Novel Results for Global Exponential Stability of Uncertain Systems with Interval Time-varying Delay

Abstract

This paper presents new results on delay-dependent global exponential stability for uncertain linear systems with interval time-varying delay. Based on Lyapunov-Krasovskii functional approach, some novel delay-dependent stability criteria are derived in terms of linear matrix inequalities (LMIs) involving the minimum and maximum delay bounds. By using delay-partitioning method and the lower bound lemma, less conservative results are obtained with fewer decision variables than the existing ones. Numerical examples are given to illustrate the usefulness and effectiveness of the proposed method.

1. Introduction

Time delays are frequently encountered in many fields such as chemical engineering system, vehicles, biological modeling, economy and other fields. Thus the stability analysis of linear time delay systems has been received considerable attention during the last decades. Also, it is well known that the existence of time delay can lead oscillation, instability or divergence in system performance [1-2]. Thus, the stability analysis and synthesis of systems with time delay have been one of the hottest topic in control society. For recent trends of the topic, see [3-19] and references therein.

More recently, the stability analysis of linear systems with interval time-varying delay, which the lower bound is not restricted to be zero, has been extensively studied by many researchers [3-11, 18, 19]. A real example of such systems is networked control systems [13]. In general, the stability analysis of time-delay systems can be classified into two categories: delay-dependent [6-8] and delayindependent approach [12, 14]. It is well known that delaydependent criteria are less conservative than the delayindependent ones when the size of time-delay is small. Hence, more attentions have been paid to the study for checking the conservatism of delay-dependent conditions. In the field of delay-dependent stability analysis, an important issue is to enlarge the feasibility region of stability criteria or to obtain a maximum allowable upper bound of time delays as large as possible. In this regard, Shao [7] provided a new delay-dependent stability criterion for linear systems with interval time-varying delay by utilizing the convex combination method. In [9], less conservative results were derived with much fewer decision variables by introducing the lower bound lemma. Liu [11] constructed a new Lyapunov functional which makes the results less conservative than the results of [7, 9]. It should be noted that these results only focused their effort on asymptotic stability.

In practice, some uncertainties in the systems are unavoidable because it is very difficult to obtain an exact mathematical model due to the environmental noise, uncertain or slowly varying parameters. Thus, it is natural to consider the parameter uncertainties in a mathematical model. In addition, fast convergence of a system is essential for real-time computation, and the exponential convergence rate is generally used to determine the speed of computations. Therefore, the exponential stability analysis for systems with time delays has received deep concern in very recent years [14-19]. However, to the best of authors’ knowledge, there are few results about the exponential stability of uncertain linear system with interval time varying delays [18, 19]. In [18], some delaydependent sufficient conditions for the exponential stabilization of the systems are established in terms of LMIs by the construction of improved Lyapunov- Krasovskii functional combined with Leibniz Newton’s formula. In [19], the authors dealt with the same problem using two novel integral equalities. But, these results have two drawbacks. One is that the results cannot be applied when the time delay is differentiable [18, 19], the other is that it involves many matrix variables [19], which increase computation burden.

With this motivation, we revisit the problem of exponential stability of uncertain linear systems with interval time-varying delay. By constructing a new Lyapunov-Krasovskii functional, novel delay-dependent stability criteria with an exponential convergence rate are derived. Our results can be applied when the delay is differentiable. Less conservative results are obtained with fewer matrix variables than [19] by using delaypartitioning technique and the lower bound lemma. Finally, three numerical examples are shown to confirm the superiority of our results

Notations: Throughout this paper, I demotes the identity matrix with appropriate dimensions, Rn denotes the n dimensional Euclidean space, and Rm×n is the set of all m× n real matrices, ‖ ⋅ ‖ refers to the Euclidean vector norm and the induced matrix norm. For symmetric matrices A and B , the notation A > B (respectively, A ≥ B ) means that the matrix A−B is positive definite (respectively, nonnegative), λmax (⋅) and λmin (⋅) stand for the largest and smallest eigenvalue of given square matrix, respectively. diag{···} denotes the block diagonal matrix. ‖ Φ ‖ = sup −hM≤s≤0 {‖ x(t + s)‖, ‖ẋ(t + s) ‖}, where hM> 0 is some constant.

 

2. Problem Statement

Construction the following uncertain linear systems with time varying delay:

where x(t)∈Rn is the state, A and A1 are known real constant matrices with appropriate dimensions; φ (t) is the initial condition of the system. The time varying delay h(t) is differentiable function satisfying

where the bounds hm, hM, μ are known positive scalars.

