Switching design for exponential stability of a class of nonlinear hybrid time-delay systems

Nonlinear Analysis: Hybrid Systems 3 (2009) 1–10

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Switching design for exponential stability of a class of nonlinear hybrid time-delay systems V.N. Phat a , T. Botmart b , P. Niamsup b,∗ a

Institute of Mathematics, 18 Hoang Quoc Viet Road, Hanoi 10307, Viet Nam

b

Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand

article

info

Article history: Received 9 September 2008 Accepted 1 October 2008 Keywords: Switching system Exponential stability Nonlinear perturbation Time-varying delay Riccati equation

a b s t r a c t This paper addresses the exponential stability for a class of nonlinear hybrid time-delay systems. The system to be considered is autonomous and the state delay is time-varying. Using the Lyapunov functional approach combined with the Newton–Leibniz formula, neither restriction on the derivative of time-delay function nor bound restriction on nonlinear perturbations is required to design a switching rule for the exponential stability of nonlinear switched systems with time-varying delays. The delay-dependent stability conditions are presented in terms of the solution of algebraic Riccati equations, which allows computing simultaneously the two bounds that characterize the stability rate of the solution. A simple procedure for constructing the switching rule is also presented. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction As an important class of hybrid systems, switched systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models, such as manufacturing, communication networks, automotive engineering control and chemical processes (see, e.g., [7,12,15,16] and the references therein). During the last decades, the stability analysis of switched time-delay systems has attracted a lot of attention [11,13,14,18,21]. The main approach for stability analysis relies on the use of Lyapunov–Krasovskii functionals and LMI approach for constructing a common Lyapunov function [4,9,21]. Under the assumption on commutative system matrices, it was shown in [10,17] that when all subsystems are asymptotically stable, the switching system is asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear systems with time delay has been studied in [20], but the result was limited to symmetric systems. In [14,18], delay-dependent asymptotic stability conditions are extended to discrete-time linear switching time-delay systems. Considering switching systems composed of a finite number of linear point time-delay differential equations, it has been shown in [4,11], that the asymptotic stability may be achieved by using a common Lyapunov function method switching rule. There are some other results concerning the stability for time-delay systems, but most of them provide conditions for linear systems for an arbitrary switching rule. Using the Lie-algebraic approach, switching design for exponential stability was proposed in [2,6,8] for nonlinear hybrid systems. On the other hand, it is worth noting that the existing stability conditions for nonlinear systems must be solved upon a grid of the parameter space, which results in testing a nonlinear Riccati-type equation or a finite number of LMIs. In this case, the results using finite gridding points are unreliable and the numerical complexity of the tests grows rapidly. Therefore, finding new conditions for the exponential stability of nonlinear switching systems with time-varying delay is of interest. In this paper, we study the problem of exponential stability for a class of nonlinear hybrid systems with time-varying delay. Compared with existing results in the literature, the novelty of our results is twofold. First, the state delay is

∗

Corresponding author. E-mail address: [email protected] (P. Niamsup).

1751-570X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2008.10.001

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V.N. Phat et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 1–10

time-varying, neither restriction on the derivative of the time-delay function nor bound restriction on nonlinear perturbations is required to design a switching rule for the exponential stability of the system. Second, the obtained conditions for the exponential stability are delay-dependent and formulated in terms of the solution of standard Riccati differential equations, which allow computing simultaneously the two bounds that characterize the stability rate of the solution. The paper is organized as follows. Section 2 presents notations, definitions and auxiliary propositions required for the proof of the main results. Switching design for the exponential stability of the system with illustrative examples is presented in Section 3. The paper ends with a conclusion followed by cited references. 2. Problem formulation The following notations will be used throughout this paper. R+ denotes the set of all non-negative real numbers; Rn denotes the n-finite-dimensional Euclidean space, with the Euclidean norm k.k and scalar product xT y of two vectors x, y. Rn×m denotes the set of all (n × m)-matrices; λmax (A)(λmin (A), resp.) denotes the maximal number (the minimum number, resp.) of the real part of eigenvalues of A; AT denotes the transpose of the matrix A; Q ≥ 0(Q > 0, resp.) means Q is semi-positive definite (positive definite, resp.), A ≥ B means A − B ≥ 0. Consider a switched nonlinear system with time-varying delay of the form

