On improved delay-dependent robust stability criteria for uncertain systems with interval time-varying delay

Abstract

This paper considers the problem of delay-dependent robust stability for uncertain systems with interval time-varying delays. By using the direct Lyapunov method, a new Lyapunov-Krasovskii (L-K) functional is introduced based on decomposition approach, when dealing with the time derivative of L-K functional, a new tight integral inequality is adopted for bounding the cross terms. Then, a new less conservative delay-dependent stability criterion is formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by optimization algorithms. Numerical examples are given to show the effectiveness and the benefits of the proposed method.

Introduction

Time-delay phenomena are ubiquitous in many practical systems such as chemical engineering systems, biological systems, economic systems and networked control systems, which are often major sources of instability and poor performance. Hence, stability analysis and stabilization of systems with time-delays have received considerable attention in the past few years (see e.g. [1–24], and the references therein). Very recently, systems with time-varying delay in a known interval have been studied in [3, 7–15, 20–22], wherein the time-delay may vary in a range for which the lower bound is not restricted to being zero.

The existing stability criteria are usually classified into the delay-independent ones and the delay-dependent ones. Since the delay-dependent ones are generally less conservative than the delay-independent ones, especially for small delays, much attention has been paid to the delay-dependent ones. An important issue in this field is to enlarge the feasible region of stability criteria. For this purpose, many approaches have been developed in the past few years, among which the free-weighting matrix method and Jensen’s integral inequality method are widely used and obtained plentiful results (see e.g. [2–14], and the references therein). For the purpose of reducing the conservatism caused by the use of system transformations and bounding techniques, the free-weighting matrix method was first proposed in [2, 3] to investigate the stability of systems with time-varying delay. By employing a convex combination technique, Park and Ko[4] further extended the free weighting matrix method to a new Lyapunov functional. Recently, motivated by the idea of L-K functional with tripleintegral terms, an extended free-weighting matrix method, i.e., double-integral inequality method was proposed in [5]. Although the free-weighting matrix method can effectively reduce the conservatism, it can also lead to the increase of computational complexity since many slack variables are introduced.

Jensen^’s integral inequality approach is another important one, which was first introduced by Gu[6] for stability analysis of time-delay systems. Soon afterwards, many researchers, such as Han[7], Zhang and Han[8], Sun et al.[9], Kwon et al.[10], Ramakrishnan and Ray[11] further extended the Jensen^’s integral inequality to some new forms. Jensen^’s integral inequality approach can provide a simple form of stability conditions without introducing redundant variables. However, how to construct a less conservative inequality is a difficult problem. Recently, a novel technique called delay-central point method was proposed in [16], by employing a central point of variation about the delay range, the time-interval is divided into two segments of equal length, and the time variation of a candidate L-K functional is evaluated individually in each subinterval. In order to further reduce the conservatism, the delay decomposition approach was proposed in [17–20], and show its improvement of maximum delay bounds. Based on decomposition technique, Ramakrishnan and Ray[21, 22] further extended the delay-central point method and proposed a less conservative stability criteria. Nevertheless, there still exists room for further improvements.

Motivated by the above discussions, we further discuss the stability of uncertain linear systems with interval time-varying delays. Based on the direct L-K approach and linear matrix inequality, new less conservative delay-dependent stability criteria for computing the maximum allowable bound of the delay range is formulated in terms of linear matrix inequalities. Less conservative results are mainly attributed to the decomposition technique for designing the L-K functional and tighter bounding technique for dealing with the cross-terms. Numerical examples are given to illustrate the effectiveness and less conservatism of the proposed method.

Notations. Throughout this paper, Rⁿ denotes the n-dimensional Euclidian space, R^n×m is the set of n × m real matrices, the notation X > 0, for X ∈ R^n×m means that the matrix X is a real symmetric positive definite. For an arbitrary matrix B and two symmetric matrices A and C, $\left[ {\begin{array}{*{20}c} A & B \\ * & C \\ \end{array} } \right]$ denotes a symmetric matrix, where * denotes the entries implied by symmetry.

