Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.     

Input-to-state stability and stabilization for switched nonlinear positive systems

Input-to-state stability (ISS) analysis and stabilization are concerned in this paper for switched nonlinear positive systems (SNPS), where the deterministic and random switching are both included. For general SNPS, switched affine nonlinear positive systems (SANPS) and switched linear positive systems (SLPS) with deterministic and some kinds of random ”slow” switching, some criterions on ISS are provided. From the criterions for SANPS and SLPS, the ISS properties can be judged just by the differential, algebraic and switching characteristics of the systems. Further, based on the criterions for SANPS and SLPS, some state feedback controllers are designed such that the closed-loop systems be positive, ISS or ISS in some stochastic senses. Four simulation examples verify the validity of our results.     

Stabilization of switched linear systems with mode-dependent time-varying delays

In this paper, we investigate the problem of stabilization via state feedback and/or state-based switching for switched linear systems with mode-dependent time-varying delays. By using the multiple Lyapunov functional method, we establish sufficient conditions that guarantee the switched system is stabilizable via state feedback and/or switching under time-varying delays with appropriate upper bounds. The main results are presented in terms of linear matrix inequalities (LMIs) which generalize some known results and can be easily tested by using the Matlab’s LMI Tool-box.     

On stability of switched homogeneous nonlinear systems

In this paper, the problem of stability of switched homogeneous systems is addressed. First of all, if there is a quadratic Lyapunov function such that nonlinear homogeneous systems are asymptotically stable, a matrix Lyapunov-like equation is obtained for a stable nonlinear homogeneous system using semi-tensor product of matrices, and Lyapunov equation of linear system is just its particular case. Following the previous results, a sufficient condition is obtained for stability of switched nonlinear homogeneous systems, and a switching law is designed by partition of state space. In particular, a constructive approach is provided to avoid chattering phenomena which is caused by the switching rule. Then for planar switched homogeneous systems, an LMI approach to stability of planar switched homogeneous systems is presented. Similar to the condition for linear systems, the LMI-type condition is easily verifiable. An example is given to illustrate that candidate common Lyapunov function is a key point for design of switching law.     

Set stabilizability of switched Boolean control networks via Ledley antecedence solution

This paper studies two kinds of set stabilizability issues of switched Boolean control networks (SBCNs) by Ledley antecedence solution, that is, pointwise set stabilizability and set stabilizability under arbitrary switching signals. Firstly, based on the state transition matrix of SBCNs, the mode-dependent truth matrix is defined. Secondly, using the mode-dependent truth matrix in every step, a switching signal and the corresponding Ledley antecedence solutions are determined. Furthermore, a state feedback switching signal and a state feedback control are obtained for the pointwise set stabilizability. Thirdly, with the help of all mode-dependent truth matrices, the Ledley antecedence solutions are derived for a set of Boolean inclusions, which admits a state feedback control for the set stabilizability under arbitrary switching signals. Finally, an example is given to show the effectiveness of the proposed results.     

Stabilization criteria of a class of switched systems

In this paper, stabilizability property for a switched system under arbitrary switching is considered from an algebraic point of view by means of the existence of a set of block‐diagonal Lyapunov solutions with common Schur complement of certain order—or, equivalently, with common block (1,1)—for the matrix bank. It is shown that the existence of that set is equivalent to the existence of solutions for some Riccati inequalities done in terms of the blocks of matrices of the bank. In addition, we conclude that a particular class of systems with matrix bank constituted by Metzler matrices—Positive Switched Systems—are stabilizable by partial state reset.     

Stability of periodically switched discrete-time linear singular systems

In this paper we present some necessary and sufficient conditions for the stability of periodically switched discrete-time linear index-1 singular system, (PSSS). In particular, it is proved that, if at least one subsystem of a PSSS is asymptotically stable, then there is a switching rule, so that the whole system is also uniformly exponentially stable. Furthermore, for a periodically switched control system with no stable subsystems, there exist a switching rule and feedback matrices, such that the obtained PSSS is uniformly exponentially stable.     

一类不确定线性切换系统二次鲁棒稳定性的充分条件

本文研究了一类不确定线性切换系统的二次鲁棒稳定性问题.首先利用矩阵集的严格完备性设计切换律,导出了二次鲁棒稳定的充分条件.同时得到了在任意切换策略下,当矩阵集的所有矩阵为负定时保证切换系统二次鲁棒稳定性.在适当的假设下,这些条件可以表示为矩阵不等式.最后,用数值例子对所得结果加以阐明,说明了文中结果的正确性.     

