Emmanuel Moulay

Finite-time stability and stabilization of time-delay systems

Finite-time stability and stabilization of retarded-type functional differential equations are developed. First, a theoretical result on finite-time stability inspired by the theory of differential equations, using Lyapunov functionals, is given. As it may appear not easily usable in practice, we show how to obtain finite-time stabilization of linear systems with delays in the input by using an extension of Artsteins model reduction to nonlinear feedback. With this approach, we give an explicit finite-time controller for scalar linear systems and for the chain of integrators with delays in the input.

Finite time stability conditions for non-autonomous continuous systems

Finite time stability is defined for continuous non-autonomous systems. Starting with a result from Haimo (1986) we then extend this result to n-dimensional non-autonomous systems through the use of smooth and non-smooth Lyapunov functions as in Perruquetti and Drakunov (2000). One obtains two different sufficient conditions and a necessary one for finite time stability of continuous non-autonomous systems.

A Global High-Gain Finite-Time Observer

A global finite-time observer is designed for nonlinear systems which are uniformly observable and globally Lipschitz. This result is based on a high-gain approach combined with recent advances on finite-time stability using Lyapunov function and homogeneity concepts.

Lyapunov Theory for 2-D Nonlinear Roesser Models: Application to Asymptotic and Exponential Stability

This technical note deals with a general class of discrete 2-D possibly nonlinear systems based on the Roesser model. We first motivate the introduction of Lyapunov type definitions of asymptotic and exponential stability. This will allow us to introduce and discuss several particularities that cannot be found in 1-D systems. Once this background has been carefully designed, we develop different Lyapunov theorems in order to check asymptotic and exponential stability of nonlinear 2-D systems. Finally we propose the first converse Lyapunov theorem in the case of exponential stability.

Brief paper: Robust output feedback sampling control based on second-order sliding mode

This paper proposes a new second-order sliding mode output feedback controller. This is developed in the case of finite sampling frequency and uses only output information in order to ensure desired trajectory tracking with high accuracy in a finite time in spite of uncertainties and perturbations. This new strategy is evaluated in simulations on an academic example.

Stabilization of nonaffine systems: a constructive method for polynomial systems

This note focuses on the stabilization of nonaffine systems described by continuous nonlinear ordinary differential equations. First, conditions of stabilization using control Lyapunov function give theoretical but nonconstructive and restrictive results. Second, some particular nonaffine systems are considered: This is polynomial system in the control variable of order two and three (a method is also given for high-order systems). The main result is a method of construction of feedbacks for this class of polynomial systems. In the end, the polynomial example of the levitation system is stabilized using an extension of this method to discontinuous feedback.

New predictive scheme for the control of LTI systems with input delay and unknown disturbances

A new predictive scheme is proposed for the control of Linear Time Invariant (LTI) systems with a constant and known delay in the input and unknown disturbances. It has been achieved to include disturbances effect in the prediction even though there are completely unknown. The Artstein reduction is then revisited thanks to the computation of this new prediction. An extensive comparison with the standard scheme is presented throughout the article. It is proved that the new scheme leads to feedback controllers that are able to reject perfectly constant disturbances. For time-varying ones, a better attenuation is achieved for a wide range of perturbations and for both linear and nonlinear controllers. A criterion is given to characterize this class of perturbations. Finally, some simulations illustrate the results.

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.

Sliding mode stabilization of the current profile in Tokamak plasmas

This paper deals with the robust stabilization of the spatial distribution of tokamak plasmas current profile using a sliding mode feedback control approach. The control design is based on the 1D resistive diffusion equation of the magnetic flux that governs the plasma current profile evolution. The feedback control law is derived in the infinite dimensional setting without spatial discretisation. Numerical simulations are provided and the tuning of the controller parameters that would reject uncertain perturbations is discussed.

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.