1016
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998
[4] R. Kumar and V. K. Garg, “State avoidance control for infinite state
systems using assignment program model,” to be published.
[5] R. Kumar, V. K. Garg, and S. I. Marcus, “Predicates and predicate transformers for supervisory control of discrete event dynamical systems,”
IEEE Trans. Automat. Contr., vol. 38, pp. 232–247, Feb. 1993.
[6] Y. Li and W. M. Wonham, “Control of vector discrete-event systems —The base model,” IEEE Trans. Automat. Contr., vol. 38, pp.
1214–1227, Aug. 1993.
[7] F. Lin and W. M. Wonham, “On observability of discrete-event systems,” Inform. Sci., vol. 44, no. 3, pp. 173–198, 1988.
[8] Y. Park and E. K. P. Chong, “Sensor assignment for invertibility in
interruptive timed discrete event systems,” in Proc. IEEE Int. Symp.
Intelligent Contr., 1994, pp. 207–212.
[9] P. J. Ramadge, “Observability of discrete event systems,” in Proc. 25th
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I
H1
One way to cope with the
control of systems with delay is
the method of Smith-predictor [5], which has some merits but also
suffers from a few drawbacks when it deals with model uncertainties.
Another way, which is more systematic on one hand, but it also leads
to more complicated controllers on the other hand, is to apply the
approach of [6] to construct estimators that include a finite interval
memory. Recently, an alternative approach to the control of timedelay systems has been proposed [7], [8] that applies the operator
Riccati approach.
Systems with state delay appear in various problems of process
control [5], especially in chemical processes with recycling. They
also appear in many other control problems where the time delay is
encountered in the measurement. If, due to simplicity reasons, memoryless controllers are required for the latter problems, the analysis of
the resulting closed loop involves stability considerations and
norm analysis of state-delayed systems. For example, consider the
system
H1
x
_ (t)
=
Ax(t)
where w(t) and u(t) are the disturbance input and control input,
respectively, and for which a controller has to be designed that is
based on the delayed measurement, C x(t 0 d), where d is the time
delay. More specifically, consider a controller of the type
u(t)
Bounded Real Criteria for Linear Time-Delay Systems
U. Shaked, I. Yaesh, and C. E. de Souza
H1
Abstract—This paper considers the problem of finding the
-norm
of a linear time-varying system having a delayed state and with an output
that comprises a linear combination of the “current” state as well as of the
“delayed” state. Both delay-independent and delay-dependent bounded real
criteria are derived which provide sufficient conditions for the system to
possess an
-norm which is less than or equal to a prescribed bound.
The stationary case for time-invariant systems is also tackled, and in
this situation the proposed bounded real criteria are given in terms of
linear matrix inequalities. Two examples are presented which compare
the guaranteed bounds that are achieved by our results with the actual
-norm of the system.
H1
H1
Index Terms—Bounded real lemma,
H1
H1 , time-delay systems.
I. INTRODUCTION
The theory of
control [1], [2] has been recently applied in
the field of process control [3]. Chemical and industrial processes are
often characterized by transport delays which are in fact reasonable
approximations to systems that have a large number of cascaded time
lags (see, e.g., [4] and the references therein).
Manuscript received January 4, 1996; revised July 29, 1996. This work
was supported by the Conselho Nacional de Desenvolvimento Cientı́fico e
Tecnológico-CNPq, Brazil, under Grant 301653/96-8/NV/PQ.
U. Shaked is with the Department of Electrical Engineering–Systems, TelAviv University, Tel-Aviv 69978, Israel.
I. Yaesh is with Israel Military Industries, Advanced Systems Div., Ramat
Hasharon, Israel.
C. E. de Souza is with the Department of Systems and Control, Lab.
Nacional de Computacao Cientifica-LNCC/CNPq, Petropolis, Rio de Janiero,
Brazil.
Publisher Item Identifier S 0018-9286(98)03596-X.
+ B1 w (t) + B2 u(t)
=
K C x(t
0
d)
that should minimize the H -norm of the transfer function Tzw ,
from w to the controlled output z , defined by
T
z (t) = C1 x(t) + D12 u(t)
with
D12 [D12
C1 ] = [I
0]:
1
We readily obtain that the closed loop is given by
0 )+
( 0 )
+ Ad x(t
x
_ (t)
=
Ax(t)
z (t)
=
C1 x(t)
+ C2 x
d
t
B1 w (t)
d
where Ad = B2 K C and C2 = D12 K C . A result which is analogous
to the bounded real lemma (see, e.g., [1]) is then required if one wants
to determine the H -norm of Tzw .
