Summary:

This video presents the ‘root locus’ of the first order model $G(s)=1∕(s-0.4)$ of a (unstable) bioreactor, when controlled with a proportional-integral (PI) regulator of type $K(s)=K_c\cdot (1+K_i∕s)$, i.e. $K_c\cdot (s+K_i)∕s$. We check and compare, in simulation, with the conclusions on how the roots of $1+G(s)K(s)=0$ change when $K_c$ varies (in the first part of the video) and when $K_i$ varies in the final part of the video.

Actually, the closed loop has order 2 and, therefore, it is not necessary to resort to the root locus methodology since said roots can be analytically obtained with the basic formula of the roots of a quadratic equation; In fact, that is what had been done in the video [bioPI1EN], which you should have watched before this one. The use of the root locus here is simply to be used as a motivating example to make the connection with previous developments; the usefulness of the methodology is better in higher-order systems, where the closed-loop poles cannot be (or are not easily) calculated with a simple analytical expression.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.