Valter J. S. Leite

H 2 guaranteed cost computation by means of parameter dependent Lyapunov functions

A linear matrix inequality approach to compute 2 guaranteed costs by means of parameter-dependent Lyapunov functions is proposed. The uncertain linear systems are supposed to belong to convex-bounded domains (polytope type uncertainty). Both continuous- and discrete-time systems are investigated and the results are illustrated by means of numerical examples.

Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models

We present a convex way to design a fuzzy dynamic output feedback compensator for locally stabilizing a class of nonlinear discrete-time systems. This class consists of the systems described by Takagi-Sugeno (T-S) models with a sector bounded nonlinear additive term and saturated control signals. The local stabilization takes into account the domain of validity of these T-S models, which is a key issue for practical applications. Two types of nonlinear fuzzy compensators are considered, one having all matrices of the controller depending on fuzzy-grade membership functions and the other with only a subset of the matrices with such a dependency. The controller design includes a fuzzy anti-windup gain that handles saturating actuators. Besides, a time-performance index based on the λ-contractivity of the level set of the fuzzy Lyapunov function is proposed regarding the closed-loop system. Examples are given to illustrate the effectiveness of this proposal.

Robust pole location by parameter dependent state feedback control

Sufficient conditions are given for the existence of a parameter dependent state feedback control assuring to a linear uncertain closed-loop system the pole location inside a circle in the complex plane. The uncertainties are supposed to belong to a polytope domain described by its vertices. The robust stabilizability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope. Extensions to cope with decentralized and output feedback parameter dependent control gains are also presented. Examples illustrate the results.

Gain scheduled state feedback control of discrete-time systems with time-varying uncertainties: an LMI approach

This paper addresses the problems of stabilization and H∞control by means of state feedback parameter-dependent gains applied to discrete-time linear systems whose matrices are affected by arbitrarily time-varying parameters belonging to a polytope. The solution of the proposed design conditions, written as a finite set of linear matrix inequalities at the polytope vertices, allows to obtain a parameter-dependent gain (i.e. a gain scheduled controller) as an analytical function of the parameters. The proposed strategy is different from similar approaches in the literature, that are based on discretizations of the space of parameters to determine interpolated control gains, or that assume special structures for the time-varying parameters or even suppose that some of the system matrices are fixed and time-invariant in order to have a convex design problem. Numerical examples illustrate the efficiency of the conditions given in the paper.

Robust control through piecewise Lyapunov functions for discrete time-varying uncertain systems

This paper is concerned with the robust stabilization by state feedback of a linear discrete-time system with time-varying uncertain parameters. An optimization problem involving a set of linear matrix inequalities and scaling parameters provides both the robust feedback gain and the piecewise Lyapunov function used to ensure the closed-loop stability. In the case of linear time-varying systems involving the convex combination of two matrices, only two scaling parameters constrained into the interval [0, 1] are needed, allowing a simple numerical solution as illustrated by means of examples.

Robust Stabilization of Discrete-Time Systems with Time-Varying Delay: An LMI Approach

Sufficient linear matrix inequality (LMI) conditions to verify the robust stability and to design robust state feedback gains for the class of linear discrete-time systems with time-varying delay and polytopic uncertainties are presented. The conditions are obtained through parameter-dependent Lyapunov-Krasovskii functionals and use some extra variables, which yield less conservative LMI conditions. Both problems, robust stability analysis and robust synthesis, are formulated as convex problems where all system matrices can be affected by uncertainty. Some numerical examples are presented to illustrate the advantages of the proposed LMI conditions.

Stability and stabilizability of discrete-time switched linear systems with state delay

The problems of stability and stabilizability of discrete-time switched linear systems whose subsystems are subject to state delays are investigated in this paper using linear matrix inequalities. The conditions, based on a Lyapunov-Krasovskii functional, assure the stability of the system for arbitrary switching functions, irrespective of the value of the time-delay. The switching function is assumed not known a priori but available in real time, the state as well as the delayed state vectors are supposed available for feedback, and the time-delay is considered unknown and unbounded. The proposed conditions can reduce the conservatism of the analysis and, thanks to extra variables, can be extended for synthesis purposes, providing switched gains for the state feedback control which encompass the fixed gains obtained from the standard quadratic stabilizability approach. Numerical examples illustrate the results.

LMI based robust stability conditions for linear uncertain systems: a numerical comparison

In the paper, several numerical experiments are performed in order to compare three linear matrix inequalities based sufficient conditions for the robust stability of linear systems in polytopic domains. The conditions are the quadratic stability and two robust stability conditions based on parameter dependent Lyapunov functions: one using extra variables and augmented equations and the other using a larger number of equations. From the examples, it is possible to infer about the conservativeness of the conditions and also about the cases in which an equivalence exists. Both continuous-time and discrete-time systems are investigated.

Improved robust ℋ ∞ control for neutral systems via discretised Lyapunov-Krasovskii functional

This paper deals with the robust stability analysis and robust control synthesis with guaranteed performance index for time-invariant uncertain neutral systems based on linear matrix inequalities (LMIs). The novelty is to extend the discretisation technique introduced by K. Gu for an appropriate parameter-dependent Lyapunov–Krasovskii functional. Then, the approach proposed improves the previous criteria from the literature that treats robust stability and control for uncertain neutral systems. The main strategy used to derive the obtained conditions is to introduce slack variables allowing one to decouple the system matrices from the Lyapunov functional matrices, without including additional dynamics. Two numerical examples are performed to support the theoretical predictions. The first example deals with stability analysis and the other illustrates how the proposed approach may provide a better solution for the ℋ ∞ guaranteed cost control problem when compared with others available in the literature.

An LMI characterization of polynomial parameter-dependent Lyapunov functions for robust stability

This paper investigates the robust stability of continuous-time, time-invariant linear uncertain systems in polytopic domains. Robust stability is checked by constructing a quadratic parameter-dependent Lyapunov function. The matrix associated with this quadratic Lyapunov function is a polynomial function of the uncertain parameters, expressed as a particular polynomial matrix involving powers of the dynamic matrix of the system and one symmetric matrix to be determined. The degree of this polynomial matrix function is arbitrary. Finsler’s Lemma is used to lift the obtained stability conditions into a larger space in which sufficient stability tests can be developed in the form of Linear Matrix Inequalities (LMIs), which must be verified at the vertices of the uncertainty polytopic domain. Examples illustrate the method, which is compared with similar results in the literature by means of random numerical experiments.