The uncertainties satisfy the following conditions:

where D, E, E1 are known constant matrices, F(t) ∈ Rn×n is the unknown real time varying matrices with Lebesgue measurable elements bounded by

Therefore, system (1) with uncertainties satisfying (4) and (5) can be written in the following form:

We need the following definition and Lemmas for deriving the main results.

Definition 2.1 [16] For a given positive scalar k , the zero solution of (6) is exponentially stable if there exist a positive scalar γ such that every solution x(t) of (6) satisfies the following condition ‖ x (t)‖ ≤ γ e −kt ‖φ‖ .

Lemma 2.1 [7] For any constant positive definite matrix M ∈ Rn and β ≤ s ≤ α , the following inequalities hold

Lemma 2.2 [9] (Lower bounds lemma) Let f1 , f2 ,···, fN : Rm ➔ R have positive values in an open subset D of Rm . Then, the reciprocally convex combination of fi over D satisfies

Subject to

 

3. Main Results

In this section, new stability criteria for system (1) will be derived by use of Lyapunov method and LMI framework

3.1 Exponential stability for nominal systems with interval time-varying delay

First, we present a delay dependent exponential stability condition for the following nominal interval time-varying delay systems with ΔA(t) = 0, ΔA1(t) = 0 .

By introducing augmented Lyapunov-Krasovskii functional, a less conservative delay-dependent stability criterion for system (7) will be proposed. In convenience, let us define

The corresponding block entry matrices are defined as ei ∈ R6n×n(i=1, 2, ⋯, 6) e.g., ∈ R6n×n. Defining , then (7) can be written as .

Denote

Theorem 3.1 For given scalars hm, hM(hM > hm), μ, k ≥ 0 , system (7) is globally exponentially stable if there exist symmetric positive definite matrices ℚ1 ∈ ℝ2n×2n, ℚ2 ∈ ℝ2n×2n, n×n dimensional positive symmetric matrices P, R,U, S,Z1 , Z2 , and appropriate dimension matrices T,T1 , N satisfying the following LMIs:

Proof. Consider the following Lyapunov-Krasoskii functional

where

Calculating the time-derivative of V(t) , we have

Here, using Lemma 2.1, it can be obtained that

Also, the following inequality is obtained from Lemma 2.2,

It should be noted that when h(t) = hm or h(t)= hM , we have or , respectively. So the relation (15) still holds.

Form (13)-(15), we obtain

Inspired by the work of [3], the following two zero inequalities hold for any appropriate dimension matrix N ,

From the above zero equalities, one can obtain

Here, using Lemma 2.1, we have

In the following discussions, the upper bound of 6 is derived by considering two different cases for (i) hm ≤ h(t) ≤ h2 and (ii) h2 ≤ h(t) ≤ hM .

When hm ≤ h(t) ≤ h2, the following inequality is satisfied by Lemma 2.1

By using the similar methods in (15), we obtain

From (18)-(21), in this cases

If (9) hold, the following inequality is satisfied with (11)-(22) by S-procedure [1],

When hm ≤ h(t) ≤ h2, the following inequality is satisfied by Lemma 2.1

By using the similar methods in (15), we obtain

From (18)-(21), in this cases

If (9) hold, the following inequality is satisfied with (11)-(19) and (26) by using S-procedure [1],

Now we can conclude that if condition (8) and (9) are satisfied, then (x)(t)) ≤ 0. Thus, V(x(t)) ≤ V(x(0)). Furthermore, from the definition of V(x(t)) and (10), we can derive the following inequalities: V(x(0)) ≤ a ||φ||2, where

On the other hand, we have Hence,

Then, the proof is completed by the Lyapunov stability theorem.

Remark 3.1 When 0 ≤ hm ≤ h(t) ≤ hM , the most attractive contribution is that we have made the best use of the lower bound (network induced delay) of the interval time-varying delay. In fact, in order to derive the less conservative stability criterion, we employ a new Lyapunov-Krasovskii functional (10), which is mainly based on the information about ,

Remark 3.2 When μ is known, Theorem 3.1 can be applied while [8, 16, 17] fails to work. If μ is unknown or h(t) is not differentiable, then the following result can be obtained from Theorem 3.1 by setting R = 0 , which will be introduced as Corollary 3.1.

Corollary 3.1 For given scalars hm, hM(hM > hm), k ≥ 0 , system (7) is globally exponentially stable if there exist symmetric positive definite matrices ℚ1 ∈ ℝ2n×2n, ℚ2 ∈ ℝ2n×2n, n×n dimensional positive symmetric matrices P, U, S, Z1 , Z2 , and appropriate dimension matrices T,T1, N satisfying the following LMIs:

where 1 = Σ1 + Σ3 + Σ4 + Σ5 + Σ6 , 2 = Σ1 + Σ3 + Σ4 + Σ5 + Σ7.

