Index Terms—Singular systems, Lyapunov equation

On the Lyapunov Theorem for Singular Systems

Jo˜ao Yoshiyuki Ishihara and Marco Henrique Terra

Abstract—In this paper we revisit the Lyapunov theory for singular sys- tems. There are basically two well known generalized Lyapunov equations used to characterize stability for singular systems. We start with the Lya- punov theorem of [6], [7]. We show that the Lyapunov equation of that theorem can lead to incorrect conclusion about stability. Some cases where that equation can be used are clari£ed. We also show that an attempt to correct that theorem with a generalized Lyapunov equation similar to the original one leads naturally to the generalized equation of [14].

Index Terms—Singular systems, Lyapunov equation.

I. INTRODUCTION

Singular systems of the form

Ex˙ = Ax (1)

y = Cx

have been of interest in the literature since they have many important applications in, for example, circuit systems [10], robotics [9], and air- craft modeling [12]. Many classical concepts and results in the usual state space theory as stability, controllability and observability have been extended to these systems. In particular, for the characteriza- tion of stability for singular systems, generalizations of the Lyapunov theorem were proposed by Lewis [6], [7] (see Theorem 1 below) and by Takaba et al. [14] (see Corollary 4 below). One can see that the Lyapunov equations of these two theorems are quite different and the connection between them is not evident.

In this paper, we £rst point out that Theorem 1 is, as stated, incor- rect. The Lyapunov equation of Theorem 1 can be used to characterize stability only with some additional restrictions on the plant(1). For a plant with the same assumptions of Theorem 1 (or even weaker as presented in Theorem 4 below), we present a corrected version of the generalized Lyapunov equation which is similar to the original one.

Then we show that this Lyapunov equation is equivalent to the Lya- punov equation of [14].

First we state here some basic de£nitions which will be used in the next section. System(1)with an×nmatrixEis called regular if det(sE−A) 6= 0for somes ∈C. We say that the regular system (1)is ([16], [3], [4], [17])

(i) stable if all roots ofdet(sE−A) = 0are in the open left half plane;

(ii) impulse-free if it exhibits no impulsive behavior;

(iii) £nite dynamics detectable if(O1)holds;

(iv) £nite dynamics observable if(O2)holds;

(v) impulse observable if(O3)holds;

(vi) S-observable (following Lewis [6] and [7], we shall say observ- able) if(O2)and(O3)hold;

(vii) C-observable if(O2)and(O4)hold where(O1)−(O4)conditions are given by:

(O1) rank

· sE−A C

¸

=n,Re(s)≥0;

(O2) rank

· sE−A C

¸

=n,for alls∈C;

The authors are with the Electrical Engineering Department - EESC - Univer- sity of S˜ao Paulo at S˜ao Carlos, Brazil. (e-mail: ishihara@sel.eesc.sc.usp.br.;

terra@sel.eesc.sc.usp.br).

This work was supported by FAPESP (S˜ao Paulo State University Council) under grant 98/12113-2.

(O3) rank



 E A

0 C

0 E



=n+rankE;

(O4) rank

· E C

¸

=n.

It is immediately seen thatC-observability impliesS-observability.

II. THE GENERALIZEDLYAPUNOVTHEOREM

Consider the usual Lyapunov function candidate for state space sys- tems

V(x) =x^TP x≥0; V˙ (x) =−x^TC^TCx.

One can conclude that for singular systems, it is ‘natural’ to consider V(Ex),V˙ (Ex)as a Lyapunov function candidate. This choice leads to the following well known generalized Lyapunov theorem [6], [7].

Theorem 1. Let(E, A) be regular and (E, A, C) be observable.

Then(E, A)is stable and impulse-free if and only if there exists a positive de£nite solutionPto the following Lyapunov equation

A^TP E+E^TP A+E^TC^TCE= 0. (2) Unfortunately, Theorem 1 is, as stated, incorrect. To see this, con- sider the system with the following values

E=





1 0 0 0 0 0 0 1 0



, A=





0 0 0

0 0 1

−1 −1 0



, C=£

2 1 2 ¤ .

