Fouad Mesquine

A separation principle for linear switching systems and parametrization of all stabilizing controllers

In this paper, we investigate the problem of designing a switching compensator for a plant switching amongst a (finite) family of given configurations (Ai, Bi, Ci). We assume that switching is uncontrolled, namely governed by some arbitrary switching rule, and that the controller has the information of the current configuration i.As a first result, we provide necessary and sufficient conditions for the existence of a family of linear compensators, each applied to one of the plant configurations, such that the closed loop plant is stable under arbitrary switching. These conditions are based on a separation principle, precisely, the switching stabilizing control can be achieved by separately designing an observer and an estimated state (dynamic) compensator. These conditions are associated with (non-quadratic) Lyapunov functions. In the quadratic framework, similar conditions can be given in terms of LMIs which provide a switching controller which has the same order of the plant. As a second result, we furnish a characterization of all the stabilizing switching compensators for such switching plants. We show that, if the necessary and sufficient conditions are satisfied then, given any arbitrary family of compensators K i,(s), each one stabilizing the corresponding LTI plant (Ai, Bi, Ci) for fixed i, there exist suitable realizations for each of these compensators, which assure stability under arbitrary switching.

Regulator problem for linear systems with constraints on control and its increment or rate

This paper discusses the problem of constraints on both control and its rate or increment, for linear systems in state space form, in both the continuous and discrete-time domains. Necessary and sufficient conditions are derived for autonomous linear systems with constrained state increment or rate (for the continuous-time case), such that the system evolves respecting incremental or rate constraints. A pole assignment technique is then used to solve the inverse problem, giving stabilizing state feedback controllers that respect non-symmetrical constraints on both control and its increment or rate. An illustrative example shows the application of the method on the double integrator problem.

The Regulator Problem for Linear Systems With Saturations on the Control and its Increments or Rate: An LMI Approach

This paper solves the problem of designing stabilizing regulators for linear systems subject to control saturations and asymmetric constraints on its increment or rate, using reduced dimension linear matrix inequalities (LMIs) developed on a reduced-order state space. 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 resolving LMIs of reduced dimensions

Stabilization of 2D saturated systems by state feedback control

The problem of stabilizability of the 2D continuous-time saturated systems under state-feedback control is solved in this paper. Two cases are considered: the first one, the control may saturate and limits may be attained. The second one, the control does not saturate and limits are avoided. Sufficient conditions of asymptotic stability are presented. The synthesis of the required controllers is given under LMIs form. Illustrative examples are treated.

Stabilization of Continuous-Time Fractional Positive Systems by Using a Lyapunov Function

The stabilization problem for continuous-time fractional linear systems with the additional condition of non negativity of the states is solved. The obtained results are based on a direct Lyapunov function. In particular, the synthesis of state-feedback controllers is obtained by giving conditions in terms of Linear Programs. The solution is also extended to stabilization by asymmetric bounded control. Illustrative examples are provided to show the usefulness of the results.

Constrained stabilization with an assigned initial condition set

The problem of stabilizing a linear discrete-time system with control constraints is considered. Necessary and sufficient conditions are given for the existence of a state feedback controller which drives the state to the origin asymptotically from every initial state in an assigned compact polyhedral set. These conditions can be checked via linear programming. It is shown that when the problem has a solution, a polyhedral function can be formed which turns out to be a Lyapunov function if a proper nonlinear feedback controller is applied. Two procedures are presented for the construction of the Lyapunov function. The first is based on the property that the stabilizing feedback compensator exists if and only if for every initial condition chosen on a vertex of the set there exists an open-loop control driving the state to its interior. The second procedure is based on the construction of the controllability regions to the given polyhedral set; this procedure can also be applied to systems with parameter u...

Delay-Dependent Stabilizability of 2D Delayed Continuous Systems with Saturating Control

This paper deals with the problem of stability and stabilization of 2D delayed continuous systems with saturation on the control. An improved delay-dependent stability condition taken from the recent literature is first extended to the case of 2D systems. Second, a delay-dependent stabilizability condition is deduced. The synthesis of stabilizing saturating state feedback controllers for such systems is then given. A set of allowed delays for both directions of the state is computed. All involved conditions are given under LMI formalism. Examples are worked to show the effectiveness of the approach.

Stabilization of positive constrained T–S fuzzy systems: Application to a Buck converter☆

Abstract This paper deals with the problem of constrained stability and tracking of Takagi–Sugeno (T–S) fuzzy positive systems. Linear programming (LP) is used to insert the constraints in the design phase while imposing positivity in closed-loop. The theoretical results are applied to the buck DC–DC converter which is widely used in the photovoltaic generators. Based on the simulation results success of the method is shown for this application.

System stabilization by unsymmetrical saturated state feedback control

This paper studies the problem of linear unsymmetrical constrained input systems. A new transformation for constrained input linear problem control is presented to deal with the asymmetry of the constraints. LMI form is selected by using the available results on symmetrical saturation. These results are obtained for the first time reducing considerably the conservatism of the results obtained with symmetrical saturations. A numerical example is given to illustrate the new results.

Robust stabilization of constrained uncertain continuous-time fractional positive systems

Abstract The robust stabilization problem is solved for uncertain fractional linear continuous-time systems having bounded asymmetric control, with additional condition of non-negativity of the states. The synthesis of state-feedback controllers is obtained by giving conditions in terms of Linear Programs. An illustrative example is provided to show the usefulness of the results.