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.     

Graph‐Based Dwell Time Computation Methods for Discrete‐Time Switched Linear Systems

Analytical computation methods are proposed for evaluating the minimum dwell time and the average dwell time guaranteeing the asymptotic stability of a discrete‐time switched linear system whose switchings are assumed to respect a given directed graph. The minimum and average dwell time can be found using the graph that governs the switchings, and the associated weights. This approach, which is used in a previous work for continuous‐time systems having non‐defective subsystems, has been adapted to discrete‐time switched systems and generalized to allow defective subsystems. Moreover, we present a novel method to improve the dwell time estimation in the case of bimodal switched systems. In this method, scaling algorithms to minimize the condition number have been used to give better minimum dwell time and average dwell time estimates.     

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.     

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.     

Adaptive tracking control for uncertain switched systems under asynchronous switching

This paper addresses the state‐tracking model reference adaptive control problem for a class of switched systems with parametric uncertainties, where switchings between subsystems and designed adaptive controller are asynchronous. First, we establish a stability criterion for a switched reference model and convert the state‐tracking problem into the stability problem of an error switched system. Then, an adaptive law is designed, and the global practical stability of the error switched system is guaranteed under a class of switching signals characterized by a dwell‐time condition. An electrohydraulic system is given as an example to demonstrate the feasibility and effectiveness of the proposed design method. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Frequency-domain stability conditions for hybrid systems

Consideration was given to a special class of the hybrid systems with switchings of time-invariant linear right-hand sides. A narrower subclass of such systems, that of connected switched linear systems, was specified among them. The necessary and sufficient frequencydomain conditions (criteria) for the existence of a common quadratic Lyapunov function providing stability of the switched systems were proposed for them. The specified subclass includes control systems with several nonstationary nonlinearities from the finite sectors that are the matter at issue of the theory of absolute stability. For the connected switched linear systems of a special kind (triangular type systems), the separate necessary and separate sufficient existence conditions were obtained for such Lyapunov functions. The interrelations between these conditions were discussed in the example.     

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.     

Region stability analysis for switched nonlinear systems with multiple equilibria

This paper investigates the region stability analysis for switched nonlinear systems whose subsystems have different equilibria, which are referred to multi-equilibrium switched nonlinear systems, and proposes a number of new results on the region stability analysis. Through investigating the dynamical behavior of the multi-equilibrium switched system, two methods are established, based on which several new results are then obtained for the region stability. It is shown that the main results obtained in this paper not only guarantee the region stability of the multi-equilibrium switched nonlinear system under arbitrary switching, but also provide several new approaches to determine the corresponding regions of convergence. Moreover, an improved average dwell time (ADT) method is proposed for the stability of switched nonlinear systems whose subsystems do not share a common Lyapunov function. Finally, an illustrative example with numerical simulation is studied by using the obtained results. The study of example shows that our results work very well in analyzing the region stability of some multi-equilibrium switched nonlinear systems.     

An impulsive‐switched‐system approach to aperiodic sampled‐data systems with time‐delay control

This paper devotes to the stability of aperiodic sampled‐data systems with time‐delay control, where the delays can impose a positive effect on the stability of the systems. The systems are modeled as impulsive switched systems with fixed switching laws. A novel separation theorem is presented to determine the Schur property of a matrix product and then used to obtain a less conservative stability criterion for the impulsive switched systems with fixed switching laws. By the separation theorem and a loop‐functional approach, some new stability and stabilization criteria for aperiodic sampled‐data systems with time‐delay control are provided in terms of linear matrix inequalities. Finally, the stability and stabilization results are tested on some classical numerical examples to illustrate the efficiency of the proposed method.     

Region stability and stabilisation of switched linear systems with multiple equilibria

This paper studies stability and stabilisation issues of switched linear time-invariant systems with stable/unstable multiple equilibria. Investigation of such switched systems is motivated by a switching economic system. The well-known common Lyapunov function method is shown to be ineffecctive in analysing region stability of switched systems with multiple equilibria via a counterexample. When every subsystem has an equilibrium point and all multiple equilibria pairwise differ, this paper proposes some sufficient conditons for region stability/instability of such switched systems with respect to a region containing all multiple equilibria under arbitrary quasi-periodical switchings. These novel results imply that there may exist stable limit cycles of such switched systems. Based on the stability results, a global asymptotic region-stabilising controller, quasi-periodical switching path, and corresponding algorithm are all designed for such switched control systems. Several illustrative examples demonstrate the effectiveness and practicality of our new results.     

Exponential stability of impulsive positive switched systems with discrete and distributed time‐varying delays

This paper studies the exponential stability problems of discrete‐time and continuous‐time impulsive positive switched systems with mixed (discrete and distributed) time‐varying delays, respectively. By constructing novel copositive Lyapunov‐Krasovskii functionals and using the average dwell time technique, delay‐dependent sufficient conditions for the solvability of considered problems are given in terms of fairly simple linear matrix inequalities. Compared with the most existing results, by introducing an extra real vector, restrictive conditions on derivative of the time‐varying delays (less than 1) are relaxed, thus the obtained improved stability criteria can deal with a wider class of continuous‐time positive switched systems with time‐varying delays. Finally, two simple examples are provided to verify the validity of theoretical results.     

