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:

**(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)**

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

Therefore, the characteristic equation is given by the following:

*s*(*s*^{2} + 2*s* + 2) = 0

**(b)** Since

*s*(*s*^{2} + 2*s* + 2) = *s*(*s* + 1 + *j*)(*s* + 1 − *j*) = 0

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

s_{1}= 0,

s_{2}= −1 −j,

s_{3}− 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 ...

Start Free Trial

No credit card required