Question 7.4: A plant has a nominal model given by Go(s) = 2/(s + 1)(s + 2...
A plant has a nominal model given by
G_{o}(s)=\frac{2}{(s + 1)(s + 2)} (7.3.9)
Synthesize a PID controller which yields a closed loop with dynamics dominated
by the factor s² + 4s + 9.
The controller is synthesized by solving the pole assignment equation, with the following quantities
A_{cl}(s)=(s² + 4s + 9)(s + 4)²; B_{o}(s) = 2; A_{o}(s)= s² + 3s + 2 (7.3.10)
where the factor (s + 4)² has been added to ensure that the pole assignment equation has a solution. Note that this factor generates modes which are faster than those originated in s² + 4s + 9.
Solving the pole assignment equation gives
C(s)=\frac{P(s)}{s\bar{L}\left(s\right) }=\frac{14s^{2} + 59s + 72}{s(s + 9)} (7.3.11)
We observe using Lemma 7.2 on page 185 that C(s) is a PID controller with
K_{p}=5.97; K_{I}=8; K_{D}=0.93; \tau _{D}=0.11 (7.3.12)
An important observation is that the solution to this problem has the structure of a PID controller for the given model G_{o}(s). For a higher order G_{o}(s), the resulting controller will not be, in general, a PID controller.