H-infinity control for cascade minimum-phase switched nonlinear systems

1Introduction H_∞control theory has become a powerful tool to solverobust stabilization or disturbance attenuation problems.Many results about linear H∞control have appeared,andlinear H∞theory has been generalized to nonlinear systems[1~5].Two major approaches have been used to providesolutions to nonlinear H∞control problems.One is basedon the dissipativity theory and differential games theory[2,6].The other is based on the nonlinear versionofclassical bounded real lemma[3~5].Both of th…     

Quadratic stabilization of switched nonlinear systems

In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated. When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law. The results of this paper are also applied to switched linear systems. Supported partially by the National Natural Science Foundation of China (Grant No. 50525721)     

一类线性切换系统具有H∞性能指标的二次稳定

研究了一类线性切换系统具有H_∞扰动衰减度二次稳定问题.这类线性切换系统由两个子系统组成,并且每个子系统都不是具有H_∞扰动衰减度二次稳定的.利用单Lyapunov函数方法,得到了线性切换系统具有H_∞扰动衰减度二次稳定的充分条件,同时由凸组合系统设计出确保线性切换系统二次稳定且具有H_∞扰动衰减度的切换律.进一步,还给出了线性切换系统具有H_∞扰动衰减度二次稳定的必要条件.最后的仿真实例表明了结论的有效性.     

一类非线性离散切换系统基于观测器的指数镇定

对一类离散非线性切换系统, 考虑了基于观测器的指数镇定问题. 借助微分中值定理(DMVT), 将非线性切换系统转化为线性参数(LPV)切换系统. 当状态变量不完全可获得时, 基于多Lyapunov函数方法, 给出系统在基于观测器的输出反馈控制器下指数镇定的充分条件. 所设计的滞后切换规则能够避免产生滑动模态. 并且将结果推广到系统方程含有不确定性的情况. 最后, 仿真例子说明了设计方法的有效性.     

一类非线性不确定时滞系统的混杂状态反馈H∞鲁棒控制

利用混杂状态反馈控制策略研究一类非线性不确定时滞系统的H∞鲁棒控制问题.假定在给定的控制器集合中有有限个备选的状态反馈控制器,并且每个单一的连续控制器都不能使系统具有鲁棒H∞性能.当控制器的增益矩阵已知时,基于单Lyapunov函数技术和凸组合条件给出控制器切换方案以确保非线性不确定时滞系统具有鲁棒H∞性能.当控制器的增益矩阵未知时,使用多Lyapunov函数技术得到了问题可解的另一个充分条件,同时还给出了混杂状态反馈H∞控制器的设计.     

Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings

This paper is concerned with the global stabilization problem for switched nonlinear systems in lower triangular form under arbitrary switchings. Two classes of state feedback controllers and a common Lyapunov function (CLF) are simultaneously constructed by backstepping. The first class uses the common state feedback controller which is independent of switching signals; the other class utilizes individual state feedback controllers for the subsystems. As an extension of the designed method, the global stabilization problem under arbitrary switchings for switched nonlinear systems in nested lower triangular form is also studied. An example is given to show the effectiveness of the proposed method.     

具有块三角结构非线性切换系统的二次稳定性

The problem of globally quadratic stability of switched nonlinear systems in block-triangular form under arbitrary switching is addressed. Under the assumption that all block-subsystems are zero input-to-state stable, a sufficient condition for the problem to be solvable is presented. A common Lyapunov function is constructed iteratively by using the Lyapunov functions of block-subsystems.     

Observer-based robust H-infinity control for uncertain switched systems

The problem of observer-based robust H-infinity control is addressed for a class of linear discrete-time switched systems with time-varying norm-bounded uncertainties by using switched Lyapunov function method. None of the individual subsystems is assumed to be robustly H-infinity solvable. A novel switched Lypunov function matrix with diagonal-block form is devised to overcome the difficulties in designing switching laws. For robust H-infinity stability analysis, two linear-matrix-inequality-based sufficient conditions are derived by only using the smallest region function strategy if some parameters are preselected. Then, the robust H-infinity control synthesis is studied using a switching state feedback and an observer-based switching dynamical output feedback. All the switching laws are simultaneously constructively designed. Finally, a simulation example is given to illustrate the validity of the results.     

H-infinity filtering for discrete-time switched linear systems under arbitrary switching

This paper is concerned with the problem of H-infinity filtering for discrete-time switched linear systems under arbitrary switching laws.New sufficient conditions for the solvability of the problem are given via switched quadratic Lyapunov functions.Based on Finsler’s lemma,two sets of slack variables with special structure are introduced to provide extra degrees of freedom in optimizing the guaranteed H-infinity performance.Compared to the existing methods,the proposed one has better performances and less conservatism.An example is given to illustrate its effectiveness.     

一类多输入级联非线性切换系统的全局镇定

研究一类带有部分线性系统的多输入级联非线性切换系统的全局镇定问题. 首先, 给出保证线性部分有一致规范型的充分条件. 其次, 利用一致规范型及其零动态的共同二次Lyapunov函数设计状态反馈使得线性部分在任意切换律下镇定. 最后, 通过构造共同Lyapunov函数能实现闭环系统在任意切换律下的全局渐近稳定性.     

Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions

Counterexamples are given which show that a linear switched system (with controlled switching) that can be stabilized by means of a suitable switching law does not necessarily admit a convex Lyapunov function. Both continuous- and discrete-time cases are considered. This fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched system. In particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a Lyapunov function.     

