Robust controllability and robust closed-loop stability with static output feedback for a class of...

Linear Algebra and its Applications 297 (1999) 133–155www.elsevier.com/locate/laa

Robust controllability and robust closed-loopstability with static output feedback for a class of

uncertain descriptor systems

Chong Lina, Jian Liang Wangb,∗, Guang-Hong Yangb,C.B. Sohb

aDepartment of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong,People’s Republic of China

bSchool of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue,Block S2, Singapore 639798, Singapore

Received 9 October 1998; accepted 27 June 1999

Submitted by R.A. Brualdi

Abstract

This paper considers robust controllability for uncertain linear descriptor systems withstructured perturbations. Necessary and sufficient conditions based on theµ-analysis are ob-tained by transforming the problem into checking the nonsingularity of a class of uncertainmatrices. Also a tight bound is obtained in terms ofµ for keeping the closed-loop systemregular, impulse-free and stable under a preconstructed static output feedback. An example isgiven to illustrate the results. © 1999 Elsevier Science Inc. All rights reserved.

Keywords: Descriptor systems; Impulse behavior; Stability; Controllability; Structured perturbations

1. Introduction

Descriptor systems are different from normal systems, where not only exponentialmodes but also impulsive modes may be involved. Consider the following linearcontinuous-time descriptor system

∗Corresponding author. Tel.: +65-790-4846; fax: +65-792-0415.E-mail address:[email protected] (J.L. Wang)

0024-3795/99/$ - see front matter © 1999 Elsevier Science Inc. All rights reserved.PII: S 0 0 2 4 - 3 7 9 5 ( 9 9 ) 0 0 1 5 0 - 0

134 C. Lin et al. / Linear Algebra and its Applications 297 (1999) 133–155

Ex(t) = Ax(t)+ Bu(t), (1)

y(t) = Cx(t),whereE,A ∈ Rn×n, B ∈ Rn×m, rank E = r 6 n. If det(αE − A) 6= 0 forsomeα ∈ C, then it is calledregular, in which case the existence and uniquenessof the solution of the system will be guaranteed. If deg det(sE − A) = rank E,then it is calledimpulse-f ree. Otherwise, it will possess impulsive modes, whichare undesired in system control. If all the eigenvalues of det(sE − A) lie in theopen left-half complex plane, then it is termedstable. The problem of controllab-ility and/or observability has been well studied [1,4,5,19,21,23]. There are severalcontrollability concepts with different meanings. (See the above references for theirdefinitions.) It is known that we can construct a state feedbacku = Kx(t) such thatthe closed-loop system(E,A+ BK) is impulse-free provided it isimpulse control-lable (I-controllable) [19,23]. If it is R-controllable [21], then we can arbitrarilyassign the finite eigenvalues. The strong controllability (S-controllability) [19] isboth I-controllability and R-controllability. System (1) is calledcomplete control-lable (C-controllable) [4,21] if for anyt1 > 0,x(0) ∈ Rn andw ∈ Rn, there exists acontrol inputu(t) such thatx(t1) = w. That is, for any initial conditionx(0) ∈ Rn,there always exists a control input such that the state response starting fromx(0)at t = 0 arrives at any prescribed position inRn in any given time period. We seethat the C-controllability is a direct generalization of the controllability concept inthe normal case. The observability concepts are dual to those of the correspondingcontrollability ones.

However, the robust controllability (observability) problem is seldom touched foruncertain descriptor systems. In [13], the authors studied the controllability of system(E,A,B) with A being an interval matrix and presented a method for checking thevarious robust controllabilities. In recent years, some efforts have been devoted to thestability robustness of uncertain descriptor systems. Qiu et al. [15] and Byers et al.[3] considered the unstructured perturbed system(E,A+D) and derived proceduresfor calculating the stability radius. Lin et al. [12] dealt with a more general casewhen the unstructured uncertainty is of the formMDN , whereM andN are knownstructured matrices. However, for the case of real perturbations, only lower boundscan be obtained. Fang et al. [7] studied the problem for the structured perturbation|D|m < αH , where| · |m denotes modulus matrix andH is a constant nonnegativematrix, and derived an upper bound of the scaling factorα for the considered un-certain system to remain regular, impulse-free and stable. Lee et al. [11] gave themaximal perturbed interval ofα ∈ R for unidirectional perturbation caseD = αH .

In this paper, we consider the system of the form (1) withA, B andC beingsubjected to the following kind of structured perturbations

A = A0+q∑i=1

αiAi, (2)

C. Lin et al. / Linear Algebra and its Applications 297 (1999) 133–155 135

B = B0+q∑i=1

αiBi, (3)

C = C0+q∑i=1

αiCi, (4)

whereAi ∈ Rn×n, Bi ∈ Rn×m andCi ∈ Rl×n (i = 0,1, . . . , q), αi ∈ R, |αi | 6 α.Here it is assumed without loss of generality that the numbers ofAi ’s, Bi ’s andCi ’sare equal. (If not the case, we can easily make them equal by adding in some zeromatrices.) It is obvious that this perturbed form is more general than those consideredin [7,11]. The purpose of this paper is to find how bigα can be such that (1)–(4)retains the required property for all|αi | 6 α. We first consider in Section 2 robustI-controllability and C-controllability, and give necessary and sufficient conditionsby using the structured singular valueµ to change the problems to the correspondingfull matrix rank problems. We will useµD(M) to denote the structured singular valueof a matrixM with respect to the set of all allowableD. (See [2,22] for the mixedµ-analysis.) We see that by using our method, the matrixE can also be subjected to thesame kind of perturbations as that ofAwhen considering the robust C-controllabilityand robust regularity. As for the observability, similar results can be derived by usingthe dual principle. Then, in Section 3 we consider the regular, impulse-free and stableproperty for the uncertain closed-loop systems under a preconstructed static outputfeedback, and a tight bound is obtained in terms ofµ. In Section 4, an illustrativeexample is given. Finally, some concluding remarks are given in Section 5.

