Robust switching of discrete-time switched linear systems

In this work, we address the problem of robust switching design, which seeks a switching signal that makes the switched system exponentially stable and robust against switching perturbations. To properly capture the sensitivity of a switching signal undergoing various switching perturbations, we define the relative distance between state-feedback path-wise switching signals. We establish that any stabilizing state-feedback path-wise switching signal is robust with respect to switching perturbations. A lower bound of the robustness margin is explicitly presented.     

A robust stabilizing law for switched linear systems

In this paper, we address some robustness issues for a class of switched linear systems with non-linear perturbations. We illustrate through an example that the well-known state-space-partition-based switching law may lead to chattering in the event that the systems undergo perturbations. To overcome this, we propose a state-feedback switching law with a non-zero level set. We prove that this switching law can prevent the system from chattering. In addition, this switching law makes the perturbed systems bounded with bounded perturbations, and practically convergent with convergent perturbations. We also propose an observer-based switching law to steer the switched system stable with nice robustness performances.     

Quadratic Stabilization of Switched Uncertain Linear Systems: A Convex Combination Approach

We consider quadratic stabilization for a class of switched systems which are composed of a finite set of continuoustime linear subsystems with norm bounded uncertainties. Under the assumption that there is no single quadratically stable subsystem, if a convex combination of subsystems is quadratically stable, then we propose a state-dependent switching law, based on the convex combination of subsystems, such that the entire switched linear system is quadratically stable. When the state information is not available, we extend the discussion to designing an output-dependent switching law by constructing a robust Luenberger observer for each subsystem.      

Asynchronous switching output feedback control of discrete-time switched linear systems

In this paper, the problem of dynamic output-feedback control synthesis is addressed for discrete-time switched linear systems under asynchronous switching. The proposed hybrid controller consists of a standard dynamic output-feedback switching control law and an impulsive reset law induced by controller state jumps. Using the average dwell time technique incorporating with multiple quadratic Lyapunov functions, the switching control synthesis conditions for asymptotic stability with guaranteed weighted ?₂-gain performance are derived as a set of linear matrix inequalities (LMIs). The proposed hybrid synthesis scheme advances existing design methods for output-feedback asynchronous switching control of switched linear systems in two important aspects: LMI formulation of the synthesis problem; and arbitrary order of the controller state. A numerical example is used to illustrate the effectiveness and advantages of the proposed design technique.     

Uniformly Stabilizing Sets of Switching Sequences for Switched Linear Systems

A set of switching sequences of a discrete-time switched linear system is said to be uniformly stabilizing if the system is uniformly exponentially stable over all switching sequences that belong to the set. Based on the recent result that a switched linear system is uniformly stable if and only if an increasing countable union of linear matrix inequality conditions is satisfied, we characterize all uniformly stabilizing sets of switching sequences. A discrete-time hybrid automaton of the automobile engine is considered to illustrate the result     

Two slow stabilizing switching laws for discrete time positive switched systems

On the basis of a linear copositive Lyapunov function (LF) and a diagonal quadratic LF, respectively, two slow stabilizing switching laws are proposed for discrete time positive switched systems composed of subsystems. Under these two stabilizing switching laws, the LFs are allowed to increase in state‐driven intervals while the stability of positive switched systems is maintained. In addition, it is shown that positive switched systems under these two slow switching laws are robust against certain classes of perturbations. Furthermore, when the states of the systems are not available, observer‐based stabilizing switching laws for positive switched systems are also proposed. Some numerical examples are finally given to illustrate the effectiveness of the proposed stabilizing switching laws. Copyright © 2013 John Wiley & Sons, Ltd.     

Design of stabilizing switching control laws for discrete- and continuous-time linear systems using piecewise-linear Lyapunov functions

In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions. In particular, we identify classes of switching sequences that result in stable trajectories. Given a switched linear system, we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system. In the proposed approach, we assume that each individual subsystem is stable and admits a piecewise linear Lyapunov function. Based on these Lyapunov functions, we compose 'global' Lyapunov functions that guarantee stability of the switched linear system. A large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space. The approach is applied to both discrete-time and continuous-time switched linear systems.     

输入有饱和的轧机液压伺服系统的多模型切换控制

针对轧机液压伺服系统随工况变化而存在的弹性负载力和外负载力跳变所引起的结构跳变问题,建立并优化了考虑控制输入饱和特性的、系统连续工作时不同工况下的轧机液压伺服系统被控对象的多模型集.对每一模型采用线性矩阵不等式(LMI)方法设计了抗饱和状态反馈控制器,并制定了整个系统的切换策略,分析了具有饱和输入和外部扰动的切换系统的...     

On the stabilization of switched linear stochastic systems with unobservable switching laws

1 Introduction Consider the following switched linear stochastic system dx(t) = Aθtx(t)dt +Bθtu(t)dt + Fθtdwt, E x0 2 < ∞ (1) where x0 = x(0). The system state x(t) ∈ Rn is completely observable and u(t) ∈ Rr is the input. The system noise {wt} is an l-dimensional standard Wiener process, which is independent of {x(s),s t}. Aθt, Bθt, Fθt are coefficient matrices with suitable dimensions. The switching law (or signal) θt : [0, ∞) → Θ is a piecewise constant function of time. I…     

On the stabilization of switched linear stochastic systems with unobservable switching laws

This paper is concerned with the stabilization problem of switched linear stochastic systems with unob- servable switching laws. In this paper the system switches among a finite family of linear stochastic systems. Since there are noise perturbations, the switching laws can not be identified in any finite time horizon. We prove that if each individual subsystem is controllable and the switching duration uniformly has a strict positive lower bound, then the system can be stabilized by using a controller that uses online state estimation.     

