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

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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.

This paper is devoted to design of interval observers for Linear Time Varying (LTV) systems and a class of nonlinear time-varying systems in the output canonical form. An interval observer design is feasible if it is possible to calculate the observer gains making the estimation error dynamics cooperative and stable. It is shown that under some mild conditions the cooperativity of an LTV system can be ensured by a static linear transformation of coordinates. The efficiency of the proposed approach is demonstrated through numerical simulations.

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.

The main results obtained in the field of input-state stable systems and systems with other similar characteristics that were published over the last two decades were reviewed.

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

The problem of output stabilization of a class of nonlinear systems subject to parametric and signal uncertainties is studied. First, an interval observer is designed estimating the set of admissible values for the state. Next, it is proposed to design a control algorithm for the interval observer providing convergence of interval variables to zero, that implies a similar convergence of the state for the original nonlinear system. An application of the proposed technique shows that a robust stabilization can be performed for linear time-varying and Linear-Parameter-Varying (LPV) systems without assumption that the vector of scheduling parameters is available for measurements. Efficiency of the proposed approach is demonstrated through two examples.