The problem of bounded realness for linear systems with a delayed
state has been recently investigated in [10]–[12], which proposed
delay-independent bounded real lemmas. Reference [10] applies a
frequency domain approach, while [11] and [12] use a time-domain
approach. Since for these results the time delay is allowed to be
arbitrarily large, these versions of the bounded real lemma are, in
general, conservative for many important applications. Very recently,
a delay-dependent bounded real lemma, i.e., one that depends on the
size of the time delay, was proposed in [13]. A common feature of
the above results is that the output is in terms of the “current” state
only and thus cannot be applied to the closed-loop system as above.
The purpose of the present paper is to develop both delayindependent and delay-dependent bounded real lemmas for linear
systems with a delayed state and where the output comprises a linear
combination of the “current” state as well as of the “delayed” state.
Bounded real lemmas for finite-time horizon and infinite-time horizon
problems will be developed.
1
II. FINITE HORIZON BOUNDED REAL CRITERIA
We consider the following linear time-varying system:
x
_ (t)
=
0018–9286/98$10.00  1998 IEEE
A(t)x(t)
+ Ad (t)x(t
0
d(t))
+ B1 (t)w (t)
(1)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998
where x(t) 2 Rn ; w(t) 2 Rm ; and A(t); Ad (t) and B1 (t) are
bounded real matrix functions of appropriate dimensions that are
piecewise continuous over [0; T ]. The scalar d(t) is a bounded
varying time delay.
Given a scalar > 0 and assuming that w 2 L2 [0; T ], we consider
the following performance index:
J = xT (T )PT x(T ) + kzk22 0 2 kwk22 ; PT = PTT 0 (2)
1017
where
y(t)
Denoting
(3)
where C1 (t) and C2 (t) are bounded real, piecewise continuous, timevarying matrices. The problem is to find conditions which will ensure
that J 0; 8 w 2 L2 [0; T ], subject to
8 0
x( ) = 0;
and
w( ) = 0;
8 < 0:
For simplicity of notation, we omit in the sequel the dependence
of the various matrices and vectors on t, and we only keep the
dependence on t 0 d.
Our first result provides a bounded real criterion that is independent
of the size of the time delay but depends on an upper bound for
the rate of change of the delay. A delay-independent criterion is
surely more conservative than a criterion that is tuned to the specific
delay length in the system. It will be seen below, however, that
due to its simplicity it may be preferred over a delay-dependent
criterion in cases where Ad is of small norm. This simplicity is most
advantageous when it is desired to perform a parametric search over
a given structure of the controller. A delay-independent criterion also
may be proven useful in cases with large uncertainty in the delay
length.
We assume that the time-delay d(t) is bounded and that it satisfies
d_(t) < 1;
8 t 2 [0; T ]
(4)
where is a given scalar. We obtain the following result.
Lemma 2.1: The performance index J of (2) is nonpositive for all
w 2 L2 [0; T ], and for all bounded time-delay d(t) that satisfies (4),
if there exists a symmetric positive definite matrix Q(t) that allows
a symmetric positive semidefinite solution P (t) over [0; T ], with
P (T ) = PT , to the following differential linear matrix inequality
(DLMI):
9
ATd P
B1T P
where
9
P Ad
0(1 0 )Q(t 0 d)
0
P B1
0
0
2
I
0;
8 t 2 [0; T ]
P_ + AT P + P A + C1T C1 + (1 0 )01 C2T C2 + Q:
(5)
(6)
Proof: For a specific time-varying d that satisfies (4), we
consider the following function:
t
U (t) xT (t)P (t)x(t) +
xT Q + (1 0 )01 C2T C2 x d:
t0d(t)
(7)
Considering (1), it can be easily established that
dU (t)
= xT P_ + AT P + P A + C1T C1 + Q
dt
+ (1 0 )01 C2T C2 x + 2xT P Ad y 0 xT C1T C1 x
0 (1 0 d_)yT Q(t 0 d)y 0 yT C2T (t 0 d)C2(t 0 d)y
_
+ 2xT P B1 w 0 0 d yT C2T (t 0 d)C2 (t 0 d)y (8)
10
[xT yT wT ]T
and
where PT is a given weighting matrix for the terminal state, x(T ),
and
C1 (t)x(t)
z=
C2 (t 0 d(t))x(t 0 d(t))
x(t 0 d(t)):
9
ATd P
B1T P
M
where
P Ad
0(1 0 d_)Q(t 0 d)
0
9 is as in (6), it follows
P B1
0
0
2
I
from (8) that
_
dU
= T M + 2 kwk2 0 kz k2 0 0 d yT C2T (t 0 d)C2 (t 0 d)y:
dt
10
(9)
Since x(t) = 0;
T dU
d
0
8 t 0, it is readily obtained from (7) that
d
= xT (T )PT x(T ) +
T
T 0d(T )
xT Q + (1 0 )01 C2T C2 x d:
By considering (9), this implies that
T
T
J = T M d 0
xT Q + (1 0 )01 C2T C2 x d
0
T 0d(T )
T
_
0 0 d yT C2T ( 0 d)C2( 0 d)y d:
(10)
0
10
Note that in view of (5), it results that M (t) 0; 8 t 2 [0; T ] and
for all bounded d(t) that satisfies (4). Hence, it follows from (10)
that J 0 for all w 2 L2 [0; T ].
