Mustapha Ait Rami

Controller Synthesis for Positive Linear Systems With Bounded Controls

This brief solves some synthesis problems for a class of linear systems for which the state takes nonnegative values whenever the initial conditions are nonnegative. In particular, the synthesis of state-feedback controllers is solved in terms of linear programming problem, including the requirement of positiveness of the controller and its extension to uncertain plants. In addition, the synthesis problem with nonsymmetrical bounds on the stabilizing control is treated

Solvability of static output-feedback stabilization for LTI positive systems

Abstract This paper addresses the stabilization problem of positive linear systems which have nonnegative states whenever the initial conditions are nonnegative. The synthesis of static output-feedback controllers that ensure the positivity and asymptotic stability of the closed-loop system is investigated. It is shown that this important problem is completely solved for single-input and single-output positive systems. The proposed approach can be applied to multi-input positive systems with controllers having one rank gains. All the provided conditions are necessary and sufficient and can be solved in terms of Linear Programming.

Control of constrained positive discrete systems

Stabilization problem of positive discrete-time systems with bounds on the inputs or the states is solved in this paper. First, the synthesis of state-feedback controllers that ensure the nonnegativity and the stability of the unconstrained closed-loop systems is studied for both the nominal and uncertain systems. These results are then extended to the case of bounded controls and also for systems with bounds on the states. All the proposed conditions are solvable in terms of Linear Programming. A cost function is then proposed to synthesis controllers with good performance. An example illustrate the feasibility of the proposed approach.

Positive observers for linear positive systems, and their implications

A treatment of the positive observation problem for positive systems is provided, for the continuous and discrete-time cases. The proposed observers are positive, that is, they ensure that the estimates are nonnegative at any time. Moreover, necessary and sufficient conditions for the existence of such positive observers are formulated in terms of linear programming. Also, we show how positive observers can provide guaranteed bounds on the real states of uncertain positive systems. In addition, some implications are discussed; especially, it is shown that it is not possible to stabilise an unstable system using these positive observers. Finally, the applicability of the proposed interval estimation approach is shown for an illustrative application from pharmacokinetics.

Stability Analysis and Synthesis for Linear Positive Systems with Time-Varying Delays

This paper provides necessary and sufficient conditions for the asymptotic stability of linear positive systems subject to time-varying delays. It introduces and initiates an original method for solving directly the proposed stability and stabilization problems without using the well-known Lyapunov theory that is commonly used in the field of stability analysis. In that way and for readers convenience, the paper avoids possible long and tedious superfluous calculus.

Characterization and Stability of Autonomous Positive Descriptor Systems

This note addresses two problems: the characterization of positivity of autonomous descriptor systems and the problem of finding conditions for their stability. For these fundamental issues, we provide necessary and sufficient conditions complemented with a numerically reliable computational approach. The given conditions can be checked by means of linear programming (LP). Also, we derive a necessary and sufficient condition for stability that is connected to the underlying eigenstructure of the system.

Estimation of linear positive systems with unknown time-varying delays

Abstract This paper considers the estimation problem for linear positive systems with time-varying unknown delays. Similar to set-valued estimation approaches, we provide a confident region within which the trajectory of the observed positive system always evolves. Guaranteed upper and lower estimates for the instantaneous states are characterized by means of a special kind of extended Luenberger-type interval observer. We provide constructive conditions for its existence and establish the asymptotic convergence of its associated interval error. In addition, we give an LP-based method which allows one to construct the proposed interval observer solely from the data of the system.

POSITIVE OBSERVATION PROBLEM FOR LINEAR TIME-LAG POSITIVE SYSTEMS

This paper deals with the problem of positive observation for linear time-delay systems for which the states take nonnegative values whenever the initial conditions are nonnegative. We focus on the design of positive observers (possibly with time-delay) which guarantee nonnegative estimates of the current states. We derive necessary and sufficient conditions for the existence of a positive observer (extended Luemberger-type) and show that the solvability of the problem can be decided via standard linear programming techniques. Moreover, on the negative side, it is shown that one cannot stabilize any unstable positive time-delay system by using extended Luenberger type positive observers. In other words, the separation principle does not hold.

Stabilization Under Constrained States and Controls of Positive Systems with Time Delays

This paper solves the stabilization problem for positive linear discrete-time systems subject to time-delays, under state and control constraints. The synthesis of statefeedback stabilizing controllers is solved in terms of Linear Programming. The proposed approach deals with the stabilization with bounded controls and states, in the presence of possibly unknown delays. Also, the robust control design is considered, and it is illustrated with an example from pharmacokinetics.

Robust Control for Uncertain Linear Systems with State and Control Constraints

Abstract The paper is devoted to the stabilization of linear uncertain systems having restricted states and controls. Two classes of uncertainties, namely norm bounded and polytopic, are studied for continuous time systems. Sufficient LMI conditions are given for the derivation of robust state-feedback controllers driving the system asymptotically to the origin without violating the constraints. Further, the determination of a large region of attraction for these systems is addressed. Moreover, numerical algorithms are provided for the enlargement of the volume of attraction region of the uncertain system. The approach is illustrated by an example for each case of uncertainties.