Adaptive neural dynamic surface control with fixed-time prescribed performance for uncertain nonstrict-feedback stochastic switched systems

An adaptive neural dynamic surface control (DSC) problem with fixed-time prescribed performance (FTPP) is investigated for a class of nonstrict-feedback stochastic switched systems. Differently from the existing works for FTPP problem, the stochastic switched systems with nonstrict-feedback form and completely unknown systems are considered in this paper, and the unknown functions are approximated by some radial basis function (RBF) neural networks (NNs). The desired adaptive neural controller is designed by using common Lyapunov function method and defining fixed-time prescribed performance function (PPF). And based on the adaptive DSC scheme with the nonlinear filter, the “explosion of complexity” problem is avoided. Besides, the constructed fixed-time PPF just need to meet the requirement of second derivative exists. According to the Lyapunov stability theory, the FTPP of output tracking error is achieved, and all signals of closed-loop system remain bounded in probability. Finally, simulation results are presented to verify the availability of the designed control strategy.     

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.     

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.     

Global fixed-time stabilization for a class of switched nonlinear systems with general powers and its application

This paper investigates the problem of global fixed-time stabilization for a class of uncertain switched nonlinear systems with the general powers, namely, the powers of the considered systems can be different odd rational numbers, even rational numbers or both odd and even rational numbers. By skillfully using the common Lyapunov function method and the adding a power integrator technique, a common state feedback control strategy is developed. It is proved that the proposed controller can guarantee that the state of the resulting closed-loop system converges to zero for any given fixed time under arbitrary switchings. Simulation results of the liquid-level system are provided to show the effectiveness of the proposed method.     

Stability analysis and uniform ultimate boundedness control synthesis for a class of nonlinear switched difference systems

In this paper, we deal with stability analysis of a class of nonlinear switched discrete-time systems. Systems of the class appear in numerical simulation of continuous-time switched systems. Some linear matrix inequality type stability conditions, based on the common Lyapunov function approach, are obtained. It is shown that under these conditions the system remains stable for any switching law. The obtained results are applied to the analysis of dynamics of a discrete-time switched population model. Finally, a continuous state feedback control is proposed that guarantees the uniform ultimate boundedness of switched systems with uncertain nonlinearity and parameters.     

Robust exponential stability of switched delay interconnected systems under arbitrary switching

The problem of robust exponential stability for a class of switched nonlinear dynamical systems with uncertainties and unbounded delay is addressed. On the assumption that the interconnected functions of the studied systems satisfy the Lipschitz condition, by resorting to vector Lyapunov approach and M-matrix theory, the sufficient conditions to ensure the robust exponential stability of the switched interconnected systems under arbitrary switching are obtained. The proposed method, which neither require the individual subsystems to share a Common Lyapunov Function (CLF), nor need to involve the values of individual Lyapunov functions at each switching time, provide a new way of thinking to study the stability of arbitrary switching. In addition, the proposed criteria are explicit, and it is convenient for practical applications. Finally, two numerical examples are given to illustrate the correctness and effectiveness of the proposed theories.     

基于LMIs的一类不确定线性切换系统的鲁棒控制

基于LMIs处理方法,研究了一类不确定线性切换系统在任意切换下的鲁棒控制问题.利用矩阵Schur补引理构造线性矩阵不等式,得到该系统的鲁棒稳定性的充要条件,同时也给出了在状态反馈下的鲁棒稳定性充要条件和在输出反馈下的充分条件.最后用数值例子对所得结果加以验证,说明了文中结果的正确性.     

On stability of switched homogeneous nonlinear systems

In this paper, the problem of stability of switched homogeneous systems is addressed. First of all, if there is a quadratic Lyapunov function such that nonlinear homogeneous systems are asymptotically stable, a matrix Lyapunov-like equation is obtained for a stable nonlinear homogeneous system using semi-tensor product of matrices, and Lyapunov equation of linear system is just its particular case. Following the previous results, a sufficient condition is obtained for stability of switched nonlinear homogeneous systems, and a switching law is designed by partition of state space. In particular, a constructive approach is provided to avoid chattering phenomena which is caused by the switching rule. Then for planar switched homogeneous systems, an LMI approach to stability of planar switched homogeneous systems is presented. Similar to the condition for linear systems, the LMI-type condition is easily verifiable. An example is given to illustrate that candidate common Lyapunov function is a key point for design of switching law.     

一类不确定线性切换系统二次鲁棒稳定性的充分条件

本文研究了一类不确定线性切换系统的二次鲁棒稳定性问题.首先利用矩阵集的严格完备性设计切换律,导出了二次鲁棒稳定的充分条件.同时得到了在任意切换策略下,当矩阵集的所有矩阵为负定时保证切换系统二次鲁棒稳定性.在适当的假设下,这些条件可以表示为矩阵不等式.最后,用数值例子对所得结果加以阐明,说明了文中结果的正确性.     

Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers

Linear Parameter-Varying (LPV) systems with jumps and piecewise differentiable parameters is a class of hybrid LPV systems for which no tailored stability analysis and stabilization conditions have been obtained so far.¹ We fill this gap here by proposing an approach based on a clock- and parameter-dependent Lyapunov function yielding stability conditions under both constant and minimum dwell-times. Interesting adaptations of the latter result consist of a minimum dwell-time stability condition for uncertain LPV systems and LPV switched impulsive systems. The minimum dwell-time stability condition is notably shown to naturally generalize and unify the well-known quadratic and robust stability criteria all together. Those conditions are then adapted to address the stabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability condition involving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps under a range dwell-time condition which is then used to provide stabilization conditions for LPV systems using a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilization conditions are all formulated as infinite-dimensional semidefinite programming problems which are then solved using sum of squares programming. Examples are given for illustration.     

