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94 Introduction to Control Theory

Linear Control Design Problem: Consider a LTI system. Design a feedback
controller that operates on the output y(t) without differentiation, and
generates an input u(t) so that the system goes from any initial state x(0) to
a specified desired final state x d in some finite time. This problem has 2 parts
to it:

1. Controllability: Can the problem be solved if y=x?
2. Observability: If so, how do I get x from y?

In the next section we introduce these concepts and find tests that will allow
us to answer the questions posed. Given is a MIMO LTI system

(2.11.1)

where , , and A, B, C, D are of the appropriate
dimensions.

DEFINITION 2.11–1 A state x 0 is controllable if there exists an input u(t),
for 0 t t 1 such that x(t 1 )=0, for some finite t 1 . The system itself is said to be
controllable if all are controllable.

Note that the input u(t) is not a feedback function of the state x(t). Let us
recall the solution of the differential equation given by

and evaluate it at t 1 assuming that x 0 is controllable so that x(t 1 )=0 to
obtain

(2.11.2)

Therefore, if x 0 is controllable, it has to satisfy the last equation. This however
is not an easy test. Fortunately, we have a much simpler test given in the
following theorem.

THEOREM 2.11–1: A necessary and sufficient condition for the complete
controllability of the LTI systems is that, The n×nm matrix

Copyright © 2004 by Marcel Dekker, Inc.