In this case, system(1)is regular, impulse-free and observable (in fact, the system isC-observable). It can be veri£ed that the Lyapunov equation (2) has a solution

P=





8 0 2 0 1 0 2 0 2



>0

but the system is not stable since it has a £nite mode ats= 0 :

det(sE−A) =det





s 0 0

0 0 −1

1 s+ 1 0



=s(s+ 1).

A careful analysis shows that the proof of [6] states, in fact, the following result.

Theorem 2. LetE, A ∈ R^n×n, andC ∈ R^p×n be given by (a Weierstrass form of some regular system

³E,e A,e Ce´ ):

E:=

· I_q 0

0 Λ

¸ , A:=

· J 0

0 In−q

¸ , C:=£

CF C∞

¤

whereIqdenotes anq×qidentity matrix,Jcorresponds to the £nite zeros ofsE−A,Λis nilpotent (Λ^k = 0,Λ^k−1 6= 0for some inte- gerk >0), and(E, A, C)is observable. Then(E, A)is stable and impulse-free if and only if there exists a positive de£nite solutionPto the following Lyapunov equation

A^TP E+E^TP A+E^TC^TCE= 0.

Moreover, ifP andP^′ are two such solutions, thenE^TP^′E = E^TP E.

The theorem above says that if(E, A, C)is already in Weierstrass form then the solution of Lyapunov equation (2) characterizes stability.

Our counterexample shows that if(E, A, C)is not in Weierstrass form then existence of a positive de£nite solution for (2) does not guarantee system stability. The reason for this discrepancy is given in the follow- ing. LetE, A∈R^n×n, andC∈R^p×nbe given with(E, A)regular.

Then the Weierstrass form can be obtained:

M EN =

· I_q 0

0 Λ

¸

, M AN=

· J 0

0 In−q

¸ , CN =£

C1 C2 ¤

whereMandNare nonsingular matrices. In this case, the Lyapunov equation (2) can be written as

N^TA^TM^TM^−TP M⁻¹M EN+N^TE^TM^TM^−TP M⁻¹M AN+ N^TE^TM^TM^−TC^TCM⁻¹M EN= 0.

Now it is easy to see that in the proof of [6] it was considered the following system

M EN=

· I_q 0

0 Λ

¸

, M AN=

· J 0

0 In−q

¸ , CM⁻¹=£

CF C∞

¤.

The counterexample shows a case where the results for CN and CM⁻¹are not the same. It shows also that even with the stronger as- sumption ofC-observability, Theorem 1 is incorrect. Although The- orem 1 is incorrect in general, in the next theorem we show that the stability test with the proposed equation (2) is a ‘natural’ and correct extension of the usual Lyapunov test for state space systems with the equation

A^TP+P A+Q= 0, Q >0.

Theorem 3. Let(E, A)be regular and consider the following gener- alized Lyapunov equation (GLE)

A^TP E+E^TP A+E^TQE= 0. (3) We have that

(a) if there exist matricesP ≥0andQ >0satisfying the GLE (3) then(E, A)is impulse-free and stable;

(b) if(E, A)is impulse-free and stable then for eachQ >0there existsP >0solution of GLE (3). FurthermoreE^TP E≥0is unique for eachQ >0.

Proof: From the regularity assumption of the system (1), we can put it in the Weierstrass form £nding nonsingular matricesM andN such that

M EN =

· Iq 0

0 Λ

¸

, M AN=

· J 0

0 In−q

¸

whereΛis nilpotent and the eigenvalues ofJ are £nite eigenvalues of(E, A). Now make a partition ofM^−TP M⁻¹andM^−TQM⁻¹ accordingly:

M^−TP M⁻¹=

· P₁₁ P₁₂ P₁₂^T P₂₂

¸

, M^−TQM⁻¹=

· Q₁₁ Q₁₂ Q^T₁₂ Q₂₂

¸ . The Lyapunov equation (3) can be written as

· J^TP11+P11J+Q11 J^TP12Λ +P12+Q12Λ P₁₂^T + Λ^TP₁₂^TJ+ Λ^TQ^T₁₂ P22Λ + Λ^TP22+ Λ^TQ22Λ

¸

= 0.