Lemma 2.1 provides a bounded real criterion in terms of the DLMI
of (5). Note that using Schur complements, (5) is equivalent to the
Riccati differential inequality (RDI)
P_ + AT P + P A + P 02 B1 B1T + (1 0 )01 Ad Q01 (t 0 d)AdT
P + C1T C1 + (1 0 )01 C2T C2 + Q 0:
(11)
The result of Lemma 2.1 is most powerful, in the sense that it
establishes a condition for the performance index J to be nonpositive
independently of the delay length. Note that the DLMI, or the RDI,
depends on the upper bound for the rate of change of the delay and
not on the length, or an upper bound, of the delay. Such a condition
should thus be, in general, conservative and will usually hold only
for Ad of small norm.
A less conservative condition that depends on the length of the
delay is derived in the next lemma, for a specific d(t).
Lemma 2.2: The performance index J of (2) is nonpositive for
all w 2 L2 [0; T ], and for a given time-delay d(t), if there exist
constant symmetric positive definite matrices P1 ; P2 ; and P3 ; with
I 0 d(t)P3 > 0; 8 t 2 [0; T ], that allow a symmetric positive semidefinite solution P (t) over [0; T ], with P (T ) = PT , to the following
RDI:
P_ + (A + Ad )T P + P (A + Ad ) + d AT P1 A
+ ATd (t + d)P2 Ad (t + d) + P 02 B1 (I 0 dP3 )01 B1T
+ 0d + 02 Ad GATd P + C1T C1 + C2T C2 0
(12)
where
G(t)
and
0d
t
t0d
B1 ( )P301 B1T ( ) d
dAd P101 + P201 ATd :
(13)
1018
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998
Proof: Considering the system of (1) subject to x( )
0 and w( ) = 0; 8 < 0, we find that for t 0
0; 8 t
x(t 0 d) = x(t) 0
Thus, for t
t0d
0
[Ax + Ad x( 0 d)] d 0
t
x_ (t) = [A(t) + Ad (t)]x(t) 0 Ad (t)
1 d 0 Ad (t)
t
t0d
B1 w d
t0d
t
t0d
=
2 (t)
t0d
t
3 (t)
t0d
=
0 d) d
xT (t)P (t)Ad (t)B1 ( )w( )d 0 xT (t)P (t)B1 (t)w(t):
2 <n and for any symmetric positive definite
t
021 dxT P Ad P101 ATd P x +
xT AT P1 Ax d
t0d
022 dxT P Ad P201 ATd P x
t
+
xT ( 0 d)AdT P2 Ad x( 0 d) d
t0d
t
+ 02 xT P Ad GATd P x
t0d
+ 2 wT P4 w + 02 xT P B1 P 01 B T P x:
023 2
(15)
(16)
wT P3 w d
4
1
(17)
Since x(t) = 0; 8 t 0 and P (T ) = PT , the integration of V_ (x)
from zero to T yields xT (T )PT x(T ). We thus find using (14)–(17)
that
T
T
J xT S1 (t)x dt + [kC2 (t 0 d)x(t 0 d)k2 0 kC2xk2 ] dt
0
T
0
T
t
0 I )w dt +
[xT AT P1 Ax
0
0 t0d
+ xT ( 0 d)AdT P2 Ad x( 0 d) + 2 wT P3 w] d dt
+
2
wT (P
4
(18)
where
S1 (t)
P_
T
0
xT S (t)x dt 0
+ 2
T
0
T
T 0d
kC2 xk2 dt
wT (dP3 + P4 0 I )w dt
(20)
where
S (t) = S1 (t) + d AT P1 A + ATd (t + d)P2 Ad (t + d) :
(21)
If there exist a matrix P (t) =
0 over [0; T ] with
P (T ) = PT and constant symmetric positive definite matrices P1 ;
P2 ; P3 ; and P4 (t) = P4T (t) > 0; 8 t 2 [0; T ] such that
P T (t)
8 t 2 [0; T ] (22)
then it follows from (20) that J 0 for all w 2 L2 [0; T ]. Finally,
02zT y zT X 01 z + yT Xy
we find that for any n 2 n constant symmetric matrices P1 > 0
and P2 > 0, and m 2 m symmetric constant matrix P3 > 0 and
P4 (t) = P4T (t) > 0; 8 t 2 [0; T ]
and
T (t) (t) dt;
S (t) 0 and P4 (t) = I 0 d(t)P3 ;
xT (t)P (t)Ad (t)A( )x( ) d
Since for any z; y
matrix X 2 <n2n
0
J
+ B1 (t)w(t):
xT (t)P (t)Ad (t)Ad ( )x(
T ( ) ( ) d dt
(18) implies that
where
t0d
t
t0d
T
8 (t) : (t) = 0; 8 t 0