3.2 Robust exponential stability for uncertain systems with interval time varying delay

Based on the result of Theorem 3.1, the following theorem provides a robust exponential stability condition of uncertain linear systems with interval time-varying delay (6).

For simplicity in Theorem 3.2, ei ∈ R7n×n(i=1,2,···, 7) are defined as block matrices. (for example, =[0 0 0 I 0 0 0]T ∈ R7n×n). The other notations for some vectors and matrices are defined as:

Theorem 3.2 For given scalars hm, hM(hM > hm), μ, k ≥ 0 , system (6) is globally exponentially stable if there exist symmetric positive definite matrices ℚ1 ∈ ℝ2n×2n , ℚ2 ∈ ℝ2n×2n, n×n dimensional positive symmetric matrices P, R,U, S, Z1 , Z2 , and appropriate dimension matrices T, T1 , N satisfying the following LMIs:

Proof. By considering the same Lyapunov-Krasovskii functional in Theorem 3.1, the upper bounds of (t) are obtained as

On the other hand, the following inequality holds From (4)-(6)

By S-procedure [1], we can derive inequalities (32) with a positive scalar ε such that

Therefore, the uncertain system (6) is exponentially stable, if LMIs (30-31) hold. This completes our proof.

The following result is obtained from Theorem 3.2 when μ is unknown or h(t) is not differentiable.

Corollary 3.2 For given scalars hm, hM(hM > hm) k ≥ 0 , system (7) is globally exponentially stable if there exist symmetric positive definite matrices ℚ1 ∈ ℝ2n×2n , ℚ2 ∈ ℝ2n×2n n×n dimensional positive symmetric matrices P,U, S, Z1 , Z2 , and appropriate dimension matrices T,T1 , N satisfying the following LMIs:

where

Remark 3.3 For system (7) with the routine delay case described by 0 ≤ h(t) ≤ hM ≤ hM(h(t) ≤ μ), the corresponding Lyapunov-Krasovskii reduces to

Similar to the proof of Theorem 3.1(Corollary 3.1) and Theorem 3.2(Corollary 3.2), one can easily derive less conservative results than some existing ones.

Remark 3.4 In most of the applications of neural networks there is a shared requirement of raising the networks convergence speed in order to cut down the time of neural computing. Since the exponential convergence rate could be used to determine the speed of neural computation [20, 21]. On the other hand, from the Dynamic simulation systems [22] and Real-time computing [23], it can be confirmed that, fast convergence of a system is essential for real-time computation, and the exponential convergence rate is generally used to determine the speed of computations.

Remark 3.5 By constructing a new augmented Lyapunov-Krasosvskii functional and using lower bound lemma, exponential stability criteria have been obtained which are expected to be less conservative than the results discussed in the recent literature [5, 8, 11]. The effectiveness of the proposed methods has been shown elaborately through the following numerical examples.

 

4. Numerical examples

Example 4.1 Consider the system given in (7) with following parameters

Case 1: When hm = 0.5, hM = 1. Table 1 gives the allowable of the maximum exponential convergence rate k for different μ. For this case, the exponential stability criteria in [8, 11] are not applicable because the criteria are only for asymptotic stability.

Case 2: For various μ and unknown μ , the allowable bound hM , which guarantee the asymptotic stability of system for given lower bounds hm are provided in Table 2. It is easy to see that our method gives improved results than the existing ones.

Example 4.2 Consider the system given in (6) with following parameters

For 0 ≤ hm ≤ h(t) ≤ hM, Table 3 presents the allowance of the upper bound when μ is unknown and the convergence k = 0 . It is obvious that our results are better than [5].

Table 1.The maximum exponential convergence rate k for various μ

Table 2.The maximum bound hM with given hm for different μ

Table 3.The maximum bound hM with given hm for unknown μ

Table 4.hm = 0.5, μ = 0.5, The maximum bound hM for various k

Example 4.3 Consider the system given in (6) with following parameters

Given hm = 0.5, μ = 0.5, Table 4 gives the maximum allowable value of hM for different convergence rate k . For this cases, it is noted that the criteria in [5, 8, 11] are not applicable for exponential stability. Therefore, our work is more general cases than in the existing results

 

5. Conclusion

In this paper, the problem of delay-dependent exponential stability of time-delay systems has been investigated. We have considered the time-varying delay in a range for which the lower bound is not restricted to be zero. By introducing a different Lyapunov functional, new delay-dependent criteria have been derived in terms of LMIs. It is shown via numerical examples that our proposed criteria are less conservative than existing ones.