(4)

(a)AsΛis nilpotent we can de£neν:=min{k >0 : Λ^k = 0}.

Now suppose thatν >1. Pre-multiplying the(2,2)block equation of (4) by¡

Λ^T¢ν−1

we have that

³ Λ^T´ν−1

P₂₂Λ = 0⇒³ Λ^T´ν−1

P₂₂Λ^ν−1 = 0⇒P₂₂Λ^ν−1 = 0.

With this, pre-multiplying the(2,2)block equation of (4) by¡ Λ^T¢ν−2

and post-multiplying byΛ^ν−2we have

³Λ^T´ν−1

Q22Λ^ν−1= 0⇒Q22Λ^ν−1= 0⇒Λ^ν−1= 0.

This contradicts the minimality ofν. Then, we must haveΛ = 0 (ν= 1)and the system is impulse-free. On the other hand, considering thatQ11 > 0andP11 ≥0, from(1,1)block of (4) it follows that P₁₁ > 0. In fact, if there existsx 6= 0such thatP₁₁x = 0, from (1,1)block we getx^TQ11x= 0, x6= 0. This contradicts the fact that Q₁₁>0. Now, asP₁₁>0is a solution ofJ^TP₁₁+P11J+Q11= 0, from usual state space Lyapunov theory we have that(E, A)is stable.

(b)Suppose that the regular system(E, A)is impulse-free and stable.

Then in the above Weierstrass form we haveΛ = 0andJstable. For eachQ >0, we can £nd a uniqueP11>0solution of

J^TP11+P11J+Q11= 0.

Now it is easy to verify that P =M^T

· P11 0 0 P22

¸ M

is a solution for the GLE for anyP22. Note that we obtainP > 0 (≥0) choosingP₂₂>0(≥0). Also,

E^TP E=N^T

· P11 0

0 0

¸ N is unique sinceP₁₁is unique. ¤

As for usual state-space case (see e.g. [11], Theorem 5.36, p.

211), it is easy to show that GLE (3) has a solutionP ≥0for some Q > 0if and only if GLE (3) is solvable for allQ > 0. Note that the assumptions in Theorem 3 are quite strong since the condition C^TC=Q >0requires an output matrixCof full column rank. We can relax a bit the rank condition onCif we already know that the system is impulse-free, as stated in the next corollary.

Corollary 1. Assume that(E, A)is regular and impulse-free and that Cis such thatrank(CE) =rankE. We have that

(a) if there exists a matrixP ≥0satisfying the GLE (2) then(E, A) is stable;

(b) if(E, A)is stable then there existsP >0solution of GLE (2).

FurthermoreE^TP E≥0is unique for eachC.

Proof: De£neQ=C^TCand consider the Lyapunov equation in Weierstrass coordinates(4). By assumption, we already haveQ₁₁>0 andΛ = 0. Then the result follows by a slight modi£cation of the proof of Theorem 3. ¤

In Theorem 3 and Corollary 1, we have considered a relationship between the solution of GLEs (2) and (3) and some system properties like regularity, impulsiveness and stability. The regularity assumption of Theorem 3 and Corollary 1 is essential for the stability analysis with GLE (2) and (3): we cannot conclude the regularity of system (1) from the solution of GLE (2) or (3). Take as example the trivial system(E, A) = (0,0). In this case, we can always £nd a positive de£nite solutionP >0,Q >0to (3) but the system is not regular. An analysis relating the solution of GLE (3) and system stability without special consideration on the presence of impulses is made in [13]. In

this paper we consider only stability without impulsive behavior. In general, for practical applications, impulses are undesirable since they may cause degradation in performance, damage components, or even destroy the system.

