IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002

On the Lyapunov Theorem for Singular Systems
João Yoshiyuki Ishihara and Marco Henrique Terra
Abstract—In this paper we revisit the Lyapunov theory for singular systems. There are basically two well known generalized Lyapunov equations
used to characterize stability for singular systems. We start with the Lyapunov 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.
Singular systems of the form
=
Ax
y
=
Cx

E A
(O3) rank  0 C  = n + rankE;
0 E
¸
·
E
= n.
(O4) rank
C
It is immediately seen that C-observability implies S-observability.
II. T HE GENERALIZED LYAPUNOV T HEOREM
Consider the usual Lyapunov function candidate for state space systems
V (x) = xT P x ≥ 0; V̇ (x) = −xT C T Cx.
One can conclude that for singular systems, it is ‘natural’ to consider
V (Ex), V̇ (Ex) as a Lyapunov function candidate. This choice leads
to the following well known generalized Lyapunov theorem [6], [7].
I. I NTRODUCTION
E ẋ
1926
(1)
have been of interest in the literature since they have many important
applications in, for example, circuit systems [10], robotics [9], and aircraft 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 characterization 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, incorrect. 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 Lyapunov equation of [14].
First we state here some basic de£nitions which will be used in the
next section. System (1) with a n × n matrix E is called regular if
det (sE − A) 6= 0 for some s ∈ C. We say that the regular system
(1) is ([16], [3], [4], [17])
(i) stable if all roots of det (sE − A) = 0 are 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 observable) if (O2) and (O3) hold;
(vii) C-observable if (O2) and (O4) hold
where (O1) − (O4) conditions are given by:
¸
·
sE − A
= n, Re (s) ≥ 0;
(O1) rank
C
·
¸
sE − A
(O2) rank
= n, for all s ∈ C;
C
The authors are with the Electrical Engineering Department - EESC - University of São Paulo at São Carlos, Brazil. (e-mail: [email protected].;
[email protected]).
This work was supported by FAPESP (São Paulo State University Council)
under grant 98/12113-2.
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 solution P to the following Lyapunov equation
AT P E + E T P A + E T C T CE = 0.
(2)
Unfortunately, Theorem 1 is, as stated, incorrect. To see this, consider the system with the following values

1
E= 0
0
0
0
1


0
0
0 ,A =  0
0
−1
0
0
−1

0
£
1 ,C = 2
0
1
2
¤
.
In this case, system (1) is regular, impulse-free and observable (in
fact, the system is C-observable). It can be veri£ed that the Lyapunov
equation (2) has a solution


8 0 2
P = 0 1 0 >0
2 0 2
but the system is not stable since it has a £nite mode at s = 0 :

s
det (sE − A) = det  0
1
0
0
s+1

0
−1  = s (s + 1) .
0
A careful analysis shows that the proof of [6] states, in fact, the
following result.
p×n
Theorem 2. Let E, A ∈ Rn×n , and C
³ ∈ R ´ be given by (a
e A,
e C
e ):
Weierstrass form of some regular system E,
E :=
·
Iq
0
0
Λ
¸
, A :=
·
J
0
0
In−q
¸
, C :=
£
CF
C∞
¤
where Iq denotes an q × q identity matrix, J corresponds to the £nite
zeros of sE − A, Λ is nilpotent (Λk = 0, Λk−1 6= 0 for some integer k > 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 solution P to
the following Lyapunov equation
AT P E + E T P A + E T C T CE = 0.
Moreover, if P and P ′ are two such solutions, then E T P ′ E =
E T P E.
The theorem above says that if (E, A, C) is already in Weierstrass
form then the solution of Lyapunov equation (2) characterizes stability.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002
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 following. Let E, A ∈ Rn×n , and C ∈ Rp×n be given with (E, A) regular.
Then the Weierstrass form can be obtained:
M EN =
·
Iq
0
0
Λ
¸
,
M AN =
CN =
·
£
0
J
0
In−q
C1
C2
¤
¸
,
where M and N are nonsingular matrices. In this case, the Lyapunov
equation (2) can be written as
1927
(a) As Λ is nilpotent we can de£ne ν := min{k > 0 : Λ k = 0}.
