Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the best choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy. (C) 2004 Elsevier B.V. All rights reserved.

A helicopter named Dolly: Behavioural cloning for autonomous helicopter control

This paper considers the pros and cons of using behavioural cloning for the development of low-level helicopter automation modules. Over the course of this project several Behavioural cloning approaches have been investigated. The results of the most effective Behavioural cloning approach are then compared to PID modules designed for the same aircraft. The comparison takes into consideration development time, reliability, and control performance. It has been found that Behavioural cloning techniques employing local approximators and a wide state-space coverage during training can produce stabilising control modules in less time than tuning PID controllers. However, performance and reliabity deficits have been found to exist with the Behavioural Cloning, attributable largely to the time variant nature of the dynamics due to the operating environment, and the pilot actions being poor for teaching. The final conclusion drawn here is that tuning PID modules remains superior to behavioural cloning for low-level helicopter automation.