The slow state variables feedback stabilization problem for semi-Markov jump discrete-time systems with slow sampling singular perturbations is discussed in this work. A new fairly comprehensive system model, semi-Markov jump system with singular perturbations, which is more general than Markov jump model, is employed to describe the phenomena of random abrupt changes in structure and parameters of the systems. Based on a slow state variables feedback control scheme, a novel technique to design the desired controller is presented and the allowed maximum of singular perturbation parameter can be calculated. With the help of the discrete-time semi-Markov kernel approach, some sojourn-time-dependent and less-conservative sufficient conditions are established via a novel matrix decoupling technique to ensure the solvability of the problem to be addressed. Finally, an illustrative example is given to show the superiority and usefulness of the proposed method.

Slow State Variables Feedback Stabilization for Semi-Markov Jump Systems With Singular Perturbations

This paper presents a new sliding mode control (SMC) design methodology for fuzzy singularly perturbed systems (SPSs) subject to matched/unmatched uncertainties. To fully accommodate the model characteristics of the systems, a novel integral-type fuzzy switching surface function is put forward, which contains singular perturbation matrix and state-dependent input matrix simultaneously. Its corresponding sliding mode dynamics is a transformed fuzzy SPSs such that the matched uncertainty/perturbation is completely compensated without amplifying the unmatched one. By adopting a $\boldsymbol \varepsilon $ -dependent Lyapunov function, sufficient conditions are presented to guarantee the asymptotic stability of sliding mode dynamics, and a simple search algorithm is provided to find the stability bound. Then, a fuzzy SMC law is synthesized to ensure the reaching condition despite matched/unmatched uncertainties. A modified adaptive fuzzy SMC law is further constructed for adapting the unknown upper bound of the matched uncertainty. The applicability and superiority of obtained fuzzy SMC methodology are verified by a controller design for an electric circuit system.

Sliding Mode Control of Fuzzy Singularly Perturbed Systems With Application to Electric Circuit

This technical correspondence considers finite-time synchronization of dynamical networks by designing aperiodically intermittent pinning controllers with logarithmic quantization. The control scheme can greatly reduce control cost and save both communication channels and bandwidth. By using multiple Lyapunov functions and convex combination techniques, sufficient conditions formulated by a set of linear matrix inequalities are derived to guarantee that all the node systems are synchronized with an isolated trajectory in a finite settling time. Compared with existing results, the main characteristics of this paper are twofold: 1) quantized controller is used for finite-time synchronization and 2) the designed multiple Lyapunov functions are strictly decreasing. An optimal algorithm is proposed for the estimation of settling time. Numerical simulations are provided to demonstrate the effectiveness of the theoretical analysis.

Finite-Time Synchronization of Networks via Quantized Intermittent Pinning Control

The aim of this paper is to establish the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption. Several special cases are also emphasized. Moreover, a novel definition of Palais–Smale condition, which is more general than the usual one, is presented and a mountain pass theorem is pointed out. As a consequence, multiple critical points theorems are then established. Finally, as an example of applications, an elliptic Dirichlet problem with critical exponent is investigated.

A critical point theorem via the Ekeland variational principle

In this paper,observer design for a class of wide-existing nonlinear systems is considered. The convergent condition of observing error is developed. Then,based on eigenstructure assignment theory,design procedure is given with a simple example to demonstrate the proposed method.

Observer Design for a Class of Nonlinear Systems

The existence of a step-like contrast structure for a class of high-dimensional singularly perturbed system is shown by a smooth connection method based on the existence of a first integral for an associated system. In the framework of this paper, we not only give the conditions under which there exists an internal transition layer but also determine where an internal transition time is. Meanwhile, the uniformly valid asymptotic expansion of a solution with a step-like contrast structure is presented.

/pdf/on-step-like-contrast-structure-of-singularly-perturbed-35bab8fgrm.pdf

On Step-Like Contrast Structure of Singularly Perturbed Systems

This brief addresses the asymptotic stability and passivity of uncertain singularly perturbed systems. Based on the Lyapunov stability theory and a singular-system approach, some sufficient conditions in terms of linear matrix inequalities are derived. Three numerical examples are presented to demonstrate the effectiveness of the proposed theoretical results.

Passivity Analysis of Uncertain Singularly Perturbed Systems

This paper investigates the controlled synchronization problem for a class of nonlinear discrete-time chaotic systems subject to limited communication capacity. A general chaotic master system and its slave system with a controller are connected via a limited capacity channel. In this case, the effect of quantization errors is considered. A practical quantized scheme is proposed so that the synchronization error is input-to-state stable with respect to the transmission error. Meanwhile, the transmission error decays to zero exponentially. This implies that the synchronization error converges to zero under a limited communication channel. A simulation example for the Fold chaotic system is presented to illustrate the effectiveness of the proposed method.

Controlled synchronization of discrete-time chaotic systems under communication constraints

This study is concerned with the dynamic output feedback control problem for fast sampling discrete-time singularly perturbed systems using the singular perturbation approach. Sufficient conditions in terms of linear matrix inequalities (LMIs) are presented to guarantee the existence of a dynamic output feedback controller for the corresponding slow and fast subsystems, respectively. The controller gains and the corresponding coefficient matrices can be obtained via solving the proposed LMIs. Thus, not only the high dimensionality and the ill condition are alleviated, but also the regularity restrictions attached to the Riccati-based solutions are avoided. The theoretical result demonstrates that the composite dynamic output feedback control designed through those of the slow and fast subsystems can stabilise the full-order discrete-time singularly perturbed systems. Finally, two real world practical examples are provided to show the effectiveness of the obtained results.

https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-cta.2016.0121

Dynamic output feedback control for fast sampling discrete-time singularly perturbed systems

This paper considers the problem of $$H \infty $$
 observer-based sliding mode control for singularly perturbed systems with input nonlinearities. First, a proper observer is designed such that the observer error system with disturbance attenuation level is asymptotically stable. Then, an observer-based sliding surface is constructed under which a criterion for the input-to-state stability (ISS) of the sliding mode dynamics with respect to the observer error is obtained via linear matrix inequality. The criterion presented is independent of the small parameter, and the upper bound for ISS can be obtained efficiently. In addition, a sliding mode control law is synthesized to guarantee the reachability of the sliding surface in the state estimation space. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed theoretical results.

Zhiming Wang

Papers

On Step-Like Contrast Structure of Singularly Perturbed Systems

Passivity Analysis of Uncertain Singularly Perturbed Systems

Controlled synchronization of discrete-time chaotic systems under communication constraints

Dynamic output feedback control for fast sampling discrete-time singularly perturbed systems

H \infty observer-based sliding mode control for singularly perturbed systems with input nonlinearity