Alexandre Kruszewski

Nonquadratic Stabilization Conditions for a Class of Uncertain Nonlinear Discrete Time TS Fuzzy Models: A New Approach

The discrete-time uncertain nonlinear models are considered in a Takagi-Sugeno form and their stabilization is studied through a non- quadratic Lyapunov function. The classical conditions consider a one- sample variation, here, the main results are obtained considering k samples variation, i.e., Deltak V(x(t)) = V(x(t + k)) - V(x(t)). The results are shown to always include the classical cases, and several examples illustrate the effectiveness of the approach.

Control Law Proposition for the Stabilization of Discrete Takagi–Sugeno Models

This paper deals with the stabilization of a class of discrete nonlinear models, namely those in the Takagi-Sugeno form; its main goal is to reduce conservatism of existing stabilization conditions using a special class of candidate Lyapunov functions and an enhanced control law. It is shown that the use of the aforementioned Lyapunov function leads to less-pessimistic solutions. The usefulness of the new control law is shown through several examples.

A membership-function-dependent approach for stability analysis and controller synthesis of Takagi--Sugeno models

This paper presents a new approach for stability analysis and controller design of Takagi-Sugeno (TS) models. The analysis considers information derived from existing or induced order relations among the membership functions. Partitioning of the state-space and the use of piecewise Lyapunov functions (PWLF) arise naturally as a consequence of induced order relations. Conditions under the novel approach can be expressed as linear matrix inequalities (LMIs) facilitating the inclusion of performance design. Examples are provided to show the advantages over the classical quadratic approach.

Discrete Tagaki-Sugeno models for control: Where are we?

Abstract This work deals with relaxed conditions for stability and stabilization of discrete-time Takagi–Sugeno (TS) models. It recalls classical results found in the literature which use quadratic Lyapunov functions leading to very conservative conditions, and various extensions based on piecewise and non-quadratic Lyapunov functions. Afterwards, a new and powerful way to enhance the previous results is depicted. The basic idea is that waiting long enough a stable model will converge towards its equilibrium and, therefore, the Lyapunov functions under consideration are not necessarily decreasing at every sample, but are allowed to decrease every k samples. Whatever it is k >1, the results are proved to include the standard one-sample case. The potential of this approach is shown through several examples in the paper.

A Triangulation Approach to Asymptotically Exact Conditions for Fuzzy Summations

Many Takagi-Sugeno (T-S) fuzzy control-synthesis problems in the literature are expressed as the problem of finding decision variables in a double convex sum (fuzzy summation) of positive definite matrices. Matricespsila coefficients in the summation take values in the standard simplex. This paper presents a triangulation approach to the problem of generating simplicial partitions of the standard simplex in order to set up a family of sufficient conditions and, in parallel, another family of necessary ones for fuzzy summations. The conditions proposed in this paper are asymptotically exact as the size of the involved simplices decreases; its conservativeness vanishes for a sufficiently fine partition (sufficiently dense mesh of vertex points). The set of conditions is in the form of linear matrix inequalities (LMIs), for which efficient software is available.

Discrete and Intersample Analysis of Systems With Aperiodic Sampling

This article addresses the stability analysis of linear time invariant systems with aperiodic sampled-data control. Adopting a difference inclusion formalism, we show that necessary and sufficient stability conditions are given by the existence of discrete-time quasi-quadratic Lyapunov functions. A constructive method for computing such functions is provided from the approximation of the necessary and sufficient conditions. In practice, this leads to sufficient stability criteria under LMI form. The inter-sampling behavior is discussed there: based on differential inclusions, we provide continuous-time methods that use the advantages of the discrete-time approach. The results are illustrated by numerical examples that indicate the improvement with regard to the existing literature.

A Switched System Approach to Exponential Stabilization Through Communication Network

This paper considers a networked control loop, where the plant is a “slave” part, and the remote controller and observer constitute the “master”. Since the performance of networked control systems (NCS) depends on the quality of service (QoS) available from the network, it is worth to design a controller that takes into account qualitative information on the QoS in realtime. The goal of the design is to provide a controller that guarantees the following two things: 1) high performances (here expressed by exponential decay rates) when the QoS remains globally the same and 2) global stability when the QoS changes. In order to guarantee the global stability, the controller will switch by respecting a dwell time constraint. The dwell time parameters are obtained by using the switched system theories and the obtained conditions are linear matrix inequalities. An experiment illustrates how the controller can be implemented for a control over Internet application (remote control of a small robot).

New Approaches for the Stabilization of Discrete Takagi-Sugeno Fuzzy Models

The work presented deals with the stability and the stabilization of discrete Takagi-Sugeno fuzzy models. Several results show that stability and stabilization conditions using a non quadratic function outperform those obtained with a quadratic one. In this work we will use some new Lyapunovs functions and show that the results obtained include all the other approaches. To show the effectiveness of the method, several examples are given.

A way to improve results for the stabilization of continuous-time fuzzy descriptor models

This paper presents a new fuzzy Lyapunov function approach for stabilization and Hinfin performance of a class of continuous-time Takagi-Sugeno fuzzy control systems in the descriptor form. The relaxation here provided includes and outperforms the existent quadratic approach. In addition, it renders in terms of linear matrix inequalities (LMIs) and does not deal with time-derivatives of membership functions. Two examples are here provided to illustrate these advantages.

Switching controller for stabilization of linear systems with switched time-varying delays

This paper considers interval time-varying delay systems with delayed estimation of the delay. This case is often encountered in the Networked Control Systems (NCS) field. Based on Lyapunov-Krasovskii functional methods and linear matrix inequality (LMI) techniques, a switching state feedback controller is designed to guarantee the exponential stability. The controller switches according to the measured time-delay which is itself delayed. The global stability of the switching closed loop is guaranteed if some dwell time conditions are satisfied.