Based on above considerations, we are now interested in obtaining a Lyapunov equation similar to (2) which can provide stability infor- mation under mild conditions (without regularity assumption, for ex- ample). For this, consider the following Lyapunov function candidate

V(x) =x^TE^TP Ex≥0, (5) whereP >0(we are interested inE^TP E≥0). Looking at its time derivative

V˙(x) = ˙x^TE^TP Ex+x^TE^TP Ex˙

we note that, for any matricesQandE0 of compatible dimensions satisfyingE^TE0= 0, we have

V˙ (x) = ˙x^TE^T(P E+E0Q)x+x^T³

E^TP+Q^TE^T₀´ Ex˙ or

V˙(x) =x^T³

A^T(P E+E0Q) + (P E+E0Q)^TA´ x.

Therefore, if the Lyapunov equation

A^T(P E+E0Q) + (P E+E0Q)^TA+C^TC= 0 has a solution(P, Q)withP >0(or, at least, withE^TP E≥0), we obtain (as in the usual state space case)

V˙(x) =−x^TC^TCx≤0.

In the next theorem we present a corrected version of the generalized Lyapunov theorem with a Lyapunov equation similar to (2).

Theorem 4. LetE, A ∈ R^n×nandC ∈ R^p×nbe such that(O1) and (O3) ((O2)and (O3)) are satis£ed. Consider also a matrix E0 ∈ R^n×(n−r) of full column rank such thatE^TE0 = 0, where r=rankE. The following statements are equivalent

(i) the system(E, A)is regular, impulse-free and stable;

(ii) there exists a solution(P, Q)∈R^n×n×R^(n−r)×nwith P≥0 (>0)to the following GLE:

A^TP E+E^TP A+C^TC+A^TE0Q+Q^TE^T₀A= 0; (6) (iii) there exists a solution (P, Q) ∈ R^n×n ×R^(n−r)×n with

E^TP E ≥ 0 (E^TP E ≥ 0andrank¡ E^TP E¢

= rankE) to GLE(6).

Proof: Consider a SVD coordinate system [1]

S^TET =

· I_r 0 0 0

¸

, S^TAT =

· A₁₁ A₁₂ A21 A22

¸ , CT=£

C₁ C₂ ¤ whereS =£

S₁ S₂ ¤

is orthogonal andT is nonsingular. In this case,(E, A)is regular and impulse-free if and only ifA22is nonsingu- lar [1]. Furthermore, ifA₂₂is nonsingular,(E, A)is stable if and only if(^A11−A₁₂A⁻¹₂₂A₂₁)is stable. AsE0 andS2 are bases ofKerE^T, there exists a nonsingular matrixW such thatE0 = S2W. De£ne accordingly

W QT =£

Q1 Q2 ¤

, and S^TP S=

· P11 P12

P₂₁ P₂₂

¸ . The Lyapunov equation(6)can be rewritten as

T^TA^TSS^T(P E+E0Q)T+T^T(P E+E0Q)^TSS^TAT+ +T^TC^TCT = 0

· A^T₁₁ A^T₂₁ A^T₁₂ A^T₂₂

¸ · P11 0 P₂₁+Q₁ Q₂

¸ + +

· P₁₁^T P₂₁^T+Q^T₁ 0 Q^T₂

¸ · A₁₁ A₁₂ A21 A22

¸ + +

· C₁^T C₂^T

¸£

C₁ C₂ ¤

= 0

· A^T₁₁P₁₁+P₁₁^TA₁₁+A^T₂₁(P₂₁+Q₁) + (P₂₁+Q₁)^TA₂₁+C₁^TC₁ A^T

12P₁₁+A^T

22(P₂₁+Q₁) +Q^T

2A₂₁+C^T

2C₁

∗

A^T₂₂Q₂+Q^T₂A₂₂+C₂^TC₂

¸

= 0 (7)

(iii) ⇒ (i)We £rst show that the system is regular and impulse- free. Indeed, considerv ∈ KerA₂₂. From the(2,2)block of(7) we havev^TC₂^TC2v= 0, which impliesC2v = 0. As(E, A, C)is impulse observable, that is,

· A₂₂ C2

¸

has full column rank, it follows thatv= 0and therefore,KerA22={0}, that is,A22is nonsingular.

