Direct state feedback optimal control of a double integrator plant implemented by an artificial neural network

The purpose of this paper is to assess the capability of an artificial neural network
(ANN) to implement a nonlinear state feedback optimal control law for a double integrator
plant. In this case, the cost function to be minimised is the settling time subject to control
saturation constraints. The reason for selection of this cost function is that the control law is
known in the analytical form and this will be used to form a benchmark. The ultimate aim is
to apply the method to form a new direct state feedback optimal position control law for
mechanisms in which the frictional energy loss is minimised. An analytical solution is not
available in this case so first the time optimal control law is studied to enable straightforward
comparison on the ANN and directly implemented closed loop control laws. Since
Pontryagin‟s method will be used to compute the optimal state trajectories for the ANN
training in the future investigation of the minimum energy loss control, this method is applied
to derive the time optimal double integrator state trajectories to illustrate the method.
Furthermore, a modification of the time optimal control law is made that avoids the control
chatter following a position change that would occur if a practical implementation of the
basic control law, which is bang-bang, were to be attempted. Training the ANN with state
and control data could be inaccurate due to the discontinuity of the control law on the
switching boundary in the state space. This problem is overcome by the authors by instead
training the ANN with state and switching function data, as the switching function is
nonlinear but continuous, the control function, i.e., the function relating the switching
function output to the control variable, being externally implemented. The simulations
confirm that the ANN can be trained to accurately reproduce the time optimal control.

Citation:
Matieni, X., Dodds, S. and Lee, S.W., (2011) "Direct state feedback optimal control of a double integrator plant implemented by an artificial neural network" Advances in Computing and Technology, University of East London, January 2011, 234-240.