A Lagrange duality characterisation for stability under arbitrary switching in switched positive linear systems

The present communication is concerned with uniform exponential stability, under arbitrary switching, in discrete-time switched positive linear systems. Lagrange duality is used in order to obtain a new characterisation for uniform exponential stability which is in terms of sets of inequalities involving each of the matrices that represent the modes of the system. These sets of inequalities are shown to generalise the classical linear Lyapunov inequality that characterises, in positive matrices, the property of being Schur. Each solution to these sets of inequalities is shown to provide a representation, in terms of a number of linear functionals, for a common Lyapunov function for the switched positive linear system. A result is further presented which conveys to, a conservative upper bound on the minimum required number of linear functionals (in the above mentioned representation), and also to a method for computing them. Our proof for the aforementioned characterisation is based on another (equivalent) characterisation, in terms of the solvability of a dynamic programming equation associated to the switched positive linear system, which is also reported in the paper. In particular, it is shown that the associated dynamic programming equation has at most one solution. And this solution is shown to be convex, monotonic, positively homogeneous, and it yields a common Lyapunov function for the switched positive linear system.     

Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time‐varying delay

In this paper, the problems of exponential stability and exponential stabilization for linear singularly perturbed stochastic systems with time‐varying delay are investigated. First, an appropriate Lyapunov functional is introduced to establish an improved delay‐dependent stability criterion. By applying free‐weighting matrix technique and by equivalently eliminating time‐varying delay through the idea of convex combination, a less conservative sufficient condition for exponential stability in mean square is obtained in terms of ε‐dependent linear matrix inequalities (LMIs). It is shown that if this set of LMIs for ε=0 are feasible then the system is exponentially stable in mean square for sufficiently small ε?0. Furthermore, it is shown that if a certain matrix variable in this set of LMIs is chosen to be a special form and the resulting LMIs are feasible for ε=0, then the system is ε‐uniformly exponentially stable for all sufficiently small ε?0. Based on the stability criteria, an ε‐independent state‐feedback controller that stabilizes the system for sufficiently small ε?0 is derived. Finally, numerical examples are presented, which show our results are effective and useful. Copyright © 2010 John Wiley & Sons, Ltd.     

A revisit to the design of switched observers for switched linear systems with unknown inputs

This paper revisits the problem of designing switched observers for switched linear systems with unknown inputs. By performing a state and output coordinates transformation that decouples the unknown input, a novel piecewise time-varying Lyapunov function is introduced to analyze the stability of the switched error dynamics. Compared with the existing time-invariant Lyapunov function method, the proposed time-varying Lyapunov function method is more suitable to exploit the structural characteristics of switched linear systems. New conditions are derived that guarantee the exponential stability of the switched error dynamics. These conditions are formulated in term of linear matrix inequalities (LMIs). By solving a set of LMIs, the switched observers can be designed. Two numerical examples are provided to illustrate the effectiveness of the proposed method.     

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.     

Exponential stabilisation for time-varying delay system with actuator faults: an average dwell time method

The issue of exponential stabilisation for a class of special time-varying delay switched systems resulting from actuator faults is considered in this article. The time-varying delay is assumed to belong to an interval and can be a slow or fast time-varying function. A hybrid state feedback strategy is redesigned to guarantee the system stable since the original controller is unavailable for some actuators failures. A class of switching laws incorporating the average dwell time method is proposed so that the special switched system with interval time-varying delay is exponentially stable. New delay-range-dependent stabilisation conditions using state feedback controllers are formulated in terms of linear matrix inequalities (LMIs) by choosing appropriate Lyapunov–Krasovskii functional without neglecting some useful knowledge on system states. Parameterised characterisations of the controllers are given in terms of the feasibility solutions to the LMIs. Two numeral examples are given to demonstrate the applicability and the effectiveness of the proposed method.     

Stability analysis for uncertain switched neutral systems with discrete time-varying delay: A delay-dependent method

This paper considers the robust stability of a class of switched neutral systems with discrete time-varying delay and time-varying structure. Sufficient conditions for exponential stability criteria are developed for arbitrary switching signal with average dwell time. The results are obtained based on Lyapunov’s stability analysis via Krasovsky–Lyapunov’s functionals and the related stability study is performed to obtain delay-dependent results. It is proved that the stabilizing switching rule is arbitrary if all the switched subsystems are exponentially stabile. These conditions are delay-dependent and are given in the form of linear matrix inequalities (LMIs). Some examples are worked out to illustrate the effectiveness of the result.     

