By Dragoslav D. Siljak

Starting with a graph-theoretic framework for structural modeling of complicated structures, the textual content provides effects relating to strong stabilization through decentralized nation suggestions. next chapters discover optimization, output suggestions, the manipulative strength of graphs, overlapping decompositions and the underlying inclusion precept, and reliability layout. Notes and references complement every one bankruptcy, and the appendix presents a beneficial diversity of effective graph algorithms.

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**Sample text**

77), which is characterized by the triple ( A , B N ,CN),the system is not decomposed into subsystems and the system matrix A is considered as a whole. It is only feedback (static or dynamic) that is decentralized. In modeling of complex systems this is not enough, and we require total decentralization of both the plant and the control. In this way, we are able to capture the effects of the essential uncertainty residing in the interconnection structure on the behavior of the overall system. Furthermore, decompositions and decentralizations, when combined together, can bring about a considerable increase in speed of computation (on- and off-line), as well as reliability of operation in control of complex systems (see Chapter 9).

Then, with M we can associate a parameter space R” such that every data point p E R” defines a matrix M E h;r(p),which is obtained by replacing the arbitrary entries %ij of M by the corresponding elements of p = ( P I , p2, . . , P , ) ~ In . a physical problem, it is important to know that if a matrix M has a certain rank at a nominal parameter vector p ” , it has the same rank at a vector p close to p ” , which corresponds to small deviations of the parameters from their nominal values. Most often, it turns out that the rank holds true for all p E R” except at the points p that lie on an algebraic surface in R” and which are, therefore, atypical.

THEOREM. A digraph D = (U U X U Y, E) is input (output) reachable if and only if its condensation D* = (U* U X' U Y*,E*) is input (output) reachable. Proof. We prove only the input reachability part of the theorem, because the output reachability part is its directional dual. If D* is input reachable then X* is the reachable set of U*, and there is a path to each point of X* from a point of U*. Since Df is the condensation of D,, it follows from reachability of components that there is a path to each point of X from a point of U.