INTERNATIONAL JOURNAL OF °c 2008 Institute for ......2008/02/09 · INTERNATIONAL JOURNAL OF c 2008 Institute for Scientiﬂc INFORMATION AND SYSTEMS SCIENCES Computing and Information

INTERNATIONAL JOURNAL OF c© 2008 Institute for ScientificINFORMATION AND SYSTEMS SCIENCES Computing and InformationVolume 4, Number 2, Pages 278–300

STABILITY IN THE SENSE OF LYAPUNOV OF GENERALIZEDSTATE SPACE TIME DELAYED SYSTEM: A GEOMETRIC

APPROACH

DRAGUTIN LJ. DEBELJKOVIC, SRETEN B. STOJANOVIC, STEVAN A. MILINKOVIC, A.

JACIC, AND NEMANJA S. VISNJIC AND MILMIR R. PJESCIC

Abstract. This paper give sufficient conditions for the stability of linear

singular continuous and descriptor time delay systems of the form Ex (t) =

A0x (t) + A1x (t− τ) and Ex (k) = A0x (k) + A1x (k − 1), respectively. These

new, delay-independent conditions are derived using approach based on Lya-

punov’s direct method. Approach that has been applied is based on crucial

idea presented in paper of Owens, Debeljkovic (1985). Numerical examples

have been working out to show the applicability of results derived.

Key Words. Singular continuous time delay systems, Descriptor discrete time

delay systems, Lyapunov stability

1. Introduction

It should be noticed that in some systems we must consider their character ofdynamic and static state at the same time. Generalized state space systems (alsoreferred to as degenerate, singular, descriptor, generalized, differential - algebraicsystems or semi - state) are those the dynamics of which are governed by a mixtureof algebraic and differential (difference) equations.Recently many scholars have paid much attention to singular and descriptor sys-tems and have obtained many good consequences. The complex nature of singularand discrete systems causes many difficulties in the analytical and numerical treat-ment of such systems, particularly when there is a need for their control.The problem of investigation of time delay systems has been exploited over manyyears. Time delay is very often encountered in various technical systems, such aselectric, pneumatic and hydraulic networks, chemical processes, long transmissionlines, etc.The existence of pure time lag, regardless if it is present in the control or/and thestate, may cause undesirable system transient response, or even instability.Consequently, the problem of stability analysis for this class of systems has beenone of the main interests for many researchers. In general, the introduction of timedelay factors makes the analysis much more complicated.We must emphasize that there are a lot of systems that have the phenomena of timedelay and singular simultaneously, we call such systems as the singular differential(difference) systems with time delay. These systems have many special characters.If we want to describe them more exactly, to design them more accurately and tocontrol them more effectively, we must paid tremendous endeavor to investigatethem, but that is obviously very difficult work.

278

STABILITY IN THE SENSE OF LYAPUNOV OF GSSTDS: A GEOMETRIC APPROACH 279

In recent references authors had discussed such systems and got some consequences.But in the study of such systems, there are still many problems to be considered.When the general time delay systems are considered, in the existing stability crite-ria, mainly two ways of approach have been adopted.Namely, one direction is to contrive the stability condition which does not includethe information on the delay, and the other is the method which takes it into ac-count. The former case is often called the delay - independent criteria and generallyprovides simple algebraic conditions.In that sense the question of their stability deserves great attention.In the short overview, that follows, we shall be familiar only with results achievedin the area of Lyapunov stability of linear, both continuous and discrete descriptortime delay systems (LCSTDS) and (LDDTDS), respectively. In that sense we willnot discuss contributions presented in papers concerned with problem of robuststability, stabilization of this class of systems with parameter uncertainty, see thelist of references, as well as with other questions in connection with stability of(LCSTDS) or (LDDTDS) being necessarily, transformed by Lyapunov - Krasovskifunctional, to the state space model in the form of differential - integral equations,Fridman (2001, 2002).To the best of our knowledge only one paper has been published on the matterof Lyapunov stability of (LDDTDS), so we shall discuss its contribution latter andmore carefully. To be familiar with other problems successfully solved, for this classof systems, check the attached list of references.To the best of our knowledge, some attempts in stability investigation of (LCSTDS)was due to Saric (2001, 2002a, 2002b) where sufficient conditions for convergenceof appropriate fundamental matrix were established.Recently, in the paper of Xu et al. (2002) the problem of robust stability andstabilization for uncertain (LCSTDS) was addressed and necessary and sufficientconditions were obtained in terms of strict LMI. Moreover in the same paper, usingsuitable canonical description of (LCSTDS) a rather simple criteria for asymptoticstability testing was also proposed.Paper of Xu et al. (2004) is addressed to the problems of robust stabilization androbust H∞ control of uncertain (LDDTDS) and parameter uncertainties. The (LD-DTDS) under consideration is not necessarily regular. Moreover presented Lemmaenables one to check systems asymptotic stability using LMI techniques. But stillthere are a lot of problems in solving complex system of inequalities.In our paper we present quite another approach to this problem. Namely, our re-sult is expressed directly in terms of matrices E, A0 and A1 naturally occurringin the system model and avoid the need to introduce any canonical form into thestatement of the Theorem or to solve complex matrix system of inequalities.The geometric theory of consistency leads to the natural class of positive definitequadratic forms on the subspace containing all solutions. This fact makes possiblethe construction of Lyapunov stability theory even for the (LCSTDS) or (LDDTDS)in that sense that asymptotic stability is equivalent to the existence of symmetric,positive definite solutions to a weak form of Lyapunov matrix equation incorporat-ing condition which refer to time delay term.In the descriptor discrete case, the concept of of smoothness has little meaningbut the idea of consistent initial conditions being these initial conditions x0 thatgenerate solution sequence (x (k) : k ≥ 0) has a physical meaning.

