Model predictive control

In the model predictive control (MPC) strategy a dynamic model of a process is applied, to simulate the future evolution of the process depending on possible simulated values of the controlled variable. Typically the future evolution will only be calculated up to a predefined prediction horizon.

Using an optimization algorithm the best value of the controlled variable is calculated using a cost function. Due to the fact that a differential equation system must be solved on-line, MPC is computationally demanding. Therefor for MPC a state estimator as well as a controller is required.

The better understanding of penicillin formation mechanisms, morphological features and the role of mycelia for the synthesis led Ashoori et al. [57] to implement a detailed unstructured model of penicillin production in a fed-batch fermenter. This model is used to implement a non-linear MPC (NMPC) for the control of the feed rate to increase the penicillin formation. As controller input they are applying the on-line measurements of pH and temperature. They propose the performance of a novel cost function applying the inverse of the product rather than the common quadratic regulation. This is implemented to avoid ordinary differential equation solver problems where it is not possible to guarantee the efficiency of set-point tracking. They are comparing the control performance to a regular auto-tuned PID controller and identify the NMPC as superior with higher process yields. The NMPC is controlling the acid as well as the base flow and the cooling water

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system. Due to the more sophisticated model the control reaches better performance than a previous work by Birol et al. [58]. To face the computational cost of this more detailed model they are proposing the application of a locally linear model tree (LoLiMoT) in order to simplify the original non-linear model, which is described in the next section.

Certainly due to the high computational power that needs to be provided for a MPC many of the NMPC approaches are still only simulation proven and not yet applied to real processes. Santos et al.

[59] are working on simulated E. coli NMPC controlled cultivations. They assume the measurement of the substrate concentration and keep the specific growth rate at maximum oxidative capacity as well as inhibiting the product formation. They applied a special NMPC scheme named min-max based robustness consideration. Another new NMPC method as well as a comparative performance assessment is applied by Kawohl et al. [10]. They are comparing the performance of NMPC and NMPC – EKF for input signal prediction to a method called on-line trajectory planning (OT). OT is basically an NMPC in which the estimation horizon is extended to the end of the cultivation. If the system is strongly disturbed, this method has certain advantages for the estimation in order to get back to optimal productivity, however at the cost of computational power. The experiments were carried out through Monte Carlo simulations, simulating experiments through disturbance scenarios.

The aim of the experiments was to maintain the optimal productivity of the product penicillin. The authors are showing the potential of this closed loop control by improving the mean productivity by 25 % for the MPC and 28 % for the OT method compared to open loop control, where these methods especially increase the minimum productivity due to disturbances.

3.6.1 ANN Fuzzy Hybrid based estimation for NMPC control

A possibility to decrease the complexity of nonlinear models in control algorithms like MPC is given by locally linear models, which are applied in a hybrid structure combining neuronal network and fuzzy logic abilities. The basic structure is displayed in Figure 3.6.2. Each neuron in the hidden layer consists of a membership function and a local linear model (LLM). The arguments of the membership function are the input value xi. The function value itself indicates the validity of the corresponding LLM, which is in fact a multi linear regression model. The estimate of this model type is the sum of the LLM output weighted by the normalized membership function.

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Figure 3.6.1: Basic structure of a Fuzzy-ANN local linear model

The algorithm was successfully applied by Ashoori et al [57] to generate a Neuro–Fuzzy model to replace equations in a mass balance model for penicillin formation. The authors assessed the resulting computational costs as very acceptable for a real time process. They show results which are comparable to results generated by the whole model. Although the method is rarely applied for biotechnological applications, it gives opportunities to overcome frequently mentioned computational limits.

Simulation studies for the optimal model training, parameter identification and comparisons between the closed loop performances are presented by Xu et al. [60-62]. Among others they employed the LOLIMOT algorithm to achieve optimal parameters for the membership function as well as for the LLM. The LOLIMOT algorithm is an incremental tree-based learning algorithm. A detailed description can be found by Nelles [63]. The algorithm adds consecutive locally linear model neurons and thereby optimizes the error of calibration. Obviously a high number of neurons will describe a trajectory best, but will possibly not decrease the computational power that is needed.

3.6.2 ANN based estimation for NMPC

Meleiro et al. [64] presented results of a MPC strategy of a fuel-ethanol fermentation process using simulations. A neural network has been applied as internal model for the controller. The authors used an optimization algorithm to determine a neural network structure as well as the shape of their activation functions guiding to parsimonious network architecture. The inputs were the feed flow rate, cells recycle rate and flash recycle rate; the output were the biomass, substrate and product concentration. The authors presented results demonstrating successfully the control of the biomass,

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substrate and ethanol concentration with varying set-points between 37 and 32 gL^-1, 10 and 3 gL^-1 and 45 and 40 gL^-1, respectively.