New stability criteria for a class of neutral systems with discrete and distributed time-delays: an LMI approach

Abstract

In this paper, a problem of the asymptotic stability for a class of neutral systems with multiple discrete and distributed time-delays is considered. Lyapunov stability theory is applied to guarantee the stability for the systems. New discrete-delay-independent and discrete-delay-dependent stability conditions are derived in terms of the spectral radius and linear matrix inequality. By mathematical analysis, the stability criteria are proved to be less conservative than the ones reported in the current literatures. A numerical example is given to illustrate the availability of the proposed results.

Introduction

In the recent years, stability analysis of time-delay systems has received considerable attention and has been one of the most interesting topics in the control systems. This is due to theoretical interests as well as a powerful tool for practical system analysis and design, since delay phenomenon is often encountered in various mechanics, physics, biology, medicine, economy, and engineering systems, such as AIDS epidemic, aircraft stabilization, chemical engineering systems, control of epidemics, distributed networks, infeed grinding and cutting model, manual control, microwave oscillator, models of lasers, neural network, nuclear reactor, population dynamic model, rolling mill, ship stabilization, and systems with lossless transmission lines [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Moreover, time-delay is frequently a source of instability and a source of generation of oscillation in many systems; for example, the trivial solution of $\text{(a−1)·[}\text{x}\text{̈}\text{(t)+}\text{x}\text{̇}\text{(t)+x(t)]=0}$, with a>1, is stable, but that of system

$$\text{x}\text{̈}\text{(t)+}\text{x}\text{̇}\text{(t)+x(t)=a·[}\text{x}\text{̈}\text{(t−h)+}\text{x}\text{̇}\text{(t−h)+x(t−h)]}$$

is unstable for any h>0 [13].

Consider the linear system

$$\text{x}\text{̇}\text{(t)=}\text{A+∑}{}_{\mathrm{i=1}}{}^{\mathrm{m}}\text{E}{}_{\mathrm{i}}\text{x(t),}\text{t⩾0,}$$

where $\text{x(t)∈}\text{R}{}^{\mathrm{n}}$, A, $\text{E}{}_{\mathrm{i}}\text{∈}\text{R}{}^{\mathrm{n\times n}}$, $\text{i∈}\text{m}\text{̄}$. The necessary and sufficient condition for asymptotic stability of system (1) is that the matrix A+∑_i=1^mE_i is Hurwitz. Recently, many reports are concentrated on systems with discrete delay [4], [8], [9], [10], [11], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. By increasing in the equation number of summands and simultaneously decreasing the differences between neighbouring argument values, one naturally arrives at equations with distributed (or continuous) and mixed (both distributed and discrete) delay arguments [2], [3], [6], [7], [13]. Now, we consider the neutral systems with discrete and distributed time-delays:

$$\text{x(t)=φ(t),}\text{t∈[−H,0],}$$

where $\text{x∈}\text{R}{}^{\mathrm{n}}$, x_t is the state at time t defined by $\text{x}{}_{\mathrm{t}}\text{(s):=x(t+s)}\text{∀s∈[−H,0]}$, $\text{H=}\text{max}{}_{\mathrm{<mtext>i\in </mtext><mtext>m</mtext><mtext>̄</mtext>}}\text{{h}{}_{\mathrm{i}}\text{,τ}{}_{\mathrm{i}}\text{}⩾0}$, with ∥x_t∥_s:=sup_−H⩽r⩽0∥x(t+r)∥, h_i and τ_i, $\text{i∈}\text{m}\text{̄}$, are both non-negative constants, which represent discrete delays and distributed delays, respectively. The matrices A, B_i, C_i, $\text{D}{}_{\mathrm{i}}\text{∈}\text{R}{}^{\mathrm{n\times n}}$, $\text{i∈}\text{m}\text{̄}$, are known, and the initial vector φ∈C₀.

In view of [2], [3], [4], [5], [6], [13], the distributed delays τ_i, $\text{i∈}\text{m}\text{̄}$, play an important role about the stability of system , , so our main results will also depend on those distributed delays τ_i, $\text{i∈}\text{m}\text{̄}$. Depending on whether the stability criterion itself contains the discrete delay argument as a parameter, stability criteria for neutral systems can be classified into two categories, namely discrete-delay-independent criteria [4], [7], [8], [16], [20], [21] and discrete-delay-dependent criteria [2], [3], [4], [6], [16], [17], [20], [21]. Generally speaking, the latter ones are less conservative than the former ones, but the former ones are also important when the effect of time delay is small. In this paper, the discrete-delay-independent and discrete-delay-dependent criteria will be proposed to guarantee the asymptotic stability for neutral systems with multiple time-delays. Many approaches have been used to searching sufficient conditions for the stability problem of time-delay systems. The stability problem for time-delay systems with distributed delay is considered using discretized Lyapunov functional [8]. A robust stability of time-delay systems was investigated by checking the Hamiltonian matrix and solving an algebraic Riccati equation [23]. Appropriate model transformations of original time-delay systems and Lyapunov theory are useful for the analysis of stability of systems [5], [11], [12], [13], [14], [15], [16], [20], [21], [22], [23]. Many sharp results are used the linear matrix inequality (LMI) approach to solve the stability problem of time-delay systems [5], [9], [14], [16], [20], [21], [22]. Furthermore, a model transformation and Lyapunov theory with LMI approach are used in this paper, less conservative criteria are proposed to guarantee the asymptotic stability for the neutral systems with discrete and distributed time-delays. A numerical example is given to illustrate the validity of the proposed results.