Robust control for incremental quadratic constrained nonlinear time-delay systems subject to actuator saturation

https://doi.org/10.1016/j.amc.2021.126271Get rights and content

Highlights

  • This paper propose the IQC nonlinear function with time delay for the first time and the nonlinear term is dealt with by the S-procedure.

  • The saturated feedback law is constructed and the considered systems are proved to be exponentially stable under the given conditions.

  • An optimization problem of maximizing the estimation of DA is derived by CHLF approach, and two simulations are provided to show that the proposed method can provide a larger estimation of DA than the QLF method.

Abstract

This article presents a saturation controller design method for a type of nonlinear time-delay systems. Different from the current works, the nonlinear systems considered herein satisfy an incremental quadratic constraint, which is a more general nonlinearity. Firstly, for the constraint in the actuator, the convex set theory is employed to transform the saturated feedback law into a parameterized linear convex hull set. Next, the convex hull Lyapunov function is constructed to obtain sufficient conditions for exponential stability of the closed-loop system. In addition, by solving an optimization problem, the maximal estimation of domain of attraction is also determined. Finally, numerical examples are provided to verify the proposed method.

Introduction

In the real world, most systems are nonlinear. Thus, the investigation of nonlinear systems is always a hot but difficult subject, whether in stabilization [1], [2] or observation [3], [4]. After the Lipschitz constraint was presented, many scholars have made outstanding achievements in the study of Lipschitz nonlinear systems [5], [6], [7], [8]. Rajamani [5] and [6] both focused on the observer design for nonlinear systems, while [7] and [8] studied the filtering problem for fuzzy nonlinear systems. Chang et al. [7] designed an H2 filter for nonlinear discrete-time systems with measurement quantization. In the case of multi-path data packet dropouts, a design method of fuzzy peak-to-peak filter for nonlinear systems was given in [8]. However, if the Lipschitz constant is relatively large, feasible solutions for sufficient conditions may not exist. Then, the one-sided Lipschitz (OSL) constraint was proposed in [9], and it was proved that the OSL constraint is more general than the Lipschitz constraint. In order to overcome the weighted bilinear form in dealing with OSL functions, Abbaszadeh and Marquez [10] introduced the condition of quadratic inner-boundedness (QIB) and designed an observer for OSL nonlinear systems satisfying QIB. Following this, Zhang et al. [11] presented a less conservative observer design method for OSL nonlinear systems. In [12], the definition of incremental quadratic constraint (IQC) was introduced to nonlinear systems, which is a more general description of nonlinearities. By introducing an incremental multiplier matrix (IMM), many nonlinear functions can be described by IQC, such as Lipschitz nonlinear functions [13], OSL nonlinear functions [14] and conic-type nonlinear functions [15], [16] (more details can be found in [17]). Li et al. [13] addressed the energy-to-peak filtering problem for Lipschitz nonlinear systems. Li and Yuan [14] discussed strong convergence problem for OSL nonlinear systems. He et al. [15] and [16] considered sliding mode control and dissipative filtering design for conic nonlinear systems, respectively. In recent years, investigation of IQC nonlinear systems has attracted significant research interest, such as [18], [19], [20], [21] herein.

As we know, time delays always appear in practical engineering, such as power systems [22], [23], neural network systems [24], multi-agent systems [25], etc. Time delay can not be ignored in the design of control systems, and it may weaken the performance of systems and even cause instability. Due to the physical restrictions of the actuator, actuator saturation phenomenon is another common problem existing in control systems [26], [27], [28], [29]. Actuator saturation has typical nonlinear characteristics, so it is necessary to deal with this nonlinearity in the design of the controller. Hu et al. [30] presented the linear convex hull method to handle saturation phenomenon by introducing an auxiliary matrix, and gave the concept of estimating the domain of attraction (DA) by quadratic Lyapunov function (QLF). In order to further enlarge the DA for saturation systems, Cao and Lin [31] extended the QLF to the saturation-dependent Lyapunov function (SDLF). However, the method in [31] is also somewhat conservative, because the estimated DA is only an intersection of a set of ellipsoids. Therefore, the convex hull Lyapunov function (CHLF) was introduced in [32] and it was proved that CHLF is less conservative than SDLF in estimating the DA. For saturation systems with time delays, many valuable achievements have also been made. By introducing a method via auxiliary time delay feedback, Chen et al. [33] investigated the saturation control problem for neutral time-delay systems. In [34], saturation controllers were designed for time-delay systems with stochastic disturbance. Recent interesting works on saturation systems with time delay can be found in [35], [36], [37], [38], [39], [40], [41]. It is noteworthy that the aforementioned results focused on the saturation control problem for linear time-delay systems. To the authors’ knowledge, the saturation controller design for IQC nonlinear time-delay systems has not yet been reported.