(Σα )



x˙ (t ) = Aα x(t ) + Dα x(t − h(t )) + fα (x(t ), x(t − h(t ))), x(t ) = φ(t ), t ∈ [−h, 0],

t ∈ R+ ,

(2.1)

where x(t ) ∈ Rn is the state, Aα , Dα ∈ Rn×n are given constant matrices, fα (x, y) : Rn × Rn → Rn is the given nonlinear perturbation; φ(t ) ∈ C ([−h, 0], Rn ) is the initial function with the norm kφk = sups∈[−h,0] kφ(s)k. α(x) : Rn → Ω := {1, 2, . . . , N } is the switching rule, which is a piecewise constant function depending on the state in each time. A switching rule is a rule which determines a switching sequence for a given switching system. Moreover, α(x) = i implies that the system realization is chosen as Σi , i = 1, 2, . . . , N. It is seen that system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state x(t ) hits predefined boundaries; i.e., the switching rule is dependent on the system trajectory. The delay function h(t ) is a continuous function satisfying either (D.1) or (D.2): (D.1)

0 ≤ h( t ) ≤ h,

(D.2)

0 ≤ h( t ) ≤ h,

h˙ (t ) ≤ δ < 1, ∀t ≥ 0, ∀t ≥ 0 . The nonlinear perturbation fi (.), i = 1, 2, . . . , N, satisfies the following condition ∃ai > 0, bi > 0 : kfi (x, y)k ≤ ai kxk + bi kyk, ∀x, y.

(2.2)

The stability problem for switched system (2.1) is to construct a switching rule that makes the system exponentially stable. Definition 2.1. Switched linear system Σα is exponentially stable if there exists a switching rule α(.) such that every solution x(t , φ) of the system satisfies the condition

∃M > 0,

β > 0 : kx(t , φ)k ≤ Me−β t kφk,

∀t ∈ R+ .

Definition 2.2. A system of symmetric matrices {Li }, i = 1, 2, . . . , N, is said to be strictly complete if for every 0 6= x ∈ Rn there is i ∈ {1, 2, . . . , N } such that xT Li x < 0. Let us define the sets

Ωi = {x ∈ Rn : xT Li x < 0},

i = 1, 2, . . . , N .

It is easy to show that the system {Li }, i = 1, 2, . . . , N, is strictly complete if and only if N [

Ωi = Rn \ {0}.

i=1

Remark 2.3. As shown in [19], a sufficient condition for the strict completeness of the system {Li } is that there exist numbers P τi ≥ 0, i = 1, 2, . . . , N, such that Ni=1 τi > 0 and N X

τi Li < 0,

i=1

and in the case if N = 2, then the above condition is also necessary for the strict completeness. Before presenting the main result, we recall the following well-known matrix inequality and the Razumikhin stability theorem. Proposition 2.1. For any 0 < W ∈ Rn×n , x, y ∈ Rn , we have

±2xT y ≤ xT Wx + yT W −1 y.

V.N. Phat et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 1–10

3

Proposition 2.2 (Razumikhin Stability Theorem [3]). Consider the functional differential equation x˙ = f (xt ), x(t ) = φ(t ), t ∈ [−h, 0]. Assume that u, v, w : R+ → R+ are non-decreasing, and u(s), v(s) are positive for s ≥ 0, v(0) = u(0) = 0, and q > 1. If there is a function V (x) : R+ × Rn → R+ such that (i) u(kxk) ≤ V (x) ≤ v(kxk), t ∈ R+ , x ∈ Rn (ii) V˙ (x(t )) ≤ −w(kx(t )k) if V (x(t + s)) < qV (x(t )), ∀s ∈ [−h, 0], then the zero solution is exponentially stable. 3. Main result For given positive numbers h, a, b, β, δ , we set µ = (1 − δ)−1 , and Li (P ) = (Ai + Di )T P + P (Ai + Di ) + 2β P + 2he2β h PDi (Ai ATi + µDi DTi )DTi P

+ 2he2β h (a2 + µb2 )PDi DTi P + (a2 + b2 e2β h µ)P 2 + 2(h + 1)I SiP = {x ∈ Rn : xT Li (P )x < 0}, S¯1P = S1P ,

S¯iP = SiP \

i−1 [

S¯jP ,

i = 2, 3, . . . , N .

(3.1)

j =1

Let us denote

s M =

λmax (P ) + h + 2h2 , λmin (P )

a = max{a1 , a2 , . . . , aN },

b = max{b1 , b2 , . . . , bN }.

