Modern Control System Theory and Design, 2nd Edition

8.5.  OBSERVABILITY

In order to introduce the concept of observability [2−4], let us again consider the simple open-loop system illustrated in Figure 8.13. A system is completely observable if, given the control and the output over the interval t₀ t T, one can determine the initial state x(t₀). Qualitatively, the system G is observable if every state variable of G affects some of the outputs in c. It is very often desirable to determine information regarding the system states based on measurements of c. However, if we cannot observe one or more of the states from the measurements of c, then the system is not completely observable. We had assumed the systems were observable in our discussion of pole placement using linear-state variable feedback in Sections 8.2 and 8.3.

A.  Observability by Inspection

An an example of a system which is not completely observable, let us consider the signal-flow diagram illustrated in Figure 8.15. This system contains four states, only two of which are observable. The states x₃(t) and x₄(t) are not connected to the output c(t) in any manner. Therefore, x₃(t) and x₄(t) are not observable and the system is not completely observable.

B.  The Observability Matrix

Let us now consider this problem more precisely and establish a mathematical criterion for ...