Emmanuel Bernuau

Verification of ISS, iISS and IOSS properties applying weighted homogeneity

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 are that, under some mild conditions, the system robustness can be established as a function of the degree of homogeneity.

Finite-time output stabilization of the double integrator

The problem of finite-time output stabilization of the double integrator is addressed applying the homogeneity approach. A homogeneous controller and a homogeneous observer are designed (for different degree of homogeneity) ensuring the finite-time stabilization. Their combination under mild conditions is shown to stay homogeneous and finite-time stable as well. The efficiency of the obtained solution is demonstrated in computer simulations.

Robust finite-time output feedback stabilisation of the double integrator

The problem of finite-time output stabilisation of the double integrator is addressed applying the homogeneity approach. A homogeneous controller and a homogeneous observer are designed (for different degrees of homogeneity) ensuring the finite-time stabilisation. Their combination under mild conditions is shown to stay homogeneous and finite-time stable as well. Robustness and effects of discretisation on the closed-loop system obtained are analysed. The efficiency of the solution obtained is demonstrated in computer simulations.

Technical communique: Retraction obstruction to time-varying stabilization

This paper addresses the problem of the global stabilization on a total space of a fiber bundle with a compact base space. We prove that, under mild assumptions (existence of a continuous section and forward unicity of solutions), no equilibrium of a continuous system defined on such a state space can be globally asymptotically uniformly stabilized using continuous time-varying feedback.

Robustness of Finite-Time Stability Property for Sliding Modes

The paper is devoted to analysis of robustness of the finite-time stability property for discontinuous systems using the homogeneity framework. A short introduction into sliding-mode systems, homogeneity and finite-time stability is given. The main result connects the homogeneity degree of a discontinuous system and its type of robust stability. For robustness analysis the input-to-state stability method is used. The proposed theory is applied to a series of well known sliding-mode control algorithms.

Homogeneous continuous finite-time observer for the triple integrator

In this paper we consider the continuous homogeneous observer defined in [1] in the case of the triple integrator. In [1], convergence of the algorithm was only proved when the degree of homogeneity was sufficiently close to 0 without more tractable information. We show here that, in the case of the triple integrator, the observer presents global finite-time stability for any negative degree under constructive conditions on the gains. This is achieved with a homogeneous Lyapunov function design. Simulations of the proposed observer are also provided.

On the robustness of homogeneous systems and a homogeneous small gain theorem

This paper is devoted to the study of the robustness properties stemming from geometric homogeneity. More precisely, we show that a continuous homogeneous system, which is asymptotically stable without perturbation, is always ISS w.r.t. a perturbation. We characterize the asymptotic gain of such a disturbed system and state a so-called homogeneous small gain theorem based on this estimation.

Robustness of Homogeneous and Homogeneizable Differential Inclusions

The chapter proposes constructive conditions for verifying the input-to-state stability property of discontinuous systems using geometric homogeneity. Two sets of conditions are developed for a class of homogeneous and homogenizable systems described by differential inclusions. The advantage of the proposed conditions is that they are not based on the Lyapunov function method, but more related to algebraic operations over the right-hand side of the system.