Problem description and preliminaries

Consider the following time-varying delay system:

$$\left\{{\matrix{{\dot x(t) = (A + \Delta (t))x(t) + (B + \Delta B(t))x(t - h(t))} \hfill \cr {x(t) = \varphi (t),\quad t \in [ - {h_2},0]} \hfill \cr}} \right.$$

(1)

where x(t) ∈ Rⁿ is the state vector, A and B are constant matrices with appropriate dimensions, h(t) is a time-varying delay satisfying the following two categories:

$${\rm{Case}}\;1:0 \leq {h_1} \leq h(t) \leq {h_2},\dot h(t) \leq \mu, \forall t \geq 0$$

(2)

$${\rm{Case}}\;2:0 \leq {h_1} \leq h(t) \leq {h_2},\forall t \geq 0.$$

(3)

ΔA(t)and ΔB(t) denote the parameter uncertainties satisfying the following condition:

$$\left[ {\Delta A(t)\quad \Delta B(t)} \right] = DF(t)\left[ {{E_a}\quad {E_b}} \right]$$

(4)

where D, E _a and E _b are constant matrices with appropriate dimensions, and F(t) is an unknown time-varying matrix, which is Lesbesgue, satisfying, F(t)^TF(t) ≤ I.

Before moving on, the following lemma is necessary in the proof of the main results.

• Lemma 1. For any scalar h(t) ≥ 0, and any constant matrix Q ∈ R^n×n, Q = Q^T > 0, the following inequality holds:

  $$\matrix{{- \int_{t - h(t)}^t {{{\dot x}^{\rm{T}}}} (s)Q\dot x(s){\rm{d}}s \leq h(t){\zeta ^{\rm{T}}}(t)V{Q^{- 1}}{V^{\rm{T}}}\zeta (t) +} \hfill \cr {\quad \quad \;\;2{\zeta ^{\rm{T}}}(t)V[x(t) - x(t - h(t))]} \hfill \cr}$$

  where

  $$\matrix{{{\zeta ^{\rm{T}}}(t) = \left[ {{x^{\rm{T}}}(t)\quad {x^{\rm{T}}}(t - {h_1})\quad \Gamma _a^{\rm{T}}(t)\quad {x^{\rm{T}}}(t - h(t))} \right.} \hfill \cr {\quad \quad \quad \left. {\Gamma _2^{\rm{T}}(t)\quad {{\dot x}^{\rm{T}}}(t)} \right]} \hfill \cr {\Gamma _2^{\rm{T}}(t) = \left[ {{x^{\rm{T}}}\left({t - {{{h_a}} \over N}} \right)\quad {x^{\rm{T}}}\left({t - 2{{{h_a}} \over N}} \right)\quad \cdots} \right.} \hfill \cr {\left. {\quad \quad \quad {x^{\rm{T}}}\left({t - (N - 1){{{h_a}} \over N}} \right)\quad {x^{\rm{T}}}(t - {h_a})} \right]} \hfill \cr {\Gamma _2^{\rm{T}}(t) = \left[ {{x^{\rm{T}}}\left({t - {{{h_2}} \over N}} \right)\quad {x^{\rm{T}}}\left({t - 2{{{h_2}} \over N}} \right)\quad \cdots} \right.} \hfill \cr {\left. {\quad \quad \quad {x^{\rm{T}}}\left({t - (N - 1){{{h_2}} \over N}} \right)\quad {x^{\rm{T}}}(t - {h_2})} \right]} \hfill \cr}$$

  in which ${h_a} = {{({h_1} + {h_2})} \over 2}$, V is free weighting matrix with appropriate dimensions, and N is a given positive integer.

