Uniform stability of switched nonlinear systems

In this paper, we investigate the stability properties of a general class of nonautonomous switched nonlinear systems. Sufficient conditions for uniform stability, uniform asymptotic stability and uniform exponential stability are derived via multiple Lyapunov functions. Our results provide stability criteria for switched systems with both stable and unstable subsystems. Particularly, our results include some existing results as special cases or improve those in the literature. Several numerical examples are worked out to illustrate our results.     

A survey of results and perspectives on stabilization of switched nonlinear systems with unstable modes

This paper surveys the recent theoretical results on the stabilization of switched nonlinear systems with unstable modes. Two cases are considered. (1) Some modes are stable, and others may be unstable. The stabilization can be achieved via the trade-off among stable modes and unstable ones. (2) All modes may be unstable. The stabilization can also be achieved via the trade-off among the potentially stable parts of all modes, or with the help of the jump dynamics at switching instants. The practical usefulness is illustrated by several applications including supervisory control, fault tolerant control, multi-agent systems, and networked control systems. Some perspectives are also provided.     

On stability of linear and weakly nonlinear switched systems with time delay

This paper studies time-delayed switched systems that include both stable and unstable modes. By using multiple Lyapunov-functions technique and a dwell-time approach, several criteria on exponential stability for both linear and nonlinear systems are established. It is shown that by suitably controlling the switching between the stable and unstable modes, exponential stabilization of the switched system can be achieved. Some examples and numerical simulations are provided to illustrate our results.     

Exponential stability analysis for neutral switched systems with interval time-varying mixed delays and nonlinear perturbations

This paper is concerned with the problem of exponential stability for uncertain neutral switched systems with interval time-varying mixed delays and nonlinear perturbations. By using the average dwell time approach and the piecewise Lyapunov functional technique, some sufficient conditions are first proposed in terms of a set of linear matrix inequalities (LMIs), to guarantee the robustly exponential stability for the uncertain neutral switched systems, where the decay estimate is explicitly given to quantify the convergence rate. Three numerical examples are finally illustrated to show the effectiveness of the proposed method.     

Stability analysis of a class of nonlinear positive switched systems with delays

This paper addresses the stability problem of delayed nonlinear positive switched systems whose subsystems are all positive. Both discrete-time systems and continuous-time systems are studied. In our analysis, the delays in systems can be unbounded. Two conditions are established to test the local asymptotic stability of the considered systems. The method to compute the domains of attraction is also proposed provided that the system is locally asymptotically stable. When reduced to general nonlinear positive systems, that is, the considered switched system consists of only one mode, an interesting conclusion follows that the proposed nonlinear positive system is locally asymptotically stable for all admissible delays and nonnegative nonlinearities which satisfy an extra condition at the origin, if and only if the system represented by the linear part is asymptotically stable for all admissible delays. Finally, a numerical example is presented to illustrate the obtained results.     

Stability analysis of switched homogeneous time-delay systems under synchronous and asynchronous commutation

In this work, stability analysis for a class of switched nonlinear time-delay systems is performed by applying Lyapunov–Krasovskii and Lyapunov–Razumikhin approaches. It is assumed that each subsystem in the family is homogeneous (of positive or negative degree) and asymptotically stable in the delay-free setting. The cases of existence of a common or multiple Lyapunov–Krasovskii functionals and a common Lyapunov–Razumikhin function are explored. The scenarios with synchronous and asynchronous switching are considered, and it is demonstrated that depending on the kind of commutation, one of the frameworks for stability analysis outperforms another, but finally leading to similar restrictions for both types of switching (despite the asynchronous one seems to be more demanded). The obtained results are applied to mechanical systems having restoring forces with real-valued powers.     

Output tracking of switched Boolean networks under open-loop/closed-loop switching signals

This paper addresses the output tracking problem of switched Boolean networks (SBNs) via the semi-tensor product method, and presents a number of new results. Firstly, the concept of switching-output-reachability is proposed for SBNs, based on which, a necessary and sufficient condition is presented for the output tracking of SBNs under arbitrary open-loop switching signal. Secondly, a constructive procedure is proposed for the design of closed-loop switching signals for SBNs to track a constant reference signal. The study of an illustrative example shows that the obtained new results are very effective.     

Improved results on fixed-time stability of switched nonlinear time-varying systems

This paper considers the problem of fixed-time stability (FTS) for switched nonlinear time-varying (NTV) systems. Firstly, three sufficient conditions are proposed to verify the FTS of NTV systems by using the improved Lyapunov function, which has a tighter upper bound of time derivative. Then, two FTS conditions are given for the switched NTV system by extending the obtained results, moreover, a switching strategy is also provided by using the minimum dwell time method. Finally, the obtained results are extended to study the FTS of impulsive NTV systems. Comparing with the existing results, the obtained conditions have two improvements: (1) provides a more accurate estimate for the upper bound of settling time of NTV systems, and (2) allows the Lyapunov function to increase at the switching instant of switched NTV (or impulsive NTV) systems. Two numerical examples are given to illustrate the theoretical results.     

