last modified Jul 2, 1996

Solving the Control Problem

Here we give an example of solving a problem using a strategy.

We will repeatedly call the NCProcess1 command and from time to time the Decompose operation will be approximated with a little human intervention.

A basic problem in systems engineering is to make a given system dissipative by designing a feedback law. We now give a demonstration of how one discovers the algebraic part of the solution to this problem.

Problem statement

Let

and

be defined as follows.

The math problem we address is:

(HGRAIL)

Let , , , , be matrices of compatible size which are given. Solve , , and for a, b, c and for , , and . When can they be solved? If these equations can be solved, find formulas for the solution.

We shall make the strong assumption that each

is invertible. While this assumption turns out to be valid, making it at this point is cheating. Ironically, we recommend strongly that the user make heavy invertibility assumptions at the outset of a session. Later, after the main ideas have been discovered, the user can selectively relax them and thereby obtain more general results.