The Separation Principle and the Dual-Certainty Equivalence Gap in Model Predictive Control

Abstract: Dual control addresses the trade-off between exploitation and exploration, where control inputs both regulate the system and generate informative data for estimation and identification. For certain problem classes, control and estimation can be designed independently without loss of optimality, a property known as the separation principle. However, in stochastic control problems with model uncertainty and constraints, this principle generally breaks down, and introduces the need for dual control. In this paper, we propose an information-weighted dual model predictive control (MPC) formulation and introduce metrics that quantify the dependence of the MPC policy on the uncertainty. We focus on parametric uncertainty in linear systems with Gaussian noise, though the metrics can be applied more broadly. Numerical results show that the dependence of the MPC policy on the posterior covariance is largest under high uncertainty and vanishes as the posterior covariance contracts, providing empirical evidence of the dual effect in closed loop. Moreover, the dual controller improves regulation performance and model accuracy compared to certainty-equivalent MPC.