On stability of discrete-time switched systems

This article deals with stability of discrete-time switched systems. Given a family of nonlinear systems and the admissible switches among the systems in the family, we first propose a class of switching signals under which the resulting switched system is globally asymptotically stable. We allow unstable systems in the family and our stability condition depends solely on asymptotic behaviour of the switching signals. We then discuss algorithmic construction of the above class of switching signals, and show that in the presence of exogenous inputs and outputs, a switching signal so constructed also ensures input/output-to-state stability for discrete-time switched nonlinear systems. We finally show that our class of switching signals that ensures global asymptotic stability also extends to the continuous-time setting with minor modifications under standard assumptions. We employ multiple Lyapunov-like functions and graph theoretic tools as the main apparatuses for our analysis.     

Exponential quasi-dissipativity and boundedness property for switched nonlinear systems

This paper is concerned with exponential quasi-dissipativity for switched nonlinear systems using multiple storage functions and multiple supply rates. First, a new concept of exponential quasi-dissipativity for a switched nonlinear system is proposed. In contrast with dissipative system which does not produce energy in some abstract sense, the supply rate of quasi-dissipativity is the sum of the conventional dissipativity supply rate and the constant supply rate. This means quasi-dissipative system can produce energy by itself. Each active subsystem is only required to be exponentially quasi-dissipative, while the energy changing of each inactive subsystem characterized by cross-supply rates is considered as the transformation of “energy” from the active subsystem. By weakening the conventional dissipation inequality, a broader class of open-loop systems and controllers are admissible, leading to broader application. Second, the ultimate boundedness property is obtained under some constraints on the energy changing of inactive subsystems. Third, the exponential quasi-dissipativity conditions for switched nonlinear systems are obtained by the design of a state-dependent switching law. In particular, the sufficient conditions of exponential quasi-passivity and exponential finite power gain for switched nonlinear systems are given thereby providing generalizations of KYP conditions and HJI inequality. Finally, the effectiveness of the obtained results is verified by two examples.     

Global finite-time stabilization of switched high-order rational power nonlinear systems

This works is concerned with the finite-time optimal stabilization problem for a class of switched non-strict-feedback nonlinear systems whose powers are possibly different positive odd rational numbers in the sense the powers of each subsystem might differ from others. It is well known that high-order nonlinear systems are intrinsically challenging as feedback linearization and backstepping method successfully developed for low-order systems fail to work. To this purpose, the nested saturation homogeneous controller is constructively devised to achieve global finite-time stability. Furthermore, the corresponding design parameters are optimized by minimizing a well-defined cost function, and thus an optimal controller being independent of switching signals is obtained. Simulation results are eventually provided to validate the effectiveness of the proposed control scheme.     

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.     

Stability of perturbed switched nonlinear systems with delays

This paper addresses the stability problems of perturbed switched nonlinear systems with time-varying delays. It is assumed that the nominal switched nonlinear system (perturbation-free system) is uniformly exponentially stable and that the perturbations satisfy a linear growth bound condition. It is revealed that there exists an upper bound of perturbation guaranteeing that the perturbed system preserves the stability property of the nominal system, locally or globally, depending on both perturbations and the nominal system itself. An example is provided to illustrate the proposed theoretical results.     

Practical finite-time stability of switched nonlinear time-varying systems based on initial state-dependent dwell time methods

This paper considers the problem of practical finite-time stability (PFTS) for switched nonlinear time-varying (SNTV) systems. Starting with nonlinear time-varying (NTV) systems, a new sufficient condition is proposed to verify the PFTS of systems by using an improved Lyapunov function. Then, the results obtained are extended to study the PFTS of SNTV systems. Two stability conditions are proposed for SNTV systems under arbitrary switching, moreover, the time and region of convergence are also given. Furthermore, an initial state-dependent dwell time method is introduced to study the PFTS of SNTV systems. Three stability conditions are proposed by using the methods of initial state-dependent minimum dwell time (ISD-MDT) and initial state-dependent average dwell time (ISD-ADT), respectively. The comparisons between the obtained results and the existing results are also given, and the obtained results are extended to impulsive switched nonlinear time-varying (ISNTV) systems. Finally, a numerical example is provided to illustrate the theoretical results.     

Robust reliable control for uncertain switched nonlinear systems with time delay under asynchronous switching

This paper investigates the problem of robust reliable control for a class of switched nonlinear systems with time delay and actuator failures under asynchronous switching. When the switching instants of the controller experience delays with respect to those of the system, a kind of reliable controller design method is proposed, and the dwell time approach is utilized for the stability analysis. Sufficient conditions for the existence of the reliable controller are formulated in terms of a set of LMIs. Then the proposed approach is extended to take into account switched delay systems with Lipschitz nonlinearities and structured uncertainties. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.     

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.     

Trajectory tracking for a class of switched nonlinear systems: Application to PMSM

This paper proposes a state-dependent switching control for a class of switched nonlinear systems, whose model describes a permanent magnet synchronous machine (PMSM) fed by a three-phase voltage source inverter. Due to its high torque density, high efficiency and wide velocity range, this electrical drive is widely used for traction and several applications in robotics, aerospace, electric vehicles among others. The proposed design conditions are based on a non-quadratic Lyapunov function, dependent on the machine shaft displacement, and assure asymptotic tracking of a pre-specified time-varying rotational velocity profile with guaranteed performance. Properties of the nonlinear system under consideration are used to derive design conditions expressed in terms of linear matrix inequalities that can be solved efficiently. Special cases involving asymptotic stability toward step and ramp velocity profiles are presented. Experimental results are used to validate the proposed technique.