Unstable 1st order system (bioreactor): 2 DoF proportional control, disturbances

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 23:25

*Enlace a Spanish version

Summary:

This video discusses the concepts of proportional control with ‘2 degrees of freedom’ and analyzes the response of such proportional control to step disturbances at the process input. The study is framed within an example of an unstable first-order bioreactor, a case study that began in the video [bio1modEN] and whose ‘basic’ proportional control was addressed in the video [bio1cPEN].

As seen in the referred video, the proportional control $u=K\cdot error$ has a position error in steady state when confronted to step setpoint changes. Here, the conclusions of the video [bio1cPEN] are quickly reviewed and said position error is theoretically calculated. Then, it is ‘intuitively’ argued that one way to solve the problem (in setpoint tracking problems) is to make the control software believe, through a ‘trick’, that the reference is different from the one entered by the operator, with a formula $u=K_c\cdot (b\cdot r-y)$, well, in this case $y=x$ is the biomass, and $b$ is an adjustable parameter to ‘deceive’ the controller into thinking that the setpoint is different to the true one. After the intuitive argument, the theoretical development is detailed to understand that, indeed, this would solve the position error problem.

The basis of the theory is that the naive control $u=K\cdot e$ is actually ‘wrongly implemented’, because the desired operating point should be ‘zero’ in linear control: the correct form should be $\mathrm{\Delta }u=-K_c\cdot \mathrm{\Delta }y$ in ‘incremental’ coordinates, so the ‘correctly formulated’ proportional control should be $u=\mathrm{\Delta }u+u_{eq}=K_c\cdot (r-y)+u_{eq}$, with the latter term $u_{eq}$ being omitted in many of the ‘naïve’ implementations of proportional control, just to be later added in refinement stages, but named as ‘input offset’, ‘input trimming’ or ‘control with 2 degrees of freedom’. This solves the stationary error when tracking constant references, but does not solve the deviations due to disturbances. The goal of the video is to explain all this in detail, using the bioreactor as an example.

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