Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems

doi:10.1016/j.jfranklin.2015.01.004

Journal of the Franklin Institute

Volume 352, Issue 4, April 2015, Pages 1378-1396

https://doi.org/10.1016/j.jfranklin.2015.01.004 Get rights and content

Abstract

Finding integral inequalities for quadratic functions plays a key role in the field of stability analysis. In such circumstances, the Jensen inequality has become a powerful mathematical tool for stability analysis of time-delay systems. This paper suggests a new class of integral inequalities for quadratic functions via intermediate terms called auxiliary functions, which produce more tighter bounds than what the Jensen inequality produces. To show the strength of the new inequalities, their applications to stability analysis for time-delay systems are given with numerical examples.

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

Gu et al. [24], Jensen inequality

For a positive definite matrix $R > 0$ , and an integrable function ${w (u) | u \in [a, b]}$ , the following inequalities hold: $\int_{a}^{b} w^{T} (α) Rw (α) d α \geq \frac{1}{b - a} {(\int_{a}^{b} w (α) d α)}^{T} R (\int_{a}^{b} w (α) d α)$ $\int_{a}^{b} \int_{β}^{b} w^{T} (α) Rw (α) d α d β \geq \frac{2}{{(b - a)}^{2}} {(\int_{a}^{b} \int_{β}^{b} w (α) d α d β)}^{T} R (\int_{a}^{b} \int_{β}^{b} w (α) d α d β)$ $\int_{a}^{b} \int_{a}^{β} w^{T} (α) Rw (α) d α d β \geq \frac{2}{{(b - a)}^{2}} {(\int_{a}^{b} \int_{a}^{β} w (α) d α d β)}^{T} R (\int_{a}^{b} \int_{a}^{β} w (α) d α d β) .$

Lemma 2.2

Park et al. [12], Reciprocal convexity lemma

For any vectors $x_{1}, x_{2}$ , matrices $R > 0, S$ , and real scalars $α \geq$ 0, $β \geq 0$ satisfying $α + β = 1$ , the following inequality holds: $- \frac{1}{α} x_{1}^{T} {Rx}_{1} - \frac{1}{β} x_{2}^{T} {Rx}_{2} \leq - {[\begin{matrix} x_{1} \\ x_{2} \end{matrix}]}^{T} [\begin{matrix} R & S \\ S^{T} & R \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]$ subject to $0 < [\begin{matrix} R & S \\ S^{T} & R \end{matrix}] .$

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 $R > 0$ , an integrable function ${w (u) | u \in [a, b]}$ , and an auxiliary scalar function ${\bar{p} (u) | u \in [a, b]}$ satisfying $\int_{a}^{b} \bar{p} (α) d α = 0$ , the following inequality holds: $\int_{a}^{b} w^{T} (α) Rw (α) d α \geq \frac{1}{b - a} {(\int_{a}^{b} w (α) d α)}^{T} R (\int_{a}^{b} w (α) d α) + {(\int_{a}^{b} {\bar{p}}^{2} (α) d α)}^{- 1} {(\int_{a}^{b} \bar{p} (α) w (α) d α)}^{T} R (\int_{a}^{b} \bar{p} (α) w (α) d α) .$

Proof

Let us find a constant vector $v$ to minimize the following energy function of ${z (u) | u \in [a, b]}$ for ${w (u) | u \in [a, b]}$

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 $R > 0$ , an integrable function ${w (u) | u \in [a, b]}$ , and an auxiliary scalar function ${\bar{p} (u) | u \in [a, b]}$ , the following inequalities hold: $\int_{a}^{b} \int_{β}^{b} w^{T} (α) Rw (α) d α d β \geq \frac{2}{{(b - a)}^{2}} {(\int_{a}^{b} \int_{β}^{b} w (α) d α d β)}^{T} R (\int_{a}^{b} \int_{β}^{b} w (α) d α d β) + {(\int_{a}^{b} \int_{β}^{b} {\bar{p}}^{2} (α) d α d β)}^{- 1} {(\int_{a}^{b} \int_{β}^{b} \bar{p} (α) w (α) d α d β)}^{T} \times R (\int_{a}^{b} \int_{β}^{b} \bar{p} (α) w (α) d α d β),$ $\int_{a}^{b} \int_{a}^{β} w^{T} (α) Rw (α) d α d β \geq \frac{2}{{(b - a)}^{2}} {(\int_{a}^{b} \int_{a}^{β} w (α) d α d β)}^{T} R (\int_{a}^{b} \int_{a}^{β} w (α) d α d β) + {(\int_{a}^{b} \int_{a}^{β} {\bar{p}}^{2} (α) d α d β)}^{- 1} {(\int_{a}^{b} \int_{a}^{β} \bar{p} (α) w (α) d α d β)}^{T}$

