Analysis and Design of Time-Delay Impulsive Systems Subject to Actuator Saturation

Chenhong Zhu ,,, Xiuping Han ,,, and Xiaodi Li ,,

Abstract—This paper investigates the exponential stability and performance analysis of nonlinear time-delay impulsive systems subject to actuator saturation.When continuous dynamics is unstable, under some conditions, it is shown that the system can be stabilized by a class of saturated delayed-impulses regardless of the length of input delays.Conversely, when the system is originally stable, it is shown that under some conditions, the system is robust with respect to sufficient small delayed-impulses.Moreover, the design problem of the controller with the goal of obtaining a maximized estimate of the domain of attraction is formulated via a convex optimization problem.Three examples are provided to demonstrate the validity of the main results.

I.INTRODUCTION

ALL actuators have limited capabilities in real control systems since practical control can only deliver limited magnitudes and rates of signals due to physical constraints.As is known to all, input saturation may cause performance deterioration and even instability [1].When input saturation is encountered, it makes sense to explore effective strategies to alleviate undesirable effects.Over the past decades many useful methods have been developed in this field.Recently, two types of methods to deal with the saturation function are widely applicable.In the first, polytopic differential inclusion is utilized to describe saturation nonlinearity [2], [3].The second main approach uses global/regional sector conditions which places saturation nonlinearity into a linear sector [4],[5].Stability analysis of nonlinear systems with input saturation has been extensively studied over the past years [6]–[9].

Impulsive systems have been extensively investigated as they provide effective mathematical models to deal with plants with discontinuous input [10]-[13].For example,impulsive phenomenon is ubiquitous in biology [14], mechanics [15] and neural networks [16].In general, there are two main kinds of impulsive effects: impulsive control and impulsive disturbance.More specifically, the first kind of impulsive effects corresponds to the case where impulses are used to control the continuous dynamics, while the second one concerned with the case where the behaviour of the system is subject to impulsive disturbance.Over the past decades, a large amount of results concerning different impulsive effects can be found in [17]-[19].In the process of transmission and sampling of the information, time delays are always inevitable[20]–[22].For instance, in the application of neural networks,time delays in dynamical nodes expresses response time and coupling delays refer to communication delays; in a financing institution, the decision of an investor is often influenced by both current transaction information and past transaction information of other investors, as shown in [23]-[26].On the other hand, saturated impulse is ubiquitous in practical applications, such as impulsive synchronization of neural networks in which signal transmission is limited due to the inherent physical constraints and instantaneous acceleration of mechanical systems in which performance is constrained by digital implementation.However, the relevant theoretical results related to impulsive saturation has been relatively less developed [27]-[29] on account of the complex coupling effects between impulses dynamics and input saturation.Recently,existence of a solution for impulsive differential equations under saturation was studied in [27].Reference [28] developed a linear differential inclusion method for exponential stability of nonlinear impulsive system with input saturation.Impulsive control of a time-delay system under input saturation was investigated in [29].However, both continuous dynamics and discrete dynamics were required to be stable/stabilized in [29].In addition, input delay was excluded in afore mentioned works.More recently, delayed-impulses control for discrete systems with input saturation was addressed in [30], where two classes of impulses, i.e., stabilizing impulses and destabilizing impulses, were studied, respectively.Nevertheless, the estimation of the stability region was excluded in [30], which is essential to the research of saturated systems.Moreover, some limitations on impulse intervals and delays were imposed, which brings more conservativeness.Therefore, the existing literature on the problems of stability and performance analysis for nonlinear systems with delayed impulses and input saturation were not effectively solved.

Motivated by the above discussion, we shall investigate the exponential stability of time-delay impulsive systems involving input saturation.With the help of Razumikhin-type technique, a relatively maximized estimate of the stability region is obtained by means of an optimization algorithm.The novelty and distinctiveness of the proposed results is that we remove the restriction on the length of input delays, i.e., the length of input delays has no implicit relationship with impulse intervals.Moreover we fully considered the relationship between impulsive actions, impulse intervals and stability region.

The remainder of the paper is organized as follows.Section II introduces the model of impulsive systems under delayed impulses and input saturation.Main results including the problems of exponential stability/stabilization and estimation of the domain of attraction are investigated in Section III.In Section IV, three numerical simulations are proposed to demonstrate the applicability of our results.Section V summarizes this paper.

II.PRELIMINARIES

Consider the following nonlinear impulsive system with saturated delayed impulses:

Consequently, s at(Kx) can be expressed as

use.

In this paper, our interests lie in stability and stabilization analysis, and the estimation of S.Specifically speaking, our objective is to establish exponential stability/stabilization criteria by Lyapunov function method and obtain a maximized estimate of S of system (1) involving saturated delayedimpulses.For this purpose, let us now employ state feedbacku(t)=Kx(t), whereK∈Rm×nis the gain matrix to be designed.

