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Each column of Dp is full-rank for all s E U. One choice for Ysl is [ i2 :] . A decoupling controller for P is

- - C ~ Z , = DqALV,5p

- -3(s - 3) ----(I- s + l [A :I)

has zero diagonal entries.

N. CONCLUSION We considered the stability of LTI, MIMO systems under stable

NLTV diagonal post-multiplicative perturbations of the plant (sensor failures) and pre-multiplicative perturbations of the plant (actuator failures). Assuming that any one of the sensors or actuators may fail one at a time, without prior knowledge of the NLTV failure and its location, we obtained necessary and sufficient conditions for stabilization of the LTI plant using LTI controllers. We developed controller design methods for two classes of plants and explicitly derived a family of controllers achieving closed-loop stability under an unknown stable NLTV failure of at most one loop.

The closed-loop stability condition for all possible unrestricted stable NLTV perturbations of one sensor or actuator is that all diagonal entries of certain transfer functions of the nominal LTI system are exactly zero. Note that such a strict condition is due to the general unconstrained nature of the NLTV perturbations. If this condition could not be met and the diagonal entries were not zero (but their gains were bounded by sufficiently small E), then as in standard robustness results based on the small-gain theorem, closed- loop stability is still guaranteed for a restricted class of perturbations (whose gains are bounded by l /~ ) .

The design goal stated above can easily be expressed as a convex design specification [ 11. Despite the obvious advantages of convex problem formulations, optimization based design approaches cannot bring an explicit answer in cases where there does not exist such a controller (in general, not for a particular finite parametrization only) or make explicit use of the plant properties in successive redesigns (each new plant will be treated as a new design problem). Hence, whenever applicable, analytical solutions should be used to weed out infeasible constraints and initialize feasible ones. The results in this note are intended for shch complementary utilization. The analysis results provide valuable necessary conditions on plants that admit such controllers. For two classes of plants, the explicit design approach developed here guarantees a desired nominal controller. The particular design approach complements the convex optimization based control design approach since it explicitly generates a family of feasible stable parameters for the controller. The freedom can then be used to satisfy other design specifications.

REFERENCES

S . P. Boyd and C. H. Barratt, Linear Controller Design: Limits of Pelfonnance. Englewood Cliffs, NJ: Prentice-Hall, 1991. C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Outpur Properiies. New York Academic, 1975. C. A. Desoer and M. G. Kabuli, “Linear stable unity-feedback system: Necessary and sufficient conditions for stability under nonlinear plant perturbations,” IEEE Trans. Auromr. Contr., vol. 34, no. 2, pp. 187-191, 1989. __, “Factorization approach to nonlinear feedback systems,” F’h.D. dissertation, Electronics Res. Lab., Memo. No. UCBERL M89/64, Univ. California, Berkeley, 1989. M. Fujita and E. Shimemura, “Integrity against arbitrary feedback-loop failure in linear multivariable control systems,” Auromatica, vol. 24, p. 765, 1988. A. N. Giindeg and C. A. Desoer, Algebraic Theory of Linear Feedbnck Sysrems Wirh Full and Decentralized Compensators, Lecture Notes in Control and Information Sciences, vol. 142, Springer-Verlag, 1990. A. N. GiindeS, “Stability of feedback systems with sensor or actuator failures: Analysis,” Inr. J. Contr., vol. 56, no. 4, pp. 135-153, 1992. __ , “Parametrization of decoupling controllers in the unity-feedback system,” IEEE Trans. Automat. Control, vol. 31, no. 10, pp. 1572-1576, 1992. C. A. Lin and T. F. Hsieh, “Decoupling controller design for linear mul- tivariable plants,’’ IEEE Trans. Automat. Contr., vol. 36, pp. 485-489, 1991. M. Vidyasagar, Conrrol System Synthesis: A Factorization Approach. Cambridge, MA: M.I.T. Ress, 1985.

Design of Observers for Descriptor Systems

M. Darouach and M. Boutayeb

Absbact-In this note simple and straightforward methods to design full- and reducedader observers for linear time-invariant descriptor systems are presented. The approach for the reducedader observer design m based on the generazed Sylvester equation. sufficient eonditiolrs for the existence of the observers are given.-An illustrative erampk is included.