Application to stability analysis of time-delay systems

Consider the following system with time-varying delay: $\dot{x} (t) = Ax (t) + A_{d} x (t - h (t)), t \geq 0, x (t) = ϕ (t), - h_{2} \leq t \leq 0,$ where the initial condition $ϕ (t)$ is a continuously differentiable function and the delay h(t) satisfies $0 \leq h_{1} \leq h (t) \leq h_{2},$ $h_{12} = h_{2} - h_{1} .$

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 $R > 0$ , and a differentiable function ${x (u) | u \in [a, b]}$ , 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) with $A = [\begin{matrix} 0.0 & 1.0 \\ - 10.0 & - 1.0 \end{matrix}], A_{d} = [\begin{matrix} 0.0 & 0.1 \\ 0.1 & 0.2 \end{matrix}] .$ The 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

References (24)

J.-P. Richard
Time-delay systemsan overview of some recent advances and open problems
Automatica
(2003)
P. Park et al.
Stability and robust stability for systems with a time-varying delay
Automatica
(2007)
J.W. Ko et al.
Delay-dependent stability criteria for systems with asymmetric bounds on delay derivative
J. Frankl. Inst.
(2011)
C. Jeong et al.
Improved approach to robust stability and $H_{\infty}$ performance analysis for systems with an interval time-varying delay
Appl. Math. Comput.
(2012)
O.M. Kwon et al.
Analysis on robust $H_{\infty}$ performance and stability for linear systems with interval time-varying state delays via some new augmented Lyapunov–Krasovskii functional
Appl. Math. Comput.
(2013)
O.M. Kwon et al.
Improved approaches to stability criteria for neural networks with time-varying delays
J. Frankl. Inst.
(2013)
D. Yue et al.
Network-based robust control of systems with uncertainty
Automatica
(2005)
H. Shao
New delay-dependent stability criteria for systems with interval delay
Automatica
(2009)
J. Sun et al.
Improved delay-range-dependent stability criteria for linear systems with time-varying delays
Automatica
(2010)
P. Park et al.
Reciprocally convex approach to stability of systems with time-varying delays
Automatica
(2011)

Y. Liu et al.

A novel approach on stabilization for linear systems with time-varying input delay

Appl. Math. Comput.

(2012)

W. Qian et al.

New stability analysis for systems with interval time-varying delay

J. Frankl. Inst.

(2013)

Cited by (683)

