Finite-time stability and stabilisation for switched linear systems

In this article, the finite-time stability analysis and stabilisation for switched continuous linear system are addressed. Several sufficient conditions for finite-time boundness and stability of switched system are proposed in this article. By comparing the proposed results, it is shown that more the information about the switching signal is known, less conservative results can be derived. Then, based on the finite-time stability analysis results, static state and dynamic output feedback controllers are designed to finite-time stabilise switched linear systems. Several numerical examples are given to illustrate the proposed results within this article.     

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.     

Modelling and stability analysis of switching impulsive power systems with multiple equilibria

This paper tries to model power systems accompanied with a series of faults in the form of switched impulsive Hamiltonian systems (SIHSs) with multiple equilibria (ME) and unstable subsystems (US), and then analyze long-term stability issues of the power systems from the viewpoint of mathematics. According to the complex phenomena of switching actions of stages and generators, impulses of state, and existence of multiple equilibria, this paper first introduces an SIHS with ME and US to formulate a switching impulsive power system composed of an active generator, a standby generator, and an infinite load. Then, based on special system structures, a unique compact region containing all ME is determined, and novel stability concepts of region stability (RS), asymptotic region stability (ARS), and exponential region stability (ERS) are defined for such SIHS with respect to the region. Third, based on the introduced stability concepts, this paper proposes a necessary and sufficient condition of RS and ARS and a sufficient condition of ERS for the power system with respect to the region via the maximum energy function method. Finally, numerical simulations are carried out for a power system to show the effectiveness and practicality of the obained novel results.     

Input and output constraints-based stabilisation of switched nonlinear systems with unstable subsystems and its application

This paper studies the problem of stabilisation of switched nonlinear systems with output and input constraints. We propose a recursive approach to solve this issue. None of the subsystems are assumed to be stablisable while the switched system is stabilised by dual design of controllers for subsystems and a switching law. When only dealing with bounded input, we provide nested switching controllers using an extended backstepping procedure. If both input and output constraints are taken into consideration, a Barrier Lyapunov Function is employed during operation to construct multiple Lyapunov functions for switched nonlinear system in the backstepping procedure. As a practical example, the control design of an equilibrium manifold expansion model of aero-engine is given to demonstrate the effectiveness of the proposed design method.     

Robust observer-driven switching stabilization of switched linear systems

In this work, we study the robust observer-driven switching stabilization problem of switched linear systems. Under the condition that each subsystem is completely observable, with the observer-driven switching law which makes the system exponentially stable for the nominal system, we prove that the overall system is robust against structural/switching perturbations and is input/output to state stable for unstructural perturbations.     

Analysis of robust stability for switched systems with multiple time-delays

The sufficient conditions of delay-dependent exponential stability for switched systems and robust exponential stability for uncertain switched systems with two time-delays are presented by using average dwell time method and free-weighting matrix method.The interaction between different time-delays is considered.The sufficient conditions do not need that every subsystem is stable.The designed methods of the switching law are also given.The sufficient conditions are given in the form of linear matrix inequalities that can be solved easily.The result is proven to be valid by the simulation at last.     

离散多平衡点正切换系统有限区间稳定与镇定

在切换事件中,外界环境的干扰或者事物自身的发展变化会导致多平衡点现象.此时,多平衡点切换系统模型比传统的切换系统模型更适合描述此类事件.因此本文研究离散多平衡点正切换线性系统在有限时间区间上的稳定性与镇定性.第1,给出离散多平衡点线性切换系统为正的充要条件.第2,提出离散多平衡点正切换线性系统在有限时间区间上稳定的概念.第3,通过构造合适的Lyapunov函数以及合理分配系统的驻留时间与切换次数,针对部分子系统不稳定的离散多平衡点正切换线性系统,建立所考虑的自治系统有限时间稳定的充分条件.第4,给出非自治多平衡点正切换线性系统的控制器设计.最后,仿真例子验证理论结果的正确性.     

Asynchronously switched control of switched linear systems with average dwell time

This paper concerns the asynchronously switched control problem for a class of switched linear systems with average dwell time (ADT) in both continuous-time and discrete-time contexts. The so-called asynchronous switching means that the switchings between the candidate controllers and system modes are asynchronous. By further allowing the Lyapunov-like function to increase during the running time of active subsystems, the extended stability results for switched systems with ADT in nonlinear setting are first derived. Then, the asynchronously switched stabilizing control problem for linear cases is solved. Given the increase scale and the decrease scale of the Lyapunov-like function and the maximal delay of asynchronous switching, the minimal ADT for admissible switching signals and the corresponding controller gains are obtained. A numerical example is given to show the validity and potential of the developed results.     

