Chains

Abstract

Ring systems can be described as closed chains. As discussed in the previous chapter, closed chains are patterned and consequently control problems are greatly simplified. Of much wider interest, however, are open chains of identical subsystems. Such systems are modeled in state space form with Toeplitz matrices. Numerous applications of open chain systems exist, including vehicle convoys, mass transit lines, serpentine manipulators, cross-directional control of continuous processes such as papermaking, and lumped approximations of p.d.e.s.

Despite the simple structure of open chains, proving the existence of Toeplitz controllers to solve control problems for general Toeplitz systems is difficult. Certainly Toeplitz matrices do not form a patterned class with a single base pattern. An interesting aside is that if a long chain can be reasonably approximated as having infinite length, then certain control problems are actually simplified. For example, Brockett and Willems [5] showed that the optimal control of infinite Toeplitz systems has an infinite Toeplitz form. The optimal control of finite-dimensional Toeplitz systems, however, can easily be shown by counterexample to not generally be Toeplitz. There does exist a method for arriving at the optimal control of symmetric Toeplitz systems through a conversion to a circulant form, which is illustrated in [74]. A simple example of this technique in the context of pole placement is included in this chapter to show how a non-patterned symmetric Toeplitz system can be converted to a patterned circulant system for the purposes of computing a controller (although the end result will not be Toeplitz).

Although Toeplitz systems do not form a patterned class, the special case of upper (or lower) triangular Toeplitz matrices is patterned, which corresponds to open chains with the property that interconnections between subsystems are only present in one direction. We call these uni-directional chains, and we examine this patterned class in this chapter.