2. Robust controllability/observability

In this section, we study the robust controllability and/or observability of thesystem (1)–(4).

2.1. Some basic lemmas

We list and review some facts in this section.

Lemma 2.1.1[21]. The system(E,A) withE,A ∈ Rn×n is regular if and only if

rank

E A

E A

.. .. . .

. . . A

E A

n2×n(n+1)

= n2. (5)


Lemma 2.1.2[5]. The system(E,A,B) is I-controllable if and only if

rank[ASE E B] = n, (6)

whereSE ∈ Rn×(n−rankE) is a maximum right annihilator matrix of E.

Lemma 2.1.3[21]. The system(E,A,B) is R-controllable if and only if

rank[sE − A B] = n (7)

holds for any finite s.

Lemma 2.1.4. The system(E,A,B) is C-controllable if and only if any of thefollowing holds:

(i) [21]: (E,A,B) is R-controllable andrank[E B] = n.(ii) [5]:

rank

A B

E A B

.. .. . .

. . .

. . . A B

E B

n2×n(n+m−1)

= n2. (8)

2.2. Results

We first consider the robust I-controllability. Our objective is to find the maximumscalarα > 0 such that(E,A,B) remains I-controllable for all|αi | 6 α provided thenominal one(E,A0, B0) is I-controllable. Note that the chosenSE (the maximumright annihilator matrix ofE) may not be unique. But this obviously does not affectour results.

Lemma 2.2.1. System(1)–(3)is I-controllable if and only if the matrix

NI (α),N0 +q∑i=1

αiNi +q∑

i,j=1

αiαjNij (9)

is invertible, where

N0 =M0MT0 , Ni = MiM

T0 +M0M

Ti , Nij = MiM

Tj ,

M0 = [A0SE E B0], Mi = [AiSE 0n Bi ], i, j = 1, . . . , q. (10)


Proof. From Lemma 2.1.2, for someαi ’s, (1)–(3) is I-controllable iff(M0+∑q

i=1 αiMi) has full row rank, which is equivalent to the matrix(

M0+q∑i=1

αiMi

)(M0+

q∑i=1

αiMi

)T

=M0MT0 +

q∑i=1

αi(MiM

T0 +M0M

Ti

)+ q∑i,j=1

αiαjMiMTj

= NI (α) (11)

being invertible. �Suppose(E,A0, B0) is I-controllable which means rankM0 = n and thus

M0MT0 = N0 is invertible. Then determining the I-controllability for all|αi | 6 α

is equivalent to determining the robust invertibility of matrixNI for all |αi | 6 α.This can be done using the way provided in [18], which gives the maximum boundin terms ofµ for keeping nonsingularity of uncertain matrices with quadraticallycoupled parameters. However, it involves quite large dimensional computations (thedimension of the computed matrix is(q + 1)qn× (q + 1)qn). We will see from theproof of the next theorem that our method to solve this problem involves much smal-ler dimensional computations (the dimension of the computed matrix is 2qn×2qn).

Theorem 2.2.1. Suppose(E,A0, B0) is I-controllable. Then system(1)–(3) isI-controllable for all |αi | 6 α if and only if

α < µ−1D (MI ), (12)

or, equivalently,

α < µ−1D (MI ), (13)

whereMI andMI are constructed as follows:

MI =

N1N−10 −N11 N1N

−10 −N12 · · · N1N

−10 −N1q

N−10 0 N−1

0 0 · · · N−10 0

N2N−10 −N21 N2N

−10 −N22 · · · N2N

−10 −N2q

N−10 0 N−1

0 0 · · · N−10 0

......

......

. . ....

...

NqN−10 −Nq1 NqN

−10 −Nq2 · · · NqN−1

0 −NqqN−1

0 0 N−10 0 · · · N−1

0 0

∈ R2qn×2qn,

(14)


MI =

N−10 N1 N−1

0 N−10 N2 N−1

0 · · · N−10 Nq N−1

0

−N11 0 −N12 0 · · · −N1q 0

N−10 N1 N−1

0 N−10 N2 N−1

0 · · · N−10 Nq N−1

0

−N21 0 −N22 0 · · · −N2q 0

......

......

. . ....

...

N−10 N1 N−1

0 N−10 N2 N−1

0 · · · N−10 Nq N−1

0

−Nq1 0 −Nq2 0 · · · −Nqq 0

∈ R2qn×2qn

(15)

and

D = diag{α1I2n, α2I2n, . . . , αqI2n} ∈ R2qn×2qn. (16)

Proof. Since the nominal triple(E,A0, B0) is I-controllable,N0 is invertible. FromLemma 2.2.1, system (1)–(3) is I-controllable for all|αi | 6 α if and only ifdet(NI (α)) 6= 0, for all |αi | 6 α, whereNI (α) is given by (9). This is equivalent to

det

N0 +[α1Iα1

q∑i=1

αiNi1 · · ·αqI αqq∑i=1

αiNiq

]N1I...

