Research On Dynamics Of Positive Systems

Addressed in this dissertation are the key dynamics of positive systems. Several relevantproblems attracting more and more attention in this field are discussed in detail, anda series of well-established, systematical, and important results are obtained. As a whole,this dissertation is consisted of the following eight parts.Part 1 introduces the backgrounds and research status of positive systems. It alsoprovides the readers with some preliminaries and important definitions and lemmas thatare frequently used in the remaining chapters.Part 2 treats the positive systems with multiple bounded delays, including four situations:continuous-and discrete-time systems with constant or time-varying delays. Stabilityanalysis and controller designing are considered. The main results lie in the followingfour aspects. 1. The necessary and sufficient stability criteria are established. 2. Thenecessary and sufficient conditions are also provided, which determines whether or not acontroller exists such that the corresponding closed-loop system is positive and asymptoticallystable. Furthermore, if such a controller exists, the method how to compute it isexplicitly provided. All the results are expressed in terms of linear programming and linearmatrix inequality. It is interesting to see that the results for systems with constant andtime-varying delays are same in form, no matter the systems are continuous- or discretetime.3. It reveals some important properties of positive systems with constant delays: ifa system is stable, then its trajectory should monotonically decrease provided that the initialconditions are properly chosen. 4. It also reveals a relationship between the solutionsto systems with identical systems matrices but with different kinds of delays: solution tosystem with time-varying delays is not greater than that to system with constant delays,providing some conditions are satisfied. One may draw a conclusion that the positivesystem is "elegant" based on the obtained results. TechnologicaUy, this chapter employsthe copositive Lyapunov functional in a flexible manner, creates some new approaches toanalyze the system stability, and makes a breakthrough in the aspect of studying stability.Part 3 is devoted to investigating the constrained control problem of positive systemswith delays(including constant delays and time-varying delays): determining two boundsand designing a controller such that the corresponding closed-loop system is positive, and its states and control input can be bounded by the two preset bounds. Therefore, in thereferences, this problem is also called bounded control one. For the general dynamicsystems, the constrained control issue is generally not a main concern and therefore themethod to treat this problem is lacking, which make our task difficult. This chapter establishessome necessary and sufficient conditions for the existence of such controllers. Bymeans of contradiction and calculus, this chapter proposes some new methods to treat thebounded control problem, resulting a set of brief and easily applied conditions, changingthe fact that no any necessary and sufficient conditions can be established before for thebounded control problem.Part 4 is concerned with the stability problem of discrete-time positive systems withunbounded delays. The main contribution lies in the following aspects: the necessary andsufficient stability condition is established, and the relationship among positive systemswith the identical system matrices but different unbounded delays is revealed. It is hopefulthat these results shed light on the stability issue of continuous-time positive systemswith unbounded delays.Part 5 performs stability analysis for switched positive systems. During the pastyears, the common linear copositive Lyapunov approach was the most popular methodto treat this problem. Obviously, this method is too conservative. In order to overcomethis shortage, the chapter proposes a new approach: switched linear copositive Lyapunovmethod. Compared with the common linear copositive Lyapunov method, ours approachis much less conservative and thus can be probably widely used in the future.Part 6 studies the stability problem of 2-D positive systems with multiple delays anddescribed by the Roesser model. Much attention is now paid to this class systems. A necessaryand sufficient stability condition is established. It can be, after simple mathematicmanipulation, applied to several delayed 2-D positive systems described by other models.In this chapter, a constructive method is used to prove the main theorem. These results,together with these method, hopefully work in the further study of n-D positive systems.Part 7 is concerned with decentralized control of large-scale systems under the positivenessconstraint, which means that a decentralized controller should be such that theclosed-loop systems are positive. The necessary and sufficient conditions determine theexistence of such controllers are established. These results solve one of the most difficultproblem in the field of decentralized control, under certain conditions. Also, this part shows that positive systems possess great potential in applications.Part 8 summarizes the main results obtained in this dissertation, and points out thefuture works that have been the author's concerns.