Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems
Introduction
Many dynamic systems in the real world inevitably have time delays, and such delays often cause poor performance, oscillation or even instability of the system [1]. Consequently, the stability issue of time-delay systems has attracted vast attention from academic research, as a result, numerous studies on the stability analysis of time-delay systems have investigated in the past few decades [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].
In the field of stability analysis, obtaining tight bounds of various functions, especially integral terms of quadratic functions in the case of continuous-time systems, has had a key part in reducing the conservatism. There have been many approaches in the literature based on various mathematical tools such as integral inequality [2], [3], [4], [5], [6] and free-weighting matrix method [7], [8], [9], but the most recent studies have been based on the Jensen inequality [10], [11], [12], [13], [14], [15], [16] because this approach requires fewer decision variables than other existing approaches having a fine performance behavior.
Recently, the conservatism of the Jensen inequality has been analyzed in [17] using the Gruss inequality. Furthermore, an alternative inequality reducing the gap of the Jensen inequality has been proposed in [18] based on the Wirtinger inequality. The proposed inequality in [18] has been successfully applied to the stability analysis of various time-delay systems [19], [20], [21], [22]. Very recently, a novel integral inequality so-called Bessel–Legendre (B–L) inequality has been developed in [23] which encompasses the Jensen inequality and the Wirtinger-based integral inequality. However, inequalities in [18], [23] only deal with single integral terms of quadratic functions while upper bounds of double integral terms should also be estimated if triple integral terms are introduced in the Lyapunov–Krasovskii functional to reduce the conservatism. Furthermore, B–L inequality has only been applied to stability analysis of the system with constant delay.
In this paper, based on the above observation, we develop a new class of integral inequalities for quadratic functions via some intermediate terms called auxiliary functions. As a special case, the proposed inequalities turn into the existing inequalities such as the Jensen inequality, the Wirtinger-based integral inequality and the B–L inequality by appropriately choosing the auxiliary functions, which means the proposed ones are more general. We develop stability criteria for systems with time-varying delays using new inequalities based on appropriate Lyapunov–Krasovskii functionals. Numerical examples are provided to illustrate the effectiveness of the proposed inequalities.
Section snippets
Preliminaries
Lemma 2.1 For a positive definite matrix , and an integrable function , the following inequalities hold: Lemma 2.2 For any vectors , matrices , and real scalars 0, satisfying , the following inequality holds: subject to Gu et al. [24], Jensen inequality
Park et al. [12], Reciprocal convexity lemma
Auxiliary function-based single integral inequalities
In this section, new single integral inequalities for quadratic functions via auxiliary functions are provided. Lemma 3.1 For a positive definite matrix , an integrable function , and an auxiliary scalar function satisfying , the following inequality holds: Proof Let us find a constant vector to minimize the following energy function of for
Auxiliary function-based double integral inequality
In this section, new double integral inequalities for quadratic functions via auxiliary functions are given. Lemma 4.1 For a positive definite matrix , an integrable function , and an auxiliary scalar function , the following inequalities hold:
Application to stability analysis of time-delay systems
Consider the following system with time-varying delay:where the initial condition is a continuously differentiable function and the delay h(t) satisfies
Before deriving the theorem, we provide the following lemma for an application of the resulting inequalities (8), (13), (16), (18) to the stability analysis of time-delay systems. Lemma 5.1 For a positive definite matrix , and a differentiable function , the
Examples
In this section, we give two numerical examples to prove the effectiveness of the proposed approaches. Example 6.1 Consider the system (20) withThe purpose is to compare the maximum allowable upper bounds of h(t) that guarantees the asymptotic stability of the above system. For given h1, Table 1 gives the maximum allowable upper bounds and the number of decision variables obtained by various methods in [12], [15], [16] and this paper. From Table 1, it is clear that the
Conclusion
In this paper, novel integral inequalities for quadratic functions have been developed via some auxiliary functions. Compared to the Jensen inequality, the proposed inequalities have had extra terms which can help to obtain much tighter bounds. By applying the new integral inequalities, the stability criteria for time-delay systems have been successfully derived based on appropriate Lyapunov–Krasovskii functionals. Two numerical examples are provided to illustrate the effectiveness of the
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