one. The above procedure can then be repeated to eliminate all zero eigenvalues and yield the required result. We are now in a position to combine Proposition ...
940
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
one. The above procedure can then be repeated to eliminate all zero eigenvalues and yield the required result. We are now in a position to combine Proposition 1 and Lemma 2 and present the main result of the paper. Theorem 1: Let Z be any set of symmetric matrices such that Z 2 Z implies that for all sufficiently small 1 and 2 ; (1+ 1 )Z + 2 I 2 Z . Let matrices R; U , and V , be given. Let the columns of U? and V? be bases for the null spaces of U 3 and V 3 , respectively. There exists a Z 2 Z and a matrix Q such that
(R + UQV 3 )3 Z (R + UQV 3 ) 0 Z < 0 if and only if there exists an invertible Z 2 Z such that U?3 (RZ 01 R3 0 Z 01 )U? < 0 V?3 (R3 ZR 0 Z )V? < 0:
(43)
(44)
Proof: The proof essentially consists of removing the assumption that 911 is invertible in the statement of Proposition 1. This is achieved by perturbing Z , as per Lemma 2. In the proof of Proposition 1, the three matrices in (26) are congruent, and thus
0 911 = P 3 Z0 U 3 (Z 01 0 RZ (45) 01 R3 )U? P ? 3 (Z 01 0 for some full-column rank P . Defining f (Z 01 ) = U? 0 1 3 RZ R )U? and applying Lemma 2 yields the required result. IV. CONCLUSIONS The inequalities in (19) are equivalent to those in [4]; the control design problems in [4] can then be trivially extended to the case where the scaling matrices are not positive definite. As an example, by relaxing the positivity constraints on the scaling matrices in [4] one can readily design suboptimal L1 controllers (as opposed to H1 controllers). It is shown in [9] that this generalization can be used to design controllers for spatially distributed systems. ACKNOWLEDGMENT The author would like to thank G. Dullerud and S. Lall for their helpful comments and suggestions. REFERENCES [1] C. Davis, W. M. Kahan, and H. F. Weinberger, “Norm-preserving dilations and their applications to optimal error bounds,” SIAM J. Numerical Anal., vol. 19, no. 3, pp. 445–469, 1982. [2] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1995. [3] S. Parrott, “On a quotient norm and the Sz.-Nagy-Foias lifting theorem,” J. Functional Anal., vol. 30, pp. 311–328, 1978. [4] A. Packard, “Gain scheduling via linear fractional transformations,” Syst. Contr. Lett., vol. 22, no. 2, pp. 79–92, 1994. [5] A. Packard, K. Zhou, P. Pandey, J. Leonhardson, and G. Balas, “Optimal constant I/O similarity scalings for full information and state feedback control problems,” Syst. Contr. Lett., vol. 19, pp. 271–280, 1992. [6] S. P. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [7] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to -infinity control,” Int. J. Robust and Nonlinear Contr., vol. 4, pp. 421–448, 1994. [8] R. D’Andrea, “A linear matrix inequality approach to decentralized control of distributed parameter systems,” in Proc. American Control Conf., 1998, pp. 1350–1354. [9] R. D’Andrea, G. Dullerud, and S. Lall, “Convex synthesis for multidimensional systems,” in Proc. IEEE Conf. Decision Control, 1998, pp. 1883–1888.
H
`
[10] A. Rantzer, “On the Kalman–Yakubovich–Popov lemma,” Syst. Contr. Lett., vol. 28, no. 1, pp. 7–10, 1996. [11] G. E. Dullerud, R. D’Andrea, and S. Lall, “Control of spatially varying distributed systems,” in Proc. IEEE Conf. Decision Control, 1998, pp. 1889–1893. [12] I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman, “The band method for positive and contractive extension problems,” J. Operator Theory, vol. 22, pp. 109–155, 1989. [13] V. Ionescu and C. Oara, “The four-block Nehari problem: A generalized Popov-Yakubovich-type approach,” IMA J. Math. Contr. Information, vol. 13, pp. 173–194, 1996. [14] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, 1992. [15] R. D’Andrea and C. Beck, “Temporal discretization of spatially distributed systems,” in Proc. IEEE Conf. Decision and Control, 1999, pp. 197–202.
Existence and Design of Functional Observers for Linear Systems M. Darouach
Abstract—This paper presents straightforward derivations of the functional observers design for linear time-invariant multivariable systems. The order of these observers is equal to the dimension of the vector to be estimated. Necessary and sufficient conditions for the existence and stability of these observers are given. Illustrative examples are included. Index Terms—Continuous-time, discrete-time, existence condition, functional observer, linear systems, stability.
