state-feedback control of continuous-time systems with state delays.
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Finite Horizon State-Feedback Control of Continuous-Time Systems with State Delays Emilia Fridman and Uri Shaked Abstract—The finite horizon control of time-invariant linear systems with a finite number of point and distributed time delays is considered. The controller is obtained by solving coupled Riccati-type partial differential equations. The solutions to these equations and the resulting controllers are approximated by series expansions in powers of the largest delay. Unlike the infinite horizon case, these approximations possess both regular and boundary layer terms. The performance of the closed-loop system under the memoryless zero-approximation controller is analyzed. –state-feedback control, Index Terms—Asymptotic approximations, Riccati type partial differential equations, singular perturbations, timedelay systems.
I. PROBLEM FORMULATION Throughout this paper we denote by j 1 j the Euclidean norm of a vector or the appropriate norm of a matrix. Given tf > 0, let L2 [0; tf ] be the space of the square integrable functions with the norm k 1 kL and let C [a; b] be the space of the continuous functions on [a; b] with the norm k1k . We denote xt = x(t + ); y t = y (t 0 ); 2 [0h; 0]. Prime denotes the transpose of a matrix and colfx; y g denotes a column vector with components x and y . Consider the system
1
x_ (t) = L(xt (1)) + Bu(t) + Dw(t) z (t) = colfCx(t); u(t)g; t0 2 [0h; 0] x() = x0 ();
R
R
R
l
L(xt (1)) =
r
i=0
Ai xt (0hi ) +
0
0h
1
(1)
where x(t) 2 is the state vector, u(t) 2 is the control signal, w(t) 2 q is the exogenous disturbance, z (t) 2 p is the observation vector, and B; C , and D are constant matrices of appropriate dimensions. The Rn -valued function L(1) which carries Rn -valued functions on [0h; 0] into Rn is defined as follows: n
and where M1 = M10 ; M2 = M20 are matrices denoting the initial weightning condition. The problem is to find a state-feedback controller which ensures that J 0 for all w 2 L2 [0; tf ] and for all initial conditions x0 2 L2 [0h; 0]. In the infinite horizon case such a controller (for zero initial condition x0 = 0) has been designed in [1]–[6] (see also references therein). In [1] and [2] the controller has been obtained by solving Riccati operator equations. In [3] and [4] delay-independent and in [6] delay-dependent memoryless controllers have been designed. In [5] the controller (with memory) has been derived from Riccati-type partial differential equations (RPDEs) or inequalities, and the solution of the RPDEs has been approximated by expansions in the powers of the delay. In [7] the gradient of J with respect to h at h = 0 has been computed. In [8] and [9] bounded real criteria have been obtained. Asymptotic series solutions of systems with small delay have been constructed in [10]–[12]. In many engineering cases (target maneuver, missile guidance, etc.) a control session of limited time length is needed. In such cases the effect of the initial conditions is most important and the results of [1]–[6] cannot provide a satisfactory control strategy. In the present paper, we generalize the results of [5] to the finite horizon case. Unlike [5], the required controllers are time-varying, they are obtained by solving coupled finite horizon RPDEs. For small delays, similarly to the case of singularly perturbed systems (see [10]–[13]), the controllers are affected by the boundary-layer phenomenon. The main contribution of the paper is the construction, for the first time, of an asymptotic solution to the important class of finite horizon RPDEs that are encountered with the finite-horizon LQ control (see [14]), and with the H control. Proofs of Theorem 1 and Lemma 2 are given in the Appendix.
R
A01 (s)xt (s) ds
II. MAIN RESULTS
1
A. H -Controller Design
Consider the following RPDEs with respect to the n 2 n-matrices P (t); Q(t; ), and R(t; ; s):
P_ (t) + A00 P (t) + P (t)A0 +
(2)
r +
where 0h = 0hr < 0hr01 < 1 1 1 < 0h1 < 0h0 = 0; A0 ; A1 1 1 1 ; Ar are constant matrices and A01 (s) is a square integrable matrix function. Denote
F (xt )( ) =
r i=1
Ai xt (0hi 0 ) i ( ) +
0h
i=1 0 +
0
+
A01 (p)xt (p 0 ) dp
and where i is the indicator function for the set [0hi ; 0], i.e., i ( ) = 1 if 2 [0hi ; 0] and i ( ) = 0 otherwise. Given > 0, and assuming that w 2 L2 [0; tf ] and x0 2 L2 [0h; 0], we consider the following performance index:
J where
=
kzk 0 kwk 0 W (x0 ) 2
W (x0 ) = x0 (0)M1 x(0) +
2 L
0
0h
(4)
F 0 (x0 )(s)M2 F (x0 )(s) ds (5)
Manuscript received July 26, 1999; revised December 23, 1999. Recommended by Associate Editor, Y. Yamamoto. The authors are with the Department of Electrical Engineering-Systems, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(00)10012-1.
0h
i=1
=
Q(t; )A01 () d A001 ()Q0 (t; ) d = 0
0[A00 + P (t)S]Q(t; ) 0 0
0
0h
A0i Q0 (t; 0hi )
Q(t; 0hi )Ai + P (t)SP (t) + C 0 C
@ @ Q(t; ) + Q(t; ) @t @
(3)
2 L
0h
r
A0
r i=1
(6)
A0i R(t; 0hi ; )
01 (s)R(t; s; ) ds
@ @ @ R(t; ; s) + R(t; ; s) + R(t; ; s) @t @ @s 0 = 0Q (t; )SQ(t; s) P (t) = Q(t; 0); Q(t; ) = R(t; 0; ); 2 [0; h]; s 2 [0; h] R(t; ; s) = R0 (t; s; ); P (tf ) = 0; Q(tf ; ) = 0; R(tf ; ; s) = 0
(7)
(8) (9) (10)
where S = 02 DD0 0 BB 0 . A solution of (6)–(10) is a triple of n 2 n-matrices fP (t); Q(t; ); R(t; ; s)g t 2 [0; tf ]; 2 [0h; 0]; s 2 [0h; 0],
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000
() ( )
(
)
where P t ; Q t; , and R t; ; s are continuous and piecewise continuously differentiable functions of their arguments that satisfy (6)–(10) for almost every t; , and s. Lemma 1: Given > . Let (6)–(10) have a solution on ; tf that for some n 2 n matrices 1 > and 2 > satisfies the following conditions:
0 1
0
1
[0 ]
0
0 5 (0) 0 0:5P 0 (0) 0 h11 0 (0 )101 1Q(0; s) 0 0 0:5h12 0 0:5 R(0; s; )1201R(0; ; s) d 0; 0h 8 s 2 [0h; 0]:
M1 0 : P M2 0 Q0 ; s
0
( ) = 0B0 P (t)x(t) +
0h
( ) ( )( )
Q t; F xt d
(11)
(12)
solves the H1 -control problem with the performance level of . Proof: Let x t be a solution of (1). Consider the following Lyapunov–Krasovskii functional [14]:
()
(
V t; xt
0
) = x(t)0 P (t)x(t) + 2x0 (t) +
0
0
0h 0h
(
0h
( ) ( )( )
Q t; F xt d
F 0 xt s R t; s; F xt ds d:
( )( ) (
) ( )( )
Remark 1: In the case of LQ problem from (15) it follows that
minu kzkL2 = V (0; x0 ) and similarly t min jzj2 dt u t = V (t0 ; x0 ); 8 t0 tf ; 8 x0 2 L2 [0h; 0]: Hence, V (t; x) 0 in the LQ case.
Remark 2: Note that a certain amount of overdesign is introduced by the conditions of (11). This overdesign stems from the bounding in (18). In the case of the zero initial conditions x ; 2 0h; the conditions of (11) are not relevant and the controller of (12) solves the H1 control problem under the sole assumption that (6)–(10) have a solution on ; tf .
