1 Hybrid Modeling Based Adaptive Neural Controller AntonAndr´aˇ sik, Alois M´ esz´aros and Sebasti˜ao Feyo de Azevedo Abstract — In this paper, a new control technique for non- linear control based on hybrid modeling is proposed. The control system utilizes the well-known gradient descent, but the learning rate is adapted in each iteration step in order to accelerate the speed of convergence. It is shown that the selection of the learning rate results in stable training in the sense of Lyapunov. Advantages of this technique are illustrated by simulations where a continuous flow stirred biochemical reactor is chosen as a case study. Keywords — neural control, PID controller, hybrid model- ing I. Introduction T HE design of most neural control schemes is based on gradient optimisation such as Back-Propagation for the training of weights. Although some nonlinear con- trol problems can be handled by using these neural control schemes, in tasks for nonlinear systems with high nonlin- earities and large uncertainties, the existing neural control schemes are severely inadequate. Major drawbacks of us- ing neural networks in controlling real systems are that the ability of the neural network to adapt to system changes is too slow and there remains some degree of error. Here we will present a new control strategy based on an indirect adaptive control, where the model of the controlled process is represented by hybrid neural network. Just a few parameters of the neural controller are adapted on-line us- ing a special, stable training algorithm in the sense of Lya- punov. To demonstrate the feasibility and the performance of this control scheme, a continuous-flow stirred biochem- ical reactor model has been chosen as a simulation case study. Simulation results demonstrate the usefulness and the robustness of the control system proposed. II. Hybrid neural networks By combining black box techniques with a physical model framework, hybrid models are obtained that com- bine first principles knowledge with the ability to deal with complex, poorly understood behavior. A partial model is derived from simple physical considerations (such as mass or energy balances), while a black box technique is used to augment the model. Hybrid models are especially suited to describe highly nonlinear behavior over a large operating domain. Examples are models of batch or fed-batch pro- cesses, cyclic processes or distributed parameter processes, such as plug flow reactors. Combining black box techniques with physical equations A. Andr´ aˇ sik and A. M´ esz´ aros are with the Slovak University of Technology, Faculty of Chemical and Food Technology, Radlinsk´ eho 9, 812 37 Bratislava, Slovakia. E-mail: andrasik@chtf.stuba.sk, ameszaro@cvt.stuba.sk S.F. de Azevedo is with the Faculdade de Engenharia de Universi- dade do Porto, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal. E-mail: sfeyo@fe.up.pt has received some attention since the early 1990’s. There are several approaches to hybrid modeling discussed in lit- erature [1]. In [2], a hybrid model of a fed-batch biore- actor is developed, in which an artificial neural network augments the performance of a parametric model that de- scribes the specific kinetic rates, such as biomass growth and substrate consumption. The combined output of the parametric model and the ANN is processed by an out- put model which calculates the system state. In compari- son with conventional approaches the hybrid modeling is a powerful tool for process modeling, particularly when lim- ited theoretical knowledge of the process is available [3]. However, for the hybrid neural network model, target outputs are not directly available. In this case, the known partial process model can be used to calculate a suitable er- ror signal that can be used to update the network’s weights. The observer error between the structured model’s predic- tions and the actual state variable measurements can be “back-propagated” through the known set of equations and translated into an error signal for the neural network com- ponent. Let’s consider a process described by differential equa- tion dy dt = f (y(t),u(t),w) (1) where f is a nonlinear vector function of the system in- puts u(t), the system outputs y(t) and some parameters w, which we assume to be represented by means of a neural network. In order to train the neural network (black-box part of the hybrid model), pair of input/output data vec- tors (u,y) must be available, e.g., as a set of past measure- ments. The training consists in adaptation of network’s weights w in such a way that the sum of the squared devi- ations between the output data predicted by the network y i and the corresponding target data y * i becomes minimal J = 1 2 N  i=1 (y i − y * i ) 2 (2) The usual way to minimize J is to use gradient procedures, like the steepest-descent algorithm. Weights in the n-th step of this iterative process are changed in the direction of gradient w ij,n+1 = w ij,n − α ∂J ∂w ij,n (3) In consideration of equation (2) the derivation of J with respect to w ij,n gives ∂J ∂w ij,n = N  i=1  (y i − y * i ) ∂y i ∂w ij,n  (4) Now the problem consists in determining the derivatives ∂y ∂w . One possibility is a method called sensitivity approach.