III.MAIN RESULTS

A. Stability and Stabilization Analysis of Nonlinear Systems Involving Delayed Impulses

and

then system (1) is LUES over the class Fmax(β).Moreover,M(P,?) is included inS.

Proof: It follows from μ1+μ2<μ that there exist positive constants λ ,?,ξ andh∈(0,1-μ) such that μ1eλξ+μ2≤μ and:

By the above analysis, according to Λ(t+θ)≤Λ(t)/μ,LML<δPand (5), we then get that

Remark 1: The idea of the proof in Theorem 1 is based on the Razumikhin technique [34].In fact, one may find from the proof of Theorem 1 thatD+Λ(t)≤σΛ(t)(σ>0), when Λ(t+θ)≤?/μ≤Λ(t)/μ, θ ∈[-r,0].Note that σ>0 means that when the behavior of the system diverges, we stabilize the system through impulsive control.Recently, exponential stability/stabilization conditions for saturated discrete-time systems were derived in [30].However, delayed impulses, based on the saturated structure, were not essentially taken into consideration during the process of stability analysis.Moreover, it requires that the size of input delays should be less than the lower bound of the impulse intervals.In addition, we remove the restrictions imposed on the input delays and impulse intervals.

and we have following corollary.

Corollary 1: Given constant scalar?and matricesH,K∈Rm×n,L=diag(lj)∈Rn×n, if there existn×nmatrixP>0,n×ndiagonal matrixM>0 and positive constants γ,β,δ andμ<1, such that (3), E (P,?/μ)?L(H),LML<δP, and

Remark 3: In general, the research on impulsive effects can be mainly divided into two categories: unstable continuous dynamics with stabilizing impulses (i.e., impulsive control)and stable continuous dynamics with destabilizing impulses(i.e., impulsive disturbance).From the perspective of impulsive control, Theorem 1 investigated the exponential stabilization problem of system (1).Note that a constraint on the upper bound of the impulsive interval length is imposed, i.e.,tk∈Fmax(β).It indicates that to guarantee the stabilization of the system, the interval length of contiguous impulse instants cannot be overlong.In addition, in the case of impulsive disturbance, Theorem 2 investigated exponential stability problem of system (1).To maintain the stability property of the system, a constraint on the lower bound of the impulsive intervals is imposed, i.e.,tk∈Fmin(β), which reveals that impulse sequences should not happen so frequently to destroy the stability of the system.

Especially, in the absence of input delays, i.e., ξk=0, and considering a special case whereC=I, (1b) can then be replaced by

As a special case, many applications involving impulses can be modelled by (23), such as multi-agent systems [35], network systems [36] and coupled dynamical systems [37].In what follows, we apply Theorem 2 to investigate the stability property of system (1) with impulse effects (23).

Corollary 2: Given a constant scalar?and matricesH,K∈Rm×n,L=diag(lj)∈Rn×n, if there existn×nmatrixP>0,n×ndiagonal matrixM>0 and positive constantsγ,β,δ,μwith μ >1, such that (15), E (P,μ?)?L(H),LML<δP, and

then system (1a) with impulse (23) is LUES over the class Fmin(β).

B. Controller Design and Estimation of the Domain of Attraction

In this section, we shall introduce an optimization approach to enlarge the estimation of the domain of attraction S by designing control gainKand choosing appropriate impulse sequences.

whereg(μ)=μ if μ >1 andg(μ)=1/μ if μ ∈(0,1).

Note that v) is bilinear since it contains two unknown decision variablesPandH, i.e., that is, it is a bilinear matrix inequality (BMI) problem.This fact makes it difficult to solve the optimization problem (24).To solve this problem, linear matrix inequality (LMI) algorithms are developed by performing a classical linearizing change of variables, which corresponds to the introduction of some auxiliary variables defined as follows.

Let η=?/α2,Γ=KW,Z=HW,W=P-1,G=M-1, andei=then we rewrite (24) as follows:

2) Given μ1,μ2, solve (25) for η,W,Z,Γ,Gas well as the maximal upper bound of the impulsive intervalβ.

Remark 5: Recently, [29] studied locally asymptotic stability of time-delay impulsive systems with input saturation.Reference [38] presented some results concerning stabilization of nonlinear time-delay system subject to input saturation via Lyapunov-Krasovskii functional technique.However, the influence of impulsive actions upon the stability region was essentially neglected in their works.Moreover, the implicit connection among impulse action, system structure and the estimate of the stability region, which is crucial to saturation impulsive control, was not specified in their results.

IV.EXAMPLES

In this section, numerical simulations are given to show the validity of the proposed results.

