265673 MPC Tuning Based On Impact of Modelling Uncertainty On Closed-Loop Performance
MPC tuning based on impact of modelling uncertainty
on closed-loop performance
Quang N. Tran, Leyla Özkan, A.C.P.M. Backx
Eindhoven University of Technology
Den Dolech 2, 5612 AZ Eindhoven, The Netherlands
firstname.lastname@example.org; email@example.com; firstname.lastname@example.org
Model predictive control (MPC) represents a class of control algorithms, in which an optimisation problem is solved online using a model of the plant. MPC has become a standard advanced controller of large process units in refining and petrochemical industries () thanks to its ability of dealing with constraints and operating the plant at its optimal performance. The basic idea of MPC is that at any given time, it solves an online open-loop optimal control problem over a finite horizon and only the first element of the optimal control sequence is actually implemented on the plant. The same procedure is implemented when the next measurement is available. The quality of MPC, like any other model-based operation support systems (such as Real-Time Optimisation or model-based measurement), is largely determined by the accuracy and the maintained calibration of the model. Due to the model-plant mismatch, the performance of MPC degrades over time, if proper supervision is not performed. Hence, the influence of the modelling uncertainty on the performance of MPC is of enormous importance.
The current tuning practice of these controllers is heuristic and there has been no standard way of tuning MPC that takes into account model-plant mismatch, especially in process industries. MPC tuning strategies that consider robustness often lead to a conservative tuning, which might be too far from the optimal trade-off between robustness and nominal performance. With this observation in mind, this research focuses on finding tuning parameters that provide this optimal balance.
In MPC systems, the controller aims to reduce the variance of the key performance variable and then pushes it towards the constraint so that the system operates at its economically optimal condition. Therefore, the variance of the key variable is a good indication of the performance of the closed-loop system. In addition, the closed loop performance, the tuning of controllers and the model accuracy are inter-related. This relationship has been extensively studied and presented in robust control theory () using frequency domain techniques. It was shown that the performance of the closed-loop system becomes sensitive the modelling uncertainty at a certain bandwidth of frequency. Increasing the bandwidth further results in closed-loop performance deterioration.
As a starting point of this research, a similar analysis was done for MPC by representing the cost function as functions of sensitivity, complementary sensitivity functions and weighting matrices assuming that the disturbance energy distribution has low-pass characteristics. The same impact of uncertainty in robust controllers was observed in MPC. In robust control theory, the sensitivity and complementary sensitivity functions are tuned to adjust the bandwidth. In case of MPC, the closed-loop bandwidth is determined by the weighting matrices on controlled variables (CV) and manipulated variables (MV). Keeping the penalty on CV constant, increasing the penalty on MV reduces the closed-loop bandwidth and vice versa. In Figure 1, the link between the closed-loop bandwidth and the variance of the key output (i.e. performance measurement) is presented. In case of no modelling uncertainty, a large bandwidth, which corresponds to small penalty on MV, leads to a low variance in output (blue dashed line). On the other hand, in case of model-plant mismatch, increasing the bandwidth further beyond a certain frequency results in a raise in output variance. In summary, there exists an optimal bandwidth which gives a good trade-off between robustness and nominal performance.
Figure 1. Relation between bandwidth and variance of key output
To find the tuning parameters corresponding to the optimal bandwidth, the method of controller matching is proposed by matching MPC controller to an controller, which is directly linked to the desired closed-loop bandwidth. This method is presented in  and other methods of controller matching are introduced in  and . Using the method of controller matching, we propose a tuning procedure to obtain the optimal closed-loop bandwidth consisting of the following steps:
· Find the corresponding MPC tuning parameters by solving inverse optimality problem.
· Compare the performance of the new tuning to the previous one. If the performance degrades, stop increasing the closed-loop bandwidth and use the previous tuning as the final one. Otherwise, repeat the previous steps until the performance degrades.
The procedure described above not only provides initial tuning during commissioning but can also be used to adjust tuning parameters when a performance degradation occurs in closed loop operation due to changes in the process. In this presentation, we will discuss the procedure and its implementation on an industrially relevant distillation column model.
 Qin, S.J. and Badgwell, T.A. A survey of industrial model predictive control technology. Control Engineering Practice, 11, 733-764, 2003.
 Skogestad, S. and Postlethwaite I. Multivariable Feedback Control. Wiley, second edition, August 2005.
 Özkan , L. , Meijs, J. and Backx, A.C. P.M., A Frequency Domain Approach For MPC Tuning, will be presented in PSE 2012.
 Rowe, C. and Maciejowski, J. Robust finite horizon MPC without terminal constraints. Proceedings of the 39th IEEE Conference on Decision and Control. December 2000.
 The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement n◦ 257059, the ‘Autoprofit' project (www.fp7-autoprofit.eu).