228
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005
[7] D. Cobb, “Controllability, observability and duality in singular systems,” IEEE Trans. Autom. Control, vol. AC-26, no. 12, pp. 1076–1082, Dec. 1984. [8] L. Dai, Singular Control Systems. New York: Springer-Verlag, 1989, Lecture Notes in Control and Information Sciences. [9] J. Huang, “Asymptotic tracking and disturbance rejection in uncertain nonlinear systems,” IEEE Trans. Autom. Control, vol. 40, no. 6, pp. 1118–1122, Jun. 1995. , “Remarks on robust output regulation problem for nonlinear sys[10] tems,” IEEE Trans. Autom. Control, vol. 46, no. 12, pp. 2028–2031, Dec. 2001. , “Nonlinear output regulation: Theory and applications,” in Ad[11] vances in Design and Control. Philadelphia, PA: SIAM, 2004. [12] J. Huang and Z. Chen, “A general framework for tackling the output regulation problem,” IEEE Trans. Autom. Control, vol. 49, no. 12, pp. 2203–2218, Dec. 2004. [13] J. Huang and C.-F. Lin, “On a robust nonlinear servomechanism problem,” IEEE Trans. Autom. Control, vol. 39, no. 7, pp. 1510–1513, Jul. 1994. [14] J. Huang and W. J. Rugh, “Stabilization on zero-error manifolds and the nonlinear servomechanism problem,” IEEE Trans. Autom. Control, vol. 37, no. 7, pp. 1009–1013, Jul. 1992. [15] J. Huang and J.-F. Zhang, “Impulse-free output regulation of singular nonlinear systems,” Int. J. Control, vol. 71, no. 5, pp. 789–806, 1998. [16] A. Isidori and C. I. Byrnes, “Output regulation of nonlinear systems,” IEEE Trans. Autom. Control, vol. 35, no. 2, pp. 131–140, Feb. 1990. [17] F. L. Lewis, “Fundamental, reachabilty and observability matrices of discrete descriptor systems,” IEEE Trans. Autom. Control, vol. 30, no. 5, pp. 502–505, May 1985.
using an augmented state space representation of dimension n. ( +1) where n is the dimension of the state of the delay system and 2 N is the value of the delay, and then apply the standard observers design techniques. However these techniques often yield large scale observers. Since large scale observers necessitate a high implementation cost, the importance of designing a low order observers is then obvious from a practical point of view. The problem of a reduced-order memoryless state observer for discrete time-delay systems has been considered in [11]. The order of this observer is equal to the number of unstable and/or poorly damped eigenvalues of the system. This result has been extended to large scale discrete time-delay systems [12]. In [15], a new method for the functional observer design for continuous-time delay systems was presented. The aim of this note is to present the discrete-time counterpart of these results. We will show that, following the approach of [15] and introducing appropriate analysis methods for the discrete-time case, we obtain new results for linear functional observers design for discrete-time delay systems. As in [15], an algebraic method for functional observers design is presented, the order of these observers is equal to r n, where r is the dimension of the functional to be estimated. Their stability is studied and linear and bilinear matrix inequalities [(LMIs) and (BMIs)] formulations for the sufficient conditions are presented. Consider the following discrete-time delay system:
( + 1) = Ax(t) + Ad x(t 0 ) + Bu(t) y (t) = Cx(t) z (t) = Lx(t)
x t
Linear Functional Observers for Systems With Delays in State Variables: The Discrete-Time Case M. Darouach
I. INTRODUCTION AND PRELIMINARIES
(1b) (1c)
where x(t) 2 n is the state vector, y (t) 2 p is the measurement output, and z (t) 2 r is the vector to be estimated, with r n. 2 is the known time-delay of the system. Matrices A, Ad , B , C , and L are real and of appropriate dimensions. Without loss of generality it is assumed that rank L = r and rank C = p. Consider the following functional observer for system (1):
Abstract—This note extends to the discrete-time case the design of linear functional state observers, recently developed for continuous-time delay systems. Sufficient conditions for the stability dependent of delays and stability independent of delays are derived using linear and bilinear matrix inequalities [(LMIs) and (BMIs)] formulations. Index Terms—Bilinear matrix inequality (BMI), discrete time-delay systems, functional observer, linear matrix inequality (LMI), stability.
