When an Approximate Model Suffices for Optimal Control

TL;DR

This paper establishes conditions under which control derived from an approximate model is optimal for the actual system, enabling robust control despite model inaccuracies, especially in quadratic control problems.

Contribution

It provides theoretical conditions ensuring that model-based optimal control matches plant-optimal control despite model mismatch, with explicit results for quadratic control effort.

Findings

Control strategies can be optimal for the plant even with approximate models.

Explicit conditions guarantee control equivalence and uniqueness in quadratic problems.

Examples demonstrate control trajectory equivalence despite significant model mismatch.

Abstract

In this paper, we develop an optimal control framework for dynamical systems when only an approximate model of the underlying plant is available. We consider a setting in which the control strategy is synthesized using a model-based optimal control problem that includes a penalty term capturing deviation from the plant trajectory, while the same control input is applied to both the model and the actual system. For a general class of optimal control problems, we establish conditions under which the control minimizing the model-based Hamiltonian coincides with the plant-optimal control, despite mismatch between the model and the true dynamics. We further specialize these results to problems with quadratic control effort, where explicit and easily verifiable sufficient conditions guarantee equivalence and uniqueness of the resulting optimal control. These results show that accurate control synthesis does not require an exact model of the underlying system, but rather alignment of the optimality conditions that govern control selection. From a learning perspective, this suggests that data-driven efforts can focus on identifying regimes in which model-based and plant-based Hamiltonian minimizers coincide, thereby providing a theoretical basis for robust model-based decision making and the effective use of digital twins under modeling error. We provide examples to illustrate the theoretical findings and demonstrate equivalence of the resulting control trajectories even in the presence of significant model mismatch.