(stability) From(2,1)and(2,2)blocks of(7)we have P21+Q1=−A⁻₂₂^T³

A^T₁₂P11+Q^T₂A21+C₂^TC1

´

Q2A⁻₂₂¹+A⁻₂₂^TQ^T₂ =−A⁻₂₂^TC₂^TC2A⁻₂₂¹. (8) With this, the(1,1)block equation of(7)can be rewritten as

³

A^T₁₁−A^T₂₁A⁻₂₂^TA^T₁₂´

P11+P₁₁^T¡

A11−A12A⁻₂₂¹A21¢ +

³

C₁^T−A^T₂₁A⁻₂₂^TC₂^T´ ¡

C1−C2A⁻₂₂¹A21¢

= 0. (9) As(E, A, C)is £nite dynamics detectable (observable),

¡A11−A12A⁻₂₂¹A21, C1−C2A⁻₂₂¹A21¢

is detectable (observable) in the usual state space sense. From P11 ≥ 0 (>0) and the usual Lyapunov theory it follows that

¡A₁₁−A₁₂A⁻₂₂¹A₂₁¢

is stable. (i) ⇒ (ii) As(E, A) is regular, impulse-free and stable,A22is nonsingular and¡

A11−A12A⁻₂₂¹A21¢ is stable. Now it is easy to verify that a solution of (6) is given by P=S

· P11 0 0 P₂₂

¸

S^T ≥0 (>0), Q=W⁻¹£

Q1 Q2 ¤ T⁻¹ where

P11≥0 (>0)is a solution of(9), P₂₂≥0 (>0)is arbitrary, Q2is a solution of(8), and Q₁ =−A⁻₂₂^T¡

A^T₁₂P₁₁+Q^T₂A₂₁+C₂^TC₁¢

.(ii)⇒(iii)Imme- diate. ¤

Theorem 4 shows that with the observability assumption(O2) and (O3), system stability is related with Lyapunov equation (6) where the termC^TC+A^TE0Q+Q^TE₀^TAis used instead of the term E^TC^TCEof Lyapunov equation(2). The extra variableQis related with the impulsive behavior of the system and we can setQ= 0only if the system is impulse-free (indeed, from(7)we should haveC2= 0 which with(O3)implies that the system is regular and impulse-free).

The next corollary follows from Theorem 4 withQ= 0.

Corollary 2. Let E, A ∈ R^n×n and C ∈ R^p×n be such that rank^{· E}_C ^¸= rankE and with (O1) and (O3) ((O2)and (O3)) satis£ed. Then the following statements are equivalent

(i) the system(E, A)is regular, impulse-free and stable;

(ii) there exists a solutionP≥0 (>0)to the following GLE:

A^TP E+E^TP A+C^TC= 0; (10) (iii) there exists a solutionP ∈R^n×nwithE^TP E≥0 (E^TP E≥

0andrank(E^TP E) =rankE)to GLE(10).

Proof: From Lemma A.1 in the Appendix, it is clear that we are already supposing that the system is regular and impulse-free.

(iii)⇒(i)Immediate from Theorem 4 withQ= 0.

(i) ⇒(ii)Consider the same SVD coordinate system used in the proof of Theorem 4. From Lemma A.1 we have thatC₂= 0. It is easy to verify that a solution of(10)is given by

P =S

· P11 −P11A12A⁻₂₂¹

−A⁻₂₂^TA^T₁₂P11 P22

¸

S^T ≥0 (>0) where

P11≥0 (>0)is a solution of(9)withC2= 0,

P₂₂ ≥ 0 (>0) is an arbitrary matrix satisfying P₂₂ ≥ A⁻₂₂^TA^T₁₂P11A12A⁻₂₂¹ ¡

P22> A⁻₂₂^TA^T₁₂P11A12A⁻₂₂¹¢

. (ii) ⇒ (iii) Immediate. ¤

Note that we haverank^{· E}_C ^¸=rankEif and only ifC=CE for some matrixC(see Lemma A.1 in the Appendix). Then the fol- lowing correction of Theorem 1 follows immediately from the above corollary.

