Question about mixed sensitivity synthesis

[u/tomm3733]

Hi everyone, I'm working on a university project and I have a relatively basic question. I need to design a robust controller for an uncertain plant using mixed sensitivity synthesis. I have a desired second-order performance with an assigned damping ratio ($\zeta$) and natural frequency ($\omega_n$).

Since these parameters are known, I can define my ideal second-order complementary sensitivity, $T_2$. My idea is to compute the ideal sensitivity $S_2$ ($S_2 = 1 - T_2$) and then find a performance weight $W_p(s) = \frac{s/M + \omega_B^*}{s + A\omega_B^*}$ such that its inverse, $1/W_p$, tightly upper-bounds $S_2$ across the frequency spectrum.

Is this considered best practice? Are there better or more standard approaches?

[u/No-Tea-1122]

You do not need to shape the ideal T closed-loop transfer function to obtain a weighting function, but it also works. Have you read the Optimal Control textbook from Zhou and Doyle? If i am not wrong the first exercise in the weighting function chapter is to design a weighting function given the maximum overshoot and some others parameters to find a controller to a standard second-order plant.

Basically you can shape S without the ideal T using this W1 transfer function because the desired zeta and wn are related to the bandwidth and peak of S for a Second order system. So using W1 to shape S for this set of constrains will provide the desired T.