( )=0
Then, the controller
u3 t
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(
B. Asymptotic Solutions to the RPDEs
( ) = P0 (t) + h[P1 (t) + 51P ( )] + h2 [P2 (t) + 52P ( )] + 1 1 1 ; Q(t; h ) = Q0 (t; ) + h[Q1 (t; ) + 51Q ( ; )] + h2 [Q2 (t; ) + 52Q ( ; )] + 1 1 1 ; R(t; h; h) = R0 (t; ; ) + h[R1 (t; ; ) + 51R ( ; ; )] + h2 [R2 (t; ; ) + 52R ( ; ; )] + 1 1 1 ; tf 0 t (19) = ; 2 [01; 0]; 2 [01; 0]: h P t
(13)
(14)
Expansion (19) has a typical for singular perturbations form: it includes the “outer expansion” (regular) terms fPi ; Qi ; Ri g; i ; 111 and the boundary-layer correction terms iP ; iQ , and iR ; i ; 1 1 1 The “outer expansion” terms constitute the major part of the solution that satisfies (6)–(9) for t 2 ; tf ; 2 0 ; ; 2 0 ; . The boundary-layer correction terms will be chosen such that (19) satisfies the terminal conditions of (10) and that
where
w3 (t) = 02 D0
( ) ( )+
P txt
0
0h
( ) ( )( )
Q t; F xt d :
(
t
) 0 V (0; x0 ) + [jzj2 0 2 jwj2 ] dt 0 = 0 2 kw 0 w3 kL + ku 0 u3 kL :
(15)
We show next that (11) implies
= W (x0 ) 0 V (0; x0 ) 0: Denoting v = x(0) and y (s) = F (x0 )(s) we have 0 d = v 0 (M1 0 P (0))v + y 0 (s)M2 y (s) ds d
0 2v0 0
0
0
0h
0
0h 0h
(16)
(0 ) ( )
)()
(17)
2
0h
v 0 Q(0; s)y (s) ds
hv0 11 v +
2
0
0h
0
0h
y 0 (s)Q0 (0; s)101 Q(0; s)y (s) ds 1
( ) (0
)()
hy0 (s)12y(s) + y0 (s)
=
and hence
y 0 s R ; s; y d
5 5
0h
(0
R ; s;
)1201R(0; ; s) dy(s)
and (11). Finally, (16), (15), and (10) imply that J
(18)
0 for u = u3 .
(20)
=
=
=
( ) = P0 (t);
_ ( )+
r
i=0
A0i P0 t
+ C 0 C = 0;
=
=
( )=0 ( 0 )= ( )
Then, from (6), we have
P0 t
5
=
(
Q0 t; 0
! 1:
The boundary-layer correction terms depend on as on the independent variable and do not depend on h. Since is a stretched-time variable around t tf , (20) asserts that iP ; iQ , and iR are essential only around t tf and they thus provide a correction to the outer expansion at the terminal point t tf . We substitute (19) in (6)–(9) and equate, separately, outer expansion and boundary-layer correction terms with the same powers of h. We notice that for t tf 0 h ; h; and s h we have @=@t 0h01 @=@ ; @=@ h01 @=@; and @=@s h01 @=@. Thus, for the zero-order terms we obtain from (7)-(9) @ Q0 t; @ @ @ R0 t; ; R0 t; ; @ @ P0 t ; R0 t; ; Q0 t; Q0 t;
( )=0 ( )+ ( 0) = ( )
Then, (16) follows from (17), the inequalities 0
as
20
= =
Q ; s y s ds
( ) (0
j5iP ( )j + sup j5iQ ( ; )j 2[01;0] + sup j5iR ( ; ; )j ! 0 ; [ 1;0]
0h
y 0 s R ; s; y d ds:
=0 1 5 5 5 = [0 ] [ 1 0] [ 1 0]
12
It follows from (14) that
V tf ; xt
= 0
For simplicity we assume that A01 further on. The H1 controller has been found above by solving a set of coupled PRDEs. Finding a solution to the latter is not an easy task and we are, therefore, looking for a solution to the RPDEs in a form of asymptotic expansion in the powers of the delay h
)
) = 0x0 (t)C 0 Cx(t) 0 2 jw(t) 0 w3 (t)j2 + 2 jw(t)j2 + ju(t) 0 u3 (t)j2 0 ju(t)j2
0]
[0 ]
Differentiating V t; xt with respect to t and integrating by parts, we obtain, similarly to [5], that
d V t; xt dt
[
( )+
R0 t; ; r
) = P0 (t):
( ) + P0 (t)SP0 (t) P0 (tf ) = 0: i=0
(21)
P0 t Ai
(22)
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The latter is the well-known Riccati differential equation (RDE) that corresponds to (1) for h = 0. Our main assumption is as follows. A1. For a specified value of > 0, the RDE of (22) has a bounded solution on [0; tf ]. Assumption A1 means that the H state-feedback control problem for (1) without delay has a solution. If this were not the case, even P0 , the zero-order term in (19), would not exist. Note that P0 = P00 . To determine the first-order terms we start with the equations for Q1
Therefore
51Q ( ; ) = 0; + > 0 51R ( ; ; ) = 51Q ( + ; 0 ) = 0 + > 0; :
1
@ Q1 (t; ) = 0M0 (t)P0 (t) 0 P_0 (t) @ Q1 (t; 0) = P1 (t);
M=
r
i=0
Ai + SP0 :
The higher order terms of the outer expansions can be similarly found. We obtain next the boundary-layer terms and show by induction that
5iP ( ) = 0; 5iQ ( ; ) = 0; 5iR ( ; ; ) = 0;
(23)
Q1 (t; ) = P1 (t) 0 [M0 (t)P0 (t) + P_0 (t)]:
+
r i=1
r i=1
gi A0i (P0 M + P_0 )
gi (M0 P0 + P_0 )Ai
P1 (tf ) + 51P (0) = 0;
gi
=0
= hi =h
0 1. We derive the
(24)
_ 1P ( ) = 0. Since 51P vanishes for ! It follows from (6) that 5 1, we have 51P ( ) 0; 0. Hence, P1 (tf ) = 0, and P1 is a solution to the linear differential equation (24) with the latter terminal condition. For 51Q ; R1 , and 51R we obtain from (7), (8), and (21) @ 51Q ( ; ) 0 @ 51Q ( ; ) = 0 Q1 (tf ; ) + 51Q (0; ) = 0 51Q ( ; 0) = 51P ( ) = 0 @ @ R1 (t; ; ) + R1 (t; ; ) = 0P0 (t)SP0 (t) 0 P_0 (t) @ @ (25) R1 ( ; 0; ) = Q1 ( ; ) @ @
and
@ @ @ 51R ( ; ; ) 0 @ 51R ( ; ; ) 0 @ 51R ( ; ; ) = 0 @ R1 (tf ; ; ) + 51R (0; ; ) = 0 51R ( ; 0; ) = 51Q ( ; ): (26) Note that Q1 (tf ; ) = 0P_ 0 (tf ) . Then, for find successively
(29)
5_ mP ( ) = fm( ) 5mP (m 0 1) = 0 @ @ 5mQ ( ; ) 0 @ 5mQ ( ; ) = m ( ; ) @ 5mQ ( ; 0) = 5mP ( ) Qm (tf ; ) + 5mQ (0; ) = 0
Substituting this expression into the equation for P1 , we obtain
P_1 + M0 P1 + P1 M +
> i01 + > i01 + > i 0 1; :
We assume that (29) is satisfied for all i m following equations for 5mP ; 5mQ , and 5mR :
Then
(28)
0 and t 2 [0; tf ], we
_ 51Q ( ; ) = ( + )P0 (tf ); if 0 if > 0 0; R1 (t; ; ) = R10 (t; ; ) = 0 [P0 (t)SP0 (t) + P_0 (t)] + Q1 (t; 0 ); 51R (0; ; ) = P_0 (tf ) 51R ( ; ; ) = 501R ( ; ; ) _ = ( + )P0 (tf ); if 0 ; (27) if > 0; : 0;
@ @
@ @ 5mR ( ; ; ) 0 @ 5mR ( ; ; ) 0 @ 5mR ( ; ; ) = m ( ; ; ) 5mR ( ; 0; ) = 5mQ ( ; ); Rm (tf ; ; ) + 5mR (0; ; ) = 0 where fm and m are known functions that vanish for > m 0 1, and m is a known function that vanishes for + > m 0 2; . From these equations we find
5mP ( ) =
0
m 1
fm (s) ds
and thus (29) holds for 5mP since fm (s) = 0 for > m 0 1. Further
5mQ (0 ; + ) + m (s; 0s + + ) ds; 5mQ ( ; ) = 5 (0 + ) mP 0 m (0s + + ; s) ds; 0
and 5mQ satisfies (29) since 5mP ( + ) = and m ( ; ) = 0 for > m 0 1. Finally
if
0
if >
0 for +
0
> m01
5mR ( ; ; ) = 50mR ( ; ; ) m (s; 0s + + ; 0s + + ) ds 0 + 5mR (0; + ; + ); if 0; = 0 m (0s + + ; s; s + 0 ) ds 0 if > 0; : + 5mQ ( + ; 0 ); (30) Conditions (29) for 5mR readily follow from (30) and the properties of 5mQ and m .