NqI

6= 0

∀|αi | 6 α,

⇐⇒ det

I +N1I...

NqI

N−10

[α1Iα1

q∑i=1

αiNi1 · · ·αqIαqq∑i=1

αiNiq

] 6= 0

∀|αi | 6 α,where we use the fact

det(I +XY) = det

[I X

−Y I

]= det(I + YX).

It is easy to check that the 2qn× 2qnmatrix (I +H) is always invertible, whereH is of the following form:


H =

0 H11 0 H12 · · · 0 H1q0 0 0 0 · · · 0 00 H21 0 H22 · · · 0 H2q0 0 0 0 · · · 0 0...

......

.... . .

......

0 Hq1 0 Hq2 · · · 0 Hqq

0 0 0 0 · · · 0 0

∀Hij ∈ Rn×n, i, j = 1, . . . , q

and its inverse is(I −H).Hence the desired equivalency is continued to be

det

I +N1I...

NqI

N−10

[α1I, α1

q∑i=1

αiNi1, . . . , αqI, αq

q∑i=1

αiNiq

] 6= 0

∀|αi | 6 α,

⇐⇒ det

I +N1I...

NqI

N−10 [α1I 0 · · · αqI 0]

×

I +

0 α1N11 · · · 0 αqN1q0 0 · · · 0 0...

......

...

0 α1Nq1 · · · 0 αqNqq0 0 · · · 0 0

6= 0 ∀|αi | 6 α,

⇐⇒ det

I −

0 α1N11 · · · 0 αqN1q0 0 · · · 0 0...

......

...

0 α1Nq1 · · · 0 αqNqq0 0 · · · 0 0

+

N1I...

NqI

N−10 [α1I 0 · · · αqI 0]

6= 0 ∀|αi | 6 α,


⇐⇒ det

I −

0 N11 · · · 0 N1q0 0 · · · 0 0...

......

...

0 Nq1 · · · 0 Nqq0 0 · · · 0 0

D

+

N1I...

NqI

N−10 [I 0 · · · I 0]D

6= 0 ∀|αi | 6 α,

⇐⇒ det(I +MID) 6= 0 ∀|αi | 6 α,⇐⇒ α < µ−1

D (MI ),

whereD is as in (16). This proves (12). The proof of (13) is analogous to that of (12),and thus is skipped. This completes the proof of the theorem.�

Remark 2.2.1. Theorem 2.2.1 also implies thatµD(MI ) is the same asµD(MI ).Indeed, for the above result, we note thatMI = MT

I sinceNl = NTl andNij = NT

ji

(l = 0,1, . . . , q andi, j = 1, . . . , q), which obviously leads toµD(MI ) = µD(MI ).

Next, we move on to the robust C-controllability analysis. For system (1)–(3), weconstruct the following matrix:

MC =

P1P−10 −P11 P1P

−10 −P12 · · · P1P

−10 −P1q

P−10 0 P−1

0 0 · · · P−10 0

P2P−10 −P21 P2P

−10 −P22 · · · P2P

−10 −P2q

P−10 0 P−1

0 0 · · · P−10 0

......

......

......

PqP−10 −Pq1 PqP

−10 −Pq2 · · · PqP−1

0 −PqqP−1

0 0 P−10 0 · · · P−1

0 0

∈ R2qn2×2qn2

,

(17)

where

P0 = R0RT0 , Pi = RiRT

0 + R0RTi , Pij = RiRT

j ∈ Rn2×n2

,


R0 =

A0 B0E A0 B0

. . .. . .

. . .

. . . A0 B0E B0

∈ Rn2×n(n+m−1),

Ri =

Ai Bi0 Ai Bi

. . .. . .

. . .

. . . Ai Bi0 Bi

∈ Rn2×n(n+m−1),

i, j = 1,2, . . . , q. (18)

Then, we have the following result for the robust C-controllability.

Theorem 2.2.2. Suppose(E,A0, B0) is C-controllable. Then system(1)–(3) isC-controllable for all|αi | 6 α if and only if

α < µ−1DC(MC), (19)

whereDC = diag{α1I2n2, α2I2n2, . . . , αqI2n2} ∈ R2qn2×2qn2.

Proof. By using Lemma 2.1.4 (ii), system (1)–(3) is C-controllable for all|αi | 6 αif and only if

rank

(R0 +

q∑i=1

αiRi

)= n2 ∀|αi | 6 α,

⇐⇒ rank

(R0 +q∑i=1

αiRi

)(R0 +

q∑i=1

αiRi

)T = n2 ∀|αi | 6 α,

⇐⇒ rank

P0+q∑i=1

αiPi +q∑

i,j=1

αiαjPij

= n2 ∀|αi | 6 α.

The rest of the proof is similar to that of Theorem 2.2.1, and is hence omitted.�

Similar to I-controllability, the result in [18] can also be used to determine C-controllability, which gives the maximum bound in terms ofµ for keeping the nonsin-gularity of the uncertainty matrix with quadratically coupled parameters. But, again,this involves quite large dimensional matrix computations (of dimension(q + 1)qn2


×(q+1)qn2). Our result in Theorem 2.2.2 reduces this dimension to a much smallervalue of 2qn2× 2qn2.

Remark 2.2.2. Due to the relationships between each observability and its corres-ponding controllability, the results for robust I-observability and C-observability arestraightforward.