280 DEBELJKOVIC, STOJANOVIC, MILINKOVIC, JACIC, AND VISNJIC AND PJESCIC

The certain aim of this paper is to present a new results concerning asymptoticstability of a particular class of linear descriptor discrete time delay systems.In order to have the best insight in these problems, the first part of paper is de-voted to new contributions in the field of linear continuous singular systems andthe second part is concerned with linear discrete descriptor time delay systems.Our aim is to derive a quite new results concerning asymptotic stability of a par-ticular class of linear singular continuous and discrete descriptor time delay systems.

2. Notations

R Real vector spaceC Complex vector spaceI Unit matrixF = (fij) ∈n×n, real matrixFT Transpose of matrix F

F > 0 Positive definite matrixF ≥ 0 Positive semi definite matrix< (F ) Range of matrix F

(F ) Null space (kernel) of matrix F

λ (F ) Eigenvalue of matrixF

σ()(F ) Singular value of matrix F

‖F‖ Euclidean matrix norm of F

FD Drazin inverse of matrix F

⇒ Follows7→ Such that

3. Linear continuous time delay systems

Generally, the linear singular continuous systems with time delay can be writtenas:

E(t)x(t) = f (t,x(t),x(t− τ),u(t)) , t ≥ 0(3.1)x(t) = ϕ(t), −τ ≤ t ≤ 0

where x(t) ∈ Rn is a state vector, u(t) ∈ Rl is a control vector, E(t) ∈ Rn×n isa singular matrix, ϕ ∈ C = C ([−τ, 0]Rn) is an admissible initial state functional,C = C ([−τ, 0]Rn) is the Banach space of continuous functions mapping the inter-val [−τ, 0] into n with topology of uniform convergence.

3.1. Some Preliminaries. Consider a linear continuous singular system withstate delay, described by

Ex (t) = A0x (t) + A1x(t− τ),(3.2)

with known compatible vector valued function of initial conditions

x(t) = ϕ(t), −τ ≤ t ≤ 0,(3.3)


where A0 and A1 are constant matrices of appropriate dimensions.Moreover we shall assume that rankE = r < n.

Definition 3.1. The matrix pair (E, A0) is said to be regular if det (sE −A0) isnot identically zero, Xu et al. (2002).

Definition 3.2. The matrix pair (E, A0) is said to be impulse free ifdeg (det(sE −A0)) = rang E, Xu et al. (2002).The linear continuous singular time delay system (3.2-3.3) may have an impul-sive solution, however, the regularity and the absence of impulses of the matrixpair (E,A0) ensure the existence and uniqueness of an impulse free solution to thesystem under consideration, which is defined in the following Lemma.

Lemma 3.1. Suppose that the matrix pair (E, A0) is regular and impulsive freeand unique on [0,∞), Xu et al (2002).

Necessity for system stability investigation makes need for establishing a properstability definition. So one can has:

Definition 3.3. a) Linear continuous singular time delay system, (3.2-3.3) is saidto be regular and impulsive free if the matrix pair (E, A0) is regular and impulsivefree.b) Linear continuous singular time delay system, (3.2-3.3), is said to be stable iffor any ε > 0 there exist a scalarδ (ε) > 0 such that, for any compatible initialconditions ϕ(t), satisfying condition: sup

−τ ≤ t≤ 0‖ϕ(t) ‖ ≤ δ(ε), the solution x(t) of

system (3.2-3.3) satisfies ‖x(t)‖ ≤ ε, ∀t ≥ 0.Moreover if lim

t→∞‖x(t)‖ → 0, system is said to be asymptotically stable, Xu et al

(2002).

3.2. Some previous result. Let us consider the case when the subspace of con-sistent initial conditions for singular time delay and singular nondelay system co-incide.

3.2.1. Pandolfi’s approach. Our result is stated as follows:

Theorem 3.1. Suppose that the system matrix A0 is nonsingular., e.i.detA0 6= 0.Then we can consider system (3.2) with known compatible vector valued functionof initial conditions and we shall assume that rank E0 = r < n.Matrix E0 is defined in the following wayE0 = A−1

0 E.The system (3.2-3.3) is asymptotically stable, independent of delay, if

‖A1‖ < σmin

(Q

12

)σ−1

max

(Q− 1

2 ET0 P

),(3.4)

and if there exist:(i) (n× n) matrix P , being the solution of Lyapunov matrix:

ET0 P + PE0 = −2IΩ,(3.5)

with the following properties:a)

P = PT(3.6)


b)

Pq(t) = 0, q(t) ∈ Λ(3.7)

c)

qT (t)Pq(t) > 0, q(t) 6= 0, q(t) ∈ Ω,(3.8)

where:

Ω =(I − EED

),(3.9)

Λ =(EED

),(3.10)

with matrix IΩ representing generalized operator on Rn and identity matrix onsubspace Ω and zero operator on subspace Λ and matrix Q being any positive definitematrix.Moreover matrix P is symmetric and positive definite on the subspace of consistentinitial conditions.Here σmax(·) and σmin( · ) have the same meaning as in the previous section.

Proof. For the sake of brevity, the proof is omitted here and can be found inDebeljkovic et al. (2005.c, 2006.a).

3.3. Main result.

3.3.1. Owens-Debeljkovic approach.

Theorem 3.2. Suppose that the matrix pair (E, A0) is regular with system matrixA0 being nonsingular., e.i. detA0 6= 0.The system (3.2-3.3) is asymptotically stable, independent of delay, if there exista positive definite matrix P , being the solution of Lyapunov matrix equation

AT0 PE + ET PA0 = −2 (S + Q) ,(3.11)

with matrices Q = QT > 0 and S = ST , such that:

xT (t) (S + Q)x(t) > 0, ∀x(t) ∈ Wk∗\ 0 ,(3.12)

is positive definite quadratic form on Wk∗\ 0, Wk∗ being the subspace of consis-tent initial conditions, and if the following condition is satisfied:

‖A1‖ < σmin

(Q

12

)σ−1

max

(Q− 1

2 ET P

),(3.13)

Here σmax(·) and σmin(·) are maximum and minimum singular values of matrix( · ),respectively.