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000
1 Control
C. Near-Optimal H
Theorem 1: Under A1 the following holds for all small enough time-delay h i) The system of (6)–(10) has a solution. This solution is approximated, for any integer m, by
P (t) = P0 (t) + Q(t; h ) = P0 (t) + R(t; h; h) = P0 (t) +
m i=1 m i=1 m i=1
hi [Pi (t) + 5iP ( )] + O(hm+1 ) hi [Qi (t; ) + 5iQ ( ; )] + O(hm+1 ) hi [Ri (t; ; ) + 5iR ( ; ; )]
+ O(hm+1 ) = tf 0 t ; 2 [01; 0]; 2 [01; 0] h
(31)
where the boundary-layer terms satisfy (29), and jO(hm+1 )j is a positive scalar which is independent of
chm+1 , where c h; t; , and . ii) If additionally
P0 (0) < M1 ;
M2 > 0
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Lemma 2: Under A1 the controller u0 (t) = 0B 0 P0 (t)x(t) for the zero initial condition x0 = 0 guarantees i) for all small enough h a performance level of ; ii) for all h, a performance level of , where
2 = tf e2t 2 2
1
(32)
D. The Zero-Order Controller Performance We study the performance of the system under the zero-order controller u0 (t) = 0B 0 P0 (t)x(t) which solves the H1 -control problem for (1) without delay. For simplicity we consider the case of x0 = 0. Applying u0 to (1), we obtain
0 0
(
0
(
)
)
t; t0 2 [0; tf ]:
(36)
z = colfx; ug
(37)
2
0
the form
P_1 + 2 tan(tf 0 t)P1 + 2 = 0;
P1 (tf ) = 0:
From the latter equation, (27), and (28) we find
P1 = tan(tf 0 t) + (t2f 0 t) ; Q1 = P1 + cos (tf 0 t) 51Q = 0( + ) (0 0 ) R1 (t; ; ) = R10 (t; ; ) = + P1 (t) 51R ( ; ; ) = 501R ( ; ; ) = 0( + ) (0 0 ); where (s) = 1 for s 0 and (s) = we obtain for 0 < tf < =2 that
and (34)
X (t; t0 ) be the transition matrix of the system of (34), i.e., X (t; t0 ) = 0 for t < t0 , X (t0 ; t0 ) = In and X (t; t0 ) satisfies (34) for t t0 . Let X0 (t; t0 ) be the transition matrix of (34) without delay, i.e., where hi = 0. Then there exist scalars 0 > 0; > 0; ; and such that for small enough h the following inequalities are valid: 0
+ 2
2
0 for s < 0. Since Q0 = P0
u0 (t) = 02 tan(tf 0 t)x(t)
Let
jX (t; t )j e t0t jX (t; t )j e t0t ;
1
2
1=4 the latter RDE has a bounded solution on > 0. Choosing = 1=5 < 1=4 we find that P = tan(tf 0 t) and thus for tf < =2 (38) has a bounded solution on [0; tf ]. It is readily seen that conditions (32) hold. Equation (24) has
Ai x(t 0 hi ) + Dw(t)
A(t) = A0 0 BB 0 P0 (t) ~ (t); C~ (t) = colfC; 0B 0 P0 (t)g: z = Cx
Ai
i=1
jAi j hi
Consider the following system:
Note that for 2 [0; tf ] for all tf
where 50P = 0; Q01 = 501;Q = 50;Q = 0; Q0 = P0 . The approximate controller um guarantees an attenuation level of + O(hm+1 ). It follows from Theorem 1 that a high-order approximate controller improves the performance polynomially in the size of the small timedelay h.
i=1
i=1
2
r
> tan tf and M2 > 0. From (22) we obtain P0 (tf ) = 0: (38) P_0 (t) + ( 02 0 4)P02 + 1 = 0;
(33)
r
r
2
and J of (4) with M1
[Qi01 (t; ) + 5i01;Q ( ; )]x(t + h ) d
x_ (t) = A(t)x(t) +
01 A+
0
x_ (t) = x(t) 0 x(t 0 h) + 2u 0 w;
i=0
01
2
E. Example
u(xt ) = um (xt ) + O(hm+1 ) m um (xt ) = 0 hi B 0 [Pi (t) + 5iP ( )]x(t)
+
2 1 + 2 0 tf e2 t
2
and where for = 0 ( = 0) one has to take limit ! 0 ( ! 0). It follows from Lemma 2 that the controller u0 guarantees a performance level for all small time delays and it guarantees a performance level for all delays. Note that ! for h ! 0. Given > 0 and h, in order to make certain that u0 leads to a performance level of one can verify conditions in terms of differential linear matrix inequalities or Riccati differential inequalities (RDI) that were formulated for the case of one delay in [8] and can be easily generalized to the case of r delays.
then the controller of (12) is approximated by
0
0 1 jDj jC j + kB0 P kL
(35a) (35b)
u1 (t) = u0 (t) 0 2h
1 P (t)x(t) + tan(tf 0 t) 1
0
01
x(t + h ) d :
Consider now the performance of (37) under u = u0 and x0 = 0 for tf = 1:1. Applying the delay-dependent criterion p of [8] on the closed-loop system we find that u0 achieves = (1= 5) for all delays h 2 (0; 0:027], since the corresponding RDIs have bounded solutions on [0, 1.1]. For h = 0:028 the solutions to the RDIs encounter escape points and the criterion of [8], which provide a sufficient p condition only, cannot therefore be used to verify the level = (1= 5).
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III. CONCLUSIONS
1
A solution to the state-feedback H control of linear time-invariant systems with state time delays in the finite horizon case is presented. The controller is obtained by solving RPDEs. An approximate solution to the RPDEs has been constructed by expansion in powers of the largest delay. The theory that has been developed in this paper shows that similarly to the case of singularly perturbed systems [13], for small delays our controllers are affected by the boundary-layer phenomenon. The high order approximate controller improves the performance of the closed-loop system polinomially in the size of the delay. The memoryless zero-approximation may, in many cases, be sufficient for robustly achieving the required performance. It is shown that the performance of the system under such a controller is robust for small time delays. Explicit formula for the guaranteed performance level is obtained for this case in terms of the coefficients of the system.