Remark 2.2.3. It is known that for a given descriptor system, the controllabilityconditions can ensure that some state feedback gainK renders the resulting closed-loop system regular. However, it is easy to check that in some cases the bound formaintaining the controllability cannot guarantee the robust regularity. Consider thesimple example(E,A+ αA1, B) with

E =[1 00 0

], A =

[0 11 0

], A1 =

[0 01 0

], B =

[01

].

We check that the nominal system is regular and C-controllable, and the uncertainsystem is C-controllable for allα ∈ R. However, the regularity is destroyed whenα = −1.

For system (1)–(3), if the robust regularity is required in order to keep the ex-istence and uniqueness of system solutions, then the result for robust regularity canalso be obtained in a similar way by using Lemma 2.1.1. We formulate the result forrobust regularity as follows. The proof is similar to that of Theorem 2.2.1 and henceomitted.

Proposition 2.2.1. Suppose(E,A0) is regular. Then system(1)–(3) is regular forall |αi | 6 α if and only if

α < µ−1Dreg(Mreg), (20)

where

Dreg=diag{α1I2n2, α2I2n2, . . . , αqI2n2} ∈ R2qn2×2qn2,

Mreg=

U1U−10 −U11 U1U

−10 −U12 · · · U1U

−10 −U1q

U−10 0 U−1

0 0 · · · U−10 0

U2U−10 −U21 U2U

−10 −U22 · · · U2U

−10 −U2q

U−10 0 U−1

0 0 · · · U−10 0

......

......

......

UqU−10 −Uq1 UqU

−10 −Uq2 · · · UqU−1

0 −UqqU−1

0 0 U−10 0 · · · U−1

0 0

∈ R2qn2×2qn2

(21)


with

U0 = V0VT0 , Ui = ViV T

0 + V0VTi , Uij = ViV T

j ∈ Rn2×n2

,

V0 =

E A0E A0

. . .. . .

. . . A0E A0

∈ Rn2×n(n+1),

Vi =0 Ai

. . .. . .

0 Ai

∈ Rn2×n(n+1)

f or i, j = 1,2, . . . , q. (22)

Remark 2.2.4. It is not hard to notice that our method can deal with the robustregularity and the robust C-controllability for descriptor systems with matrixE alsobeing of the perturbed form as those ofA, B andC. For this case, similar results canbe obtained by using Lemmas 2.1.1 and 2.1.4.

3. Robust closed-loop stability

If a given system is I-controllable and I-observable, then an output feedback lawcan be constructed to eliminate impulse modes. Moreover, if it is also R-controllableand R-observable, then its eigenvalues can be assigned arbitrarily by output feedback(see [5]). For system (1)–(4), its closed-loop system under a static output feedback

u(t) = Ky(t) (23)

is as follows

Ex(t) =A0+

q∑i=1

αiAi +q∑

i,j=1

αiαj Aij

x(t), (24)

where

A0 = A0+ B0KC0, Ai = Ai + BiKC0 + B0KCi, Aij = BiKCj ,i, j = 1, . . . , q. (25)


In this section, the problem under study is as follows. Suppose that a static out-put feedback matrixK renders the nominal closed-loop system(E,A0 + B0KC0)

regular, impulse-free and stable. Find the maximum scalarα such that the uncertainclosed-loop system (24) remains regular, impulse-free and stable for all|αi | 6 α.So, in the sequel, we will concentrate on the analysis of system (24).

3.1. Result by using a sweeping parameter

The following well-known lemma is useful for concerning impulse-free robust-ness.

Lemma 3.1.1. The pair(E,A) is impulse-free if and only if

rank(LEASE) = n− r, (26)

wherer = rankE, andLE ∈ R(n−r)×n andSE ∈ Rn×(n−r) are the maximum leftand right annihilator matrices of E, respectively, which satisfy

LEE = 0 and rankLE + rankE = n,ESE = 0 and rankSE + rankE = n.

It should be noted that the choice ofLE andSE may not be unique, but this willnot make any difference to our results.

From now on, for later convenience, we definef :Fn×n −→F2qn×2qn as

f (Z),f (Z0, Zi, Zij , i, j = 1, . . . , q),

,

Z1Z−10 −Z11 Z1Z

−10 −Z12 · · · Z1Z

−10 −Z1q

Z−10 0 Z−1

0 0 · · · Z−10 0

Z2Z−10 −Z21 Z2Z

−10 −Z22 · · · Z2Z

−10 −Z2q

Z−10 0 Z−1

0 0 · · · Z−10 0

......

......

......

ZqZ−10 −Zq1 ZqZ

−10 −Zq2 · · · ZqZ−1

0 −ZqqZ−1

0 0 Z−10 0 · · · Z−1

0 0

∈F2qn×2qn,

(27)

whereF is either realR or complexC, Z0 ∈ Fn×n is invertible andZi , Zij ∈Fn×n.

Theorem 3.1.1. Suppose that(E, A0) is impulse-free. Then(24) is impulse-free forall |αi | 6 α if and only if


α < µ−1DIF(f (F )), (28)

where

DIF = diag{α1I2(n−r), α2I2(n−r), . . . , αqI2(n−r)} ∈ R2q(n−r)×2q(n−r),r = rankE

and

F0 = LEA0SE, Fi = LEAiSE, Fij = LEAij SE ∈F(n−r)×(n−r),i, j = 1, . . . , q.

