Invertibility of switched linear systems

We address the invertibility problem for switched systems, which is the problem of recovering the switching signal and the input uniquely given an output and an initial state. In the context of hybrid systems, this corresponds to recovering the discrete state and the input from partial measurements of the continuous state. In solving the invertibility problem, we introduce the concept of singular pairs for two systems. We give a necessary and sufficient condition for a switched system to be invertible, which says that the individual subsystems should be invertible and there should be no singular pairs. When the individual subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included.     

Incremental passivity and output regulation for switched nonlinear systems

This paper studies incremental passivity and global output regulation for switched nonlinear systems, whose subsystems are not required to be incrementally passive. A concept of incremental passivity for switched systems is put forward. First, a switched system is rendered incrementally passive by the design of a state-dependent switching law. Second, the feedback incremental passification is achieved by the design of a state-dependent switching law and a set of state feedback controllers. Finally, we show that once the incremental passivity for switched nonlinear systems is assured, the output regulation problem is solved by the design of global nonlinear regulator controllers comprising two components: the steady-state control and the linear output feedback stabilising controllers, even though the problem for none of subsystems is solvable. Two examples are presented to illustrate the effectiveness of the proposed approach.     

Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings

This paper is concerned with the global stabilization problem for switched nonlinear systems in lower triangular form under arbitrary switchings. Two classes of state feedback controllers and a common Lyapunov function (CLF) are simultaneously constructed by backstepping. The first class uses the common state feedback controller which is independent of switching signals; the other class utilizes individual state feedback controllers for the subsystems. As an extension of the designed method, the global stabilization problem under arbitrary switchings for switched nonlinear systems in nested lower triangular form is also studied. An example is given to show the effectiveness of the proposed method.     

任意相对阶下非线性切换系统的事件触发漏斗控制

针对一类具有任意相对阶且带有部分非输入到状态稳定逆动态的非线性切换系统, 提出一种动态事件触 发漏斗跟踪控制方案. 首先, 引入一个虚拟输出将任意相对阶的非线性切换系统转换为相对阶为一的非线性切换系 统. 其次, 设计各子系统的事件触发漏斗控制器和切换的动态事件触发机制, 解决候选事件触发漏斗控制器和子系 统之间的异步切换问题, 所提方案消除已有文献中为所有子系统设计共同控制器带来的保守性. 在一类具有平均驻 留时间切换信号的作用下, 保证切换闭环系统的所有信号都是有界的, 且跟踪误差一直在预设的漏斗内演化, 并排 除采样中的奇诺现象. 最后, 一个仿真例子验证方案的实用性和有效性.     

Stabilization of a class of cascade nonlinear switched systems with application to chaotic systems

In this study, the problem of finite‐time practical control of a class of nonlinear switched systems in the presence of input nonlinearities is investigated. The subsystems of the switched system are considered as complex nonlinear systems with a cascade structure. Each subsystem is fluctuated by lumped uncertainties. Moreover, some parts of the system's dynamics are considered to be unknown in advance. This paper sets no restrictive assumption on the switching logic of the system. Therefore, the aim is to propose a controller to work under any arbitrary switching signals. After providing a smooth sliding manifold, a simple adaptive control input is developed such that the system trajectories approach the prescribed sliding mode dynamics in finite‐time sense. The adopted control signal does not use the upper bounds of the lumped uncertainties, and it is robust against unknown nonlinear parts of the subsystems. It is proved that the origin is the (practical) finite‐time stable equilibrium point of the overall closed‐loop system. Subsequently, the proposed control rule is modified to handle the same switched system with no input nonlinearities. Computer simulations via 2 chaotic electric direct current machines demonstrate the robust performance of the derived variable structure control algorithm against system fluctuations and nonlinear inputs.     

Non-approximation-based outputs stabilization for switched large-scale nonlinear systems with quantized inputs and sensor uncertainties

A non-approximation-based output feedback control strategy for a class of switched large-scale nonlinear systems with quantized inputs and sensor uncertainties is proposed. A dynamic gain, which is shared by the state observers and controllers of all the subsystems, is designed so that the effects of sensor uncertainties, quantized inputs, unknown parameters, and external disturbances can be compensated. By constructing some common Lyapunov functions (CLFs) shared by the switched systems, it is proved that with the proposed scheme, the closed-loop system stability can be guaranteed under arbitrary switching, and the outputs of all the subsystems can be steered to within arbitrarily small neighborhoods of the origin.     

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.     

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.     

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.     

Global uniform input-to-state stabilization of large-scale interconnections of MIMO generalized triangular form switched systems

We solve the problem of global uniform input-to-state stabilization with respect to external disturbance signals for a class of large-scale interconnected nonlinear switched systems. The overall system is composed of switched subsystems each of which has the nonlinear MIMO generalized triangular form, which (in contrast to strict-feedback form) has non-invertible input–output maps. The switching signal is an arbitrary unknown piecewise constant function and the feedback constructed does not depend on the switching signal.     

