Robust stabilization of a class of singularly perturbed discrete bilinear systems

This correspondence is concerned with asymptotic properties on the codeword length of a fixed-to-variable length code (FV code) for a general source {X/sup n/}/sub n=1//sup /spl infin// with a finite or countably infinite alphabet. Suppose that for each n /spl ges/ 1 X/sup n/ is encoded to a binary codeword /spl phi//sub n/(X/sup n/) of length l(/spl phi//sub n/(X/sup n/)). Letting /spl epsiv//sub n/ denote the decoding error probability, we consider the following two criteria on FV codes: i) /spl epsiv//sub n/ = 0 for all n /spl ges/ 1 and ii) lim sup/sub n/spl rarr//spl infin///spl epsiv//sub n/ /spl les/ /spl epsiv/ for an arbitrarily given /spl epsiv/ /spl isin/ [0,1). Under criterion i), we show that, if X/sup n/ is encoded by an arbitrary prefix-free FV code asymptotically achieving the entropy, 1/nl(/spl phi//sub n/(X/sup n/)) - 1/nlog/sub 2/ 1/PX/sup n/(X/sup n/) /spl rarr/ 0 in probability as n /spl rarr/ /spl infin/ under a certain condition, where P/sub X//sup n/ denotes the probability distribution of X/sup n/. Under criterion ii), we first determine the minimum rate achieved by FV codes. Next, we show that 1/nl(/spl phi//sub n/(X/sup n/)) of an arbitrary FV code achieving the minimum rate in a certain sense has a property similar to the lossless case.

In this paper, the robust stability problem for a class of uncertain two-time-scale neutral systems subject to unstructured perturbations is investigated. The properties of H/sub /spl infin//-norm are used throughout this paper to derive the robust stability criterion. A frequency-domain delay-dependent criterion for the asymptotic stability of the slow neutral subsystem and the fast subsystem of the nominal two-time-scale neutral system is first presented. Under the condition that the slow neutral and fast subsystems of the nominal system are both asymptotically stable, we then propose the allowable bounds for the system uncertainties such that the slow neutral subsystem and the fast subsystem of the original uncertain two-time scale neutral system are both asymptotically stable. Finally, a stability bound /spl epsi//sup */ of the small parameter /spl epsi/ is given such that the original uncertain two-time scale neutral system is asymptotically stable for any /spl epsi/ /spl isin/ (0, /spl epsi//sup */).

Deals with the problem of singularly perturbed systems when the parameter /spl epsi/, representing the ratio speed between the fast and slow dynamics, is time-varying. We show that using an adequate fuzzy partition of /spl epsi/, we can combine techniques of fuzzy control and singular perturbations to control the system. The parameter /spl epsi/, may be only partially known (known only on some particular "representative" points). The lack of information is recovered using a method of fuzzy approximation, that provides in the same time the partition required to build the control law. The paper ends with an illustrative example.

In this paper the H/sub /spl infin//-optimal control and risk-sensitive control of linear singularly perturbed, discrete-time systems are considered. It is shown that the Riccati equation associated with the solution of the H/sub /spl infin//-optimal control problem, can be approximated by an outer series solution, and a boundary-layer correction series solution. As the singular perturbation parameter /spl epsiv/ tends to zero, it is shown that the full-order Riccati equation can be approximated by the zeroth-order outer-series solution. It is also recognized that the outer-series solution and the boundary-layer solution, are equivalent to the resulting asymptotic series expansion in powers of /spl epsiv/, of the Riccati equation associated with a certain risk-sensitive control problem. When a two-time-scale decomposition is considered, it is shown that an approximate composite control can be formed, as the sum of the slow and fast subsystems. Unlike the asymptotic expansion approach, it is shown that the original system can be approximated by the maximum of the performance associated with the slow and fast subsystems. Finally, it is shown that the slow and fast subsystems, are equivalent to certain slow and fast subsystems, respectively, resulting from applying a two-time-scale decomposition to a certain risk-sensitive control problem. >

This paper introduces a new epsilon-insensitive fuzzy c-regression models (epsilonFCRM), that can be used in fuzzy modeling. To fit these regression models to real data, a weighted epsilon-insensitive loss function is used. The proposed method make it possible to exclude an intrinsic inconsistency of fuzzy modeling, where crisp loss function (usually quadratic) is used to match real data and the fuzzy model. The epsilon-insensitive fuzzy modeling is based on human thinking and learning. This method allows easy control of generalization ability and outliers robustness. This approach leads to c simultaneous quadratic programming problems with bound constraints and one linear equality constraint. To solve this problem, computationally efficient numerical method, called incremental learning, is proposed. Finally, examples are given to demonstrate the validity of introduced approach to fuzzy modeling.

