三阶切换系统的镇定

研究了三阶切换系统的镇定问题. 给出三阶线性控制系统，以给定的正定矩阵[WTHX]M[WTBZ]作为二次李亚普诺夫函数的新的充分条件. 然后, 证明了在适当条件下三阶切换系统是二次可镇定的. 文中设计出镇定三阶切换系统的反馈控制律, 并提出了解决二次镇定问题的直接算法.     

Nonlinear versus linear control in the robust stabilizability oflinear uncertain systems via fixed-order output feedback

This paper considers the problem of robustly stabilizing an uncertain system via fixed-order output feedback control. This problem is considered by defining a notion of robust stabilizability with a quadratic storage function. This notion is closely related to the notion of quadratic stabilizability. The main result of the paper shows that if an uncertain system is robustly stabilizable with a quadratic storage function via a nonlinear time-varying controller, then it will also be robustly stabilizable with a quadratic storage function via a linear time-invariant controller of the same order     

Convex necessary and sufficient stabilisability conditions in switched linear systems with rank-one modes

ABSTRACT

A novel convex necessary and sufficient condition for state-feedback exponential stabilisability in discrete-time switched linear systems, whose modes are described by rank-one matrices, is reported and proved in the present communication. A switched linear system, of this class, is shown to be state-feedback exponentially stabilisable if and only if a set of linear matrix inequalities (LMIs) associated to the system is feasible. And the solvability of this set of LMIs associated to the system is shown to be equivalent to that of a set of (standard) linear inequalities associated to the system. It is also proved that each solution to this set of LMIs (associated to the system) yields, through explicit formulas, to an exponentially stabilising state-feedback mapping, and also to a Lyapunov function for the exponential stability of the trivial solution of the corresponding closed-loop system (obtained by means of that feedback mapping). And such a Lyapunov function is always represented by a number of quadratic functionals that equals the number of modes composing the switched system.     

On the Value Functions of the Discrete-Time Switched LQR Problem

In this paper, we derive some important properties for the finite-horizon and the infinite-horizon value functions associated with the discrete-time switched LQR (DSLQR) problem. It is proved that any finite-horizon value function of the DSLQR problem is the pointwise minimum of a finite number of quadratic functions that can be obtained recursively using the so-called switched Riccati mapping. It is also shown that under some mild conditions, the family of the finite-horizon value functions is homogeneous (of degree 2), is uniformly bounded over the unit ball, and converges exponentially fast to the infinite-horizon value function. The exponential convergence rate of the value iterations is characterized analytically in terms of the subsystem matrices.      

Quadratic Stabilizability for Polytopic Uncertain Switched Linear Systems via Switched Observer

In this paper, we study a quadratic stabilizability problem via switched observer for uncertain continuous‐time switched linear systems in the sense that the subsystem's matrices are represented as a polytopic linear combination of vertex matrices. Firstly, sufficient conditions for polytopic uncertain continuous‐time switched linear systems to be quadratically stabilizable via state feedback are summarized. Next, sufficient conditions for polytopic uncertain switched linear systems to be quadratically stabilizable via switched observer are given under the assumption that output matrices of subsystems have no uncertainties. Further, a numerical example is also investigated.     

Quadratic stabilizability of switched linear systems with polytopic uncertainties

In this paper, we consider quadratic stabilizability via state feedback for both continuous-time and discrete-time switched linear systems that are composed of polytopic uncertain subsystems. By state feedback, we mean that the switchings among subsystems are dependent on system states. For continuous-time switched linear systems, we show that if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices, then the switched system is quadratically stabilizable via state feedback. For discrete-time switched linear systems, we derive a quadratic stabilizability condition expressed as matrix inequalities with respect to a family of non-negative scalars and a common positive definite matrix. For both continuous-time and discrete-time switched systems, we propose the switching rules by using the obtained common positive definite matrix.     

Stability of switched systems: a Lie-algebraic condition

We present a sufficient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems.     