and considering that x(t) = 0; 8 t 0 and w(t) = 0; 8 t < 0,
dV (x)
= xT [P_ +(A + Ad )T P + P (A + Ad )]x 0 2(1 + 2 + 3 )
dt
(14)
1 (t)
t
d
[Ax + Ad x( 0 d)]
T
0
B1 w d:
Defining the function V (x)
xT (t)P (t)x(t), where P (t)
T
P (t) 0 over [0; T ] and P (T ) = PT , we obtain
t
Since
+ (A + Ad )T P + P (A + Ad )
+ P 0d + 02 B1 P401 B1T + 02 Ad GATd P + C1T C1
+ C2T C2 :
(19)
we note that, in view of (19) and (21), the conditions of (22) are
equivalent to the RDI (12).
Lemma 2.2 provides a delay-dependent criterion of bounded realness for linear systems having delayed state and with an output that
comprises a linear combination of the “current” state as well as of
the “delayed” state. It should be noted that this criterion applies for
a specific time-varying delay d(t), and no restriction is imposed on
the rate of change of the time delay.
Observe that in the case where d(t) 0, Lemma 2.2 coincides
with the standard bounded real lemma for linear systems without a
delayed state.
In view of the monotonicity of the matrices in (11) with respect
to d and the well-known monotonicity properties of RDI’s, Lemma
2.2 leads to the following result.
Corollary 2.1: Consider the system of (1) where the matrix Ad
is assumed time-invariant. Given a scalar d 0, the performance
index J of (2) is nonpositive for all w 2 L2 [0; T ], and for all timedelay d(t) satisfying 0 d(t) d, if there exist constant symmetric
3 > 0 that
positive definite matrices P1 ; P2 ; and P3 , with I 0 dP
allow a symmetric positive semidefinite solution P (t) over [0; T ],
with P (T ) = PT , to the following RDI:
P_ + (A + Ad )T P + P (A + Ad ) + d AT P1 A + ATd P2 Ad
3 )01 B1T + dA
d P101 + P201 ATd
+ P 02 B1 (I 0 dP
+ 02 Ad GATd P + C1T C1 + C2T C2 0:
(23)
In the case where the matrix B1 is time-invariant, using Schur
complements, (12) and (23) can be easily transformed into DLMI’s.
We readily obtain from Lemma 2.2 the following result.
Corollary 2.2: Consider the system of (1) where the matrices Ad
and B1 are assumed time-invariant. Given a scalar d > 0, the
performance index J of (2) is nonpositive for all w 2 L2 [0; T ],
and for all time-delay d(t) satisfying 0 d(t) d, if there exist
constant symmetric positive definite matrices P1 ; P2 ; and P3 ; with
3 > 0, and a matrix P (t) = P T (t) 0; 8 t 2 [0; T ], with
I 0 dP
P (T ) = PT , that solve the following DLMI:
Ad
Ad B1
Ad dP
dP
8(P; P1 ; P2 ; d) dP
P B1
Td P
1
dA
0
dP
0
0
0
2
Td P
0
0
dP
0
0
dA
3)
B1T P
0
0
0 2 (I 0 dP
0
T
T
3
0
0
0
0 2 dP
dB1 Ad P
0;
8 t 2 [0; T ] (24)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998
where
8(P; P1 ; P2 ; d) = P_ + (A + Ad )T P + P (A + Ad )
+ d AT P1 A + ATd P2 Ad + C1T C1 + C2T C2 :
III. INFINITE HORIZON BOUNDED REAL CRITERIA
This section addresses the problem of bounded realness on infinitehorizon. In this case, the matrices of the state-space model of (1) and
(3) are constant; however, the time-delay d is still allowed to be timevarying. The input signal w is assumed to be an arbitrary signal in
L2 [0; 1) and the performance index J of (2) is replaced by
J0 = kzk22 0 2 kwk22
(25)
where is a given positive scalar and now k1k denotes the L2 [0; 1)-
norm.
