Data-based parameter refinement

The file Deme_016.m is found in IQClab’s folder demos In this demo we demonstrate the data-based parameter refinement tools presented here which are based on [28]. The demo consists of a 2DoF control design problem for a plant that is affected by parametric uncertainties. The plant, $G=\tfrac{1}{k^2s^2+2dks+1}$ is a simple oscillator model which is interconnected as:

$\left(\begin{array}{c}e\\u\\y_2\\ \hline y_1\\y_2\end{array}\right)= \left(\begin{array}{cc|c}0&1&-G\\0&0&1\\1&0&G\\ \hline 0&1&0\\1&0&G\end{array}\right)\left(\begin{array}{c}e\\u\\y_2\\y_1\\y_2\end{array}\right),\ \ u=K\left(\begin{array}{c}y_1\\y_2\end{array}\right).$

Here the parameters $k(\delta_k)=k_0(1+k_d\delta_k)$ and $d(\delta_d)=d_0(1+d_d\delta_d)$ are assumed to be uncertain with $\delta_k\in[-1,1]$ and $\delta_d\in[-1,1]$ .

The design proceeds in 3 steps:

Initial design: Perform a nominal $H_\infty$ -controller synthesis using hinfsyn. This controller works well for the nominal scenario, while the performance degrades for various samples of $k(\delta_k)$ and $d(\delta_d)$ . The synthesis procedure is not further detailed here.
Parameter refinement: Collect input-output data from an experiment and run the function fRefine with the aim to reduce the uncertainty bounds as much as possible.
Retuned/calibrated design: Based on the reduced parameter bounds, synthesize a new nominal (mean value of the uncertainties) controller. The new controller will then be insensitive to variations of the uncertainties due to the (much) reduced parameter bounds.

The following figure depicts the tracking error for a step response for the nominal closed-loop system as well as for various samples of $\delta_k\in[-1,1]$ and $\delta_d\in[-1,1]$ . As can be seen, the control performance significantly degrades for some values of $\delta_k$ and $\delta_d$ .

Let us now generate data for the true system (i.e., for $\delta_k=0.635$ and $\delta_d=-0.398$ ) and perform a data-based parameter refinement using the function fRefine. Here we set the option nlift=3 for improved accuracy which yields the following refinements of the parameters $\delta_k$ and $\delta_d$ .

As can be seen, the the refinement algorithm is capable of significantly tightening the uncertainty intervals to $\delta_k\in[0.634,0.636]$ and $\delta_d\in[-0.402,-0394]$ .

By now performing another nominal $H_\infty$ -controller synthesis for the mean values of the refined intervals of $\delta_k$ and $\delta_d$ (and using the same weighting functions) we obtain the following results (again for the nominal system as well as for different samples of the uncertainties on their refined intervals. As can be seen, we obtain a nice system response, even for different values of the uncertainties.

All this demonstrates the usefulness and potential of the data-based refinement algorithms. Further demonstrations are found in [28].