(309h) Robust Optimization Approximation of Chance Constrained Model Predictive Control

Uncertainty is an important issue to be considered in the control of process systems. Uncertainties in the system include parameter uncertainty, system disturbance and measurement noise. Conventional model predictive control (MPC) can have severe problems if the uncertainty is not properly dealt with. The problems include: severe constraint violation, instability of the system, unrealistic results, etc. In stochastic MPC, chance constraints are introduced to handle the uncertainty. A probability level of constraint satisfaction is defined by the user to seek a trade-off between robustness and optimality.

There are generally two approaches to handle the chance constraints: analytical methods and sampling methods. Typical analytical methods include inverse cumulative distribution function method, ellipsoidal method, and risk allocation method. Most of the aforementioned analytical methods require that the uncertainty distribution being Gaussian. Inverse cumulative distribution function method only applies under the condition of individual chance constraints. For one-dimensional Gaussian distribution, the inverse cumulative distribution function can be calculated directly. The resulting deterministic constraints are then equivalent to the chance constraints [1]. For joint chance constraints, the problem is much more complicated as there is no closed form of cumulative distribution function for multidimensional Gaussian distribution. But closed form exists if the integration area is an ellipsoid. However, the ellipsoidal method leads to very conservative results [2]. Risk allocation method is based on the Booleâ??s inequality: the probability that at least one of the events happens is not greater than the sum of the probabilities of the individual events. So the joint chance constraint can be decomposed into individual chance constraints, which are much easier to solve [3]. Risk allocation can also be improved by optimizing the risk allocation to each individual chance constraint [4]. Sampling method has the advantage that it is not restricted to certain types of distribution. It includes mixed integer programming based sample average approximation approach and scenario optimization approach. However, those methods are computationally expensive [5,6] and scenario approach may lead to very conservative result [7].

In this work, we propose robust optimization approximation method to handle both the individual and the joint chance constraint in MPC problem. By introduction of the uncertainty set, the individual and joint chance constrained problem can be converted to tractable robust optimization problem. Various types of uncertainty set can be introduced to deal with different types of distribution. Box and polyhedral types result in linear constraints and ellipsoidal type results in second order cone constraints. Robust optimization approximation method preserves the simplicity of analytical methods and it is not restricted to certain type of probability distribution. Compared with other classical methods in the case study, robust optimization method leads to good performance in numerical case studies. The proposed method is also applied to the steam assisted gravity drainage (SAGD) process in the oil sands industry. Recursive data-driven modeling methods are applied to build models for the process. Chance constrained MPC is applied and the technique is tested with a reservoir simulator: Petroleum Experts. The robust optimization approximation based chance constrained MPC leads to satisfactory results.

Reference:

[1] Schwarm, Alexander T., and Michael Nikolaou. "Chance�constrained model predictive control." AIChE Journal 45.8 (1999): 1743-1752.

[2] Van Hessem, D. H., C. W. Scherer, and O. H. Bosgra. "LMI-based closed-loop economic optimization of stochastic process operation under state and input constraints." Decision and Control, 2001. Proceedings of the 40th IEEE Conference on. Vol. 5. IEEE, 2001.

[3] Nemirovski, Arkadi, and Alexander Shapiro. "Convex approximations of chance constrained programs." SIAM Journal on Optimization 17.4 (2006): 969-996.

[4] Ono, Masahiro, and Brian C. Williams. "Iterative risk allocation: A new approach to robust model predictive control with a joint chance constraint."Decision and Control, 2008. CDC 2008. 47th IEEE Conference on. IEEE, 2008.

[5] RuszczyÅ?ski, Andrzej. "Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra." Mathematical Programming 93.2 (2002): 195-215.

[6] Blackmore, Lars. "A probabilistic particle control approach to optimal, robust predictive control." Proceedings of the AIAA Guidance, Navigation and Control Conference. No. 10. 2006.

[7] Calafiore, Giuseppe C., and Marco C. Campi. "The scenario approach to robust control design." IEEE Transactions on Automatic Control 51.5 (2006): 742-753.