时滞状态反馈下离散时滞系统H∞控制器失效时间分析

针对控制器存在短暂失效的情形,研究一类时变时滞离散系统在时滞状态反馈控制下的H_∞控制器失效时间分析问题.本文的目标是寻求控制器正常工作时间与失效时间的比率应满足的条件以确保系统指数镇定且具有加权l_2增益.为此,基于切换的思想,所考虑的系统被转化为一个仅含有两个子系统的切换系统,其中一个子系统是控制器失效时的不稳定子系统,另一个是控制器未失效时的稳定子系统.通过使用多Lyapunov函数及平均驻留时间方法,给出问题可解的充分条件及时滞状态反馈H_∞切换控制器的设计方案.仿真算例表明了所得结果的有效性.     

Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions

This paper shows that the matrix inequality conditions for stability/stabilizability of linear differential inclusions derived from two classes of composite quadratic functions are not conservative. It is established that the existing stability/stabilizability conditions by means of polyhedral functions and based on matrix equalities are equivalent to the matrix inequality conditions. This implies that the composite quadratic functions are universal for robust, possibly constrained, stabilization problems of linear differential inclusions. In particular, a linear differential inclusion is stable (stabilizable with/without constraints) iff it admits a Lyapunov (control Lyapunov) function in these classes. Examples demonstrate that the polyhedral functions can be much more complex than the composite quadratic functions, to confirm the stability/stabilizability of the same system.     

Critical dwell time of switched linear systems

In this paper, we consider the relation between the switching dwell time and the stabilization of switched linear control systems. First of all, a concept of critical dwell time is given for switched linear systems without control inputs, and the critical dwell time is taken as an arbitrary given positive constant for a switched linear control systems with controllable switching models. Secondly, when a switched linear system has many stabilizable switching models, the problem of stabilization of the overall system is considered. An on-line feedback control is designed such that the overall system is asymptotically stabilizable under switching laws which depend only on those of uncontrollable subsystems of the switching models. Finally, when a switched system is partially controllable (While some switching models are probably unstabilizable), an on-line feedback control and a cyclic switching strategy are designed such that the overall system is asymptotically stabilizable if all switching models of this uncontrollable subsystems are asymptotically stable. In addition, algorithms for designing switching laws and controls are presented.     

Critical dwell time of switched linear systems

In this paper, we consider the relation between the switching dwell time and the stabilization of switched linear control systems. First of all, a concept of critical dwell time is given for switched linear systems without control inputs, and the critical dwell time is taken as an arbitrary given positive constant for a switched linear control systems with controllable switching models. Secondly, when a switched linear system has many stabilizable switching models, the problem of stabilization of the overall system is considered. An on-line feedback control is designed such that the overall system is asymptotically stabilizable under switching laws which depend only on those of uncontrollable subsystems of the switching models. Finally, when a switched system is partially controllable （While some switching models are probably unstabilizable）, an on-line feedback control and a cyclic switching strategy are designed such that the overall system is asymptotically stabilizable if all switching models of this uncontrollable subsystems are asymptotically stable. In addition, algorithms for designing switching laws and controls are presented.     

不确定线性时滞系统的稳定化控制器设计*

本文研究了不确定线性时滞系统的稳定化鲁棒控制器设计问题,给出了一类不确定线性时滞系统稳定化鲁棒控制器的设计方法,对于一般的不确定线性时滞系统,如果它们的标称系统是二次型能稳的,则该不确定线性时滞系统也是能稳的,且给出了其稳定化控制器的设计方法。     

Quadratic stabilizability of uncertain linear systems: Existence of a nonlinear stabilizing control does not imply existence of a linear stabilizing control

This note is concerned with the problem of stabilizing an uncertain linear system via state feedback control. An uncertain system which admits a stabilizing state feedback control and some associated quadratic Lyapunov function is said to be quadratically stabilizable. In a number of recent papers, conditions are given under which quadratic stabilizability via nonlinear control implies quadratic stabilizability via linear control. These papers restrict the manner in which the uncertain parameters are permitted to enter structurally into the state equation in order to establish this result. This note presents an example which shows that this implication is not true for more general uncertain linear systems. To this end, we describe an uncertain linear system which is quadratically stabilizable via nonlinear control but not quadratically stabilizable via linear control.     