• Proof. For any real vectors a, b and any matrix Q > 0 with appropriate dimensions, we know that the following inequality holds

  $$ \pm 2{a^{\rm{T}}}b \leq {a^{\rm{T}}}Qa + {b^{\rm{T}}}{Q^{- 1}}b.$$

  (5)

  From (5), we obtain

  $$\matrix{{- 2\int_{t - h(t)}^t {{{({v^{\rm{T}}}\zeta (t))}^{\rm{T}}}\dot x(s){\rm{d}}s \leq} \int_{t - h(t)}^t {{{\dot x}^{\rm{T}}}(s)Q\dot x(s){\rm{d}}s} +} \hfill \cr {\int_{t - h(t)}^t {{\zeta ^{\rm{T}}}} (t)V{Q^{- 1}}{V^{\rm{T}}}\zeta (t){\rm{d}}s.} \hfill \cr}$$

  So we have

  $$\matrix{{- \int_{t - h(t)}^t {{{\dot x}^{\rm{T}}}} (s)Q\dot x(s){\rm{d}}s \leq 2\int_{t - h(t)}^t {{{({V^{\rm{T}}}\zeta (t))}^{\rm{T}}}\dot x(s){\rm{d}}s +}} \hfill \cr {\int_{t - h(t)}^t {{\zeta ^{\rm{T}}}} (t)V{Q^{- 1}}{V^{\rm{T}}}\zeta (t){\rm{d}}s =} \hfill \cr {h(t){\zeta ^{\rm{T}}}(t)V{Q^{- 1}}{V^{\rm{T}}}\zeta (t) + 2{\zeta ^{\rm{T}}}(t)V[x(t) - x(t - h(t))].} \hfill \cr}$$

  □

• Lemma 2[25]. Suppose γ₁ ≤ γ(t) ≤ γ₂, where γ(·): R₊ → R₊. Then, for any constant matrices Ξ₁,Ξ₂ and Ω with proper dimensions, the following matrix inequality

  $$\Omega + (\gamma (t) - {\gamma _1}){\Xi _1} + ({\gamma _2} - \gamma (t)){\Xi _2} < 0$$

  holds, if and only if

  $$\Omega + ({\gamma _2} - {\gamma _1}){\Xi _1} < 0,\quad \Omega + ({\gamma _2} - {\gamma _1}){\Xi _2} < 0.$$

• Lemma 3[26]. Given matrices Q = Q^T, H, E and R = R^T with appropriate dimensions, the inequality

  $$Q + HFE + {E^{\rm{T}}}{F^{\rm{T}}}{H^{\rm{T}}} < 0$$

  for all F satisfying F^T F ≤ R, if and only if there exists some scalar ε > 0, such that

  $$Q + \varepsilon H{H^{\rm{T}}} + {\varepsilon ^{- 1}}{E^{\rm{T}}}RE < 0.$$

Main results

In this section, we shall establish our main results based on LMI framework. First, we will consider nominal system, whereafter, we study the robust stability of the uncertain system. Consider the nominal system of (1):

$$\left\{{\matrix{{\dot x(t) = Ax(t) + Bx(t - h(t))} \hfill \cr {x(t) = \varphi (t),t \in [ - {h_2},0].} \hfill \cr}} \right.$$

(6)

• Theorem 1. For given values of h₁, h₂ and μ, system (6) is asymptotically stable, if there exist real symmetric positive definitive matrices P₁, P₂, P₃, Z₁, Z₂; symmetric matrices Q, S and free matrix variables L _j , M _j , V _j , T _j (j = 1, 2) with appropriate dimensions such that the following LMIs hold:

  $$\left[ {\begin{array}{*{20}c} \Phi & {\sqrt {\frac{{h_a }} {N}} L} & {\sqrt {h_\delta } M} \\ * & { - Z_1 } & 0 \\ * & * & { - Z_2 } \\ \end{array} } \right] < 0$$

  (7)
  $$\left[ {\begin{array}{*{20}c} \Phi & {\sqrt {\frac{{h_a }} {N}} L} & {\sqrt {h_\delta } V} \\ * & { - Z_1 } & 0 \\ * & * & { - Z_2 } \\ \end{array} } \right] < 0$$

