6. Continuous-Time Gauss—Markov Systems: Continuous-Time Kalman Filter, Stationarity, Power Spectral Density, and the Wiener Filter
Society for Industrial and Applied Mathematics eBooks · 2008
In this chapter, we derive the continuous linear minimum variance filter using the orthogonal projection lemma of Chapter 4 and the stochastic Gauss-Markov processes of Chapter 5. With the assumption that the additive noise is Gaussian, the filter becomes a continuous-time conditional mean estimator. If the additive noise is uncorrelated, the best linear filter is obtained and is equivalent to the Kalman filter in structure. We then specialize to infinite-time, time-invariant systems. In the stochastic context we study what is called stationary processes. This gives us the opportunity to introduce transform techniques on time-correlated processes such as autocorrelation matrix functions to obtain power spectral matrix functions in the transform frequency variable. In this way the frequency derivation of the infinite-time time-invariant Kalman filter, known as the Wiener filter, can be obtained. 6.1 The Continuous-Time Kalman Filter (Kalman-Bucy Filter) One year after he introduced the Kalman filter, Kalman cowrote a paper with Bucy that introduced a continuous-time version of the filter [27]. For this reason, the continuous-time filter is sometimes called the Kalman-Bucy filter. Historically, this filter has been used more for theory than practice, but advances in computer technology and speeds are enabling engineers to directly use continuous-time designs on microprocessors, and some of this may find its way into Kalman filter applications. Consider a continuous-time system represented by a system of Itô stochastic differentials: dx (t)=F (t)x (t)dt+G (t)d βt , 6.1 dz (t)=H (t)x (t)dt+d ηt . 6.2 The inputs, βt and ηt, are Brownian motion processes.