Naceur Benhadj Braiek

Observer-based fault tolerant control design for a class of LPV descriptor systems

This paper presents a new Fault Tolerant Control (FTC) methodology for a class of LPV descriptor systems that are represented under a polytopic LPV form. The aim of this FTC strategy is to compensate the effects of time-varying or constant actuator faults by designing an Adaptive Polytopic Observer (APO) which is able to estimate both the states of the system and the magnitude of the actuator faults. Based on the information provided by this APO, a new state feedback control law is derived in order to stabilize the system. Stability conditions of the designed observer and the state-feedback control are provided and solved through a set of Linear Matrix Inequalities (LMI) under equality constraints. The performance of the proposed Fault Tolerant Control scheme is illustrated using a two-phase flash system.

The Tensor Product Based Analytic Solutions of Nonlinear Optimal Control Problem

In this paper, the attention is focused on the optimization of a particular class of nonlinear systems. The optimum linear solution is not the best one so the problem of determining a nonlinear state feedback optimal control law with quadratic performance index over infinite time horizon is considered. It isnt an easy task and the most discouraging obstacle is the resolution of the Hamilton-Jacobi equation. Thus our contribution, based on the use of the tensor product and its algebraic laws, provide analytic solutions of the studied optimal control problem. The polynomial state feedback solution is computed through a numerical procedure. A numerical example is treated to illustrate the proposed solutions and some conclusions are drawn.

On the Estimation of Asymptotic Stability Region of Nonlinear Polynomial Systems: Geometrical Approaches

In recent years, the problem of determining the asymptotic stability region of autonomous nonlinear dynamic systems has been developed in several researches. Many methods, usually based on approaches using Lyapunov’s candidate functions (Davidson & Kurak, 1971) and (Tesi et al., 1996) which altogether allow for a sufficient stability region around an equilibrium point. Particularly, the method of Zubov (Zubov, 1962) is a vital contribution. In fact, it provides necessary and sufficient conditions characterizing areas which are deemed as a region of asymptotic stability around stable equilibrium points. Such a technique has been applied for the first time by Margolis (Margolis & Vogt, 1963) on second order systems. Moreover, a numerical approach of the method was also handled by Rodden (Rodden, 1964) who suggested a numerical solution for the determination of optimum Lyapunov function. Some applications on nonlinear models of electrical machines, using the last method, were also presented in the Literature (Willems, 1971), (Abu Hassan & Storey, 1981), (Chiang, 1991) and (Chiang et al., 1995). In the same direction, the work presented in (Vanelli & Vidyasagar, 1985) deals with the problem of maximizing Lyapunov’s candidate functions to obtain the widest domain of attraction around equilibrium points of autonomous nonlinear systems. Burnand and Sarlos (Burnand & Sarlos, 1968) have presented a method of construction of the attraction area using the Zubov method. All these methods of estimating or widening the area of stability of dynamic nonlinear systems, called Lyapunov Methods, are based either on the Characterization of necessary and sufficient conditions for the optimization of Lyapunov’s candidate functions, or on some approaches using Zubov’s digital theorem. Equally important, however, they also have some constraints that prevented obtaining an exact asymptotic stability domain of the considered systems. Nevertheless, several other approaches nether use Lyapunov’s functions nor Zubov’s which have been dealt with in recent researches. Among these works cited are those based on topological considerations of the Stability Regions (Benhadj Braiek et al., 1995), (Genesio et al., 1985) and (Loccufier & Noldus, 2000). Indeed, the first method based on optimization approaches and methods using the consideration of Lasalle have been developed to ensure a practical continuous stability

Time Optimal Control Laws for Bilinear Systems

The aim of this paper is to determine the feedforward and state feedback suboptimal time control for a subset of bilinear systems, namely, the control sequence and reaching time. This paper proposes a method that uses Block pulse functions as an orthogonal base. The bilinear system is projected along that base. The mathematical integration is transformed into a product of matrices. An algebraic system of equations is obtained. This system together with specified constraints is treated as an optimization problem. The parameters to determine are the final time, the control sequence, and the states trajectories. The obtained results via the newly proposed method are compared to known analytical solutions.

Minimum time control synthesis for high order LTI systems

This paper proposes a generalization of the minimum time control problem for systems with real poles. Two cases are considered, namely, systems with only real poles and systems having in addition real double poles. The problem is written in the temporal domain in a new form, the switching times are computed numerically. Several examples are given to illustrate the proposed approach of optimal control.