Safety stabilization of switched systems with unstable subsystems

This paper is concerned with the problem of safety stabilization for switched systems where the solvability of the problem under study for individual subsystems is not assumed. A new state-dependent switching strategy with guaranteed dwell-time for switched systems is constructed, and a sufficient condition for absence of Zeno behavior is derived. Also, a novel switched control design method is proposed to simultaneously guarantee the safety of the switched closed-loop system and stabilize the system based on the union of a common barrier function and a single Lyapunov function, which effectively handles the conflict between safety and stability objectives. Finally, two examples are presented to demonstrate the effectiveness of the proposed design approach.     

Quadratic stabilization of switched nonlinear systems

In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated. When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law. The results of this paper are also applied to switched linear systems. Supported partially by the National Natural Science Foundation of China (Grant No. 50525721)     

Input-to-state stabilization of switched nonlinear systems

This note derives some sufficient conditions to ensure that the whole switched nonlinear system is input-to-state stabilizable (ISS) when each mode is ISS. Both cases, that the switchings of system modes coincide exactly and that they do not coincide with those of the corresponding controllers, are considered. For the latter, a model-based identification scheme is used to identify the system modes. The proposed scheme can overcome the finite escape time that may happen in this case     

带有不稳定子系统的切换非线性系统的积分输入状态稳定

本文针对带有不稳定子系统的切换非线性系统研究了系统的积分输入状态稳定性问题. 应用导数不定的 类Lyapunov函数得出切换非线性系统的积分输入状态稳定. 导数不定的类Lyapunov函数方法比传统的导数正定 的Lyapunov函数的方法更具有一般性. 文中包含两种情况: 当所有子系统为积分输入状态稳定时, 切换非线性系统 是积分输入状态稳定的; 当部分子系统为非积分输入状态稳定时, 本文证明了切换非线性系统的积分输入状态稳 定. 最后应用一个仿真例子描述了所提结果的有效性.     

Stability and asynchronous stabilization for a class of discrete-time switched nonlinear systems with stable and unstable subsystems

The stability analysis and asynchronous stabilization problems for a class of discrete-time switched nonlinear systems with stable and unstable subsystems are investigated in this paper. The Takagi-Sugeno (T-S) fuzzy model is used to represent each nonlinear subsystem. Through using the T-S fuzzy model, the studied systems are modeled into the switched T-S fuzzy systems. By using the switching fuzzy-basis-dependent Lyapunov functions (FLFs) approach and mode-dependent average dwell time (MDADT) technique, the stability conditions for the open-loop switched T-S fuzzy systems with unstable subsystems and asynchronous stabilization conditions for the closed-loop switched T-S fuzzy systems with unstable subsystems are obtained. Both the stability results and asynchronous stabilization results are derived in terms of linear matrix inequalities (LMIs). Finally two numerical examples are provided to illustrate the effectiveness of the results obtained.     

Exponential stability of switched positive systems with all modes being unstable

This article studies the exponential stability of continuous‐time switched positive systems consisting of unstable subsystems. Different from the existing results, both stabilizing and destabilizing switching behaviors act in the switching sequences. By employing multiple composite copositive Lyapunov functions, sufficient condition is derived to ensure the exponential stability of the system, which evaluates the ratio of stabilizing switching behaviors to compensate the state divergence caused by either unstable subsystems or destabilizing switching behaviors. Simulations demonstrate the effectiveness of the result.     

Stabilization by forwarding design for switched feedforward systems with unstable modes

The problem of global stabilization is investigated for a class of switched nonlinear feedforward systems in this paper where the solvability of the stabilization problem for individual subsystem is not assumed. Some sufficient condition for the stabilization problem to be solvable is derived for the first time by exploiting the multiple Lyapunov functions method and the forwarding technique. Also, we design a switching law and construct bounded state feedback controllers of subsystems explicitly by a recursive design algorithm to achieve global asymptotic stability. The provided technique permits removal of a common restriction in which all subsystems in switched nonlinear feedforward systems are globally asymptotically stable. Finally, a numerical example is provided to demonstrate the feasibility of the theoretical result. Copyright © 2017 John Wiley & Sons, Ltd.     

Absolute exponential stability and stabilization of switched nonlinear systems

This paper is concerned with the problems of absolute exponential stability and stabilization for a class of switched nonlinear systems whose system matrices are Metzler. Nonlinearity of the systems is constrained in a sector field, which is bounded by two odd symmetric piecewise linear functions. Multiple Lyapunov functions are introduced to deal with the stability of such nonlinear systems. Compared with some existing results obtained by the common Lyapunov function approach in the literature, the conservatism of our results is reduced. All present conditions can be solved by linear programming. Furthermore, the absolute exponential stabilization for the considered systems is designed by the state-feedback and average dwell time switching strategy. Two examples are also given to illustrate the validity of the theoretical findings.     

Stabilization of switched continuous-time systems with all modes unstable via dwell time switching

Stabilization of switched systems composed fully of unstable subsystems is one of the most challenging problems in the field of switched systems. In this brief paper, a sufficient condition ensuring the asymptotic stability of switched continuous-time systems with all modes unstable is proposed. The main idea is to exploit the stabilization property of switching behaviors to compensate the state divergence made by unstable modes. Then, by using a discretized Lyapunov function approach, a computable sufficient condition for switched linear systems is proposed in the framework of dwell time; it is shown that the time intervals between two successive switching instants are required to be confined by a pair of upper and lower bounds to guarantee the asymptotic stability. Based on derived results, an algorithm is proposed to compute the stability region of admissible dwell time. A numerical example is proposed to illustrate our approach.     

