Fixed-time Stabilization with a Prescribed Constant Settling Time by Static Feedback for Delay-Free and Input Delay Systems

A static non-linear homogeneous feedback for a fixed-time stabilization of a linear time-invariant (LTI) system is designed in such a way that the settling time is assigned exactly to a prescribed constant for all nonzero initial conditions. The constant convergence time is achieved due to a dependence of the feedback gain of the initial state of the system. The robustness of the closed-loop system with respect to measurement noises and exogenous perturbations is studied using the concept of Input-to-State Stability (ISS). Both delay-free and input delay systems are studied. Theoretical results are illustrated by numerical simulations

The Alestle - Vol. 58 No. 22 - 11/08/2005

Vol. 58 No. 2

Generalized Homogeneous Rigid-BodyAttitude Control

The attitude tracking problem for a full-actuated rigid body in 3D is studied using a impulsive system model based on Lie algebra so(3). A nonlinear homogeneous controller is designed to globally track a smooth attitude trajectory in a finite or a (nearly) fixed time. A global settling time estimate is obtained, which is easily adjustable by tuning the homogeneity degree. The local input-to-state stability is proven. Simulations illustrating the performance of the proposed algorithm are presented

Input-to-State Stability of Homogeneous Infinite Dimensional Systems with Locally Lipschitz Nonlinearities

International audienceThe Input-to-State Stability (ISS) of homogeneous evolution equations in Banach spaces with unbounded linear operators and locally Lipschitz nonlinearities in the right-hand sides is studied. A new homogeneous converse Lyapunov theorem is presented. Similarly to finite-dimensional models, it is shown that the uniform asymptotic stability of an unperturbed homogeneous evolution equation implies its ISS with respect to homogeneously involved exogenous inputs

On Optimal Control Problems for Generalized Homogeneous Systems

The paper deals with optimal control problems for plants modeled by non-linear ordinary differential equations with homogeneous (in a generalized sense) right hand sides. The classical tools of optimal control theory, namely, Dynamic Programming and Maximum Principle are refined for generalized homogeneous control systems

On Homogeneous Approximations, Stability and Robustness of Infinite Dimensional Systems

The paper generalizes the concept of homogeneous approximations to a class of unbounded operators satisfying certain regularity assumptions. Stability and robustness of locally homogeneous abstract control systems are studied. The viscous Burgers equation and its nonlinear modifications are considered as illustrative examples

Finite/fixed-time Stabilization of Linear Systems with States Quantization

This paper develops a homogeneity-based approach to finite/fixed-time stabilization of linear time-invariant (LTI) system with quantized measurements. A sufficient condition for finite/fixed-time stabilization of multi-input LTI system under states quantization is derived. It is shown that a homogeneous quantized state feedback with logarithmic quantizer can guarantee finite/fixed-time stability of the closed-loop system provided that the quantization is sufficiently dense. Theoretical results are supported with numerical simulations

Sliding Mode Control Design Using Canonical Homogeneous Norm

International audienceThe problem of sliding mode control design for nonlinear plant is studied. Necessary and sufficient conditions of quadratic-like stability (stabi-lizability) for 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