Control of switched systems with actuator saturation

1 Introduction Switched systems are composed of a set of subsystems and a switching rule designating which subsystem to be actuated at each moment. Switched systems are commonly found in various engineering practice, such as in auto- motive engine control systems, chemical process control, robotic manufacture, multiple-model systems and so on. In the past decade, many literatures have been devoted to the study of switched systems. Stability is an important issue in the analysis and synthesis o…     

L 2-gain analysis and control synthesis of uncertain switched linear systems subject to actuator saturation

This article addresses the L ₂-gain analysis and control synthesis problem for a class of uncertain switched linear systems with saturating actuators and external disturbances. First, when the controllers are pre-given, an analysis condition on disturbance tolerance is established under which the state trajectory starting from the origin will remain inside a bounded set. By this condition, the problem of estimating disturbance tolerance capability is formulated as a constrained optimisation problem. Then, the restricted L ₂-gain property is analysed over the set of tolerable disturbances. An upper bound on the restricted L ₂-gain is estimated by solving a constrained optimisation problem. Furthermore, when controller gain matrices are design variables, these optimisation problems can be adapted for controller design. All the results are achieved by utilising the multiple Lyapunov functions method and presented in terms of an LMI optimisation-based approach. A numerical example is given to show the effectiveness of the proposed method.     

Stability analysis for linear discrete-time systems subject to actuator saturation

In this paper, stability of discrete-time linear systems subject to actuator saturation is analyzed by combining the saturation-dependent Lyapunov function method with Finsler’s lemma. New stability test conditions are proposed in the enlarged space containing both the state and its time difference which allow extra degree of freedom and lead to less conservative estimation of the domain of attraction. Furthermore, based on this result, a useful lemma and an iterative LMI-based optimization algorithm are also developed to maximize an estimation of domain of attraction. A numerical example illustrates the effectiveness of the proposed methods.     

Robust stabilization of switched discrete-time systems with actuator saturation

This paper studies the robust stabilization problem of switched discrete-time linear systems subject to actuator saturation. New switched saturation-dependent Lyapunov functions are exploited to design a robust stabilizing state feedback controller that maximizes an estimation of the domain of attraction. The design problem of controller （coefficient matrices） is then reduced to an optimization problem with linear matrix inequality （LMI） constraints. A numerical example is given to show the effectiveness of the proposed method.     

L 2-gain analysis and anti-windup design of discrete-time switched systems with actuator saturation

This paper investigates L2-gain analysis and anti-windup compensation gains design for a class of discrete-time switched systems with saturating actuators and L2 bounded disturbances by using the switched Lyapunov function approach.For a given set of anti-windup compensation gains,we firstly give a sufficient condition on tolerable disturbances under which the state trajectory starting from the origin will remain inside a bounded set for the corresponding closed-loop switched system subject to L2 bounded disturbances.Then,the upper bound on the restricted L2-gain is obtained over the set of tolerable disturbances.Furthermore,the antiwindup compensation gains aiming to determine the largest disturbance tolerance level and the smallest upper bound of the restricted L2-gain are presented by solving a convex optimization problem with linear matrix inequality(LMI) constraints.A numerical example is given to illustrate the effectiveness of the proposed design method.     

Anti-windup strategies for discrete-time switched systems subject to input saturation

This paper deals with the design of anti-windup compensator for discrete-time switched systems subject to input saturation. The cases of static and dynamic anti-windup controllers are addressed aiming at maximising the estimate of the basin of attraction of the origin for the closed-loop system. Two aspects of the switching law are taken into account during the design: either it is arbitrary or it is a part of the complete control law. Theoretical conditions allowing to synthesise the anti-windup compensator are mainly described through linear matrix inequalities. Computational oriented conditions are then provided to solve convex optimisation problems that are able to give a constructive solution.     

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.     

L 2-Gain analysis and control synthesis of uncertain discrete-time switched linear systems with time delay and actuator saturation

The problem of L ₂-gain analysis and control synthesis is studied for a class of uncertain discrete-time switched linear systems with time delay and saturating actuators by using the multiple Lyapunov functions method. With the state feedback controllers adopted beforehand, an analysis condition on disturbance tolerance is derived under which the operating state trajectory with zero initial condition will remain inside a bounded set in its proximity. Then with this condition at hand, the largest disturbance tolerance level is determined via the solution of a constrained optimisation problem. Then, a sufficient condition for the restricted L ₂-gain over the set of tolerable disturbances is derived. The smallest upper bound on the restricted L ₂-gain is also obtained by solving a constrained optimisation problem. Furthermore, when controller gain matrices are design variables, these optimisation problems are adjusted for solving the control design task. An illustrative numerical example is given to demonstrate the feasibility and effectiveness of the proposed method.     

On analysis and design of discrete-time constrained switched systems*

This paper presents novel design techniques for discrete-time constrained switched systems. Two problems are considered: the first one is centred at determining a set of switching rules for which the system is unconditionally stable; the second one focuses on the design of a stabilising state-dependent switching function that is subject to structural constraints. Convex optimisation is the kernel of the devised procedures and, thus, these techniques are computationally viable.     

Analysis and design for discrete-time linear systems subject to actuator saturation

We present a method to estimate the domain of attraction for a discrete-time linear system under a saturated linear feedback. A simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. Moreover, the condition can be expressed as linear matrix inequalities (LMIs) in terms of all the varying parameters and hence can easily be used for controller synthesis. The following surprising result is revealed for systems with single input: suppose that an ellipsoid is made invariant with a linear feedback, then it is invariant under the saturated linear feedback if and only if there exists a saturated (nonlinear) feedback which makes the ellipsoid invariant. Finally, the set invariance condition is extended to determine invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements.     

