descriptor system
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Ex( k + 1) = Ax( k) + Bw( k)
(2.1)
z( k) = Cx( k) ,
where the nonnegative integer k denotes the discrete time, x( k) ∈ R n is the descriptor
variable, w( k) ∈ R p is the disturbance input, z( k) ∈ R q is the controlled output, E ∈ R n× n, A ∈ R n× n, B ∈ R n× p and C ∈ R q× n are constant matrices. The matrix E may be singular and we denote its rank by r = rank E ≤ n.
Definition 1: Consider the linear descriptor system (2.1) with w = 0. The system has a unique
solution for any initial condition and is called regular, if | zE − A| ≡ 0. The finite eigenvalues
of the matrix pair ( E, A), that is, the solutions of | zE − A| = 0, and the corresponding
(generalized) eigenvectors define exponential modes of the system. If the finite eigenvalues lie
in the open unit disc of z, the solution decays exponentially. The infinite eigenvalues of ( E, A)
with the eigenvectors satisfying the relations Ex 1 = 0 determine static modes. The infinite
eigenvalues of ( E, A) with generalized eigenvectors xk satisfying the relations Ex 1 = 0 and
Exk = xk−1 ( k ≥ 2) create impulsive modes. The system has no impulsive mode if and only if
rank E = deg | sE − A| (deg | zE − A|). The system is said to be stable if it is regular and has
only decaying exponential modes and static modes (without impulsive modes).
Lemma 1 (Weiertrass Form)[1, 2] If the descriptor system (2.1) is regular, then there exist two
nonsingular matrices M and N such that
Id 0
Λ 0
MEN =
, MAN =
(2.2)
0 J
0 In− d
where d = deg | zE − A|, J is composed of Jordan blocks for the finite eigenvalues. If the
system (2.1) is regular and there is no impulsive mode, then (2.2) holds with d = r and J = 0.
If the system (2.1) is stable, then (2.2) holds with d = r, J = 0 and furthermore Λ is Schur
stable.
Let the singular value decomposition (SVD) of E be
E 11 0
E = U
VT , E 11 = diag{ σ 1, · · · , σr}
(2.3)
0 0
where σi’s are the singular values, U and V are orthonormal matrices ( UTU = VTV = I).
With the definitions
A 11 A 12
¯ x = VTx =
¯ x 1 , UTAV =
,
(2.4)
¯ x 2
A 21 A 22
the difference equation in (2.1) (with w = 0) takes the form of
E 11 ¯ x 1( k + 1) = A 11 ¯ x 1( k) + A 12 ¯ x 2( k)
(2.5)
0 = A 21 ¯ x 1( k) + A 22 ¯ x 2( k) .
It is easy to obtain from the above that the descriptor system is regular and has not impulsive
modes if and only if A 22 is nonsingular. Moreover, the system is stable if and only if A 22 is
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nonsingular and furthermore E−1 A
A
11
11 − A 12 A−1
22
21
is Schur stable. This discussion will
be used again in the next sections.
Definition 2: Given a positive scalar γ, if the linear descriptor system (2.1) is stable and satisfies
k
k
∑ zT( j) z( j) ≤ φ( x(0)) + γ 2 ∑ wT( j) w( j)
(2.6)
j=0
j=0
for any integer k > 0 and any l 2-bounded disturbance input w, with some nonnegative definite
function φ(·), then the descriptor system is said to be stable and have L2 gain less than γ.
The above definition is a general one for nonlinear systems, and will be used later for switched
descriptor systems.
3. Problem formulation
In this article, we consider the switched system composed of N linear discrete-time descriptor
subsystems described by
Ex( k + 1) = Aix( k) + Biw( k)
(3.1)
z( k) = Cix( k) ,
where the vectors x, w, z and the descriptor matrix E are the same as in (2.1), the index i
denotes the i-th subsystem and takes value in the discrete set I = {1, 2, · · · , N }, and thus the
matrices Ai, Bi, Ci together with E represent the dynamics of the i-th subsystem.