Now suppose that ν > 1. Pre-multiplying the (2, 2) block equation of
¡ ¢ν−1
(4) by ΛT
we have that
³
ΛT
´ν−1
³ ´ν−1
P22 Λ = 0 ⇒ ΛT
P22 Λν−1 = 0 ⇒ P22 Λν−1 = 0.
¡ ¢ν−2
With this, pre-multiplying the (2, 2) block equation of (4) by ΛT
and post-multiplying by Λν−2 we have
³ ´ν−1
ΛT
Q22 Λν−1 = 0 ⇒ Q22 Λν−1 = 0 ⇒ Λν−1 = 0.
This contradicts the minimality of ν. Then, we must have Λ = 0 (ν =
1) and the system is impulse-free. On the other hand, considering
N T AT M T M −T P M −1 M EN + N T E T M T M −T P M −1 M AN + that Q11 > 0 and P11 ≥ 0, from (1, 1) block of (4) it follows that
N T E T M T M −T C T CM −1 M EN = 0. P11 > 0. In fact, ifTthere exists x 6= 0 such that P11 x = 0, from
(1, 1) block we get x Q11 x = 0, x 6= 0. This contradicts the fact that
T
Now it is easy to see that in the proof of [6] it was considered the Q11 > 0. Now, as P11 > 0 is a solution of J P11 + P11 J + Q11 = 0,
from usual state space Lyapunov theory we have that (E, A) is stable.
following system
(b) Suppose that the regular system (E, A) is impulse-free and stable.
Then in the above Weierstrass form we have Λ = 0 and J stable. For
·
¸
·
¸
Iq 0
J
0
each Q > 0, we can £nd a unique P 11 > 0 solution of
M EN =
, M AN =
,
0 Λ
0 In−q
£
¤
J T P11 + P11 J + Q11 = 0.
C
CM −1 = C
.
F
∞
The counterexample shows a case where the results for CN and
CM −1 are not the same. It shows also that even with the stronger assumption of C-observability, Theorem 1 is incorrect. Although Theorem 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
AT P + P A + Q = 0, Q > 0.
Theorem 3. Let (E, A) be regular and consider the following generalized Lyapunov equation (GLE)
AT P E + E T P A + E T QE = 0.
(3)
We have that
(a) if there exist matrices P ≥ 0 and Q > 0 satisfying the GLE (3)
then (E, A) is impulse-free and stable;
(b) if (E, A) is impulse-free and stable then for each Q > 0 there
exists P > 0 solution of GLE (3). Furthermore E T P E ≥ 0 is
unique for each Q > 0.
Proof: From the regularity assumption of the system (1), we can
put it in the Weierstrass form £nding nonsingular matrices M and N
such that
¸
¸
·
·
J
0
Iq 0
, M AN =
M EN =
0 In−q
0 Λ
where Λ is nilpotent and the eigenvalues of J are £nite eigenvalues
of (E, A). Now make a partition of M −T P M −1 and M −T QM −1
accordingly:
·
¸
·
¸
P11 P12
Q11 Q12
M −T P M −1 =
, M −T QM −1 =
.
T
T
P12 P22
Q12 Q22
The Lyapunov equation (3) can be written as
¸
·
J T P11 + P11 J + Q11
J T P12 Λ + P12 + Q12 Λ
= 0.
T
T
J + ΛT QT12 P22 Λ + ΛT P22 + ΛT Q22 Λ
+ ΛT P12
P12
(4)
Now it is easy to verify that
P = MT
·
P11
0
0
P22
¸
M
is a solution for the GLE for any P22 . Note that we obtain P > 0
(≥ 0) choosing P22 > 0 (≥ 0). Also,
¸
·
P11 0
N
ET P E = N T
0
0
is unique since P11 is 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 solution P ≥ 0 for some
Q > 0 if and only if GLE (3) is solvable for all Q > 0.
Note
that the assumptions in Theorem 3 are quite strong since the condition
C T C = Q > 0 requires an output matrix C of full column rank. We
can relax a bit the rank condition on C if 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
C is such that rank (CE) = rankE. We have that
(a) if there exists a matrix P ≥ 0 satisfying the GLE (2) then (E, A)
is stable;
(b) if (E, A) is stable then there exists P > 0 solution of GLE (2).
