disturbance decoupling of linear time-varying singular systems

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 2, FEBRUARY 2002 335

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Disturbance Decoupling of Linear Time-Varying SingularSystems

Xiaoping Liu and Daniel W. C. Ho

Abstract—The problem of disturbance decoupling by state feedback isdefined for a linear time-varying singular system. It is required that theclosed-loop system has a unique impulse-free solution and its output is notaffected by disturbances. An algorithm, namely disturbance decoupling al-gorithm, is proposed. It is proved that the feasibility of the disturbance de-coupling algorithm is invariant under any regular feedback control law.Based on the disturbance algorithm, a constructive method is provided todesign a disturbance decoupling feedback control law. Sufficient conditionsfor the solvability of the disturbance decoupling problem are derived. It isproved that one of the sufficient conditions is also necessary provided thatother conditions are satisfied.

Index Terms—Disturbance decoupling, singular systems, state feedback,time-varying systems.

I. INTRODUCTION

Consider the following linear time-varying singular system:

_x1 = A11(t)x1 + A

12(t)x2 +B1(t)u+H

1(t)w

0 = A21(t)x1 + A

22(t)x2 +B2(t)u+H

2(t)w

y = C1(t)x1 + C

2(t)x2 +D(t)u+ L(t)w (1)

where t 2 [0; Tf ] 2 R; xi 2 Rn with i = 1; 2; u 2 Rm isthe vector of inputs,y 2 Rr is the vector of outputs,w 2 Rl isthe vector of disturbances,A11(t); A12(t); A21(t); A22(t); B1(t);B2(t);H1(t);H2(t);C1(t);C2(t);D(t) and L(t) are analyticmatrices of dimensionsn1 � n1; n1 � n2; n2 � n1; n2 � n2; n1 �

m;n2 � m;n1 � l; n2 � l; r � n1; r � n2; r � m, and r � l,respectively. The model (1) can be easily obtained from a generalmodelE(t) _x = A(t)x + B(t)u + H(t)w by using a nonsingular

change of coordinatesx = M(t)[x1

x2] with M being a nonsingular

matrix and a premultiplication with a nonsingular matrixN(t) underthe assumption rankE(t) = constant. The theory for linear time-in-variant singular systems has been well established from the analytical,geometric and numerical point of view, see, e.g., [4] and [5].

The objective of this note is to find a regular feedback

u = F1(t)x1 + F

2(t)x2 +G(t)v (2)

with G(t) nonsingular for anyt 2 [0; Tf ] andv the new input, suchthat the closed-loop system

_x1 = [A11(t) +B1(t)F 1(t)]x1 + [A12(t) +B

1(t)F 2(t)]x2

+B1(t)G(t)v+H

1(t)w

0 = [A21(t) +B2(t)F 1(t)]x1 + [A22(t) +B

2(t)F 2(t)]x2

+B2(t)G(t)v+H

2(t)w

y = [C1(t) +D(t)F 1(t)]x1 + [C2(t) +D(t)F 2(t)]x2

+D(t)G(t)v+ L(t)w (3)

Manuscript received January 24, 2000; revised February 20, 2001. Recom-mended by Associate Editor L. Dai. This work was supported by the NationalScience Foundation of China under Grant 69974007, the Trans-CenturyTraining Programme Foundation for the Talents by the State EducationCommission of China, and also supported by the Research Grants Councilof Hong Kong SAR under Grant CityU 1073/97E and by CityU under Grant7100210.

X. Liu is with the Department of Electrical Engineering, Lakehead University,Thunder Bay, ON P7B 5E1, Canada (e-mail: [email protected]).

D. W. C. Ho is with the Department of Mathematics, City University of HongKong, Hong Kong (e-mail: [email protected]).

Publisher Item Identifier S 0018-9286(02)02076-7.

0018–9286/02$17.00 © 2002 IEEE

336 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 2, FEBRUARY 2002

has the following properties: (1) it has a unique solution; (2) its solutionis not impulsive for anyx1(0) = x10; (3) its output is not affected bydisturbances. Such a feedback is called a disturbance decoupling feed-back. If there exists a disturbance decoupling feedback for a system,then the disturbance decoupling problem (DDP) is said to be solvablefor the system. It follows from [10] that the following assumption isnecessary for the closed-loop system to be free of impulse.

1) A): [A22(t) B2(t)] has full row rankn2 for anyt 2 [0; Tf ].

The disturbance decoupling problem is one of the major control de-sign problems since it aims to eliminate the effects of disturbances onoutputs by imposing an appropriate feedback. For linear time-invariantsingular systems, the disturbance decoupling problem was first formu-lated and solved by Fletcher and Aasaraai [6]. It is required that theoutput is independent of the input disturbance in the sense that there isa set of admissible initial conditions such that the system’s response iszero. However, it is not clear how a given initial state can be qualifiedas an admissible initial condition due to the lack of information on thedisturbance input. This limitation was overcome by Ailon [1] by pro-viding a simple criterion for the solvability of the DDP and developingan efficient algorithm for solving DDP. Another approach based ontransfer function was developed by [9] for constructing the disturbancedecoupling feedback controller, which makes the closed-loop systemstable. The same problem was also addressed by Chu and Mehrmann[3] by introducing a numerically stable procedure based on orthog-onal matrix transformations. Necessary and sufficient conditions werederived for the existence of a solution to the disturbance decouplingproblem with/without stability. Both proportional and derivative feed-backs were constructed so that the resulting closed-loop system is reg-ular, of index at most one, stable, and decoupled from the disturbances.For discrete-time descriptor systems, DDP has been discussed by Ba-naszuk et al. [2] and Lebret [8]. To the authors knowledge, no workhas been done for the DDP of time-varying singular systems. This isthe first note to give theoretical results in this area.

