Study For Structure Properties Of Positive Descriptor Systems

Positive systems are systems that their state and output variables remain in the nonnegative orthant for any nonnegative initial state and any nonnegative input. Positive systems appear naturally in the modeling of a lot of practical applications. For instance, in the fields of population model, economics, engineering, biology, pharmacokinetics and so on. In these models, the variables represent population quantities, quantities of output of goods, numbers of bacterias or cells, etc, that have no meaning unless nonnegative. Due to the nonnegative restrictions on both variables and inputs, positive linear systems are defined on cones and not on linear spaces. Consequently, many well-known results of linear systems cannot be applied to positive linear systems. Recently, positive standard systems have drawn much attention of researchers because of their wide applications, and many fundamental results have been obtained. Among the great number of presented results for positive standard systems, much attention has been devoted to descrete-time positive standard systems. So far, the papers for positive descriptor systems are rather few, specially, for continuous-time positive descriptor systems. Therefore, there are still many open problems for positive descriptor systems. Analysis and synthesis for positive descriptor systems will receive considerable attention and interest of researchers.This thesis has systematically analyzed positive descriptor systems. Some basic characteristics of positive descriptor systems:positive reachable cones, positive reachability, positive controllability, stability and admissibility have been investigated.Firstly, basic notions of positive descriptor systems are introduced. According to the solutions in terms of Drazin inverse, we study motion characteristics of positive descriptor systems.Secondly, we investigate the positive reachable cones, positive reachability, positive controllability and positive null-controllability of positive descriptor systems. Much attention has been devoted to discrete-time positive descriptor systems.Thirdly, we analyze stability of positive descriptor systems. We mainly study asymptotic stability, componentwise asymptotic stability, exponential stability of positive descriptor systems. Some necessary and sufficient conditions are given. Meanwhile, the equilibrium points of positive descriptor systems are studied. And some numerical examples are given to illustrate the validity of the results presented.Finally, we consider the admissibility of positive descriptor systems. We show LMIs criteria for admissibility of positive descriptor systems based on LMIs criteria for stability of positive descriptor systems existed. Meanwhile, some criteria for admissibility of a class of positive descriptor systems are given in terms of Lyapunov equations. Some examples are given to demonstrate the effectiveness of the results obtained.