The main result of this paper is summarized in the following theorem. Theorem 3.1. Assume the conditions (D.1) and (2.2). Switched nonlinear system (2.1) is exponentially stable if there exist a positive number β and a symmetric positive definite matrix P ∈ Rn×n such that one of the following conditions holds. (i) The system of matrices {Li (P )}, i = 1, 2, . . . , N, is strictly complete. PN (ii) There exists τi ≥ 0, i = 1, 2, . . . , N, with i=1 τi > 0 such that N X

τi Li (P ) < 0.

(3.2)

i =1

The switching rule is chosen in case (i) as α(x(t )) = i whenever x(t ) ∈ S¯i , and in case (ii) as

α(x(t )) = arg min{xT (t )Li (P )x(t )}. Moreover, the solution x(t , φ) of the system satisfies

kx(t , φ)k ≤ Me−β t kφk,

t ∈ R+ .

In the case N = 2, conditions (i) and (ii) are equivalent. Proof. We utilize the following Newton–Leibniz formula x(t − h(t )) = x(t ) −

Z

t t −h(t )

x˙ (s)ds,

system (2.1) is transformed into x˙ (t ) = (Aα + Dα )x(t ) − Dα Aα

Z + Dα

Z

t

t −h(t )

x(s)ds − Dα Dα

Z

t t −h(t )

x(s − h(s))ds

t t −h(t )

fα (x(s), x(s − h(s)))ds + fα (x(t ), x(t − h(t ))).

(3.3)

Note that system (3.3) requires initial function ψ(t ) on [−2h, 0] : ψ(s) = φ(s + h(0)), −h − h(0) ≤ s ≤ −h(0), ψ(s) = x(t + s), −h(0) ≤ s ≤ 0, and as shown in Hale and Lunnel [4], p.156, it is a special case of system (2.1) such that the stability property of system (3.3) will ensure the stability property of system (2.1). Therefore, we will consider the stability of system (3.3) in order to ascertain the stability of system (2.1). For the closed-loop system of (3.3), we consider the following Lyapunov–Krasovskii functional V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ) + V4 (xt ),

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V.N. Phat et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 1–10

where V1 = hPx(t ), x(t )i t

Z V2 =

t −h(t ) 0

Z

e2β(s−t ) kx(s)k2 ds, t

Z

V3 = 0

Z

e2β(τ −t ) kx(τ )k2 dτ ds,

t +s

−h

t

Z

V4 =

t +s−h(t +s)

−h

e2β(τ −t ) kx(τ )k2 dτ ds.

The derivative of V1 along the trajectory of system (3.3) is given by V˙ 1 = 2hP x˙ (t ), x(t )i

= h[(Ai + Di )T P + P (Ai + Di )]x(t ), x(t )i +

t −h(t )

t

Z

−2hPDi Di x(s − h(s)), x(t )ids +

+ t −h(t )

t

Z Z

−2hPDi Ai x(s), x(t )ids

t t −h(t )

−2hPDi fi (s, x(s), x(s − h(s))), x(t )ids

+ 2hPfi (t , x(t ), x(t − h(t ))), x(t )i. Using Proposition 2.1, and the condition h(t ) ≤ h, h˙ (t ) ≤ δ < 1, we have

Z

t t −h(t )

−2hPDi Ai x(s), x(t )ids =

Z

t

−2hx(s), ATi DTi Px(t )ids

t −h(t )

≤ 2he2β h hPDi Ai ATi DTi Px(t ), x(t )i + 0.5e−2β h

Z

≤ 2he2β h hPDi Ai ATi DTi Px(t ), x(t )i + 0.5e−2β h

Z

≤ 2he

2β h

PDi Ai ATi DTi Px

h

−2β h

(t ), x(t )i + 0.5e

t t −h(t ) 0

−h(t )

Z

kx(s)k2 ds

kx(t + s)k2 ds

0

kx(t + s)k2 ds, −h

and similarly,

Z

t t −h(t )

−2hPDi Di x(s − h(s)), x(t )ids ≤ 2he2β h µhPDi Di DTi DTi Px(t ), x(t )i −2 β h

+ 0.5e

(1 − δ)

Z

0

kx(t + s − h(t + s))k2 ds, −h

Z

t t −h(t )

−2hPDi fi (x(s), x(s − h(s))), x(t )ids ≤ t

Z ≤

2ak t −h(t )

(t )kkx(s)kds +

DTi Px

Z

Z

t t −h(t )