Proof. Let us consider functional

V (x (t)) = xT (t)ET PEx (t) +

t∫

t−τ

xT (γ)Qx (γ) dγ,(3.14)

Note that and Lemma A1 and Theorem A1 indicates that

V (x (t)) = xT (t) ET PEx (t) ,(3.15)

is positive quadratic form on Wk∗ , and it is obvious that all smooth solutions x(t)evolve in Wk∗ , so V (x (t)) can be used as a Lyapunov function for the system underconsideration, Owens, Debeljkovic (1985).


It will be shown that the same argument can be used to declare the same propertyof another quadratic form present in (3.14).Clearly, using the equation of motion of (3.2-3.3), we have

V (x (t)) = xT (t)(AT

0 PE + ET PA0 + Q)x (t)

+2xT (t)(ET PA1

)x (t− τ)− xT (t− τ) Qx (t− τ) ,

(3.16)

and after some manipulations, yields to

V (x (t)) = xT (t)(AT

0 PE + ET PA0 + 2Q + 2S)x (t)

+2xT (t)(ET PA1

)x (t− τ)

−xT (t)Qx (t)− xT (t)Sx (t)− xT (t− τ)Qx (t− τ)(3.17)

From (3.16) and the fact that the choice of matrix S, can be done, such that

xT (t)Sx (t) ≥ 0, ∀x (t) ∈ Wk∗\ 0 ,(3.18)

one can have the following result

V (x (t)) ≤ 2xT (t)(ET PA1

)x (t− τ)

−xT (t) Qx (t)− xT (t− τ) Qx (t− τ)(3.19)

and based on well known inequality.

2xT (t)ET PA1x (t− τ) = 2xT (t)(ET PA1Q

− 12 Q

12

)x (t− τ)

≤ xT (t)ET PA1Q−1AT

1 PET x (t) + xT (t− τ)Qx (t− τ),(3.20)

and by substituting into (3.19), it yields

V (x (t)) ≤ −xT (t) Qx (t) + xT (t)ET PA1Q−1AT

1 PEx (t) ,(3.21)

or

V (x (t)) ≤ −xT (t) Q12 Γ∗Q

12 x (t) ,(3.22)

with matrix Γ∗ defined by

Γ∗ =(I −Q−

12 ET PA1Q

− 12 Q− 1

2 AT1 PEQ−

12

).(3.23)

V (x (t)) is negative definite if

1− λmax

(Q−

12 ET PA1Q

− 12 Q−

12 AT

1 PEQ−12

)> 0,(3.24)

which is satisfied if

1− σ2max

(Q−

12 ET PA1Q

− 12

)> 0.(3.25)

Using the properties of the singular matrix values, Amir-Moez (1956), the condition(3.25) holds if

1− σ2max

(Q−

12 ET P

)σ2

max

(A1Q

− 12

)> 0,(3.26)


1−‖A 1‖ 2

σ2max

(Q−

12 ET P

)

σ2min

(Ω

12

) > 0,(3.27)

what completes Proof, Debeljkovic et al. (2007). Q.E.D. ¤


Remark 3.1. Equations (3.11-3.12) are, in modify form, taken from Owens, De-beljkovic (1985).

Remark 3.2. If the system under consideration is just ordinary time delay, e.g.E = I, we have result identical to that presented in Tissir, Hmamed (1996).

Remark 3.3. Let us discuss first the case when the time delay is absent.Then the singular (weak) Lyapunov matrix equation (3.11) is natural generalizationof classical Lyapunov theory.In particulara) If E is nonsingular matrix, then the system is asymptotically stable if and onlyif A = E−1A0 Hurwitz matrix.Equation (3.11) can be written in the form

AT ET PE + ET PEA = −Q(3.28)

with matrix Q being symmetric and positive definite, in whole state space, sincethen Wk∗ = Wk∗ = < (

Ek∗)

= Rn.In this circumstances ET PE is a Lyapunov function for the system.b) The matrix A0 by necessity is nonsingular and hence the system has the form

E0x (t) = x (t) , x (0) = x0 .(3.29)

Then for this system (3.2-3.3) to be stable must hold also, and has familiar Lya-punov structure

ET0 P + PE0 = −Q,(3.30)

where Q is symmetric matrix but only required to be positive definite on Wk∗ .

Remark 3.4. There is no need for system, given (3.2-3.3), to posses propertiesgiven in 3.2, since this is obviously guaranteed by demand that all smooth solutionsx (t) evolve inWk∗ .

In what follows we give an example to show the effectiveness of proposed method.

Example 3.1. Consider the linear continuous singular time delay system withmatrices as follows:

E =

1 0 00 1 00 0 0

, A 0 =

−1 0 00 −1 00 −1 −1

, A 1 =

0, 1 0 00 0 00 0 0

Based on the above presented procedure, one can easily find the following data,Debeljkovic et. al.(1996.b):

E = (λE + A0)−1|λ=0 · E = E = A−1

0 E =

−1 0 00 −1 00 1 0

, ⇒

ED =

−1 0 00 −1 00 1 0

(I − EED

)=

(I − EED

)x0 =

0 0 00 0 00 1 1

x0 = 0, ⇒


(I − EED

)= Wk∗ = x : x1 ∈, x2 ∈, x2 = −x3 .

detA 0 6= 0, ∃λ 7→ det (λE −A 0) 6= 0, rang E = 2, deg det (sE −A 0) = 2 .