一类不确定切换系统的容错控制与极点配置

分析一类非线性不确定切换系统的容错控制与基于状态反馈的极点配置。系统包含有界未知结构的不确定性和未知非线性项。在各子系统不稳定的前提下,设计切换系统的状态反馈控制器,基于Lyapunov稳定性理论和LMI方法,保证在任意切换下不确定系统在传感器和执行器同时失效情况下具有鲁棒容错控制性能的充分条件。在此基础上研究该系统的极点配置在左半复平面选定圆域内以理想速度渐近衰减。文中得到了容错控制切换系统可状态反馈镇定的充分条件,然后用易于求解的线性矩阵不等式形式给出结果,最后通过仿真验证所设计的切换系统的极点配置在圆域内,在状态反馈控制器下渐近稳定。     

Stabilization by forwarding design for switched feedforward systems with unstable modes

The problem of global stabilization is investigated for a class of switched nonlinear feedforward systems in this paper where the solvability of the stabilization problem for individual subsystem is not assumed. Some sufficient condition for the stabilization problem to be solvable is derived for the first time by exploiting the multiple Lyapunov functions method and the forwarding technique. Also, we design a switching law and construct bounded state feedback controllers of subsystems explicitly by a recursive design algorithm to achieve global asymptotic stability. The provided technique permits removal of a common restriction in which all subsystems in switched nonlinear feedforward systems are globally asymptotically stable. Finally, a numerical example is provided to demonstrate the feasibility of the theoretical result. Copyright © 2017 John Wiley & Sons, Ltd.     

Multiple Lyapunov functions approach to observer-based H ∞ control for switched systems

This article investigates the issue of H _∞ control for a class of continuous-time switched Lipschitz nonlinear systems. None of the individual subsystems is assumed to be stabilisable with H _∞ disturbance attenuation. Based on a generalised multiple Lyapunov functions (GMLFs) approach, which removes the nonincreasing requirement at switching points, a sufficient condition for the solvability of the H _∞ control problem under a state estimation-dependent switching law is presented. Observers, controllers and a switching law are simultaneously designed. As an extension, a sufficient condition for exponential stabilisability is also given.     

Stabilizing controllers design for switched nonlinear systems in strict-feedback form

This paper considers the stabilization problem for a class of switched nonlinear systems under arbitrary switching. Based on the backstepping method and the control Lyapunov function approach, it is shown that, under a simultaneous domination assumption, a switched nonlinear system in strict-feedback form can be globally uniformly asymptotically stabilized by a continuous state feedback controller. A universal formula for constructing stabilizing feedback laws is presented. One example is included for verifying the obtained results.     

Dual approach to stability and stabilisation of uncertain switched positive systems

This paper addresses two kinds of dual approaches to stability and stabilisation of uncertain switched positive systems under arbitrary switching and average dwell-time switching, respectively. The uncertainties in systems refer to polytopic ones. A new parameter-dependent switched linear copositive Lyapunov function is first proposed for uncertain switched positive systems. By using the new Lyapunov function associated with arbitrary switching and average dwell-time switching, respectively, sufficient conditions for the stability of the systems are established. Two alternative stability criteria based on two kinds of dual approaches are addressed. It is shown that the alternative criteria hold for not only the primal switched positive system but also its dual system. Then, the stabilisation of primal and dual switched positive systems under arbitrary switching and average dwell-time switching is solved, respectively. All present conditions are solvable in terms of linear programming. By some comparisons with existing results, the less conservativeness of the obtained results is verified. Finally, a practical example is provided to illustrate the effectiveness of the theoretical findings.     

Robust stability analysis of uncertain switched linear systems with unstable subsystems

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.     

Robust Stabilization and Tracking Control for a Class of Switched Nonlinear Systems

This paper addresses the problem of robust stabilization and tracking control for a class of switched nonlinear systems via the multiple Lyapunov functions (MLFs) approach. First, a state feedback controller and a state dependent switching law are designed to globally asymptotically stabilize the switched system via linear matrix inequalities (LMIs). The main objective of this paper is to develop a tracking control approach that assures global asymptotic output and state tracking with zero tracking error in the steady state. Then, the tracking control is formulated such that the robust H_∞ tracking performance is achieved. Finally, a simulation example is provided to demonstrate the effectiveness of the main method.     

切换状态反馈控制系统的稳定性分析

研究了一类切换控制系统的稳定性分析问题，切换控制器由多个线性状态反馈控制器组成。运用Lyapunov方法得出了稳定性的时域判据，通过求解一个线性矩阵不等式的方法分析切换控制系统的稳定性，利用KYP引理得到了等价的频域判据；并且研究了一类含有参数不确定性的切换控制系统的鲁棒稳定性分析问题，给出了稳定性时域判据和频域判据。稳定性分析的主要结果是基于公共二次Lyapunov函数的，适用于切换系统任意切换的情形，该方法在工程实例中的应用表明了其有效性。     

Design proportional‐integral‐derivative/proportional‐derivative controls for second‐order time‐varying switched nonlinear systems

Based on proportional‐integral‐derivative (PID)/PD controls, we in the article investigate the tracking problem of a class of second‐order time‐varying switched nonlinear systems. To start with, for tracking a given point under arbitrary switching signals, we propose a sufficient condition about PID controller parameters, which can be implicitly described as semialgebraic sets. Successively, we consider the tracking problem under average dwell time (ADT)‐based switching signals and propose an alternative sufficient condition about PID controller parameters. Especially, for tracking an equilibrium point of the system without controls, we can further simply utilize the proportional‐derivative control and similarly construct corresponding semialgebraic conditions about proportional‐derivative controller parameters under arbitrary switching signals and ADT‐based switching signals. Finally, two examples are given to show the applicability of our theoretical results.