Exercise 2.5

Exercise 2.5#

Analysis ∙ Linear dynamical systems Analysis ∙ Feedback control

We are trying to design the controller for the system:

\[ G(s) = \frac{5}{s^3 + 8s^2 + 20s + 25} \]

Assume we go for a family of PI controllers described by the following transfer function, where \(K > 0\) is a parameter to be designed.

\[ F(s) = \frac{K(s+1)}{s} \]

  1. Compute the closed-loop transfer function \(G_c(s)\).

  2. Apply the Routh-Hurwitz criterion to find for what values of \(K\) the system is stable.

Hint

Question 2 can be solved using the Routh-Hurwitz criterion.

  1. The characteristic polynomial \(\Phi(s) = a^0 s^n + b_0 s^{n-1} + a_1 s^{n-2} + b_1 s^{n-3} + ...\) is the denominator of the system we are assessing the stability of. In this case, it is the denominator of \(G_c(s)\).

  2. Note that the coefficients \(a_i\) and \(b_i\) are defined starting from the highest degree, \(s^n\), and they are “alternating”. \(b_n\) might be zero if \(n\) is an even number.