Proof of Theorem 1: i) To prove the validity of (31) we consider the equations for the remainders
h
Pm+1
=P 0
m
i=0
= Q(t; ) 0
m
i=0
hi Qi t; h01
hm+1 Rm+1 (t; ; s)
= R(t; ; s) 0
m
i=0
+ 5iQ h01 (tf 0 t); h01
hi Ri t; h01 ; h01 s
P_m+1 + Pm+1 M + M0 Pm+1
+
i=1 r i=1
t t r
i=1
8(t; p)
r i=1
0 (p; 0hi ) 0 Q0 (p; 0)] A0i [Qm +1 m+1
[Qm+1 (p; 0hi ) 0 Qm+1 (p; 0)]Ai
+ E m (p; h; hPm+1 (p)) 80 (t; p) dp Qm+1 (t; )
0 (t; 0)] A0i [Q0m+1 (t; 0hi ) 0 Qm +1
[Qm+1 (t; 0hi ) 0 Qm+1 (t; 0)]Ai
+ Em (t; h; hPm+1 (t)) = 0 @ @ Qm+1 (t; ) + Qm+1 (t; ) @t @ = 0M0 Qm+1 (t; ) 0
r
i=1
(39)
A0i [Rm+1 (t; 0hi ; )
0 Rm+1 (t; 0; )] + h01 gm (t; ) + Gm (t; h; hPm+1 (t); hQm+1 (t; )) @ Rm+1 (t; ; s) @t @ + @ Rm+1 (t; ; s)
@ + @s Rm+1 (t; ; s) + h01 km (t; ; s) + Km (t; h; ; s; hQm+1 (t; ); hQm+1 (t; s)) = 0 Pm+1 (t) = Qm+1 (t; 0 Qm+1 (t; ) = Rm+1 (t; 0; ) 0 (t; s; ) Rm+1 (t; ; s) = Rm +1 Pm+1 (tf ) = 0; Qm+1 (tf ; ) = 0; R(tf ; ; s) = 0:
8(t; p)Gm (p; p 0 t + ) dp; if t 0 tf = 8(t; t 0 )Pm+1 (t 0 ) + 8(; p)Gm (p + t 0 ; p) dp; 0 if t 0 < tf Rm+1 (t; ; s) = Rm0 +1 (t; s; ) t = K m (p; p + 0 t; p + s 0 t) dp; t if tf 0 t 0; s Rm+1 (t; ; s) = Rm0 +1 (t; s; ) = 0 K m (p 0 + t; p; p + s 0 ) dp 0 + Qm+1 (t 0 ; s 0 ); if tf 0 t > 0; s : t
in the following expansions: r
=0
t
+ 5iR h01 (tf 0 t); h01 ; h01 s)
+
Pm+1 (t)
+
hi Pi
hm+1 Qm+1 (t; )
E m (t) = Em (t; h; hPm+1 (t)) Gm (t; ) = h01 gm (t; ) + Gm (t; h; hPm+1 (t); hQm+1 (t; )) K m (t; ; s) = h01 km (t; ; s) + Km (t; h; ; s; hQm+1 (t; ); hQm+1 (t; s)): Then, the system of (39)–(42) implies the following integral system for the determination of Pm+1 ; Rm+1 , and Qm+1 :
APPENDIX
m+1
Note that Pm+1 ; Qm+1 , and Rm+1 depend on h. The known matrix functions Em ; Gm , and Km are continuous on t; h; ; s and contain linear and quadratic terms in hPm+1 and hQm+1 . The known matrix functions gm and km are continuous on t; ; s. Let 8(t; s) be the transition matrix of the system x_ (t) = 0M0 (t)x(t). Denote
(40)
(41)
(42)
Applying the contraction principle argument on the latter system, one can show that for all small enough h > 0 this system has a unique solution Pm+1 ; Qm+1 , and Rm+1 , uniformly bounded and continuously depending on h > 0; t; s, and . Hence, the approximation of (31) is uniform on h; t; , and . ii) Equation (33) follows from (31) and the rest of ii) is similar to [5]. Proof of Lemma 2: i) Applying u0 to (1), we obtain the system of ~ (t) are time-varying (34). Note that in (34) only the matrices A(t) and C and thus the corresponding F (xt ) is given by (3) and is time-invariant. Similarly to Remark 2 it can be shown that for this closed-loop system kzk2L 2 kwkL2 for all w 2 L2 [0; tf ] and x0 = 0 if the cor~ (t), and responding RPDEs of (6)–(10), where A0 = A(t); C = C
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000
S = DD0 = 2 , have a solution. Similarly to i) of Theorem 1 it can be
proved that the resulting RPDEs have a solution, approximated by
P (t) = P0 (t) + O(h) Q(t; h ) = P0 (t) + O(h) R(t; h; h) = P0 (t) + O(h)
1 1
where P0 (t) satisfies (22). ii) Let x(t) be a solution of (1) with u = u0 and with h > 0 and let y (t) be a solution of (1) with u = u0 and with h = 0. Then, v(t) = x(t) 0 y(t) satisfies the following equation:
v_ (t) = A(t)v(t) + r +
r
Ai v(t 0 hi )
i=1
r
2
i=1 1+
jAi j jhi j jDj 2
A+
t 0
t p
2 2 tf e2t
1
1+
2
r
2
2
Ai
0 1 tf 2 2 0
e2 t
i=1 1 t (2t02 ) e d dtjw(p)j2 dp
0
01
r
2
i=1
t e2 t f 2 2 0
jAi j hi jDj 2
2
01 A+
r
2
i=1
Ai
kwkL : 2
1
Hence,
Ai [y(t 0 hi ) 0 y(t)];
i=1
2411
v0 = 0
(43)
kzkL kCvkL + kB0 P vkL + kCykL 0 + kB P y kL kw kL : 2
2
2
0
2
0
2
2
2
where
t
y(t) 0 y(t 0 hi ) =
(A(s) +
t0h
r i=1
Ai )y(s) + Dw(s) ds: (44)
From (44) and (35b) it follows that
jy(t) 0 y(t 0 hi )j t jDj jw(s)j + A(s) +
r
t0h
i=1
Ai jy(s)j ds:
(45)
By the variation of constants formula [15], (43) is equivalent to the integral equation
t
v(t) =
r
X (t; s)
0
Ai [y(s 0 hi ) 0 y(s)] ds:
i=1
(46)
Applying (35a) and changing the order of integration we find
kykL 2
t
=
t
0
t
t
dt ds jDj e t0s jw(s )j 2 e t0s jw(s )j ds t t t jDj dt ds e t0s jw(s )j ds jDj t t e t 0 1 jw(s )j ds dt = 0
0 (
)
2 0
2
2 0
2
2
1
2
0
2
(
1
2 (
)
2
0
2
0
1
2
0
2
0
2 2 02jD j e2 t
)
0
2
2
0 1 tf kwkL : 2
From (45), (46), (35b), and the latter inequality, we obtain
kvkL 2
2
r
jAi j
t
2
i=1
t 0
0
e(t0s)
1 jDj jw( )j + A( ) + 2 +
t 0
e(t0p)
A(r) +
r i=1
p p0h
r i=1
s s0h
Ai jy( )j d ds
jDj jw(r)j
Ai jy(r)j dr dp dt
2
[1] A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter, Representation and Control of Infinite Dimensional Systems. Boston, MA: Birkhauser, 1992, vol. 2. [2] B. van Keulen, H -Control for Distributed Parameter Systems: A StateSpace Approach. Boston, MA: Birkhauser, 1993. [3] J. Lee, S. Kim, and W. Kwon, “Memoryless controllers for delayed systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 159–162, 1994. [4] J. Ge, P. Frank, and C. Lin, “ control via output feedback for state delayed systems,” Int. J. Contr., vol. 64, pp. 1–7, 1996. [5] E. Fridman and U. Shaked, “ -state-feedback control of linear systems with small state-delay,” Syst. Contr. Letters, vol. 33, no. 3, pp. 141–150, 1998. [6] C. de Souza and X. Li, “Delay-dependent robust control of uncertain linear state-delayed systems,” Automatica, vol. 35, pp. 1313–1321, 1999. [7] P. Ndiaye and M. Sorine, “Delay sensitivity of quadratic controllers—A singular perturbation approach,” SIAM J. Contr. Optim. [8] U. Shaked, I. Yaesh, and C. de Souza, “Bounded real criteria for linear time-delay systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 1016–1022, July 1998. [9] E. Fridman and U. Shaked, “ -norm and invariant manifolds of systems with state delays,” Syst. Contr. Letters, vol. 36, no. 2, pp. 157–165, 1999. [10] A. Vasilieva, “Asymptotic solutions of differential-difference equations in the case of a small deviation in the argument,” Comput. Meth. Math. Phys., vol. 2, no. 5, pp. 869–893, 1962. [11] R. O’Malley, Introduction to Singular Perturbations. New York: Academic, 1974. [12] P. Sannuti and P. Reddy, “Asymptotic series solution to optimal systems with small time delay,” IEEE Trans. Automat. Contr., vol. 18, pp. 250–259, 1973. [13] Z. Pan and T. Basar, “ -optimal control for singularly perturbed systems. Part I: Perfect state measurements,” Automatica, vol. 2, pp. 401–424, 1993. [14] M. Delfour, “The linear quadratic optimal control problem with delays in state and control variables,” SIAM J. Contr. Optim., vol. 24, pp. 835–883, 1986. [15] J. Hale, Functional Differential Equations. New York: SpringerVerlag, 1977.