Robust stabilization for uncertain switched impulsive control systems with state delay: An LMI approach

This paper deals with the problem of robust H_∞ state feedback stabilization for uncertain switched linear systems with state delay. The system under consideration involves time delay in the state, parameter uncertainties and nonlinear uncertainties. The parameter uncertainties are norm-bounded time-varying uncertainties which enter all the state matrices. The nonlinear uncertainties meet with the linear growth condition. In addition, the impulsive behavior is introduced into the given switched system, which results a novel class of hybrid and switched systems called switched impulsive control systems. Using the switched Lyapunov function approach, some sufficient conditions are developed to ensure the globally robust asymptotic stability and robust H_∞ disturbance attenuation performance in terms of certain linear matrix inequalities (LMIs). Not only the robustly stabilizing state feedback H_∞ controller and impulsive controller, but also the stabilizing switching law can be constructed by using the corresponding feasible solution to the LMIs. Finally, the effectiveness of the algorithms is illustrated with an example.     

The set of stable switching sequences for discrete-time linear switched systems

In this paper we study the characterization of the asymptotical stability for discrete-time switched linear systems. We first translate the system dynamics into a symbolic setting under the framework of symbolic topology. Then by using the ergodic measure theory, a lower bound estimate of Hausdorff dimension of the set of asymptotically stable sequences is obtained. We show that the Hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence. The obtained result reveals an underlying fundamental principle: a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them, provided that the switching is arbitrary. We also develop frequency and density indexes to identify those asymptotically stable switching sequences of the system.     

Robust Stability and Stabilization of a Class of Nonlinear Switched Discrete-Time Systems with Time-Varying Delays

In this paper, we investigate the problems of robust delay-dependent ℒ₂ gain analysis and feedback control synthesis for a class of nominally-linear switched discrete-time systems with time-varying delays, bounded nonlinearities and real convex bounded parametric uncertainties in all system matrices under arbitrary switching sequences. We develop new criteria for such class of switched systems based on the constructive use of an appropriate switched Lyapunov-Krasovskii functional coupled with Finsler’s Lemma and a free-weighting parameter matrix. We establish an LMI characterization of delay-dependent conditions under which the nonlinear switched delay system is robustly asymptotically stable with an ℒ₂-gain smaller than a prescribed constant level. Switched feedback schemes, based on state measurements, output measurements or by using dynamic output feedback, are designed to guarantee that the corresponding switched closed-loop system enjoys the delay-dependent asymptotic stability with an ℒ₂ gain smaller than a prescribed constant level. All the developed results are expressed in terms of convex optimization over LMIs and tested on representative examples.     

Robust stability of Switched Boolean Networks with function perturbation

In genetic regulatory networks, gene mutations are one of natural phenomena, which attract much attention by biological researchers. In modeling gene networks using switched Boolean networks (SBNs), gene mutations can be described by function perturbations, which is a meaningful issue in analyzing function perturbation of SBNs. This paper studies robust stability of SBNs with function perturbation. With the help of semi-tensor product (STP) of matrices, one equivalent algebraic form of SBNs is established. By constructing two state sets, a criterion for global stability of SBNs under arbitrary switching signals is proposed. In order to relax the conditions of global stability, pointwise stabilizability and consistent stabilizability of SBNs are further considered. Based on state reachable sets, several criteria are established for the proposed kinds of stability. Finally, the obtained results are verified by two examples and lac operon in the Escherichia coli, respectively.     

Periodical stabilization of switched linear systems

Periodical stabilization problems for switched linear systems are investigated in this paper. For autonomous switched systems, if there exists a stable convex combination of the subsystems, then a periodically switching signal can be constructed such that the overall system is asymptotically stable. Based on this fact, for switched control systems, corresponding sufficient conditions are presented under which constant/switching direct/observer-based state feedback controller can be designed such that the corresponding closed-loop systems are asymptotically stable under some periodically switching signal. Some numerical examples are given to illustrate our results.     

Sampled-data control of uncertain switched neutral nonlinear systems under asynchronous switching

The robust exponential stabilization for a class of the uncertain switched neutral nonlinear systems with time-varying delays based on the sampled-data control is investigated in this paper. The closed-loop system with sampled-data control is modeled as a continuous time system with a time-varying piecewise continuous control input delay. Considering the relationship between the sampling period and the dwell time of two switching instants, sampling interval with no switching and sampling interval with one switching are discussed, respectively. By Wirtinger-based inequality, Wirtinger-based double integral inequality, and free-weighting matrix technique, some delay-dependent sufficient conditions are given to guarantee the exponential stability of uncertain switched neutral nonlinear systems under asynchronous switching. In addition, sampled-data controllers can also be designed by special operations of matrices. Finally, two numerical examples are used to show the effectiveness of the approach proposed in this paper.     