Proof. Since(E, A0) is impulse-free, we see from Lemma 3.1.1 thatF−10 is defined.

Then, by using Lemma 3.1.1 again, (24) is impulse-free for all|αi | 6 α if and onlyif

det

LEA0+

q∑i=1

αiAi +q∑

i,j=1

αiαj Aij

SE 6= 0

for all |αi | 6 α, or

det

F0+q∑i=1

αiFi +q∑

i,j=1

αiαjFij

6= 0

for all |αi | 6 α. The rest of the proof is similar to that of Theorem 2.2.1.�

For keeping robust impulse-free and stable property, the following lemma is re-quired. The proof is similar to Theorem 3.2 of [12], hence omitted here.

Lemma 3.1.2. Suppose that(E, A0) is impulse-free and stable, and(24) is impulse-free for all |αi | 6 α. Then(24) is impulse-free and stable for all|αi | 6 α if and onlyif

det

jωE − A0−q∑i=1

αiAi −q∑

i,j=1

αiαj Aij

6= 0, (29)

for all |αi | 6 α and for allω > 0.

Using the above Lemma 3.1.2, we give the following result without proof.

Theorem 3.1.2. Suppose that(E, A0) is impulse-free and stable. Then(24) isimpulse-free and stable for all|αi | 6 α if and only if

α < min{µ−1

DIF(f (F )), inf

ω>0µ−1

DS(f (G(ω)))

}, (30)


where

DS = diag{α1I2n, α2I2n, . . . , αqI2n} ∈ R2qn×2qn, r = rankE

and

G0(ω)=jωE − A0, Gi(ω) = −Ai, Gij (ω) = −Aij ∈Fn×n,i, j=1, . . . , q.

We notice that there is a sweeping parameterω in Theorem 3.1.2, which bringsmuch difficulty to the computation ofµ using the existingMATLAB tools. In thefollowing section, we develop another way to eliminate the sweeping parameter.

3.2. Result without the sweeping parameter

We first present a basic lemma for the development.

Lemma 3.2.1. Suppose that(E, A0) is impulse-free and stable. Then system(24) isregular, impulse-free and stable for all|αi | 6 α if and only if

rank

(E ⊗

(A0 +

q∑i=1

αiAi +q∑

i,j=1

αiαj Aij

+A0+

q∑i=1

αiAi +q∑

i,j=1

αiαj Aij

⊗ E = 2nr − r2 (31)

holds for all|αi | 6 α.

Proof. See the Appendix for its proof. �

To proceed, let the two nonsingular matricesT1 andT2 render

T1ET2 =[Ir 00 0

], T1AlT2 =

[A(1)l A

(2)l

A(3)l A

(4)l

], l = 0,1, . . . , q,

T1Aij T2 =[A(1)ij A

(2)ij

A(3)ij A

(4)ij

], i, j = 1, . . . , q (32)

and define

Hk = A(k)0 +q∑i=1

αiA(k)i +

q∑i,j=1

αiαjA(k)ij , k = 1,2,3,4, i, j = 1, . . . , q.

(33)


Let A(α) = A0 +∑q

i=1 αiAi +∑q

i,j=1 αiαj Aij . By appropriate exchanges ofmatrix rows and columns, we arrive at

rank(E ⊗ A(α)+ A(α)⊗ E)= rank((T1ET2)⊗ (T1A(α)T2)+ (T1A(α)T2)⊗ (T1ET2))

= rank

H1⊗ Ir + Ir ⊗H1 Ir ⊗H2 H2⊗ Ir 0

Ir ⊗H3 Ir ⊗H4 0 0H3⊗ Ir 0 H4⊗ Ir 0

0 0 0 0

. (34)

Note that the above equations hold for the ranks only. However, the correspondingmatrices may not be equal. Denote byH the nonzero matrix in the upper left block.Then,

H = H0+q∑i=1

αiHi +q∑

i,j=1

αiαj Hij ∈ R(2nr−r2)×(2nr−r2), (35)

whereH0, Hi andHij ∈ R(2nr−r2)×(2nr−r2) are as follows:

Hl=A

(1)l ⊗ Ir + Ir ⊗A(1)l Ir ⊗ A(2)l A

(2)l ⊗ Ir

Ir ⊗ A(3)l Ir ⊗ A(4)l 0A(3)l ⊗ Ir 0 A

(4)l ⊗ Ir

, l = 0,1, . . . , q,

Hij =A

(1)ij ⊗ Ir + Ir ⊗A(1)ij Ir ⊗ A(2)ij A

(2)ij ⊗ Ir

Ir ⊗ A(3)ij Ir ⊗ A(4)ij 0

A(3)ij ⊗ Ir 0 A

(4)ij ⊗ Ir

, i, j = 1, . . . , q.

(36)

Now, we are in a position to give our main result concerning the maximumα forkeeping the system (24) regular, impulse-free and stable.

Theorem 3.2.1. Suppose that(E, A0) is impulse-free and stable. Then(24) is reg-ular, impulse-free and stable for all|αi | 6 α if and only if

α < µ−1DS(f (H )), (37)

whereH0, Hi andHij are given by(36),and

DS = diag{α1I2(2nr−r2), . . . , αqI2(2nr−r2)} ∈ R2q(2nr−r2)×2q(2nr−r2).

