State Feedback Stablization of Positive Systems

Elarbi Achhab, Professeur au Département de Mathématiques de Faculté des Sciences El Jadida (Université Chouaib Doukkali, Maroc) donnera un séminaire le 24 janvier 2018 à 10h, dans la salle de réunion du MIS

Positive linear systems are linear dynamical systems whose state trajectories are nonnegative for every nonnegative initial state and for every admissible nonnegative input function. Equivalently a linear dynamical system is positive whenever the corresponding cone (which defines the considered order) of the state-space is invariant under the state transition map (positive invariance). The posivity (or, more precisely, nonnegativity) property occurs quite frequently in practical applications where the state variables correspond to quantities that do not have real meaning unless they are nonnegative.  We first give a brief overview of the positivity property and the positive stabilization problem for finite dimensional systems. Then a short tutorial is given on the positivity of infinite dimensional linear systems. Some new points of view and perspectives are described. Algebraic conditions of positivity for dynamical systems defined on an ordered Banach space whose positive cone has an empty interior are reported. Sufficient conditions are reported for the positive stabilization of a class of distributed parameter systems, such that the closed loop system is stable and positive. More specifically, for positive unstable infinite-dimensional linear systems, conditions are established for positive stabilizability and a method is described for computing a positively stabilizing state feedback, which guarantees that the stable closed loop dynamics are nonnegative for specific initial states. A feedback control is designed such that the unstable finite-dimensional spectrum of the dynamics generator is replaced by the eigenvalues of the stable input dynamics and such that the resulting input trajectory remains in an affine cone. Moreover a synthesis methodology of a positively stabilizing state feedback will be presented.