具有一步随机通讯时延的网络化切换系统H∞滤波

针对数据包在网络中传输的一步随机时延情形,研究任意切换律作用下的离散切换系统的H∞滤波问题.时延数据包包含系统的测量输出信号和切换信号,其时延由一个满足Bernoulli分布的随机变量来描述.选用切换Lyapunov函数,设计全阶和降阶的切换滤波器使得滤波误差系统在均方意义下指数稳定且具有扰动衰减度γ.滤波器的参数通过...     

一类非线性切换时延系统的鲁棒稳定性

本文通过利用平均驻留时间方法,研究一类具有不确定性非线性切换时延系统的指数稳定性问题。给出非切换系统的候选李雅普诺夫函数的衰减估计分析,然后以线性矩阵不等式的形式给出使系统保持指数稳定及鲁棒指数稳定的充分条件,同时也给出了系统状态指数衰减的具体的估计形式。     

Switched state-feedback control for continuous time-varying polytopic systems

This article deals with switched state-feedback ?₂ control design of continuous time-varying polytopic systems. More specifically, the main goal is to determine, simultaneously, a set of state-feedback gains and a switching rule to orchestrate them, rendering the closed-loop system globally asymptotically stable for all time-varying uncertain parameter under consideration and assuring a guaranteed ?₂ cost. A contribution of the present switched control technique compared to the gain scheduling, widely used in the literature, is that the online measurement of the uncertain parameter is not required and no assumption on its time derivative is imposed. The conditions are based on modified Lyapunov–Metzler inequalities and can be solved by line search coupled with LMIs. An academic example illustrates the theoretical results and compares the present technique with other techniques from literature.     

Constrained quadratic stabilization of discrete-time uncertain non-linear multi-model systems using piecewise affine state-feedback

In this paper a method for non-linear robust stabilization based on solving a bilinear matrix inequality (BMI) feasibility problem is developed. Robustness against model uncertainty is handled. In different non-overlapping regions of the statespace known as clusters the plant is assumed to be an element in a polytope whose vertices (local models) are affine systems. In the clusters containing the origin in their closure, the local models are restricted to being linear systems. The clusters cover the region of interest in the state-space. A n affine state-feedback is associated with each cluster. By utilizing the affinity of the local models and the state-feedback, a set of linear matrix inequalities (LMIs) combined with a single non-convex BMI are obtained which, if feasible, guarantee quadratic stability of the origin of the closed loop. The feasibility problem is attacked by a branch-and-bound-based global approach. If the feasibility check is successful, the Lyapunov matrix and the piecewise affine state-feedback are given directly by the feasible solution. Control constraints are shown to be representable by LMIs or BMIs, and an application of the control design method to robustify constrained non-linear model predictive control is presented. In addition, the control design method is applied to a simple example.     

Robust reliable sampled-data control for switched systems with application to flight control

This paper addresses the robust reliable stabilisation problem for a class of uncertain switched systems with random delays and norm bounded uncertainties. The main aim of this paper is to obtain the reliable robust sampled-data control design which involves random time delay with an appropriate gain control matrix for achieving the robust exponential stabilisation for uncertain switched system against actuator failures. In particular, the involved delays are assumed to be randomly time-varying which obeys certain mutually uncorrelated Bernoulli distributed white noise sequences. By constructing an appropriate Lyapunov–Krasovskii functional (LKF) and employing an average-dwell time approach, a new set of criteria is derived for ensuring the robust exponential stability of the closed-loop switched system. More precisely, the Schur complement and Jensen's integral inequality are used in derivation of stabilisation criteria. By considering the relationship among the random time-varying delay and its lower and upper bounds, a new set of sufficient condition is established for the existence of reliable robust sampled-data control in terms of solution to linear matrix inequalities (LMIs). Finally, an illustrative example based on the F-18 aircraft model is provided to show the effectiveness of the proposed design procedures.     

Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> static output feedback control of discrete-time switched polytopic linear systems with average dwell-time

This paper investigates the problem of robust exponential H _∞ static output feedback controller design for a class of discrete-time switched linear systems with polytopic-type time-varying parametric uncertainties. The objective is to design a switched static output feedback controller guaranteeing the exponential stability of the resulting closed-loop system with a minimized exponential H _∞ performance under average dwell-time switching scheme. Based on a parameter-dependent discontinuous switched Lyapunov function combined with Finsler’s lemma and Dualization lemma, some novel conditions for exponential H _∞ performance analysis are first proposed and in turn the static output feedback controller designs are developed. It is shown that the controller gains can be obtained by solving a set of linear matrix inequalities (LMIs), which are numerically efficient with commercially available software. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approaches.     