In this note, we consider the disturbance decoupling problem forlinear time-varying singular systems. First, the disturbance decouplingalgorithm is proposed for constructing a set of new coordinatesin which the system admits a simple form. Second, based on thedisturbance decoupling algorithm, a disturbance decoupling feedbackcontroller is designed. Finally, sufficient conditions A1)–A4) areprovided for the solvability of the problem, and it is shown that A4)is also necessary if A1)–A3) hold.

Notations for the Note:The notationsy = [y1

y2]; [C1(t) C2(t)] =

[C110 (t) C21

0 (t)

C120 (t) C22

0 (t)]; D(t) = [

D10(t)

D20(t)

]; L(t) = [L10(t)

L20(t)]; C1i

k+1(t) =

(d�ik=dt) + �i

k(t)A11(t); C2i

k+1(t) = �ik(t)A

12(t);Dik+1(t) =

�ik(t)B

1(t);Lik+1(t) = �i

k(t)H1(t) are used in Algorithm 1. Fori =

0; 1; 2; . . . k; �i(t);i(t); �ij(t) for j = 0; . . . ; i;�i(t); �i(t); �ij(t)for j = 0; . . . ; i are matrices to be defined in Algorithm 1 throughStep 0 tok. Also �i(t) = rank(i(t)) and�i(t) is rank of a matrixdefined in Stepi of the Algorithm 1.

In Lemma 1,XF! Y implies thatX becomesY after imposing the

feedback (2) on system (1). In Section III, the following notations areused:

C11

j (t) = C11

j (t)� C21

j (t) +D1

j (t)F2(t)

� [A22(t) +B2(t)F 2(t)]A21(t)

D1

j (t) = D1

j (t)� C21

j (t) +D1

j (t)F2(t)

� [A22(t) +B2(t)F 2(t)]�1B2(t)

L1j (t) = L1j (t)� C21

j (t) +D1

j (t)F2(t)

� [A22(t) +B2(t)F 2(t)]�1H2(t)

C21(t) =

C210 (t)

C211 (t)

...C21k (t)

D1(t) =

D10(t)

D11(t)...

D1k (t)

L1(t) =

L10(t)

L11(t)...

L1k (t)

D1(t) =

D10(t)

D11(t)...

D1k (t)

:

For some nonsingular matrcesP (t); Q(t)we also have the followingnotations and partition in Remark 4 of Section III,

�A22(t) = P (t)A22(t) =�A2211(t) �A22

12(t)�A2221(t) �A22

22(t)

�B2(t) = P (t)B2(t) =�B21(t)

�B22(t)

F 2(t) = F 2

1 (t) F 2

2 (t)

L1(t)Q(t) =L111(t) L112(t)

L121(t) L122(t)

C21(t) =C2111(t) C21

12(t)

C2121(t) C21

22(t)D1(t) =

D111(t)

D121(t)

:

II. DISTURBANCE DECOUPLINGALGORITHM

Based on the block decoupling algorithm developed in [7], this sec-tion will introduce the disturbance decoupling algorithm, which playsan important role in discussing the problem in question.

A. Algorithm 1 (Disturbance Decoupling Algorithm)

Step 0: Suppose[A22(t) B2(t) H2(t)

C2(t) D(t) L(t)] has constant rank, say�0,

in [0; Tf ]. Without loss of generality, assume that its first�0 rows arelinearly independent. Then,[C1(t) C2(t) D(t) L(t)] can be parti-tioned into

[C1(t) C2(t) D(t) L(t)]

=C110 (t) C21

0 (t) D10(t) L10(t)

C120 (t) C22

0 (t) D20(t) L20(t)

(4)

so that[A22(t) B2(t) H2(t)

C210 (t) D1

0(t) L10(t)

] has full row rank�0. Therefore, there

exist analytic matrices�0(t) and�00(t) such that

C22

0 (t) D2

0(t) L20(t) = �0(t) A22(t) B2(t) H2(t)

+�00(t) C21

0 (t) D1

0(t) L10(t) (5)

Now let0(t) = C120 (t)��0(t)A

21(t)��00(t)C110 (t). Suppose that

0(t) has constant rank, say�0, in [0; Tf ]. If �0 = 0, then terminatethe algorithm. Otherwise, without loss of generality, assume that its

first �0 rows are linearly independent. Let0(t) = [�0(t)

�0(t)] where

�0(t) is the first�0 rows of0(t). Then there exists a matrix�00(t)so that�0(t) = �00(t)�0(t). Setk = 0 and go to next step.

