Eigenvalue assignment procedure for delayed interconnected system by expansion

Abstract

A procedure for constructing the decentralized and the hierarchical control that achieves precise eigenvalue assignment for dynamically interconnected system is presented. Delays in signals transmitted through the interconnecting channels is teen into consideration. It will be shown that the problem can be solved by reformulating it also as an assignment problem for an independent of delay augmented system. Emphasis is made on the manner in which eigenvectors interact to achieve desired spectrum by coordinating subsystem-interconnections solutions. I. IiTTRCDUCTIONLarge scale systems have been under consideration for a considerable time resulting in many approches for synthesizing decentralized and/or hierarchical controllers.for eigenvalue assignment for large scale complex systems that may be partitioned into a number of interconnected subsystems, Architecture of the system together with its interconnections without any possible delay was previously studied from stabilization point of view .Here,due to physical characteristics or structural properties and constraints the multilevel concept will be adopted to design both the decentralized and the higher hierarchical level controllers. LIoreover,delay in interconnections is introduced to resemble a realistic and physical constraints. delays, the system is expanded in a larger state space where delay disappears. The original problem is reformulated as a n eigenvalue assignment problem for the augmented system. Throughout the procedure, eigenvectors plays an important role both to reduce the conplexity of the analysis,and to obtain better information about the nodal interaction among subsystems to achieve the desired spectrum.Interestingly, controllers' design is executed for the augmented system,but their inplementation is performed for the original system.Here we outline an effective procedure 1 To by-pass difficulties associated with Consider a large scale system which is assumed to be composed of S interconnected subeystems,each is described as; yr(k> = Cr xr(k) (1) where xr(k)€ R is the state,s(k)€ R% is the input ,and yr(k)€ Rpr is the output of the r-th subsystem. The matrices are of appropriate dimensions. are considered;each is described by; zr(k+l) = b$ zr(k) + Lrq Yq(k) n In this nodel S interconnections S q=l S ( 2 1 wr(k> = Pr zr(k) + Qrq yq(k) (r=l, ..., S ) . Here zr(k)'? Xar is the state, and wr(k) f Rq. is the output of the r-th subsystem. systems is that its subsystems are sparse, leading to emergence of delay in signals transmitted through the interconnecting channe Is, Let q = l A conmon feature in large scale complex s ( k > = vr(k) + Wr(k-l> (3 1where vr(k> is the external input, Koreover, let vr(k) be generated through a two level control structure;instantaneously decentralized followed by another higher hierarchical delayed,namely; S vr(k> = k, x,&-) + krq xq(k-l) (4 1 q=l where the feed-back gains kr and k are constants. rq 95