Elsevier

Journal of Process Control

Volume 105, September 2021, Pages 214-222
Journal of Process Control

Robust adaptive stabilization of nonlinear systems with mismatched time delays

https://doi.org/10.1016/j.jprocont.2021.08.005Get rights and content

Highlights

  • We introduce control gain function ρ() and embed it into the Lyapunov–‘Krasovskii functional.

  • Continuous memoryless control functions are designed.

  • The theoretical result is applied to chemical reactor.

Abstract

For a class of nonlinear systems with mismatched time delays and disturbances, robust adaptive stabilization control is studied. The control law is split into two terms. One is used to cope with the delayed states, and the other is to handle the disturbance. The improved robust adaptive laws with control gain function and time-varying σ-modification are constructed to estimate the unknown integrated parameters. It is proved that the designed adaptive controller can ensure the asymptotic convergence of the closed-loop system states. Another adaptive algorithm is also proposed to ensure the asymptotic stability of the closed delayed nonlinear system without external disturbance.

Introduction

Time delay is a common phenomenon in control engineering, and it exists widely in biological system, networked control system, process control system, and so on, see, e.g. [1], [2]. Due to the existence of modelling error, measurement error and external disturbance, the significant uncertainties are usually encountered in practical systems, see, e.g. [3], [4]. Specially, [5] studied the asymptotic stability of two kinds of strict-feedback nonlinear delayed systems. Adaptive output-feedback control scheme was developed for linear and nonlinear dynamic systems with time delays within the model reference adaptive control framework in [6], [7]. Adaptive fuzzy control was applied to nonlinear delayed systems in [8], [9]. The dynamic gain based backstepping approach was developed for strict-feedback nonlinear delayed systems, cascade nonlinear delayed systems and high-order nonlinear delayed systems in [10], [11], respectively. By adding a dynamic subsystem, smooth output-feedback controller was designed for multiple delayed uncertain systems in [12]. A novel Lyapunov–Krasovskii functional with two integral-type functions was applied to the adaptive control problem of a class of large-scale delayed systems with dead-zone inputs in [13]. In [14], effective adaptive memoryless controller was designed for uncertain delayed nonlinear systems.

In this paper, we consider the following nonlinear time-delay system: dx(t)dt=F(x(t),t)+i=1v1ξ1i(x(th1i(t)),t)+η1(x(t),t)d1(t)+G(x(t),t)(u(t)+j=1v2ξ2j(x(th2j(t)),t)+η2(x(t),t)d2(t)),x(t)=φ(t),t[t0h̄,t0], where x()Rn is the state variable; u(t)Rm is the control vector; ξ1i():Rn×RRn, ξ2j():Rn×RRm are the delayed state perturbations; h1i():RR, h2j():RR, are the time-varying delays satisfying 0h1i(t)h̄1i,ḣ1i(t)h1i<1,0h2j(t)h̄2j,ḣ2j(t)h2j<1, with unknown constants h̄1i,h1i,h̄2j,h2j, and h̄=max{h̄1i, h̄2j,1iv1,1jv2}; d1()Rl1, d2()Rl2 are bounded disturbances; φ()Rn is the initial continuous function; F():Rn×RRn, G():Rn×RRn×m, η1():Rn×RRn×l1, η2():Rn×RRm×l2 are known and continuous.

Assumption 1

The uncertain functions ξ1i(),ξ2j() satisfy: ξ1i(x(th1i(t)),t)(θ1i)Tψ1i(x(th1i(t))),ξ2j(x(th2j(t)),t)(θ2j)Tψ2j(x(th2j(t))), where 1iv1, 1jv2, denotes Euclidean norm, θ1i=[θ1i,1,θ1i,2,,θ1i,p1]T,θ2j=[θ2j,1,θ2j,2,,θ2j,p2]T,ψ1i()=[ψ1i,1(),ψ1i,2(),,ψ1i,p1()]T,ψ2j()=[ψ2j,1(),ψ2j,2(),,ψ2j,p2()]T, θ1iRp1,θ2jRp2 are unknown constant vectors, and ψ1i,k(),1kp1, ψ2j,k(),1kp2, are known, continuous, nondecreasing and nonnegative functions.

