in fact, from the small gain theorem [16] which can also be applied to our case of retarded systems. The
1( )
(j )
Q2
+Q j T + ( ) 2
3 3 3
2
i=1
j
4^ j
( )
(j )T
( )
hi Zi1
h1 Q2
0Q j 0 Q j T + i 3 3 ( ) 3
( ) 3
2
=1
j
( )
hi Zi3
(j )T
h1 Q3
0h R 3 1
1
=12
(j )T
h2 Q2
(j )T
h2 Q3
0
0h R 2
2
