Yanzheng Zhu

Resilient Asynchronous

This paper is concerned with the resilient H∞ filtering problem for a class of discrete-time Markov jump neural networks (NNs) with time-varying delays, unideal measurements, and multiplicative noises. The transitions of NNs modes and desired mode-dependent filters are considered to be asynchronous, and a nonhomogeneous mode transition matrix of filters is used to model the asynchronous jumps to different degrees that are also mode-dependent. The unknown time-varying delays are also supposed to be mode-dependent with lower and upper bounds known a priori. The unideal measurements model includes the phenomena of randomly occurring quantization and missing measurements in a unified form. The desired resilient filters are designed such that the filtering error system is stochastically stable with a guaranteed H∞ performance index. A monotonicity is disclosed in filtering performance index as the degree of asynchronous jumps changes. A numerical example is provided to demonstrate the potential and validity of the theoretical results.

H_{\infty }

This chapter is concerned with the stability and stabilization problems of a class of continuous-time and discrete-time Markov jump linear system (MJLS) with partially unknown transition probabilities (TPs). It will be proved that the system under consideration is more general, which covers the systems with completely known and completely unknown TPs as two special cases, the latter is hereby the switching linear systems under arbitrary switching. Moreover, in contrast with the uncertain TPs, the concept of partially unknown TPs proposed in this chapter does not require any knowledge of the unknown elements. Firstly, the sufficient conditions for stochastic stability and stabilization of the underlying systems are derived via linear matrix inequality (LMI) formulation, and the relationship between the stability criteria currently obtained for the usual MJLS and switching linear systems under arbitrary switching is exposed by the proposed class of hybrid systems. Further, the necessary and sufficient criteria are obtained by fully considering the properties of the transition rates matrices (TRMs) and transition probabilities matrices (TPMs), and the convexity of the uncertain domains. We will show by comparison the less conservatism of the methodologies for obtaining the necessary and sufficient conditions, but note that in the next chapters of Part I, we prefer the ones in the sufficient stability conditions to carry out other studies. The extensions to less conservative results are relatively straightforward and we leave them to readers who are interested.

Filtering for Markov Jump Neural Networks With Unideal Measurements and Multiplicative Noises

In this note, the stabilization problem for a class of discrete-time switched linear systems with additive disturbances is investigated. The considered switching signals are of mode-dependent persistent dwell-time (MPDT) property and the disturbances are assumed to be amplitude-bounded. By constructing a quasi-time-varying (QTV) Lyapunov function, a QTV stabilizing controller is designed for the nominal system such that the resulting closed-loop system is globally uniformly asymptotically stable. In the presence of bounded additive disturbances, a MPDT robust positive invariant set is determined for the error system between the nominal system and disturbed system. A concept of generalized robust positive invariant (GRPI) set under admissible MPDT switching is further proposed for the error system. It is demonstrated that the disturbed system is also asymptotically stable in the sense of converging to the MPDT GRPI set that can be regarded as the cross section of a uniform tube of the disturbed system. A numerical example is provided to verify the theoretical findings.

Stability and Stabilization

This book focuses on the basic control and filtering synthesis problems for discrete-time switched linear systems under time-dependent switching signals. Chapter 1, as an introduction of the book, gives the backgrounds and motivations of switched systems, the definitions of the typical time-dependent switching signals, the differences and links to other types of systems with hybrid characteristics and a literature review mainly on the control and filtering for the underlying systems. By summarizing the multiple Lyapunov-like functions (MLFs) approach in which different requirements on comparisons of Lyapunov function values at switching instants, a series of methodologies are developed for the issues on stability and stabilization, and l2-gain performance or tube-based robustness for l disturbance, respectively, in Chapters 2 and 3. Chapters 4 and 5 are devoted to the control and filtering problems for the time-dependent switched linear systems with either polytopic uncertainties or measurable time-varying parameters in different sense of disturbances. The asynchronous switching problem, where there is time lag between the switching of the currently activated system mode and the controller/filter to be designed, is investigated in Chapter 6. The systems with various time delays under typical time-dependent switching signals are addressed in Chapter 7

Uniform Tube Based Stabilization of Switched Linear Systems With Mode-Dependent Persistent Dwell-Time

In this paper, the resilient H∞ filtering problem for a class of discrete-time Markov jump systems with unideal measurements is investigated. The unideal measurements contain both quantization and missing measurements simultaneously, which occur randomly satisfying two mutually independent Bernoulli distribute white sequences. A unified model is used to describe the unideal measurements phenomena, and a norm-bounded additive gain perturbation is introduced to model the resilient filter. A mode-dependent full-order filter is designed such that the filtering error system is stochastically stable with an ensured H∞ performance index. An application on a single-link robot arm is provided to verify the theoretical results.

Time-Dependent Switched Discrete-Time Linear Systems: Control and Filtering

This chapter is concerned with the robust stability problem for a class of discrete-time uncertain Markov jump systems (MJSs) with both partially unknown and uncertain transition probabilities (TPs). Therefore, the scenario is more practical and such TPs comprise three sorts: known, uncertain and unknown. Moreover, the system considered in this chapter is specifically meant to be a class of Markov jump neural networks (MJNNs) with uncertainties and perturbations. The parameters uncertainties are considered to be norm-bounded and the stochastic perturbations are described in forms of the Brownian motion. By invoking the property of the transition probabilities matrix (TPM) and the convexity of uncertain domains, a sufficient stability criterion for the underlying system is derived. Furthermore, a monotonicity is observed in concern of the maximum value of a given scalar which bounds the stochastic perturbation that the system can tolerate as the level of the defectiveness varies.

