New Absolute Stability Conditions of Lur ’ e Systems with Time-Varying Delay

This paper is focused on the absolute stability of Lur’e systems with time-varying delay. Based on the quadratic separation framework, a complete delay-decomposing Lyapunov-Krasovskii functional is constructed. By considering the relationship between the time-varying delay and its varying interval, improved delay-dependent absolute stability conditions in terms of linear matrix inequalities (LMIs) are obtained. Moreover, the derived conditions are extended to systems with time-varying structured uncertainties. Finally, a numerical example is given to show the advantage over existing literatures.


Introduction
Since the concept of absolute stability and the Lur' e problem was introduced in 1940s, the absolute stability of Lur' e control systems has received much attention and many rich results have been proposed [1][2][3][4][5][6].As time-delay is frequently encountered in practical systems and is often a source of instability and poor performance, the problem of absolute stability of Lur' e systems with time-delay has been attracting much attention [7][8][9][10][11][12][13][14].
In [10,11], the absolute stability of Lur' e systems with time-invariant delay was addressed.In the case that the delay is time-varying, the problem of absolute stability has also been investigated in [12].In addition, by retaining some useful information and employing an improved free-matrix approach to consider the relationship between the timevarying delay and its upper bound, some less conservative criteria are obtained in [13].Nevertheless, the results obtained in [13] are based on simple Lyapunov-Krasovskii functionals and are still conservative.In [14], the absolute stability of Lur' e systems with interval-varying delay has been investigated via a delay decomposition approach.However, there is room for further improvement.
Recently, a delay decomposition method was proposed in [15], which significantly reduced the conservatism of the derived stability conditions for systems with time-varying delay.However, as pointed out in [16], the term () − ( − 1) with ( − 1) ≤ () ≤  was enlarged as  and another term  − () was also regarded as  in [15], leading to the conservatism.Moreover, in the construction of Lyapunov-Krasovskii functional, the term ∫  −()   ()() has been universally employed in [15]; that is,  ≥ 0 is kept on the whole delay interval [ − (), ], which is also a source of conservatism for systems with time-varying delay.
In this paper, a complete-decomposing Lyapunov-Krasovskii functional is employed to investigate the absolute stability of Lur' e systems with a time-varying delay.By considering the relationship between the time-varying delay and its varying interval, improved delay-dependent absolute stability conditions are presented in the linear matrix inequality (LMI) setting.Finally, a numerical example is given to demonstrate the effectiveness and the merits of the presented method.
Notation.Throughout this paper, the superscripts "−1" and "" stand for the inverse and transpose of a matrix, respectively;   denotes the -dimensional Euclidean space;  × is the set of all  ×  real matrices;  > 0 means that the matrix  is symmetric and positive definite;  is an appropriately

Problem Statement
Consider the following system: where () ∈   , () ∈   , and () ∈   are the state, input, and output vectors of the system, respectively; , , , , and  are constant matrices with appropriate dimensions; the initial condition () is a continuous vectorvalued function of  ∈ [−ℎ,0].(, ()) ∈   is a nonlinear function, which is piecewise continuous in , globally Lipschitz in (), (, 0) = 0, and satisfies the following sector condition ∀ ≥ 0 and ∀() ∈   : where where ℎ and  are constants.First, we introduce the following definition of absolute stability.
In this paper, we investigate not only the absolute stability of nominal system (1), but also the following system with time-varying structured uncertainties: where the time-varying structured uncertainties are of the form ,   , and   are appropriately dimensioned constant matrices, and () is an unknown real and possibly time-varying matrix satisfying Before presenting our main results, we first introduce two lemmas, which are useful in the stability analysis of the considered system.
Then, for any scalar  > 0
For the absolute stability of system (1), we have the following result.

Numerical Example
In this section, we will present a numerical example to show the effectiveness of the proposed method.
For different  and , the maximum allowable time-delay bounds (MATB) obtained by Theorem 6 and those methods in [12][13][14] are listed in Table 1.It is clear that our results are less conservative than those in [12][13][14].Furthermore, it can be concluded that the larger the  is, the less conservative results we obtain and the more computing time we consume.As a compromise between less conservative results and CPU computing time, taking  = 2 or  = 3 is a good choice.

Conclusions
In this paper, the problem of absolute stability of Lur' e systems with a time-varying delay has been investigated.A complete-decomposing Lyapunov-Krasovskii functional has been proposed, in which all integral terms including delay are decomposed.By introducing some free-weighting matrices to consider the relationship between time-varying delay and its varying interval, some improved conditions have been derived.A numerical example has been given to demonstrate the superiority over the existing ones.

Example 1 .
Consider the robust absolute stability of the uncertain system (5) with the following parameters: 1and  2 are constant real matrices of appropriate dimensions, and  =  2 −  1 is a symmetric positive definite matrix.Customarily, the nonlinear function, (, ()), is said to belong to the sector [ 1 ,  2 ].

Table 1 :
Maximum allowable time-delay bounds ℎ for different .