*This is the M.Sc. thesis of Mr. Kalle Halmevaara, graduated in 2004*

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The goal of the master's thesis was to test in practice a novel method for simultaneous tuning of multiple controllers, initially proposed by Hyötyniemi in [1]. The thesis reports the first industrial scale application of the technique and the obtained results.

Let us examine the block diagram below that illustrates a process and a system for its performance evaluation. If an input signal *u* is introduced to the system, it evokes the response signal *y* (the dynamic system can be a MIMO system). If this system is viewed in a somewhat wider scope and one concentrates on the quality of the performance rather than the actually resulting output signals, the concept of *quality measure* can be introduced. Quality measures, *q*, are characteristic figures that measure how acceptable or desirable the performance of the system is. For instance, a quality measure could be defined as the variance of the end product properties, the efficiency in power production or the setpoint tracking ability of a controlled variable.

In a statistical sense, the resulting performance expressed by means of quality measures is more or less the same as long as the system parameters are held constant. In other words, the performance is assumed to be a function of the parameters, *q* = *f*(*θ*). This gives us the opportunity to improve the performance of the system by optimizing the values of the parameters *θ*. Further, it turns out that under certain assumptions the dependency is linear in the maximum likelihood sense. The numerical problems typically plaguing the efficient estimation of high dimensional linear models are avoided by using *statistical multivariate methods*.

Since the dependency of the parameters and the quality measures can be approximated with a linear model only locally, the optimization has to be performed with iterative algorithms, such as *gradient descent method*. The figure below illustrates the iterative optimization procedure. The parameter values are slightly varied around the prevailing controller tuning, and the resulting performance is recorded. A local linear model is estimated based on this parameter - quality measure data and the parameters are updated towards the steepest gradient descent direction. These iteration steps are repeated until the performance of the system meets its objectives or a local minimum is reached.

Distinctive to this parameter tuning method is the massive utilization of computation. Iterations are performed in three different levels when applying this method: First, a dynamic simulator is used for obtaining the data, i.e., the state of the simulator is solved in every simulation time step by iteration. Secondly, similar simulation runs are repeated to obtain enough data points around the current parameter values, and, finally, the values of the parameters are iteratively tuned towards a local minimum of the optimization cost function. The tuning procedure can be summarized as a pseudocode as follows:

while (stopping condition not fulfilled) { for (kappa = 1 … k) { change the parameter values randomly; run the predefined simulation run; calculate the quality measures; } if (the data is not Gaussian) { notify the user and ask further instructions; } center and scale the data; construct the local parameter - quality measure model; determine the gradient direction; update values of the parameters; }

The examined power plant process was modelled with Apros simulation software [2], a professional tool for modeling and simulation of combustion and nuclear power plants, and pulp and paper mills.

Instead of introducing the examined process in detail, it is only remarked here that the applied model was a complex and fairly realistic system. The simulation model consisted of turbine, feed water and boiler sections of the power plant and the related controllers. Grasping the general view on a process model of that scale is a demanding task. That is, however, the very motivation behind the new tuning methodology: Since human mind is unable to comprehend the underlying interdependencies as the size of a system increases, advanced statistical multivariate methods are required to capture the emerging higher abstraction level concepts.

In the case study, the tuning concentrated in three PI(D) controllers, i.e., seven parameters altogether were tried to optimize simultaneously. The three tuning objectives, the quality measures, that were minimized were defined as

- the settling time of the produced active power after a setpoint change (into 4 percent error margin),
- the overshoot of the produced active power after a setpoint change, and
- the variance of the produced active power caused by a pressure drop in the combustion.

The figure below presents the behavior of the system before (blue) and after (red) the tuning (18 iterative tuning steps were performed). Obviously the tuning has succeeded in achieving the goals: the overshoot and the settling time after the setpoint change have become notably smaller as well as the effect of the disturbance.

In future, the research will concentrate on, e.g., tuning of ramping coefficients that determine grade change situations on paper machines, and, tuning of model parameters in order to improve the estimation accuracy of simulation models.

- Hyötyniemi, H., Towards New Languages for Systems Modeling. Proceedings of 42’nd Scandinavian Simulation Conference SIMS’02. Oulu, Finland, September 26-27, 2002.
- APROS - The Advanced Process Simulation Environment, VTT.