(1a)
( + 1) = N (t) + Nd (t 0 ) + Dy(t) + Dd y(t 0 ) + Eu(t) z ( t) = ( t) + F y ( t)
t
(2a) (2b)
()2
()2
r is the state vector and z t r is the estimate of where t z t . N , Nd , D , Dd , E , and F are unknown matrices of appropriate dimensions, which must be determined such that z t asymptotically converges to z t .
()
()
()
The problem of state estimation of dynamic systems that include time delays in their models are problems of recurring interest. This is justified by the fact that the time-delays are generally causes of instability, and are frequently encountered in various engineering systems [1]–[3] and [13]. In recent years, the problem of observers design for continuous time-delay systems has been a subject of intensive research (see [15] and the references therein). Little attention has been paid toward discrete-time systems with delays. Contrary to the continuous-time case, the discrete-time delays systems are of finite dimension. The approach to design an observer for these systems consists in
N
Manuscript received September 2, 2003; revised July 12, 2004. Recommended by Associate Editor L. E. Holloway. The author is with the CRAN-CNRS (UMR 7039), Université Henri Poincaré NancyI, IUT de Longwy, 54400 Cosnes et Romain, France (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2004.841932
then we have the following theorem (see [15] for the proof). Theorem 1: The r th-order observer (2) will estimate (asymptotically) z (t) if and only if the following conditions hold: i) e(t + 1) = N e(t) + Nd e(t 0 ) is asymptotically stable; ii) 9A 0 N 9 0 DC = 0; iii) 9Ad 0 Nd 9 0 Dd C = 0; iv) E = 9B .
Now, define the following matrix:
9 = L 0 FC and define e(t) as the error between z (t) and its estimate z (t) as
( ) = z(t) 0 z(t) = Lx(t) 0 z(t):
e t
(3)
Its derivative is given by e(t + 1)
= N e(t) + Nd e(t 0 ) + (9A 0 9 0 DC )x(t)+(9Ad 0 Nd 9 0 Dd C )x(t 0 ) + (9B 0 E )u(t).
0018-9286/$20.00 © 2005 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005
From Theorem 1, the design of the observer(2) is reduced to find the matrices N , Nd , D , Dd , E , and F so that conditions i)–iv) are satisfied. Now, using notations and results from [15] (Lemma 1, Lemma 2, and Corollary 1), we have the following results. Let T1 be a full row matrix of dimension (n 0 r ) 2 n such that
L = [ H1 E1 ]01 T1
0 CE1 CAE1 CAd E1 and 2 = CE1 0 [ LAE1 LAd E1 ]. Then, the unknown matrices T , F , and K satisfy and define the following matrices 6 = the following equation:
[T F K ]6 = 2
(4)
where D = K + NF and Dd = T + Nd F . The necessary and sufficient condition for the existence of a solution to equations ii)–iv) of Theorem 1 or, equivalently, to matrix equation (4) can then be given by the following lemma. Lemma 1: The necessary and sufficient condition for (4) to be solvable is
0 C 0 C CA CAd CA CAd C 0 rank = rank C 0 : LA LAd 0 L 0 L L 0 L 0
(5)
Under (5), the general solution of (4) is given by
[ T F K ] = 26+ + Z (I 0 66+ )
229
Before giving the stability conditions of the obtained observer, we can give the following lemmas [15]. Lemma 2: The matrix N = N + Nd is Hurwitz if and only if
L 0 L(A + Ad ) C (A + A d ) rank C
where
3 = LAH1 C
0 26+
L 0 LA CA rank C
26+ CAd H1 , 0 = (I 0 (I
0 66+ )
0 66+ )
0
CA H1 , and = C
C CAd H1 , and where Z is an arbitrary matrix of
0
appropriate dimension. Using the result of Lemma 1 and relations (7) and (8), the observer error dynamics can be written as
e(t + 1) = Ne(t) + Nd e(t 0 ) = (3 0 Z 0)e(t) + (1 0 Z )e(t 0 ): Now, the problem of the functional observer (2) design is reduced to the determination of the free matrix parameter Z such that condition i) of theorem 1 is satisfied. We can distinguish two different stability criteria, the first one is delay-independent and the second one is delaydependent. These criteria are related to the stability of matrices N and N = N + Nd .
