Structural Analysis And H_∞ Control For Some Classes Of Positive Systems

Positive systems are a new branch in systems theory. During recent years, due to the widespread applications in science and technology, positive systems have been of great interest to many researchers. The nonnegativity restriction on variables results in the fact that positive systems are defined on cones and not on linear space, which implies that many results for general systems are not available for positive systems. Therefore, positive systems which are now the focus of attention have become a new and challenging research field. A great number of novel and straightforward theoretical results have been reported by capturing the nonnegativity characteristic of variables.Based on the previous results on positive systems, this dissertation is concerned with bounded real lemmas for positive descriptor systems, positivity and stability for descrip-tor systems with time delays, state feedback H∞ control for discrete positive systems, dynamic output feedback H∞ control problems for positive systems with time delays. The main contributions of this dissertation are summarized as follows:(1) The bounded real lemmas for positive descriptor systems are investigated. Two novel bounded real lemmas in the form of strictly linear matrix inequalities(LMIs) are developed for positive descriptor systems in continuous case as well as discrete case. It is pointed out that if continuous and discrete positive descriptor systems are admissible, the exact values of H∞ norms are given by the maximal singular values of transfer function matrices at ω=0 and θ=0, respectively. At the same time, a model reduction method is introduced. It is shown that the reduced systems are also positive, asymptotically stable and preserve the H∞ norms of the original systems. Such method can reduce the dimen-sion of the system quickly and effectively and therefore simplify analysis provided that a positive descriptor system satisfies the given condition.(2) Positivity of continuous descriptor systems with time delays is analyzed. The definition of continuous time-delay positive descriptor systems is given. Then a neces-sary and sufficient condition to check positivity is established based on which a criterion for positivity of impulse-free descriptor systems is obtained. Moreover, considering a descriptor system with two assumptions, a new time-delay system is established. It is shown that positivity of the new system is equivalent to that of the original system. Then a necessary and sufficient condition is given to check the positivity of the new system.(3) Positivity and stability for discrete descriptor systems with time delays are inves-tigated. Firstly, an explicit solution to the state equation is derived using z-transform and inverse z-transform. Secondly, based on the solution, a necessary and sufficient condition to check positivity is established. Furthermore, a new notion named strict positivity is proposed, and then a necessary and sufficient condition is given to guarantee such pos-itivity. Thirdly, asymptotic and exponential stability are studied by constructing linear Lyapunov functions. A necessary and sufficient condition for asymptotic stability, a suffi-cient condition and a necessary condition for exponential stability are established in terms of linear programming, respectively. The proposed results show that the magnitude of de-lay and index of matrix pair have nothing to do with asymptotic stability but have impact on exponential stability.(4) State feedback H∞ control problem for discrete positive systems is studied. First-ly, an alternative proof of the bounded real lemma is given by using the separating hyper-plane theorem and the Perron-Frobenius theorem. Secondly, H∞ control problems using positive state feedback, negative state feedback and state feedback without sign restriction are investigated, respectively. Necessary and sufficient conditions are given to ensure the existence of controllers. It is pointed out that there is no solution to H∞ control problem using positive state feedback if the open-loop system is unstable or its H∞ norm is greater than or equal to the prescribed H∞ performance.(5) Dynamic output H∞ control problems for positive systems with time delays are considered. Two novel bounded real lemmas are first established for continuous and dis-crete positive systems with time delays in state and output equations, which provide nec-essary and sufficient conditions to check H∞ performance. The new lemmas imply that H∞ norms are independent of the magnitudes of delays but dependent on systems’ma-trices, and show that the numerical complexity of linear matrix inequalities to be solved is obviously lower than that of LMIs for general systems. Then, H∞ control problems using dynamic output feedback are investigated. Necessary and sufficient conditions to-gether with matrix equalities which can be solved by cone complementary linearization techniques, are proposed to ensure the existence of controllers. In addition, the obtained results are extended to discrete interval uncertain positive systems with time delays. A novel bounded real lemma and a necessary and sufficient condition for H∞ control prob- lem are established, respectively.