One can adopt

Q =

1 0 00 1 00 0 1

= QT > 0, S = ST =

0 1 11 0 −11 −1 −1

,

(S + Q) =

1 1 11 1 −11 −1 0

⇒

xT (t)Sx (t) =[

x1 x2 x3

]

0 1 11 0 −11 −1 −1

x1

x2

x3

=(2x1x2 + 2x1x3 − 2x2x3 − x2

3

)x2=−x3

=(2x1 (x2 + x3) + 2x2

3 − x23

)x2=−x3

= x23 > 0, ∀x (t) ∈ Wk∗\ 0

xT (t)Qx (t) =[

x1 x2 x3

]

1 0 00 1 00 0 1

x1

x2

x3

=(x2

1 + x22 + x2

3

)x2=−x3

= x21 + 2x2

2 > 0, ∀x (t) ∈ Wk∗\ 0Moreover we have

xT (t) Qx (t) = x21 + x2

2 + x23 > 0, ∀x (t) ∈ Wk∗\ 0 , det Q = 1 6= 0.

and

xT (t) (S + Q)x (t) =[

x1 x2 x3

]

1 1 11 1 −11 −1 0

x1

x2

x3

=(x2

1 + x22 + 2 (x2 + x3) + 2x2

2

)x2=−x3

= x21 + 3x2

2 > 0,

x (t) ∈ Wk∗\ 0Also, one can compute

1 0 00 1 00 1 1

p11 p12 p13

p12 p22 p23

p13 p23 p33

−1 0 00 −1 00 0 0

+

+

−1 0 00 −1 00 0 0

p11 p12 p13

p12 p22 p23

p13 p23 p33

1 0 00 1 10 0 1

= −2

1 1 11 1 −11 −1 0

P =

1 0 20 3 −22 −2 p33

, ⇒ ∆3 (p33) > 0, ⇒ p33 >

163

⇒ P = PT > 0.


Generally we can adopt

P =

1 0 20 3 −22 −2 6

= PT > 0.

Finally, we have to check condition (3.13)

‖A 1‖ = 0, 10 σ Q = 1, 1, 1 ,

Q12 =

1 0 00 1 00 0 1

,

Q−12 =

1 0 00 1 00 0 1

, Q− 1

2 ET P =

−1 − 12 1

− 12 −3 2

0 0 0

σmin

(Q

12

)= 1 , σmax =

(Q−

12 ET P

)= 3.82

0.10 = ‖A 1‖ <σmin

(Q

12

)

σmax

(Q−

12 ET P

) < 0.26,

so, the system under consideration is asymptotically stable.For the sake of further investigation let us, for the previous case, adopt situationwhen E = I.Then one can calculate

σ A 0 = λ 1, λ 2, λ 3 = −1, −1, −1 , λmax = −1.

‖A0‖ = σmax (A0) =√

λmax

(AT

0 A0

),

A0 + AT0 =

−2 0 00 −2 −10 −1 −2

,

λi

(A0 + AT

0

)= λ1, λ2, λ3 = −1, −2, −3 ,

µ (A0) = 12λmax

(A0 + AT

0

)= − 1

2 .

Following Mori et al. (1981):

µ (A0) + ‖A1‖ < 0, 7→ −0.5 + 0.10 = −0.4 < 0.

asymptotic stability of the non-delay system under consideration is confirmed.Moreover we have

xT (t)ET PEx (t) =[

x1 x2 x3

]

1 0 00 3 00 0 0

x1

x2

x3

=

=(x2

1 + 3x22

)x2=−x3

> 0, ∀x (t) ∈ Wk∗\ 0,


so V (x(t)) can be used as a Lyapunov function for the system (3.2-3.3).A simple calculation, also shows:

f(s) = (s + 1) (s + 1− 0.1e−s) = 0i) (s + 1) = 0 ⇒ s = −1,ii) (s + 1− 0.1e−s) = 0 ⇒ t = s + 1, t− 0.1e−t = 0, tet = e/10

⇒ t = lambert (e/10)⇒ s = t− 1 = lambert (e/10)− 1,

t = σ + jω = wnejϕ,tet = wnejϕeσ+jω = wneσej(ϕ+ω) = e/10ej2kπ√

σ2 + ω2eσ = e/10, ϕ + ω = 2kπ, k = 0,±1,±2 · · ·k = 0 ⇒ t = 0.2185

⇒ s = t− 1 = −0.7815k = ±1 ⇒ t = −2.9173± j4.0932

⇒ s = t− 1 = −3.9173± j4.0932k = ±2 ⇒ t = −3.7267± j10.6592

⇒ s = t− 1 = −4.7267± j10.6592

and so on.

4. Linear discrete descriptor systems

4.1. Some preliminaries. (LDDTDS) is described by

E x(k + 1) = A0x(k) + A1x(k − 1),(4.1)

where x (k) ∈ Rn is a state vector.The matrix E ∈ Rn×n is a necessarily singular matrix, with property rank E = r <n and with matrices A0 and A1 of appropriate dimensions.For a (LDDTDS), (4.1), we present the following definitions taken from, Xu et al.(2004).

Definition 4.1. The (LDDTDS) is said to be regular if det(z2E − zA0 −A1

), is

not identically zero.

Definition 4.2. The (LDDTDS) is said to be causal if is regular and

deg(zn det

(zE −A 0 − z−1A 1

))= n + rang E.

Definition 4.3. The (LDDTDS) is said to be stable if it is regular andρ (E, A0, A1) ⊂ D (0, 1), where

ρ (E, A0, A 1) =z | det

(z2E − zA0 −A1

)= 0

.

Definition 4.4. The (LDDTDS) is said to be admissible if it is regular, causal andstable.

Lemma 4.1. ?? The (LDDTDS) is admissible if there exist a matrix Q > 0 andan invertible symmetric matrix P such thati) ET PE ≥ 0ii) AT

0PA 0 − ET PE + AT0PA 1

(Q−AT

1PA 1

)−1AT

1PA 0 + Q < 0iii) Q−AT

1PA 1 > 0,Xu et al. (2004).

Proof. See, Xu et al. (2004).


4.2. Main results.

Theorem 4.1. Suppose that (LDDTDS) is regular and causal with system matrixA0 being nonsingular, e.i. det A0 6= 0.Moreover, suppose matrix

(Q−AT

1 PA1

)is regular.

The system (4.1) is asymptotically stable, independent of delay, if

‖A1‖ <σmin

((Q−AT

1 PA1

) 12)

σmax

(Q− 1

2 AT0 P

) ,(4.2)

and if there exist a symmetric positive definite matrix P , being the solution ofdiscrete Lyapunov matrix equation

AT0 PA0 − ET PE = −2 (S + Q) ,(4.3)

with matrices Q = QT > 0 and S = ST , such that

xT (k) (S + Q)x (k) > 0, ∀x (k) ∈ W dk∗\ 0 ,(4.4)

is positive definite quadratic form on W dk∗\ 0, W d

k∗ being the subspace of consistentinitial conditions for both time delay and non-time delay discrete descriptor system.