Proof. The proof can be followed by Lemma 3.2.1 and the above analysis, using asimilar process to that of Theorem 2.2.1.�


Remark 3.2.1. Comparing Theorem 3.1.2 with Theorem 3.2.1, we see that the di-mension in Theorem 3.1.2 is lower but a sweeping parameter is involved, and thedimension in Theorem 3.2.1 is larger but no sweeping parameter is involved.

Remark 3.2.2. As a special case whenq = 1 (orαi = α1), let

M ,[−H1H

−10 I

−H11H−10 0

].

Then, H is invertible iff (I − α1M) is invertible, i.e., there are only a finitenumber ofα1 which render (24) not regular, impulse-free and stable, which arethe reciprocals of all non-zero real eigenvalues ofM. Furthermore, ifB0 andC0are not subject to perturbations and only the autonomous system of the system (1)–(4) is considered, then the maximum interval ofα for keeping the system regular,impulse-free and stable is(1/λ−min(−H1H

−10 ), 1/λ+max(−H1H

−10 )), whereλ−min(M)

andλ+max(M) denote the smallest negative real eigenvalue and the largest positivereal eigenvalue ofM, respectively. However, in Theorem 1 in [11], the maximuminterval ofα for keeping the system regular, impulse-free and stable is(

1/λ−min

(−H1H

−10

), 1/λ+max

(−H1H

−10

))⋂(1/λ−min

(−A(4)1

(A(4)0

)−1),

1/λ+max

(−A(4)1

(A(4)0

)−1))

.

So, our result simplifies this intersection of two intervals to the first interval only,hence providing a simpler result.

4. An illustrative example

Although approximating the structured singular values in our formulas using theMATLAB µ Toolbox [2] may result in conservative bounds, it does provide us a toolto get approximate values. In the following, we consider a numerical example toillustrate the use of the presented methods. Given a system of the form (1)–(4) with

E=1 0 0

0 1 00 −1 0

, A0 =−3 1 0−2 1 22 −1 −2

, A1 =0 1 0

0 0 01 0 0

,A2=

0 0 11 0 −10 0 1

, B0 =0

11

, B1 =1

00

, B2 =1

01

,C0=

[1 0 00 0 1

], C1 =

[0 0 01 0 1

], C2 =

[0 1 00 0 0

].


It is easy to see that the nominal system(E,A0) is not regular and not impulse-free. ChooseLE = [0 1 1] andSE = [0 0 1]T. We check that rankM0 =rank[A0SEE B0] = 3, which means(E,A0, B0) is I-controllable. Next we computethe allowableα using the formula in Theorem 2.2.1 for keeping robust I-controlla-bility.

We calculate that the upper and lower bounds ofµD(MI ) are 0.7101 and 0.6325,respectively, whereD = diag{α1I6, α2I6}. So, adopting the upper one, we get thatif α < µ−1

D (MI ) = 1.4083, then the given uncertain system is I-controllable for all|αi | 6 α (i = 1,2).

Now, we further check that the nominal system(E,A0, B0, C0) is also I-obser-vable, R-controllable and R-observable. Thus, there is an output feedback matrixK ∈ R1×2 such that(E,A0+B0KC0) is regular, impulse-free and stable. Specially,with K = [−3 − 2], the system(E,A0 + B0KC0) is impulse-free and has stableeigenvalues−1±j . Next we compute the allowableα using the formulas in Theorem3.1.2 and Theorem 3.2.1 for keeping regular, impulse-free and stable property of theuncertain closed-loop system under such aK.

We calculate that the upper and lower bounds ofµDIF (f (F )) are 1.0242 and1.0000, respectively, whereDIF = diag{α1I2, α2I2}. So, adopting the upper one,we have that ifα < µ−1

DIF(f (F )) = 0.9764, then the closed-loop uncertain system

under thisK is impulse-free for all|αi | 6 α (i = 1,2).Letting

T1 =1 0 0

0 1 0

0 1 1

andT2 = I3, which renders

T1ET2 =1 0 0

0 1 0

0 0 0

,we compute that the upper and lower bounds ofµDS (f (H )) are 6.2835 and 6.2729,respectively, whereDS = diag{α1I16, α2I16}. Then the closed-loop uncertain systemunder thisK is regular, impulse-free and stable for all|αi | 6 α (i = 1,2) if α <0.1591.

5. Conclusion

In this paper, we consider the robust controllability and closed-loop stability undera preset output feedback for linear continuous-time descriptor systems with struc-tured perturbations. Necessary and sufficient conditions are obtained usingµ ana-lysis and Kronecker product.


Appendix A.

Proof of Lemma 3.2.1.The necessity follows immediately from Lemma 1 of [15]with a little modification. For sufficiency, we prove by contradiction. Suppose thatsystem (24) is not regular, impulse-free and stable for all|αi | 6 α. Denote

A(α) ,

A = A0+q∑i=1

αiAi +q∑

i,j=1

αiαj Aij

∣∣∣ αi ∈ R, |αi | 6 α .

Then there are three cases:

Case 1. There existsA(α0) ∈ A(α) such that(E,A(α0)) is not regular. For thiscase, it is obvious thatA(α0) is not invertible. Let two nonsingular matricesT1 andT2 render

T1ET2 =[Ir 00 0

], T1A(α0)T2 =

[A1(α0) A2(α0)

A3(α0) A4(α0)

](A.1)

and

A1(α0)=a

(1)11 · · · a(1)1r...

...

a(1)r1 · · · a(1)rr

, A2(α0) =

a(2)11 · · · a(2)1,n−r...