I. INTRODUCTION The problem of observing the state vector of a deterministic linear time-invariant multivariable system has been the object of numerous studies ever since the original work of Luenberger [4], [5] first appeared. Algorithms for designing a reduced-order observer which can give an estimate of the entire state vector or of a linear functional have been reported in many papers [1], [6] and in the book [8]. The Observers problem is to reconstruct (or to estimate) the state or a linear combination of the states of the system using the input and output measurements. The problem of the functional observer design was related to the constrained or unconstrained Sylvester equations [10], [11]. This equation was also obtained in the unbiased optimal filtering [2], [3], [7], [12]. Generally to solve this problem, many authors have proposed to transform the initial system to an equivalent one (by using some regular transformations) of reduced-order and to design an observer for this system. Only sufficient conditions for the existence of these observers were given [2], [3], [7]. Recently the authors of [12] have given the necessary and sufficient conditions for the existence of the optimal unbiased functional filter (or functional observer); unfortunately, as we can see these conditions are only sufficient in general. In this paper, a new functional observer of dimension r is introduced, and a straightforward method for its design is derived. The necessary
Manuscript received April 23, 1999; revised October 1, 1999. Recommended by Associate Editor, K. Zhou. The author is with C.R.A.N-ACS, IUT de Longwy, Université Henri Poincaré Nancy I, 54400 Cosnes et Romain, France. Publisher Item Identifier S 0018-9286(00)04177-5.
0018–9286/00$10.00 © 2000 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
and sufficient conditions for the existence and stability of this observer are given. Continuous-time as well as discrete-time systems are considered.
941
Using the definition of matrix P , (6) leads to
=
N
+ 0 K C L+
(7)
P AL
and
=
II. CONTINUOUS-TIME SYSTEMS
PA
Consider the linear time-invariant multivariable system described by
R
_= = =
+
(1-a)
x
Ax
y
Cx
(1-b)
z
Lx
(1-c)
R
Bu
where x 2 n and y 2 p are the state vector, the output vector of the system, u 2 m is the input vector of the system, and z 2 r is the vector to be estimated where r n. A; C , and L are known constant matrices of appropriate dimensions. We assume, without loss of generality that rank C = p and rank L = r . Our aim is to design an observer of the form
R
R
_= + ^= +
w
Nw
z
R
w
Jy
+
0
0
JC H
=0 =
z
z
Px
Lx
_=
Ne
+(
PA
0
0
02 1 0 1 03 1 0 0 01
=
L
= [1 1 0]
;
rank
0
C
) +(
PB
JC
)[ + L
N E:
(10)
:
C
Proof: Equation (8) can be written as
[
E
where 6 is given by
K
]6 =
6=
(11)
LA
CA
:
C
From the general solution of linear matrix equations [9], there exists a solution of (11) if and only if
( 0 6+ 6) = 0
rank 6 = rank(6) LA
0 )
H u:
(5)
= 10 01 00
;
I
0
L
since [L+ I 0 L+ L] Moore–Penrose generalized inverse of matrix L.
(12)
:
In this case the general solution of (11) is given by
[
E
K
] = 6+ + ( 0 66+ ) LA
(13)
Z I
where Z is an arbitrary matrix of appropriate dimension. Now the left-hand side of (10) is equal to LA
rank
CA C
LA
= rank
L
[ +
CA
L
C L
+ + C AL + CL LAL
= rank
and
+ L] = 0 (6) + is of full row rank, where L denotes the
0
J
L
I
0
+ L]
L
0
LA
CA C
I
= + rank 6 LA
r
NP
= 0
LA I
One can easily verify that the necessary and sufficient condition of [12] (which is also equivalent to rank [ L ] = rank[ Lc ]) is not satisfied, and the functional observer (or the unbiased filter) exists. This example shows the limitations of the observer considered in [12]. In the sequel we shall give the necessary and sufficient conditions for the existence and stability of the functional observer (2). Equation (3) is equivalent to PA
K
CA
= rank
CA
L
:
(
and
L ;
or equivalently
JC x
C
L
LA
z
The rest of the proof is similar to that of [13, Ths. 7–9]. Equation (3) is a Sylvester one and generalizes also the condition (1) of [13, Ths. 7–9] and the condition for the unbiasedness of the filter in the stochastic case [2], [3], [7], [12] (where E = 0). In [2], [3], and [7] it was shown that this equation is difficult to solve and only sufficient conditions for the existence of the solutions were given. Recently, the authors of [12] (for the case E = 0) have given the necessary and sufficient conditions for the existence of the observer (2) (or unbiased filter); unfortunately, these conditions are very restrictive as we can see from the following example. Consider the linear continuous system described by the following matrices: A
C I
Necessary and sufficient conditions for the existence of the observer are given by the following lemma. Lemma 1: There exists an r th-order observer (2) for the system (1) if and only if
0^
w:
NP
C
(9)
Then, the dynamics of this estimation error are given by e
= ( 0 + )
L ;
(4)
is Hurwitz, i.e., has all its eigenvalues in the left-hand side of the complex plane, where P = L 0 E C . Proof: The observer reconstruction error is
= 0^= = 0
L
(3)
P B:
N
e
A I
(2-b)
Ey
NP
= ( 0 + )
A
(2-a)
Hu
where z^ 2 r is the estimate of z . N; J; H , and E are constant matrices of appropriate dimensions to be designed. We have the following theorem. Theorem 1: The state z^(t) in (2) is an asymptotic estimate of z (t) for any x(0); z^(0), and u(t) if and only if PA
where
(8)
KC
(14)
and the right-hand side is equal to
rank
CA C
= rank
L
CA
[ + L
C L
= rank
+ + CL
C AL
I
= + rank(6) r
Using (12), the lemma is proved.