2kfi (.)kkDTi Px(t )kds

t t −h(t )

2bkDTi Px(t )kkx(s − h(s))kds

≤ 2he2β h (a2 + b2 µ)hPDi DTi Px(t ), x(t )i Z 0 Z + 0.5e−2β h kx(t + s)k2 ds + 0.5e−2β h (1 − δ) −h

0

kx(t + s − h(t + s))k2 ds,

−h

2hPfi (x(t ), x(t − h(t ))), x(t )i ≤ 2kPx(t )kkf(.) k ≤ 2akPx(t )kkx(t )k + 2bkPxkkx(t − h(t ))k



2

≤ a +

b2 e2β h 1−δ



hP 2 x(t ), x(t )i + kx(t )k2 + e−2β h (1 − δ)kx(t − h(t ))k2 .

Then we have V˙ 1 ≤ h[(Ai + Di )T P + P (Ai + Di )]x(t ), x(t )i + 2he2β h hPDi (Ai ATi + Di DTi µ)DTi Px(t ), x(t )i

+ 2he2β h (a2 + b2 µ)hPDi DTi Px(t ), x(t )i + (a2 + b2 e2β h µ)hP 2 x(t ), xi + kx(t )k2 + e−2β h (1 − δ)kx(t − h(t ))k2 Z 0 Z 0 + e−2β h kx(t + s)k2 + e−2β h (1 − δ)ds kx(t + s − h(t + s))k2 ds. −h

−h

V.N. Phat et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 1–10

5

Similarly, the derivatives of V2 , V3 , V4 are given as V˙ 2 ≤ −2β V2 + kx(t )k2 − e−2β h (1 − δ)kx(t − h(t ))k2 , V˙ 3 ≤ −2β V3 + hkx(t )k2 − e−2β h

0

Z

kx(t + s)k2 ds, −h

V˙ 4 ≤ −2β V4 + hkx(t )k2 − e−2β h (1 − δ)

Z

0

kx(t + s − h(t + s))k2 ds. −h

Therefore, V˙ (xt ) ≤ −2β V + h[(Ai + Di )T P + P (Ai + Di ) + 2β P + 2he2β h PDi (Ai ATi + µDi DTi )DTi P

+ 2he2β h (a2 + µb2 )PDi DTi P + (a2 + b2 e2β h µ)P 2 + 2(h + 1)I ]x(t ), x(t )i, and hence V˙ (xt ) + 2β V ≤ hLi (P )x(t ), x(t )i

(3.4)

(i) Let us assume condition (i). The system of matrices {Li (P )} is strictly complete. We have N [

SiP = Rn \ {0}.

i =1

Based on the sets SiP , we construct the sets S¯iP by (3.1) and we can verify that S¯iP

\

i 6= j,

S¯jP = ∅,

S¯iP

[

S¯jP = Rn \ {0}.

(3.5)

We then construct the following switching rule: α(x(t ))i, whenever x(t ) ∈ S¯iP (this switching rule is well-defined due to condition (3.5)). Thus, from (3.4) we obtain V˙ (xt ) + 2β V (xt ) = xT (t )Lβ (P )x(t ) < 0,

∀t ∈ R+ ,

(3.6)

which implies that the solution x(t ) of system (3.2), is exponentially stable. To define the stability factor M, using the expression of V (xt ) and estimation (3.6), we have

λmin (P )kx(t )k2 ≤ V (xt ) ≤ V (0, x0 )e−2β t , where the estimation of V (0, x0 ) is easily verified by V (0, x0 ) ≤ (λmax (P ) + h + 2h2 )kφk2 . Therefore,

kx(t )k ≤ Me−β t kφk,

∀t ≥ 0.

(ii) We now assume condition (ii), then we have N X

τi Li (P ) < 0.

i =1

where τi ≥ 0, i = 1, 2, . . . , N, i=1 τi > 0. Since the numbers τi are non-negative and number 1 > 0 such that for any non-zero x(t ) we have

PN

N X

PN

i=1

τi > 0, there is always a

τi xT (t )Li (P )x(t ) ≤ −1 xT (t )x(t ).

i =1

Therefore, N X i =1

τi min {xT (t )Li (P )x(t )} ≤ i=1,...,N

N X

τi xT (t )Li (P )x(t ) ≤ −1 xT (t )x(t ).

i=1

The Lyapunov–Krasovskii functional V (.) is defined as above and the switching rule is designed as follows

