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### 8.14.  ILLUSTRATIVE PROBLEMS AND SOLUTIONS

This section provides a set of illustrative problems and their solutions to supplement the material presented in Chapter 8.

I8.1. The state and output equations of a second-order control system are given by the following:

where x1(t) and x2(t) represent the system states, c(t) is the system’s output, and u(t) represents its input.

(a)  Determine whether the system is controllable.

(b)  Determine whether the system is observable.

SOLUTION: (a) From Eq. (8.65) controllability can be determined for this second-order system from:

D = [B PB].

The phase variable canonical form of the state and output equations can be written as:

Therefore the companion matrix, P is given by

and the input vector, B, is given by

and the output matrix is given by:

L = [1   0]

Therefore, the matrix D is given by:

and the system is controllable.

(b)  Observability can be determined for this second-order control system from Eq. (8.75) where

U = [LT    PTLT]

Therefore, ...

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