Step k + 1: Assume that C11j (t); C21

j (t);D1j (t); L

1j (t), and

�j(t); j = 0; 1; . . . ; k, have been defined through steps 0 tok.Now calculateC1

k+1(t) = (d�k)=(dt) + �k(t)A11(t);C2

k(t) =�k(t)A

12(t);Dk(t) = �k(t)B1(t), andLk(t) = �k(t)H

1(t).Suppose the matrix

A22(t) B2(t) H2(t)

C210 (t) D1

0(t) L10(t)...

......

C21k (t) D1

k(t) L1k(t)

C2k+1(t) Dk+1(t) Lk+1(t)


has constant rank, say�k+1, in [0; Tf ]. Without loss of generality, as-sume that its first�k+1 rows are linearly independent. Then,�k(t) can

be expressed as�k(t) = [�1k(t)

�2k(t)

] so that

A22(t) B2(t) H2(t)

C210 (t) D1

0(t) L10(t)...

......

C21k (t) D1

k(t) L1k(t)

C21k+1(t) D1

k+1(t) L1k+1(t)

has full-row rank�k+1. Thus there exist analytic matrices�k+1(t) and�k+1;j(t); j = 0; . . . ; k + 1, so that

C22

k+1(t) D2

k+1(t) L2

k+1(t)

= �k+1(t) A22(t) B

2(t) H2(t)

+

k+1

j=0

�k+1;j(t) C21

j (t) D1

j (t) L1

j (t) : (6)

Now set k+1(t) = C12k+1(t) � �k+1(t)A

21(t) �

k+1

j=0�k+1;j(t)C

11j (t). Suppose that the matrix

[�0(t)� �1(t)

�� k(t)

� k+1(t)]� has constant rank, say�k+1,

in [0; Tf ]. If �k+1 = �k, then terminate the algorithm. Otherwise,without loss of generality, assume that its first�k+1 rows are linearly

independent. Letk+1(t) = [�k+1(t)

�k+1(t)] where�k+1(t) is the first

�k+1 � �k rows of k+1(t). Then there exists a matrix�k+1;j(t)so that�k+1(t) = k+1

j=0�k+1;j(t)�j(t). Setk = k + 1 and go

to next step.Remark 1: The aim of Step 0 in the above algorithm is to decom-

posey into two parts, namelyy1 andy2, in such a way thaty2 canbe expressed by a function�0(t)x1 andy1, which is independent ofx2; u, andw. In fact, after carrying out the first step of the algorithm,we end up with

y1 = C

11

0 (t)x1 + C21

0 (t)x2 +D1

0(t)u+ L1

0(t)w

y2 =

�0(t)x1�00(t)�0(t)x1

+ �00(t)y1 (7)

It will be seen later thaty1 can be easily decoupled fromw by simplychoosing the feedback of the formC11

0 (t)x1+C210 (t)x2+D1

0(t)u+L10(t)w = v0. So, what we need to do next is to decoupley2 fromw,which is equivalent to decouple�0(t)x1 from w. This is done in thenext step.

Remark 2: Differentiating�k(t)x1 with respect to time gives

d

dt[�k(t)x1] = C

1

k+1(t)x1 + C2

k+1(t)x2

+Dk+1(t)u+ Lk+1(t)w: (8)

What Stepk + 1 does is to decompose�k+1 into �1k+1 and�2

k+1 sothat

d

dt�1

k(t)x1

= C11

k+1(t)x1 + C21

k+1(t)x2 +D1

k+1(t)u+ L1

k+1(t)w

d

dt�2

k(t)x1

=�k+1(t)x1

k+1

j=0�k+1;j(t)�j(t)x1

+ �k+1;0(t)y1 +

k

j=1

�k+1;j(t)d

dt�1

j�1(t)x1 (9)

Similar to Step 0,�1k(t)x1 can be decoupled fromw by choosing

C11k+1(t)x1 + C21

k+1(t)x2 + D1k+1(t)u + L1k+1(t)w = vk+1. Next

step is to handle�k+1(t) in the same way.Algorithm 1 is said to be feasible if the constant rank assumptions

are satisfied at every step of the algorithm. It follows from [7] thatany feasible algorithm will terminate after finite steps bounded byn1.Therefore, we have the following assumptions.

A2): Algorithm 1 is feasible and terminates atk�.Remark 3: The condition A2) implies that all constant assumptions

hold in the algorithm, which is not restrictive because they are satisfiedautomatically for the time-invariant case. In addition, it also impliesthat the matrix

A22(t) B2(t) H2(t)

C210 (t) D1

0(t) L10(t)...

......

C21k (t) D1

k (t) L1k (t)

has full row rank in[0; Tf ].The following lemma shows that the feasibility of the algorithm

above is invariant under the regular feedback of the form (2).Lemma 1: Suppose Assumption A1) is satisfied. Then, matrices and

integers produced in Algorithm 1 possess the following properties, fori = 0; 1; . . . ; k�:

1) A2i(t)F! A2i(t) + B2(t)F i(t) for i = 1; 2; B2(t)

F!