Throughout this section, (1) is said to be globally uniformly
asymptotically stable (g.u.a.s.) if the equilibrium point x(t) 0 of
the functional differential equation (1), with w(t) 0, is g.u.a.s.
The problem we address here is to find conditions which will ensure
that the system of (1) is g.u.a.s. and J0 0; 8 w 2 L2 [0; 1),
subject to zero initial conditions and w( ) 0; 8 < 0.
A delay-independent criterion for J0 to be nonpositive, which is
the stationary counterpart of Lemma 2.1, is as follows.
Lemma 3.1: Consider the system of (1) and (3) under the assumption that all the matrices are constant. Given a scalar d > 0,
this system is g.u.a.s., and the performance index J0 of (25) is
nonpositive for all w 2 L2 [0; 1) and for all bounded time-delay d(t)
that satisfies (4) if there exist symmetric positive definite matrices P
and Q satisfying the following linear matrix inequalities (LMI’s):
90
ATd P
P Ad
0(1 0 )Q < 0
90 + C1T C1 + (1 0 )01 C2T C2
(26)
P Ad
P B1
0
0 2 I
0(1 0 )Q
ATd P
B1T P
0
0
(27)
where
90
AT P
+ P A + Q:
(28)
1019
The time-derivative of V (t) along the state trajectory of system
(1) is
V_ (t) = xT (t)(AT P + P A + Q)x(t)
+ 2xT (t)P Ad x(t 0 d) 0 (1 0 d_)x(t 0 d)T Qx(t 0 d)
[xT (t) xT (t 0 d)] A9T0P 0(1P0Ad )Q x(xt (0t)d) < 0
d
where 90 is as in (28) and the last inequality follows from the LMI
of (26). This, together with (29), implies that the system of (1) is
g.u.a.s.; see e.g., [9].
In view of the global uniform asymptotic stability of (1), it follows
that z belongs to L2 [0; 1) and then J0 is well defined. Thus, the
proof that J0 0 can be carried out similarly to that of Lemma
2.1.
Similarly to Lemma 2.1, the result of Lemma 3.1 is independent
of the size of the time delay, i.e., the delay is allowed to be any
bounded piecewise continuous function of time with a rate of change
satisfying (4).
The criterion of Lemma 3.1 is given in terms of LMI’s and thus has
the following advantages: 1) it does not involve any parameter tuning
and 2) it can be tested numerically very efficiently using interior point
algorithms that have been recently developed for solving linear matrix
inequalities; see, e.g., [14] and [15].
Remark 3.1: We note that the problem of finding the smallest
> 0, namely 0 , can be computed by solving the following convex
optimization problem in P; Q; and = 2 :
minimize subject to P > 0;
t
0
t d
xT ()Qx() d
Ad
Ad dP
80 (P; P1 ; P2 ; d) dP
Td P
1 0
dA
0dP
Td P
2
dA
0 0dP
<0
(30)
with (31), as shown at the bottom of the page, and
3>0
I 0 dP
where P and Q are symmetric positive definite matrices to be chosen.
Note that
min (P )kx(t)k2 V (t)
[max (P ) + dmax(Q)]
p
The bound 0 is given by 0 = 3 , where 3 is the optimal value
of the optimization problem. Note that this optimization problem can
be solved numerically very efficiently (see [14] and [15]).
A delay-dependent bounded real criterion is presented in the next
lemma.
Lemma 3.2: Consider the system of (1) and (3) under the assumption that all the matrices are constant. Given a scalar d > 0, this
system is g.u.a.s. and the performance index J0 of (25) is nonpositive
for all w 2 L2 [0; 1), and for all constant time-delay d satisfying
0 d d, if there exist symmetric positive definite matrices P; P1 ;
P2 ; and P3 satisfying the following LMI’s:
Proof: Let the following Lyapunov functional candidate exist
for the system of (1):
V (t) = xT (t)P x(t) +
Q > 0; > 0; (26); and (27):
(32)
where
sup kx()k
2 0
[t d;t]
2
:
(29)
80 (P; P1 ; P2 ; d) = (A + Ad )T P + P (A + Ad )
+ d AT P1 A + ATd P2 Ad :
Ad dP
Ad
Ad B1
dP
80 (P; P1 ; P2 ; d) + C1T C1 + C2T C2 dP
P B1
dATd P
0dP1 0
0
0
Td P
2
dA
0 0dP
0
0
0
3)
0
0 0 2 (I 0 dP
0
B1T P
3
1T ATd P
0
0
0
0 2 dP
dB
(33)
(31)
1020
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998
Fig. 1. The amplitude Bode plot of
Tzw
for the first example.