Furthermore E T P E ≥ 0 is unique for each C.
Proof: De£ne Q = C T C and consider the Lyapunov equation in
Weierstrass coordinates (4). By assumption, we already have Q11 > 0
and Λ = 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 solution P > 0, Q > 0 to (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
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002
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 information under mild conditions (without regularity assumption, for example). For this, consider the following Lyapunov function candidate
T
T
V (x) = x E P Ex ≥ 0,
1928
T T AT SS T (P E + E0 Q) T + T T (P E + E0 Q)T SS T AT +
+T T C T CT = 0
·
+
(5)
·
AT11
AT12
T
P11
0
where P > 0 (we are interested in E T P E ≥ 0). Looking at its time
derivative
V̇ (x) = ẋT E T P Ex + xT E T P E ẋ
we note that, for any matrices Q and E0 of compatible dimensions
satisfying E T E0 = 0, we have
³
´
V̇ (x) = ẋT E T (P E + E0 Q) x + xT E T P + QT E0T E ẋ
or
T
V̇ (x) = x
³
´
A (P E + E0 Q) + (P E + E0 Q) A x.
T
T
Therefore, if the Lyapunov equation
AT (P E + E0 Q) + (P E + E0 Q)T A + C T C = 0
has a solution (P, Q) with P > 0 (or, at least, with E T P E ≥ 0), we
obtain (as in the usual state space case)
In the next theorem we present a corrected version of the generalized
Lyapunov theorem with a Lyapunov equation similar to (2).
Theorem 4. Let E, A ∈ Rn×n and C ∈ Rp×n be such that (O1)
and (O3) ((O2) and (O3)) are satis£ed. Consider also a matrix
E0 ∈ Rn×(n−r) of full column rank such that E T E0 = 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) ∈ Rn×n ×R(n−r)×n with P ≥ 0
(> 0) to the following GLE:
T
T
T
T
A P E + E P A + C C + A E0 Q + Q
n×n
E0T A
= 0; (6)
(n−r)×n
(iii) there exists a solution (P, Q) ∈ R ¡ × R¢
with
E T P E ≥ 0 (E T P E ≥ 0 and rank E T P E = rankE)
to GLE (6).
Proof: Consider a SVD coordinate system [1]
·
¸
·
¸
Ir 0
A11 A12
S T ET =
, S T AT =
,
0 0
A21 A22
¤
£
CT = C1 C2
¤
£
where S = S1 S2 is orthogonal and T is nonsingular. In this
case, (E, A) is regular and impulse-free if and only if A22 is nonsingular [1]. Furthermore, if A22 is nonsingular, (E, A) is stable if and only
T
if (A11 −A12 A−1
22 A21 ) is stable. As E0 and S2 are bases of KerE ,
there exists a nonsingular matrix W such that E0 = S2 W . De£ne
accordingly
·
¸
£
¤
P11 P12
W QT = Q1 Q2 , and S T P S =
.
P21 P22
The Lyapunov equation (6) can be rewritten as
T
T
T
T
AT
11 P11 + P11 A11 + A21 (P21 + Q1 ) + (P21 + Q1 ) A21 + C1 C1
T
T
T
AT
P
+
A
(P
+
Q
)
+
Q
A
+
C
C
11
21
1
21
1
12
22
2
2
∗
T
T
AT
22 Q2 + Q2 A22 + C2 C2
¸
=0
(7)
(iii) ⇒ (i) We £rst show that the system is regular and impulsefree. Indeed, consider v ∈ KerA22 . From the (2, 2) block of (7)
we have v T C2T C2 v = 0, which
implies
C2 v = 0. As (E, A, C) is
·
¸
A22
impulse observable, that is,
has full column rank, it follows
C2
that v = 0 and therefore, KerA22 = {0}, that is, A22 is nonsingular.
(stability) From (2, 1) and (2, 2) blocks of (7) we have
´
³
AT12 P11 + QT2 A21 + C2T C1
P21 + Q1 = −A−T
22
−1
−T T
−T T
Q2 A−1
22 + A22 Q2 = −A22 C2 C2 A22 .
