Minimum Dwell Time for Global Exponential Stability of a Class of Switched Positive Nonlinear Systems

This paper will investigate global exponential stability analysis for a class of switched positive nonlinear systems under minimum dwell time switching, whose nonlinear functions for each subsystem are constrained in a sector field by two odd symmetric piecewise linear functions and whose system matrices for each subsystem are Metzler. A class of multiple time-varying Lyapunov functions is constructed to obtain the computable sufficient conditions on the stability of such switched nonlinear systems within the framework of minimum dwell time switching. All present conditions can be solved by linear/nonlinear programming techniques. An example is provided to demonstrate the effectiveness of the proposed result.      

Stabilization of Discrete‐time Switched Systems with State Constraints Based on Mode‐Dependent Average Dwell Time

This paper considers the stabilization problem for a class of switched systems with state constraints based on mode‐dependent average dwell time (MDADT) in discrete‐time context. An improved average dwell time method is proposed, which is less conservative than the common average dwell time method. The sufficient conditions and stabilizing state feedback controllers for stabilization of discrete‐time switched systems with state constraints under MDADT switching are derived. Finally, the simulation results show that the approach designed by this paper is effective.     

Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems

Exponential stability and robust exponential stability relating to switched systems consisting of stable and unstable nonlinear subsystems are considered in this study. At each switching time instant, the impulsive increments which are nonlinear functions of the states are extended from switched linear systems to switched nonlinear systems. Using the average dwell time method and piecewise Lyapunov function approach, when the total active time of unstable subsystems compared to the total active time of stable subsystems is less than a certain proportion, the exponential stability of the switched system is guaranteed. The switching law is designed which includes the average dwell time of the switched system. Switched systems with uncertainties are also studied. Sufficient conditions of the exponential stability and robust exponential stability are provided for switched nonlinear systems. Finally, simulations show the effectiveness of the result.     

Equivalence of several stability conditions for switched linear systems with dwell time

In this paper, several equivalent stability conditions for switched linear systems with dwell time are presented. Both continuous‐time and discrete‐time cases are considered. For the continuous‐time case, the conditions that are convex in system matrices are presented in terms of infinite‐dimensional linear matrix inequalities (LMIs), which are not numerically testable. Then, by adopting the sum of square (SOS) and piecewise linear approach, computable conditions are formulated in terms of SOS program and LMIs. Compared to the literature, less conservative results can be obtained through solving these conditions for the same polynomial degree or discretized order. For the discrete‐time case, the stability conditions, which are convex in system matrices, are numerically testable. The convexity comes at the price of increment of computational complexity. Furthermore, by adopting the convexification approach, sufficient stability conditions of switched linear systems with polytopic uncertainties are derived, both for continuous‐time and discrete‐time cases. At last, several examples are given to demonstrate the correctness and advantages of our results.     

Dwell time-based stabilisation of switched delay systems using free-weighting matrices

In this paper, we present a quasi-convex optimisation method to minimise an upper bound of the dwell time for stability of switched delay systems. Piecewise Lyapunov–Krasovskii functionals are introduced and the upper bound for the derivative of Lyapunov functionals is estimated by free-weighting matrices method to investigate non-switching stability of each candidate subsystems. Then, a sufficient condition for the dwell time is derived to guarantee the asymptotic stability of the switched delay system. Once these conditions are represented by a set of linear matrix inequalities , dwell time optimisation problem can be formulated as a standard quasi-convex optimisation problem. Numerical examples are given to illustrate the improvements over previously obtained dwell time bounds. Using the results obtained in the stability case, we present a nonlinear minimisation algorithm to synthesise the dwell time minimiser controllers. The algorithm solves the problem with successive linearisation of nonlinear conditions.     

Output regulation for a class of switched nonlinear systems: an average dwell‐time method

This paper studies the problem of output regulation for a class of switched nonlinear systems. Sufficient conditions for the problem to be solved are presented in terms of the average dwell‐time scheme. These conditions are obtained based on full information feedback laws and error feedback laws, respectively. The results extend the output regulation theory for non‐switched nonlinear systems to switched nonlinear systems. A simulation example also shows the validity of the results.Copyright © 2011 John Wiley & Sons, Ltd.     

Robust Output Feedback Stabilization of Switched Nonlinear Systems with Average Dwell Time

For some switched nonlinear systems, stabilization can be achieved under arbitrary switching with state feedback control. Due to switching zero dynamics, output feedback stabilization for some switched nonlinear systems needs dwell time between switching to guarantee system stability. In this paper, we consider a class of switched nonlinear systems with unknown parameters and unknown switching signals. We design a robust output feedback controller that stabilizes the system under a class of switching signals with average dwell time (ADT) where the value of ADT can be reduced by adjusting the control gain. For some special cases, common quadratic Lyapunov functions of the closed‐loop systems can be found and the value of ADT is further relaxed. Some examples and simulations are provided to validate the results.     