α(x(t )) = arg min {xT (t )Li (P )x(t )}. i=1,...,N

Combining (3.4) and (3.7) gives V˙ (t , xt ) + 2β V (t , xt ) ≤ −ηkx(t )k2 ≤ 0,

∀t ∈ R+ ,

(3.7)

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V.N. Phat et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 1–10

where η = 1 ( i=1 τi )−1 . This implies that the zero solution x(t ) of system (3.3) is exponentially stable. The proof is then completed by the same way as in part (i). We now consider the stability problem for system (2.1) with no restriction on the derivative of the time-varying delay function. For given positive numbers h, a, b, λ,  , we set

PN

Wi (P , Q , S , λ) = (Ai + Di )T P + P (Ai + Di ) + hPDi [Qi + Si

+ λ−1 (a2 + b2 )I ]DTi P + (a2 + λ−1 b2 )P 2 + (4h + 1)(1 + )P + I SiP = {x ∈ Rn : xT [Wi (P , Q , S , λ)]x < 0, xT [λI − P ]x ≤ 0, xT [ATi Qi−1 Ai − P ]x ≤ 0, xT [DTi Si−1 Di − P ]x ≤ 0}, S¯1P = S1P ,

S¯iP = SiP \

i −1 [

S¯jP ,

i = 2, 3, . . . , N .

j =1

a = max{a1 , a2 , . . . , aN },

b = max{b1 , b2 , . . . , bN }.

Theorem 3.2. Assume the conditions (D.2) and (2.2). Switched nonlinear system (2.1) is exponentially stable if there exist positive numbers λ,  and symmetric positive definite matrices P , Qi , Si ∈ Rn×n , i = 1, 2, . . . , N such that one of the following conditions holds. (i) The system of matrices {Wi (P , Q , S , λ)}, i = 1, 2, . . . , N, is strictly complete. PN (ii) There exists τi ≥ 0, i = 1, 2, . . . , N, with i=1 τi > 0 and N X

τi Wi (P , Q , S , λ) < 0.

(3.8)

i=1

The switching rule is chosen in case (i) as α(x(t )) = i whenever x(t ) ∈ S¯i , and in case (ii) as

α(x(t )) = arg min{xT (t )Wi (P , Q , S , λ)x(t )}. In the case N = 2, conditions (i) and (ii) are equivalent. Proof. For system (3.3), we consider the following Lyapunov functional V (t , x(t )) = hPx(t ), x(t )i. The derivative of V (t , x) along the trajectory of (3.3) gives V˙ (t , x(t )) = 2hP x˙ (t ), x(t )i

= h[(Ai + Di ) P + P (Ai + Di )]x(t ), x(t )i + T

Z

t

+ t −h(t )

t

Z

−2hPDi Di x(s − h(s)), x(t )ids +

t −h(t )

−2hPDi Ai x(s), x(t )ids

t

Z

t −h(t )

−2hPDi fi (x(s), x(s − h(s))), x(t )ids

+ 2hPfi (x(t ), x(t − h(t ))), x(t )i. Take a number q = 1 +  , such that V (t + s, x(t + s)) < (1 + )V (t , x(t )),

∀s ∈ [−2h, 0],

and choose the number λ > 0 and symmetric positive definite matrices Qi , Si , i = 1, 2, . . . , N such that

λI ≤ P ,

ATi Qi−1 Ai ≤ P ,

DTi Si−1 Di ≤ P .

Then using the Razumikhin theorem, Proposition 2.2, by the same arguments used in the proof of Theorem 3.1 for appropriate estimations, we have

Z

t t −h(t )

 −2 hPDi Ai x(s), x(t )i ds ≤ h PDi Qi DTi Px(t ), x(t ) +

Z

≤ h PDi Qi DTi Px(t ), x(t ) +

Z





0

t −h(t )

Ai Qi−1 ATi x(s), x(s) ds

0

−h(t )

V (t + s, x(t + s))ds

≤ h PDi Qi DTi Px(t ), x(t ) + h(1 + ) hPx(t ), x(t )i ,



and similarly

Z

t t −h(t )



−2hPDi Di x(s − h(s)), x(t )ids ≤ hhPDi Si DTi Px, xi + h(1 + )hPx, xi.