B2(t)G(t);H2(t)F! H2(t);

2) Cj1i (t)

F! C

j1i (t) + D1

i (t)Fj(t) for j = 1; 2; D1

i (t)F!

D1i (t)G(t); L1i (t)

F! L1i (t);

3) �i(t)F! �i(t); �i(t)

F! �i(t); �ij(t)

F! �ij(t)

for j = 0; . . . ; i;i(t)F! i(t); �i(t)

F! �i(t);

�i(t)F! �i(t); �i(t)

F! �i(t); �ij(t)

F! �ij(t) for

j = 0; 1; . . . ; i.Proof: See Appendix A.

For convenience in notations, let�kj(t) = [�1kj(t) �2kj(t)]with �1kj(t) being the first �k+1 columns of �kj(t). And let�ik = �i

k(t)x1; k = 0; . . . ; k� � 1; i = 1; 2. Then, it follows from (7)thaty takes the form of:

y1 = C

11

0 (t)x1 + C21

0 (t)x2 +D1

0(t)u+ L1

0(t)w

y2 =

�10

�20

�100(t)�10 + �200(t)�

20

+ �00(t)y1D0(t)u+ �L0(t)w]: (10)

By taking the time derivatives of�1k and�2k and considering (9), it fol-lows that:

_�1k = C11

k+1(t)x1 + C21

k+1(t)x2 +D1

k+1(t)u+ L1

k+1(t)w

_�2k =

�1k+1�2k+1

k+1

j=0�1k+1;j(t)�

1j + �2k+1;j(t)�

2j

+ �k+1;0(t)y1 +

k+1

j=1

�k+1;j(t) _�1

j�1 (11)

for k = 0; 1; . . . ; k� � 2. Similarly, it is easily deduced that

_�1k �1 = C11

k (t)x1 + C21

k (t)x2 +D1

k (t)u+ L1

k (t)w

_�2k �1 =

k �1

j=0

�1

k j(t)�1

j + �2

k j(t)�2

j

+ �k 0(t)y1 +

k

j=1

�k j(t) _�1

j�1 (12)


which follows from �k = �k �1, that is,�k (t) = k (t) =k �1j=0 �k j(t)�j(t).

III. D ESIGN OFDECOUPLINGFEEDBACK

This section is devoted to designing of the decoupling feedback. Firstof all, let us make the following assumption, which makes the feedback(18) regular due to (17):

A3): The matrix

A22(t) B2(t)

C210 (t) D1

0(t)...

...C21k (t) D1

k (t)

has full-row rank in[0; Tf ].The design of the decoupling feedback consists of two steps. First,

choose any matrixF 2(t) so thatA22(t)+B2(t)F 2(t) is invertible foranyt 2 [0; Tf ], which is always possible because of A1). By imposingthe feedback of the form

u = F2(t)x2 + u (13)

and solving the corresponding algebraic equation forx2, it follows that(10)–(12) become:

y1 = C

110 (t)x1 + D

10(t)u+ L

10(t)w

y2 =

�10

�20

�100(t)�10 + �200(t)�

20

+ �00(t)y1 (14)

_�1k = C11k+1(t)x1 + D

1k+1(t)u+ L

1k+1(t)w

_�2k =

�1k+1�2k+1

k+1j=0 �1k+1;j(t)�

1j + �2k+1;j(t)�

2j

+ �k+1;0(t)y1 +

k+1

j=1

�k+1;j(t) _�1j�1

k = 0; 1; . . . ; k� � 2 (15)_�1k �1 = C

11k (t)x1 + D

1k (t)u+ L

1k (t)w

_�2k �1 =

k �1

j=0

�1k j(t)�

1j + �

2k j(t)�

2j

+ �k ;0(t)y1 +

k

j=1

�k j(t) _�1j�1 (16)

whereC11j (t); D1

j (t); L1j(t); j = 0; 1; 2; . . . ; k� have already been de-

fined in Section I.The second step is to design a feedback foru to cancel the first terms

in the equations abouty1; _�1k, and _�1k �1. Now, it is not difficult to checkthe relation

A22(t) B2(t)

C21(t) D1(t)

I 0

F 2(t) I

�I �[A22(t) +B2(t)F 2(t)]�1B2(t)

0 I

=A22(t) +B2(t)F 2(t) 0

C21(t) +D1(t)F2(t) D1(t)(17)

is true for anyF 2(t) such that is invertible. Note that the second andthird matrices on the left-hand side of (17) are nonsingular. Accordingto A3), the first matrix on the left-hand side of (17) is of full row rank.As a result, the matrix on the right-hand side of (17) is also of full row

rank, which implies thatD1(t) is of full row rank. For simplicity andwithout loss of generality, assume that first�k columns ofD1(t) arenonsingular in[0; Tf ]. Then, the feedback law

C110 (t)x1 + D

10(t)u = v0

...