Assumption 2

There exists a C1 function V0():Rn×RR+, K-class functions γ1():RR, γ2():RR, and K-class function γ3():RR, such that the following inequalities hold: γ1(x)V0(x,t)γ2(x),V0(x,t)t+xTV0Fγ3(x),

Assumption 3

For 1iv1, 1jv2, the functions ψ1i,k(), ψ2j,k(), γ1(), γ2(), γ3() satisfy ψ1i,k(0)=0,ψ2j,k(0)=0, ψ1i,k2(γ11())γ3(γ21()), ψ2j,k2(γ11())γ3(γ21()), which imply that ψ1i,k2(γ11()) and ψ2j,k2(γ11()) are the same order infinitely small with γ3(γ21()), that is, limχ0ψ1i,k2(γ11(χ))γ3(γ21(χ))=ϱ1i,k0,k=1,2,,p1,limχ0ψ2j,k2(γ11(χ))γ3(γ21(χ))=ϱ2j,k0,k=1,2,,p2, where ϱ1i,k0>0,ϱ2j,k0>0 are constants.

Assumption 4

If x(t) is bounded, the functions G(x,t), xV0(x(t),t), η1(x(t),t),η2(x(t),t), F(x(t),t) are bounded for all (x(t),t)Rn×R.

Instead of dynamic gain approach, we propose the static gain function method to deal with the delayed state perturbations. New control gain function and Lyapunov–Krasovskii functional are proposed. When the delayed state perturbations are counteracted by one term in the derivative of the Lyapunov–Krasovskii functional, another nonnegative term in the derivative of the Lyapunov–Krasovskii functional appears. In view of this, we introduce control gain function ρ() and embed it into the Lyapunov–Krasovskii functional. Based on the construction of ρ(), the stability of the closed-loop delayed system can be achieved.

Section snippets

Adaptive control design

We first develop a continuous function ϱ(χ), χ0. When χ=0,ϱ(χ)=ϱ0; when χ>0, ϱ(χ)=i=1v1k=1p1η1i12ψ1i,k2(γ11(χ))(1k0)γ3(γ21(χ))+j=1v2k=1p2η2j12ψ2j,k2(γ11(χ))(1k0)γ3(γ21(χ)), where ϱ0=i=1v1k=1p1η1i12(1k0)ϱ1i,k0+j=1v2k=1p2η2j12(1k0)ϱ2j,k0, η1i>0,η2j>0,0k01, are design constants, and ϱ1i,k0,ϱ2j,k0 are defined in Assumption 3. The control gain ρ:[0,)R is defined by ρ(χ)=c+ϱ(χ),χ0,with positive design constant c.

Assumption 5

The design constants c,k0,η1i,1iv1, η2j,1jv2, satisfy: k0γ3

Simulation study

A. Example 1

We consider an uncertain nonlinear system: ẋ1(t)=x1(t)g(t)x2(t)+v1(t)x1(th1(t))+x2(t)v2(t)x2(th2(t))+u1(t)+x2(t)u2(t)+x2(t)d10(t)+x2(t)d20(t),ẋ2(t)=x1(t)x2(t)+u1(t)+u2(t)+v2(t)x2(th2(t))+x1(t)sin(t)d10(t)+d20(t), where g(t)=0.1/(1+t). Let us choose x(t)=x1(t)x2(t),u(t)=u1(t)u2(t),F=x1(t)g(t)x2(t)x1(t)x2(t),G=1x2(t)11, ξ1(x(th1(t)),t)=v1(t)x1(th1(t))0,ξ2(x(th2(t)),t)=0v2(t)x2(th2(t)),η1=x2(t)0x1(t)sin(t)0,η2=I2, d1(t)=d10(t)0,d2(t)=0d20(t).For simulation, we set the

Conclusion

Under some system assumptions, the control law and adaptive laws with explicit control gain function are constructed. By employing appropriate Lyapunov–Krasovskii functional with explicit control gain function, asymptotic convergence of the closed-loop system states is proved. Simulation results are given to demonstrate the design procedure for the presented memoryless state feedback controller. For further research, it is interesting to extend the proposed control strategy to large-scale

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 61873330, the Taishan Scholarship Project of Shandong Province, China under Grant tsqn20161032 and Central Government Guides Local Funds for Science and Technology Development of Shandong Province, China under Grant YDZX20203700001633.

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