Proof: First, note that by [13, Th. 3.1], the global uniform
asymptotic stability of system (1) follows immediately from (30).
This implies that z belongs to L2 [0; 1) and J0 is well defined. The
proof that J0 0 then follows, similarly to that of Lemma 2.2.
It should be noted that the assumption of a constant time delay is
only required for the stability proof. Similarly to Lemma 2.1, the part
of the proof of Lemma 3.2 related to the condition J0 0 also holds
for a time-varying delay, as long as (1) is guaranteed to be g.u.a.s.
In view of this, we have the following corollary.
Corollary 3.1: Consider the system of (1) and (3) under the
assumption that all the matrices are constant and (1) is g.u.a.s. Given
a scalar d > 0, the performance index J0 of (25) is nonpositive for all
w 2 L2 [0; 1) and for all time-delay d(t) satisfying 0 d(t) d if
there exist a symmetric matrix P 0 and symmetric positive definite
matrices P1 ; P2 ; and P3 satisfying the LMI’s of (31) and (32).
Remark 3.2: Similarly to Lemma 3.1, the criteria of Lemma 3.2
and Corollary 3.1 are given in terms of linear matrix inequalities.
They have the advantages that no parameter tuning is required and
can be tested numerically very efficiently using recently developed
algorithms for solving linear matrix inequalities [14] and [15].
It should be observed that the problem of finding the largest d for
a fixed , or the smallest for a fixed d, can easily solved without
the need of tuning any parameter. Indeed, the largest value of d for a
fixed > 0 can be computed by solving the following quasiconvex
optimization problem in P; P1 ; P2 ; P3 ; and d:
maximize
subject to
d
P > 0; P1 > 0; P2 > 0; P3 > 0; d > 0;
and (30)–(32).
On the other hand, the smallest > 0, namely 0 , for a fixed d > 0
can be found by solving the following quasiconvex optimization
problem in P; P1 ; P2 ; P3 ; and = 2 :
minimize
subject to
P > 0; P1 > 0; P2 > 0; P3 > 0; > 0;
and (30)–(32):
p
The bound 0 is given by 0 = 3 , where 3 is the optimal
value of the optimization problem. Note that the above optimization
problems have the form of a generalized eigenvalue problem, which
is known to be solvable numerically very efficiently; see, e.g., [14]
and [15].
IV. EXAMPLES
We give two examples that demonstrate the use of the above theory.
The first example compares the delay-independent result with the
corresponding delay-dependent solution. It shows that for Ad of small
norm, the former result is not much more conservative than the latter
and may thus be preferred due to its simplicity. In the second example,
we treat a case that cannot be solved by a delay-independent method.
Example 4.1: Consider the system of (1)–(3) with a constant time
delay and the following constant matrices (this is a modified version
of an example of [3]):
A=
B1 =
0
1
02 03
0
;
1
;
Ad =
C1 = [1 0];
0
0:1
00:2 00:3
C2 = 0;
d = 0:4:
Applying Lemma 3.1 to this system we find that the minimum
achievable value of is 0 = 0:556. To compare this 0 with the
actual H1 -norm of the system, we depicted the Bode-plot (the system
is single-input/single-output) of the transfer function
Tzw (s) = C1 (sI 0 A 0 Ad e0sd )01 B1 :
The result for jT (jw)j; w 2 <; and the bound 0 are illustrated in
Fig. 1. We observe there that the peak value of jTzw (j!)j is 0.4545.
The application of Lemma 3.2 on the same system for d = 0:4
leads to 0 = 0:462.
The reason why we were able to solve this example by Lemma 3.1
was the “small” Ad that was taken by [3]. If we increase Ad to
Ad =
0
0:9
01:3 01:9
we find that there exist no 0 such that the conditions of
Lemma 3.1 are satisfied. Applying, however, Lemma 3.2, with d =
0:4, we obtain the minimum value of 0 = 0:71.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998
Fig. 2. The amplitude Bode plot of
Tzw
for the second example.
Example 4.2: We consider the example of [16] where
02
0
;
Ad =
0 00:9
00:5
B1 =
C1 = [1 0];
1
A=
01 0
01 01
C2 = 0:
Note that the matrix A+Ad is Hurwitz stable, but A 0 Ad is Hurwitz
unstable. It was shown in [16] that in such a case the corresponding
time-delay system is not asymptotically stable independently of the
size of the delay. Hence, the stability of the above system is delaydependent. We note that, by the delay-dependent stability criterion in
[17], it was found that the system in this example is guaranteed to
be g.u.a.s. for any time-delay d(t) satisfying 0 d(t) 0:8571.