H H
2
H
H
H
2
X0 (t; s)Dw(s) ds dt
0
t
2 0
REFERENCES
H
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000
Finite Horizon State-Feedback Control of Continuous-Time Systems with State Delays Emilia Fridman and Uri Shaked Abstract—The finite horizon control of time-invariant linear systems with a finite number of point and distributed time delays is considered. The controller is obtained by solving coupled Riccati-type partial differential equations. The solutions to these equations and the resulting controllers are approximated by series expansions in powers of the largest delay. Unlike the infinite horizon case, these approximations possess both regular and boundary layer terms. The performance of the closed-loop system under the memoryless zero-approximation controller is analyzed. –state-feedback control, Index Terms—Asymptotic approximations, Riccati type partial differential equations, singular perturbations, timedelay systems.
I. PROBLEM FORMULATION Throughout this paper we denote by j 1 j the Euclidean norm of a vector or the appropriate norm of a matrix. Given tf > 0, let L2 [0; tf ] be the space of the square integrable functions with the norm k 1 kL and let C [a; b] be the space of the continuous functions on [a; b] with the norm k1k . We denote xt = x(t + ); y t = y (t 0 ); 2 [0h; 0]. Prime denotes the transpose of a matrix and colfx; y g denotes a column vector with components x and y . Consider the system
1
x_ (t) = L(xt (1)) + Bu(t) + Dw(t) z (t) = colfCx(t); u(t)g; t0 2 [0h; 0] x() = x0 ();
R
R
R
l
L(xt (1)) =
r
i=0
Ai xt (0hi ) +
0
0h
1
(1)
where x(t) 2 is the state vector, u(t) 2 is the control signal, w(t) 2 q is the exogenous disturbance, z (t) 2 p is the observation vector, and B; C , and D are constant matrices of appropriate dimensions. The Rn -valued function L(1) which carries Rn -valued functions on [0h; 0] into Rn is defined as follows: n
and where M1 = M10 ; M2 = M20 are matrices denoting the initial weightning condition. The problem is to find a state-feedback controller which ensures that J 0 for all w 2 L2 [0; tf ] and for all initial conditions x0 2 L2 [0h; 0]. In the infinite horizon case such a controller (for zero initial condition x0 = 0) has been designed in [1]–[6] (see also references therein). In [1] and [2] the controller has been obtained by solving Riccati operator equations. In [3] and [4] delay-independent and in [6] delay-dependent memoryless controllers have been designed. In [5] the controller (with memory) has been derived from Riccati-type partial differential equations (RPDEs) or inequalities, and the solution of the RPDEs has been approximated by expansions in the powers of the delay. In [7] the gradient of J with respect to h at h = 0 has been computed. In [8] and [9] bounded real criteria have been obtained. Asymptotic series solutions of systems with small delay have been constructed in [10]–[12]. In many engineering cases (target maneuver, missile guidance, etc.) a control session of limited time length is needed. In such cases the effect of the initial conditions is most important and the results of [1]–[6] cannot provide a satisfactory control strategy. In the present paper, we generalize the results of [5] to the finite horizon case. Unlike [5], the required controllers are time-varying, they are obtained by solving coupled finite horizon RPDEs. For small delays, similarly to the case of singularly perturbed systems (see [10]–[13]), the controllers are affected by the boundary-layer phenomenon. The main contribution of the paper is the construction, for the first time, of an asymptotic solution to the important class of finite horizon RPDEs that are encountered with the finite-horizon LQ control (see [14]), and with the H control. Proofs of Theorem 1 and Lemma 2 are given in the Appendix.
R
A01 (s)xt (s) ds
II. MAIN RESULTS
1
A. H -Controller Design
Consider the following RPDEs with respect to the n 2 n-matrices P (t); Q(t; ), and R(t; ; s):
P_ (t) + A00 P (t) + P (t)A0 +
(2)
r +
where 0h = 0hr < 0hr01 < 1 1 1 < 0h1 < 0h0 = 0; A0 ; A1 1 1 1 ; Ar are constant matrices and A01 (s) is a square integrable matrix function. Denote
F (xt )( ) =
r i=1
Ai xt (0hi 0 ) i ( ) +
0h
i=1 0 +
0
+
A01 (p)xt (p 0 ) dp
and where i is the indicator function for the set [0hi ; 0], i.e., i ( ) = 1 if 2 [0hi ; 0] and i ( ) = 0 otherwise. Given > 0, and assuming that w 2 L2 [0; tf ] and x0 2 L2 [0h; 0], we consider the following performance index:
J where
=
kzk 0 kwk 0 W (x0 ) 2
W (x0 ) = x0 (0)M1 x(0) +
2 L
0
0h
(4)
F 0 (x0 )(s)M2 F (x0 )(s) ds (5)
Manuscript received July 26, 1999; revised December 23, 1999. Recommended by Associate Editor, Y. Yamamoto. The authors are with the Department of Electrical Engineering-Systems, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(00)10012-1.
0h
i=1
=
Q(t; )A01 () d A001 ()Q0 (t; ) d = 0
0[A00 + P (t)S]Q(t; ) 0 0
0
0h
A0i Q0 (t; 0hi )
Q(t; 0hi )Ai + P (t)SP (t) + C 0 C
@ @ Q(t; ) + Q(t; ) @t @
(3)
2 L
0h
r
A0
r i=1
(6)
A0i R(t; 0hi ; )
01 (s)R(t; s; ) ds
@ @ @ R(t; ; s) + R(t; ; s) + R(t; ; s) @t @ @s 0 = 0Q (t; )SQ(t; s) P (t) = Q(t; 0); Q(t; ) = R(t; 0; ); 2 [0; h]; s 2 [0; h] R(t; ; s) = R0 (t; s; ); P (tf ) = 0; Q(tf ; ) = 0; R(tf ; ; s) = 0
(7)
(8) (9) (10)
where S = 02 DD0 0 BB 0 . A solution of (6)–(10) is a triple of n 2 n-matrices fP (t); Q(t; ); R(t; ; s)g t 2 [0; tf ]; 2 [0h; 0]; s 2 [0h; 0],
0018–9286/00$10.00 © 2000 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000
() ( )
(
)
where P t ; Q t; , and R t; ; s are continuous and piecewise continuously differentiable functions of their arguments that satisfy (6)–(10) for almost every t; , and s. Lemma 1: Given > . Let (6)–(10) have a solution on ; tf that for some n 2 n matrices 1 > and 2 > satisfies the following conditions:
0 1
0
1
[0 ]
0
0 5 (0) 0 0:5P 0 (0) 0 h11 0 (0 )101 1Q(0; s) 0 0 0:5h12 0 0:5 R(0; s; )1201R(0; ; s) d 0; 0h 8 s 2 [0h; 0]:
M1 0 : P M2 0 Q0 ; s
0
( ) = 0B0 P (t)x(t) +
0h
( ) ( )( )
Q t; F xt d
(11)
(12)
solves the H1 -control problem with the performance level of . Proof: Let x t be a solution of (1). Consider the following Lyapunov–Krasovskii functional [14]:
()
(
V t; xt
0
) = x(t)0 P (t)x(t) + 2x0 (t) +
0
0
0h 0h
(
0h
( ) ( )( )
Q t; F xt d
F 0 xt s R t; s; F xt ds d:
( )( ) (
) ( )( )
Remark 1: In the case of LQ problem from (15) it follows that
minu kzkL2 = V (0; x0 ) and similarly t min jzj2 dt u t = V (t0 ; x0 ); 8 t0 tf ; 8 x0 2 L2 [0h; 0]: Hence, V (t; x) 0 in the LQ case.
Remark 2: Note that a certain amount of overdesign is introduced by the conditions of (11). This overdesign stems from the bounding in (18). In the case of the zero initial conditions x ; 2 0h; the conditions of (11) are not relevant and the controller of (12) solves the H1 control problem under the sole assumption that (6)–(10) have a solution on ; tf .