Robust stability and stabilization analysis for discrete-time randomly switched fuzzy systems with known sojourn probabilities

The stability and stabilization analysis problem is considered in this paper for a class of discrete-time switched fuzzy systems with known sojourn probabilities. By using Lyapunov functional, new delay-dependent sufficient conditions are derived to ensure the stability of the system. Convex combination technique is incorporated and the stability criteria are presented in terms of Linear matrix inequalities (LMIs). Furthermore, the developed approach is extended to address the robust stability and stabilization analysis of the delayed discrete-time switched fuzzy systems with randomly occurring uncertainties. Finally numerical examples are exploited to substantiate the theoretical results.     

Equivalence between the Lyapunov–Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems

The stability of discrete-time systems with time varying delay in the state can be analyzed by using a discrete-time extension of the classical Lyapunov–Krasovskii approach. In the networked control systems domain a similar delay stability problem is treated using a switched system transformation approach. The paper aims to establish a relation between the switched system transformation approach and the classical Lyapunov–Krasovskii method. It is shown that using the switched systems transformation is equivalent to using a general delay dependent Lyapunov–Krasovskii functionals. This functional represents the most general form that can be obtained using sums of quadratic terms. Necessary and sufficient LMI conditions for the existence of such functionals are presented.     

A new class of fixed-time bipartite flocking protocols for multi-agent systems

This work is concerned with the fixed-time stability theorem and the fixed-time bipartite flocking with collision avoidance for multi-agent systems. Under the framework of Filippov solution, a new theorem of fixed-time stability is established and a high-precision estimation of settling time is given. As an important application, the fixed-time bipartite flocking protocol of nonlinear multi-agent systems is proposed. Employing this fixed-time stability theorem and the structurally balanced signed graph theorem, the bipartite flocking without collision is achieved within a fixed-time. Moreover, the convergent time of the bipartite flocking is merely depending on the parameters of the protocol and the network connectivity. In addition, the upper bound of the size for each disjoint cluster can be estimated by the parameters of the protocol, the network connectivity and the initial states of the system. These results are novel, which are illustrated by both theoretical analysis and numerical simulations.     

Stability analysis of time-varying switched systems via indefinite difference Lyapunov functions

Asymptotic stability of time-varying switched systems is investigated in this paper. The less conservative sufficient criteria for asymptotic stability of time-varying discrete-time switched systems are proposed via common indefinite difference Lyapunov functions and multiple indefinite difference Lyapunov functions introduced in this note, respectively. Common indefinite difference Lyapunov functions can be used to analyze stability of a switched system with asymptotic stable subsystems and arbitrary switching signal. Multiple indefinite difference Lyapunov functions can be used to investigate stability of a switched system with unstable subsystems and a given switching signal. The difference of the proposed Lyapunov function may be positive at some instants for an asymptotically stable subsystem. We compare these main results and illustrate the effectiveness of the obtained theorems by three numerical examples.     

Polynomial stability of positive switched homogeneous systems with time delay

Based on the logarithm contraction average dwell-time method, this paper investigates the polynomial stability of positive switched homogeneous time-delay systems whose vector fields are of different degrees with respect to a dilation map. Using the analytical skills developed in positive systems, an explicit polynomial stability criterion is established for the first time for the involved system under the logarithm contraction average dwell-time switching. Moreover, the main result is applied to the polynomial stability of Persidskii-type switched systems.     

Robust reliable stabilization of uncertain switched neutral systems with delayed switching

This paper investigates the problem of robust reliable control for a class of uncertain switched neutral systems under asynchronous switching, where the switching instants of the controller experience delays with respect to those of the system and the parameter uncertainties are assumed to be norm-bounded. A state feedback controller is proposed to guarantee exponential stability and reliability for switched neutral systems, and the dwell time approach is utilized for the stability analysis and controller design. A numerical example is given to illustrate the effectiveness of the proposed method.     

Adaptive compensation of uncertain Euler–Lagrange systems with multiple time-varying actuator faults

This work proposes the command tracking problem for uncertain Euler–Lagrange (EL) systems with multiple partial loss of effectiveness (PLOE) actuator faults. Compared to existing fault-tolerant controllers for EL systems, the proposed adaptive controller accounts for parametric uncertainties in the system and multiple time-varying actuator fault parameters. The proposed method can also handle an infinite number of fault cases. The closed-loop fault-tolerant system is treated as a switched dynamical system, and a switched system stability is established using multiple Lyapunov functions. It is shown that all signals are bounded in each sub-interval and at the switching instances, and asymptotic tracking can be obtained only for a finite number of fault occurrences, whereas tracking error is bounded for the infinite case. Finally, a simulation example on a robotic manipulator is presented to show the effectiveness of the proposed method.     

Adaptive projective synchronization for a class of switched chaotic systems

This paper addresses the problem of projective synchronization of chaotic systems and switched chaotic systems by adaptive control methods. First, a necessary and sufficient condition is proposed to show how many state variables can realize projective synchronization under a linear feedback controller for the chaotic systems. Then, accordingly, a new algorithm is given to select all state variables that can realize projective synchronization. Furthermore, according to the results of the projective synchronization of chaotic systems, the problem of projective synchronization of the switched chaotic systems comprised by the unified chaotic systems is investigated, and an adaptive global linear feedback controller with only one input channel is designed, which can realize the projective synchronization under the arbitrary switching law. It is worth mentioning that the proposed method can also realize complete synchronization of the switched chaotic systems. Finally, the numerical simulation results verify the correctness and effectiveness of the proposed method.