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.     

Design of two-layer switching rule for stabilization of switched linear systems with mismatched switching

A two-layer switching architecture and a two-layer switching rule for stabilization of switched linear control systems are proposed, under which the mismatched switching between switched systems and their candidate hybrid controllers can be allowed. In the low layer, a state-dependent switching rule with a dwell time constraint to exponentially stabilize switched linear systems is given; in the high layer, supervisory conditions on the mismatched switching frequency and the mismatched switching ratio are presented, under which the closed-loop switched system is still exponentially stable in case of the candidate controller switches delay with respect to the subsystems. Different from the traditional switching rule, the two-layer switching architecture and switching rule have robustness, which in some extend permit mismatched switching between switched subsystems and their candidate controllers.     

On Padé approximations, quadratic stability and discretization of switched linear systems

In this note we consider the stability preserving properties of diagonal Padé approximations to the matrix exponential. We show that while diagonal Padé approximations preserve quadratic stability when going from continuous-time to discrete-time, the converse is not true. We discuss the implications of this result for discretizing switched linear systems. We also show that for continuous-time switched systems which are exponentially stable, but not quadratically stable, a Padé approximation may not preserve stability.     

Global stabilisation for a class of switched high-order stochastic nonlinear systems

This paper is devoted to the problem of global stabili sation for a class of switched high-order stochastic nonlinear systems under arbitrary switchings. By extending the adding -a -power -integrator technique and choosing an appropriate common Lyapunov function, a common state-feedback controller is explicitly constructed to render the closed-loop system globally asymptotically stable in probability. The effectiveness of the proposed approach is demonstrated by a simulation example.     

Finite-time stability and stabilisation of singular discrete-time linear positive systems with time-varying delay

ABSTRACT

In this paper, the problem of finite-time stability and stabilisation for positive singular discrete-time linear systems with time-varying delay is investigated. We first present novel delay-dependent sufficient conditions for positivity and finite-time stability of the unforced systems. We then apply the obtained results to solve finite-time stabilisation problem of the considered systems. The sufficient conditions for the positivity and finite-time stabilisable of such systems are formulated in terms of a standard linear programming (LP) problem. Numerical examples are provided to illustrate the effectiveness and advantages of our results.     

Improved results on stability of continuous-time switched positive linear systems

In this paper, the problems of stability for switched positive linear systems (SPLSs) under arbitrary switching are investigated in a continuous-time context. The so-called “copositive polynomial Lyapunov function” (CPLF) giving a generalization of copositive types of Lyapunov function is first proposed, which is formulated in a higher order form of the positive states of the underlying systems. It is illustrated in this paper that some classical types of Lyapunov functions can be seen as special cases of the proposed CPLF. Then, new stability conditions are developed by the new Lyapunov function approach. It is also proved that the conservativeness of the obtained criteria can be further reduced as the degree of the Lyapunov function increases. A numerical example is given to demonstrate the effectiveness and less conservativeness of the developed techniques.     

Excitability, stability, and sign of equilibria in positive linear systems

The relationship between sign and stability of equilibria in positive linear systems is investigated in this paper. In particular it is proved that the well-known result “stability and irreducibility imply strict positivity” can be better specified by introducing the notion of excitability. In fact stability, excitability and strict positivity of the equilibria are three perfectly balanced properties since any one of them is implied by the two others.     

线性切换容错控制系统稳定性的新判据*

研究了线性切换容错控制系统的稳定性问题。利用分段李雅普诺夫函数方法,结合梅茨勒矩阵的性质和矩阵不等式的分析技巧,得到了基于李雅普诺夫—梅兹勒线性矩阵不等式判定系统稳定的新结果。设计依赖于状态的切换规则便于计算、易于检验。最后利用MATLAB工具箱得到的仿真实例验证了本结果的可行性。     

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.     

Stability of switched positive linear delay systems with mixed impulses

ABSTRACT

This paper investigates the global exponential stability of both continuous-time and discrete-time switched positive linear delay systems with the coexistence of destabilising and stabilising impulses. Based on the comparison principle, with the help of discretised multiple linear copositive Lyapunov functions, some delay-independent stability results are obtained. It should be noted that it is the first time that discrete-time switched delay systems with asynchronous switching and impulses are considered, and the switching effect can be stabilising in the obtained results of this paper. Some numerical examples are provided to demonstrate the effectiveness and superiority of the derived results.     

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.