(186e) MPC Tuning Based On Impact of Modelling Uncertainty On Closed-Loop Performance

MPC
tuning based on impact of modelling uncertainty

on
closed-loop performance[1]

Quang
N. Tran, Leyla Özkan, A.C.P.M. Backx

Eindhoven
University of Technology

Den Dolech 2,
5612 AZ Eindhoven, The Netherlands

n.q.tran@tue.nl;
l.ozkan@tue.nl; a.c.p.m.backx@tue.nl

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 ([1]) 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 ([2]) 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 [3] and other methods of controller matching are introduced in [4]
and [5]. Using the method of controller matching, we propose a tuning procedure
to obtain the optimal closed-loop bandwidth consisting of the following steps:

·       Design
a suitable  controller to ensure
robustness using the model and its uncertainty bound.

·       Find
the corresponding MPC tuning parameters by solving inverse optimality problem.

·       Shrink
the uncertainty bound; raise the closed-loop bandwidth of the controller. Find the
corresponding MPC tuning parameters.

·       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.

REFERENCE

[1] Qin, S.J.
and Badgwell, T.A. A survey of industrial model predictive control technology. Control
Engineering Practice, 11, 733-764, 2003.

[2] Skogestad,
S. and Postlethwaite I. Multivariable Feedback Control. Wiley, second
edition, August 2005.

[3] Özkan , L. ,
Meijs, J. and Backx, A.C. P.M., A
Frequency Domain Approach For MPC Tuning, will be presented in PSE 2012.

[4] Rowe, C. and
Maciejowski, J. Robust finite horizon MPC without terminal constraints.
Proceedings of the 39^th IEEE Conference on Decision and Control.
December 2000.

[5] Rowe, C. and
Maciejowski, J. Tuning MPC using loop
shaping. Proceedings of the American Control Conference, June 2000.