Corollary 3. Let(E, A, CE)be regular, impulse-free and £nite dy- namics observable. Then(E, A)is stable if and only if there exists a positive de£nite solutionPto Lyapunov equation(2).

Proof: For matricesE, A ∈ R^n×nandC ∈ R^p×nit is easy to show that the rank conditions(O2)and (O3)are satis£ed with C replaced byCEif and only if the system(E, A, CE)is regular, impulse-free and £nite dynamics observable. ¤

Until now we have seem that Lyapunov equation(2)can be used to characterize stability only if we consider some additional assump- tions (cf. Theorem 2, Theorem 3, Corollary 1, and Corollary 3). The Lyapunov equation(6)can be used for more general situations but it involves two unknown matricesPandQ. The matrixPis related with the £nite dynamics behavior and the matrixQis related with regular- ity and the impulsive behavior. The Lyapunov equation(6)with two unknown matrices can be rewritten as a system of two equations with one unknown matrix. Indeed, suppose that in GLE(6)we de£ne an auxiliary variable

X :=P E+E₀Q.

In this case, since

0≤E^TP E=E^T(P E+E0Q) =E^TX, the Lyapunov function candidate(5)is rewritten as

V(x) =x^TE^TXx.

Then we may consider the following sets X₁={X∈R^n×n:E^TX=X^TE, E^TX≥0};

X₂={X∈R^n×n:E^TX=X^TE, E^TX≥0, rank(E^TX) =r};

X1={X=P E+E₀Q:P ∈R^n×n, P≥0, Q∈R^(n−r)×n};

X2={X=P E+E0Q:P ∈R^n×n, P >0, Q∈R^(n−r)×n};

X1={X=P E+E0Q:P ∈R^n×n, E^TP E≥0, Q∈R^(n−r)×n};

X2={X=P E+E0Q:P∈R^n×n, E^TP E≥0,

rank(E^TP E) =r, Q∈R^(n−r)×n},

whereE0 ∈ R^n×(n−r) is a matrix of full column rank such that E^TE0= 0, andr=rankE.

It is easy to verify that

X₁=X1=X1 and X₂=X2=X2. (11) Thus, the next result due to Takaba et al. [14] follows immediately from Theorem 4 (here the regularity assumption, considered in [14], is eliminated).

Corollary 4. LetE, A ∈ R^n×nandC ∈ R^p×nand consider the following system of equations:

E^TX=X^TE≥0,

A^TX+X^TA+C^TC= 0. (12) If(O1)and(O3) ((O2)and(O3))are satis£ed, the system(E, A) is regular, impulse-free and stable if and only if there exists a so- lutionX ∈ R^n×n to the GLE (12) (GLE (12) with the restriction rank(E^TX) =rankE).

The equalitiesX_i=Xi=Xipresented in(11)show that Lyapunov equations(6)and(12)are equivalent. The equality between these sets can also be useful when we consider Lyapunov inequality tests for stability (cf. [8], [15]).

Corollary 5. LetE, A ∈ R^n×n,r = rankE and considerE0 ∈ R^n×(n−r) of full column rank such thatE^TE₀ = 0. The following statements are equivalent

(i) the system(E, A)is regular, impulse-free and stable;

(ii) there exists a solution(P, Q)∈R^n×n×R^(n−r)×nwithP >0 to the following Lyapunov inequality:

A^T(P E+E0Q) + (P E+E0Q)^TA <0; (13) (iii) there exists a solutionX ∈ R^n×nto the following system of

inequalities:

A^TX+X^TA <0,

E^TX≥0, (14)

E^TX=X^TE.

Proof: (ii) ⇒ (i) De£ne L = −A^T(P E+E0Q) − (P E+E0Q)^TA. We have that³

E, A, L^1/2´

satis£es conditions (O2)and(O3)and

A^T(P E+E0Q) + (P E+E0Q)^TA+³ L^1/2´T

L^1/2= 0.

