(309g) On the Moment-Based Robust MPC Formulations

The control technique commonly used in process industry for constrained multivariable optimal control is the model predictive control (MPC) [1]. The performance of these controllers depends to a large extent on the quality/validity of process models ([2]). In the mean time, more and more rigorous process models that cover wider operating conditions have become available. A valid hypothesis is then the use of this type of models for MPC. The practice, on the other hand, is conflicting since the model based estimation and control algorithms require relatively simple explicit models for efficient and robust calculations in their algorithms. This practical drawback has led to research directions in the area of model simplifications ([3] and references therein) or multiple model MPC approaches ([4]). In addition to the plant-model mismatch as a result of inaccurate identification of model parameters and its dynamics, such simplifications introduce uncertain behavior between the observations induced from the true process and the model predictions. In order to limit the effect of uncertainty on the closed loop performance, robust control techniques are required.

In the last couple of decades, different MPC algorithms have been introduced to achieve the desired control objectives while reducing the effect of uncertainty ([5]). We can broadly classify these algorithms into two classes; i) the worst case based techniques (WC-MPC) ([6]), including min-max based, tube based or input-to-state stability based arguments, where under any effect of predefined maximum level of uncertainty, the process variables should not violate the specifications; or ii) chance based MPC algorithms ([7]), including quantile based or scenario based formulations, where the specifications are softened, meaning that the constraints are allowed to be violated, but the chance of being off the specification is kept small, thus violations are rare events. Both of these formulations have their own drawbacks. The WC-MPC implementations are known to be pessimistic, resulting in deterioration of the actual closed loop performance compared to the nominal case where the uncertainties are realistically described with the nominal model. Furthermore the formulation of the problem tends to be overly complex for large-scale process models. The chance-based formulation reduces the conservatism inherent to WC-MPC by discarding the severe-but-rare events out of consideration. This reduces the pessimism of WC-MPC however the computational problems should be expected to occur, due to the tedious numerical integrations over the process trajectories. It is straightforward to see that the stochastic approach to the robustified MPC problem is more powerful than the WC-MPC. The ability to discard rare and severe realizations, by a tuning parameter, leads to balancing the resulting performance and constraint satisfaction tradeoff.

In this contribution we make a first attempt to apply stochastic MPC approach with a robustfication step different than the worst case or chance based MPC formulations. Instead of worst case realization or the probability distribution function of state, we consider the statistics, (finite order) moments, of the state predictions to calculate the control actions. We introduce Mean-MPC (M-MPC) and Mean-Variance MPC (MV-MPC), which consider the expectations and variances of the state trajectories and their functions, cost and constraints. By making use of variance of state predictions, one can effectively back-off the operating conditions, as a function of the process dynamics and the uncertainty model. We consider different types of uncertainty models and show parallelism between moment based MPC and aforementioned MPC techniques.

We distinguish multiple observations from the M-MPC and the MV-MPC results. Under several technical assumptions we show that the M-MPC is equivalent with nominal MPC. This explains the inherent robustness aspects of the nominal MPC, commented in other publications, e.g., [8]. In case of the MV-MPC formulation we observe that when the cost and constraint functions are mapped to robustified equivalents, the structure of these functions (linear, quadratic etc.) and the uncertainty model effect the resulting MPC problem. For additive disturbance case with quadratic cost and linear constraint functions, the case in this contribution, the resulting MPC problem consists of a cost function weighting both the process variablesâ?? current/predicted values and the spread of the state trajectories. The cost term relating to the variance is growing monotonically with the prediction horizon, which in turn allows us to establish stability without terminal cost function, decreasing the effect of artificial terminal cost. Furthermore, the robustified constraints are improving the chance of satisfying the constraints, by making use of the covariance matrix of the predicted states, hence the process dynamics, explicitly.

One another important aspect of the moment based MPC is the computational benefits. Through the use of moment operators, one obtains a MPC problem that is explicit in the optimization variables, meaning that no maximization or integration step is required. Hence for processes with medium to large scale dynamics, the moment based MPC formulation remains computationally tractable. This property leads to adjustable performance vs. robustness specifications, allowing the practitioners to implement moment based MPC for process systems.

Acknowledgement:

This work has been done within the project â??Improved Process Operation via Rigorous Simulation Models (IMPROVISE)â?? in the Institute for Sustainable Process Technology (ISPT)

References:

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