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.     

Nonfragile robust exponential stabilization of nonlinear uncertain switched systems with actuator saturation

For nonlinear uncertain switched systems, the problem of how to overcome the controller vulnerability is studied when the actuator saturation is considered. The sufficient condition for guaranteeing nonfragile robust exponential stabilization of the system is derived by using the method of minimum dwell time. Then, a switching law and the nonfragile state feedback controllers are designed such that the closed-loop system can be robustly exponentially stabilized at the origin. Next, when some scalar parameters of the closed-loop system are given, the design issue of the nonfragile state feedback controllers, which aim at enlarging the estimation of domain of attraction for closed-loop system, is transformed into a convex optimization issue with linear matrix inequalities (LMI) constraints. Finally, an example is given to verify the effectiveness of the proposed method.     

Global stabilization for a class of switched nonlinear feedforward systems

The problem of global stabilization for a class of switched nonlinear feedforward systems under arbitrary switchings is investigated in this paper. Based on the integrator forwarding technique and the common Lyapunov function method, we design bounded state feedback controllers of individual subsystems to guarantee asymptotic stability of the closed-loop system. A common coordinate transformation of all subsystems is exploited to avoid individual coordinate transformations for subsystems that are required when applying the forwarding recursive design scheme. An example is provided to demonstrate the effectiveness of the proposed design method.     

Practical output tracking of switched nonlinear systems in p-normal form with unstable subsystems

This paper studies practical output tracking of switched nonlinear systems in p-normal form. No solvability of the practical output tracking problem for subsystems is required. A constructive scheme to solve the problem for a switched nonlinear system is set up by exploiting the single Lyapunov function method and the tool of adding a power integrator. Also, we design a proper switching law and construct state-feedback controllers of subsystems. A two inverted pendulums as a practical example, which cannot be handled by the existing approaches, illustrates our theoretical result.     

On reachability and stabilization of switched linear systems

Reachability and stabilization issues for switched linear control systems are addressed. A necessary and a sufficient condition are presented for reachability. Under mild assumptions, we prove that the switched linear control systems are stabilizable. In addition, we show that an element-driven switching stabilization strategy can be explicitly constructed     

Adaptive stabilization of a class of uncertain switched nonlinear systems with backstepping control

In this paper, we focus on the problem of adaptive stabilization for a class of uncertain switched nonlinear systems, whose non-switching part consists of feedback linearizable dynamics. The main result is that we propose adaptive controllers such that the considered switched systems with unknown parameters can be stabilized under arbitrary switching signals. First, we design the adaptive state feedback controller based on tuning the estimations of the bounds on switching parameters in the transformed system, instead of estimating the switching parameters directly. Next, by incorporating some augmented design parameters, the adaptive output feedback controller is designed. The proposed approach allows us to construct a common Lyapunov function and thus the closed-loop system can be stabilized without the restriction on dwell-time, which is needed in most of the existing results considering output feedback control. A numerical example and computer simulations are provided to validate the proposed controllers.     

On stability of switched nonlinear systems with time-scaling transformation

This paper considers a class of switched nonlinear systems with time-scaling transformation. A newconcept named ‘time-scaling transformational average dwell time’ is proposed, based on which two stability conditions for the original switched system are established that rely on each mode's dynamics of the transformational switched system and the characteristic of the time-scaling transformation. The new observation significantly extends existing stability results, especially for the case with unstable modes. A numerical example is given to illustrate the effectiveness of the proposed method.     

Adaptive stabilization of a class of nonlinear systems with unstable internal dynamics

This note considers global stabilization of a class of uncertain nonlinear output feedback systems with unstable internal dynamics. The coefficients which characterize the internal dynamics are allowed to be functions of system output, and the uncertainty of the system is characterized by an unknown constant parameter vector. The key step in the proposed control design is to estimate the internal state variables and impose control over them. The control design is presented first for systems without unknown parameters, and then for the systems with unknown parameters using adaptive control techniques.     

一类多输入级联非线性切换系统的全局镇定

研究一类带有部分线性系统的多输入级联非线性切换系统的全局镇定问题. 首先, 给出保证线性部分有一致规范型的充分条件. 其次, 利用一致规范型及其零动态的共同二次Lyapunov函数设计状态反馈使得线性部分在任意切换律下镇定. 最后, 通过构造共同Lyapunov函数能实现闭环系统在任意切换律下的全局渐近稳定性.     

Output feedback stabilization of switched stochastic nonlinear systems under arbitrary switchings

This paper is concerned with the problem of global output feedback stabilization in probability for a class of switched stochastic nonlinear systems under arbitrary switchings. The subsystems are assumed to be in output feedback form and driven by white noise. By introducing a common Lyapunov function, the common output feedback controller independent of switching signals is constructed based on the backstepping approach. It is proved that the zero solution of the closed-loop system is fourth-moment exponentially stable. An example is given to show the effectiveness of the proposed method.     

一类非线性离散切换系统基于观测器的指数镇定

对一类离散非线性切换系统, 考虑了基于观测器的指数镇定问题. 借助微分中值定理(DMVT), 将非线性切换系统转化为线性参数(LPV)切换系统. 当状态变量不完全可获得时, 基于多Lyapunov函数方法, 给出系统在基于观测器的输出反馈控制器下指数镇定的充分条件. 所设计的滞后切换规则能够避免产生滑动模态. 并且将结果推广到系统方程含有不确定性的情况. 最后, 仿真例子说明了设计方法的有效性.