C11k (t)x1 + D

1k (t)u = vk

0(m�� )�� I(m�� )�(m�� ) u = ~v (18)

is a regular feedback, which renders (14)–(16) to take the form of

y1 = v0 + L

10(t)w

y2 =

�10

�20

�100(t)�10 + �200(t)�

20

+ �00(t)y1 (19)

_�1k = vk+1 + L1k+1(t)w

_�2k =

�1k+1�2k+1

k+1j=0 �1k+1;j(t)�

1j + �2k+1;j(t)�

2j

+ �k+1;0(t)y1 +

k+1

j=1

�k+1;j(t) _�1j�1

k = 0; 1; . . . ; k� � 2 (20)_�1k �1 = vk + L

1k (t)w

_�2k �1 =

k �1

j=0

�1k j(t)�

1j + �

2k j(t)�

2j

+ �k ;0(t)y1 +

k

j=1

�k j(t) _�1j�1: (21)

It is easily seen from (19)–(21) thatL1j (t) � 0 for j = 0; 1; . . . ; k�

guarantee the solvability of the disturbance decoupling problem. There-fore, the decoupling feedback can be easily determined from (13) and(18) provided that the following assumption is also satisfied

A4): There exists a matrixF 2(t) such thatA22(t) + B2(t)F 2(t)is invertible for anyt 2 [0; Tf ] andL1

j(t) = 0 for anyt 2 [0; Tf ] andfor any0 � j � k�.

Up to now, we have shown that Assumptions A1), A2), A3), andA4) are sufficient for the solvability of the DDP. Furthermore, it can beproved that A4) is also necessary provided that the first three assump-tions hold. Such results are summarized by the following theorem.

Theorem 1: Assume that A1), A2), A3), and A4) are satisfied. Then,the DDP is solvable, and the disturbance decoupling feedback is givenby (13) and (18). Moreover, under the conditions of A1), A2), and A3),the DDP is solvable only if A4) is true.

Proof: The first part has been proved before the theorem. Theproof for the second part is given in Appendix B.

Note that for linear time-invariant singular systems, the disturbancedecoupling algorithm is always feasible because all constant rank as-sumptions in every step are satisfied automatically. As a result, A2) isalways true. Therefore, the following corollary can be obtained fromTheorem 1.

Corollary 1: Consider a linear time-invariant singular system of theform (1) and suppose A1) is satisfied. Then, the disturbance decouplingproblem is solvable if and only if A4) holds.

Remark 4: Note that Assumption A4) plays an important role insolving the DDP, which results in a nonlinear algebraic equation of theform

L1(t) = [C21(t) +D

1(t)F2(t)]

� [A22(t) +B2(t)F 2(t)]�1H2(t): (22)


To the authors knowledge, there is no systematic method to solve (22).However, such an equation can be easily handled for the case ofH

2(t)with constant rank�. In this case, there exist nonsingular matricesP (t)

andQ(t) so thatP (t)H2(t)Q(t) = [H2

1 (t) 0

0 0]. Substituting this into

(22) gives

L1(t)Q(t) =

L1

11(t) L1

12(t)

L1

21(t) L1

22(t)

= [C21(t) +D1(t)F 2(t)]

� [ �A22(t) + �B2(t)F 2(t)]�1H2

1 (t) 0

0 0

with �A22(t) = P (t)A22(t) and �B2(t) = P (t)B2(t). So, we candeduce thatL1

12(t) � 0; L1

22(t) � 0

L1

11(t)

L1

21(t)= [C21(t) +D

1(t)F2(t)]

� [ �A22(t) + �B2(t)F 2(t)]�1H2

1 (t)

0: (23)

After appropriately partitioningC21(t);D1(t); �A22(t); �B2(t) andF 2(t), (23) can be expressed as

L1

11(t)

L1

21(t)=

C21

11(t) C21

12(t)

C21

21(t) C21

22(t)+

D1

11(t)

D1

21(t)F2

1 (t) F2

2 (t)

�

�A22

11(t) �A22

12(t)�A22

21(t) �A22

22(t)+

�B2

1(t)�B2

2(t)F2

1 (t) F2

2 (t)�1

�

H2

1 (t)

0: (24)

Now chooseF 2

2 (t) so thatA = �A22

22(t) + �B2

2(t)F2

2 (t) is invertible (ifnot, one may reorder the components ofx2 to achieve this objective).Then, by the inverse of partitioned matrix, it follows from (24) thatF 2

1 (t) satisfies the following linear equation:

P1(t) = P2(t)F2

1 (t)

where

P1(t) =L1

11(t)

L1

21(t)H

2

1 (t)�1 �A22

11(t)

��A22

12(t) + �B2

1(t)F2

2 (t) A�1 �A22

21(t)

�

C21

11(t)

C21

21(t)�

C21

12(t) +D1

11(t)F2

2 (t)

C21

22(t) +D1

21(t)F2

2 (t)

� A�1 �A22

21(t)

P2(t) = �L1

11(t)

L1

21(t)H

2

1 (t)�1 �B2

1(t)

��A22

12(t) + �B2

1(t)F2

2 (t) A�1 �B2

2(t)

+D1

11(t)

D1

21(t)�

C21

12(t) +D1

11(t)F2

2 (t)

C21

22(t) +D1

21(t)F2

2 (t)

� A�1 �B2

2(t)

IV. CONCLUSION

The disturbance decoupling problem has been considered for lineartime-varying singular systems. A new algorithm has been proposed, bywhich the system can be put into a simple form. Sufficient conditionsfor the solvability of the disturbance decoupling problem have beenderived. It has been proved that A4) is also necessary for the problemto have a solution provided that A1)–A3) are satisfied. A regular statefeedback has been designed so that the closed-loop system has a uniquesolution without impulses and is decoupled from disturbances.