基于驻留时间切换下离散切换系统的异步控制器设计

针对一类切换线性系统,本文提出了一种基于系统状态的驻留时间策略.这种切换策略不仅使异步状态反馈切换系统稳定,而且缩短了系统的运行时间.对于异步切换系统的稳定性和增益问题,本文主要的结论是在子系统运行期间Lyapunov函数允许增加,同时又没有的限制.通过利用基于系统状态的驻留时间策略,推导出了切换线性系统的控制器设计的充分条件.得出的结论也可以推广到非线性切换系统.本文中最后给出的算例用于说明该方法的有效性.     

Finite‐Time Stabilization of a Class of Cascade Nonlinear Switched Systems

This paper investigates the finite‐time stabilization problem for a class of cascade nonlinear switched systems. Using the average dwell time and multiple Lyapunov function technologies, some sufficient conditions to guarantee that the corresponding closed‐loop system is finite‐time stabilized are derived for the switched systems. Via multiple Lyapunov functions, the state feedback controller is designed to finite‐time stabilize a cascade nonlinear switched system, and the conditions are formulated in terms of linear matrix inequalities. An example is given to illustrate the efficiency of the proposed methods.     

切换非线性正系统的有限时间稳定性

研究模型依赖平均驻留时间(MDADT)切换信号下一类齐次度为1的切换非线性正系统的有限时间稳定问题.首先,通过构造恰当的切换最大分离Lyapunov函数,借助于Dini导数,基于MDADT切换信号,给出切换非线性正系统有限时间稳定的充分条件.与已有的指数稳定性结果相比,进一步说明有限时间稳定与指数稳定的区别.其次,将所得结论应用于切换线性正系统,得到切换线性正系统在MDADT或平均驻留时间(ADT)切换信号下有限时间稳定的充分条件.最后,通过仿真算例验证所得结论的有效性.     

H∞ Control for a Class of Continuous‐time Switched Systems with State Constraints

This paper investigates the H_∞ control problem for a class of continuous‐time switched systems subject to state constraints with an improved average dwell time. The state constraints are converted into state saturations by limiting the state in a unit hypercube. By employing this proposed improved average dwell time method, which takes into account different decay rates of a Lyapunov function related to an active subsystem according to whether the saturations occur or not, the resulting minimum of admissible average dwell time is smaller than the one of the standard average dwell time, which only considers the same decay rate regardless of whether the saturations occur or not. In addition, switched state feedback controllers are designed such that the closed‐loop switched system, subject to state constraints, is asymptotically stable and achieves a weighted L₂‐gain. Sufficient conditions for H_∞ control of the switched system with state constraints are derived before an application to a Pulse‐Width‐Modulation (PWM)‐driven boost converter is given to illustrate the applicability and the effectiveness of the proposed method.     

Necessary and sufficient conditions for constraint satisfaction in switched systems using switch‐robust control invariant sets

This paper studies the control of constrained systems whose dynamics and constraints switch between a finite set of modes over time according to an exogenous input signal. We define a new type of control invariant sets for switched constrained systems, called switch–robust control invariant (switch‐RCI) sets, that are robust to unknown mode switching and exploit available information on minimum dwell‐time and admissible mode transitions. These switch‐RCI sets are used to derive novel necessary and sufficient conditions for the existence of a control‐law that guarantees constraint satisfaction in the presence of unknown mode switching with known minimum dwell‐time. The switch‐RCI sets are also used to design a recursively feasible model predictive controller (MPC) that enforces closed‐loop constraint satisfaction for switched constrained systems. We show that our controller is nonconservative in the sense that it enforces constraints on the largest possible domain, ie, constraints can be recursively satisfied if and only if our controller is feasible. The MPC and switch‐RCI sets are demonstrated on a vehicle lane‐changing case study.     

Nussbaum gain adaptive fuzzy control for switched nonlinear systems with predefined output and full time-varying states constraints

In this paper, the problem of adaptive fuzzy tracking control for a class of uncertain switched nonlinear systems with unknown control direction is studied. Aiming at the problem, an adaptive control scheme with Nussbaum gain technology is constructed by using the average dwell time (ADT) method and the backstepping method to overcome the unknown control direction, and time-varying asymmetric barrier Lyapunov functions (ABLFs) are adopted to ensure the full-state constraints satisfaction. The proposed control scheme guarantees that all closed-loop signals remain bounded under a class of switching signals with ADT, while the output tracking error converges to a small neighborhood of the zero. An important innovation of this design method is that the unknown control direction, asymmetric time-varying full state constraints, and predefined time-varying output requirements are simultaneously considered in uncertain switched nonlinear systems for the first time. We set a moment in advance, and make the systems comply with the constraint conditions before running the moment by the shift function nested in the first time-varying ABLF. Finally, a simulation example verifies the effectiveness of the proposed scheme.     