As expected, for this example there exist no P = P T > 0 and
Q = QT > 0 such that the conditions of Lemma 3.1 are satisfied for
any 0. If we apply, however, Lemma 3.2 for d = 0:846 (which is
very close to the maximum delay of 0.8571 for which [17] predicted
global uniform asymptotic stability of the systems), the minimum
achievable is 0 = 2. If we repeat the calculations for = 3, we
find that the largest d for which the conditions of Lemma 3.2 are
satisfied is 0.893, and for = 100 the largest achievable d is 0.98.
The bound 0 and the corresponding Bode plot of Tzw for d =
0:846 are depicted in Fig. 2, where it is seen that in this example
the bound of 0 is less tight than the one achieved in Example 4.1.
We only note here that one of the reasons for the large difference
between 0 and 0.2364, the actual peak of jTzw (j!)j, is the fact
that as the system is guaranteed to be global uniform asymptotic
stable, Corollary 3.1 guarantees the bound of 0 also for arbitrarily
fast time-varying delays.
V. CONCLUSION
Two lemmas have been introduced for evaluating the H -norm
of linear time-varying systems with time delays. The first provides a
sufficient condition for a given system to possess an H -norm that is
less than or equal to a prescribed value. This condition guarantees, in
fact, that in the time-invariant case the system will satisfy the required
H -norm bound irrespective of the delay length. It would thus be
satisfied only in cases where the state matrix Ad for the delayed state
is of small norm.
1
1
1
1021
The second lemma introduces another sufficient condition which
depends linearly on three positive definite matrix parameters that have
to be found. Unlike the first lemma, the condition of this second
lemma depends strongly on the length of the time delay, and for a
delay that tends to zero this condition coincides with the standard
bounded real lemma that has been derived for systems without delay.
Both lemmas are given in terms of an RDI.
The criteria that have been obtained for the finite horizon case reduce to linear matrix inequalities in the infinite horizon time-invariant
case. An additional linear strict inequality should be solved, however,
to secure the global uniform asymptotic stability of the system.
REFERENCES
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[3] M. Morari and E. Zafiriou, Robust Process Control. Englewood Cliffs,
NJ: Prentice-Hall, 1989.
[4] M. Malek-Zavarei and M. Jamshidi, Time Delay Systems: Analysis
Optimization and Applications. Amsterdam, The Netherlands: North
Holland, 1987.
[5] O. J. M. Smith, “Closer control of loops with dead time,” Chem. Eng.
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[6] A. W. Pila, U. Shaked, and C. E. de Souza, “
filtering for
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Congr., San Francisco, CA, July 1996, vol. J, pp. 49–54.
[7] A. Kojima and S. Ishijima, “Explicit formulas for operator Riccati
equations arising in
control with delays,” in Proc. 34th IEEE Conf.
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[8] G. Tadmor, “
control in systems with a single input delay,” in Proc.
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[9] J. Hale and S. M. V. Lunel, Introduction to Functional Differential
Equations. New York: Springer-Verlag, 1993.
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Netherlands, July 1993.
H
H1
H1
H1
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H1
H1
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H1
[13] X. Li and C. E. de Souza, “Robust stabilization and
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uncertain linear time-delay systems,” in Proc. 13th IFAC World Congr.,
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[16] S. I. Niculescu, L. Dugard, and J. M. Dion, “Stabilité et stabilization
robustes des systèmes à retard,” in Proc. Journées Robustesse, Toulouse,
France, Feb. 1995.
[17] C. E. de Souza and X. Li, “Delay dependent stability of linear timedelay systems: An LMI approach,” in Proc. 3rd IEEE Mediterranean
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tackled in [10]. The design of robust filters and controllers for both
uncertain linear and nonlinear continuous-time systems using sampled
measurements has been investigated in [11] and [12].