  (8)

  and

  $$\begin{array}{*{20}c} {Q = \left[ {\begin{array}{*{20}c} {Q_{11} } & {Q_{12} } & \cdots & {Q_{1N} } \\ * & {Q_{22} } & \cdots & {Q_{2N} } \\ \vdots & \vdots & \ddots & \vdots \\ * & * & {Q_{N - 1,N - 1} } & {Q_{N - 1,N} } \\ * & * & * & {Q_{NN} } \\ \end{array} } \right] \geqslant 0} \\ {S = \left[ {\begin{array}{*{20}c} {S_{11} } & {S_{12} } & \cdots & {S_{1N} } \\ * & {S_{22} } & \cdots & {S_{2N} } \\ \vdots & \vdots & \ddots & \vdots \\ * & * & {S_{N - 1,N - 1} } & {S_{N - 1,N} } \\ * & * & * & {S_{NN} } \\ \end{array} } \right] \geqslant 0} \\ \end{array} $$

  where

  $$\begin{array}{*{20}c} {\Phi = \left[ {\begin{array}{*{20}c} {\Phi _{11} } & {\Phi _{12} } & {\Phi _{13} } & {\Phi _{14} } \\ * & {\Phi _{22} } & {\Phi _{23} } & {B^T T_2^T } \\ * & * & {\Phi _{33} } & 0 \\ * & * & * & {\Phi _{44} } \\ \end{array} } \right]} \\ {\Phi _{11} = \left[ {\begin{array}{*{20}c} {\Phi _1 } & 0 & {\Phi _2 } & {Q_{13} } & \cdots & {Q_{1N} } & 0 \\ * & {P_3 - P_2 } & 0 & 0 & \cdots & 0 & 0 \\ * & * & {\Phi _3 } & {Q_{23} - Q_{12} } & \cdots & {Q_{2N} - Q_{1,N - 1} } & { - Q_{1N} } \\ * & * & * & {Q_{33} - Q_{22} } & \cdots & {Q_{3N} - Q_{2,N - 1} } & { - Q_{2N} } \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ * & * & * & * & * & {Q_{NN} - Q_{N - 1,N - 1} } & { - Q_{N - 1,N} } \\ * & * & * & * & * & * & {V_1 + V_1^T - Q_{N,N} } \\ \end{array} } \right]} \\ {\Phi _{33} = \left[ {\begin{array}{*{20}c} {S_{22} - S_{11} } & {S_{23} - S_{12} } & \cdots & {S_{2N} - S_{1,N - 1} } & { - S_{1N} } \\ * & {S_{33} - S_{22} } & \cdots & {S_{3N} - S_{2,N - 1} } & { - S_{2N} } \\ * & * & \ddots & \vdots & \vdots \\ \vdots & \vdots & \cdots & {S_{NN} - S_{N - 1,N - 1} } & { - S_{N - 1,N} } \\ * & * & * & * & { - S_{NN} - M_2 - M_2^T } \\ \end{array} } \right]} \\ {\Phi _{13} = \left[ {\begin{array}{*{20}c} {S_{12} } & {S_{13} } & \cdots & {S_{1N} } & 0 \\ 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 0 & 0 \\ \end{array} } \right]} \\ {\Phi _1 = P_2 + Q_{11} + S_{11} + L_1 + L_1^T + T_1 A + A^T T_1^T } \\ {\begin{array}{*{20}c} {\Phi _2 = Q - L_1 + L_2^T } & {\Phi _3 = Q_{22} - Q_{11} - L_2 } \\ \end{array} - L_2^T } \\ {\Phi _{12} = \left[ {(T_1 B)^T 0 \cdots 0 ( - V_1 + V_2^T )^T } \right]^T } \\ {\Phi _{14} = \left[ {(P_1 - T_1 + A^T T_2^T )^T 0 \cdots 0} \right]^T } \\ {\Phi _{22} = - (1 - \mu )P_3 - V_2 - V_2^T + M_1 + M_1^T } \\ {\Phi _{23} = \left[ {0 \cdots 0 - M_1 + M_2^T } \right]} \\ {\Phi _{44} = \frac{{h_a }} {N}Z_1 + h_\delta Z_2 - T_2 - T_2^T } \\ {\begin{array}{*{20}c} {h_a = \frac{{(h_1 + h_2 )}} {2},} & {h_\delta = \frac{{(h_2 - h_1 )}} {2}} \\ \end{array} } \\ {L = [L_1^T 0 L_2^T 0 0 \cdots 0 0]^T } \\ {V = [0 \cdots 0 V_1^T V_2^T 0 \cdots 0]^T } \\ {T = [T_1^T 0 0 0 \cdots 0 0 T_2^T ]^T } \\ {M = [0 \cdots 0 M_1^T 0 \cdots M_2^T 0]^T .} \\ \end{array} $$

