A general approach to control non-linear uncertain systems is to apply a pre-computed nominal optimal control, and use a pre-computed LQG compensator to generate control corrections from the on-line measured data. If the non-linear model, on which the optimal control and LQG compensator design is based, is of sufficient quality, and when the LQG compensator is designed appropriately, the closed-loop control system is approximately optimal. This paper contributes to the selection and computation of the time-varying LQG weighting and noise matrices, which determine the LQG compensator design. It is argued that the noise matrices may be taken time-invariant and diagonal. Three very important considerations concerning the selection of the time-varying LQG weighting matrices are turned into a concrete computational scheme. Thereby, the selection of the time-varying LQG weighting matrices is reduced to selecting three scalar design parameters, each one weighting one consideration. Although the three considerations seem straightforward they may oppose one another. Furthermore, they usually result in time-varying weighting matrices that are indefinite, rather than positive (semi) definite as required for the LQG design. The computational scheme presented in this paper addresses and resolves both problems. By two numerical examples the benefits of our optimal closed-loop control system design are demonstrated and evaluated using Monte Carlo simulation.
van Willigenburg, L. G., & de Koning, W. L. (2006). On the synthesis of time-varying LQG weights and noises along optimal control and state trajectories. Optimal Control Applications and Methods, 27(3), 137-160. https://doi.org/10.1002/oca.770