Robust performance case study (Matlab) 2: generalised plant, norm-based performance PID certification

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 16:20

*Enlace a Spanish version

Summary:

This video continues the case study started in video [cerp1EN], dealing with the discussion of a robust performance analysis of a given trial-and-error tuned PID controller for the process with transfer function $G(s)=5∕{(s+1)}^2$. In the first video, a simulation-based analysis of the achieved performance was pursued.

Now, we aim to obtain theoretical guarantees of robust performance. So, first a setpoint-to-error gain bound is defined in the frequency domain. Second, a generalised plant with uncertainty is built. Last, the robust performance certification via $H_{\infty }$ norm bound (video [prth1EN]) is used.

We see that expressing the “size 0.2 uncertainty” as either $0.2\cdot \mathrm{\Delta }$ or $\mathrm{\Delta }\cdot 0.2$ gives different results. The second choice allows for proving robust performance, but that inspires expressing the uncertainty as $S^{-1}\cdot \mathrm{\Delta }\cdot S\cdot 0.2$ and considering $S$ to be an additional degree of freedom. Indeed, that’s a scaling that allows to generalise the robust performance check to the so-called scaled small gain theorem. The idea is further explored in video [cerp3EN], where $H_{\infty }$ optimal controllers are also pursued instead of a hand-tuned PID we were using up to this moment for simplicity in the illustration of key ideas.

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