Jean-Claude Hennet

(A,B)-invariant polyhedral sets of linear discrete-time systems

The problem of confining the trajectory of a linear discrete-time system in a given polyhedral domain is addressed through the concept of (A, B)-invariance. First, an explicit characterization of (A, B)-invariance of convex polyhedra is proposed. Such characterization amounts to necessary and sufficient conditions in the form of linear matrix relations and presents two major advantages compared to the ones found in the literature: it applies to any convex polyhedron and does not require the computation of vertices. Such advantages are felt particularly in the computation of the supremal (A, B)-invariant set included in a given polyhedron, for which a numerical method is proposed. The problem of computing a control law which forces the system trajectories to evolve inside an (A, B)-invariant polyhedron is treated as well. Finally, the (A, B)-invariance relations are generalized to persistently disturbed systems.

Eigenstructure assignment for state constrained linear continuous time systems

Abstract Linear state constraints define a polyhedron in the state space of a dynamical system. If this polyhedron can be made positively invariant by state feedback, the constraints are satisfied all along the trajectory for any initial point in this polyhedron. In the context of continuous-time linear systems, some algebraic positive invariance conditions are established for unbounded symmetrical polyhedra. The existence of a solution to the corresponding set of equations and inequalities depends on the structural properties of an associated input-output system. In particular, the number of zeros of this system should be appropriate and they should be stable. If these conditions are satisfied, the problem can easily be solved by eigenvalue and eigenvector assignment.

Supply chain coordination: A game-theory approach

In a supply chain organized as a network of autonomous enterprises, the main objective of each partner is to optimize his production and supply policy with respect to his own economic criterion. Conflicts of interests and the distributed nature of the decision structure may induce a global loss of efficiency. Contracts can then be used to improve global performance and decrease risks. The purpose of the paper is to evaluate the efficiency of different types of contracts between the industrial partners of a supply chain. Such an evaluation is made on the basis of the relationship between a producer facing a random demand and a supplier with a random lead-time. The model combines queuing theory for evaluation aspects and game theory for decisional purposes.

A bimodal scheme for multi-stage production and inventory control

This study considers a multi-stage multi-item production plant with its supply chain and customer environment. The production, supply and inventory plan is optimized on a dual-mode basis, under two different information patterns. The short-term plan relies on firm orders received from customers. On the contrary, the long-term plan is based on predicted demands represented by random sequences. In this study, the role of the long-term plan is mainly to impose a final condition set to the short-term plan.

A class of invariant regulators for the discrete-time linear constrained regulation problem

Abstract Stable dynamic systems admit positively invariant domains associated to their Lyapunov functions. Conversely, some domains can be made positively invariant for systems with state feedback controllers designed in such a way that some associated non-negative definite functions are bound to decrease. In particular, this approach can be used to establish conditions on the gain matrix for Linear Constrained Regulation Problems (LCRP). We construct fixed and variable regulators easy to compute through linear programming, for a class of constrained linear systems.

STABILITY CONDITIONS OF CONSTRAINED DELAY SYSTEMS VIA POSITIVE INVARIANCE

A delay system is represented by a linear difierence equation. The system parameters and the delays are assumed to be unperfectly known. The output vector is perturbed by a bounded external disturbance vector. The addressed problem is to characterize conditions which guarantee that the output vector remains in a given domain deflned by a set of symmetrical linear constraints. This problem is solved by imposing positive invariance conditions. These conditions also imply delay independent asymptotic stability of the associated deterministic system. The notion of distance to instability is then analyzed through the concept of stability radius. The possible use of these new robust stability conditions for controlling an input-output delay model is then presented. An application is flnally proposed ; it concerns an inventory control problem for a simple production loop subject to constraints on inventory levels.

(A, B)-Invariance Conditions of Polyhedral Domains for Continuous-Time Systems

This paper provides an algebraic characterization of the (A, B)-invariance property of polyhedral sets with respect to linear continuous-time systems. The family of control laws which is investigated is the set of continuous and Lipschitz functions. Some particular conditions of existence of linear state feedback laws are also presented.

Stability and stabilization of delay differential systems

The delay systems considered here are represented by linear delay differential equations. The system parameters and the delays are assumed to be imperfectly known. The instantaneous state vector is perturbed by a bounded external disturbance vector. The problem addressed is that of characterizing conditions that guarantee that the trajectory of the instantaneous state vector remains in a domain defined by a set of symmetrical linear constraints. It is shown that the positive invariance property can be used to solve this problem, and that positive invariance of a compact domain of the instantaneous state space implies delay-independent asymptotic stability of the associated deterministic system. The possible use of these results for the control of a multiple-delay MIMO differential model is then presented. Finally, an example is given.

On (A, B)-invariance of polyhedral domains for discrete-time systems

The problem of confining the trajectory of a discrete-time linear system in a given polyhedral set is addressed using the positive invariance approach. The property of (A, B)-invariance of polyhedral domains is introduced. It is then geometrically and analytically characterized. In particular, this property is verified by the maximal admissible set included in a given polyhedral domain. An important class of (A, B)-invariant polyhedral domains admitting a linear state feedback invariant control law is exhibited. The results are extended to systems subject to additive disturbances.

On invariant polyhedra of continuous-time linear systems

The author presents some conditions of existence of positively invariant polyhedra for linear continuous-time systems. These conditions are described algebraically and then interpreted on the basis of the system eigenstructure. A simple state-feedback placement method is described for solving some linear regulation problems under constraints. The proposed solution describes the structural conditions (on the invariant zeros) under which the domain of constraints can be made positively invariant by a simple eigenstructure assignment algorithm.<<ETX>>