For the above switched system, we consider the stability and L2 gain properties under the
assumption that all subsystems in (3.1) are stable and have L2 gain less than γ. As in the case
of stability analysis for switched linear systems in state space representation, such an analysis
problem is well posed (or practical) since a switched descriptor system can be unstable even if
all the descriptor subsystems are stable and there is no variable (state) jump at the switching
instants. Additionally, switchings between two subsystems can even result in impulse signals,
even if the subsystems do not have impulsive modes themselves. This happens when the
variable vector x( kr), where kr is a switching instant, does not satisfy the algebraic equation
required in the subsequent subsystem. In order to exclude this possibility, Ref. [19] proposed
an additional condition involving consistency projectors. Here, as in most of the literature,
we assume for simplicity that there is no impulse occurring with the variable (state) vector at
every switching instant, and call such kind of switching impulse-free.
Definition 3: Given a switching sequence, the switched system (3.1) with w = 0 is said to
be stable if starting from any initial value the system’s trajectories converge to the origin
exponentially, and the switched system is said to have L2 gain less than γ if the condition
(2.6) is satisfied for any integer k > 0.
In the end of this section, we state two analysis problems, which will be dealt with in Section
4 and 5, respectively.
Stability Analysis Problem: Assume that all the descriptor subsystems in (3.1) are stable.
Establish the condition under which the switched system is stable under impulse-free
arbitrary switching.
L2 Gain Analysis Problem: Assume that all the descriptor subsystems in (3.1) are stable and
have L2 gain less than γ. Establish the condition under which the switched system is also
stable and has L2 gain less than γ under impulse-free arbitrary switching.
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Remark 1: There is a tacit assumption in the switched system (3.1) that the descriptor matrix
E is the same in all the subsystems. Theoretically, this assumption is restrictive at present.
However, as also discussed in [17, 18], the above problem settings and the results later can
be applied to switching control problems for linear descriptor systems. This is the main
motivation that we consider the same descriptor matrix E in the switched system. For
example, if for a single descriptor system Ex( k + 1) = Ax( k) + Bu( k) where u( k) is the control input, we have designed two stabilizing descriptor variable feedbacks u = K 1 x, u = K 2 x, and
furthermore the switched system composed of the descriptor subsystems characterized by
( E, A + BK 1) and ( E, A + BK 2) are stable (and have L2 gain less than γ) under impulse-free arbitrary switching, then we can switch arbitrarily between the two controllers and thus can
consider higher control specifications. This kind of requirement is very important when we
want more flexibility for multiple control specifications in real applications.
4. Stability analysis
In this section, we first state and prove the common quadratic Lyapunov function (CQLF)
based stability condition for the switched descriptor system (3.1) (with w = 0), and then
discuss the relation with the existing commutation condition.
4.1 CQLF based stability condition
Theorem 1: The switched system (3.1) (with w = 0) is stable under impulse-free arbitrary
switching if there are nonsingular symmetric matrices Pi ∈ R n× n satisfying for ∀ i ∈ I that
ETPiE ≥ 0
(4.1)
ATP
i
i Ai − ET PiE < 0
(4.2)
and furthermore
ETPiE = ETPjE , ∀ i, j ∈ I, i = j.
(4.3)
Proof: The necessary condition for stability under arbitrary switching is that each subsystem
should be stable. This is guaranteed by the two matrix inequalities (4.1) and (4.2) [20].
Since the rank of E is r, we first find nonsingular matrices M and N such that
Ir 0
MEN =
.
(4.4)
0 0
Then, we obtain from (4.1) that
Pi 0
( NTETMT)( M− TP
11
i M−1)( MEN) =
≥ 0 ,
(4.5)
0 0
where
Pi
Pi
M− TP
11
12
i M−1 =
.
(4.6)
( Pi ) T Pi
12
22
Since Pi (and thus M− TPiM−1) is symmetric and nonsingular, we obtain Pi > 0.
11
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Discrete Time Systems
Again, we obtain from (4.3) that
( NTETMT)( M− TPiM−1)( MEN) = ( NTETMT)( M− TPjM−1)( MEN) ,
(4.7)
and thus
Pi 0
j
11
= P 0
11
(4.8)
0 0
0 0
j
which leads to Pi = P , ∀ i, j ∈ I. From now on, we let Pi = P
11
11
11
11 for notation simplicity.