Global Robust Output Regulation for a Class of Switched Nonlinear Systems with Nonlinear Exosystems

The problem of global robust output regulation for a class of switched nonlinear systems with nonlinear exosystems is investigated in this paper where not all subsystems are stabilizable. First, as an extension of the concept of non‐switched internal model, this paper defines a switched internal model, which together with the plant is called the switched augmented system. Also, we show that the problem of output regulation for the switched nonlinear system can be converted into a global robust regulation problem of the switched augmented system. Second, based on the average dwell time method, the global robust regulation problem under a class of switching signals with average dwell time is solved, which leads to the solution of the problem of global robust output regulation. Finally, an example is provided to demonstrate the effectiveness of the proposed design approach.     

Input/output-to-state stability of switched nonlinear systems with an improved average dwell time approach

This paper investigates the input/output-to-state stable (IOSS) property of the switched systems under average dwell time (ADT) switching signals in two cases: 1) all of the subsystems are IOSS, 2) parts of the subsystems are IOSS, and proposes a number of new results on stability analysis. First, we present a new IOSS result for the switched nonlinear systems whose subsystems are IOSS with an improved ADT method. Second, extending the improved ADT method to unforced nominal switched nonlinear systems in which parts of subsystems are stable, we establish a new stability analysis result. IOSS property of switched nonlinear systems in which parts of subsystems are IOSS, we show that if the average dwell time is large enough and if the fraction of time where one of the non-IOSS system is active is not too big, then IOSS property of the switched system can be established. It should be pointed that the main results obtained in this paper have some advantages over the exiting ones. Finally, two illustrative examples with simulation verify the correctness and validity of our results.     

Stability of a class of switched stochastic nonlinear systems under asynchronous switching

The stability of a class of switched stochastic nonlinear retarded systems with asynchronous switching controller is investigated. By constructing a virtual switching signal and using the average dwell time approach incorporated with Razumikhin-type theorem, the sufficient criteria for pth moment exponential stability and global asymptotic stability in probability are given. It is shown that the stability of the asynchronous stochastic systems can be guaranteed provided that the average dwell time is sufficiently large and the mismatched time between the controller and the systems is sufficiently small. This result is then applied to a class of switched stochastic nonlinear delay systems where the controller is designed with both state and switching delays. A numerical example illustrates the effectiveness of the obtained results.     

Output tracking control of cascade switched nonlinear systems

The output tracking problem for cascade switched nonlinear systems is studied based on the average dwell time method in this paper. Sufficient conditions for this problem to be solvable are derived, where the zero dynamics is unstabilisable under an arbitrary switching. When the switching signal satisfies an average dwell time scheme, H_∞ output tracking is achieved by virtue of a designed switched integral controller. A numerical example is illustrated to show the effectiveness of the main result.     

基于模型依赖驻留时间的异步切换控制

研究在模型依赖平均驻留时间切换策略下切换线性系统的异步切换控制问题,同时考虑模型依赖的控制器滞后时间的约束问题.在实际情况下,信号传输和系统检测等原因会导致控制器的切换滞后于子系统.基于这类情况,首先将子系统运行的区间划分为子系统与控制器相匹配的区间和非匹配的区间,根据模型依赖的驻留时间策略设计出各子系统的控制器;然后,结合模型依赖的控制器滞后时间、系统参数和Lyapunov稳定条件推导出合适的驻留时间设计参数,且使得异步切换系统全局一致指数稳定;最后通过数值仿真验证了所提出方法的有效性.     

至少有一个增长无源的子系统切换非线性系统的输出跟踪控制

本文利用平均驻留时间方法解决至少有一个增长无源的子系统的切换非线性系统的输出跟踪问题.对于给定的无源率,设计子系统控制器使得平均驻留时间变小.所得到的平均驻留时间依赖于增长无源子系统的增长指数小时间范数可观性.最后,通过数值例子验证所提出方法的有效性.     

Stability of switched positive nonlinear systems

This paper considers the stability problem for a class of switched positive nonlinear systems (SPNSs), which includes switched positive linear systems as a special case. We derive a necessary and sufficient condition for stability of continuous‐time SPNSs defined by homogeneous and cooperative vector fields under average dwell time switching. A corresponding necessary and sufficient condition is also given for stability of discrete‐time SPNSs defined by homogeneous and order‐preserving vector fields under average dwell time switching. The stability results for switched positive linear systems, which have been studied in the literature, can be easily obtained. We also extend the results to general switched linear systems. Finally, a numerical example is provided to illustrate the effectiveness of our results. Copyright © 2015 John Wiley & Sons, Ltd.