Robust Feedforward Control Based on Probabilistic Description of Model Errors

Mikael Sternad and Anders Ahlén

Swedish Control Meeting ``Reglermöte'',
Gothenburg, Sweden, November 4-5, 1992.

In Pdf.

Outline:

We present a simple and flexible method for building performance robustness into the design of linear filters for open loop problems. Such problems include signal estimation, state estimation and feedforward control. The key is a novel design method for Wiener filters. It can be regarded as a generalization of Kucera's polynomial approach. Here, we briefly describe how the method is applied to the design of robust feedforward links from disturbances or from command signals.

Abstract:

Can the performance robustness of a feedforward controller be improved? For any such compensator, Bodes relative sensitivity function is =1. This means that a x % parameter deviation will result in a y % deviation of the transfer function at a given frequency, regardless of the applied filter. However, the effect of the y % deviation on e.g. the step response will very much depend on the magnitude of the transfer function. (The absolute sensitivity function describes this effect, while the commonly used relative one misses it completely!) When the performance of a nominal LQG controller is sensitive to model errors, our robustified control achieves a significant (absolute) sensitivity reduction. There is only a small performance deterioration in the nominal case, if the assumed modelling errors are not extremely large.

Modelling errors will be described by sets of systems, parametrized by random variables with known covariances. Robust control is then obtained by minimizing a quadratic criterion, averaged both with respect to model errors and the noise. A polynomial solution, based on averaged spectral factorizations and a Diophantine equation, is presented.

Robust design turns out to be no more complicated than the design of an ordinary LQG feedforward controller. The proposed method is very simple to use, and it also has two other advantages. First, probabilistic descriptions of model uncertainties may have soft bounds . These are more readily obtainable in a noisy environment than the hard bounds required for e.g. minimax design. Furthermore, not only the range of uncertainties, but also their likelihood is taken into account; common model deviations will have a greater impact on a controller design than do very rare ``worst cases''. The conservativeness is thus reduced, compared to worst case design.

Related publications:

Paper in Automatica 1993, with robust Wiener design and a feedforward design example.
Paper in IEEE Trans. AC 1995, on robust MIMO Wiener filters and feedforward controllers.
PhD Thesis by Kenth Öhrn May 1996.
Paper in Automatica 1988, on nominal LQG design of feedforward regulators.