,

I. INTRODUCTION The problem of designing observers for descriptor systems has

received considerable attention in the last two decades [ 11-[7]. Many approaches exist to design observerg for descriptor systems. In [l], a method based on the singular value decomposition and the concept of matrix generalized inverse to design a reduced-order observer has been proposed. In [2], the generalized Sylvester equation was used to develop a procedure for designing reduced-order observers. In [3], a method based on the generalized inverse, which extend the method developed in [SI, was presented. Full- and reduced-order observers for discrete-time descriptor systems have been presented in [5] and [7].

All these works have been done under the assumptions of regularity and generalized observability. Theae conditions are very restrictive and, as can be seen in [6], the design of the observer can be done under less restrictive conditions. In [6], the conditions of regularity and modal observability have been assumed. In [lo], a new method

Manuscript received August 26, 1993; revised July 6, 1994. The authors are with the CRAN-EAIkAL-CNRS UA 821 - Universitd de

IEEE Log Number 9412366. Nancy I, 54400 Cosnes et Romain, France.

0018-9286/95$04,00 0 1995 IEEE

1324 BEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 7, JULY 1995

is developed for the reduced-order observer design for descriptor systems where the regularity assumption (A*, E* are square and det (XE' - A*) # 0) is not required and a condition less restrictive than the impulse observability is used.

In this note we present simple new methods to design full- and reduced-order observers for continuous-time descriptor systems. For the full-order observer the approach is based on a method developed recently in [9], and extends it to descriptor systems, while the reduced-order observer design method is based on a new resolution method of the generalized Sylvester equation. These approaches do not require any coordinate transformation. As in [lo], the regularity assumption is not required; however, a condition which generalizes the impulse observability is assumed as in [12].

It is well known that a solution to nonregular descriptor systems does not exist or may not be unique for given initial conditions and time piecewise continuously differentiable input [lo]. According to [ 111, one can note that even if the descriptor vector of a nonregular system cannot be uniquely determined, it is possible to obtain a unique input-output description. As can be seen in [lo], the observers design for nonregular descriptor systems can be realized, since measurements provide additional information to the descriptor vector equation. One can note that the descriptor vector can be uniquely determined from the descriptor vector equation and the measurement vector if and only if the matrix pencil [""',l"*] is of full column rank.

II. FULL-ORDER OBSERVER DESIGN Consider the linear time-invariant descriptor system

E'X = A * x + B*u (1.1) y* = c + x (1.2)

where r E R". U E Rk, and y* E Rp are the state vector, the input vector, and the output vector, respectively. E* E R" n , A* E R" ", B' E R" ', and C8 E RP are known constant matrices. We assume that rank E* = T < n, and without loss of generality rank C* = p.

In the sequel we assume that

(2)

Then, since rank E* = T , there exists a nonsingular matrix P such that

PE' = [f] PA' =.[.1]

and

PB* = [i] (3)

with E E R'." and rank E = I T .

Following [?I. system (1) is a restricted system equivalent (r.s.e) to

Ei = A x + Bu y =Cx

where

T .

Then, using the above transformations, we can easily prove that

(5.1)

or equivalently

= n

since E is a full row rank matrix.

such that Now since rank [E] = n , there exists a nonsingular matrix [z j]

aE + bC = I , cE+dC=O.

Our aim is to design an observer of fixed order n satisfying

i = N z + L ~ ~ + L Z ~ + G U (7.1) 2 =t + by + K d y (7.2)

where N . L1, Lz, G, and A- are unknown matrices of appropriate dimensions, which must be determined such that 2 will asymptotically converge to T .

From (4), (6), and (7) the estimate error

e = x - z = ( a + Kc)Ez - 2 (8)

satisfies the linear state equation

P = N e + [ ( a + h-c)A - N(a + Kc)E - L I C - L z C ] ~ + [ ( U + K c ) B - G]u.

Quation (9) reduces to the homogeneous equation

i = Me

provided the matrices G, K , L1. Lz , and N satisfy

G = (a + K c ) B

and

(9)

( a + Kc)A - N ( a + 1ic)E - L1C - LzC = 0 (11)

then 3 will asymptotically converge to z if and only if N is a stability matrix.

The problem of designing a full-order observer of the singular system (1) is reduced to find matrices K , L1, Lz, and a stability matrix N such that (11) is satisfied.

Using (6), (1 1) can be written as

( a + Kc)A - N ( I n - bC - K d C ) - LiC - LzC = 0 or

X = ( U + Kc)A - LzC + [ N ( b + Kd) - Ll]C. (12)

Take

L1 = N ( b + Kd) (13)

then

N = ail + KcA - LzC = aA + [ K - Lz] . ['."I (14)

which must be a stability matrix. Now Y is a stability matrix if and only if the pair ( a A , rg]) is

detectable. In this case, matrices K and LZ can be determined from the standard observer design problem.

