Stability analysis for continuous system with additive time-varying delays: A less conservative result

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Abstract

This paper presents a less conservative result for stability analysis of continuous-time systems with additive delays by constructing a new Lyapunov–Krasovskii functional and utilizing free matrix variables in approximating certain integral quadratic terms in obtaining the stability condition in terms of linear matrix inequalities. Numerical example is provided to show the effectiveness of the proposed method compared to some recent results.

Introduction

All physical systems possess inherently time-delay in it. However, they are often mathematically modeled neglecting the effect of these delays leading to a simplified model of the system which is actually not the situation in the real world. Presence of time-delay makes the dynamical model as infinite dimensional one and neglecting it leads to an approximation to finite dimension whose analysis is easier but yields approximated results. According to [6], [14], [15], [17], the presence of time-delay in the model can be a source of instability and poor performance. From engineering and applied science perspective it is required to consider a dynamic model that can behave close to the real world system. This requirement necessitates that the information of the states at both present and past time instants are needed to be included in the dynamic model. In context to the control problem of a physical system, if the considered model is accurate then the designed control action for the system will also be accurate. Hence, one can find many results concerning stability analysis and stabilization of time-delay systems, e.g., [3], [13], [16], [17], [19], [20]. In [2], [3], [9], [10], [11], [13], [16], [17], [18], the system with time-delays have been modeled in state-space forms with the delay appearing in the states in a singular or simple form as x˙(t)=Ax(t)+Adx(t-τ(t)), where, τ(t) is the time-varying delay in the states (x(t)) that is unknown and needs to be estimated by deriving stability conditions.

Recently in [5], [7] it was pointed out that, in networked controlled system (NCS), if the signal transmitted from one point to another passes through few segments of networks then successive delays are induced with different properties due to variable transmission conditions, thus it is appropriate to consider time-delays in the dynamical model as x˙(t)=Ax(t)+BKx(t-τ1(t)-τ2(t)) where, τ1(t) is the time-delay induced from sensor to controller and τ2(t) is the delay induced from controller to the actuator. The stability analysis of such system was earlier carried out by adding up all the successive delays into single delay i.e., τ1(t)+τ2(t)=τ(t) to develop a sufficient stability condition. A less conservative delay-dependent stability analysis has been formulated in [7] and subsequently its improvement can be found in [5] by considering the delays separately.

In this work, considering the time-delay model of [5], [7] we derive a new and improved delay-dependent stability condition for system with multiple additive delay components. The sufficient stability condition is derived using Lyapunovs’ second method by proposing a new Lyapunov–Krasovskii functional candidate and making use of improved techniques for formulating quadratic state-space dynamics [2], [18] in the LMI framework [1], the obtained condition can be solved as convex optimization problem using LMI toolbox of MATLAB® [4]. We provide an illustrative example of [7] to show that the new stability condition proposed in this paper is less conservative in terms of the estimation of upper bound of the delays then some existing LMI conditions.

Notations

P=PT>0 means that matrix P is a symmetric positive definite matrix and the symmetric term in the symmetric matrix is denoted by ‘*’, as for examplePQR=PQQTR.

Section snippets

Problem definition

Consider a time-delay system,x˙(t)=Ax(t)+Adx(t-τ(t)),where τ(t) is the time-varying delay obtained due to successive addition of various delay factors in the system of the same nature, satisfying certain conditions like0τ(t)τ¯<,τ˙(t)d<.

In [5], [7] a new mathematical model with multiple additive time-delays has been proposed, which relates to the practical situation in networked control system (NCS) has the following form:x˙(t)=Ax(t)+Adxt-i=1sτi(t),0τi(t)τ¯i<,τ˙i(t)di<.

To make the

Main result

A new Lyapunov–Krasovskii functional is proposed in this section for stability analysis of system (4) satisfying (6).

Theorem 1

The system described in (4) satisfying conditions (6) is asymptotically stable if there exists P=PT>0,Q1=Q1T>0,Q2=Q2T>0,Q3=Q3T>0,R1=R1T>0,R2=R2T>0,R3=R3T>0 and G1,G2,G3,G4,M1,M2,M3,M4,N1,N2,N3andN4 are free matrices with Q2Q3 satisfying following LMI,Ω11Ω12Ω13Ω14L1M1N1Ω22Ω23Ω24L2M2N2Ω33Ω34L3M3N3Ω44L4M4N4-1τ¯R100-1τ¯1R20-1τ¯2R3<0,whereΩ11=Q1+Q2+G1A+ATG1T+L1+L1

Illustrative example

We consider the example in [5], [7] to show that the proposed theorem provides less conservative delay bound compared to the existing result. The system in (4) is considered with the following parameters given in [7], [8].A=-200-0.9andAd=-10-1-1,assumingτ˙1(t)0.1,τ˙2(t)0.8

We calculate the delay upper bound τ¯1 of τ1(t) or τ¯2 of τ¯2(t), when other is known. By combining the two delay factors the results by some existing stability theorems have been provided in Table 1. The results validates

Conclusions

A new stability condition for linear system with two additive time-varying delays in the states has been proposed in the LMI framework. The proposed LMI has been obtained by utilizing a new Lyapunov–Krasovskii functional candidate which eliminates the over design in the Lyapunov–Krasovskii functional considered in [7], [5], thus providing less conservative delay upper bound that is validated from the result presented in Table 1. Also it must be emphasized here that, the less number of matrix

Acknowledgement

This work is supported by All India Council of Technical Education, Govt. of India under Research Promotion Scheme (RPS), vide Grant No. 8023/BOR/RID/RPS-229/2008-2009.

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Supported by All India Council of Technical Education, Govt. of India (Grant No. 8023/BOR/RID/RPS-229/2008-2009).

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