V (x (k)) = xT (k)ET PEx (k) + xT (k − 1)Qx (k − 1) .(4.5)

with matrices P = PT > 0 and Q = QT > 0.

Remark 4.1. Equations (4.3 - 4.4) are, in modify form, taken from Owens, De-beljkovic (1985).

Note that Lemma B1 and Theorem B1 indicates that

V (x(k)) = xT (k)ET PEx(k),(4.6)

is positive quadratic form on W dk∗ , and it is obvious that all solutions x (k) evolve

in W dk∗ , so V (x (k)) can be used as a Lyapunov function for the system under

consideration, Owens, Debeljkovic (1985).It will be shown that the same argument can be used to declare the same propertyof another quadratic form present in (4.5).Clearly, using the equation of motion of (4.1), we have

∆V (x (k)) = V (x (k + 1))− V (x (k)) == xT (k)

(AT

0 PA0 − ET PE + Q)x (k)

+2xT (k)(AT

0 PA1

)x (k − 1)− xT (k − 1)Qx (k − 1)

= xT (k)(AT

0 PA0 − ET PE + 2Q + 2S)x (k)

−xT (k)Qx (k) − 2xT (k)Sx (k)++2xT (k)

(AT

0 PA1

)x (k − 1) − xT (k − 1)

(Q−AT

1PA 1

)x (k − 1)

(4.7)

From (4.3) and the inequality:

2xT (k) AT0 PA 1x (k − 1) =

= 2xT (k)(AT

0 PA 1

(Q−AT

1PA 1

)− 12

(Q−AT

1PA 1

) 12)x (k − 1)

≤ xT (k)AT0 PA 1

(Q−AT

1PA 1

)− 12

(Q−AT

1PA 1

)− 12 AT

1 PA0x (k)

+xT (k − 1)(Q−AT

1PA 1

) 12

(Q−AT

1PA 1

) 12 x (k − 1)

(4.8)


one can obtain

∆V (x (k)) = −xT (k)Qx (k)− xT (k)Sx (k) + xT (k)××

(AT

0 PA1

(Q−AT

1PA 1

)− 12

(Q−AT

1PA 1

)− 12 AT

1 PA0

)x (k)

≤ −xT (k)Sx (k)

−xT (k)Q12

(I −Q−

12 AT

0 PA1×× (

Q−AT1 PA1

)−1AT

1 PA0Q− 1

2

)Q

12 x (k)

(4.9)

From the fact that the choice of matrix S, can be done, such that

xT (k) Sx (k) ≥ 0, ∀x (k) ∈ W dk∗ \ 0 ,(4.10)

and after some manipulations, (4.9) yields to

∆V (x (k)) ≤ −xT (k) Q12 Π Q

12 x (k) ,(4.11)

with matrix Π defined by

Π =(I −Q−

12 AT

0PA 1Q− 1

2 Q− 12 AT

1PA 0Q− 1

2

).(4.12)

and following the procedure presented in Tissir, Hmamed (1996), V (x (k)) is neg-ative definite if

1− λmax

(Q−

12 AT

0PA 1

(Q−AT

1PA 1

)− 12 ×

× (Q−AT

1PA 1

)− 12 AT

1PA 0Q− 1

2

)> 0,(4.13)


1− σ2max

(Q−

12 AT

0 PA 1

(Q−AT

1 PA 1

)Q−

12

)> 0.(4.14)

Using the properties of the singular matrix values, Amir-Moez (1956), the condition(4.14) holds if

1− σ2max

(Q−

12 AT

0 P)× σ2

max

(A1

(Q−AT

1 PA1

)Q−

12

)> 0,(4.15)


1−‖A 1‖ 2

σ2max

(Q−

12 AT

0 P)

σ2min

((Q−AT

1 PA 1

)− 12) > 0,(4.16)

what completes proof. Q.E.D. ¤

In the sequel we give an example to show the effectiveness of proposed method.

Example 4.1. Consider the linear discrete descriptor time delay system with ma-trices as follows:

E =

2 0 00 0 00 0 0

, A 0 =

0.10 0 01 1 00 0 1

, A 1 =

0.10 0 00 0 00 0 0

.


Based on the above presented procedure, one can easily find the following data,Debeljkovic et. al. (1996.b):

E = (zE + A0) −1|z=0 · E = E = A−1

0 E =

−20 0 020 0 00 0 0

, ⇒

⇒ ED =

−0.05 0 00, 05 0 0

0 0 0

(I − EED

)=

(I − EED

)x0 =

0 0 01 1 00 0 1

x0 = 0, ⇒

(I − EED

)= x ∈n: x10 + x20 = 0 ∧ x30 = 0 .

det A0 6= 0, rang E = 1

det(z2E − zA0 −A1

)= z

(2z2 − 0.01z − 1

)(z − 0.10) 6= 0

deg(zn det

(zE −A0 − z−1A1

))= n + rang E ⇒

n = 3, 7→ deg(z4

)= 4, n + rank E = 3 + 1 = 4

One can adopt

Q =

1 1 01 2 00 0 1

= QT > 0, S = ST =

0 −2 0−2 −3 00 0 −2

,

(S + Q) =

1 −1 0−1 −1 00 0 −1

⇒

xT (k)Sx (k) =[

x1 x2 x3

]

0 −2 0−2 −3 00 0 −2

x1

x2

x3

=(−2x1x2 − 2x1x2 − 3x2

2 − 2x23

)x1=−x2x3=0

=

= x21 > 0, ∀x (k) ∈ W d

k∗\ 0

.

xT (k)Qx (k) =[

x1 x2 x3

]

1 1 01 2 00 0 1

x1

x2

x3

=((x1 + x2)

