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 *x*_{1}(*t*) and *x*_{2}(*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**^{T}**L**^{T}]

Therefore, ...

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