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96 Introduction to Control Theory

Then by Ackermann’s formula

(1)

Next, we introduce the concept of observability and find an observability
test.

DEFINITION 2.11–2 The state x(0)=x 0 is said to be observable if knowledge
of u(t), y(t); 0 t 1 is enough to uniquely determine x 0 . Note that once x(0) is
known, so is x(t) for t 0. The system is said to be completely observable if
every initial state is observable.

This problem is a dual problem to the controllability problem as will
become obvious in the following.

THEOREM 2.11–3: A necessary and sufficient condition for the complete
observability of the LTI systems is that The np×n matrix

(1)

is full rank, i.e.

EXAMPLE 2.11–2: Observer-Controller Design
Consider the system

Copyright © 2004 by Marcel Dekker, Inc.