(284e) Model Predictive Control Under Model Structural Uncertainty

Model predictive control is a widely used control strategy in the process control industry. Model predictive control, as the name suggests uses a model to predict the future behaviour of the process and compute optimal input moves in a receding horizon fashion. Models that are used in the controller may be developed based on first principles or from experimental data or a combination of both. However, models are inherently uncertain due to model simplification or lack of knowledge. In order to handle the uncertainty, many different forms of MPC have been developed in the past, such as robust MPC, stochastic MPC, feedback min-max MPC and adaptive MPC. Almost all the works in the literature on MPC under uncertainty, focuses on model parameters and external disturbances as the primary source of uncertainty. However, more commonly, plant-model mismatch occurs due to model simplification and/or lack of knowledge. Alternatively, step response models may be identified using experimental data with the model order chosen rather arbitrarily. This leads to model structural uncertainty, where the underlying model structure itself is uncertain.

Structural uncertainty however, is often disregarded in model based control design and has received little attention in the past. This is due to the fact that, unlike model parametric uncertainty, it is more challenging to handle structural uncertainty. A robust MPC framework considering structural uncertainty for linear systems has been presented in [3] and very recently, an explicit dual control framework for handling structural uncertainty was presented in [4]. However, in this work, we consider the scenario-based MPC framework presented in [1].

In this work, we show that scenario-based MPC is an efficient tool to handle model structural uncertainty. Scenario-based MPC or feedback min-max MPC is based on the evolution of the uncertainty represented by a scenario tree. It was first introduced as a means to handle uncertainty in the model parameters and future disturbances [1],[2]. Traditionally in this approach, the uncertain parameters, either belonging to a compact set or described by a probability distribution function, are sampled to form a discrete set of scenarios. The scenario tree is then used to predict a cone of state trajectories to represent the propagation of uncertainty. An optimization problem is then solved to compute optimal control trajectories. The notion of feedback is introduced by letting the computed optimal input vary according to the different scenarios. The closed-loop optimization nature of the scenario-based MPC has led to reduced conservativeness compared to robust MPC methods [2]. Similarly, we apply the same principle to handle model structure uncertainty. In the case of model structure uncertainty, the cone of predicted state trajectories is based on different model structures, instead of a single model structure with sampled parameters. The optimization problem is then solved over the entire scenario tree, hence taking into account the different model structures. Each model can be given different weights to prioritize between the different models.

Previous work considering scenario-based MPC either assumes uncertainty in the form of constant model parameters as in [2] or future external disturbances as in [1]. In this work, we show that scenario-based MPC is not just restricted to these forms of uncertainty, but is also an efficient tool to handle model structural uncertainty, hence opening up a new class of uncertainty that can be handled in MPC problems.

An offshore oil and gas production optimization problem is used as a case study to demonstrate the proposed method. Oil well models are often subject to structural uncertainty due to lack of knowledge or model simplifications. Consider the reservoir inflow performance model that describes the flow rate of fluids from the reservoir that enters the well. Due to the lack of measurements in the wellbore, it is challenging to develop empirical model based on data and thus, one usually relies on physical models. Different physical models exist that describe the inflow performance. For example, the linear PI model [5], Vogel inflow model [6], Fektovich inflow model [7] and Jones inflow model [8] are often used in the oil and gas industry. Such model structure uncertainty is often ignored, which may lead to suboptimal performance or even constraint violations. Scenario-based MPC can be used in such cases to account for this uncertainty.

References

[1] Scokaert, P., Mayne, D., 1998. Min-max feedback model predictive control for constrained linear systems. IEEE Transactions on Automatic control 43 (8), 1136–1142.

[2] Lucia, S., Finkler, T., Engell, S., 2013. Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty. Journal of Process Control 23 (9), 1306–1319.

[3] Feng L, Wang JL, Poh EK. Improved robust model predictive control with structured uncertainty. Journal of Process Control. 2007 Sep 30;17(8):683-8.

[4] Heirung TA, Mesbah A. Perspectives On Stochastic Predictive Control With Autonomous Model Adaptation For Model Structure Uncertainty. FOCAPO-CPC, 2017.

[5] Evinger, H.H. and Muskat, M. 1942. Calculation of Theoretical Productivity Factor. Trans., AIME 146: 126.

[6] Vogel, J.V. 1968. Inflow Performance Relationships for Solution-Gas Drive Wells. J Pet Technol 20 (1): 83–92. SPE 1476-PA.

[7] Fetkovich, M.J.: “The Isochronal Testing of Oil Wells,” paper SPE 4529 presented at the 1973 SPE Annual Meeting, Las Vegas, Nevada

[8] Jones, L.G., Blount, E.M., and Glaze, O.H. 1976. Use of Short Term Multiple Rate Flow Tests To Predict Performance of Wells Having Turbulence. Presented at the SPE Annual Fall Technical Conference and Exhibition, New Orleans, Louisiana, 3-6 October 1976. SPE-6133-MS