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I have a series of MIMO LTIs representing the controllers for my MIMO LTI plants, which are the linearization of my nonlinear model. What is the best way to interpolate the controllers to come up with the nonlinear controller?

Interpolating the elements of $A,B,C,D$ representing the controllers seems terribly wrong.

I can interpolate over the output $y$ of each controller according to the scheduling parameter, but I was wondering whether there is a more correct / elegant way that is not a LPV synthesis.

Thanks!

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If you would like to interpolate the matrices of a state space model you have to make sure the state associated with each model has a similar meaning. Namely, otherwise you could interpolate similar state space models (with the same frequency response) and get different intermediate frequency responses when interpolating. For example you could use the controllable canonical form for each state space model.

Is is also worth noting that in general interpolating stabilizing controllers, which stabilize the respective linearized system operating point, does not ensure that the nonlinear would be stable.

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  • $\begingroup$ Thanks for your reply. Actually that’s the case, but I was wondering whether there is a better approach, like interpolating the poles or something similar... $\endgroup$ – venom Apr 10 at 11:59
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    $\begingroup$ @venom Your suggestion wouldn't be well defined, since you would then also have to choose pole pairings. It can be noted that when using the controllable canonical form you are essentially interpolating the coefficients of the characteristic polynomial. I also looked interpolating the transfer functions, but this yielded intermediate transfer functions of potentially double the order and also added initially not present anti-resonances. $\endgroup$ – Kwin van der Veen Apr 11 at 6:39
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    $\begingroup$ @venom you could maybe split all controllers into a series of blocks of first and second order transfer functions, such that it the pole matching would be well defined. Though, for the second order blocks you might have to interpolate the natural frequency and damping coefficient. $\endgroup$ – Kwin van der Veen Apr 11 at 6:42
  • $\begingroup$ Thank you!! I’ll have a try! $\endgroup$ – venom Apr 12 at 8:29

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