• Proof. Construct an L-K functional candidate as

  $$V(t) = {V_1}(t) + {V_2}(t) + {V_3}(t)$$

  (9)

  where

  $$\matrix{{{V_1}(t) = {x^{\rm{T}}}(t){P_1}x(t)\int_{t - {h_1}}^t {{x^{\rm{T}}}} (s){P_2}x(s){\rm{d}}s +} \hfill \cr {\quad \quad \int_{t - h(t)}^{t - {h_1}} {{x^{\rm{T}}}} (s){P_3}x(s){\rm{d}}s} \hfill \cr {{V_2}(t) = \int_{t - {\textstyle{{{h_a}} \over N}}}^t {\xi _1^{\rm{T}}} (s)Q{\xi _1}(s){\rm{d}}s + \int_{t - {\textstyle{{{h_2}} \over N}}}^t {\xi _2^{\rm{T}}} (s)S{\xi _2}(s){\rm{d}}s} \hfill \cr {{V_3}(t) = \int_{- {\textstyle{{{h_a}} \over N}}}^0 {} \int_{t + \theta}^t {{{\dot x}^{\rm{T}}}} (s){Z_1}\dot x(s){\rm{d}}s{\rm{d}}\theta +} \hfill \cr {\int_{- {h_2}}^{- {h_a}} {} \int_{t + \theta}^t {{{\dot x}^{\rm{T}}}} (s){Z_2}\dot x(s){\rm{d}}s{\rm{d}}\theta} \hfill \cr {{\xi _1}(t) = {{\left[ {{x^{\rm{T}}}(t)\quad {x^{\rm{T}}}\left({t - {{{h_a}} \over N}} \right)\;\; \cdots \;\;{x^{\rm{T}}}\left({t - (N - 1){{{h_a}} \over N}} \right)} \right]}^{\rm{T}}}} \hfill \cr {{\xi _2}(t) = {{\left[ {{x^{\rm{T}}}(t)\quad {x^{\rm{T}}}\left({t - {{{h_2}} \over N}} \right)\;\; \cdots \;\;{x^{\rm{T}}}\left({t - (N - 1){{{h_2}} \over N}} \right)} \right]}^{\rm{T}}}.} \hfill \cr}$$

The time-derivative of the L-K functional along the trajectory of (6) is given by

$$\dot V(t) = {\dot V_1}(t) + {\dot V_2}(t) + {\dot V_3}(t)$$

(10)

where

$$\matrix{{{{\dot V}_1}(t) \leq 2{x^{\rm{T}}}(t){P_1}\dot x(t) + {x^{\rm{T}}}(t){P_1}x(t) +} \hfill \cr {\quad \quad \;\;{x^{\rm{T}}}(t - {h_1})({P_3} - {P_2})x(t - {h_1}) -} \hfill \cr {(1 - \mu){x^{\rm{T}}}(t - h(t)){P_3}x(t - h(t))} \hfill \cr {{{\dot V}_2}(t) = \xi _1^{\rm{T}}(t)Q{\xi _1}(t) - \xi _1^{\rm{T}}\left({t - {{{h_a}} \over N}} \right)Q{\xi _1}\left({t - {{{h_a}} \over N}} \right) +} \hfill \cr {\quad \quad \;\;\xi _2^{\rm{T}}(t)S{\xi _2}(t) - \xi _2^{\rm{T}}\left({t - {{{h_2}} \over N}} \right)S{\xi _2}\left({t - {{{h_2}} \over N}} \right)} \hfill \cr {{{\dot V}_3}(t) = {{{h_a}} \over N}{{\dot x}^{\rm{T}}}(t){Z_1}\dot x(t) + ({h_2} - {h_a}){{\dot x}^{\rm{T}}}(t){Z_2}\dot x(t) -} \hfill \cr {\quad \quad \;\;\int_{t - {\textstyle{{{h_a}} \over N}}}^t {{{\dot x}^{\rm{T}}}} (s){Z_1}\dot x(s){\rm{d}}s - \int_{t - {h_2}}^{t - {h_a}} {{{\dot x}^{\rm{T}}}(s){Z_2}\dot x(s){\rm{d}}s.}} \hfill \cr}$$