Complete controllability and contractibility in multimodal systems

The concept of a strictly positive definite set of Hermitian matrices is introduced. It is shown that a strictly positive definite set is always a positive definite set, and conditions are found under which a positive definite set is strictly positive definite. We also show that a set of Hermitian matrices is strictly positive definite if and only if some nonnegative linear combination of these matrices is a positive definite matrix. For state dimension two, we use this concept to find necessary and sufficient conditions for a two-mode completely controllable irreducible multimodal system to be contractible relative to an elliptic norm. For general state dimensions, we give necessary and sufficient conditions for a special-type two-mode completely controllable irreducible system to be contractible relative to a weakly monotone norm. Applying the above results, we show that, for state dimension two, there exists a completely controllable two-mode system which is not contractible relative to either an elliptic or a weakly monotone norm. We leave open the question whether or not complete controllability implies contractibility, relative to some norm, for multimodal systems of two or more modes.     

Stability analysis of switched ARX models and application to learning with guarantees

The main subject of this work is the stability analysis of Switched Auto-Regressive models with eXogenous inputs (SARX), which constitute a reference class for switched and hybrid system identification. The work introduces novel conditions for the arbitrary switching stability of multiple-input multiple-output SARX models which exploit the peculiar structure of their state-space realization. The analysis relies on the properties of block companion matrices, and partly leverages results from the theory of non-negative matrices, without nevertheless asking for an input–output positive behavior of the model. The novel stability conditions have a simple formulation in terms of linear co-positive common Lyapunov functions, and come at a remarkably low computational cost, being solvable by Linear Programming. The low computational burden is particularly attractive in an identification context, as it allows to efficiently constrain learning procedures in order to obtain SARX models with stability guarantees. The latter is itself a contribution of the work, as it fills a gap in the literature on the estimation of SARX models. The results are validated on a particular learning technique based on Regression Trees – a well known machine learning algorithm – which has shown remarkable accuracy in experimental environments.     

Adaptive event-triggered dynamic distributed control of switched positive systems with switching faults

This paper designs the dynamic output-feedback controller of switched positive systems subject to switching faults using an improved adaptive event-triggering mechanism. An adaptive event-triggering condition is addressed in the form of 1-norm by virtue of the measurable outputs of distributed sensors and the corresponding error. An error-based closed-loop control system whose dynamic variable relies on a state observer is obtained. A multiple copositive Lyapunov function is constructed to deal with the positivity and stability of the systems. The matrix decomposition and linear programming approaches are used to design and compute the controller and observer gains. An improved average dwell time scheme is proposed to handle the switching faults. The contributions of this paper lie in that: (i) An adaptive event-triggering mechanism is established for switched positive systems, (ii) A framework on the fault of switching signal is constructed, and (iii) A dynamic distributed controller is proposed for the considered systems. Finally, two illustrative examples are given to verify the effectiveness of the obtained results.     

Exponential stability and applications of switched positive linear impulsive systems with time-varying delays and all unstable subsystems

The global uniform exponential stability of switched positive linear impulsive systems with time-varying delays and all unstable subsystems is studied in this paper, which includes two types of distributed time-varying delays and discrete time-varying delays. Switching behaviors dominating the switched systems can be either stabilizing and destabilizing in the new designed switching sequence. We design new linear programming algorithm process to find the feasible ratio of stabilizing switching behaviors, which can be compensated by unstable subsystems, destabilizing switching behaviors, and impulses. Specically, we add a kind of nonnegative impulses which is consistent with the switching behaviors for the systems. Employing a multiple co-positive Lyapunov-Krasovskii functional, we present several new sufficient stability criteria and design new switching sequence. Then, we apply the obtained stability criteria to the exponential consensus of linear delayed multi-agent systems, and obtain the new exponential consensus criteria. Three simulations are provided to demonstrate the proposed stability criteria.     

基于LMIs的一类不确定线性切换系统的鲁棒控制

基于LMIs处理方法,研究了一类不确定线性切换系统在任意切换下的鲁棒控制问题.利用矩阵Schur补引理构造线性矩阵不等式,得到该系统的鲁棒稳定性的充要条件,同时也给出了在状态反馈下的鲁棒稳定性充要条件和在输出反馈下的充分条件.最后用数值例子对所得结果加以验证,说明了文中结果的正确性.