From the definition of matrices E, A, and C it is easy to prove the following lemma.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 7, JULY 1995

Lemma 2.1: For systems (1) and (3) we have

1325

Our aim is to design an observer in the form

rankpEL:A'] =rank[ S E - A 1, V s E C. (15)

Now we can give the sufficient conditions for the existence of the observer (7) for system (1).

Theorem 2.1: Sufficient conditions for the existence of the ob- server (7) are

a) rank 0 E* = n + r a n k E * , [: 3 Proof: From (2)-(5) condition a) is equivalent to rank [ g ] = n,

which yields (6), (8), and (12). Now, note that

SE - '4 rank "'1 =rank [ ]

0 0 S E - A =rank [:; I , -;:I [ $ ]

0 0 SE - A

=rank[ 5 ] (17)

then using (a) or equivalently (6) we obtain

S E - A b 0 S E - A rank[ 2 ] =rank[+ --: i] [ 2 ]

SI , - nA =rank[ 1,

Vs E C, Re(s) 2 0 (18)

or ( a d . ['e]) is detectable. Remark I: Matrices a, b , c, and d such that (6) is verified can

be found by using the singular value decomposition (S.V.D.) of the matrix [ F ] . In fact, since rank [ E ] = R there exist two orthogonal matrices U and 1' such that

[:] = P:]. where

C =diag (a l ? U Z , a,)

and I

IT, > 0 ( i = 1, n )

then

111. REDUCED-ORDER OBSERVER DE~IGN In this section we present a new method to design a reduced-

order observer for singular system (1). We assume without loss of generality that rank C = q.

6 =nu, + Ly + H u (19.1) i = M w + F y (19.2)

where U' E R n P q . The problem of the observer design is reduced to find matrices

II, L , H, M . and F such that the estimate j: converges acymptot- i c d y to s.

The solution of this problem is given in the following theorem. Theorem 3.1: Let T be an (n - q ) x r matrix such that

T A - IITE = LC where

det r:] f 0 .

Then for

and

H =TB (21)

we have ?( t ) - ~ ( t ) = bfe"'[w(O) - T E z ( O ) ] . The convergence of the reduced-order observer is obtained when

II is a stability matrix. This theorem is easy to prove, hence its proof is omitted.

We will next give a new method to find the matrix T. Define the following nonsingular matrix:

K TE [:I = piq I q ] [ c ] where K is an (T I - q).q arbitrary matrix and R is an (n - q) .n matrix of full row rank.

Then, we have

(T h-) [E] = R

since rank [E] = n, (23) has the following solution

where [ : I + = A ( E T C T ) : arbitrary matrix of appropriate dimension.

A = ( E T E + CTC)- l , and 2 is an

From (24), we have

T = R A E T + Z -CAET ['. -

and from (20) we obtain

11 = T A M L = T A F .

The problem is reduced to find the matrix T such that II is a stability matrix.

1326 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40. NO. 7. JULY 1995

Substituting (25) into (26) gives

where

r = R A E ~ A M and

If the pair (I?, a) is detectable, one can design a reduced-order

In the sequel, we will give the sufficient conditions for the existence

Theorem 3.2: For system (l), the reduced-order observer (19)

Pro08 Condition a) can be obtained from (23), (2), and (5).

observer from the standard methods, in the form (19).

of the reduced-order observer.

exists if conditions a) and b) in Theorem 2.1 are satisfied.

Now define the following matrices

r R A E T ( ~ E - A I F 1 - C A E T ( ~ E -Ay

-(rr - E A E ~ ) ( ~ E - A ) F - C A E ~ ( ~ E - A ) F

172 = [ M F ] E R"'" and

where U 2 is nonsingular matrix and rank S = n + q , since P,. = Ker[F] and rank Cil = T + 9 .

Then we have

S E - A

s~,,-q - R A E ~ A M o -CA A M

I

which gives

R e ( s ) 2 0 ( V S E C).

Remarks: 1) If rank [$:I = n the above results can be applied directly to

2) The reduced order p of the observer is given by: j i = n -

. system (1) without any transformation.

rank C.

We can now propose the following design procedure: 1) Choose an (n - q ) x n matrix R such that [ g ] is nonsingular,

2) The matrix A4 can be obtained from (22) and (23) this can be always done, since C is of full row rank.

