Dual adaptive controllers find the optimal inputs for uncertain systems. Dual controllers trade off control against anticipated learning by taking into account the value of future information gained by excitation. As a result, the model error is actively minimized while the performance objective is optimized. In other words, the dual tasks of an optimal control for an unknown system are (1) directing the output toward the reference using caution while (2) investigating the system for learning (“exploration” or “probing”). The uncertainty is reduced over time and the controller gradually shifts more focus to optimizing for the classic performance objective. From a Bayesian perspective it follows that the dual controls actually are the optimal controls for control under uncertainty.
Significant progress has been made in designing algorithms that solve approximations to the dual control in the last few years. This presentation gives and overview some recent advances in dual receding-horizon control, or dual model-predictive control (DMPC). While solutions to dual control problems have been possible to obtain with dynamic programming (DP) for very small systems (one parameter), the “curse of dimensionality” prevents DP from being a practical method for dual control synthesis. Recent novel re-formulations using the MPC approach enable efficient computation using nonlinear programming.
The class of methods we review is based on reformulating the expected output error with respect to the future decision sequence. The resulting deterministic optimal-control problem can be cast as a quadratically constrained-quadratic program (QCQP), a problem class for which there exists efficient solvers capable of finding global optima. This formulation also permits efficient implementation of probabilistic (chance) output constraints. The approach is particularly relevant for industrial MPC implementations, where FIR and step-response models are common. While exact reformulation of the objective function into deterministic form is not possible for all systems classes, the results we present here give insight into how uncertain systems are optimally excited during normal operation and can aid the design of approximate and heuristic dual controllers for more general systems.
The approaches we discuss rely on least-squares estimation of the unknown parameters, a method that provides the first two statistical moments of the stochastic variables: the means and the covariances. For the class of systems we focus on here the future error covariances of the parameters are deterministic functions of the control inputs, which means it is possible to propagate the uncertainty forward in time. This enables exact uncertainty prediction in the finite-horizon optimal-control problems, and is the foundation of exact optimal dual control.
The presentation contains demonstrations of how dual control can improve closed-loop performance, as well as some examples of control problems that can be solved with dual controllers (like the admissibility problem of adaptive control). We conclude the presentation with some thoughts on future directions for dual control research and applications.