Safe Output Feedback Improvement with Baselines

TL;DR

It is seen that minimizing the baseline regret under model uncertainty can guarantee safe controller improvement with less conservatism and variance in the resulting controllers.

Abstract

In data-driven control design, an important problem is to deal with uncertainty due to limited and noisy data. One way to do this is to use a min-max approach, which aims to minimize some design criteria for the worst-case scenario. However, a strategy based on this approach can lead to overly conservative controllers. To overcome this issue, we apply the idea of baseline regret, and it is seen that minimizing the baseline regret under model uncertainty can guarantee safe controller improvement with less conservatism and variance in the resulting controllers. To exemplify the use of baseline controllers, we focus on the output feedback setting and propose a two-step control design method; first, an uncertainty set is constructed by a data-driven system identification approach based on finite impulse response models; then a control design criterion based on model reference control is used. To solve the baseline regret optimization problem efficiently, we use a convex approximation of the criterion and apply the scenario approach in optimization. The numerical examples show that the inclusion of baseline regret indeed improves the performance and reduces the variance of the resulting controller.

Figures (3)

• Figure 1: Controller gain versus performance criterion. Left: The black dashed line shows the worst-case cost under model uncertainty represented by shaded area and the dot indicates the min-max $K$, see \ref{['eq:min-max']}. The blue dashed line shows the baseline $K_b$. The red dashed line indicates the optimal K for the unknown system $G_{\circ}$. Right: The black dashed line shows worst-case baseline regret and the dot indicates the proposed controller, see \ref{['eq:min-max:baseline']}. With high uncertainty, the min-max strategy chooses a conservative controller with a low gain while our method selects the baseline controller.
• Figure 2: Experiment comparison between the proposed method without and with different baseline controllers. The optimization problem is solved with $M=1523$ sampled scenarios from $\mathcal{G}$ represented by the colored quadratic lines. The black dashed line denotes the worst-case cost over these scenarios and the dot is the solution. The red dashed line indicates the optimal controller and the blue dashed line indicates the baseline controllers. The results demonstrate that incorporating a baseline reduces conservatism and results in a controller gain closer to the optimal value, particularly with limited data.
• Figure 3: Performance comparison under 100 runs w.r.t $F_W$ in \ref{['eq:FW']} and $F_C$ in \ref{['eq:FC']} between $\rho_{\text{nom}}$ in \ref{['eq:rho_nom']}, $\rho_{\text{MMB}}$ (min-max with baseline) and $\rho_{\text{MM}}$ (min-max without baseline controller) with the different number of samples to identify the model parameters. The optimization problem is solved with $M=6138$ sampled scenarios. The blue dash line is the performance of the baseline controller. The box shows the quantiles of the runs, while the whiskers extend to the 99.3% coverage and the black dots represent the outliers.

Theorems & Definitions (3)

• Proposition 1
• proof
• Theorem 2