V.N. Phat et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 1–10

7

To estimate the last two terms of V˙ (.) we use condition (3.7), we obtain

Z

t t −h(t )

−2hPDi fi (x(s), x(s − h(s))), x(t )ids ≤

Z

t t −h(t )

[2akDTi Px(t )kkx(s)k + 2bkDTi Px(t )kkx(s − h(s))k]ds

t

Z ≤

t −h(t )

[λkx(s)k2 + λ−1 a2 hPDi DTi Px(t ), x(t )i]ds +

≤ λ−1 (a2 + b2 )hhPDi DTi Px(t ), x(t )i +

Z

Z

t t −h(t )

[λkx(s − h(s))k2 + λ−1 b2 hPDi DTi Px(t ), x(t )i]ds

0

−h(t )

V (t + s, x(t + s))ds +

0

Z

−h(t )

V (t + s − h(s), x(t + s − h(s)))ds.

Applying Proposition 2.2, we have

Z

t t −h(t )

−2hPDi fi (x(s), x(s − h(s))), x(t )ids ≤ λ−1 h(a2 + b2 )hPDi DTi Px(t ), x(t )i + 2h(1 + )hPx(t ), x(t )i,

and similarly, 2hPfi (x(t ), x(t − h(t ))), x(t )i ≤ 2akPx(t )kkx(t )k + 2bkPx(t )kkx(t − h(t ))k

≤ a2 hP 2 x, xi + kxk2 + λ−1 b2 hPx, xi + λkx(t − h(t ))k2 ≤ (a2 + λ−1 b2 )hP 2 x, xi + kxk2 + V (t − h(t ), x(t − h(t ))) ≤ (a2 + λ−1 b2 )hP 2 x, xi + kxk2 + (1 + )hPx, xi. Therefore, we obtain V˙ (xt ) ≤ h[(Ai + Di )T P + P (Ai + Di ) + hPDi [Ai Qi ATi + Di Si DTi

+ λ−1 (a2 + b2 )I ]DTi P + (a2 + λ−1 b2 )P 2 + (4h + 1)(1 + )P + I ]x(t ), x(t )i ≤ hWi (P , Q , S , λ)x(t ), x(t )i. Based on the Razumikhin theorem, Proposition 2.2, the proof of the theorem is completed by the same arguments used in the proof of Theorem 3.1. Remark 3.1. Note that the derived conditions involve the solution of some algebraic Riccati equation, which can be solved by various numerical methods given in [1,5]. The following simple procedure can be applied to construct a switching rule. Step 1. Define the matrices Li (P ). Step 2. Find the solution P of the Riccati equation (3.2). Step 3. Construct the sets SiP , and then S¯iP , by (3.1) and verify condition (3.5). Step 4. The switching signal α(.) is chosen as α(x) = i, whenever x ∈ S¯iP or as

α(x(t )) = arg min{xT (t )Li (P )x(t )}. 4. Numerical examples Example 4.1. Consider the following switched nonlinear system x˙ (t ) = Ai x(t ) + Di x(t − h(t )) + fi (t , x(t ), x(t − h(t ))),

i = 1, 2

where

(A1 , D1 ) = (A2 , D2 ) =

0.1 −0.001



0.15 −0.001



  −0.001 −0.7 , 0.15 0.001   −0.001 −0.6 , 0.05 0.001

0.001 −0.7



0.001 −0.6



h(t ) = 0.0833 sin2 (3t ) and

kfi (x(t ), x(t − h(t )))k ≤ 0.1 kx(t )k + 0.0667kx(t − h(t ))k, i = 1, 2. We have h = 0.0833, δ = 0.5, ai = 0.1, bi = 0.0667, i = 1, 2. By choosing the positive definite matrix P as

 P =

3.5 0.01

0.01 3.5



8

V.N. Phat et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 1–10

Fig. 1. The regions S1P (left) and S2P (right).

and β = 0.05, we obtain

(L1 (P ), L2 (P )) =

 −0.2666 −0.0091

  −0.0091 0.3299 , 0.096 −0.0076

 −0.0076 . −0.3849

Therefore, the sets S1P and S2P are defined as S1P = {(x1 , x2 ) : x21 + 0.0685x1 x2 − 0.3601x22 > 0}, S2P = {(x1 , x2 ) : x21 − 0.0463x1 x2 − 1.1665x22 < 0}. These sets are equivalent to S1P = {(x1 , x2 ) : (x1 − 0.5668x2 )(x1 + 0.6353x2 ) > 0}, and S2P = {(x1 , x2 ) : (x1 − 1.1035x2 )(x1 + 1.0571x2 ) < 0}; see Fig. 1. Obviously, the union of these sets is equal to R2 . We can verify condition (ii) of Theorem 3.1 with τ1 = 0.6667 and τ2 = 0.3333. Moreover, the sum

