Discrete-Time Smoothing

Abstract

1 IntroductionObservations are invariably accompanied by measurement noise and optimal filters are the usual solution of choice. Filter performances that fall short of user expectations motivate the pursuit of smoother solutions. Smoothers promise useful mean-square-error improvement at mid-range signal-to-noise ratios, provided that the assumed model parameters and noise statistics are correct.In general, discrete-time filters and smoothers are more practical than the continuous-time counterparts. Often a designer may be able to value-add by assuming low-order discretetime models which bear little or no resemblance to the underlying processes. Continuoustime approaches may be warranted only when application-specific performance considerations outweigh the higher overheads.This chapter canvasses the main discrete-time fixed-point, fixed-lag and fixed interval smoothing results [1] - [9]. Fixed-point smoothers [1] calculate an improved estimate at a prescribed past instant in time. Fixed-lag smoothers [2] -[3] find application where small end-to-end delays are tolerable, for example, in press-to-talk communications or receiving public broadcasts. Fixed-interval smoothers [4] - [9] dispense with the need to fine tune the time of interest or the smoothing lags. They are suited to applications where processes are staggered such as delayed control or off-line data analysis. For example, in underground coal mining, smoothed position estimates and control signals can be calculated while a longwall shearer is momentarily stationary at each end of the face [9]. Similarly, in exploration drilling, analyses are typically carried out post-data acquisition.The smoother descriptions are organised as follows. Section 7.2 sets out two prerequisites: time-varying adjoint systems and Riccati difference equation comparison theorems. Fixedpoint, fixed-lag and fixed-interval smoothers are discussed in Sections 7.3, 7.4 and 7.5, respectively. It turns out that the structures of the discrete-time smoothers are essentially the same as those of the previously-described continuous-time versions. Differences arise in the calculation of Riccati equation solutions and the gain matrices. Consequently, the treatment "An inventor is simply a person who doesn't take his education too seriously. You see, from the time a person is six years old until he graduates from college he has to take three or four examinations a year. If he flunks once, he is out. But an inventor is almost always failing. He tries and fails maybe a thousand times. If he succeeds once then he's in. These two things are diametrically opposite. We often say that the biggest job we have is to teach a newly hired employee how to fail intelligently. We have to train him to experiment over and over and to keep on trying and failing until he learns what will work." Charles Franklin Kettering www.intechopen.com is somewhat condensed. It is reaffirmed that the above-mentioned smoothers outperform the Kalman filter and the minimum-variance smoother provides the best performance. Prer...