Note that

$$\matrix{{- \int_{t - {h_2}}^{t - {h_a}} {{{\dot x}^{\rm{T}}}} (s){Z_2}\dot x(s){\rm{d}}s = - \int_{t - {h_2}}^{t - h(t)} {{{\dot x}^{\rm{T}}}(s)} {Z_2}\dot x(s){\rm{d}}s -} \hfill \cr {\quad \quad \quad \quad \quad \quad \quad \quad \quad \int_{t - h(t)}^{t - {h_a}} {{{\dot x}^{\rm{T}}}(s){Z_2}\dot x} (s){\rm{d}}s.} \hfill \cr}$$

Using Lemma 1, one can obtain

$$\matrix{{- \int_{t - {\textstyle{{{h_a}} \over N}}}^t {{{\dot x}^{\rm{T}}}(s){Z_1}\dot x(s){\rm{d}}s \leq {{{h_a}} \over N}{\zeta ^{\rm{T}}}(t)LZ_1^{- 1}{L^{\rm{T}}}\zeta (t) +}} \hfill \cr {\quad \quad 2{\zeta ^{\rm{T}}}(t)L\left[ {x(t) - x(t - {{{h_a}} \over N})} \right]} \hfill \cr}$$

(11)

$$\matrix{{- \int_{t - h(t)}^{t - {h_a}} {{{\dot x}^{\rm{T}}}} (s){Z_2}\dot x(s){\rm{d}}s \leq (h(t) - {h_a}){\zeta ^{\rm{T}}}(t)VZ_2^{- 1}{V^{\rm{T}}}\zeta (t) +} \hfill \cr {\quad \quad \quad 2{\zeta ^{\rm{T}}}(t)V[x(t - {h_a}) - x(t - h)]} \hfill \cr}$$

(12)

$$\matrix{{- \int_{t - {h_2}}^{t - h(t)} {{{\dot x}^{\rm{T}}}} (s){Z_2}\dot x(s){\rm{d}}s \leq ({h_2} - h(t)){\zeta ^{\rm{T}}}(t)MZ_2^{- 1}{M^{\rm{T}}}\zeta (t) +} \hfill \cr {\quad \quad \quad 2{\zeta ^{\rm{T}}}(t)M[x(t - h(t)) - x(t - {h_2})].} \hfill \cr}$$

(13)

On the other hand, for any matrices T with appropriate dimensions, and from system (6), we have

$$0 = 2{\zeta ^{\rm{T}}}(t)T[Ax(t) + Bx(t - h(t)) - \dot x(t)]$$

(14)

by substituting (11)–(14) in (10), the time derivative $\dot V(t)$ can be expressed as

$$\dot V(t) \leq {\zeta ^{\rm{T}}}(t)\Lambda \zeta (t)$$

(15)

where

$$\matrix{{\Lambda = \Phi + {{{h_a}} \over N}LZ_1^{- 1}{L^{\rm{T}}} + ({h_2} - h(t)MZ_2^{- 1}{M^{\rm{T}}}) +} \hfill \cr {\quad \quad (h(t) - {h_a})VZ_2^{- 1}{V^{\rm{T}}}.} \hfill \cr}$$

If ∀h(t) ∈ [h1, h2]

$$\Lambda < 0.$$

(16)

Then $\dot V(t) < - \varepsilon {\left\Vert {x(t)} \right\Vert^2}$ for some scalar ε > 0, from which we conclude that the nominal system (6) is asymptotically stable according to Lyapunov stability theory.