( )=0
Then, the controller
u3 t
2407
(
B. Asymptotic Solutions to the RPDEs
( ) = P0 (t) + h[P1 (t) + 51P ( )] + h2 [P2 (t) + 52P ( )] + 1 1 1 ; Q(t; h ) = Q0 (t; ) + h[Q1 (t; ) + 51Q ( ; )] + h2 [Q2 (t; ) + 52Q ( ; )] + 1 1 1 ; R(t; h; h) = R0 (t; ; ) + h[R1 (t; ; ) + 51R ( ; ; )] + h2 [R2 (t; ; ) + 52R ( ; ; )] + 1 1 1 ; tf 0 t (19) = ; 2 [01; 0]; 2 [01; 0]: h P t
(13)
(14)
Expansion (19) has a typical for singular perturbations form: it includes the “outer expansion” (regular) terms fPi ; Qi ; Ri g; i ; 111 and the boundary-layer correction terms iP ; iQ , and iR ; i ; 1 1 1 The “outer expansion” terms constitute the major part of the solution that satisfies (6)–(9) for t 2 ; tf ; 2 0 ; ; 2 0 ; . The boundary-layer correction terms will be chosen such that (19) satisfies the terminal conditions of (10) and that
where
w3 (t) = 02 D0
( ) ( )+
P txt
0
0h
( ) ( )( )
Q t; F xt d :
(
t
) 0 V (0; x0 ) + [jzj2 0 2 jwj2 ] dt 0 = 0 2 kw 0 w3 kL + ku 0 u3 kL :
(15)
We show next that (11) implies
= W (x0 ) 0 V (0; x0 ) 0: Denoting v = x(0) and y (s) = F (x0 )(s) we have 0 d = v 0 (M1 0 P (0))v + y 0 (s)M2 y (s) ds d
0 2v0 0
0
0
0h
0
0h 0h
(16)
(0 ) ( )
)()
(17)
2
0h
v 0 Q(0; s)y (s) ds
hv0 11 v +
2
0
0h
0
0h
y 0 (s)Q0 (0; s)101 Q(0; s)y (s) ds 1
( ) (0
)()
hy0 (s)12y(s) + y0 (s)
=
and hence
y 0 s R ; s; y d
5 5
0h
(0
R ; s;
)1201R(0; ; s) dy(s)
and (11). Finally, (16), (15), and (10) imply that J
(18)
0 for u = u3 .
(20)
=
=
=
( ) = P0 (t);
_ ( )+
r
i=0
A0i P0 t
+ C 0 C = 0;
=
=
( )=0 ( 0 )= ( )
Then, from (6), we have
P0 t
5
=
(
Q0 t; 0
! 1:
The boundary-layer correction terms depend on as on the independent variable and do not depend on h. Since is a stretched-time variable around t tf , (20) asserts that iP ; iQ , and iR are essential only around t tf and they thus provide a correction to the outer expansion at the terminal point t tf . We substitute (19) in (6)–(9) and equate, separately, outer expansion and boundary-layer correction terms with the same powers of h. We notice that for t tf 0 h ; h; and s h we have @=@t 0h01 @=@ ; @=@ h01 @=@; and @=@s h01 @=@. Thus, for the zero-order terms we obtain from (7)-(9) @ Q0 t; @ @ @ R0 t; ; R0 t; ; @ @ P0 t ; R0 t; ; Q0 t; Q0 t;
( )=0 ( )+ ( 0) = ( )
Then, (16) follows from (17), the inequalities 0
as
20
= =
Q ; s y s ds
( ) (0
j5iP ( )j + sup j5iQ ( ; )j 2[01;0] + sup j5iR ( ; ; )j ! 0 ; [ 1;0]
0h
y 0 s R ; s; y d ds:
=0 1 5 5 5 = [0 ] [ 1 0] [ 1 0]
12
It follows from (14) that
V tf ; xt
= 0
For simplicity we assume that A01 further on. The H1 controller has been found above by solving a set of coupled PRDEs. Finding a solution to the latter is not an easy task and we are, therefore, looking for a solution to the RPDEs in a form of asymptotic expansion in the powers of the delay h
)
) = 0x0 (t)C 0 Cx(t) 0 2 jw(t) 0 w3 (t)j2 + 2 jw(t)j2 + ju(t) 0 u3 (t)j2 0 ju(t)j2
0]
[0 ]
Differentiating V t; xt with respect to t and integrating by parts, we obtain, similarly to [5], that
d V t; xt dt
[
( )+
R0 t; ; r
) = P0 (t):
( ) + P0 (t)SP0 (t) P0 (tf ) = 0: i=0
(21)
P0 t Ai
(22)
2408
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000
The latter is the well-known Riccati differential equation (RDE) that corresponds to (1) for h = 0. Our main assumption is as follows. A1. For a specified value of > 0, the RDE of (22) has a bounded solution on [0; tf ]. Assumption A1 means that the H state-feedback control problem for (1) without delay has a solution. If this were not the case, even P0 , the zero-order term in (19), would not exist. Note that P0 = P00 . To determine the first-order terms we start with the equations for Q1
Therefore
51Q ( ; ) = 0; + > 0 51R ( ; ; ) = 51Q ( + ; 0 ) = 0 + > 0; :
1
@ Q1 (t; ) = 0M0 (t)P0 (t) 0 P_0 (t) @ Q1 (t; 0) = P1 (t);
M=
r
i=0
Ai + SP0 :
The higher order terms of the outer expansions can be similarly found. We obtain next the boundary-layer terms and show by induction that
5iP ( ) = 0; 5iQ ( ; ) = 0; 5iR ( ; ; ) = 0;
(23)
Q1 (t; ) = P1 (t) 0 [M0 (t)P0 (t) + P_0 (t)]:
+
r i=1
r i=1
gi A0i (P0 M + P_0 )
gi (M0 P0 + P_0 )Ai
P1 (tf ) + 51P (0) = 0;
gi
=0
= hi =h
0 1. We derive the
(24)
_ 1P ( ) = 0. Since 51P vanishes for ! It follows from (6) that 5 1, we have 51P ( ) 0; 0. Hence, P1 (tf ) = 0, and P1 is a solution to the linear differential equation (24) with the latter terminal condition. For 51Q ; R1 , and 51R we obtain from (7), (8), and (21) @ 51Q ( ; ) 0 @ 51Q ( ; ) = 0 Q1 (tf ; ) + 51Q (0; ) = 0 51Q ( ; 0) = 51P ( ) = 0 @ @ R1 (t; ; ) + R1 (t; ; ) = 0P0 (t)SP0 (t) 0 P_0 (t) @ @ (25) R1 ( ; 0; ) = Q1 ( ; ) @ @
and
@ @ @ 51R ( ; ; ) 0 @ 51R ( ; ; ) 0 @ 51R ( ; ; ) = 0 @ R1 (tf ; ; ) + 51R (0; ; ) = 0 51R ( ; 0; ) = 51Q ( ; ): (26) Note that Q1 (tf ; ) = 0P_ 0 (tf ) . Then, for find successively
(29)
5_ mP ( ) = fm( ) 5mP (m 0 1) = 0 @ @ 5mQ ( ; ) 0 @ 5mQ ( ; ) = m ( ; ) @ 5mQ ( ; 0) = 5mP ( ) Qm (tf ; ) + 5mQ (0; ) = 0
Substituting this expression into the equation for P1 , we obtain
P_1 + M0 P1 + P1 M +
> i01 + > i01 + > i 0 1; :
We assume that (29) is satisfied for all i m following equations for 5mP ; 5mQ , and 5mR :
Then
(28)
0 and t 2 [0; tf ], we
_ 51Q ( ; ) = ( + )P0 (tf ); if 0 if > 0 0; R1 (t; ; ) = R10 (t; ; ) = 0 [P0 (t)SP0 (t) + P_0 (t)] + Q1 (t; 0 ); 51R (0; ; ) = P_0 (tf ) 51R ( ; ; ) = 501R ( ; ; ) _ = ( + )P0 (tf ); if 0 ; (27) if > 0; : 0;
@ @
@ @ 5mR ( ; ; ) 0 @ 5mR ( ; ; ) 0 @ 5mR ( ; ; ) = m ( ; ; ) 5mR ( ; 0; ) = 5mQ ( ; ); Rm (tf ; ; ) + 5mR (0; ; ) = 0 where fm and m are known functions that vanish for > m 0 1, and m is a known function that vanishes for + > m 0 2; . From these equations we find
5mP ( ) =
0
m 1
fm (s) ds
and thus (29) holds for 5mP since fm (s) = 0 for > m 0 1. Further
5mQ (0 ; + ) + m (s; 0s + + ) ds; 5mQ ( ; ) = 5 (0 + ) mP 0 m (0s + + ; s) ds; 0
and 5mQ satisfies (29) since 5mP ( + ) = and m ( ; ) = 0 for > m 0 1. Finally
if
0
if >
0 for +
0
> m01
5mR ( ; ; ) = 50mR ( ; ; ) m (s; 0s + + ; 0s + + ) ds 0 + 5mR (0; + ; + ); if 0; = 0 m (0s + + ; s; s + 0 ) ds 0 if > 0; : + 5mQ ( + ; 0 ); (30) Conditions (29) for 5mR readily follow from (30) and the properties of 5mQ and m .