In this paper, we investigate an
-filtering problem for a class of
uncertain continuous-time systems under sampled measurements. The
class of uncertain systems is described by a state-space model with
real time-varying norm-bounded parameter uncertainties, unknown
time delays in the state equation. Here attention is focused on the
design of linear dynamic filters using local sampled measurements
which guarantee the robust stability as well as a prescribed
performance for the filtering error dynamics, irrespective of parameter
uncertainties and unknown time delays. The performance measure we
use is defined directly in the continuous-time context. It is assumed
that the initial state of the uncertain system is unknown and both
the cases of finite and infinite horizon filtering are considered. We
show that the robust
sampled-data filtering problems can be
solved via scaled output feedback
sampled-data control problems
that incorporate unknown initial states and involve neither parameter
uncertainties nor unknown time delays. Therefore, existing results on
sampled-data control such as those in [6], [13], and [10] can be
applied to obtain solutions to the problem of robust
sampleddata filtering for the uncertain systems. Throughout this paper <n and
<n2m denote respectively, the n-dimensional Euclidean space and
the set of all n 2 m real matrices. The superscript “T ” denotes matrix
transposition and the notation X Y (respectively, X > Y ) where
X and Y are symmetric matrices, meaning that X 0 Y is positive
semidefinite (respectively, positive definite). L2 [0; T ] stands for the
space of square integrable vector functions over [0; T ]; `2 (0; T ) is
the space of square summable vector sequences over (0; T ); k1k[0;T ]
will refer to the L2 [0; T ] norm over [0; T ]; and k 1 k(0;T ) is the
`2 (0; T ) norm over (0; T ). T is allowed to be 1 and in this case
by the notation [0; T ] we mean [0; 1). k 1 k refers to either the
Euclidean vector norm or the induced matrix two-norm. F (0 ) and
F (+ ) stand for the left limit and right limit of a function F (),
respectively.
H1
H1
H1
Filtering on Sampled-Data Systems
with Parametric Uncertainty
H1
H1
Peng Shi
H1
Abstract— This paper is concerned with the problem of robust
filtering for a class of systems with parametric uncertainties and unknown
time delays under sampled measurements. The parameter uncertainties
considered here are real time-varying and norm-bounded, appearing in
the state equation. An approach has been proposed for the designing
of
filters, using sampled measurements, which would guarantee a
prescribed
performance in the continuous-time context, irrespective
of the parameter uncertainties and unknown time delays. Both cases of
finite and infinite horizon filtering are studied. It has been shown that the
-filtering problem can be solved in terms of differential
above robust
Riccati inequalities with finite discrete jumps.
H1
H1 filtering and control for linear sampled-data systems has been
H1
H1
Index Terms— Riccati equation, sampled measurements, time-delay
systems, uncertain systems.
H1
I. INTRODUCTION
Control of time-delay systems is a subject of great practical
importance which has attracted a great deal of interest for several
decades; see e.g., [1]. Recently, some research work has been
concentrated on the design of robust control for uncertain systems
with time delay. In [2], nonlinear time-delay state feedback controllers
have been considered, whereas Shen et al. [3] have focused on
memoryless linear state feedback. A multirate controller design for a
linear periodic system with multiple delays at input and output has
been presented in [4]. More recently, much attention has been paid to
the problem of
filtering. Several techniques have been proposed
to solve this problem, including a polynomial equation approach [5]
and a Riccati equation approach [6]–[8]. Meanwhile, the
-filtering
problem for systems with parameter uncertainty has also been studied
in [9], which addresses the design of filters guaranteeing both the
robust stability and a prescribed
performance for the filtering
error dynamics in the face of parameter uncertainty. Furthermore,
II. PROBLEM FORMULATION AND PRELIMINARIES
Consider the following class of time-varying uncertain sampleddata delay systems:
(61 ): x_ (t) = [A(t) + 1A(t)]x(t)
+ [A (t) + 1A (t)](x; t; d) + B(t)w(t);
t 2 [0; T ];
x(0) = x0
z (t) = L(t)x(t);
t 2 [0; T ]
y(ih) = C (ih)x(ih) + D(ih)v(ih);
ih 2 (0; T )
H1
H1
H1
Manuscript received August 29, 1995; revised March 12, 1996. This work
was supported in part by the Australian Research Council under Grant
A49532206.
The author is with the Centre for Industrial and Applicable Mathematics,
School of Mathematics, The University of South Australia, The Levels, SA
5095, Australia (e-mail: matsp@zarniwoop.levels.unisa.edu.au).
Publisher Item Identifier S 0018-9286(98)03603-4.
(1)
(2)
(3)
where
d = (d1 ; d2 ; 1 1 1 ; ds )T
(x; t; d) = xi t 0 dr xi t 0 dr
ik 2 [1; 2; 1 1 1 ; n];
1 1 1 xi t 0 dr T ;
rk 2 [1; 2; 1 1 1 ; s];
k = 1; 2; 1 1 1 ; g: (4)
In system (61 ), we also define x(t) = 0; t 2 [0dmax ; 0), where
dmax := maxfdi g; i = 1; 2; 1 1 1 ; s.
In the above, x(t) 2 <n is the state, x0 is an unknown initial state,
w(t) 2 <p is the process noise which is from L2 [0; T ]; v(ih) 2 <q
is the measurement noise which belongs to `2 (0; T ); y (ih) 2 <m is
0018–9286/98$10.00  1998 IEEE