V̇ (x) = −xT C T Cx ≤ 0.
T
·
¸·
¸
P11
0
+
P21 + Q1 Q2
¸
·
¸
T
P21
+ QT1
A11 A12
+
A21 A22
QT2
· T ¸
¤
£
C1
C1 C2 = 0
+
T
C2
AT21
AT22
With this, the (1, 1) block equation of (7) can be rewritten as
´
³
¢
−1
T ¡
T
AT11 − AT21 A−T
22 A12 P11 + P11 A11 − A12 A22 A21 +
´
³
¢
T ¡
C1 − C2 A−1
C1T − AT21 A−T
22 A21 = 0.
22 C2
(8)
(9)
As (E, A, C) is £nite dynamics detectable (observable),
¡
−1
A11 − A12 A−1
22 A21 , C1 − C2 A22 A21
¢
is detectable (observable) in the usual state space sense. From
P
¢ the usual Lyapunov theory it follows that
¡ 11 ≥ 0 (>−10) and
A11 − A12 A22 A21 is stable. (i) ⇒ (ii) ¡As (E, A) is regular,¢
impulse-free and stable, A22 is nonsingular and A11 − A12 A−1
22 A21
is stable. Now it is easy to verify that a solution of (6) is given by
·
¸
£
¤
P11
0
P =S
S T ≥ 0 (> 0) , Q = W −1 Q1 Q2 T −1
0
P22
where
P11 ≥ 0 (> 0) is a solution of (9),
P22 ≥ 0 (> 0) is arbitrary,
Q2 is a solution
¢
¡ ofT (8), and T
A12 P11 + Q2 A21 + C2T C1 . (ii) ⇒ (iii) ImmeQ1 = −A−T
22
diate. ¤
Theorem 4 shows that with the observability assumption (O2)
and (O3), system stability is related with Lyapunov equation (6)
where the term C T C +AT E0 Q+QT E0T A is used instead of the term
E T C T CE of Lyapunov equation (2). The extra variable Q is related
with the impulsive behavior of the system and we can set Q = 0 only
if the system is impulse-free (indeed, from (7) we should have C2 = 0
which with (O3) implies that the system is regular and impulse-free).
The next corollary follows from Theorem 4 with Q = 0.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002
n×n
Corollary
and C ∈ Rp×n be such that
¸2. Let E, A ∈ R
·
E
rank 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 solution P ≥ 0 (> 0) to the following GLE:
AT P E + E T P A + C T C = 0;
(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 with Q = 0.
(i) ⇒ (ii) Consider the same SVD coordinate system used in the
proof of Theorem 4. From Lemma A.1 we have that C2 = 0. It is easy
to verify that a solution of (10) is given by
¸
·
P11
−P11 A12 A−1
22
S T ≥ 0 (> 0)
P =S
−T T
P22
−A22 A12 P11
where
P11 ≥ 0 (> 0) is a solution of (9) with C2 = 0,
P22 ≥ 0 (> 0) ¡ is an arbitrary matrix satisfying
P22 ≥
¢
−T T
−1
−1
T
A
A−T
A
P
>
A
P
A
A
.
(ii)
⇒ (iii)
P
A
A
22
11
12
11
12
12
12
22
22
22
22
Immediate. ¤
¸
·
= rankE if and only if C = CE
Note that we have rank E
C
for some matrix C (see Lemma A.1 in the Appendix). Then the following correction of Theorem 1 follows immediately from the above
corollary.
Corollary 3. Let (E, A, CE) be regular, impulse-free and £nite dynamics observable. Then (E, A) is stable if and only if there exists a
positive de£nite solution P to Lyapunov equation (2).
p×n
Proof: For matrices E, A ∈ R
and C ∈ R
it is easy
to show that the rank conditions (O2) and (O3) are satis£ed with
C replaced by CE if 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 assumptions (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 matrices P and Q. The matrix P is related with
the £nite dynamics behavior and the matrix Q is related with regularity 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 + E0 Q.
In this case, since
0 ≤ E T P E = E T (P E + E0 Q) = E T X,
the Lyapunov function candidate (5) is rewritten as
T
where E0 ∈ Rn×(n−r) is a matrix of full column rank such that
E T E0 = 0, and r = rankE.