:
I
0
+ L]
L
0
CA C
(15)
942
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
From the definition of P , (7) can be written as N
+ 0 [E
=
LAL
K
2) The condition given in [12] is only sufficient for the existence of the functional observer (2), since if
+ + CL
]
C AL
(16)
and using (13), we obtain F
(17)
ZG
where F
+ + CL
+ 0 LA6+
=
= rank
C
C AL
LAL
(18)
rank + + CL
= ( 0 66+ )
C AL
I
(19)
:
The condition for N to be Hurwitz is given by the following lemma. Lemma 2: The matrix N given by (17) is Hurwitz (or the pair (G; F ) is detectable) if and only if sL
rank
0
LA
CA
CA
= rank
C
rank
;
s
:
sL
0
LA
[ + L
CA C
sI
0
+ + CL
+
LAL
C AL
I
6+ 0 66+ 66+
I
0
sI
66+
sI
0
G
+ +
C AL CL
0
F
+ + CL
G
66+
= rank
F
G
sI
= rank
rank
sL
0
LA
CA
E
]=
L
0 ]
= rank [
E
= rank
sI
01
C
0
0
M
sL
sM I
A
0
LA
CA C
:
C
With the above results and under conditions (10) and (20), we can propose the following design method. 1) Matrices F and G can be obtained from (18), (19). 2) The matrix gain Z can be obtained by any pole placement method (since the pair (G; F ) is detectable). Then we obtain N given by (17). 3) From the obtained values of Z and N , we can deduce the matrices E and K from (13), and the matrix J from (9). 4) Matrix H is given by (4).
C AL
F
0
I
+ L]
L
LA
6
sL
0
+ + CL
+
LAL
C AL
0 0 6 0 0 6
0 LA
6
III. DISCRETE TIME SYSTEMS In this section we present the extension of the above results to the discrete-time systems. Let us consider the following system:
0
A
CA
06+
0
+ + CL
I
C AL
I
:
C
= rank
I
0
C
I
0 0
sI
x t
Ax t
y t
Cx t
z t
Lx t
; u t
Bu t
y t
z t
+ rank 6
= rank
( + 1) = ( ) + ( ) (21-a) ( )= ( ) (21-b) ( )= ( ) (21-c) where ( ) 2 Rn ( ) 2 Rm , and ( ) 2 Rp are the state vector, the input vector, the output vector of the system, and ( ) 2 Rr is the x t
From (15), the lemma is proved. From the above lemmas, we have the following theorem. Theorem 2: The necessary and sufficient conditions for the existence and stability of the functional observer (2) for system (1) are: conditions (10) and (20). Remarks: 1) If L = I , the observer is a full-order one, in this case the conditions of the theorem reduce to the well-known result: the detectability of the pair (C; A) since sI
rank
M
Design of the functional observer
0
LA
= rank 0 0 = rank
n
LA
C
= rank
then
(20)
CA
= rank
[
L
Proof: The left-hand side of (20) can be written as
0
L
C
8 2 C Re( ) 0
=
C
then (10) is verified, and condition (20) becomes: the pair (C; A) detectable. In fact, let
;
C
s
sL
L
then (10) is verified, but the inverse is not true. In this case, it is )
