Dual adaptive model predictive control

Abstract

We present an adaptive dual model predictive controller (dmpc) that uses current and future parameter-estimate errors to minimize expected output error by optimally combining probing for uncertainty reduction with control of the nominal model. 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. 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. 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. 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. The paper demonstrates the application of dmpc to a single-input single-output (siso) system with unknown parameters. 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.

Linear robust adaptive model predictive control: Computational complexity and conservatism

“…Neglecting the scalar additions and the projection, this algorithm requires the solution of 2p LPs with p optimization variables each. Thus, the computational demand of Algorithm 1 is typically small compared to the MPC optimization problem (14).…”

“…, 2 p , where X f is a terminal set to be specified later. The solution to (14) is denoted by x * ·|t ,x * ·|t , v * ·|t ,û * ·|t , u * ·|t , w * ·|t , s * ·|t . The proposed scheme ensures robust constraint satisfaction by predicting a polytopic tube X k|t = {z| H i (z − x k|t ) ≤ s k|t } that contains all possible future trajectories of the uncertain system (1) subject to the input trajectory v ·|t .…”

“…Update ρ θt using (6). Solve MPC optimization problem (14). Apply control input u t = v * (0|t) + Kx t .…”

“…Robust adaptive MPC methods for nonlinear systems can be found in [12]. Methods to explicitly incentivize or enforce learning of the unknown parameters can be found in [13], [14], [15], [16], which is, however, not the goal of this paper.…”

Self-Reflective Model Predictive Control

“…A related approach toward dual MPC has been proposed in [24,25], where the MPC objective is augmented by a term that-similar to optimal experiment design [19,46,54]-penalizes the predicted parameter error variance. In [26] the same authors develop a way to predict and optimize future parameter estimation errors for single-input-single-output finite impulse response systems based on non-convex quadratically constrained quadratic programming formulations. A different way of taking predicted parameter estimation errors into account has been suggested in [32], where an MPC scheme is augmented by an adaptive parameter estimation algorithm.…”