Exponential stabilization of discrete-time switched linear systems

This article studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise quadratic control-Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier synthesis methods that have adopted piecewise quadratic Lyapunov functions for convenience or heuristic reasons. In addition, it is also proved that if a switched linear system is exponentially stabilizable, then it must be stabilizable by a stationary suboptimal policy of a related switched linear-quadratic regulator (LQR) problem. Motivated by some recent results of the switched LQR problem, an efficient algorithm is proposed, which is guaranteed to yield a control-Lyapunov function and a stabilizing policy whenever the system is exponentially stabilizable.     

Disturbance rejection of singularly perturbed switched systems subject to actuator saturation

This paper addresses the disturbance rejection problem for singularly perturbed switched systems subject to actuator saturation. We focus our attention on the synthesis of feedback control laws to achieve disturbance rejection for the systems. By using the average dwell‐time approach together with the piecewise Lyapunov function technique, sufficient conditions for the existence of a controller are derived in terms of LMIs, which are independent of the singular perturbation parameter ε. Some convex optimization problems are formulated to get a larger estimate of the basin of attraction and a better disturbance rejection ability. The state‐feedback controller depends on the singular perturbation parameter ε, but it is shown to be well posed for all ε∈(0,ε₀]. In addition, if ε is sufficiently small, the ε‐dependent controller can be reduced to an ε‐independent one. Finally, a hydraulic servoposition system is used to show the feasibility and effectiveness of the obtained results.     

饱和不确定离散切换系统的静态输出反馈控制

研究一类具有饱和输入的多胞不确定性离散切换系统在平均驻留时间切换方式下的静态输出反馈控制问题.在平均驻留时间切换信号下,基于参数依赖间断切换Lyapunov函数和几个基本引理,通过求解LMI最优问题设计出使闭环系统达到局部指数稳定的静态输出反馈控制器,并给出了系统稳定的吸引域.仿真结果表明了所提出的设计方法的有效性.     

Output regulation for linear discrete-time systems subject to input saturation

The purpose of this paper is to examine the problem of controlling a linear discrete-time system subject to input saturation in order to have its output track (or reject) a family of reference (or disturbance) signals produced by some external generator. It is shown that a semi-global framework, rather than a global framework, for this problem is a natural one. Within this framework, a set of solvability conditions are given and feedback laws which solve the problem are constructed. The theory developed in this paper parallels the one we developed earlier for the continuous-time system. © 1997 by John Wiley & Sons, Ltd.     

Switching signal design for asymptotic stability of uncertain multiple equilibrium switched systems with actuator saturation

This paper addresses the problem of robust control for a class of multiple equilibrium switched systems with actuator saturation and parameter uncertainties. The parameter uncertainties are described in norm‐bounded form. By combining the parametric Lyapunov equation approach and the gain‐scheduling technique, stabilization conditions are established in terms of linear matrix inequalities. State‐feedback controllers can be designed to guarantee the robust stability and performance of the closed‐loop system in the presence of parameter uncertainties and actuator saturation. The proposed method is to increase the convergence rate during the convergence of the states. Numerical examples and simulations are worked out to illustrate the feasibility of the proposed results. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and performance analysis of switched linear systems subject to time delays and actuator saturation

This paper is devoted to the stability and the performance for the switched linear systems with time delays and actuator saturation. Problems to be considered include the reachable set under a class of bounded energy disturbances and time delays. This treatment yields two forms of differential inclusions, a polytopic differential inclusion (PDI) and a norm-bounded differential inclusion (NDI) that contain the original system. Using the PDI and the NDI describing actuator saturation systems, sufficient conditions for the output feedback control laws are derived to stabilise the systems. The estimation of the attraction domain is to ensure that the state remains inside the level set of a certain Lyapunov function where the PDI and NDI are valid. The sufficient conditions for stability and performances are derived as linear matrix inequalities. A example based on jitterbug tools is used to illustrate the effectiveness of the proposed approach.     

Stability analysis for discrete-time switched time-delay systems

The stability analysis problem is studied in this paper for a class of discrete-time switched time-delay systems. By using a newly constructed Lyapunov functional and the average dwell time scheme, a delay-dependent sufficient condition is derived for the considered system to be exponentially stable. The obtained results provide a solution to one of the basic problems in discrete-time switched time-delay systems, that is, to find a switching signal for which the switched time-delay system is exponentially stable. Two illustrative examples are given to demonstrate the effectiveness of the proposed results.     

Stability analysis of discrete-time systems with actuator saturation by a saturation-dependent Lyapunov function

In this paper, stability of discrete-time linear systems subject to actuator saturation is analyzed using a saturation-dependent Lyapunov function. This saturation-dependent Lyapunov function captures the real-time information on the severity of actuator saturation and leads to less conservative estimate of the domain of attraction, which is based on the solution of an LMI optimization problem. Numerical examples are presented to show the effectiveness of the proposed method.     

Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.     

Nonfragile robust exponential stabilization of nonlinear uncertain switched systems with actuator saturation

For nonlinear uncertain switched systems, the problem of how to overcome the controller vulnerability is studied when the actuator saturation is considered. The sufficient condition for guaranteeing nonfragile robust exponential stabilization of the system is derived by using the method of minimum dwell time. Then, a switching law and the nonfragile state feedback controllers are designed such that the closed-loop system can be robustly exponentially stabilized at the origin. Next, when some scalar parameters of the closed-loop system are given, the design issue of the nonfragile state feedback controllers, which aim at enlarging the estimation of domain of attraction for closed-loop system, is transformed into a convex optimization issue with linear matrix inequalities (LMI) constraints. Finally, an example is given to verify the effectiveness of the proposed method.