Some Structural Properties Of Singular Systems And Observers Problem

A considerable amount of research concerning singular systems has been reported because of their extensive applications. It is well known that a singular system has complicated structures and contains not only finite poles but also infinite poles which may generate undesired impulsive behaviors. The finite fixed modes of system is poles that can not be changed by output feedbacks. The impulsive fixed modes of system are the least infinite poles under output feedbacks. Basing on paper [1], In this paper, we analyse some structural properties of LTI systems.The observer theory plays an important part in control theory all the time. In this paper, we mainly discuss how to design observers and the rank of observers. Basing on the theory of dynamic observers which was proposed by Goodwin and Middleton in 1989 and the dynamic observers for continuous-time systems, we give an algorithm to design dynamic observers for singular systems. We also analyze some properties of them. It is a matter of fact common knowledge that the observer design is the dual of the feedback design. In this paper, we shift the lower order observers of continuous-time to discrete-time ones. We also discuss the choosen of gain matrix that reconstruct the state of system.The main conclusions in the paper are:(1) we continue the research in paper [l],derive an equality on the rank of variable matrices. Basing on the equality, we show the degree of output feedback variable polynomial of LTI singular systems. Then we generalize the theorem in [3] which is about the pole assign-ability of LTI singular systems under output feedback, we generalize the matrix K to almost any K which is compatible. At last we give an algorithm to characterize the algebraic multiplicity of the IFM in terms of only the system matrices of single input singular systems.(2) We discuss a fixed parameter under feedback and coordinate transformation for discrete-time systems. It is shifted from the continuous-time case. In view of the duality between state feedback controllers and state observers, we give another way of choosing gain matrix to reconstruct the state in a special case.(3) For LTI singular systems, we give a kind of dynamic system which differ from classical observers in that it contains dynamics in its state. We can see that, if the dynamic observer is designed as proposed, the separation property holds. Then wederive necessary and sufficient conditions for making a singular system both regular and impulse-free by decentralized output feedback control laws.