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A new definition of homogeneity for discrete-time systems is introduced. As in the continuous-time case, the property can be verified algebraically in the transition map of the system, and it implies that a dilation of the initial conditions leads to a scaling of the trajectory. Stability properties and convergence rates of the system's solutions can be established by considering only the homogeneity degree. The existence of homogeneous Lyapunov and Lyapunov-like functions is proven.

International audience; A nonlinear control law is designed for finite-time stabilization of a chain of integrators. The method is based on Implicit Lyapunov Function (ILF) technique and homogeneity properties. The scheme of control parameters selection is presented by a Linear Matrix Inequality (LMI). The method is simple in implementation and does not assume on-line procedure for computation of the ILF value at the current state that is an improvement with respect to [9], [13]. The control law is presented in an explicit form and allows to find the values of all control parameters, that make the solution one of the most constructive. The theoretical results are supported by numerical example.

A solution to the problem of global fixed-time output stabilization and estimation of a chain of integrators is proposed. A nonlinear scheme comprising a state feedback controller and a dynamic observer are designed in order to guarantee both fixed-time estimation and fixed-time control. Robustness with respect to exogenous disturbances and measurement noises is established and a parameter optimization algorithm is provided. The performance of the obtained control and estimation algorithms are illustrated by numeric experiments.

The problem of sliding mode control design for nonlinear plant is studied. Necessary and sufficient conditions of quadratic-like stability (stabi-lizability) for nonlinear homogeneous (control) system are obtained. Sufficient conditions of robust stability/stabilizability are deduced. The results are supported with academic examples of sliding mode control design.

Stabilization rates of power-integrator chains are easily regulated. It provides a framework for acceleration of uncertain multi-input multi-output (MIMO) dynamic systems of known relative degrees (RDs). The desired rate of the output stabilization (sliding-mode (SM) control) is ensured for an uncertain system, if its RD is known, and a rough approximation of the high-frequency gain matrix is available. The uniformly bounded convergence time (fixed-time stability) is obtained as a particular case. The control can be kept continuous everywhere accept the SM set, if the partial RDs are equal. Similarly uncertain smooth systems of complete MIMO RDs (i.e. lacking zero dynamics) are stabilized by continuous control at their equilibria in finite time and also accelerated. Output-feedback controllers are constructed. Computer simulation demonstrates the efficiency of the proposed approach.

The challenging problem of active control of separated flows is tackled in the present paper using model-based design principles, and applied to data issued from a two-dimensional separated flow experiment. First, a bilinear state and input delay model of the system has been obtained from experimental data by means of a modified identification procedure. Adequacy and precision of the obtained model are demonstrated and compared with existing results. Next, two control problems (setpoint tracking and optimal control) have been formulated and studied using sliding mode control methodology and averaging analysis. The theoretical control results are supported with numerical simulations.

This paper presents feedback sensitivity functions analysis of implicit Lyapunov function-based control system in case of finite-time stabilization problem. The Gang of Four is chosen as a feedback sensitivity tool. The results can be used for parametric tuning of control algorithms in order to guarantee desired closed-loop sensitivity specifications. The obtained results are supported by numerical examples.

This paper deals with the design of a robust control for linear systems with external disturbances using a homogeneous differentiator-based observer based on a implicit Lyapunov function approach. Sufficient conditions for stability of the closed-loop system in the presence of external disturbances are obtained and represented by linear matrix inequalities. The parameter tuning for both controller and observer is formulated as a semi-definite programming problem with linear matrix inequalities constraints. Simulation results illustrate the feasibility of the proposed approach and some improvements with respect to the classic linear observer approach.

International audience; SUMMARY The control design problem for finite-time and fixed-time stabilization of linear multi-input system with nonlinear uncertainties and disturbances is considered. The control design algorithm based on block decomposition and Implicit Lyapunov Function (ILF) technique is developed. The robustness properties of the obtained control laws with respect to matched and unmatched uncertainties and disturbances are studied. Procedures for tuning of control parameters are presented in the form of Linear Matrix Inequalities (LMI). Aspects of practical implementation of developed algorithms are discussed. Theoretical results are supported by numerical simulations.

Sufficient conditions for a robust relay delayed non-local stabilization of linear systems are found, which relate the upper bound of an uncertainty in a time delay and the maximum of the real part of system spectrum. Algorithm of delayed relay control gain adaptation for non-local stabilization is suggested. The proposed algorithm suppresses bounded uncertainties in the time delay: once this relay delayed control law for the upper bound of uncertainty in the time delay for given system is designed, we ensure non-local stabilization for all values of the time delay less than the upper bound even in the case of a variable delay. Copyright © 2003 John Wiley & Sons, Ltd.