2 + x22 + x2

3

)x1=−x2x3=0

=

= x22 > 0, ∀x (k) ∈ W d

k∗\ 0Moreover we have

xT (k) Qx (k) = (x1 + x2)2 + x2

2 + x23 > 0, ∀x (k) ∈n,

detQ = 1 6= 0,


and

xT (k) (S + Q)x (k) =[

x1 x2 x3

]

1 −1 0−1 −1 00 0 −1

x1

x2

x3

=(x2

1 + 2x22 − x2

2 − x23

)x1=−x2x3=0

=

= x21 + x2

2 > 0, ∀x (k) ∈ W dk∗\ 0

Also, one can compute

0.10 1 00 1 00 0 1

p11 p12 p13

p12 p22 p23

p13 p23 p33

0.10 0 01 1 00 0 1

−

−

2 0 00 0 00 0 0

p11 p12 p13

p12 p22 p23

p13 p23 p33

2 0 00 0 00 0 0

=

= −2 (S + Q) = −2

1 −1 0−1 −1 00 0 −1

⇒

P =

p11 p12 p13

p12 p22 p23

p13 p23 p33

=

=

− 3.99p11 + 0.2p12 + p22 0.1p12 + p22 0.1p13 + p23

0.1p12 + p22 p22 p23

0.1p13 + p23 p23 p33

with solution

P =

0.501 0 00 2 00 0 6

= PT > 0.

Moreover we have

xT (k)ET PEx (k) =[

x1 x2 x3

]

0.501 0 00 2 00 0 2

x1

x2

x3

=(0.501x2

1 + 2x22 + 2x2

3

)x1=−x2x3=0

=

=(2.501x2

1

)x1=−x2x3=0

> 0, ∀x (k) ∈ W dk∗\ 0

so V (x (k)) can be used as a Lyapunov function for the system (4.1).Finally, we have to check condition (4.3)

‖A 1‖ = 0.10 σ Q = 0.382, 1.00, 2.618

Ω = Q−AT1PA 1 =

0.995 1 01 2 00 0 1

σmin

((Q−AT

1PA 1

) 12)

= 0.618 , σmax =(Q−

12 AT

0 P)

= 1.42

0, 10 = ‖A 1‖ <σmin

((Q−AT

1PA 1

) 12)

σmax(Q− 12 AT

0 P )= 0.4361,


so, the system under consideration is asymptotically stable.Moreover it should be noticed that there are eight different solutions for Q

12 , Q−

12

and of course for expression Q−12 AT

0 P .But solutions for σmin

(Q

12

)and σmax =

(Q− 1

2 AT0 P

)are unique in all cases.

In particular case, when E = I, we have result identical to that presented in De-beljkovic et al. (2005.c).

Now we investigate the general case, namely when one can let that basic systemmatrix is singular, e.g. det A0 = 0.Note that this is impossible case for linear continuous singular system see, 3.1 and3.2.

Theorem 4.2. Suppose that (LDDTDS) is regular and causal.Moreover, suppose matrix

(Qλ −AT

1 PλA1

)is regular, with Qλ = QT

λ > 0.The system (4.1) is asymptotically stable, independent of delay, if:

‖A1‖ <σmin

((Q−AT

1 PλA1

) 12)

σmax

(Q− 1

2λ (A0 − λE)T

Pλ

) ,(4.17)

and if there exist real positive scalar λ∗ > 0 such that for all λ within the range0 < |λ| < λ∗ there exist symmetric positive definite matrix Pλ , being the solutionof discrete Lyapunov matrix equation:

(A 0 − λE)TPλ (A 0 − λE)− ET PλE = −2 (Sλ + Qλ) ,(4.18)

with matrix Sλ = STλ , such that:

xT (k) (Sλ + Qλ)x (k) > 0, ∀x (k) ∈ W dk∗\ 0 ,(4.19)

is positive definite quadratic form on W dk∗\ 0 , W d

k∗ being the subspace of consis-tent initial conditions for both time delay and non-time delay discrete descriptorsystem .Here σmax(·) and σmin(·) are maximum and minimum singular values of matrix ( · ), respectively.


V (x (k)) = xT (k)ET PλEx (k) + xT (k − 1) Qλx (k − 1) .(4.20)

with matrices Pλ = PTλ > 0 and Qλ = QT

λ > 0.

Remark 4.2. Equations (4.18-4.19) are, in modify form, taken from Owens,Debeljkovic (1985).

Note that Lemma B1 and Theorem B1 indicates that:

V (x (k)) = xT (k)ET PλEx (k) ,(4.21)

is positive quadratic form on W dk∗ , and it is obvious that all solutions x (k) evolve

in W dk∗ , so V (x (k)) can be used as a Lyapunov function for the system under

consideration, Owens, Debeljkovic (1985).It will be shown that the same argument can be used to declare the same property


of another quadratic form, present in 4.20.Clearly, using the equation of motion of 4.1, we have

∆V (x (k)) = V (x (k + 1))− V (x (k)) == +xT (k)

((A0 − λE)T

Pλ (A0 − λE))x (k)

−xT (k)(Qλ − ET PλE

)x (k)+

+2xT (k)((A0 − λE)T

PλA 1

)x (k − 1)

−xT (k − 1)(Qλ −AT

1PλA 1

)x (k − 1)

= xT (k)((A0 − λE)T

Pλ (A0 − λE))x (k)

+xT (k)(2Qλ − ET PλE + 2Sλ

)x (k)−

−xT (k)Qλx (k) − 2xT (k)Sλx (k)+2xT (k)

((A0 − λE)T

PλA1

)x (k − 1)

−xT (k − 1)(Qλ −AT

1PλA 1

)x (k − 1)

(4.22)

Using the same procedure, as in the previous case, one can get:(4.23)2xT (k) (A0 − λE)T

PλA 1x (k − 1) =

= 2xT (k)((A0 − λE)T

PλA 1

(Qλ −AT

1PA 1

)− 12

(Qλ −AT

1PλA 1

) 12)x (k − 1)