Relaxed criteria on admissibility analysis for singular Markovian jump systems with time delay
2024, Nonlinear Analysis: Hybrid Systems
This article studies the problem of admissibility analysis for singular Markovian jump systems with time delay. Without introducing more variables, an augmented Lyapunov–Krasovskii functional (LKF) is presented, which does not require all matrices to satisfy the positive definite constraint. Inspired by free-matrix-based inequality, an improved version is proposed to obtain more accurate estimation of the integral term. Based on the novel LKF and the improved inequality, some relaxed stochastic admissibility criteria are proposed, which are less conservative. Finally, several examples are offered to verify the effectiveness of the proposed results.
Stability and stabilization of systems with a cyclical time-varying delay via delay-product-type looped-functionals
2024, Journal of the Franklin Institute
In this paper, the stability analysis and synthesis of systems with a cyclical time-varying delay are investigated. At first, the entire delay interval is divided into intervals of monotonically increasing and monotonically decreasing alternating cycles. Then, a Lyapunov-Krasovskii functional (LKF) with two delay-product-type looped-functionals is constructed based on different delay monotone intervals. For the generated quadratic function with respect to delay in the derivative of LKF, a matrix-injection-based method is utilized to obtain the negative-definite conditions of LKF’s derivative. A delay-dependent stability criterion with less conservatism is derived and the corresponding state-feedback control strategy is given consequently. Finally, a numerical example is utilized to show the superiority and the effectiveness of the proposed method, and a single-area power system under load frequency control is applied to verify the feasibility of the proposed control strategy.
Hierarchical stability conditions and iterative reciprocally high-order polynomial inequalities for two types of time-varying delay systems
2024, Automatica
This paper investigates the stability problem of linear systems with a time-varying delay that the delay’s derivative has an upper bound or no constraint. Based on the state vectors and multiple integral state vectors involved in the delay related Bessel–Legendre inequalities (BLIs), the hierarchical Lyapunov–Krasovskii functionals (LKFs) are proposed. In order to deal with the fractions of the delay introduced by the BLIs, a novel iterative reciprocally high-order polynomial (IRHP) combination lemma is developed, which encompasses the existing reciprocally convex inequalities as special cases. Then, by introducing the specific matrix valued negative definite conditions (NDCs) of the odd number high degree polynomials, the hierarchical stability conditions are expressed in the form of linear matrix inequalities (LMIs). Eventually, the validity and advantages of the proposed conditions are illustrated through some classical numerical examples.
Improved looped-functional approach for dwell-time-dependent stability analysis of impulsive systems
2024, Nonlinear Analysis: Hybrid Systems
This paper studies the dwell-time-dependent stability analysis of impulsive systems by using a new time-square-dependent looped-functional. Based on the Lyapunov theory and two-sided looped-functional method, a time-square-dependent looped-functional is proposed, which fully utilizes the information of both the intervals $[t_{k}, t]$ and $[t, t_{k + 1}]$ . Then, by applying Jensen’s inequality and free-matrix-based inequalities to deal with integral terms in the functional derivatives, sufficient stability conditions in the form of linear matrix inequality are derived for periodic and aperiodic impulsive systems. Finally, numerical examples and simulation tests are given to illustrate the effectiveness and superiority of the proposed method.
Almost sure finite-time synchronization of Markov jump complex dynamical networks with asynchronous switching
2024, Neurocomputing
The almost sure finite-time synchronization for Markovian jump complex dynamical networks (CDNs) with time-varying delay is studied. The designed controller is supposed to be asynchronous with the CDNs states. The existing almost sure synchronization result of networks is explored within a finite time interval. The sampled-data control approach is employed to improve the control precision. Some sufficient criteria are obtained under the definition of almost sure finite-time synchronization for the CDNs with Markovian switching topology. To determine the asynchronous controller gain, the linear matrix inequality-based conditions are proposed. Then, two examples are given to illustrate the effectiveness.
General single/multiple integral inequalities and their applications to stability of time-delay systems
2024, Journal of the Franklin Institute
This paper is concerned with the stability of time delay systems. A new type of multiple integral inner product and a new class of multiple integral inner product space are firstly introduced. Based on them, general single/multiple integral inequalities are developed, and some previous integral inequalities including single and multiple integral inequalities can be regarded as their special cases. Secondly, suitable Lyapunov–Krasovskii functionals (LKF) are constructed. On the basis of the proposed integral inequalities, the derivative of the LKF can be gauged more accurate than those in the literature. As a result, some improved stability criteria of time-delay systems are formulated in terms of linear matrix inequalities. Through two numerical examples, it is shown that these criteria can provide larger delay upper bounds than some existing ones, showing the effectiveness and superiority of the proposed method.

View all citing articles on Scopus

View full text

Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems

Abstract

Introduction

Section snippets

Preliminaries

Gu et al. [24], Jensen inequality

Park et al. [12], Reciprocal convexity lemma

Auxiliary function-based single integral inequalities

Auxiliary function-based double integral inequality

Application to stability analysis of time-delay systems

Examples

Conclusion

Automatica

Automatica

J. Frankl. Inst.

Appl. Math. Comput.

Appl. Math. Comput.

J. Frankl. Inst.

Automatica

Automatica

Automatica

Automatica

Appl. Math. Comput.

J. Frankl. Inst.