Hybrid control of continuous systems with actuator saturation and state time-delay

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(School of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China)

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Hybrid control of continuous systems with actuator saturation and state time-delay

JINGLi,HANShuangwei

(School of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China)

In controlling of practical systems actuator saturation and time-delay problems often happen. These badly make effects on the stability of systems. Thus systems may face serious accidents. To solve these problems the paper studies the robust stabilization problem of continuous systems with actuator saturation and state time-delay. By introducing auxiliary matrices, control input of the system is limited in convex polyhedron. So the nonlinear saturation function is dealt with into linear function. Hybrid controllers are designed and the closed loop of the system is a kind of switched system. Then single Lyapunov function based approach is used to stabilize it. A numerical simulation is done. The results we get include: a sufficient condition for the stability of continuous system, the switching strategy for the systems under hybrid control. The paper also proposes an estimation scheme of the maximum attraction domain for the system and expresses it into LMI. The results show the method given in this paper is less conservative comparing to convex combination method. The simulation shows a system can be asymptotically stable at the origin in a short time and the domain of attraction is larger.

continuous system; actuator saturation; time-delay; hybrid control

0 Introduction

In practical engineering, systems with actuator saturation or state saturation are common. Saturation function is nonlinear, and the existence of it in the system will cause the system complex and difficult to handle. So research on saturation is important. Since 1990 scholars has studied the saturation problems of continuous and discrete time systems, mainly on the stability and controller design, and achieved fruitful results by using LMI technique. They has also taken into account the partially saturated case[1-2]. State feedback and dynamic output feedback methods have been once used to design the controller[3-5]. In 2001, HU Tingshu used convex combination to deal with saturation part of systems[4], and gave an estimation scheme of attraction domain in 2002[5]. CAO Yongyan (2003), ZUO Zhiqiang (2008) focused on the saturated systems, by using a similar approach they obtained less conservative sufficient conditions of stability and an estimation scheme of attraction domain[6-7]. KAR H and SINGH V proposed a stable condition for discrete systems by utilizing the error existing between variable and its saturation function[1].Systems with saturation and time-delay were also studied. Related researches see references[8-12]. Currently, researches on saturation problems of T-S fuzzy systems and switched systems are being done[13-19].

In 2012, CHEN Yonggang considered the control synthesis problem for discrete-time switched linear systems with input saturation. The convex combination method was used to deal with the saturation nonlinearity and the minimumdwell time approach was used to stabilize the switched system. A state feedback controller and a dynamic output feedback controller were designed, and the domain of attraction of the system was estimated[16]. In 2014, stability and control problem for a class of cascade switched nonlinear systems with actuator saturation was studied[17]. In the paper, the actuator saturation was equivalent to a convex combination formula, and the dwell time approach and the state feedback control method were used to stabilize the system. Reference[18]usedthe antiwindup method to research the control problem of the discrete-time switched system subject to actuator saturation. The paper first designed the dynamic output controllers for the system under neglecting the existence of the saturation. Then antiwindup compensation gains were proposed to maximize the domain of attraction of the system.

The paper analyzes the stability of continuous systems with actuator saturation and time-delay.It first suggests designing a hybrid controller for the system, and the system turns into a switched system. By single quadratic Lyapunov function method the switched system is stabilized. The paper obtains a sufficient condition of stability for continuous system with actuator saturation and time-delay. The paper also proposes an estimation scheme of attraction domain of the system. In the end of the paper a simulation is given.

1 Problem statement

The system model with actuator saturation and state time-delay is described by

(1)

wherex∈Rn,u∈Rm,u=[u1,u2,…,um]T,σ(u) is the standard saturation function.

Define

(2)

where matrixF∈Rm×n,fiis the ith row ofF.

Define

(3)

Definition 2[5]LetV(x,t)=xTPxbe the Lyapunov function of system (1). The ellipsoidε(P,ρ) is said to be contractively invariant if for allx∈ε(P,ρ),

Clearly ifε(P,ρ) is contractively invariant,ε(P,ρ) is a subset of the attraction domain of system (1).

The paper studies the stabilization problem of system (1), which has saturation nonlinearity and state time-delay. The paper uses hybrid control scheme to stabilize system (1).

Thecontrollerofsystem(1)canbechosenasfollowing.For?x∈ε(P,ρ),

(4)

Otherwise,

Further according to the definition of saturation function, formula (2) and (4), the closed-loop system of system (1) is

(5)

So system (5) is a switched system. In the next section the paper uses single Lyapunov function method to stabilize it.

