Finite-time Stabilization Using Implicit Lyapunov Function Technique

International audienceThe Implicit Lyapunov Function (ILF) method for finite-time stability analysis is introduced. The control algorithm for finite-time stabilization of a chain of integrators is developed. The scheme of control parameters selection is presented by a Linear Matrix Inequality (LMI). The robustness of the finite-time control algorithm with respect to system uncertainties and disturbances is studied. The new high order sliding mode (HOSM) control is derived as a particular case of the developed finite-time control algorithm. The settling time estimate is obtained using ILF method. The algorithm of practical implementation of the ILF control scheme is discussed. The theoretical results are supported by numerical simulations

Stabilization of Chain of Integrators with Arbitrary Order in Finite-time

International audienceA control algorithm for finite-time stabilization of a chain of integrators with arbitrary order is introduced. The method is based on Implicit Lyapunov Function (ILF) approach with applying properties of homogeneous systems. Scheme of control parameter tuning is presented in Linear Matrix Inequality (LMI) form. The method is simple in implementation and does not assume any additional computational on-line procedures that is an improvement with respect to [8], [11]. The theoretical results are supported by numerical simulations

Implicit Lyapunov-Krasovski Functionals for Time Delay Systems

International audienceThe method of Implicit Lyapunov-Krasovski Functional (ILKF) for stability analysis of time-delay systems is introduced. Theorems on Lyapunov, asymptotic, fiite-time,fixed-time and (hyper-) exponential stability analysis using ILKF are presented. The hyper exponential stabilization algorithm for a time-delay system is developed. The theoretical results are supported by numerical simulations

Suboptimal LMI-based Solution of Minimum Time Control Problem

International audienceThe problem of time optimal control design is considered for a chain of integrators. The suboptimal solution based on Implicit Lyapunov Function (ILF) method is presented in the form of continuous stabilizing feedback. The Semi-Definite Programming (SDP) problem with the constraints in the form of Linear Matrix Inequalities (LMI) is obtained for tuning the optimal parameters. The suboptimal solution is compared with the minimum discontinuous feedback on numerical example

Sliding Mode Control Design for MIMO Systems: Implicit Lyapunov Function Approach

International audienceA new approach to robust control design is considered. It is based on Implicit Lyapunov Function method. An algorithm of robust finite-time stabilization for a quasi- linear multi-input disturbed system is developed. A new high- order sliding mode control algorithm is deduced. Procedures for tuning of control parameters are presented. They are based on Linear Matrix Inequalities (LMI). Aspects of practical implementation of developed algorithms are discussed. A scheme for chattering reduction is proposed. Theoretical results are supported by numerical simulations

On Homogeneous Distributed Parameter Systems

International audienceA geometric homogeneity is introduced for evolution equations in a Banach space. Scalability property of solutions of homogeneous evolution equations is proven. Some qualitative characteristics of stability of trivial solution are also provided. In particular, finite-time stability of homogeneous evolution equations is studied. Theoretical results are illustrated on important classes of partial differential equations

Finite-time Attractive Ellipsoid Method: Implicit Lyapunov Function Approach

http://dx.doi.org/10.1080/00207179.2015.1118660International audienceA finite-time version, based on Implicit Lyapunov Functions (ILF), for the Attractive Ellipsoid Method (AEM) is developed. Based on this, a robust control scheme is presented to ensure finite-time convergence of the solutions of a chain of integrators with bounded output perturbations to a minimal ellipsoidal set. The control parameters are obtained by solving a minimization problem of the " size " of the ellipsoid subject to a set of Linear Matrix Inequalities (LMI's) constraints, and by applying the implicit function theorem. A numerical example is presented to support the implementability of these theoretical results

A polytopic strategy for improved non-asymptotic robust control via implicit Lyapunov functions

International audienceThis paper is concerned with finite-and fixed-time robust stabilization of uncertain multi-input nonlinear systems via the implicit Lyapunov function method. Instead of splitting the system into a linear nominal model and an additive perturbation which gathers nonlinearities, parametric uncertainties , and exogenous disturbances, the methodology hereby proposed preserves some nonlinear terms in the nominal system via an exact polytopic representation which leads to design conditions in the form of linear matrix inequalities. As a result, feasible solutions are found where former approaches fail; these solutions have more accurate settling-time estimates with reduced control effort. The corresponding control law includes well-known high-order sliding modes as a particular case. Numerical simulations are provided to illustrate the advantages of the proposal

Теорема Харитонова та робастна стабiлiзацiя, заснованi на ортогональних полiномах

Представлена теорема Харитонова для iнтервальних полiномiв у термiнах ортогональних полiномiв на $[0, +\infty )$  та їх полiномiв другого роду. Запропонований клас керувань, якi робастно стабiлiзують канонiчну систему

Finite-time Stabilization Using Implicit Lyapunov Function Technique

International audienceThe Implicit Lyapunov Function (ILF) method for finite-time stability analysis is introduced. The control algorithm for finite-time stabilization of a chain of integrators is developed. The scheme of control parameters selection is presented by a Linear Matrix Inequality (LMI). The robustness of the finite-time control algorithm with respect to system uncertainties and disturbances is studied. The new high order sliding mode (HOSM) control is derived as a particular case of the developed finite-time control algorithm. The settling time estimate is obtained using ILF method. The algorithm of practical implementation of the ILF control scheme is discussed. The theoretical results are supported by numerical simulations