Note that the design approach developed in this note is also new forapplying to the time-invariant case. Compared with the methods pro-posed in [1] and [9], the method here is easier because Algorithm 1 canbe easily implemented and A4) can be easily solved according to Re-mark 4. Also those existing methods in [1] and [9] cannot be extendedto time-varying case.

APPENDIX ATHE PROOF OFLEMMA 1

Item 1 follows from the application of (2) to the second equation of(1). The proof of both items 2 and 3 fori = 0 is given below. It followsfrom (3) and (4) that the equation at the bottom of the page holds true,which implies that item 2 is true fori = 0 and�0

F! �0. In addition,

it follows from (5) that:

C22

0 (t) D2

0(t) L2

0(t)

F! C

22

0 (t) +D2

0(t)F2(t) D

2

0(t)G(t) L2

0(t)

= �0(t)[A22(t) +B

2(t)F 2(t) B2(t)G(t) H

2(t)]

+ �00(t) C21

0 (t) +D1

0(t)F2(t) D

1

0(t)G(t) L1

0(t) :

(25)

As a result,�0(t)F! �0(t) and�00(t)

F! �00(t). By the definition of

0(t) and (5), a simple calculation shows that

0(t)F! C

12

0 (t) +D2

0(t)F1(t)

� �0(t)[A21(t) +B

2(t)F 1(t)]

� �00(t) C11

0 (t) +D1

0(t)F1(t) = 0(t)

C11

0 (t) C21

0 (t) D1

0(t) L1

0(t)

C12

0 (t) C22

0 (t) D2

0(t) L2

0(t)

F!

C11

0 (t) +D1

0(t)F1(t) C21

0 (t) +D1

0(t)F2(t) D1

0(t)G(t) L1

0(t)

C12

0 (t) +D2

0(t)F1(t) C22

0 (t) +D2

0(t)F2(t) D2

0(t)G(t) L2

0(t)

=C11

0 (t) C21

0 (t) D1

0(t) L1

0(t)

C12

0 (t) C22

0 (t) D2

0(t) L2

0(t)

I 0 0 0

0 I 0 0

F 1(t) F 2(t) G 0

0 0 0 I


which implies that�0F! �0;�0(t)

F! �0(t), and�0(t)

F! �0(t).

Therefore,�00(t)F! �00(t).

Now suppose both items 2 and 3 hold fori = 0; 1; . . . ; k. In whatfollows, we shall prove that they also hold fori = k + 1. By thedefinitions ofC11

k+1(t); C21k+1(t);D

1k+1(t) andL1k+1(t), taking (3) into

consideration, it follows from�k(t)F! �k(t) that item 2 is true for

i = k + 1, from which it is not difficult to see that�k+1F! �k+1. By

the relation

C22

k+1(t) D2

k+1(t) L2

k+1(t)

F! C

22

k+1(t) +D2

k+1(t)F2(t) D

2

k+1(t)G(t) L2

k+1(t)

= �k+1(t) A22(t) +B

2(t)F 2(t) B2(t)G(t) H

2(t)

+

k+1

j=0

�k+1;j(t)

� C21

j (t) +D1

j (t)F2(t) D

1

j (t)G(t) L1

j(t)

which is from (6), it is easily seen that�k+1(t)F! �k+1(t) and

�k+1;j(t)F! �k+1;j(t). By taking (6) into consideration, the

construction ofk+1(t) results in

k+1(t)F!

d�2k

dt+�2

k(t)[A11(t) +B

1(t)F 1(t)]

� �k+1(t)[A21(t) +B

2(t)F 1(t)]

�

k+1

j=0

�k+1;j(t) C11

j (t) +D1

j (t)F1(t)

=d�2

k

dt+�2

k(t)A11(t)� �k+1(t)A

21(t)

�

k+1

j=0

�k+1;j(t)C11

j (t)

+ D2

k+1(t)� �k+1B2(t)

�

k+1

j=0

�k+1;j(t)D2

j (t) F1(t) = k+1(t):

Therefore,�k+1F! �k+1;�k+1(t)

F! �k+1(t); �k+1(t)

F!

�k+1(t), and�k+1;j(t)F! �k+1;j(t). By induction, both items 1 and

2 are true.

