HW11 unstable solution

HW11 unstable solution

by Masopust Ondřej -

Hello,

while trying to solve the HW11 I ended up with an unstable closed loop system. Although I expect that hinfsyn should give a stabilizing controller, my assumption is that maybe the weigting filters are set up incorrectly? My question is if there is some general guideline for what to focus on in the design? Currently my approach is just to randomly change numbers in the weighting filters. Is there a better approach?

Best regards,

Ondrej Masopust

Re: HW11 unstable solution

by Hurák Zdeněk -
Well, the role of the weigting filters is pretty much identical to the role of the weighting matrices Q and R in LQ-optimal control. The higher the value, the more penalty on the corresponding signal, and therefore the smaller the resulting signal (provided the optimization succeeds in finding a stabilizing controller). However, in contrast the the Q and R matrices, the weighting filters allow to have different penalties at different frequencies. Typically when it comes to disturbances and references, these are significant at low frequencies (including the zero frequency aka steady state) up to some middle frequencies. Some disturbances may be confined to some frequency band. Measurement noises are typical at high(er) frequencies. This gives some general guidance. So, not completely random filters but mostly some low-pass filters. The tuning knobs are then then the gains and cut-off frequencies.

Re: HW11 unstable solution

by Masopust Ondřej -
I figured out the problem which was causing the instability. When I tried to simulate the system using the Pade delay approximation, the solution was stable as expected. The problem is in simulating the system with the original delay transfer function. I tried different approximation degrees for the calculation part but none of them seemed to solve the problem. Is there a hint on how to tackle this problem generally?

Re: HW11 unstable solution

by Šíp Václav -
I have the same problem. I figured out that if you model the disturbance "at the beginning" as it is shown in the assignment, the result will diverge exactly as your solution, but if I model the disturbance as two signals at the end (one as signal adding to y1 and the another to y2) it works and regulates it according to the requirements. But I can't figure out why it is like that.

Re: HW11 unstable solution

by Hurák Zdeněk -
I am afraid it is difficult to give a general guidance here. Indeed, systems with delays are troublemakers. I my solution I did satisfactorily with the order of pade approximation below 10 (some 7 seems fine). And then it all boils down to parameterizing the weighting filters. It certainly requires some iterations of trials and errors. These iterations can hardly be avoided in any engineering design.