−0.0678 τ1 L1 (P ) + τ2 L2 (P ) = −0.0086 

 −0.0086 −0.0643

is negative definite; i.e. the first entry in the first row and the first column −0.0678 is negative and the determinant of the matrix is positive. The switching regions are given as P

S 1 = {(x1 , x2 ) : x21 + 0.0685x1 x2 − 0.3601x22 > 0}, P

S 2 = {(x1 , x2 ) : x21 + 0.0685x1 x2 − 0.3601x22 < 0}. P

According to Theorem 3.1, the system with the switching rule α(x(t )) = i, if x(t ) ∈ S i is exponentially stable. Moreover, the solution x(t , φ) of the system satisfies

kx(t , φ)k ≤ 1.0167e−0.05t kφk,

t ∈ R+ .

Example 4.2. Consider the switched linear system defined by x˙ (t ) = Ai x(t ) + Di x(t − h(t )) + fi (x(t ), x(t − h(t ))),

i = 1, 2

where

(A1 , D1 ) = (A2 , D2 ) =

0.075 −0.005



0.01 −0.01



   −0.005 −1.15 0.005 , , 0.07 0.005 −1.22    −0.01 −1.15 0.01 , , 0.045 0.01 −1.05

h(t ) is any continuous function satisfying h(t ) ≤ h = 0.0667 and

kfi (x(t ), x(t − h(t )))k ≤ 0.1429 kx(t )k + 0.1429 kx(t − h(t ))k ,

V.N. Phat et al. / Nonlinear Analysis: Hybrid Systems 3 (2009) 1–10

9

i = 1, 2. We have h = 0.0667, ai = bi = 0.1429, i = 1, 2. By choosing the positive definite matrix P, Q1 , Q2 , S1 , S2 as

 P =

1.5 0.01

 S1 =

1 0

0.01 , 1.5





0 , 1

 S2 =

0.005 0

 Q1 = 0.885 0



0 , 0.005

0.004 0

 Q2 =



0 , 0.004



0 , 0.885

λ = 1.25, and  = 0.01, one can verify all conditions of Theorem 3.2 with τ1 = 0.7 and τ2 = 0.3. The switching regions are given as P

S 1 = {(x1 , x2 ) : x21 − 0.3682x1 x2 − 2.3907x22 < 0}, P

S 2 = {(x1 , x2 ) : x21 − 0.3682x1 x2 − 2.3907x22 > 0}. P

According to Theorem 3.2, the system with the switching rule α(x(t )) = i if x(t ) ∈ S i is exponentially stable. In this case, it can be shown that

(L1 (P ), L2 (P )) =

0.0455 −0.0084



  −0.0084 −0.1796 , −0.1087 −0.0076

 −0.0076 . 0.1535

Moreover, the sum

 −0.0221 τ1 L1 (P ) + τ2 L2 (P ) = −0.0081

 −0.0081 −0.0301

is negative definite; i.e. the first entry in the first row and the first column −0.0221 is negative and the determinant of the matrix is positive. The sets S1P and S2P are given as S1P = {(x1 , x2 ) : x21 − 0.3682x1 x2 − 2.3907x22 < 0}, S2P = {(x1 , x2 ) : x21 + 0.0845x1 x2 − 0.8542x22 > 0}. These sets are equivalent to S1P = {(x1 , x2 ) : (x1 − 1.7412x2 )(x1 + 1.373x2 ) < 0}, and S2P = {(x1 , x2 ) : (x1 − 0.8829x2 )(x1 + 0.9675x2 ) > 0}. The union of these sets is equal to R2 .

5. Conclusions We have presented new delay-dependent conditions for the exponential stability of nonlinear switched systems with improved time-varying delay. Using the Lyapunov functional approach combined with the Newton–Leibniz formula, neither restriction on the derivative of time-delay function nor bound restriction on nonlinear perturbations is required to design a switching rule for the exponential stability of the system. The conditions are formulated in terms of the solution of some algebraic Riccati equation. A simple procedure for constructing the switching rule has been given.

Acknowledgments This work was supported by the Basic Program in Natural Sciences, Vietnam, the Thailand Research Fund and the Commission for Higher Education, Thailand.

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