It is easy to verify that
X1 = X1 = X1 and X2 = X2 = X2 .
(iii) there exists a solution P ∈ Rn×n with E T P E ≥ 0 (E T P E ≥
0 and rank(E T P E) = rankE) to GLE (10).
n×n
1929
T
V (x) = x E Xx.
Then we may consider the following sets
X1 = {X ∈ Rn×n : E T X = X T E, E T X ≥ 0};
X2 = {X ∈ Rn×n : E T X = X T E, E T X ≥ 0, rank(E T X) = r};
(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. Let E, A ∈ Rn×n and C ∈ Rp×n and consider the
following system of equations:
E T X = X T E ≥ 0,
AT X + X T A + C T C = 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 solution X ∈ Rn×n to the GLE (12) (GLE (12) with the restriction
rank(E T X) = rankE).
The equalities Xi = Xi = Xi presented 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. Let E, A ∈ Rn×n , r = rankE and consider E0 ∈
Rn×(n−r) of full column rank such that E T E0 = 0. The following
statements are equivalent
(i) the system (E, A) is regular, impulse-free and stable;
(ii) there exists a solution (P, Q) ∈ Rn×n ×R(n−r)×n with P > 0
to the following Lyapunov inequality:
AT (P E + E0 Q) + (P E + E0 Q)T A < 0;
(13)
(iii) there exists a solution X ∈ Rn×n to the following system of
inequalities:
AT X + X T A < 0,
E T X ≥ 0,
(14)
T
E X = X T E.
T
(ii) ⇒ (i) De£ne
³ L = −A
´ (P E + E0 Q) −
1/2
(P E + E0 Q) A. We have that E, A, L
satis£es conditions
Proof:
T
(O2) and (O3) and
³
´T
AT (P E + E0 Q) + (P E + E0 Q)T A + L1/2 L1/2 = 0.
Then, from Theorem 4 it follows that (E, A) is regular, impulse-free
and stable.
(i) ⇒ (ii) For every nonsingular matrix C ∈ Rn×n , we have that
(E, A, C) satis£es conditions (O2) and (O3). Then, from Theorem 4
there exists a solution (P, Q) with P > 0 to
AT (P E + E0 Q) + (P E + E0 Q)T A = −C T C < 0.
The equivalence (ii) ⇔ (iii) is immediate since X2 = X2 . ¤
In the above corollary, note that although (ii) and (iii) are equivalent, inequality (13) is easier solved via software than (14) since (13)
does not have equality restrictions ([2], [5]).
X1 = {X = P E + E0 Q : P ∈ Rn×n , P ≥ 0, Q ∈ R(n−r)×n };
X2 = {X = P E + E0 Q : P ∈ Rn×n , P > 0, Q ∈ R(n−r)×n };
X1 = {X = P E + E0 Q : P ∈ Rn×n , E T P E ≥ 0, Q ∈ R(n−r)×n };
X2 = {X = P E + E0 Q : P ∈ Rn×n , E T P E ≥ 0,
rank(E T P E) = r, Q ∈ R(n−r)×n },
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.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002
Theorem 5. Let E, A ∈ Rn×n , C ∈ Rp×n be such that the system (E, A) is regular, impulse-free and stable and consider the statements
T
(i) the Lyapunov¡ equation
¢ (2) has a solution P such that E P E ≥
0 and rank E T P E = r;
T
(ii) the Lyapunov¡ equation
¢ (6) has a solution P such that E P E ≥
T
0 and rank E P E = r;
(iii) the Lyapunov equation
¡ (10) ¢has a solution P such that
E T P E ≥ 0 and rank E T P E = r;
T
(iv) the Lyapunov¡ equation
¢ (12) has a solution X such that E X ≥
T
0 and rank E X = r.
Then
(1) (E, A, CE) is observable ((O2) and (O3) hold with C replaced by CE) if and only if (i) holds;
(2) (E, A, C) is observable if and only if any one of the above statements (ii) − (iv) holds.
Proof: The result for GLE (2) can be derived from the result for
GLE (10) with C replaced by CE. 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 ∈ C and v 6= 0 such that
·
¸
λ0 E − A
v = 0.