...

a(2)r1 · · · a(2)r,n−1

,

A3(α0)=

a(3)11 · · · a

(3)1r

......

a(3)n−r,1 · · · a(3)n−r,r

, A4(α0) =

a(4)11 · · · a

(4)1,n−r

......

a(4)n−r,1 · · · a(4)n−r,n−r

.Due to the singularity ofA(α0), without loss of generality, we assume that the last

row of T1A(α0)T2 in (A.1) can be expressed as the linear combination of other rows,i.e., there are(n− 1) real scalarsγ1, . . . , γr , β1, . . . , βn−r−1 such that

−[a(3)n−r,1 · · · a(3)n−r,r a(4)n−r,1 · · · a(4)n−r,n−r

]= γ1

[a(1)11 · · · a(1)1r a

(2)11 · · · a(2)1,n−r

]+ · · · + γr

[a(1)r1 · · · a(1)rr a(2)r1 · · · a(2)r,n−r

]+ β1

[a(3)11 · · · a(3)1r a

(4)11 · · · a(4)1,n−r

]+ · · ·

+ βn−r−1

[a(3)n−r−1,1 · · · a(3)n−r−1,r a

(4)n−r−1,1 · · · a(4)n−r−1,n−r

].

By appropriate exchanges of matrix rows and columns, we have the followingrank equalities

rank(E ⊗ A(α0)+ A(α0)⊗ E)= rank((T1ET2)⊗ (T1A(α0)T2)+ (T1A(α0)T2)⊗ (T1ET2))


= rank

A1(α0)⊗ Ir + Ir ⊗ A1(α0) Ir ⊗ A2(α0) A2(α0)⊗ Ir 0

Ir ⊗A3(α0) Ir ⊗ A4(α0) 0 0A3(α0)⊗ Ir 0 A4(α0)⊗ Ir 0

0 0 0 0

= rank

A1(α0)+ a(1)11 Ir a(1)12 Ir · · · a

(1)1r Ir A2(α0) 0 · · · 0 a

(2)11 Ir · · · a(2)1,n−r Ir

a(1)21 Ir A1(α0)+ a(1)22 Ir · · · a

(1)2r Ir 0 A2(α0) · · · 0 a

(2)21 Ir · · · a

(2)2,n−r Ir

.

.

.

.

.

.. . .

.

.

.

.

.

.

.

.

.. . .

.

.

.

.

.

.. . .

.

.

.

a(1)r1 Ir a

(1)r2 Ir · · · A1(α0)+ a(1)rr Ir 0 0 · · · A2(α0) a

(2)r1 Ir · · · a

(2)r,n−r Ir

A3(α0) 0 · · · 0 A4(α0) 0 · · · 0 0 · · · 0

0 A3(α0) · · · 0 0 A4(α0) · · · 0 0 · · · 0

.

.

.

.

.

.. . .

.

.

.

.

.

.

.

.

.. . .

.

.

.

.

.

.. . .

.

.

.

0 0 · · · A3(α0) 0 0 · · · A4(α0) 0 · · · 0

a(3)11 Ir a

(3)12 Ir · · · a

(3)1r Ir 0 0 · · · 0 a

(4)11 Ir · · · a

(4)1,n−r Ir

.

.

.

.

.

.. . .

.

.

.

.

.

.

.

.

.. . .

.

.

.

.

.

.. . .

.

.

.

a(3)n−r,1Ir a

(3)n−r,2Ir · · · a

(3)n−r,r Ir 0 0 · · · 0 a

(4)n−r,1Ir · · · a(4)n−r,n−r Ir

.

Note that the above ranks are equal, but their matrices may not be equal. We seethat there arer+ r+ (n− r) = n+ r “block rows” in the last matrix. Multiply blockrow 1, . . . , r, 2r+1, . . . , n+ r −1 byγ1, . . . , γr , β1, . . . , βn−r−1, respectively, andadd them to the last block row; multiply block rowr+1, . . . ,2r−1 byγ1, . . . , γr−1,respectively, and add them to the block row 2r which is multiplied byγr . Thus, weobtain that the above rank is equal to the rank of matrix

A1(α0)+ a(1)11 Ir a(1)12 Ir · · · a

(1)1r Ir A2(α0) 0 · · · 0 a

(2)11 Ir · · · a

(2)1,n−r Ir

a(1)21 Ir A1(α0)+ a(1)22 Ir · · · a

(1)2r Ir 0 A2(α0) · · · 0 a

(2)21 Ir · · · a

(2)2,n−r Ir

.

.

.

.

.

.. . .

.

.

.

.

.

.

.

.

.. . .

.

.

.

.

.

.

.

.

.

a(1)r1 Ir a

(1)r2 Ir · · · A1(α0)+ a(1)rr Ir 0 0 · · · A2(α0) a

(2)r1 Ir · · · a

(2)r,n−r Ir

A3(α0) 0 · · · 0 A4(α0) 0 · · · 0 0 · · · 0

0 A3(α0) · · · 0 0 A4(α0) · · · 0 0 · · · 0

.

.

.

.

.

.. . .

.

.

.

.

.

.

.

.

.. . .

.

.

.

.

.

.

.

.

.

γ1A3(α0) γ2A3(α0) · · · γrA3(α0) γ1A4(α0) γ2A4(α0) · · · γrA4(α0) 0 · · · 0

a(3)11 Ir a

(3)12 Ir · · · a

(3)1r Ir 0 0 · · · 0 a

(4)11 Ir · · · a

(4)1,n−r Ir

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

γ1A1(α0) γ2A1(α0) · · · γrA1(α0) γ1A2(α0) γ2A2(α0) · · · γrA2(α0) 0 · · · 0

.


