This section provides a set of illustrative problems and their solutions to supplement the material presented in Chapter 6.
A control system can be represented by the following companion matrix:
(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?
Using minors along the first column, we obtain the following:
Therefore, the characteristic equation is given by the following:
s(s2 + 2s + 2) = 0
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:
(a) Determine the characteristic equation of this control ...