Now, we apply Lemma 2 to (16) to yield the following inequalities:

$$\Phi + {{{h_a}} \over N}LZ_1^{- 1}{L^{\rm{T}}} + ({h_2} + {h_a})MZ_2^{- 1}{M^{\rm{T}}} < 0$$

(17)

$$\Phi + {{{h_a}} \over N}LZ_1^{- 1}{L^{\rm{T}}} + ({h_2} + {h_a})VZ_2^{- 1}{V^{\rm{T}}} < 0.$$

(18)

Schur complement on (17) and (18) yields the LMIs stated in the Theorem 1. □

• Remark 1. Different from the decomposition approach used in [22], when dealing with the time derivative of L-K functional, we proposed a new tight integral inequality (Lemma 1) for bounding the cross terms, hence, it yields a less conservative result. The comparisons of conservatism with some existing methods will be presented in Section 4.

• Remark 2. When the information of the time derivative h(t) is unknown, by choosing P₃ = 0, we can get delay-dependent and rate-independent stability criterion from Theorem 1.

Next, we study the robust stability of the uncertain system (1).

• Theorem 2. For given values of h₁, h₂ and μ, system (1) is asymptotically stable, if there exist a scalar ε _i > 0(i = 1, 2), and real symmetric positive definitive matrices P₁, P₂, P₃, Z₁, Z₂, symmetric matrices Q, S and free matrix variables L _j , M _j , V _j , T _j (j = 1, 2), with appropriate dimensions such that the following LMIs hold:

  $$\left[ {\begin{array}{*{20}c} {\hat \Phi _1 } & {\Gamma _1^T D} & {\varepsilon _1 \Gamma _2^T } \\ * & { - \varepsilon _1 I} & 0 \\ * & * & { - \varepsilon _1 I} \\ \end{array} } \right] < 0$$

  (19)
  $$\left[ {\begin{array}{*{20}c} {\hat \Phi _2 } & {\Gamma _1^T D} & {\varepsilon _2 \Gamma _2^T } \\ * & { - \varepsilon _2 I} & 0 \\ * & * & { - \varepsilon _2 I} \\ \end{array} } \right] < 0$$

  (20)

  and

  $$\begin{array}{*{20}c} {Q = \left[ {\begin{array}{*{20}c} {Q_{11} } & {Q_{12} } & \cdots & {Q_{1N} } \\ * & {Q_{22} } & \cdots & {Q_{2N} } \\ \vdots & \vdots & \ddots & \vdots \\ * & * & {Q_{N - 1,N - 1} } & {Q_{N - 1,N} } \\ * & * & * & {Q_{NN} } \\ \end{array} } \right] \geqslant 0} \\ {S = \left[ {\begin{array}{*{20}c} {S_{11} } & {S_{12} } & \cdots & {S_{1N} } \\ * & {S_{22} } & \cdots & {S_{2N} } \\ \vdots & \vdots & \ddots & \vdots \\ * & * & {S_{N - 1,N - 1} } & {S_{N - 1,N} } \\ * & * & * & {S_{NN} } \\ \end{array} } \right] \geqslant 0} \\ \end{array} $$

  where

  $$\begin{array}{*{20}c} {\hat \Phi _1 = \left[ {\begin{array}{*{20}c} \Phi & {\sqrt {\frac{{h_a }} {N}} L} & {\sqrt {h_\delta } M} \\ * & { - Z_1 } & 0 \\ * & * & { - Z_2 } \\ \end{array} } \right]} \\ {\hat \Phi _2 = \left[ {\begin{array}{*{20}c} \Phi & {\sqrt {\frac{{h_a }} {N}} L} & {\sqrt {h_\delta } V} \\ * & { - Z_1 } & 0 \\ * & * & { - Z_2 } \\ \end{array} } \right]} \\ {\Gamma _1 = \left[ {\begin{array}{*{20}c} {T_1^T } & 0 & \cdots & 0 & {T_2^T } & 0 & 0 \\ \end{array} } \right]} \\ {\Gamma _2 = \left[ {\begin{array}{*{20}c} {E_a } & 0 & \cdots & 0 & {E_b } & 0 & \cdots & 0 \\ \end{array} } \right].} \\ \end{array} $$

• Proof. Replacing A and B in Theorem 1 with A + ΔA, B + ΔB, respectively, and using Lemma 3 completes the proof. □

Numerical examples

In this section, three numerical examples are given to show that the proposed approach reduces the conservativeness compared with some of the existing ones.