APPENDIX BTHE PROOF OF THESECOND PART OF THEOREM 1

In order to prove A4), is necessary for the solvability of DDP. Letus suppose that there exists a disturbance decoupling feedback of theform (2), which implies that the response of the closed-loop system(3) is impulse-free and its output is unaffected byw. It follows fromLemma 1 that the algebraic equation in (1) admits the form of:

0 = [A21(t) +B2(t)F 1(t)]x1 + [A22(t) +B

2(t)F 2(t)]x2

+B2(t)G(t)v +H

2(t)w (26)

and (10)–(12) assume the form of

y1 = C

11

0 (t) +D1

0(t)F1(t) x1

+ C21

0 (t) +D1

0(t)F2(t) x2 +D

1

0(t)G(t)v+ L1

0(t)w

y2 =

�10

�20

�100(t)�10 + �200(t)�

20

+ �00(t)y1 (27)

_�1k = C11

k+1(t) +D1

k+1(t)F1(t) x1

+ C21

k+1(t) +D1

k+1(t)F2(t) x2

+D1

k+1(t)G(t)v+ L1

k+1(t)w

_�2k =

�1k+1�2k+1

k+1

j=0�1k+1;j(t)�

1j + �2k+1;j(t)�

2j

+ �k+1;0(t)y1 +

k+1

j=1

�k+1;j(t) _�1

j�1

k = 0; 1; . . . ; k� � 2 (28)_�1k �1 = C

11

k (t) +D1

k F1(t) x1

+ C21

k (t) +D1

k F2(t) x2

+D1

k (t)G(t)v+ L1

k (t)w

_�2k �1 =

k �1

j=0

�1

k j(t)�1

j + �2

k j(t)�2

j + �k ;0(t)y1

+

k

j=1

�k j(t) _�1

j�1: (29)

It follows from [10] that it is necessary to chooseF 2(t) such thatA22(t) + B2(t)F 2(t) is nonsingular in[0; Tf ] in order to make thesystem impulse-free. As a result,x2 can be uniquely determined from(26) as

x2 = �[A22(t) +B2(t)F 2(t)]�1f[A21(t) +B

2(t)F 1(t)]x1

+B2(t)G(t)v+H

2(t)wg

Substituting this into (27)–(29) produces

y1 = ~C11

0 (t)x1 + ~D1

0(t)v + L1

0(t)w

y2 =

�10

�20

�100(t)�10 + �200(t)�

20

+ �00(t)y1 (30)

_�1k = ~C11

k+1(t)x1 + ~D1

k+1(t)v + L1

k+1(t)w

_�2k =

�1k+1�2k+1

k+1

j=0�1k+1;j(t)�

1j + �2k+1;j(t)�

2j

+ �k+1;0(t)y1 +

k+1

j=1

�k+1;j(t) _�1

j�1

k = 0; 1; . . . ; k� � 2 (31)_�1k �1 = ~C11

k (t)x1 + ~D1

k (t)v + L1

k (t)w

_�2k �1 =

k �1

j=0

�1

k j(t)�1

j + �2

k j(t)�2

j + �k ;0(t)y1

+

k

j=1

�k j(t) _�1

j�1 (32)

where ~C11j (t) = C11

j (t) + D1j (t)F

1(t) and ~D1j (t) = D1

j (t)G(t).After the application of the feedback similar to (18), (30)–(32)

admit the form of (19)–(21). Sincey = [y1

y2] has been decoupled from

w; [y1

y2]; [

_y1

_y2]; [

�y1

�y2]; . . . are not affected byw. As a result, it follows

from (19) thatL10(t) � 0 and[�10

�20]; [

_�10_�20]; [

��10��20]; . . . are not affected by

w, which, according to (20) withk = 1, implies thatL11(t) � 0. Byinduction, it is not difficult to draw a conclusion thatL1k(t) � 0 foranyt 2 [0; Tf ] and for any0 � k � k�.


REFERENCES

[1] A. Ailon, “A solution to the disturbance decoupling problem in singularsystems via analogy with state-space systems,”Automatica, vol. 29, pp.1541–1545, 1993.

[2] A. Banaszuk, M. Kociecki, and K. M. Przyluski, “The disturbance de-coupling problem for implicit linear discrete-time systems,”SIAM J.Control Optim., vol. 28, pp. 1270–1293, 1990.

[3] D. L. Chu and V. Mehrmann, “Disturbance decoupling for descriptorsystems by state feedback,”SIAM J. Control Optim., vol. 38, pp.1830–1858, 2000.

[4] S. L. Campbell,Singular Systems of Differential Equations II. London,U.K.: Pitman, 1982.

[5] L. Dai, Singular Control Systems. New York, NY: Springer, 1989.[6] L. R. Fletcher and A. Aasaraai, “On disturbance decoupling in descriptor

systems,”SIAM J. Control Optim., vol. 27, pp. 1319–1332, 1989.[7] X. P. Liu, X. H. Wang, and D. W. C. Ho, “Input-output block decoupling

of linear time-varying singular systems,”IEEE Trans. Automat. Contr.,vol. 45, pp. 312–318, 2000.

[8] G. Lebret, “Structural solution of the disturbance decoupling problemfor implicit discrete-time systems,” inProc. 2nd Int. Symp. Implicit Ro-bust Syst., Warsaw, Poland, 1991, pp. 127–130.

[9] P. N. Paraskevopoulos, F. N. Koumboulis, and K. G. Tzierakis, “Distur-bance rejection of left-invertible generalized state space systems,”IEEETrans. Automat. Contr., vol. 39, pp. 185–190, 1994.

