Abdellah Benzaouia

Introduction to Two-Dimensional Systems

This chapter presents the backgrounds on two-dimensional (2-D) systems: first, 2-D representations are discussed, then definitions of 2-D stability are provided and complemented with some previous results that will be useful in the rest of the book; then some techniques used for dealing with actuator saturation are discussed, as used in Chaps. 2 and 4.

Positive observation of Takagi-Sugeno systems

This paper presents conditions for design of observers for nonlinear systems represented by a Takagi-Sugeno-type model, whose state variables take only nonnegative values at all times t for any nonnegative initial state (positive systems). The derived observers have a Takagi-Sugeno structure and ensure the nonnegativity of the observed states. The conditions are derived as Linear Programming and Linear Matrix Inequalities, and tested on a simulated practical problem.

Constrained Control and Rate or Increment: An LMI Approach

This chapter solves the problem of designing stabilizing regulators for linear system subject to input saturation or magnitude input constraints together with asymmetric constraints on its increment or rate. Reduced dimension LMIs developed on a reduced-order state space are given. Compared with previous approaches, the proposed technique is valid for asymmetric constraints on the increment or rate of the control, while the computing time is improved by solving reduced dimension LMI.

Regulator Problem for Singular Linear Systems with Constrained Control

This chapter solves the problem of controlling singular continuous-time systems with constraints on the control magnitude and its rate, by using state and derivative feedback (proportional-derivative controllers). Necessary and sufficient conditions for positive invariance are proposed. The synthesis of the controller is presented using two methods: an exact method based on non-symmetric Riccati equations and an approximate method based on the solution of equation \(XA+XBX=HX\). An example illustrates the feasibility of the proposed approach. As a particular case, the extension to non-singular systems is also presented.

Output Feedback Stabilization for Constrained Control Systems

This chapter presents sufficient conditions of asymptotic stability for both discrete-time and continuous-time linear systems subject to actuator saturation with an output feedback law. The obtained results are given under LMI’s formulation. A numerical example is used to illustrate this technique by using a linear optimization problem subject to LMI’s constraints.

Delay Systems with Saturating Control

This chapter is devoted to the stabilization of linear time-delay systems containing saturating actuators. Independent delay conditions and delay dependent conditions are derived. These conditions are given under LMI formalism. Hence, less conservative stabilizability conditions are deduced. First, the partitioning of the time-delay interval onto r sub-intervals enables one to introduce delay independent less conservative conditions. Second, an improved delay dependent stabilizability condition is proposed. Stabilization conditions are worked out to obtain the stabilizability conditions enabling one to design the stabilizing memoryless state feedback. Hence, these lasts are easily obtained as the solution to optimization problems via LMIs. Examples are given in both cases to show the applicability of the obtained conditions.

Stabilization of Unsymmetrical Saturated Control Systems

This chapter deals with the regulator problem for discrete-time and continuous-time linear systems with asymmetric saturation on the control. The main contribution of this work is to extend the available results for symmetrical saturation to the case of unsymmetrical saturation. The results are expressed in term of LMIs. New less conservative results on saturation are obtained. Symmetrization of the problem is obtained via some change of variables. First, an approach where the regularity of the system matrix is needed is presented. Second, this assumption is removed leading to an improved approach based on the same symmetrization techniques. An example is presented, in each case, to illustrate the obtained results.

Robust Stabilization of Two-Dimensional Uncertain Systems

Uncertainties are one of the main causes for instabilities and poor performance in feedback systems. Thus, robust stability is an important issue for any control design, so this chapter incorporates uncertainty into the control methodologies for two-dimensional systems studied in Chap. 2.

Stabilization of Saturated Systems

Two-dimensional (2-D) saturated systems are studied in this chapter. More precisely, the stabilization under state-feedback control is solved in two different situations: first, when the control may take the saturation value, and second, when control limits are avoided. Sufficient conditions of asymptotic stabilization are presented that make it possible to synthesize the required controllers solving LMIs. At the end of the chapter some examples are solved to illustrate the approach.

Stabilization of Two-Dimensional Takagi–Sugeno Systems with Attenuation of Stochastic Perturbations

This chapter investigates the stabilization of two-dimensional (2-D) Takagi–Sugeno (T–S) systems with stochastic perturbations. Our aim is to develop state feedback controllers that ensure mean-square stabilization and a guaranteed level of attenuation for the perturbations expressed as an \(H_{\infty }\) norm. Sufficient conditions are derived for the existence of such controls in terms of linear matrix inequalities (LMIs) that make it possible to solve the corresponding control synthesis problem. Numerical examples are given to demonstrate the effectiveness of the proposed method.