0 C CA CAd = rank C 0 0 0 L C L 0 L 8 2 ; j j 1 :
CAd
(10)
Remark 1: The above results are independent of the choice of the
L T2 is nonsingular, then there exists a regular matrix Q1 of appropriate L L L 01 dimension such that = Q1 , its inverse is = T2 T1 T2 0 1 [ H1 E1 ] Q1 , then by postmultiplying (4) and (5) of [15] by this matrix T1 , in fact let T2 be another full-row rank matrix such that
matrix, we obtain (4), which proves that the results are independent the choice of T1 . Remark 2: Another approach to design a functional observer for the discrete-time delay system can be obtained from the augmented system, for the simplicity of the presentation we consider the case where u(t) = 0, in this case, (1) can be written as
(6)
0
0LAd
0 0
X (t + 1) = X (t) Y ( t) = X ( t) z ( t) = X ( t)
(8)
CA H1 , 1 = LAd H1 C
(9)
Lemma 3: The matrix N is Hurwitz if and only if
(7)
0
0 C CA CAd = rank C 0 0 0 L C L 0 L 8 2 ; j j 1 :
CAd
0 0
and matrices N and Nd are given by [15]
N = 3 0 Z0 Nd = 1 0 Z
0LAd
where X (t) =
x ( t) x(t 0 1)
...
x(t 0 + 1) x (t 0 )
, Y ( t) =
(11a) (11b) (11c)
y ( t) , y(t 0 1)
0 0 . . . Ad 0 0 ... 0 = ... ... ... ... ... 0 0 0 ...I 0 0 0 0 ... I A I
= [ L 0 0 . . . 0 ], and
=
C 0 0 ... 0 . In this 0 0 0 ... C
case, by applying the results of [16], one can see that the conditions for the reduction of the order of the observer are very restrictive compared to those given by (5) and (10). II. MAIN RESULTS
In the sequel, the orthogonal complement A? for a real n 2 p matrix A with rank q is defined as the (n 0 q) 2 n matrix such that A? A = 0 and A? A?T > 0. The independent of delay conditions for the stability of the obtained observer are given by the following theorem.
230
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005
Theorem 2: Under conditions (5) and (10) the observer (2) is asymptotically stable if there exist 0 < P = P T and 0 < Q = QT satisfying the following LMIs:
0P + Q 0 3T P T? [ 00 0 0 ] 0 0Q 1T P P3 P 1 0P 2 [ 00 0 0 ]? < 0 0 P + Q < 0:
which can be written as
0 5 0 0 Z [ 0 0 ] 0 [ 0 0 ]T Z T [ 0 0 P ] < 0
(12a) (12b)
0P + Q 0 3 T P where 5 = 0 0Q 1T P P3 P 1 0P
0 ? 0 5 [ 0 0 P ]? < 0
form
0
j =t
eT (j )Qe(j )
and
where P > 0 and Q > 0. Then, along the solution of equation i) of Theorem 1, we obtain
1V = V (e(t + 1)) 0 V (e(t)) = eT (t)[N T P N 0 P + Q]e(t) + e T ( t) N T P N d e ( t 0 ) + eT (t 0 )NdT P Ne(t) + eT (t 0 ) NdT P Nd 0 Q e(t 0 ) e ( t) T e (t 0 ) T 2 N PNNT 0P NP + Q d e ( t) 2 e (t 0 ) :
0
N P Nd Q 0 NdT P Nd (13)
N T P Nd Q 0 NdT P Nd
N
Nd
NT NdT 0P 01
bP and q 2 aP Q 0 abP 2 < , one can see that there exist P > and Q > satisfying these
