A neural network controller for systems with unmodeled dynamics with applications to wastewater treatment

This paper considers the use of neural networks (NNs) in controlling a nonlinear, stochastic system with unknown process equations. The approach here is based on using the output error of the system to train the NN controller without the need to assume or construct a separate mode (NN or other type) for the unknown process dynamics. To implement such a direct adaptive control approach, it is required that connection weights in the NN be estimated while the system is being controlled. As a result of the feedback of the unknown process dynamics, however, it is not possible to determine the gradient of the loss function for use in standard (back-propagation-type) weight estimation algorithms. In principle, stochastic approximation algorithms in the standard (Kiefer–Wolfowitz) finite-difference form can be used for this weight estimation since they are based on gradient approximations from available system output errors. However, these algorithms will generally require a prohibitive number of observed system outputs. Therefore, this paper considers the use of a new stochastic approximation algorithm for this weight estimation, which is based on a ‘simultaneous perturbation’ gradient approximation. It is shown that this algorithm can greatly enhance the efficiency over more standard stochastic approximation algorithms based on finite-difference gradient approximations. The approach will be illustrated on a simulated wastewater treatment system with stochastic effects and nonstationary dynamics.