Then, from Theorem 4 it follows that(E, A)is regular, impulse-free and stable.

(i)⇒(ii)For every nonsingular matrixC∈R^n×n, we have that (E, A, C)satis£es conditions(O2)and(O3). Then, from Theorem 4 there exists a solution(P, Q)withP >0to

A^T(P E+E0Q) + (P E+E0Q)^TA=−C^TC <0.

The equivalence(ii)⇔(iii)is immediate sinceX₂=X2. ¤ In the above corollary, note that although(ii)and(iii)are equiva- lent, inequality(13)is easier solved via software than(14)since(13) does not have equality restrictions ([2], [5]).

Until now we have considered Lypunov equations to characterize stability. As for usual state space systems, we can also use Lyapunov equations to give necessary and suf£cient conditions for observability.

In the next theorem we present the converse of Corollary 3, Theorem 4, Corollary 2, and Corollary 4.

Theorem 5. LetE, A ∈ R^n×n, C ∈ R^p×nbe such that the sys- tem(E, A)is regular, impulse-free and stable and consider the state- ments

(i) the Lyapunov equation (2) has a solutionPsuch thatE^TP E≥ 0andrank¡

E^TP E¢

=r;

(ii) the Lyapunov equation (6) has a solutionPsuch thatE^TP E≥ 0andrank¡

E^TP E¢

=r;

(iii) the Lyapunov equation (10) has a solution P such that E^TP E≥0andrank¡

E^TP E¢

=r;

(iv) the Lyapunov equation (12) has a solutionXsuch thatE^TX≥ 0andrank¡

E^TX¢

=r.

Then

(1) (E, A, CE) is observable ((O2) and (O3) hold with C re- placed byCE) if and only if(i)holds;

(2) (E, A, C)is observable if and only if any one of the above state- ments(ii)−(iv)holds.

Proof: The result for GLE(2)can be derived from the result for GLE (10) withCreplaced byCE. As any solution of GLE (10) is a solution of GLE (6) and GLE(12)is equivalent to GLE (6), we need only to prove the result with Lyapunov equation (6). The suf£ciency is proved in Theorem 4. To prove the necessity of observability, suppose that(O2)does not hold, that is,∃λ0∈Candv6= 0such that

· λ0E−A C

¸

v= 0. (15)

Multiplying (6) from the left byv^∗and from the right byvwe obtain 2Re(λ0)v^∗E^TP Ev= 0.

SinceRe(λ0)<0,E^TP E≥0, andrank¡ E^TP E¢

=r, it follows thatEv = 0which, by (15), impliesAv = 0. Then we have that (E, A)is not regular. By contradiction one concludes that(O2)holds.

The statement that(O3)holds (impulse observability) is immediate since(E, A)is regular and impulse-free. ¤

Theorem 5 is valid not only for P such that E^TP E ≥ 0, rank¡

E^TP E¢

= rbut also forP > 0(use equalityX2 =X2 of (11)in statement(ii)of Theorem 5).

As it was not our intent to give a complete treatment on the subject, we do not consider here the relationship between Lyapunov equations and inertia of(E, A)or observability/controllability Gramians. For some particular GLEs, these relationships can be found in [13].

III. CONCLUSION

We revisited the Lyapunov theory for singular systems. Although the Lyapunov equation of [6], [7] appears ‘naturally’, it can be used for stability analysis only with some restrictive assumptions on the plant.

A corrected generalized Lyapunov equation is presented. It makes a

‘connection’ between the generalized Lyapunov equations of [6] and of [14].

APPENDIX

We state here a technical lemma.

Lemma A. 1. LetE, A ∈ R^n×nandC ∈R^p×nand consider the same SVD coordinate system used in the proof of Theorem 4. Then the following statements are equivalent

(a) C₂= 0;

(b) rank

· E C

¸

=rankE;

(c) C=CEfor some matrixC.

Furthermore, ifrank

· E C

¸

= rankE then the system(E, A)is regular and impulse-free if and only if(O3)is satis£ed.

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