According to the Lyapunov stability theorem and the bounds of solutions of the Lyapunov equation, the state feedback gain matrices can be determined to guarantee the stability of the singularly perturbed systems for all /spl epsiv//spl isin/(0,/spl infin/). A numeral example and an operational amplifier circuit are given to illustrate the effectiveness of the proposed scheme.

Robust multivariable controllers for stable infinite-dimensional systems in the Callier-Desoer (CD) algebra are discussed. In particular, the following robust regulation problem is solved. Given reference and disturbance signals, which are linear combinations of signals of the form t/sup j/ sin(/spl omega//sub k/t+/spl phi//sub k/), j/spl ges/0, k=0, /spl middot//spl middot//spl middot/, n, find a low-order finite-dimensional controller so that the outputs asymptotically track the reference signals, asymptotically reject the disturbance signals, and the closed-loop system is stable and robust with respect to a class of perturbations in the plant. The proposed controller consists of a positive scalar gain /spl epsi/ and certain polynomial matrices K/sub k/(s) for k=0, /spl middot//spl middot//spl middot/, n. The main result of the paper shows that the matrices K/sub k/(s) must satisfy certain stability conditions involving the values of the plant transfer function only at the reference and disturbance signal frequencies /spl omega//sub k/ for k=0, /spl middot//spl middot//spl middot/, n. The controller has unstable poles on the imaginary axis. The behavior of these poles as a function of the scalar parameter /spl epsi/ in the form of a Puiseux series is given.

In this paper, a novel robust position controller is proposed for two-mass resonant systems. It is designed by using Differential Flatness (DF) and Disturbance Observer (DOb) in state space. Firstly, DF-based trajectory tracking controller is designed by neglecting plant uncertainties and external disturbances. Since a two-mass resonant system is controllable, state and control input references are generated in terms of differentially flat output. However, the trajectory tracking controller is sensitive so the stability and performance may significantly change due to the robustness issues in practice. Secondly, the robustness is achieved by treating estimated disturbances, which are obtained via DOb, in the design of the controller. A two-mass resonant system includes matched and mismatched disturbances; therefore, the robustness cannot be achieved by directly feeding-back their estimations. The states of the system are re-constructed by using estimated disturbances so that the mismatched disturbance is automatically cancelled via state feed-back control. The matched disturbance is simply cancelled by feeding-back its estimation. Hence, the robust trajectory tracking controller is designed for two-mass resonant systems. The validity of the proposal is verified by giving simulation results.

In this paper, we propose a fuzzy singular perturbed unified model which can be used in both continues-time domain and discrete-time domain. The sampling period should be the same order small amount of the singular perturbation parameter ε and ε ≪ 1 in fuzzy singular perturbed systems. Hence, the unified model makes use of delta operator which takes numerical advantages in high speed sampling system. In the proposed model framework, the problem of robust control is addressed. Furthermore, the singular perturbation parameter independent solution is also obtained. Some simulation results are given to show the effectiveness of the proposed approach.

Robust linear controller design of uncertain discrete systems with time-varying delay

Controllability of bilinear systems

The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional Grünwald-Letnikov-type backward difference operator with the equivalent Riemann-Liouville-type nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result.

The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional Grunwald-Letnikov-type backward difference operator with the equivalent Riemann-Liouville-type nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result.

From the Publisher: Singular perturbations and time-scale techniques were introduced to control engineering in the late 1960s and have since become common tools for the modeling, analysis, and design of control systems. In this SIAM Classics edition of the 1986 book, the original text is reprinted in its entirety (along with a new preface), providing once again the theoretical foundation for representative control applications. This book continues to be essential in many ways. It lays down the foundation of singular perturbation theory for linear and nonlinear systems, it presents the methodology in a pedagogical way that is not available anywhere else, and it illustrates the theory with many solved examples, including various physical examples and applications. So while new developments may go beyond the topics covered in this book, they are still based on the methodology described here, which continues to be their common starting point. Audience Control engineers and graduate students who seek an introduction to singular perturbation methods in control will find this text useful. The book also provides research workers with sketches of problems in the areas of robust, adaptive, stochastic, and nonlinear control. No previous knowledge of singular perturbation techniques is assumed. About the Authors Petar Kokotovic is Director of the Center for Control Engineering and Computation at the University of California, Santa Barbara. Hassan K. Khalil is Professor of Electrical and Computer Engineering at Michigan State University. John O'Reilly is Professor of Electronics and Electrical Engineering at the University of Glasgow, Scotland.

This paper presents the stabilization problem of discrete event systems modeled by Petri nets using the vector Lyapunov theory. After recalling some known stability information about discrete event systems modeled by Petri nets, the stabilization problem is addressed. A new promising methodology based on vector Lyapunov functions shows that it is possible to restrict the systems state space in such a way that boundedness is guaranteed. Moreover, the technique described can be used for regulation.