Bounds on Estimation Error

Abstract

In this chapter, we show that our preceding analysis can be applied to estimation problems as well. The results can be viewed as implications of the performance bounds on power gain and in variance minimization presented in the previous two chapters. In particular, we derive fundamental estimation bounds for estimation systems that are not necessarily LTI with noises that are not necessarily white Gaussian. The bounds are seen to be tight in the particular case of a scalar LTI system with white Gaussian noises, as verified by the benchmark given by the renowned Kalman filter.

It can also be shown that the Kalman filter extracts the maximum possible information about output data. If we form the residual between the measured output and the estimated output, we can show that for the Kalman filter the error (residual) is a white noise process, so there is no remaining dynamic information content in the error.

—K. J. Åström, R. M. Murray, “Feedback Systems: An Introduction for Scientists and Engineers,” 2010 [7]