2 Main results

2.1 System stability

In this part we propose the sufficient conditions of stability of system (5), and design the controller of the system.

Let

be the Lyapunov function of system (5).

For ?x∈ε(P,ρ), ifx∈L(F), the subsystem (5) is active, so

2xT(t)P(A+BF)x(t)+2xT(t)PAdx(t-d)+xT(t)Qx(t)-xT(t-d)Qx(t-d)=

xT(t)(PA+ATP+Q+PBF+FTBTP)x(t)+2xT(t)PAdx(t-d)-xT(t-d)Qx(t-d)=

Suppose

(6)

Assume

(7)

In order to solve inequalities (6) and (7) with LMI Tools, multiply both right and left side of (6) and (7) respectively bydiag(P-1,I). We obtain

LetX=Q-1,Y=P-1,Z=FP-1,Mi=HiP-1, the above inequalities are further rewritten as

(8)

The above analysis can be summarized into the theorem.

Theorem Consider system (1). If there exist positive definite matricesX>0,Y>0, matricesZ∈Rm×nandMi∈Rm×n(i=1,2,…,s), which satisfy inequalities (8), system (1) is local asymptotic stabilization at the origin under controllers (4).

2.2 The estimation of the attraction domain of system (1)

Define[20]

The estimation problem of the attraction domain of system (1) can be formulated as

(9)

The above optimization problem can be changed into following inequalities problem.

Constraint (a) is equivalent to[5]

Constraint (c) is equivalent to[5]

wherehijis thejthrow of matrixHi, (j=1,2,…,m).

Letγ=α-2,ρ=1 and replaceP-1withY,hijP-1withmij. We obtain the equivalence of (9)

(10)

Problem (10) can be solved with LMI Tools.

3 Simulation

Consider system (1):

Let matrix

We solve the optimization problem (10) with LMI Tools for a feedback with the objective of maximizing the domain of attraction. The result is,

The invariant ellipsoid isε(P,1), see Fig.1.

If we choose (0.9,2)T. as the initial state, the system is asymptotically stable at the origin, see Fig.2.

Fig.1 Attraction domain of the system

Fig.2 State response of the system

4 Conclusion

The stabilization problem of continuous systems with actuator saturation and state time-delay is first considered. A hybrid controller is designed and the system is stabilized based on single Lyapunov function method. As a result we propose a sufficient condition of the stability of the system. Then we estimate the attraction domain of the systems. A simulation is given at last to show the effectiveness of the result of the paper.

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1673-5862(2016)02-0154-06

具有執(zhí)行器飽和及狀態(tài)時(shí)滯系統(tǒng)的混雜控制

景 麗, 韓雙微

(沈陽(yáng)師范大學(xué) 數(shù)學(xué)與系統(tǒng)科學(xué)學(xué)院, 沈陽(yáng) 110034)

實(shí)際系統(tǒng)在控制中容易發(fā)生執(zhí)行器飽和及狀態(tài)時(shí)滯問題,這影響系統(tǒng)的穩(wěn)定性能,給系統(tǒng)帶來不可預(yù)想的嚴(yán)重后果。為了解決這類問題,研究具有執(zhí)行器飽和及時(shí)滯的連續(xù)系統(tǒng)魯棒控制問題。通過引進(jìn)輔助矩陣,使系統(tǒng)的控制輸入被限制在凸多面體內(nèi),系統(tǒng)的飽和非線性函數(shù)因而得以處理:采用混雜控制的方法設(shè)計(jì)了系統(tǒng)控制輸入,控制輸入與系統(tǒng)構(gòu)成的閉環(huán)系統(tǒng)是切換系統(tǒng):進(jìn)一步依據(jù)單Lyapunov函數(shù)方法鎮(zhèn)定系統(tǒng):最后進(jìn)行了Matlab數(shù)值仿真。提出混雜控制下系統(tǒng)穩(wěn)定的充分條件及系統(tǒng)切換方案,給出估計(jì)系統(tǒng)的最大吸引域方案。結(jié)論相比凸組合方法具有較少的保守性,仿真顯示所提出的方法可以使系統(tǒng)在較短時(shí)間內(nèi)漸近穩(wěn)定于原點(diǎn),且系統(tǒng)具有較大的吸引域。

連續(xù)系統(tǒng); 執(zhí)行器飽和; 時(shí)滯系統(tǒng); 混雜控制

TP273 Document code: A

10.3969/ j.issn.1673-5862.2016.02.006

過籌學(xué)與控制論