Example 1: Consider the following nonlinear time-delay system:

where τ (t)∈[0,0.1], ξk∈[0,0.4],k∈Z+, and

It is worth noting that in the absence of impulses, the continuous dynamics of system (26) is diverging (see Fig.1(a)).In this case, we shall stabilize system (26) with appropriate control gainKand estimate the maximal domain of attraction.Choose reference set Ξ=co{?1,-?1}, where ?1=(-0.8,0.8)T, and parameters γ=1.1,δ=0.5,μ=0.49,μ1=0.25 andμ2=0.2.Then by using the LMI Toolbox in Matlab, some feasible solutions for can be derived from the optimization problem (25).We have the admissible upper bound of β ≤0.2016, the optimum value α?=6.9881 and corresponding matrices

By Theorem 1, for any bounded impulse input delayξ, system (26) is LUES within E (P?,1) over the class Fmax(0.2016).

In simulations, let impulse interval β =0.2.Fig.1(b) shows the simulation result of system (26) with initial value φ(θ)=(2.4,0.8)T,(-2.7,1.6)T,(-0.7,-1.8)T,θ ∈[-0.4,0],respectively.It depicts that, under saturated impulsive control,the state trajectories starting from the initial state setE(P?,1)(the inner ellipsoid) may enter the permissive state set E(P?,1/u)?L(H) (the outer ellipsoid) keeping inside of it and finally converge to the origin.

Fig.1.(a) State responses of system (26) in the absence of stabilizing impulses; (b) State responses of system (26) with saturated stabilizing impulses.

Under same conditions, consider another case ofβ=0.23>0.2016 , we shall apply (25) to find the upper bound of μ1andμ2.One can verify that the maximum of μ1,μ2,μ is 0.1740,0.1859and 0.3786, respectively.By Theorem 1, for anyboundedimpulseinputdelayξ, system(26)isLUES with inE(P?′,1)overtheclassFmax(0.23) withcontrol gainK?′=(0.2616 0.1680), where

Example 2: Consider another nonlinear system with destabilizing impulses

where ξ ,τ ≥0 and

Choose parameters δ=0.5, γ=1.1, μ=1.8, μ1=0.32, μ2=

In simulations, let ξ=1, ?=0.1 and impulse interval β=0.9466, where the simulation of (27) with initial value φ(θ)=(3,3.5),θ ∈[0,1]is displayed in Fig.2.When the continuous dynamics of system (27) is stable (blue curve), it shows that the stability property can be still maintained with certain saturated delayed-impulses disturbance (red curve).

Fig.2.State trajectory of system (27) under bounded delayed impulsive perturbation.

Example 3: Consider a two-neuron network as follows:

where τ =1 and

It was shown in [39] that system (28) admits chaotic behavior withf(x(t))=(tanh(x1(t)),tanh(x2(t)))Tand initial condition φ=(0.4,0.6)T, see Fig.3(a).Reference [40] pointed out that the chaotic time-delay neural network (28) realizes synchronization under certain stabilizing impulses.In the case that the states are subjected to uncertain input delays, a novel impulsive control strategy was established to guarantee the synchronization of system (28).

Consider the slave system

Fig.3.(a) Chaotic phenomenon of system (28) with initial condition φ=(0.4,0.6)T; (b) State trajectory of error system (31) with initial condition φ=(2,1)Tunder saturated impulsive control.

wherek∈Z+,ξ=0.2 andC,Dare two known parameter matrices given by

Define synchronization error as U(·):=ν(·)-x(·).Then we have the following error system:

wherek∈Z+,u(·)=KU(·).In fact, in view of impulsive saturation, system (30) can be modified as

Choose parameters δ=0.1,γ=1,μ=0.8,μ1=0.3,μ2=0.4 and reference set Ξ=co{?1,-?1}, where ?1=(0.2, -0.2)T.By solving the optimization problem (25), we have the feasible solution η?=0.6944, β ≤βmax=0.364, and gain matrixK=[0.6489 0.5990].By Theorem 1 system (28) achieves exponential synchronization under saturated impulsive control over the class Fmax(0.364).In simulations, take the impulse time sequence {tk}∈Fmax(0.364) as follows:tk=0.36k-0.12,k∈Z+.State trajectoryof error system(31) with initial condition φ =(2,1)Tisdepicted in Fig.3(b).

V.CONCLUSION

In this paper, LUES of nonlinear systems with saturated delayed impulses have been considered.Our results show that under actuator saturation, the time-delay systems processing destabilizing continuous dynamics become stable by choosing appropriate impulsive time sequences.On the other hand,a nonlinear system subject to input saturation has robust stability with respect to sufficiently small impulsive disturbance.ThenLMI-based methods have been established for enlarging the estimation of the stability region as well as for control design in a convex optimization problem.Finally, the proposed control method was validated by simulation results.In the future, we will try to extend the obtained results to impulsive sequences satisfying average-type dwell time conditions or those with eventually uniformly bounded impulsive frequencies.