Stability and stabilisation of Itô stochastic systems with piecewise homogeneous Markov jumps

This paper focuses on the problem of stability and stabilisation for continuous-time Itô stochastic Markovian jump linear systems with time-varying transition rates. The time-varying property of the transition rates is considered to be finite piecewise homogeneous. Firstly, the stability conditions of the piecewise homogeneous Itô stochastic Markovian jump linear systems are given in terms of linear matrix inequalities (LMIs). Especially, a novel stability criterion is developed for the considered systems by the existence of the unique positive-definite solution of the corresponding coupled Lyapunov matrix equations. Secondly, two state-feedback controllers are designed via LMIs to stabilise the systems. Finally, a practical example is provided to illustrate the effectiveness of the presented theoretical results.     

Guaranteed cost control of switched systems with neutral delay via dynamic output feedback

This article is concerned with the problem of cost-guaranteed dynamic output feedback (DOF) control for a class of continuous-time linear switched system with both discrete and neutral delays. By using the average dwell time approach and the piecewise Lyapunov function technique, a sufficient condition is first proposed, in terms of a set of linear matrix inequalities (LMIs), to guarantee the exponential stability and a certain bound for the cost function of the closed-loop system, where the decay estimate is explicitly given to quantify the convergence rate. Then, the corresponding solvability conditions for a desired DOF controller under guaranteed cost are established by using the approach of linearising variable transforms. Since these obtained conditions are not all expressed by strict LMIs, the cone complementary linearisation method is exploited to cast them into sequential minimisation problems subject to LMI constraints, which can be easily solved numerically. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theory.     

Persistent bounded disturbance rejection for discrete-time delay systems

In this article, we provide a novel solution to the problem of persistent bounded disturbance rejection in linear discrete-time systems with time-varying delays. The solution is developed based on the tools of invariant set analysis and Lyapunov-function method. As an integral part of the solution, we derive less conservative sufficient conditions on robust attractor for discrete-time systems with delays in terms of strict linear matrix inequalities to guarantee the desired ?₁-performance. A robust state-feedback controller is designed and the associated gain is determined using strict LMIs. The developed results are tested on a representative example.     

一类含有输入时滞的不确定切换系统的鲁棒指数镇定

针对一类含有输入时滞的不确定切换系统,讨论了鲁棒指数镇定问题,提出了可行的切换控制器和切换信号的设计方法．首先,运用还原法将具有时滞的切换系统转化为无时滞的切换系统;然后,基于线性矩阵不等式,给出了系统在切换控制器和切换信号作用下鲁棒指数可镇定的充分条件,并利用多Ｌｙａｐｕｎｏｖ函数方法给出了定理的详细证明;最后,利用数值仿真例子验证了该方法的有效性．     

Exponential H∞ filtering for uncertain discrete‐time switched linear systems with average dwell time: A µ‐dependent approach

In this paper, the problem of exponential H_∞ filter problem for a class of discrete‐time polytopic uncertain switched linear systems with average dwell time switching is investigated. The exponential stability result of the general discrete‐time switched systems using a discontinuous piecewise Lyapunov function approach is first explored. Then, a new µ‐dependent approach is proposed, which means the analysis or synthesis of the underlying systems is dependent on the increase degree µ of the piecewise Lyapunov function at the switching instants. A mode‐dependent full‐order filter is designed such that the developed filter error system is robustly exponentially stable and achieves an exponential H_∞ performance. Sufficient existence conditions for the desired filter are derived and formulated in terms of a set of linear matrix inequalities, and consequently the minimal average dwell time and the corresponding filter are obtained from such conditions for a given system decay degree. A numerical example is presented to demonstrate the potential and effectiveness of the developed theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

一类线性不确定切换系统的非脆弱控制器设计方法

针对一类线性不确定切换系统,在控制器增益存在加性摄动的情况下,给出了非脆弱状态反馈控制器的两种设计方案：方案1是利用多Lyapunov函数方法,通过子系统之间的切换,使系统对所有可允许的不确定性保持渐近稳定;方案2是基于平均驻留时间方法,给出了非脆弱状态反馈控制器存在的一个充分条件．相应的结果都是以线性矩阵不等式的形式给出．最后以一个数值例子说明了上述两种方法的有效性.     

Stability analysis of piecewise discrete-time linear systems

Presents a stability analysis method for piecewise discrete-time linear systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the system can be established if a piecewise Lyapunov function can be constructed and, moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that is numerically feasible with commercially available software.