Note that dual effect and neutrality are properties of the system rather than the control algorithm. Informally, a control that with nonzero probability affects not only the system state but also the uncertainty (specifically, error covariances or higher-order central moments) has a dual effect on the system systems in which the control cannot affect this uncertainty are called neutral (see Bar-Shalom & Tse, 1974 for a rigorous definition). These controllers learn from normal operating data, which can contain very little information. Most adaptive control algorithms are passively adaptive in the sense that learning takes place only as a side-effect of the control action. Using data to progressively reduce uncertainty is often framed as a learning process, in the control community primarily studied in the field of adaptive control. Dual control problems include the mechanisms for both control and learning in the formulation, and the solution optimally incorporates both aspects in the input to the process. Dual control, as introduced by Feldbaum (1961), is the optimal control under decision-relevant, reducible uncertainty. This paper addresses the problem of optimal control and learning in the context of stochastic systems and models with stochastic parametric uncertainty and probabilistic constraints.
In the simulation example, the parameter estimates converge quickly and the probing vanishes with increasing accuracy and precision of the estimates, improving the future control performance. The paper demonstrates the application of dmpc to a single-input single-output ( siso) system with unknown parameters. The adaptive dmpc solves this qcqp at each sampling time on a receding horizon the adaptation is a result of updating the parameter estimates used by the dmpc to decide the control input. We further reformulate the nonlinear deterministic problem to pose an equivalent quadratically-constrained quadratic-programming ( qcqp) problem that state-of-the-art algorithms can solve efficiently, providing the exact solution to the probabilistically constrained finite-horizon dual control problem.
Propagating the exact future statistics enables reformulating the original stochastic problem into a deterministic equivalent that illustrates the dual nature of the optimal control but is nonlinear and nonconvex. Our novel approach relies on orthonormal basis-function models to derive expressions for the predicted distributions for the output and unknown parameters, conditional on the future input sequence. We present an adaptive dual model predictive controller ( dmpc) that uses current and future parameter-estimation errors to minimize expected output error by optimally combining probing for uncertainty reduction with control of the nominal model.
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