Based on the above discussion, this article investigates the saturation control problem for nonlinear systems satisfying IQC with time delay. The main challenge of this article includes the following two aspects: One is using a proper differential inclusion to describe the saturated control system, and employing the CHLF to derive less conservative conditions under which the closed-loop system is exponentially stable. The other is dealing with the nonlinear function satisfying IQC properly, and obtaining sufficient conditions formulated by the form of linear matrix inequalities. The contributions are mainly reflected in three components:

(i) The IQC nonlinear function with time delay is proposed for the first time and the nonlinear term is dealt with by the S-procedure.

(ii) The saturated feedback law is constructed and the considered systems are proved to be exponentially stable under the given conditions.

(iii) An optimization problem of maximizing the estimation of DA is derived by CHLF approach, which is less conservative than that derived by QLF.

The remainder of this paper is organized as follows: Section 2 gives the problem statement and preliminaries. In Section 3, sufficient conditions are proposed to ensure that the closed-loop system satisfies exponential stability and an optimization problem is given for maximizing the estimation of DA. Section 4 presents an algorithm for computing IMMs of some commonly encountered nonlinearities. In Section 5, the validity of the method is demonstrated by two examples, and the comparisons with existing works illustrate that the proposed method can reduce the conservatism in estimating the DA.

The following standard notations will be adopted in this article. sign(·) represents the symbol function and I[1,n] stands for the set of integers {1,2,,n}. Let x=[x1,x2,,xn]T, and |x| is the -norm of xRn, which means that |x|=maxiI[1,n]|xi|. LV(ρ)={x:V(x)ρ} is the ρ-level set of V(x) and ε(P,ρ)={x:xTPxρ} is an ellipsoid.

Section snippets

Problem statement and preliminaries

In this article, the following system is taken into consideration:x˙(t)=Ax(t)+Adx(th)+ϕ(x(t))+ψ(x(th))+BNsat(u(t)),where x(t)Rn, x(th)Rn are the state vectors, and h is the time delay. ϕ(x(t))Rn and ψ(x(th))Rn represent nonlinear terms. Nsat(u(t))Rm is the saturated input. A,AdRn×n and BRn×m are given matrices. For the sake of simplicity, the variable t is omitted and the variable x(th) is defined as xh. Therefore, (1) is equivalent tox˙=Ax+Adxh+ϕ(x)+ψ(xh)+BNsat(u).Denote the

Saturated linear feedback for robust stabilization

In this section, the CHLF is employed to obtain sufficient conditions to ensure that system (3) is exponentially stable. Furthermore, the maximal estimation of DA is determined by solving an optimization problem.

Theorem 3.1

Assume that nonlinear functions ϕ(x) and ψ(xh) satisfy the condition of IQC. Let QlRn×n>0 for lI[1,K], ρ>0. If there exist F,HRm×n, γikl0 for iI[1,2m], k,lI[1,K], ξ1>0, ξ2>0 and c2>c10, such that[1ρziρziTρQl]0,where zi is the ith row of HQl,

andΠ=[Π11AdQlI+ξ1QlM12I*Π220ξ2QlN12**ξ1

An algorithm of computing IMM

In this section, a method for calculating IMMs of some common nonlinear functions satisfying IQC is introduced. Based on the results in [12], the nonlinear functions are assumed to be continuously differentiable. Denote X=co{θ1,θ2,,θv} as a convex set of vertex matrices θi and the nonlinear function ϕ(x) satisfies ϕ(x˜)ϕ(x)=θi(x˜x). According to the expression of (4), one can get[Iθi]TM[Iθi]0,θiX.By substituting the decomposition of M into (51), we haveM11+M12θi+θiTM12T+θiTM22θi0,θiX.

