A plant has a nominal model given by Go(s) = 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.

Question

A plant has a nominal model given by

[latex]G_{o}(s)=\frac{2}{(s + 1)(s + 2)}[/latex] (7.3.9)

Synthesize a PID controller which yields a closed loop with dynamics dominated
by the factor s² + 4s + 9.

Accepted Answer

The controller is synthesized by solving the pole assignment equation, with the following quantities

[latex]A_{cl}(s)=(s² + 4s + 9)(s + 4)²; B_{o}(s) = 2; A_{o}(s)= s² + 3s + 2[/latex] (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

[latex]C(s)=\frac{P(s)}{s\bar{L}\left(s ight) }=\frac{14s^{2} + 59s + 72}{s(s + 9)}[/latex] (7.3.11)

We observe using Lemma 7.2 on page 185 that C(s) is a PID controller with

[latex]K_{p}=5.97; K_{I}=8; K_{D}=0.93; au _{D}=0.11[/latex] (7.3.12)

An important observation is that the solution to this problem has the structure of a PID controller for the given model [latex]G_{o}(s)[/latex]. For a higher order [latex]G_{o}(s)[/latex], the resulting controller will not be, in general, a PID controller.

Question 7.4: A plant has a nominal model given by Go(s) = 2/(s + 1)(s + 2...

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