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Modern Control System Theory and Design, 2nd Edition by Stanley M. Shinners

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6.23.  ILLUSTRATIVE PROBLEMS AND SOLUTIONS

This section provides a set of illustrative problems and their solutions to supplement the material presented in Chapter 6.

I6.1.

A control system can be represented by the following companion matrix:

Image

(a)  Determine the characteristic equation.

(b)  Determine the location of the route of the characteristic equation.

(c)  Is the control system operating in a stable region?

SOLUTION: (a)

Image

Using minors along the first column, we obtain the following:

Image

Therefore, the characteristic equation is given by the following:

s(s2 + 2s + 2) = 0

(b)  Since

s(s2 + 2s + 2) = s(s + 1 + j)(s + 1 − j) = 0

the roots of the characteristic equation are given by the following:

s1 = 0,

s2 = −1 − j,

s3 − 1 + j.

(c)  Because the complex-conjugate roots have negative real parts, and the third root is located at the origin, none of the roots are located in the right half-plane and the system is operating in a stable region.

I6.2. A second-order control system can be represented by the following companion matrix:

Image

(a)  Determine the characteristic equation of this control ...

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