Next, let
¯
Ai
¯
Ai
MA
11
12
i N =
(4.9)
¯
Ai
¯
Ai
21
22
and substitute it into the equivalent inequality of (4.2) as
( NT ATiMT)( M− TPiM−1)( MAiN) − ( NTETMT)( M− TPiM−1)( MEN) < 0
(4.10)
to reach
Λ11 Λ12 < 0,
(4.11)
Λ T Λ
12
22
where
Λ
¯
¯
¯
11 = ( ¯
Ai ) TP Ai − P
) T( Pi ) T ¯ Ai + ( ¯ Ai ) TPi Ai + ( ¯ Ai ) TPi Ai
11
11
11
11 + ( ¯
Ai 21
12
11
11
12
21
21
22
21
Λ
¯
¯
¯
12 = ( ¯
Ai ) TP Ai + ( ¯
Ai ) TPi Ai + ( ¯
Ai ) T( Pi ) T ¯
Ai + ( ¯
Ai ) TPi Ai
(4.12)
11
11
12
11
12
22
21
12
12
21
22
22
Λ
¯
¯
¯
22 = ( ¯
Ai ) TP Ai + ( ¯
Ai ) T( Pi ) T ¯
Ai + ( ¯
Ai ) TPi Ai + ( ¯
Ai ) TPi Ai
12
11
12
22
12
12
12
12
22
22
22
22 .
At this point, we declare ¯
Ai is nonsingular from Λ
22
22 < 0. Otherwise, there is a nonzero
vector v such that ¯
Ai 22 v = 0. Then, vTΛ22 v < 0. However, by simple calculation,
vTΛ
¯
22 v = vT ( ¯
Ai ) T
12
P 11 Ai 12 v ≥ 0
(4.13)
since P 11 is positive definite. This results in a contradiction.
I −( ¯
Ai ) T( ¯
Ai )− T
Multiplying the left side of (4.11) by the nonsingular matrix
21
22
and the
0
I
right side by its transpose, we obtain
( ˜ Ai ) TP ˜ Ai − P
11
11
11
11
∗
< 0 ,
(4.14)
(∗) T
Λ22
where ˜
Ai = ¯
Ai − ¯
Ai ( ¯
Ai )−1 ¯
Ai .
11
11
12
22
21
With the same nonsingular transformation ¯ x( k) = N−1 x( k) = [ ¯ xT( k) ¯ xT( k)] T, ¯ x 1
2
1 ( k) ∈ R r, all
the descriptor subsystems in (3.1) take the form of
¯ x 1( k + 1) = ¯
Ai ¯ x
¯ x
11 1 ( k) + ¯
Ai 12 2( k)
(4.15)
0 = ¯
Ai ¯ x
¯ x
21 1 ( k) + ¯
Ai 22 2( k) ,
Stability and L Gain Analysis of Switched Linear Discrete-Time Descriptor Systems
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which is equivalent to
¯ x 1( k + 1) = ˜
Ai 11 ¯ x 1( k)
(4.16)
with ¯ x 2( k) = −( ¯
Ai )−1 ¯
¯
22
Ai x
21 1 ( k). It is seen from (4.14) that
( ˜ Ai ) T
˜
−
11
P 11 Ai 11
P 11 < 0 ,
(4.17)
which means that all ˜
Ai ’s are Schur stable, and a common positive definite matrix P
11
11 exists
for stability of all the subsystems in (4.16). Therefore, ¯ x 1( k) converges to zero exponentially
under impulse-free arbitrary switching. The ¯ x 2( k) part is dominated by ¯ x 1( k) and thus also
converges to zero exponentially. This completes the proof.
Remark 2: When E = I and all the subsystems are Schur stable, the condition of Theorem
1 actually requires a common positive definite matrix P satisfying ATPA
i
i − P < 0 for ∀ i ∈
I, which is exactly the existing stability condition for switched linear systems composed of
x( k + 1) = Aix( k) under arbitrary switching [12]. Thus, Theorem 1 is an extension of the
existing result for switched linear state space subsystems in discrete-time domain.
Remark 3: It can be seen from the proof of Theorem 1 that ¯ xT P
1 11 ¯
x 1 is a common quadratic
Lyapunov function for all the subsystems (4.16). Since the exponential convergence of ¯ x 1
results in that of ¯ x 2, we can regard ¯ xTP
1 11 ¯
x 1 as a common quadratic Lyapunov function for the
whole switched system. In fact, this is rationalized by the following equation.
xT ETPiEx = ( N−1 x) T( MEN) T( M− TPi M−1)( MEN)( N−1 x)
T
¯ x
I
P
I
¯ x
=