Modern Control System Theory and Design, 2nd Edition

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 = [L^T    P^TL^T]

Therefore, ...