Example 1

Consider system (2) with the following parametersA=[22730.6],Ad=[0.90.40.30.6],B=[32],h=2,and the nonlinear functions are given byϕ(x)=[sin(x1(t))sin(x2(t))],ψ(x)=[tanh(x1(t2))tanh(x2(t2))].Using the results in Section 4, the vertex matrices of ϕ(x) and ψ(xh) areXϕ=co{[1001],[1001],[1001],[1001]},Xψ=co{[2002],[2000],[0002],[0000]}.By substituting the expressions of Xϕ and Xψ into LMIs (53), the IMMs of the nonlinear functions ϕ(x) and ψ(xh) are determined byM=[15.0089000015.00890000

Conclusion

In this article, we have studied the saturation control problem for nonlinear systems satisfying IQC with time delay. By using the theory of convex set, the saturated nonlinear system is transformed into the form of convex set, and CHLF has been constructed to analyze the stability of the closed-loop system. Meanwhile, we have presented an optimization problem to maximize the estimation of DA and demonstrated that the maximal ellipsoid of DA derived by the CHLF approach is larger than those

References (49)

  • Y. Wei et al.

    Composite control for switched impulsive time-delay systems subject to actuator saturation and multiple disturbances

    Nonlinear Anal. Hybrid Syst.

    (2020)
  • J. Zhang et al.

    Finite-time dissipative control of uncertain singular T-S fuzzy time-varying delay systems subject to actuator saturation

    Comput. Appl, Math

    (2020)
  • S. Song et al.

    Adaptive hybrid fuzzy output feedback control for fractional-order nonlinear systems with time-varying delays and input saturation

    Appl. Math. Comput

    (2020)
  • T. Hu et al.

    Stability and performance for saturated systems via quadratic and nonquadratic lyapunov functions

    IEEE Trans. Autom. Control

    (2006)
  • T. Hu et al.

    Properties of the composite quadratic Lyapunov functions

    IEEE Trans. Autom. Control

    (2004)
  • J. Luo et al.

    Robust control for a class of uncertain switched fuzzy systems with saturating actuators

    Asian J. Control

    (2015)
  • X. Lin et al.

    Output-feedback stabilization for planar output-constrained switched nonlinear systems

    Int. J. Robust Nonlin

    (2020)
  • C. Ren et al.

    Finite-time L2-gain asynchronous control for continuous-time positive hidden Markov jump systems via T-S fuzzy model approach

    IEEE T. Cybern.

    (2020)
  • H. Che et al.

    Functional interval observer for discrete-time systems with disturbances

    Appl. Math. Comput.

    (2020)
  • R. Rajamani

    Observers for Lipschitz nonlinear systems

    IEEE Trans. Autom. Control

    (1998)
  • F. Zhu et al.

    A note on observers for Lipschitz nonlinear systems

    IEEE Trans. Autom. Control

    (2002)
  • X. Chang et al.

    Fuzzy generalized H2 filtering for nonlinear discrete-time systems with measurement quantization

    IEEE Trans. Syst. Man Cybern. Syst.

    (2018)
  • X. Chang et al.

    Fuzzy peak-to-peak filtering for networked nonlinear systems with multipath data packet dropouts

    IEEE Trans. Fuzzy Syst.

    (2019)
  • G. Hu

    Observers for one-sided Lipschitz nonlinear systems

    IMA J. Math. Control Inf.

    (2006)
  • Cited by (13)

    • Stochastic stability analysis of nonlinear semi-Markov jump systems with time delays and incremental quadratic constraints

      2023, Journal of the Franklin Institute
      Citation Excerpt :

      Recently, [40] proposed the nonlinearity with the constraint of incremental quadratic constraints (IQC). Most of the current research results focus on determination systems [41–43]. But as far as the writers know, there seem to be no related results on the research of time-delayed SMJSs with nonlinearities satisfying IQC.

    • Observer design for semi-Markov jump systems with incremental quadratic constraints

      2021, Journal of the Franklin Institute
      Citation Excerpt :

      The nonlinearities satisfying IQC can include many common nonlinearities, such as monotone, incremental sector bounded, Lipschitz, one-sided Lipschitz nonlinearities, and so on. Then, some works on IQC [19,20,29,30] have been accomplished. Açıkmese and Corless [19] studied the observer of nonlinear systems with IQC.

    View all citing articles on Scopus

    This work is supported by National Natural Science Foundation of China (61403267).

    View full text