[10] C. J. Wang, “State feedback impulse elimination of linear time-varyingsingular systems,”Automatica, vol. 32, pp. 133–136, 1996.

Common Issues in Discrete Optimization andDiscrete-Event Simulation

Sheldon H. Jacobson and Enver Yücesan

Abstract—This note studies and exploits common issues betweendiscrete-event simulation models and algorithms for discrete optimizationproblems to prove that two discrete-event simulation search problemsare NP-hard. More specifically, NEIGHBORHOOD, seeks a sequenceof events such that two distinct states can be accessed, one state afterexecuting all but the last events and another state after executing all theevents, while INITIALIZE seeks a sequence of events such that executingthe sequence with one particular initial event results in a particular statebeing reached, while for a second initial event, that particular state cannotbe reached. Implications of these results for discrete-event simulationmodeling and analysis (e.g., assessing when steady state, terminationconditions have been reached, or optimal input parameters values forsimulation optimization have been established) as well as for discreteoptimization problems (e.g., assessinga priori the effectiveness of aneighborhood for simulated annealing or tabu search) are discussed.

Index Terms—Computational complexity, discrete-event simulation, dis-crete optimization, NP-hard.

I. INTRODUCTION

Discrete optimization and discrete-event simulation are two areas ofstudy in the field of operations research. Discrete optimization prob-lems are characterized by a countable set of solutions and an objective

Manuscript received September 28, 2001; revised October 12, 2001.Recommended by Associate Editor L. Dai. The work of the first author wassupported in part by the Air Force Office of Scientific Research under GrantF49620-01-1-0007, and in part by the National Science Foundation underGrant DMII-9907980.

S. H. Jacobson is with the Department of Mechanical and Industrial Engi-neering, University of Illinois, Urbana, IL 61801 USA.

E. Yücesan is with the INSEAD, Technology Management Area, 77305Fontainebleau Cedex, France.

Publisher Item Identifier S 0018-9286(02)02077-9.

function value associated with each solution with the goal of finding thesolution for which the objective function is minimized or maximized.Discrete event simulation is a powerful modeling tool for studyingcomplex systems (such as manufacturing and service networks), typ-ically referred to asdiscrete event dynamic systems, which cannot bestudied analytically.

The interaction between the two areas has been limited. Path searchtechniques have been applied for crude simulation optimization [6].Monte Carlo simulation (or other randomization techniques) has beenused to address difficult discrete optimization problems. However, thetwo areas contain many similarities. For example, both involve discreteobjects (solutions for discrete optimization problems, and events orevent sequences for discrete-event simulation); both employ solutionprocedures that process these discrete objects (algorithms for discreteoptimization problems, and model implementations for discrete-eventsimulation); both seek to find optimal values for output measures (op-timal solutions for discrete optimization problems, and minimum vari-ance output estimators for discrete-event simulation); both can be eitherstochastic or deterministic (algorithms for discrete optimization prob-lems, and the presence or absence of random variates for discrete-eventsimulation). Exploiting such similarities, [7] uses a computational com-plexity approach to assess the difficulty of simulation modeling andanalysis problems. In particular, they formulate four search problemsassociated with validation and verification of discrete-event simula-tion models and prove them to be NP-hard [3]. The implications ofthese four problems explain, for example, why automated model vali-dation and verification tools have not been forthcoming [19]. [20] alsoshows how algorithms, including variations of simulated annealing, canbe used to address one of these problems. They provide results thatbridge discrete-event simulation and discrete optimization by modelingdiscrete-event simulation modeling problems as discrete optimizationproblems, showing their difficulty within the computational complexityframework, and applying algorithms for these problems.

There are also a number of key differences between the twoareas. Most notably, discrete optimization problems are typicallystatic, while discrete-event simulation models can be either static ordynamic. Nonetheless, with an appropriate formulation and algorithm,a discrete optimization problem can be depicted as a discrete-eventsimulation model. This is a particular illustration of a general resultthat establishes the equivalence between Turing machines and eventgraphs [21]. The key implication of this equivalence is thatanythingTuring computable is computable on an event graph. In particular, withan appropriate formulation, any discrete optimization algorithm canbe depicted as a discrete-event model specification. This observationis further discussed in Section III.

This note exploits these observations by modeling and analyzingcertain discrete optimization algorithms as discrete event simulationmodels with the objective of gaining insight into how these algorithmsexecute on discrete optimization problems. From a theoretical perspec-tive, the mapping between discrete event simulation models and dis-crete optimization algorithms is a particular instance of the more gen-eral result on computability, as noted in [21], [22], and [23]. Morespecifically, we formulate two search problems for discrete-event sim-ulation models and prove them to be NP-hard. These intractability re-sults have implications on issues concerning neighborhoods (e.g., de-termining whether a neighborhood rule will be effective), tabu lists(e.g., how the tabu list should be designed), and initial conditions (e.g.,how to initialize simulated annealing so as to reach an optimal solu-tion with the fewest number of iterations) for simulated annealing andtabu search. Note that although discrete event simulation models typ-ically have a time component, this factor is neither needed nor used

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disturbance decoupling of linear time-varying singular systems

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