Control of a time-variant system

I am working in a control project in which I need to control two variables $$\x\$$ and $$\y\$$ according to their reference values.

Since I don't have an advanced knowledge in modern control theory, I usually tackle these problems using PI, PID, lead/lag controllers.

I have simplified my problem in the following image. My system has a matrix with time varying coefficients (and it adds coupling to the variables), and I am not being able to write the open-loop transfer function to find the good parameters of my PI controllers.

Do I need to use a fancier control structure than PI control or can I obtain an approximation of the transfer function and keep the topology of this solution?

• The time varying matrix, I think you may need to explain with a little more detail, The visual I have on my head would be to just plot the correction vector for the proportional response of both outputs vs both inputs on a 2D graph to reverse out the transfer function. – Reroute Sep 12 at 8:53
• This time varying matrix is actually a rotation matrix \begin{bmatrix}cos(\theta) & -sin(\theta)\\sin(\theta) & cos(\theta)\end{bmatrix} with $\theta = \omega t$. – Vinícius Lopes Simões Sep 12 at 9:01
• Can I ask what you are trying to control, this seems like an odd approach, I'm left wondering if its an X-Y problem, meta.stackexchange.com/questions/66377/what-is-the-xy-problem – Reroute Sep 12 at 9:26
• This is a very specific problem, but I am trying to suppress some harmonics from an electrical motor according to ieeexplore.ieee.org/document/8481417 and ieeexplore.ieee.org/document/6780706. In order to identify these harmonics, the variables (currents in dq model) need to go through this rotation matrix with the respective frequency $\omega$ of the harmonics. – Vinícius Lopes Simões Sep 12 at 9:41
• This seems to be a rundown on what your looking for, It is a bit beyond my normal control systems knowledge, citeseerx.ist.psu.edu/viewdoc/… It mainly looks like they are using a variation of FFT to get the phase and frequency information and apply corrections in the frequency domain. – Reroute Sep 12 at 9:55