(430f) Study of Moment-Based MPC Formulations and Their Connection to Classical Control

R. Zhang*, M. B. Saltik*, L. Özkan*

* Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands

Model predictive control (MPC) is a well-established modern control technology used in diverse applications to provide (sub)optimal operating conditions while incorporating safety and performance constraints. In MPC, the control action at the current time instant is obtained by solving a finite horizon optimal control problem according to the forecasts of the future process behavior. Hence the quality/validity of predictions generated from process models determine the performance of these controllers ([1]). Although models based on first principles are expected to provide better predictions for a wider range of operating conditions ([2]), the model predictions should also incorporate the effects of uncertain deviations to overcome the adverse effects. To this end, robust model predictive control techniques are developed in order to reduce the effect of uncertainty ([3,4]) in dynamical processes.

Several distinct robust model predictive control techniques have established over the last decades, which can be classified according to the modeling and treatment of uncertainty realizations. A very well known technique of establishing robustness is the worst-case optimization techniques, e.g., ([5]). In this technique, a min-max problem is constructed which suppresses the effect of all possible uncertainty realizations over the prediction horizon. Recently, the so-called chance constrained, e.g., ([6]), and the scenario based optimization ([7]) paradigms have been introduced. In these approaches the MPC specifications or the constraints are allowed to be violated, but one controls the chance of specification violations. However, all of these mentioned approaches are yet to be adopted into the industrial applications. The essential reason for this gap between industry and academia is the huge deterioration of the closed-loop performance and the required computational power to solve the robust optimization problem. In order to surpass these problems, relatively recently, moment based MPC formulations are introduced ([8,9]) which consider the linear combinations of the (centralized) moments of the uncertain (stochastic) predictions within the MPC problem. In this approach, the ability to discard rare-and-severe realizations is adjusted by tuning parameters weighting the moments of the cost function. Hence, these parameters balance the suppression of uncertain effects versus the on average the closed-loop behavior.

In this contribution, we discuss the effects of tuning parameters in moment based MPC problems on the closed-loop time and frequency domain characteristics with the help of simulations. We present the effects resulting from Mean-MPC, for the plant-model mismatch case, and the Mean-Variance MPC, for the additive perturbations case, while comparing them with the nominal MPC response. Since the effect of moments of cost function leads to an MPC problem with stage-wise changing state and input weights, one can also construct the closed-loop transfer function of the unconstrained MPC controller, thus compare the effects the moment parameters in the frequency domain. More specifically, we first demonstrate that the MV-MPC introduces a virtual (low pass) filter into the prediction model. This leads to introduction of cross terms of states (over the prediction stages) with state (equivalently output) weighting matrix accumulating the effects of uncertainty. This re-adjusts the spread (variance) versus on average (mean) performance of the MPC controller. Moreover, by adjusting the weight parameter, we show improvements in the bandwidth or the phase response of the closed-loop system. Lastly, we demonstrate the backing-off effect the mean-MPC for plant-model mismatch case. In this case the resulting MPC controller reduces the aggressiveness of the controller more and more as the trust in model predictions reduces. This effect is adjusted via additional terms accumulating in the input actionsÕ weighting matrix. We demonstrate the proposed moment based MPC techniques on a quadruple tank system.

Acknowledgement:

We acknowledge Bahadir Saltik for valuable discussions.
References:

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