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000
1 Control
C. Near-Optimal H
Theorem 1: Under A1 the following holds for all small enough time-delay h i) The system of (6)–(10) has a solution. This solution is approximated, for any integer m, by
P (t) = P0 (t) + Q(t; h ) = P0 (t) + R(t; h; h) = P0 (t) +
m i=1 m i=1 m i=1
hi [Pi (t) + 5iP ( )] + O(hm+1 ) hi [Qi (t; ) + 5iQ ( ; )] + O(hm+1 ) hi [Ri (t; ; ) + 5iR ( ; ; )]
+ O(hm+1 ) = tf 0 t ; 2 [01; 0]; 2 [01; 0] h
(31)
where the boundary-layer terms satisfy (29), and jO(hm+1 )j is a positive scalar which is independent of
chm+1 , where c h; t; , and . ii) If additionally
P0 (0) < M1 ;
M2 > 0
2409
Lemma 2: Under A1 the controller u0 (t) = 0B 0 P0 (t)x(t) for the zero initial condition x0 = 0 guarantees i) for all small enough h a performance level of ; ii) for all h, a performance level of , where
2 = tf e2t 2 2
1
(32)
D. The Zero-Order Controller Performance We study the performance of the system under the zero-order controller u0 (t) = 0B 0 P0 (t)x(t) which solves the H1 -control problem for (1) without delay. For simplicity we consider the case of x0 = 0. Applying u0 to (1), we obtain
0 0
(
0
(
)
)
t; t0 2 [0; tf ]:
(36)
z = colfx; ug
(37)
2
0
the form
P_1 + 2 tan(tf 0 t)P1 + 2 = 0;
P1 (tf ) = 0:
From the latter equation, (27), and (28) we find
P1 = tan(tf 0 t) + (t2f 0 t) ; Q1 = P1 + cos (tf 0 t) 51Q = 0( + ) (0 0 ) R1 (t; ; ) = R10 (t; ; ) = + P1 (t) 51R ( ; ; ) = 501R ( ; ; ) = 0( + ) (0 0 ); where (s) = 1 for s 0 and (s) = we obtain for 0 < tf < =2 that
and (34)
X (t; t0 ) be the transition matrix of the system of (34), i.e., X (t; t0 ) = 0 for t < t0 , X (t0 ; t0 ) = In and X (t; t0 ) satisfies (34) for t t0 . Let X0 (t; t0 ) be the transition matrix of (34) without delay, i.e., where hi = 0. Then there exist scalars 0 > 0; > 0; ; and such that for small enough h the following inequalities are valid: 0
+ 2
2
0 for s < 0. Since Q0 = P0
u0 (t) = 02 tan(tf 0 t)x(t)
Let
jX (t; t )j e t0t jX (t; t )j e t0t ;
1
2
1=4 the latter RDE has a bounded solution on > 0. Choosing = 1=5 < 1=4 we find that P = tan(tf 0 t) and thus for tf < =2 (38) has a bounded solution on [0; tf ]. It is readily seen that conditions (32) hold. Equation (24) has
Ai x(t 0 hi ) + Dw(t)
A(t) = A0 0 BB 0 P0 (t) ~ (t); C~ (t) = colfC; 0B 0 P0 (t)g: z = Cx
Ai
i=1
jAi j hi
Consider the following system:
Note that for 2 [0; tf ] for all tf
where 50P = 0; Q01 = 501;Q = 50;Q = 0; Q0 = P0 . The approximate controller um guarantees an attenuation level of + O(hm+1 ). It follows from Theorem 1 that a high-order approximate controller improves the performance polynomially in the size of the small timedelay h.
i=1
i=1
2
r
> tan tf and M2 > 0. From (22) we obtain P0 (tf ) = 0: (38) P_0 (t) + ( 02 0 4)P02 + 1 = 0;
(33)
r
r
2
and J of (4) with M1
[Qi01 (t; ) + 5i01;Q ( ; )]x(t + h ) d
x_ (t) = A(t)x(t) +
01 A+
0
x_ (t) = x(t) 0 x(t 0 h) + 2u 0 w;
i=0
01
2
E. Example
u(xt ) = um (xt ) + O(hm+1 ) m um (xt ) = 0 hi B 0 [Pi (t) + 5iP ( )]x(t)
+
2 1 + 2 0 tf e2 t
2
and where for = 0 ( = 0) one has to take limit ! 0 ( ! 0). It follows from Lemma 2 that the controller u0 guarantees a performance level for all small time delays and it guarantees a performance level for all delays. Note that ! for h ! 0. Given > 0 and h, in order to make certain that u0 leads to a performance level of one can verify conditions in terms of differential linear matrix inequalities or Riccati differential inequalities (RDI) that were formulated for the case of one delay in [8] and can be easily generalized to the case of r delays.
then the controller of (12) is approximated by
0
0 1 jDj jC j + kB0 P kL
(35a) (35b)
u1 (t) = u0 (t) 0 2h
1 P (t)x(t) + tan(tf 0 t) 1
0
01
x(t + h ) d :
Consider now the performance of (37) under u = u0 and x0 = 0 for tf = 1:1. Applying the delay-dependent criterion p of [8] on the closed-loop system we find that u0 achieves = (1= 5) for all delays h 2 (0; 0:027], since the corresponding RDIs have bounded solutions on [0, 1.1]. For h = 0:028 the solutions to the RDIs encounter escape points and the criterion of [8], which provide a sufficient p condition only, cannot therefore be used to verify the level = (1= 5).
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000
III. CONCLUSIONS
1
A solution to the state-feedback H control of linear time-invariant systems with state time delays in the finite horizon case is presented. The controller is obtained by solving RPDEs. An approximate solution to the RPDEs has been constructed by expansion in powers of the largest delay. The theory that has been developed in this paper shows that similarly to the case of singularly perturbed systems [13], for small delays our controllers are affected by the boundary-layer phenomenon. The high order approximate controller improves the performance of the closed-loop system polinomially in the size of the delay. The memoryless zero-approximation may, in many cases, be sufficient for robustly achieving the required performance. It is shown that the performance of the system under such a controller is robust for small time delays. Explicit formula for the guaranteed performance level is obtained for this case in terms of the coefficients of the system.