≤ xT (k) (A0 − λE)TPλA 1

(Qλ −AT

1PλA 1

)− 12

(Qλ −AT

1PλA 1

)− 12 AT

1PλA0x (k)

+xT (k − 1)(Qλ −AT

1PλA 1

) 12

(Qλ −AT

1PλA 1

) 12 x (k − 1)

so that:

(4.24)

∆V (x (k)) = −xT (k) Qλx (k)− xT (k)Sλx (k)

+xT (k)

((A0 − λE)T

PλA1

(Qλ −AT

1PλA 1

)− 12 ×

× (Qλ −AT

1PλA 1

)− 12 AT

1PλA0

)x (k)

≤ −xT (k) Sλx (k)− xT (k)Q12λ×

×(

I −Q− 1

2λ (A0 − λE)T

PλA 1×× (

Q−AT1PλA 1

)−1AT

1Pλ (A0 − λE)Q− 1

2λ

)Q

12λ x (k)

From the fact that the choice of matrix Sλ , can be done, such that:

xT (k)Sλx (k) ≥ 0, ∀x (k) ∈ W dk∗ \ 0 ,(4.25)

and after some manipulations, 4.24 yields to:

∆V (x (k)) ≤ −xT (k) Q12λ Υλ Q

12λ x (k) ,(4.26)

with matrix Υλ defined by:

Υλ =(I −Q

− 12

λ (A0 − λE)TPλA 1Q

− 12

λ Q− 1

2λ AT

1Pλ (A0 − λE)Q− 1

2λ

)(4.27)

and following the procedure presented in Tissir, Hmamed (1996), V (x (k)) is neg-ative definite if

(4.28) 1− λmax

(Q− 1

2λ (A0 − λE)T

PλA1

(Qλ −AT

1PλA 1

)− 12 ×

× (Qλ −AT

1PλA 1

)− 12 AT

1 Pλ (A0 − λE)Q− 1

2λ

)> 0


(4.29) 1− σ2max

(Q− 1

2λ (A0 − λE)T

PλA 1

(Qλ −AT

1 PλA 1

)Q− 1

2λ

)> 0


Using the properties of the singular matrix values, Amir-Moez (1956), the condition4.29 holds if:

1− σ2max

(Q− 1

2λ (A0 − λE)T

Pλ

)×

×σ2max

(A 1

(Qλ −AT

1 PλA 1

)Q− 1

2λ

)> 0

,(4.30)

which is satisfied if:

1−‖A 1‖ 2

σ2max

(Q− 1

2λ (A0 − λE)T

Pλ

)

σ2min

((Qλ −AT

1 PλA 1

)− 12) > 0,(4.31)

what completes proof. Q.E.D. ¤

In particular case, when E = I and det A 0 6= 0 , we have result identical to thatpresented in Debeljkovic et al. (2005.c).In particular case, when det A 0 6= 0 , we have result identical to that presented inDebeljkovic et al. (2006.b).

Example 4.2. Consider the linear discrete descriptor time delay system with ma-trices as follows:

E =

−1 0 01 0 00 0 0

, A 0 =

0 0 0−1 −1 00 0 −1

, A 1 =

110 0 00 0 00 0 0

In comparison with the ??, it should be underlined that here we have fact thatmatrix A 0 singular, e.g. detA0 = 0 .

Based on the above presented procedure, one can easily find the following data,Debeljkovic et. al. (2006.b):

E = (zE + A0) −1|z=1 · E = Ez=1 = I · E =

=

−1 0 01 0 00 0 0

, ⇒ ED =

−1 0 01 0 00 0 0

N(I − EED

)=

(I − EED

)x0 =

0 0 01 1 00 0 1

x0 = 0, ⇒

N(I − EED

)= x ∈ Rn : x10 + x20 = 0 ∧ x30 = 0 .

detA 0 = 0, rang E = 1,

Def. 1

det(z2E − zA 0 −A 1

)= −z

(z2 + 0.10

) 6= 0,det

(zE −A 0 − z−1A 1

)= − (

z − 0.10z

)

Def. 2

deg(zn det

(zE −A0 − z−1A1

))= n + rang E ⇒


n = 3, 7→ deg(z4

)= 4, n + rank E = 3 + 1 = 4

One can adopt:

Qλ =

1 1 01 2 00 0 1

= QT

λ > 0,

Sλ = STλ =

0 −2 0−2 −3 00 0 −2

, Sλ + Qλ =

1 −1 0−1 −1 00 0 −1

xT (k)Sλx (k) =[

x1 x2 x3

]

0 −2 0−2 −3 00 0 −2

x1

x2

x3

=(−2x1x2 − 2x1x2 − 3x2

2 − 2x23

)x1=−x2x3=0

=

= x21 > 0, ∀x (k) ∈ W d

k∗\ 0

.

xT (k) Qλx (k) =[

x1 x2 x3

]

1 1 01 2 00 0 1

x1

x2

x3

=((x1 + x2)

2 + x22 + x2

3

)x1=−x2x3=0

=

= x22 > 0, ∀x (k) ∈ W d

k∗\ 0

,

Moreover we have:

xT (k)Qλx (k) = (x1 + x2)2 + x2

2 + x23 > 0, ∀x (k) ∈ Rn

detQ = 1 6= 0,

and:

xT (k) (Sλ + Qλ)x (k) =[

x1 x2 x3

]

1 −1 0−1 −1 00 0 −1

x1

x2

x3

=(x2

1 + 2x22 − x2

2 − x23

)x1=−x2x3=0

=

= x21 + x2

2 > 0, ∀x (k) ∈ W dk∗\ 0

Also, one can compute:

(A0 − λE)Tλ=2 Pλ (A0 − λE)λ=2 =

=

4p11 − 12p12 + 9p22 3p22 − 2p12 3p23 − 2p13

3p22 − 2p12 p22 p23

3p23 − 2p13 p23 p33

ET PλE =

p11 − 2p12 + p22 0 00 0 00 0 0

,

so finally, we have:


4p11 − 14p12 + 8p22 3p22 − 2p12 3p23 − 2p13

3p22 − 2p12 p22 p23

3p23 − 2p13 p23 p33

=

= −2 (Qλ + Sλ) =

−2 2 02 2 00 0 2

with solution:

Pλ =

103 0 00 2 00 0 2

= PT

λ > 0.

