The first seven chapters were an extended introduction to the next topic: Kalman filtering. Recall from Chapter 1 that there are two primary differences between least-squares estimators and the Kalman filter: (1) the Kalman filter models stochastic (possibly nonstationary) systems while least squares is used for deterministic systems, and (2) the Kalman filter processes measurements recursively while most least-squares implementations use batch processing. However, the second difference is not unique as Chapter 7 described recursive implementations of least-squares estimators. Hence the ability of Kalman filters to model nonstationary dynamic systems driven by random process noise is its most important characteristic.
Much of Kalman filter theory has already been presented in previous chapters. Chapter 2 discussed stochastic dynamic models, integrated a continuous stochastic dynamic model over a finite time interval (eq. 2.2-14), defined the state transition matrix (eq. 2.2-7), and derived the state noise covariance matrix (eq. 2.2-20). Chapter 4 discussed the minimum variance estimator and showed how the recursive measurement update (eqs. 4.3-34 and 4.3-35) can be derived by setting the cost gradient to zero or using the orthogonality principle. We only need to connect these concepts to derive the discrete Kalman filter. Many of the solution techniques in Chapter 5 also apply to Kalman filtering.
Most of the material in this chapter is well covered ...