Up to this point, we have discussed what Kalman ®lters are and how they are supposed to behave. Their theoretical performance has been shown to be characterized by the covariance matrix of estimation uncertainty, which is computed as the solution of a matrix Riccati differential equation or difference equation. However, soon after the Kalman ®lter was ®rst implemented on computers, it was discovered that the observed mean-squared estimation errors were often much larger than the values predicted by the covariance matrix, even with simulated data | Kalman Filtering Theory and Practice Using MATLAB Second Edition. Mohinder S. Grewal Angus P. Andrews Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-39254-5 Hardback 0-471-26638-8 Electronic 6 Implementation Methods There is a great difference between theory and practice. Giacomo Antonelli 1806-1876 1 CHAPTER FOCUS Up to this point we have discussed what Kalman filters are and how they are supposed to behave. Their theoretical performance has been shown to be characterized by the covariance matrix of estimation uncertainty which is computed as the solution of a matrix Riccati differential equation or difference equation. However soon after the Kalman filter was first implemented on computers it was discovered that the observed mean-squared estimation errors were often much larger than the values predicted by the covariance matrix even with simulated data. The variances of the filter estimation errors were observed to diverge from their theoretical values and the solutions obtained for the Riccati equation were observed to have negative variances an embarrassing example of a theoretical impossibility. The problem was eventually determined to be caused by computer roundoff and alternative implementation methods were developed for dealing with it. This chapter is primarily concerned with 1. how computer roundoff can degrade Kalman filter performance 2. alternative implementation methods that are more robust against roundoff errors and 3. the relative computational costs of these alternative implementations. in a letter to the Austrian Ambassador as quoted by Lytton Strachey in Eminent Victorians 101 Cardinal Antonelli was addressing the issue of papal infallibility but the same might be said about the infallibility of numerical processing systems. 202 CHAPTER FOCUS 203 Main Points to Be Covered The main points to be covered in this chapter are the following 1. Computer roundoff errors can and do seriously degrade the performance of Kalman filters. 2. .
