Compact Closed-Loop Response Representations for Controller–Task Matching

Reservoir computing enables robot controllers to exploit temporal information while training only an output readout. However, different trained reservoirs can interact differently with the same plant, making their suitability dependent on the requested motion. We introduce Closed-Loop Response Representations (CLRR) to compare trained nonlinear echo state network controllers through compact, reusable descriptions of their feedback behavior. For each plant, weighted principal component analysis learns a shared basis for local closed-loop response residuals from development controllers, without nonlinear performance labels. In the complex-response formulation, each finite-window multisine tracking task defines an analytical quadratic functional that evaluates the reconstructed response directly in its learned coordinates. After response characterization and encoding within a fixed operating regime, the same controller code supports new task queries without retraining or additional task-specific nonlinear rollouts. A distortion bound relates compression error to response-based task-cost error and controller-comparison margins. In near-equilibrium Cart-Pole simulations, eight coordinates achieve macro Kendall correlation 0.936 across 43 held-out controllers and 36 tasks, outperforming the tested compact spectral summaries. On a separate 16-controller test bank for a simulated seven-joint Panda arm, eight response coordinates plus seven equilibrium offsets achieve correlation 0.538, compared with 0.919 for the full response, yet select the best or second-best controller on all eight tasks. Full and sparse responses select the best throughout. These findings support compact response representations for task-dependent reservoir-controller comparison, while showing that useful controller choices can survive substantial losses in overall ranking fidelity. The quality of the representation therefore depends on the controller distinctions it preserves, not reconstruction accuracy alone.