Nonlinear control algorithms of two types are presented for uncertain linear plants. Controllers of the first type are stabilizing polynomial feedbacks that allow to adjust a guaranteed convergence time of system trajectories into selected neighborhood of the origin independently on initial conditions. The control design procedure uses block control principles and finite-time attractivity properties of polynomial feedbacks. Controllers of the second type are modifications of the second order sliding mode control algorithms. They provide global finite-time stability of the closed-loop system and allow to adjust a guaranteed settling time independently on initial conditions. Control algorithms are presented for both single-input and multi-input systems. Theoretical results are supported by numerical simulations.

International audienceTheorems on Implicit Lyapunov Functions (ILF) for finite-time and fixed-time stability analysis of nonlinear systems are presented. Based on these resutls, new nonlinear control laws are designed for robust stabilization of a chain of integrators. High order sliding mode (HOSM) algorithms are obtained as particular cases. Some aspects of digital implementations of the presented algorithms are studied, it is shown that they possess a chattering reduction ability. Theoretical results are supported by numerical simulations

Summary The control design problem for finite‐time and fixed‐time stabilizations of linear multi‐input system with nonlinear uncertainties and disturbances is considered. The control design algorithm based on block decomposition and implicit Lyapunov function 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. Aspects of practical implementation of developed algorithms are discussed. Theoretical results are supported by numerical simulations. Copyright © 2015 John Wiley & Sons, Ltd.

International audienceThe paper surveys mathematical tools required for stability and convergence analysis of modern sliding mode control systems. Elements of Filippov theory of differential equations with discontinuous right-hand sides and its recent extensions are discussed. Stability notions (from Lyapunov stability (1982) to fixed-time stability (2012)) are observed. Concepts of generalized derivatives and non-smooth Lyapunov functions are considered. The generalized Lyapunov theorems for stability analysis and convergence time estimation are presented and supported by examples from sliding mode control theory

Several conditions are proposed to check different robustness properties (ISS, iISS, IOSS and OSS) for generic nonlinear systems applying the weighted homogeneity concept (global or local). The advantages of this result is that, under some mild conditions, the system robustness can be established as a function of the degree of homogeneity.

International audienceThe paper is reviewing the tools to handle high-order sliding mode design and robustness. The main ingredient is homogeneity which can be checked using an algebraic test and which helps us in obtaining one of the most desired properties in sliding mode control that is finite-time stability. This paper stresses some recently obtained results about homogeneity for differential inclusions and robustness with respect to perturbations in the context of input-to-state stability. Lastly within this framework, most of the popular high-order sliding mode control schemas are analysed

Summary The problem of sliding mode control design for a nonlinear plant is studied. The necessary and sufficient conditions of quadratic‐like stability (stabilizability) for a 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.

International audienceThis monograph introduces a newly developed robust control design technique for a wide class of continuous-time dynamical systems called the “attractive ellipsoid method.” Along with a coherent introduction to the proposed control design and related topics, the monograph studies nonlinear affine control systems in the presence of uncertainty and presents a constructive and easily implementable control strategy that guarantees certain stability properties. The authors discuss linear-style feedback control synthesis in the context of the above-mentioned systems.The development and physical implementation of high-performance robust-feedback controllers that work in the absence of complete information is addressed, with numerous examples to illustrate how to apply the attractive ellipsoid method to mechanical and electromechanical systems. While theorems are proved systematically, the emphasis is on understanding and applying the theory to real-world situations.Attractive Ellipsoids in Robust Control will appeal to undergraduate and graduate students with a background in modern systems theory as well as researchers in the fields of control engineering and applied mathematics