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

On Convex Embedding and Control Design for Nonlinear Homogeneous Systems

International audienceThe paper presents methods for nonlinear homogeneous systems representation in a canonical form allowing stability conditions to be given by a linear matrix inequality. The main restriction for a system to admit the required representation is that its right-hand side has to be bounded on a unit sphere. It is shown that some nonhomogeneous systems can also be presented in the canonical form. Based on canonical representation a stabilizing control algorithm for affine in control nonlinear systems is presented with LMI-based tuning procedure. The results are supported with numerical examples

An exact robust hyperexponential differentiator

International audienceA simple differentiator is proposed, which is modeled by a second order time-varying linear differential equation. It is shown that for any signal of interest, whose second derivative is an essentially bounded function of time, the differentiation error converges to zero with a hyperexponential rate (faster than any exponential). An implicit discretization scheme of the differentiator is given, which preserves all main properties of the continuous-time counterpart. In addition, the differentiation error is robustly stable with respect to the measurement noise with a linear gain. The efficiency of the suggested differentiator is illustrated through comparison in numeric experiments with popular alternatives

On dynamical feedback control design for generalized homogeneous differential inclusions

International audienceIn the present paper, the stability criterion for generalized homogeneous differential inclusions is obtained that may simplify the control design procedure in some particular cases. Robust dynamical feedback control law is designed for homogeneous differential inclusions. Performance of the resulting dynamical feedback is illustrated by numerical simulation

Robust Stabilization of Control Affine Systems with Homogeneous Functions

International audienceThe stabilization problem of the affine control system $\dot x = f_0 (x) + u_1 f_1(x)+..+u_m f_m(x)$ with homogeneous functions $f_i$ is studied. This class of systems is of interest due to the robust properties of homogeneity and the fact that many affine systems can be approximated by or transformed to the class under consideration. An advantage of the introduced design method is that the tuning rules are presented in the form of linear matrix inequalities. Performance of the approach is illustrated by a numerical example

On Finite-Time Robust Stabilization via Nonlinear State Feedback

International audienceA 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

Finite-time and fixed-time stabilization for integrator chain of arbitrary order

International audienceIn the present paper, homogeneous control laws are designed for finite-time and fixed-time stabilization of integrator chains of arbitrary order. Provided analysis is based on Lyapunov function method and homogeneity concept. Fixed-time convergence is achieved by use of hybrid control algorithm with homogeneity degree changing. Performance of the resulting finite-time and fixed-time feedbacks is illustrated by numerical simulations

Control of Systems with Arbitrary Bounded Input Delay Using Implicit Lyapunov Function Technique

International audienceThe paper presents control algorithms for systems with input delay. There are two main results based on using Implicit Lyapunov Function (ILF) technique: 1) an LMI-based approach is presented to evaluate the domain of attraction of a finite-time stable control in the case of the arbitrary bounded delayed control input; 2) a uniting control is designed with commutation between two laws providing a global boundedness of all trajectories of systems with any input delay, and convergence to the origin for a sufficiently small one. The results are also preserved for the time-varying delay case. The theoretical results are supported by numerical examples

On Notions of Output Finite-Time Stability

International audienceLyapunov characterizations of output finite-time stability are presented for the system $x' = f (x), y = h(x)$ which is locally Lipschitz continuous out of the set $Y = {x ∈ R n : h(x) = 0}$ and continuous on $R^n$. The definitions are given in the form of $K$ and $KL$ functions. Necessary and sufficient conditions for output finite-time stability are given using Lyapunov functions. The theoretical results are supported by numerical examples

On necessary and sufficient conditions for output finite-time stability

International audienceOutput global finite-time stability of locally Lipschitz continuous autonomous systems is characterized by means of smooth Lyapunov functions. The so-called output-Lagrange stable systems are studied with details. Influence of a kind of continuity of the settling-time function is considered. Necessary and sufficient conditions of output finite-time stability are presented. The theoretical results are supported by academic examples and numerical simulations