Structure of expansion-contraction matrices in the inclusion principle for dynamic systems. (English) Zbl 0979.93006

Bakule, Lubomír; Rodellar, José; Rossell, Josep M.

SIAM J. Matrix Anal. Appl. 21, No. 4, 1136-1155 (2000).

In the analysis and control of complex and large-scale, time-invariant linear dynamic systems, the “inclusion principle” proposed in the early eighties [see, e.g. D. D. Siljak, “Decentralized control of complex systems”, Academic Press, New York (1990; Zbl 0728.93004), and references therein found] has become a useful tool since it enables to establish a mathematical framework in which two dynamic systems with different dimensions may have an equivalent behaviour. This means that a “big” system \({\mathbf S}(1):\dot{\widetilde x}= \widetilde A\widetilde x+ \widetilde B\widetilde u\); \(\widetilde y=\widetilde B\widetilde x\), can be built from a “small” one \({\mathbf S}(2):\dot x= Ax+ Bu\); \(y= Cx\), through an expansion process, in such a way, that \({\mathbf S}(1)\) contains the essential information about the behaviour of \({\mathbf S}(2)\), and on the other hand, this information can be extracted from the “big” system by a contraction process.
Here, the authors characterize (main result, Theorem 3.9) the block structure of the complementary matrices, involved in the expansion-contraction process as well as the conditions to be satisfied by blocks to guarantee the inclusion principle. In this respect, the work under review represents an improvement of previous result given by M. Ikeda and D. D. Siljak [“Overlapping decentralized control with input, state and output inclusion”, Control Theory Adv. Tech. 2, 155-172 (1986)] where the necessary and sufficient conditions on the complementary matrices are difficult to verify in practice.

Reviewer: P.Gonzalez-Vera

MSC:

93A15	Large-scale systems
93A14	Decentralized systems
93B11	System structure simplification
93B17	Transformations

Keywords:

inclusion principle; large scale systems; decentralized control; decomposition; overlapping; block structure; complementary matrices; expansion-contraction process