Moreover we have:

xT (k)ET PλEx (k) =[

x1 x2 x3

]

103 0 00 2 00 0 2

x1

x2

x3

=(

1009 x2

1 + 2x22 + 2x2

3

)x1=−x2x3=0

=

=(3.11x2

1

)x1=−x2x3=0

> 0, ∀x (k) ∈ W dk∗\ 0

so V (x(k)) can be used as a Lyapunov function for the system 4.1.Finally, we have to check condition ??

‖A 1‖ = 0.10 σ Qλ = 0.382, 1.00, 2.618

Ωλ = Qλ −AT1PλA 1 =

0.997 1 01 2 00 0 1

σmin

((Qλ −AT

1PλA 1

) 12)

= 0.616 , σmax =(

Q− 1

2λ AT

0 Pλ

)= 1.422

0, 10 = ‖A 1‖ <σmin

((Qλ −AT

1PλA 1

) 12)

σmax

(Q− 1

2λ AT

0 Pλ

)(Q− 1

2λ AT

0 Pλ

) = 0.481,

so, the system under consideration is asymptotically stable.

The same conclusion can be derived directly from the locations of roots of char-acteristic equation.

5. Conclusion

A quite new sufficient delay–independent criteria for asymptotic stability of (LD-DTDS) is presented. In some sense this result may be treated as the further exten-sion of results derived in Debeljkovic et. al (2006.b).

In that sense it seems as the extension of weak Lyapunov equation (4.3) and 4.18to discrete descriptor time delay systems.One should say that this approach gives an insight in the structure of singularcontinuous and discrete descriptor systems under consideration, so it may be called:a geometric approach to the Lyapunov stability of this particular class of systems.In comparison with some other papers on this matter, there is no need for linear


transformations of basic system, as well there is no need of solving the systems ofhigh order linear matrix inequalities.Two numerical examples are presented to show the applicability of results derived

6. Appendix A

The fundamental geometric tool in the characterization of the subspace of con-sistent initial conditions, for linear singular system without delay, is the subspacesequence

W0 = Rn(6.1)...

Wj+1 = A−10 (EWj) , j ≥ 0,(6.2)

where A−10 ( · ) denotes inverse image of ( · ) under the operator A 0.

Lemma 6.1. The subsequence W0, W1, W2, ........ is nested in the sense that:

W0 ⊃ W1 ⊃ W2 ⊃ W3 ⊃ ....(6.3)

Moreover:

N (A) ⊂ Wj , ∀j ≥ 0,(6.4)

and there exist an intager k ≥ 0 , such that:

Wk+j = Wk.(6.5)

Then it is obvious that:

Wk+j = Wk, ∀j ≥ 1.(6.6)

If k∗ is the smallest such integer with this property, then:

Wk ∩ N (E) = 0 , k ≥ k∗,(6.7)

provide that (λE −A 0) is invertible for some λ ∈ R.

Theorem 6.1. Under the conditions of 6.1, x0 is a consistent initial condition forthe system under consideration if and only if x0 ∈ Wk∗ .Moreover x0 generates a unique solution x (t) ∈ Wk∗ , t ≥ 0 , that is real analyticon t : t ≥ 0 .

Proof. See Owens, Debeljkovic (1985). ¤7. Appendix B

The fundamental geometric tool in the characterization of the subspace of con-sistent initial conditions, for linear discrete descriptor system without delay 4.1, isthe subspace sequence

W d0 = Rn,(7.1)

...

W dj+1

= A−10

(EW d

j

), j ≥ 0,(7.2)

where A−10 ( . ) denotes inverse image of ( · ) under the operator A0.


Lemma 7.1. 7.1 is identical to the 6.1.One should, only, change W0 = W d

0 , W1 = W d1 , ......, Wk = W d

k .Final result follows as:

W dk ∩ N (E) = 0 , k ≥ k∗,(7.3)

provide that (zE −A 0) is invertible for some z ∈ C .

Proof. See Owens, Debeljkovic (1985). ¤

Theorem 7.1. Under the conditions of 7.1, x0 is a consistent initial condition forthe system under consideration, e.g. linear discrete singular system without delayif and only if x0 ∈ W d

k∗ .

Moreover x0 generates a discrete solution sequence (x (k) : k ≥ 0) such thatx (k) ∈ W d

k∗ , ∀k ≥ 0 .

Proof. See Owens, Debeljkovic(1985). ¤

8. Appendix C

Definition 8.1. Let E be a square matrix, if there exists a matrix ED satisfying:1. EED = EDE2.

EDEED = ED(8.1)

3. E℘+1ED = E℘

we call the Drazin inverse matrix of matrix E , simply D - inverse matrix.℘ is the index of the matrix E , it is the smallest nonnegative integer which

makes:

rank E℘+1 = rank E℘,(8.2)

be true.

Lemma 8.1. ?? For any square matrix E , its Drazin inverse matrix is existentand unique.

If the Jordan normalized form of E is

E = T

(J 00 N

)T−1(8.3)

then:

ED = T

(J−1 00 0

)T−1(8.4)

Here N is a nilpotent matrix, and T are invertible matrices.


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Email: [email protected]

University of Belgrade, Faculty of Mechanical Eng. Department of Control Eng., 11000 Bel-grade, Serbia

E-mail : [email protected]

University of Belgrade, Faculty of Mechanical Eng Department of Control Eng., 11000 Bel-grade, Serbia

E-mail : [email protected]

Faculty of Computer Science Knez Mihajlova 6, 11000 Belgrade, SerbiaE-mail : [email protected]

High Technical School Nemanjina 2,12000 Pozarevac, SerbiaE-mail : [email protected]

University of Belgrade, Faculty of Mechanical Eng, Department of Control Eng., 11000 Bel-grade, Serbia

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