Proof of Theorem 1: i) To prove the validity of (31) we consider the equations for the remainders
h
Pm+1
=P 0
m
i=0
= Q(t; ) 0
m
i=0
hi Qi t; h01
hm+1 Rm+1 (t; ; s)
= R(t; ; s) 0
m
i=0
+ 5iQ h01 (tf 0 t); h01
hi Ri t; h01 ; h01 s
P_m+1 + Pm+1 M + M0 Pm+1
+
i=1 r i=1
t t r
i=1
8(t; p)
r i=1
0 (p; 0hi ) 0 Q0 (p; 0)] A0i [Qm +1 m+1
[Qm+1 (p; 0hi ) 0 Qm+1 (p; 0)]Ai
+ E m (p; h; hPm+1 (p)) 80 (t; p) dp Qm+1 (t; )
0 (t; 0)] A0i [Q0m+1 (t; 0hi ) 0 Qm +1
[Qm+1 (t; 0hi ) 0 Qm+1 (t; 0)]Ai
+ Em (t; h; hPm+1 (t)) = 0 @ @ Qm+1 (t; ) + Qm+1 (t; ) @t @ = 0M0 Qm+1 (t; ) 0
r
i=1
(39)
A0i [Rm+1 (t; 0hi ; )
0 Rm+1 (t; 0; )] + h01 gm (t; ) + Gm (t; h; hPm+1 (t); hQm+1 (t; )) @ Rm+1 (t; ; s) @t @ + @ Rm+1 (t; ; s)
@ + @s Rm+1 (t; ; s) + h01 km (t; ; s) + Km (t; h; ; s; hQm+1 (t; ); hQm+1 (t; s)) = 0 Pm+1 (t) = Qm+1 (t; 0 Qm+1 (t; ) = Rm+1 (t; 0; ) 0 (t; s; ) Rm+1 (t; ; s) = Rm +1 Pm+1 (tf ) = 0; Qm+1 (tf ; ) = 0; R(tf ; ; s) = 0:
8(t; p)Gm (p; p 0 t + ) dp; if t 0 tf = 8(t; t 0 )Pm+1 (t 0 ) + 8(; p)Gm (p + t 0 ; p) dp; 0 if t 0 < tf Rm+1 (t; ; s) = Rm0 +1 (t; s; ) t = K m (p; p + 0 t; p + s 0 t) dp; t if tf 0 t 0; s Rm+1 (t; ; s) = Rm0 +1 (t; s; ) = 0 K m (p 0 + t; p; p + s 0 ) dp 0 + Qm+1 (t 0 ; s 0 ); if tf 0 t > 0; s : t
in the following expansions: r
=0
t
+ 5iR h01 (tf 0 t); h01 ; h01 s)
+
Pm+1 (t)
+
hi Pi
hm+1 Qm+1 (t; )
E m (t) = Em (t; h; hPm+1 (t)) Gm (t; ) = h01 gm (t; ) + Gm (t; h; hPm+1 (t); hQm+1 (t; )) K m (t; ; s) = h01 km (t; ; s) + Km (t; h; ; s; hQm+1 (t; ); hQm+1 (t; s)): Then, the system of (39)–(42) implies the following integral system for the determination of Pm+1 ; Rm+1 , and Qm+1 :
APPENDIX
m+1
Note that Pm+1 ; Qm+1 , and Rm+1 depend on h. The known matrix functions Em ; Gm , and Km are continuous on t; h; ; s and contain linear and quadratic terms in hPm+1 and hQm+1 . The known matrix functions gm and km are continuous on t; ; s. Let 8(t; s) be the transition matrix of the system x_ (t) = 0M0 (t)x(t). Denote
(40)
(41)
(42)
Applying the contraction principle argument on the latter system, one can show that for all small enough h > 0 this system has a unique solution Pm+1 ; Qm+1 , and Rm+1 , uniformly bounded and continuously depending on h > 0; t; s, and . Hence, the approximation of (31) is uniform on h; t; , and . ii) Equation (33) follows from (31) and the rest of ii) is similar to [5]. Proof of Lemma 2: i) Applying u0 to (1), we obtain the system of ~ (t) are time-varying (34). Note that in (34) only the matrices A(t) and C and thus the corresponding F (xt ) is given by (3) and is time-invariant. Similarly to Remark 2 it can be shown that for this closed-loop system kzk2L 2 kwkL2 for all w 2 L2 [0; tf ] and x0 = 0 if the cor~ (t), and responding RPDEs of (6)–(10), where A0 = A(t); C = C
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000
S = DD0 = 2 , have a solution. Similarly to i) of Theorem 1 it can be
proved that the resulting RPDEs have a solution, approximated by
P (t) = P0 (t) + O(h) Q(t; h ) = P0 (t) + O(h) R(t; h; h) = P0 (t) + O(h)
1 1
where P0 (t) satisfies (22). ii) Let x(t) be a solution of (1) with u = u0 and with h > 0 and let y (t) be a solution of (1) with u = u0 and with h = 0. Then, v(t) = x(t) 0 y(t) satisfies the following equation:
v_ (t) = A(t)v(t) + r +
r
Ai v(t 0 hi )
i=1
r
2
i=1 1+
jAi j jhi j jDj 2
A+
t 0
t p
2 2 tf e2t
1
1+
2
r
2
2
Ai
0 1 tf 2 2 0
e2 t
i=1 1 t (2t02 ) e d dtjw(p)j2 dp
0
01
r
2
i=1
t e2 t f 2 2 0
jAi j hi jDj 2
2
01 A+
r
2
i=1
Ai
kwkL : 2
1
Hence,
Ai [y(t 0 hi ) 0 y(t)];
i=1
2411
v0 = 0
(43)
kzkL kCvkL + kB0 P vkL + kCykL 0 + kB P y kL kw kL : 2
2
2
0
2
0
2
2
2
where
t
y(t) 0 y(t 0 hi ) =
(A(s) +
t0h
r i=1
Ai )y(s) + Dw(s) ds: (44)
From (44) and (35b) it follows that
jy(t) 0 y(t 0 hi )j t jDj jw(s)j + A(s) +
r
t0h
i=1
Ai jy(s)j ds:
(45)
By the variation of constants formula [15], (43) is equivalent to the integral equation
t
v(t) =
r
X (t; s)
0
Ai [y(s 0 hi ) 0 y(s)] ds:
i=1
(46)
Applying (35a) and changing the order of integration we find
kykL 2
t
=
t
0
t
t
dt ds jDj e t0s jw(s )j 2 e t0s jw(s )j ds t t t jDj dt ds e t0s jw(s )j ds jDj t t e t 0 1 jw(s )j ds dt = 0
0 (
)
2 0
2
2 0
2
2
1
2
0
2
(
1
2 (
)
2
0
2
0
1
2
0
2
0
2 2 02jD j e2 t
)
0
2
2
0 1 tf kwkL : 2
From (45), (46), (35b), and the latter inequality, we obtain
kvkL 2
2
r
jAi j
t
2
i=1
t 0
0
e(t0s)
1 jDj jw( )j + A( ) + 2 +
t 0
e(t0p)
A(r) +
r i=1
p p0h
r i=1
s s0h
Ai jy( )j d ds
jDj jw(r)j
Ai jy(r)j dr dp dt
2
[1] A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter, Representation and Control of Infinite Dimensional Systems. Boston, MA: Birkhauser, 1992, vol. 2. [2] B. van Keulen, H -Control for Distributed Parameter Systems: A StateSpace Approach. Boston, MA: Birkhauser, 1993. [3] J. Lee, S. Kim, and W. Kwon, “Memoryless controllers for delayed systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 159–162, 1994. [4] J. Ge, P. Frank, and C. Lin, “ control via output feedback for state delayed systems,” Int. J. Contr., vol. 64, pp. 1–7, 1996. [5] E. Fridman and U. Shaked, “ -state-feedback control of linear systems with small state-delay,” Syst. Contr. Letters, vol. 33, no. 3, pp. 141–150, 1998. [6] C. de Souza and X. Li, “Delay-dependent robust control of uncertain linear state-delayed systems,” Automatica, vol. 35, pp. 1313–1321, 1999. [7] P. Ndiaye and M. Sorine, “Delay sensitivity of quadratic controllers—A singular perturbation approach,” SIAM J. Contr. Optim. [8] U. Shaked, I. Yaesh, and C. de Souza, “Bounded real criteria for linear time-delay systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 1016–1022, July 1998. [9] E. Fridman and U. Shaked, “ -norm and invariant manifolds of systems with state delays,” Syst. Contr. Letters, vol. 36, no. 2, pp. 157–165, 1999. [10] A. Vasilieva, “Asymptotic solutions of differential-difference equations in the case of a small deviation in the argument,” Comput. Meth. Math. Phys., vol. 2, no. 5, pp. 869–893, 1962. [11] R. O’Malley, Introduction to Singular Perturbations. New York: Academic, 1974. [12] P. Sannuti and P. Reddy, “Asymptotic series solution to optimal systems with small time delay,” IEEE Trans. Automat. Contr., vol. 18, pp. 250–259, 1973. [13] Z. Pan and T. Basar, “ -optimal control for singularly perturbed systems. Part I: Perfect state measurements,” Automatica, vol. 2, pp. 401–424, 1993. [14] M. Delfour, “The linear quadratic optimal control problem with delays in state and control variables,” SIAM J. Contr. Optim., vol. 24, pp. 835–883, 1986. [15] J. Hale, Functional Differential Equations. New York: SpringerVerlag, 1977.
H H
2
H
H
H
2
X0 (t; s)Dw(s) ds dt
0
t
2 0
REFERENCES
H
