L₁-Gain Performance Analysis Of Discrete-Time Positive Linear Time-Delay Systems And Its Applications

Positive systems are dynamic systems which include in almost all fields, to name a few, engineering, ecology, network communication and so on. A common property of the positive dynamic systems is that its state and output are nonnegative for any nonnegative initial state and nonnegative input. On the other hand, the reaction of practical systems to exogenous signals is never instantaneous and always infected by certain time-delays. The presence of delay may result in instability or bad performances of the closed-loop system. Therefore, studying the analysis and synthesis problem of positive systems with time-delays play an essential role in the cybernetics.In this thesis, the problems of l1-gain performance analysis and its ap-plication in the interconnected system are discussed for discrete-time positive linear time-delay systems. Necessary and sufficient conditions are presented in the linear programming (LP) framework which guarantee that the discrete-time positive linear time-delay system is internally stable and has the prescribed l1-gain performance. Then the above results are applied to the stability anal-ysis of interconnected positive linear time-delay systems, and it turns out that an interconnected discrete-time positive linear time-delay system is stable if and only if there exists a set of weighting vectors that render the l1-gain of each positive subsystem less than unity. The thesis contains the following three chapters.In chapter1, the background of positive linear time-delay systems is in-troduced, and some important achievements on such systems are traced. The advantage of l1-orm in the performance characterization of positive systems is also pointed out. Moreover, the notations used in this thesis are listed.In chapter2, the problem of l1-ain performance with weighting is in-vestigated for discrete-time positive linear time-delay systems. Two necessary and sufficient conditions are established to guarantee that the system is in-ternally stable and its l1-gain with weighting is less than γ. In addition, the corresponding results about the standard l1-gain are obtained under the degradation condition. Furthermore, a numerical example is given to verify the result.Chapter3is devoted to the stability analysis of discrete-time intercon-nected positive linear time-delay systems and a class of uncertain discrete-time interconnected positive linear time-delay systems where the uncertainty is defined by the l1-gain performance. Necessary and sufficient conditions are provided which guarantee that the interconnected system is admissible and asymptotically stable. At last, numerical examples are also given to illustrate the effectiveness of the proposed results.