L₁（l₁）-gain Performance Analysis And Positive Controller Design Of Positive Linear Delay Systems

Positive systems are dynamic systems whose states and outputs are non-negative whenever the initial conditions and inputs are nonnegative. Positive systems abound in almost all fields, for instance, biomedicine, population mod-el, engineering and so on. Since the states of positive systems are confined within a "cone" located in the positive orthant rather than in linear space, many results for general linear systems can not be readily applied to positive systems.In this paper, the problems of L1(l1)-gain performance analysis and pos-itive controller design are investigated. Necessary and sufficient conditions are given to guarantee that the positive linear delay system is asymptotically internally stable and satisfies the prescribed L1(l1)-gain index. And neces-sary and sufficient conditions are provided for the existence of the positive L1(l1)-controller. Then some numerical examples are presented to show the effectiveness and correctness of the theoretical results. This paper contains the following three chapters.In Chapter1, the background of positive systems and delay systems are reviewed, and some advances in the study of L1-gain performance analysis and positive controller design for positive systems are recalled. Then some open problems on L1-gain performance analysis for positive systems with delays are pointed out. Finally, the notations used in this paper are listed.In Chapter2, the stability and performance analysis problem is studied for a class of continuous-time positive linear systems with delays. For system-s without control input, based on Lyapunov theory, Laplace Transform and strong alternative for linear inequalities, three equivalent conditions are given to guarantee that the positive linear delay system is asymptotically internally stable and satisfies the prescribed L1-gain index. And the condition (iv) is presented in terms of linear program (LP). For systems with single-input, by using the singular value decomposition (SVD) approach, necessary and suf-ficient conditions are provided for the existence of the positive L1-controller. And this condition can be described in terms of nonlinear program (NLP). The proposed approach can be extended to the multi-input case, and the condition is provided for the controller with special form. Then the positive controller design problem can be converted into NLP problem.In Chapter3, the stability and performance analysis problem is considered for a class of discrete-time positive linear systems with delays. Necessary and sufficient conditions are provided to guarantee that the positive linear delay system is asymptotically internally stable and satisfies the prescribed l1-gain index. And a positive l1-controller is designed for this system.