Resilient estimation for a class of Markov jump linear systems with unideal measurements and its application to robot arm systems

This chapter first investigates the stability problem of a class of discrete-time linear switched systems with cyclic switching and state delays, and a numerical searching algorithm is explored to compute the feasible values of dwell time of the subsystems. Then, the problem of \( H_{\infty }\) output feedback control for discrete-time switched linear systems with time delays is investigated. The time delay is assumed to be time-varying and has minimum and maximum bounds, which covers the constant delay and mode-dependent constant delay as two special cases. By constructing a switched quadratic Lyapunov function for the underlying system, both static and dynamic \(H_{\infty }\) output feedback controllers are designed respectively such that the corresponding closed-loop switched system under arbitrary switching signals is asymptotically stable and guarantees a prescribed \(H_{\infty }\) noise attenuation level bound. Moreover, under the arbitrary switching, the problem of robust \(l_{2}-l_{\infty }\) filtering is studied for discrete-time switched linear systems with polytopic uncertainties and time-varying delays. The robust switched linear filters are designed based on the mode-dependent idea and parameter-dependent stability approach, and the existence conditions of such filters, dependent on the upper and lower bound of time-varying delays, are formulated in terms of a set of linear matrix inequalities. Finally, the state estimation problem is studied for a class of discrete-time switching neural networks (NNs) with persistent dwell time (PDT) switching regularities and mode-dependent time-varying delays in \(H_{\infty } \) sense. The random packet dropouts, which are governed by a Bernoulli distributed white sequence, are considered to exist together for the estimator design of underlying switching NNs. The desired mode-dependent estimators are designed such that the resulting estimation error system is exponentially mean-square stable and achieves a prescribed \( H_{\infty }\) level of disturbance attenuation. The effectiveness and the superiority of the developed results are demonstrated through numerical examples.

Composite TPs Case

This chapter first investigates the stability and \(l_2\)-gain analysis problems for a class of discrete-time switched systems with average dwell time (ADT) switching by allowing the Lyapunov-like functions to increase during the running time of subsystems. The obtained results then facilitate the studies on the issues of asynchronous control, where “asynchronous ” means the switching of the controllers has a lag to the switching of system modes. The basic asynchronous stabilization and asynchronous \(H_\infty \) control problem are both studied and the case for the system with time-varying parameter is further addressed under the modal average dwell time (MADT). Finally, the asynchronous \(H_\infty \) filter design problem is dealt with for the underlying switched linear systems with ADT switching. The phenomenon of “asynchronous” switching will unavoidably deteriorate the control performance such as the \(H_\infty \) noise attenuation index. However, it can be verified that the designed controller/filter considering the synchronous switching will be not necessarily valid in the presence of asynchronous switching. Several examples are provided to show the potential of the developed results.

Time-Delay Switched Systems

This chapter concerns the problem of model reduction for a class of Markov jump linear system (MJLS) with time-varying (or nonhomogeneous) transition probabilities (TPs) in discrete-time domain. The time-varying character of TPs is considered as finite piecewise homogeneous and the variations in the finite set are considered as two types: arbitrary variation and stochastic variation, respectively. The latter means that the variation is subject to a higher-level transition probability matrix (TPM). Invoking the idea in the recent studies of partially unknown TPs for the traditional MJLS with homogeneous TPs, a generalized framework covering the two kinds of variation is proposed. The model reduction results for the underlying systems are obtained in \(H_{\infty }\) sense. A numerical example is presented to illustrate the effectiveness and potential of the developed theoretical results.

Asynchronous Switched Systems: ADT Switching

This chapter concerns the problem of \(H_{\infty }\) estimation for a class of Markov jump linear system (MJLS) with time-varying transition probabilities (TPs) in discrete-time domain. The time-varying character of TPs is also considered as finite piecewise homogeneous and the variations in the finite set are considered as two types: arbitrary variation and stochastic variation, respectively. The latter means that the variation is subject to a higher-level transition probability matrix (TPM). The mode-dependent and variation-dependent \(H_{\infty }\) filter is designed such that the resulting closed-loop systems are stochastically stable and have a guaranteed \(H_{\infty }\) filtering error performance index. Using the idea of partially unknown TPs for the traditional MJLS with homogeneous TPs, a generalized framework covering the two kinds of variation is proposed. Then, the derived results are extended to the study of the resilient \(H_{\infty }\) filtering problem for a class of discrete-time Markov jump neural networks (MJNNs) with time-varying delays, unideal measurements and multiplicative noises. The transitions of neural networks modes and desired mode-dependent filters are considered to be asynchronous, and a nonhomogeneous mode TPM of filters is used to model the asynchronous jumps to different degrees that are also mode-dependent. The unknown time-varying delays are also supposed to be mode-dependent with lower and upper bounds known a priori. The unideal measurements model includes the phenomena of randomly occurring quantization and missing measurements in a unified form. The desired resilient filters are designed such that the filtering error system is stochastically stable with a guaranteed \(H_{\infty }\) performance index. A monotonicity is disclosed in filtering performance index as the degree of asynchronous jumps changes. Numerical examples are provided to demonstrate the potential and validity of the theoretical results.