Stability and Stabilization of Block-cascading Switched Linear Systems

The main purpose of this paper is to investigate the problem of quadratic stability and stabilization in switched linear systems using reducible Lie algebra. First, we investigate the structure of all real invariant subspaces for a given linear system. The result is then used to provide a comparable cascading form for switching models. Using the common cascading form, a common quadratic Lyapunov function is (QLFs) is explored by finding common QLFs of diagonal blocks. In addition, a cascading Quaker Lemma is proved. Combining it with stability results, the problem of feedback stabilization for a class of switched linear systems is solved.     

On Asymptotic Stabilizability of Linear Systems With Delayed Input

This paper examines the asymptotic stabilizability of linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open-loop system is not exponentially unstable (i.e., all the open-loop poles are on the closed left-half plane). A simple example shows that such results would not be true if the open-loop system is exponentially unstable. It is further shown that such systems, when subject to actuator saturation, are semiglobally asymptotically stabilizable by linear state or output feedback.     

线性多变量系统的控制Lyapunov函数构造

提出线性多变量系统控制Lyapunov函数(CLF)构造的一般方法. 先证明可以通过解一类Lyapunov方程, 得到线性系统二次型的CLF. 接着证明了对于线性系统, 这种方法可以提供所有二次型的CLF. 最后证明了若线性系统存在CLF, 那么必存在二次型的CLF. 由此完全解决了线性系统的CLF构造问题.     

A new optimal multirate control of linear periodic andtime-invariant systems

The optimal multirate design of linear, continuous-time, periodic and time-invariant systems is considered. It is based on solving the continuous linear quadratic regulation (LQR) problem with the control being constrained to a certain piecewise constant feedback. Necessary and sufficient conditions for the asymptotic stability of the resulting closed-loop system are given. An explicit multirate feedback law that requires the solution of an algebraic discrete Riccati equation is presented. Such control is simple and can be easily implemented by digital computers. When applied to linear time-invariant systems, multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly. Compared to the classical single-rate sampled-data feedback in which the state is always sampled at the same rate, the multirate system can provide a better response with a considerable reduction in the optimal cost. In general, the multirate scheme offers more flexibility in choosing the sampling rates     

一类离散时间切换混杂系统鲁棒控制

由于切换规则的存在使得切换混杂控制系统的稳定性研究变得极为复杂,如何针对给定的系统设计适当的控制器和切换规则没有统一的方法.本文考虑一类线性不确定离散时间切换混杂系统的鲁棒二次镇定和渐近镇定问题.利用公共李雅普诺夫函数方法和多李雅普诺夫函数方法,分别设计了切换混杂系统鲁棒状态反馈控制器和鲁棒输出反馈控制器,保证了切换混杂系统的二次稳定性和渐近稳定性.仿真结果验证了所提算法的正确有效性.     

Stability and stabilisation of planar switched linear systems via LaSalle's invariance principle

This paper investigates the problem of uniformly asymptotical stability (UAS) and stabilisation of planar switched linear systems using LaSalle's invariance principle of switched systems. First, we show that a common weak quadratic Lyapunov function (WQLF) is enough to assure the UAS of a switched linear system with stable modes. Then the necessary and sufficient conditions for the existence of common WQLF are obtained. Secondly, we consider the problem of uniformly asymptotical stabilisation (UASZ) of single-input planar switched linear systems. Necessary and sufficient conditions for the closed-loop system with proper feedback to share a common WQLF are presented. It is also proved that a common WQLF is enough to assure the UAS of the closed-loop system.     

Event-triggered Control of Discrete-time Switched Linear Systems with an Arbitrary Sampling Period

In this paper, the event-triggered control problem for discrete-time switched linear systems with an arbitrary sampling period is considered. At each sampling instant, only the sampled information of system state and switching signal is available to the controller. Particularly, the sampling period is arbitrary in this paper and frequent switching is allowed to happen in an inter-event period. Based on that, by constructing a time- and mode-dependent quadratic piecewise Lyapunov function, a new globally exponentially stability (GES) result under modal dwell time (MDT) criteria is obtained. By the novel Lyapunov function and the state variable transformation technique, a statefeedback controller is designed for the switched linear system. At last, a numerical example is proposed to illustrate our approach.