SinceT1A(α0)T2 in (A.1) is not invertible, it is easy to show that[γ1A3(α0) γ2A3(α0) · · · γrA3(α0) γ1A4(α0) γ2A4(α0) · · · γrA4(α0)

γ1A1(α0) γ2A1(α0) · · · γrA1(α0) γ1A2(α0) γ2A2(α0) · · · γrA2(α0)

]

is not of full row rank. Hence, rank(E ⊗A(α0)+A(α0)⊗E) < 2nr − r2. This is acontradiction.Case 2. System (24) is regular for allA ∈ A(α), but for someA(α0) ∈ A(α),

the pair(E,A(α0)) is not impulse-free. In this situation, let two nonsingular matricesT1 andT2 transform(E,A(α0)) to the following Weierstrass decomposition

T1ET2 =[I1 00 N

], T1A(α0)T2 =

[A1(α0) 0

0 I2

], (A.2)

whereN 6= 0 is nilpotent. Let

A1(α0)=a11 · · · a1r1...

...

ar11 · · · ar1r1

∈ Rr1×r1,

N=n11 · · · n1r2...

...

nr21 · · · nr2r2

∈ Rr2×r2. (A.3)

Then we haver1+ r2 = n, rankN = r − r1 > 0. So

rank(E ⊗ A(α0)+ A(α0)⊗ E)= rank((T1ET2)⊗ (T1A(α0)T2)+ (T1A(α0)T2)⊗ (T1ET2))

= rank

[A1(α0) 0

0 I2

]. . . [

A1(α0) 00 I2

]n11

[A1(α0) 0

0 I2

]· · · n1r2

[A1(α0) 0

0 I2

]...

. . ....

nr21

[A1(α0) 0

0 I2

]· · · nr2r2

[A1(α0) 0

0 I2

]


+

a11

[I1 00 N

]· · · a1r1

[I1 00 N

]...

.

.

.

ar11

[I1 00 N

]· · · ar1r1

[I1 00 N

][I1 00 N

]. . . [

I1 00 N

]

= rank

[I1 ⊗ A1(α0)+ A1(α0)⊗ I1 0 0 0

0 I1 ⊗ I2 + A1(α0)⊗ N 0 00 0 N ⊗A1(α0)+ I2 ⊗ I1 00 0 0 N ⊗ I2 + I2 ⊗N

]6 r2

1 + r1r2 + r1r2 + rank(N ⊗ I2 + I2⊗N).Next we show that rank(N ⊗ I2+ I2⊗N) < 2r2(r − r1)− (r − r1)2.

Indeed, forJ = diag{J1, . . . , Jt } ∈ Rn×n with rank J = ρ > 1 and eachJibeing a Jordan block, we have rank(J ⊗ In + In ⊗ J ) < 2nρ − ρ2. To prove this,without loss of generality, assume thatJ contains only one Jordan block (for the caseof more than one Jordan block, the proof is similar), i.e., rankJ = ρ = n − 1 > 1.Then

rank(J ⊗ In + In ⊗ J )=rank

J In. . .

. . .

. . . InJ

=n(n− 1)

<n2 − 1 (sincen > 2)

=2nρ − ρ2.

This completes the proof of the above fact.So far, we can get

rank(E ⊗ A(α0)+ A(α0)⊗ E)6r21 + r1r2 + r1r2 + rank(N ⊗ I2 + I2⊗N)

<r21 + 2r1r2+ 2r2(r − r1)− (r − r1)2

=2nr − r2.

Hence, this is also a contradiction.Case 3. System (24) is impulse-free (thus regular) for allA ∈A(α), but for some

A(α0) ∈A(α), the pair(E,A(α0)) is not stable. By assumption, (24) is impulse-freefor all |αi | 6 α. Hence the roots of


det

sE −A0+

q∑i=1

αiAi +q∑

i,j=1

αiαj Aij

are continous with respect toαi . Noting that(E, A0) is impulse-free and stable,there must exist someα∗i ’s satisfying|α∗i | 6 α, or sayA∗ = A0 +∑q

i=1 α∗i Ai +∑q

i,j=1 α∗i α∗j Aij , such that(E,A∗) has imaginary eigenvalues, say±λj . Now, let

two nonsingular matricesT1 andT2 render

T1ET2 =[Ir 00 0

], T1A∗T2 =

[Ar 00 In−r

](A.4)

with±λj being eigenvalues ofAr . Thus, 0 is an eigenvalue ofAr ⊕Ar = Ar ⊗ Ir +Ir ⊗ Ar , i.e.,

rank(Ar ⊕ Ar) < r2. (A.5)

It is not hard to check that

rank(E ⊗ A∗ + A∗ ⊗E)= rank((T1⊗ T1)(E ⊗ A∗ + A∗ ⊗E)(T2⊗ T2))

= rank((T1ET2)⊗ (T1A∗T2)+ (T1A∗T2)⊗ (T1ET2))

= rank

Ar ⊗ Ir + Ir ⊗ Ar 0 00 I2nr−2r2 00 0 0

< r2+ (2nr − 2r2) = 2nr − r2 (A.6)

which is a contradiction again.This completes the proof of the lemma.�

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