Bayesian Optimization Based Grid Point Allocation for LPV and Robust Control

TL;DR

The paper tackles the challenge of selecting informative grid points for grid-based LPV and robust controller synthesis, where global stability and performance must be ensured across scheduling or uncertainty spaces. It introduces a formal informativity notion, and deploys Bayesian optimization with Gaussian process surrogates to efficiently identify the most informative grid points that shape the complete closed-loop behavior. A three-step grid-point allocation algorithm is proposed, leveraging an acquisition function such as expected improvement, and practical interpolation schemes to realize controllers from scattered grid points. The method is validated on three case studies (unbalanced disk, satellite with flexible arrays, and a 2-DOF robotic arm), showing that it can achieve comparable or superior performance with far fewer local-model evaluations than dense grids or traditional worst-case approaches, while accommodating budgetary constraints. The work highlights the potential for scalable, data-efficient grid design in industrial LPV and robust control contexts and outlines avenues for formal global guarantees and alternative optimization strategies.

Abstract

This paper investigates systematic selection of optimal grid points for grid-based Linear Parameter-Varying (LPV) and robust controller synthesis. In both settings, the objective is to identify a set of local models such that the controller synthesized for these local models will satisfy global stability and performance requirements for the entire system. Here, local models correspond to evaluations of the LPV or uncertain plant at fixed values of the scheduling signal or realizations of the uncertainty set, respectively. Then, Bayesian optimization is employed to discover the most informative points that govern the closed-loop performance of the designed LPV or robust controller for the complete system until no significant further performance increase or a user specified limit is reached. Furthermore, when local model evaluations are computationally demanding or difficult to obtain, the proposed method is capable to minimize the number of evaluations and adjust the overall computational cost to the available budget. Lastly, the capabilities of the proposed method in automatically obtaining a sufficiently informative grid set are demonstrated on three case-studies: a robust controller design for an unbalanced disk, a multi-objective robust attitude controller design for a satellite with uncertain parameters and two flexible rotating solar arrays, and an LPV controller design for a robotic arm.

Figures (9)

• Figure 1: Example of a generalized plant $P$, containing an uncertain system $G$, the generalized disturbance channels $w_{\{\bullet\}}$, the generalized performance channels $z_{\{\bullet\}}$, the weighting filters $W_{\{ \bullet\}}$ that capture the desired closed-loop behaviour specifications and the to-be-synthesized controller $K$.
• Figure 2: Structure of the generalized plant for controller synthesis in the unbalanced disk example.
• Figure 5: The left plot shows the points allocated in $\Theta_0$, $\Theta_{\mathrm{dense}}$, $\Theta_{\mu}$, and $\Theta_{\mathrm{BO}}$. The right plot shows the frequency response of $K_0$, $K_{\mathrm{dense}}$, $K_{\mu}$, and $K_{\mathrm{BO}}$, which are synthesized with the sets $P_{\Theta_0}$, $P_{\Theta_{\mathrm{dense}}}$, $P_{\Theta_\mu}$, and $P_{\Theta_{\mathrm{BO}}}$, respectively.
• Figure 6: A posteriori robust performance analysis of the generalized plant $P$ in feedback with $K_0$, $K_{\mathrm{dense}}$, $K_\mu$, and $K_{\mathrm{BO}}$ for the unbalanced disc system.
• Figure 7: Time-domain simulation results of the nonlinear unbalanced disk system \ref{['eq:unbalDiskNL']} with the $K_0$, $K_{\mathrm{dense}}$, $K_\mu$, and $K_{\mathrm{BO}}$ in feedback configuration. The off-centered mass $M$ is set at different value within the defined uncertainty range at each simulation.

Theorems & Definitions (2)

• definition 1: $N_\theta$-optimal selection
• definition 2: Most informative grid point w.r.t. $\Theta$