(15)
C
Multiplying (6) from the left by v ∗ and from the right by v we obtain
2Re(λ0 )v ∗ E T P Ev = 0.
¡
¢
Since Re(λ0 ) < 0, E T P E ≥ 0, and rank E T P E = r, it follows
that Ev = 0 which, by (15), implies Av = 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. ¤
T
¡Theorem¢ 5 is valid not only for P such that E P E ≥ 0,
rank E T P E = r but also for P > 0 (use equality X2 = 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. C ONCLUSION
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].
A PPENDIX
We state here a technical lemma.
Lemma A. 1. Let E, A ∈ Rn×n and C ∈ Rp×n and consider the
same SVD coordinate system used in the proof of Theorem 4. Then the
following statements are equivalent
(a) C2 = ·0;
¸
E
(b) rank
= rankE;
C
(c) C = CE for some matrix C.
·
¸
E
Furthermore, if rank
= rankE then the system (E, A) is
C
regular and impulse-free if and only if (O3) is satis£ed.
1930
R EFERENCES
[1] D. J. Bender and A. J. Laub, “The linear-quadratic optimal regulator for
descriptor systems,” IEEE Trans. Automat. Contr., vol. AC-32, no. 8, pp.
672–688, 1987.
[2] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix
Inequalities in System and Control Theory, SIAM, Philadelphia, 1994.
[3] D. Cobb, “Controllability, observability, and duality in singular systems,”
IEEE Trans. Automat. Contr., vol. AC-29, no.12, pp. 1076–1082, 1984.
[4] L. Dai, “Impulsive modes and causality in singular systems,” Int. J.
Contr., vol. 50, no. 4, pp. 1267- 1281, 1989.
[5] P. Gahinet, A. Nemirovski, A. J. Laub and M. Chiali, LMI Control Toolbox, The Math Works Inc., 1995.
[6] F. L. Lewis, “Preliminary notes on optimal control for singular systems,”
Proceedings of 24th IEEE Conference on Decision and Control, pp. 266272, 1985.
[7] F. L. Lewis, “A survey of linear singular systems,” Circuits, Syst. Sig.
Proc., vol. 5, no. 1, pp. 3-36, 1986.
[8] I. Masubuchi, Y. Katamine, A. Ohara, and N. Suda. “H∞ control for
descriptor systems. A matrix inequalities approach,” Automatica, vol. 33,
no. 4, pp. 669-673, 1997.
[9] J. K. Mills and A. A. Goldenberg, “Force and position control of manipulators during constrained motion tasks,” IEEE Trans. Robot. Automat.,
vol. 38, pp. 30-46, 1989.
[10] R. W. Newcomb and B. Dziurla, “Some circuits and systems applications
of semistate theory,” Circuits, Syst. Sig. Proc., vol. 8, pp. 235-260, 1989.
[11] S. Sastry, Nonlinear Systems: Analysis, Stability and Control, SpringerVerlag, New York, 1999.
[12] B. L. Stevens and F. L. Lewis, Aircraft Modelling, Dynamics and Control.
New York: Wiley, 1991.
[13] T. Stykel, Generalized Lyapunov equations for descriptor systems: stability and inertia theorems. Preprint SFB393/00-38, Fakltät für Mathematik,
TU Chemnitz, DO9107 Chemnitz, Germany, 2000.
[14] K. Takaba, N. Morihira and T. Katayama, “A generalized Lyapunov theorem for descriptor system,” Systems and Control Letters, vol. 24, no. 1,
pp. 49-51, 1995.
[15] E. Uezato and M. Ikeda. “Strict LMI conditions for stability, robust stabilization, and control of descriptor systems,” Proceedings of 38th IEEE
Conference on Decision and Control, pp. 4092-4097, 1999.
[16] G. C. Verghese, B. C. Lévy, and T. Kailath, “A generalized state-space
for singular systems,” IEEE Trans. Automat. Contr., vol. AC-26, no. 4,
pp.811-831,1981.
[17] E. L. Yip and R. F. Sincovec, “Solvability, controllability, and observability of continuous descriptor systems,” IEEE Trans. Automat. Contr., vol.
AC-26, no.3, pp. 702-707, 1981.