Unified dwell time–based stability and stabilization criteria for switched linear stochastic systems and their application to intermittent control

This paper considers the stability and stabilization problems for the switched linear stochastic systems under dwell time constraints, where the considered systems can be composed of an arbitrary combination of stable and unstable subsystems. First, a time‐varying discretized Lyapunov function is constructed based on the projection of a linear Lagrange interpolant and a switching‐time‐dependent “weighted” function. The “weighted” function not only enforces the Lyapunov function to decrease at switching instants but also coordinates the dynamical behavior of the subsystems. As a result, some unified criteria for mean square stability and almost sure stability of the switched stochastic systems are established in terms of linear matrix inequalities. Based on the obtained stochastic stability criteria, 2 types of state feedback controllers for the systems are designed. Moreover, the novel results are applied to solve the intermittent control or the controller failure problems. Finally, conservatism analysis and numerical examples are provided to illustrate the effectiveness of the established theoretical results.     

Backstepping Robust H ∞ Control for a Class of Uncertain Switched Nonlinear Systems Under Arbitrary Switchings

This paper addresses global robust H _∞ control for a class of switched nonlinear systems with uncertainty under arbitrary switchings. Each subsystem is in lower triangular form. The uncertainties are assumed to be in a known compact set. The backstepping design technique is used to design a smooth state feedback controller that renders the associated closed‐loop switched system globally robustly asymptotically stable and imposes a pre‐specified upper bound to the L ₂‐gain under arbitrary switchings. An example is provided to demonstrate the efficacy of the design approach.     

On the stochastic finite-time stability for switched stochastic nonlinear systems

In this paper, the stochastic finite-time stability (SFTS) criteria for switched stochastic nonlinear systems (SSNS) with nonlocal Lipschitz coefficients are established. The main tools that we used here are mode-dependent limiting average dwell time (MD-LADT), stopping time technology and the stochastic analysis theory. The established criteria show that the diffusion coefficient, the drift coefficient and the switching law play crucial roles in obtaining the SFTS for the SSNS. Besides, our criteria provide some approaches on how to design the switching law to achieve the SFTS if a switched stochastic system contains SFTS subsystems and stochastic finite-time instable (SFTI) subsystems simultaneously. Finally, an example is given to illustrate the importance and usefulness of the theoretical results.     

Robust state‐feedback control of stochastic state‐multiplicative discrete‐time linear switched systems with dwell time

Linear discrete‐time switched stochastic systems are considered, where the problems of mean square stability, stochastic l₂‐gain and state‐feedback control design are treated and solved. Solutions are obtained for both nominal and polytopic‐type uncertain systems. In all these problems, the switching obeys a dwell time constraint. In our solution, to each subsystem of the switched system, a Lyapunov function is assigned that is nonincreasing at the switching instants. The latter function is allowed to vary piecewise linearly, starting at the end of the previous switch instant, and it becomes time invariant after the dwell. In order to guarantee asymptotic stability, we require the Lyapunov function to be negative between consecutive switchings. We thus obtain Linear Matrix Inequalities conditions. Based on the solution of the stochastic l₂‐gain problem, we derive a solution to the state‐feedback control design, where we treat a variety of special cases. Being affine in the system matrices, all the aforementioned solutions are extended to the uncertain polytopic case. The proposed theory is demonstrated by a practical example taken from the field of flight control. Copyright © 2015 John Wiley & Sons, Ltd.     

Weighted H ∞ performance analysis of switched linear systems with mode-dependent average dwell time

This article is concerned with the disturbance attenuation properties of a class of switched linear systems by using a mode-dependent average dwell time (MDADT) approach. The proposed switching law is less strict than the average dwell time (ADT) switching in that each mode in the underlying system has its own ADT. By using the MDADT approach, a sufficient condition is obtained to guarantee the exponential stability with a weighted H _∞ performance for the underlying systems. A numerical example is given to show the validity and potential of the developed results on improving the disturbance attenuation performance.     

Design proportional‐integral‐derivative/proportional‐derivative controls for second‐order time‐varying switched nonlinear systems

Based on proportional‐integral‐derivative (PID)/PD controls, we in the article investigate the tracking problem of a class of second‐order time‐varying switched nonlinear systems. To start with, for tracking a given point under arbitrary switching signals, we propose a sufficient condition about PID controller parameters, which can be implicitly described as semialgebraic sets. Successively, we consider the tracking problem under average dwell time (ADT)‐based switching signals and propose an alternative sufficient condition about PID controller parameters. Especially, for tracking an equilibrium point of the system without controls, we can further simply utilize the proportional‐derivative control and similarly construct corresponding semialgebraic conditions about proportional‐derivative controller parameters under arbitrary switching signals and ADT‐based switching signals. Finally, two examples are given to show the applicability of our theoretical results.     

Stability of Switched Positive Linear Systems with Actuator Saturation under Mode-dependent Average Dwell Time

The stability problem is investigated for switched positive linear systems (SPLS) with unstabilized subsystem and actuator saturation. By using Lyapunov functions method and mode-dependent average dwell time (MDADT) strategy, sufficient conditions of the stability for the considered systems are established. We also give an estimate of the attraction domain. The convex hull represent approach is used to deal with actuator saturation. Finally, two examples are carried out to show the validity of results, which is less conservative than the average dwell time (ADT) switching.

     

Backstepping design for global robust stabilisation of switched nonlinear systems in lower triangular form

This article deals with the global robust stabilisation for a class of switched nonlinear systems under arbitrary switchings. The system under consideration is in lower triangular form and contains uncertainty. Both common Lyapunov function and state feedback controller are simultaneously constructed by backstepping such that the closed-loop system is globally robustly asymptotically stable under arbitrary switchings. Lastly, the design method proposed is extended to the uncertain switched nonlinear systems in nested lower triangular form to solve the global robust stabilisation problem under arbitrary switchings. Two examples are given to show the effectiveness of the proposed methods.