M = [a]-' ["F']. 3) If the pair (I?, 12) is detectable, we can find 2 such that the

observer is asymptotically stable, then we can deduce T from (25).

4) The matrix F can be obtained from (22)

5 ) Matrices H and L can be obtained from (21) and (27).

IV. EXAMPLE The following example illustrates the above design methods. Example: Consider the singular system (1) described by

and

E'

A *

B'

C'

1 0 0 0 = l x j =[-a H 4 y] = [i]

0 0 1 0

0

= [0 1 0 01.'

In this case

and E' A*

0 c* rank [ 0 E * ] - r d E ' = 4 .

If we choose the transformation matrix P = 1 4 , we obtain the r.s.e singular system (4) described by

E = [ ; : x x ] .=[1 0 0 01

. I = [ 0 1 1 11

0 0 1 0

- 1 0 0 1

- 1 0 0 1

0 1 0 0 C = [ " 1 1 '1

EEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 7, JULY 1995 1321

and

y = [ - F t f ] , thenwehaverank =1. [:I The above result can then be applied to the transformed system.

a) Full-order observer: Matrices a, b . c, and d are

1 / = [ -1 %] 0

/ _ [ -1 ; 1 -1 ;] c = [0 11

and

and 0

I L = [ s’i -112

For II = -4, we can choose 2 = [0 -3 0 0 31, then T = [l -31

F = [ ! -1 1 -4 i] L = [ - l 1 -131

and H = O .

d = [O 0 -11. Thus the reduced-order observer is

Now, let the observer eigenvalues be specified by -4. -3, -2, and ii = -4w - 13y’ + U

-1, then we can find

N = diag (-4, -3, -2, -1)

and

G = O .

Thus the full-order observer is

I

b) Reduced-order observer: Choose I3 = [l 0 0 01, then

1

M = [-;I 0 112 -112

-1 -112 7/2 1 1 0 -2 21

r = - 1

V. CONCLUSION We have presented systematic design methods of full- and reduced-

order observers for linear descriptor systems. The reduced-order observer design method is based on the new resolution method of the constrained generalized Sylvester equation. It was shown that the existence conditions of the observer generalize those adopted in [12] for the observer design of square descriptors systems problem. An extension to a less restrictive conditions is under study.

m ACKh’OWLEDGMENT

The authors would like to thank the anonymous reviewers for their helpful and constructive comments.

REFERENCES

[l] M. El-Tohanu, V. Lovass-Nagy, and R. Mukundan, “On the design of observers for generalized state space systems using singular value decomposition,” Int. J. Contr., vol. 38, pp. 673-683, 1983.

[Z] M. Verhaegen and P. Vandooren, “A reduced observer for descriptor systems,” Syst. Contr. Lett., vol. 8, no. 5, pp. 29-37, 1986.

[3] B. Shafai and R. L. Caroll, “Design of minimal order observers for singular systems,” Int. J. Contr., vol. 45, no. 3, pp. 1075-1081, 1987.

[4] M. M. Fahmy and J. O’Reilly, “Observer for descriptor systems,” Int. J. Contr,, vol. 49, pp. 2013-2028. 1989.

[5] L. Dai, “Observer for discrete singular systems,” IEEE Trans. Automat. Contr.. vol. 33, pp. 187-191, 1988.

[6] P. N. Paraskevopoulos and F. N. Kounboulis, “Observers for singular systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 1211-1215, 1992.

[7] L. Dai, Singular Control Systems. [8] G. Das and T. K. Goshal, “Reduced-order observer construction by

generalized matrix,” Int. J. Contr., vol. 33, pp. 371-378, 1981. [9] M. Darouach, M. Zasadzinski, and S. J. Xu, “Full-order observers for

linear systems with unknown inputs,” IEEE Trans. Automat. Contr., vol. 39, pp. 606609, 1994.

[lo] P. C. Muller and M. Hou, “On the observer design for descriptor systems,” IEEE Trans. Automat. Contr., vol. 38, pp. 1666-16671, 1993.

[ l l ] M. Hou and P. C. Muller “A singular matrix pencil theoty for linear descriptor systems,” in Symp. on Implicit and Nonlinear Systems, Ft. Worth, TX, Dec. 14-15, 1992.

[12] D. N. Shields, “Observers for descriptor systems,” Inr. J . Contr., vol. 55, pp. 249-256, 1992.

Berlin: Springer, 1989.

Y