Incremental passivity and output tracking of switched nonlinear systems

This article studies incremental passivity and the output tracking problem for switched nonlinear systems. The concept of incremental passivity for switched systems is given using the proposed weak-storage functions. Furthermore, conditions for a switched nonlinear system to be incrementally passive are obtained without the requirement of the incremental passivity conditions for subsystems. It is shown that once the incremental passivity is assured, the output tracking problem for switched systems is solvable via the designed controllers, even though the problem for none of subsystems is solvable.     

Sufficient and necessary conditions for the stability of second-order switched linear systems under arbitrary switching

Many practical systems can be modelled as switched systems, whose stability problem is challenging even for linear subsystems. In this article, the stability problem of second-order switched linear systems with a finite number of subsystems under arbitrary switching is investigated. Sufficient and necessary stability conditions are derived based on the worst-case analysis approach in polar coordinates. The key idea of this article is to partition the whole state space into several regions and reduce the stability analysis of all the subsystems to analysing one or two worst subsystems in each region. This article is an extension of the work for stability analysis of second-order switched linear systems with two subsystems under arbitrary switching.     

Adaptive finite‐time control for nonlinear teleoperation systems with asymmetric time‐varying delays

This paper addresses the adaptive finite‐time control problem of nonlinear teleoperation system in the presence of asymmetric time‐varying delays. To achieve the finite‐time position tracking, a novel adaptive finite‐time coordination algorithm based on subsystem decomposition is developed. By introducing a switching‐technique‐based error filtering into our design framework, the complete closed‐loop master (slave) teleoperation system is modeled as a special class of switched system, which is composed of two subsystems. To analyze such system, a finite‐time state‐independent input‐to‐output stability criterion is first developed for some normal switched nonlinear delayed systems. Then based on the classical Lyapunov–Krasovskii method, the stability of complete closed‐loop systems is obtained. It is shown that the proposed scheme can make the position errors converge into a deterministic domain in finite time when the robots continuously contact with human operator and/or the environment in the presence of asymmetric time‐varying delays. Finally, the simulation results are given to demonstrate the effectiveness. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Input-to-state stability of switched nonlinear systems

The input-to-state stability （ISS） problem is studied for switched systems with infinite subsystems. By using multiple Lyapunov function method, a sufficient ISS condition is given based on a quantitative relation of the control and the values of the Lyapunov functions of the subsystems before and after the switching instants. In terms of the average dwell-time of the switching laws, some sufficient ISS conditions are obtained for switched nonlinear systems and switched linear systems, respectively.     

Stability analysis and control synthesis of switched impulsive systems

In this paper, we investigate the stability analysis problem of switched impulsive nonlinear systems and several stabilization problems of switched discrete‐time linear systems are studied. First, sufficient conditions ensuring globally uniformly asymptotically stability of switched nonlinear impulsive system under arbitrary and DDT (dynamical dwell time which defines the length of the time interval between two successive switchings) switching are derived, respectively. In the DDT switching case, we first consider the switched system composed by stable subsystems, then we extend the results to the case where not all subsystems are stable. The stabilizations of switched discrete‐time linear system under arbitrary switching, DDT switching and asynchronous switching are investigated respectively. Based on the stability analysis results, the control synthesis consists of controller design for each subsystem and state impulsive jumping generators design at switching instant. With the aid of the state impulsive jumping generators at switching instant, the ‘energy’ produced by switching can be minimized, which leads to less conservative results. Several numerical examples are given to illustrate the proposed results within this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

不确定非线性切换系统的鲁棒H∞控制

讨论了一类不确定非线性切换系统的鲁棒H∞控制问题．首先,基于多Lyapunov函数方法,设计状态反馈控制器以及切换律,使得对于所有允许的不确定性．相应的闭环系统渐近稳定又具有指定的L2-增益．该问题可解的充分条件以一组含有纯量函数的偏微分不等式形式给出,此偏微分不等式较一般Hamilton-Jacobi不等式更具可解性．所提出的方法不要求任何一个子系统渐近稳定．接着作为应用,借助混杂状态反馈策略讨论了非切换不确定非线性系统的鲁棒H∞控制问题．最后通过一个简单例子说明了控制设计方法的可行性．     

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/output‐to‐state stability for switched nonlinear systems with unstable subsystems

In this paper, a couple of sufficient conditions for input/output‐to‐state stability (IOSS) of switched nonlinear systems with non‐IOSS subsystems are derived by exploiting the multiple Lyapunov functions (MLFs) method. A state‐norm estimator–based small‐gain theorem is also established for switched interconnected nonlinear systems under some proper switching laws, where the small‐gain property of individual connected subsystems is not required in the whole state space instead only in some subregions of the state space. The state‐norm estimator for the switched system under study is explicitly designed via a constructive procedure by exploiting the MLFs method and the classical small‐gain technique. The presented results permit removal of a technical condition in existing literature, where all subsystems in switched systems are IOSS or some are IOSS. An illustrative example is also provided to illustrate the effectiveness of the theoretical results.     

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.