一类离散时间切换线性系统的无差拍控制

研究一类带多控制器和多传感器离散时间线性系统的无差拍控制.对能控系统,通过适当的状态坐标变换获得系统矩阵的块三角结构,再设计状态反馈和周期切换策略使得状态反馈矩阵在有限周期内为零,从而保证闭环系统的无差拍稳定.进一步,对能观系统,设计具有有限时间精确估计的动态输出反馈,通过适当的周期切换策略实现闭环系统的无差拍稳定.最后,给出一个例子以验证所提设计方法的有效性.     

Nonfragile control for a class of uncertain switching fuzzy time-delay systems

A sufficient condition for the existence of nonfragile state-feedback controllers and a switching law for a class of uncertain switching fuzzy time-delay systems are proposed based on the multiple Lyapunov functions technique, under the assumption that the controller gain to be designed has variations. The corresponding results are all formulated in terms of linear matrix inequalities (LMIs). Simulation results show the effectiveness of the design method.     

Stability and stabilization of switched stochastic systems under asynchronous switching

This paper studies the stability and stabilization problems for a class of switched stochastic systems under asynchronous switching. The asynchronous switching refers to that the switching of the candidate controllers does not coincide with the switching of system modes. Two situations are considered: (1) time-delayed switching situation, that is, the switching of the candidate controllers has a lag to the switching of the system modes; (2) mismatched switching situation, the switching of the candidate controllers does not match the switching of the system modes. Using average dwell time and Lyapunov-like function, sufficient conditions are established for stochastic input-to-state stability of the whole system. Also, the stabilizing controller design approach is proposed for switched stochastic linear systems. The minimal average dwell time and the controller gain are achieved. Finally, a numerical example is used to demonstrate the validity of the developed results.     

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.     

Tracking control for switched linear systems with time‐delay: a state‐dependent switching method

Tracking control for switched linear systems with time‐delay is investigated in this paper. Based on the state‐dependent switching method, sufficient conditions for the solvability of the tracking control problem are given. We use single Lyapunov function technique and a typical hysteresis switching law to design a tracking control law such that the H_∞ model reference tracking performance is satisfied. The controller design problem can be solved efficiently by using linear matrices inequalities. Since convex combination techniques are used to derive the delay independent criteria, some subsystems are allowed to be unstable. It is highly desirable that a non‐switched time‐delay system can not earn such property. Simulation example shows the feasibility and validity of the switching control law. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Observer-based stabilization of switching linear systems

In this paper, we present a “deep pole assignment method” to study the observer-based stabilization of switching linear systems where the dynamics of each mode are known a priori but the switching times of modes are arbitrary. The design can be used for both finite and infinite switched linear systems. We emphasize our paper on the case where the switchings of the observer and controller do not coincide with those of the system.     

LQR optimization of linear system switching

Considers offline optimization of a switching sequence for a given finite set of linear control systems, together with joint optimization of control laws. A linear quadratic full information criterion is optimized and dynamic programming is used to find an optimal switching sequence and control law. The main result is a method for efficient pruning of the search tree to avoid combinatoric explosion. A method to prove optimality of a found candidate switch sequence and corresponding control laws is presented.     

Control discrete-time switched singular systems with state delays under asynchronous switching

This article is concerned with the problem of state feedback control for a class of discrete-time switched singular systems with time-varying state delays under asynchronous switching. The asynchronous switching considered here means that the switching instants of the candidate controllers lag behind those of the system modes. The concept of mismatched control rate is introduced. By using the multiple Lyapunov function approach and the average dwell time technique, a sufficient condition for the existence a stabilising switching law is first derived to guarantee the regularity, causality and exponential stability of the closed-loop system in the presence of asynchronous switching. The stabilising switching law is characterised by a upper bound on the mismatched control rate and a lower bound on the average dwell time. Then, the corresponding solvability condition for a set of mode-dependent state feedback controllers is established by using the linear matrix inequality (LMI) technique. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.     

Frequency-Domain Stability Conditions for Discrete-Time Switched Systems

We consider discrete-time switched systems with switching of linear time-invariant right-hand parts. The notion of a connected discrete switched system is introduced. For systems with the connectedness property, we propose necessary and sufficient frequency-domain conditions for the existence of a common quadratic Lyapunov function that provides the stability for a system under arbitrary switching. The set of connected switched systems contains discrete control systems with several time-varying nonlinearities from the finite sectors, considered in the theory of absolute stability. We consider the case of switching between three linear subsystems in more details and give an illustrative example.     

切换线性系统的聚合优化

分析了切换线性系统的稳定性和优化问题,提出了一个Armijo步长优化的共轭梯度算法来寻找适当代价函数下的优化切换时间点集.为了确保优化切换路径是可压缩的,提出了代价函数需要满足的受限表达式.设计了几种优化分段状态反馈切换律来搜索聚合系统的最优切换路径,同时这些切换路径就是对应原始切换线性系统的次优化切换路径.最后,一个实例演示了不同切换律下的切换策略和优化代价.