• Example 1[3]. Consider a nominal time-delay system with the following parameters:

  $$\begin{array}{*{20}c} {A = \left[ {\begin{array}{*{20}c} { - 2} & 0 \\ 0 & { - 0.9} \\ \end{array} } \right],} & {B = \left[ {\begin{array}{*{20}c} { - 1} & 0 \\ { - 1} & { - 1} \\ \end{array} } \right].} \\ \end{array} $$

  (21)

  For given μ and unknown μ, Tables 1 and 2 provides the maximal allowable bounds of the delay h₂ for given lower bounds h₁, respectively. From the tables, it is clear that the proposed stability criterion is less conservative than those in [3, 14, 16, 19–22] even when we use small delay partitioning number. Furthermore, with the increase of the delay partitioning number, the conservatism will gradually reduce. The less conservative results are mainly attributed to the use of the tighter integral inequalities (Lemma 1) for dealing with the cross-terms.

• Example 2[13]. Consider an uncertain system described by the matrices as

  $$\begin{array}{*{20}c} {A = \left[ {\begin{array}{*{20}c} { - 2 + \delta _1 } & 0 \\ 0 & { - 1 + \delta _2 } \\ \end{array} } \right],} & {B = \left[ {\begin{array}{*{20}c} { - 1 + \delta _3 } & 0 \\ { - 1} & { - 1 + \delta _4 } \\ \end{array} } \right]} \\ \end{array} $$

  (22)

  where

  $$\vert {\delta _1}\vert \leq 1.6,\vert {\delta _2}\vert \leq 0.05,\vert {\delta _3}\vert \leq 0.1,\vert {\delta _4}\vert \leq 0.3.$$

  When there is no restriction on the delay derivative, Table 3 shows the obtained maximum allowable delay bound for a varying h₁. From the table, it is clear to see that for this example, some existing results have been improved.

• Example 3[16]. Consider another uncertain system described by the matrices

  $$\begin{array}{*{20}c} {\begin{array}{*{20}c} {A = \left[ {\begin{array}{*{20}c} { - 0.5} & { - 2} \\ 1 & { - 1} \\ \end{array} } \right],} & {B = } \\ \end{array} \left[ {\begin{array}{*{20}c} { - 0.5} & { - 1} \\ 0 & {0.6} \\ \end{array} } \right],} \\ {\begin{array}{*{20}c} {D = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right],} & {E_a = } \\ \end{array} \left[ {\begin{array}{*{20}c} {0.2} & 0 \\ 0 & {0.2} \\ \end{array} } \right],} \\ {E_b = \left[ {\begin{array}{*{20}c} {0.2} & 0 \\ 0 & {0.2} \\ \end{array} } \right].} \\ \end{array} $$

Table 1 Maximum allowable delay bound h₂ for given h₁ and μ

Table 2 Maximum allowable delay bound h₂ for given h₁ and unknown μ

Table 3 Upper delay bound h₂ for given h₁

When h₁ = 0, Table 4 lists the maximum allowable delay bounds for different values of the delay derivative. It is found that the maximum allowable upper bounds of time-delay are 0.5594, 0.5599, 0.5601 respectively when N = 2, 3, 4, which are larger than those in [16, 21, 22]. Obviously, our criterion leads to much less conservative results.

Table 4 Maximum allowable delay bound h₂ for h₁ =0

Conclusions

This paper proposes a new approach for delay-dependent robust stability analysis of uncertain system with interval time-varying delay. The key features of the approach include delay decomposition technique for designing the L-K functional and a tighter integral inequality for bounding the cross-terms. As a result, less conservative results are achieved. Numerical examples have illustrated the effectiveness of the proposed method.