The Transitivity In Simultaneous Stabilization And Reliability Of Systems

Throughout this paper, C denotes the complex plane and H will always denote a complex separable infinite dimensional Hilbert space. Let B(H) denote the algebra of all bounded linear operators on H. L, J. S denote the algebra of all linear systems, time-invariant linear systems and stable systems, respectively.The simultaneous stabilization is concerned with the problem of designing a sin-gle controller that stabilizes a finite family of linear systems, which has been used widely in the field of robust control. The problem was first introduced by Saeks and Murray, Vidyasagar and Viswanadham. In 1982. The link between strong stabiliza-tion(stabilization by a stable controller) and simultaneous stabilization of two plants was discovered, and used, by Saeks and Murray, the simultaneous stabilization ques-tion for two systems can be reformulated into one of strong stabilization. A system is stabilizable by a stable controller if and only if it has an even number of real unsta-ble zeros between each pair of real unstable poles! Such systems are said to have the parity interlacing property. This solves the problem of simultaneous stabilization of two systems. In 1982, this result was further extended to a multi-input, multi-output setting by Vidyasagar and Viswanadham.The situation is different for three systems. About thirty years ago, many results have been obtained for simultaneous stabilization in various frameworks. In 1994, Blondel et al. proved that simultaneously stabilization of k systems is equivalent to bistable stabilization (stabilization with a stable and inverse-stable controller) of as-sociatedκ- 2 systems. Within the framework of nest algebras, Feintuch and Yufeng give some necessary and sufficient conditions for (strongly) simultaneous stabilization, respectively. The presently available results are in the form of necessary conditions sufficient conditions, or untractable necessary and sufficient conditions. Despite all these efforts, at present no tractable necessary and sufficient conditions exist for si-multaneous stabilization except for the case of two systems. In 1993. Blondel and Gevers proved that simultaneous stabilization of three (or more) plants represents an undecidable problem. So, in the paper, we study a very interesting related problem-the transmission problem in simultaneous stabilization within the framework of nest algebras.The Transitivity in Simultaneous Stabilization:Given linear time-varying systems L0,L1 and L2, if Lo and L1 can be simultaneously stabilized, L1 and L2 can be simultaneously stabilized, does there exist a single controller C that stabilizes Lo and L2?If the answer is Yes, it is natural to ask more:The Strong Transitivity in Simultaneous Stabilization:If Lo and L1 can be simultaneously stabilized, L1 and L2 can be simultaneously stabilized, does there exist a single controller C that stabilizes L0,L1和L2?The first part of this thesis mainly deals with the transmission problem and strong transmission problem in simultaneous stabilization for time-varying linear systems, we establish criteria for the transitivity in simultaneous stabilization. Furthermore we obtain the following result. be a strong right representation of Li for i= 0,1,2. Assume that Co stabilizes L0 and L1, C2 stabilizes L1 and L2. If M2÷C0N2= 0, then Lo and L2 can be simultaneously stabilized, be a strong right representation of Li for i= 0,1,2. Assume that Co stabilizes Lo and L1, C2 stabilizes L1 and L2,where C0,C2∈J. If then L0 and L2 can be simultaneously stabilized. We now turn to the strong transmission problem of strongly simultaneous stabi-lization. We give the following result.Theorem 0.3 Let[(?)]be a strong right representation of Li and Co∈S such that Mi+C0Ni= I for i= 0,1. Assume that L1, L2 are strongly simultaneously stabilized by C2∈S. If N0= N1, then L0, L1 and L2 are strongly simultaneously stabilizable.Based on the results above, we consider the strong transmission problem of simul-taneous stabilization, that is, if L0 and L1 can be simultaneously stabilized, L1 and L2 can be simultaneously stabilized, does there exist a single controller C that stabilizes L0,L1和L2? In fact, we give a necessary and sufficient condition for certain linear systems to satisfy a strengthened version of transitivity.Theorem 0.4 Let L1∈S. Assume that L0 and L1 are simultaneously stabilizable, and L1,L2 are simultaneously stabilizable. Then L0, L1and L2 are simultaneously stabilizable if and only if there exists T E S such that N0X2+M0Y2+(M0L2 - N0)T is invertible in S, where Yi,Xi, Ni, Mi, are associated with Li as those in equation (1.1) for i= 0,1,2.The strongly transmission problem of the simultaneous stabilization of three time-varying linear systems is addressed. We establish a criterion for the strong transitivity in simultaneous stabilization.Theorem 0.5 Let L0,Li,L2∈L and [(?)] be a strong right representation of Li for i = 0,2. Assume that Lo, L1 are simultaneously stabilizable, and L1, L2 are simultane-ously stabilizable. If there existsε> 0 such that for every x,y∈l2, and n= 0,1,…, then L0, L1 and L2 are simultaneously stabilizable.In the view of transitivity, our research focuses on the design of simultaneously sta-bilizing controllers, which simultaneously stabilizes L0,L1 and L2, we get the following result. Theorem 0.6 Suppse that C0 stabilizes L0,L1,and C2 stabilizes L1,L2.Define A= N1X0+M1Y0,B=N1M0-M1N0 and T=(M0+C0N0)-1(C0Y0-X0) where Mi,Ni,Yi,Xi,Ni,Mi,Xi,Yi are associated with Li as those in equation (1,1) for i= 0,1,2.Then the controller C(Q) for L0,L1 and L2 can be characterized as where R(Q)=(I+QB)-1(T-QA)with Q∈S is such that and‖(M1+C0N1)-1(C0-C2)(I+L2C2)-1(L2M1-N1)-Q(N2M1-M2N1)‖<1.(8)In the view of transitivity,we consider the simultaneous robust stabilization of time-varying linear systems,that is,if Co stabilizes L0 and L1,C2 stabilizes L1 and L2, does there exist a single controller C,which simultaneously robust stabiljzes L0 and L2? Based on the Theorem 0.1,Theorem 0.7 Suppose that C0 admits coprime factorizations C0=M(?)NC0,De-fine Then for all [△N,△M] such that(i)△N,△M∈S,(ii)‖[△N,△M]‖<1/r,C simultaneously stabilizes[-(N0+△N),M0+△M] and [-(N2+△N),M2+△M, where[-Ni,Mi] is a strong left representation of Li(i=0,2).The other part of our research focuses on the reliable stabilization problem for discrete time-varying linear systems using a two-controller configuration in the frame-work of nest algebras. We establish some suggicient conditions for the existence of solutions of the reliable stabilization problem and obtain the parametrization of the reliable controller pairs for strongly stabilizable linear systems. Theorem 0.8 Suppose L is strongly stabilizable. Then there exist two controllers C1,C2 such that C1 and C2 together solve the reliable stabilization problem.Theorem 0.9 Let L∈￡L can be strongly stabilized byC1∈S.Suppose that L admits a doubly coprime factorization of the form L=NM-1N.If S(L,C1,C2)::{(C1,C2):C2=(I-QN)-1(C1+QM),Q∈S,(I+C1N-QNC1N)-1∈S｝, then each element of S(L,C1,C2) is a solution of reliable stabilization problem.