CHAPTER FOCUS Estimation Problem This is the problem of estimating the state of a linear stochastic system by using measurements that are linear functions of the state. We suppose that stochastic systems can be represented by the types of plant and measurement models (for continuous and discrete time) shown as Equations ± in Table , with dimensions of the vector and matrix quantities as shown in Table . | Kalman Filtering Theory and Practice Using MATLAB Second Edition Mohinder S. Grewal Aigus P. Andrews Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-39254-5 Hardback 0-471-26638-8 Electronic 4 Linear Optimal Filters and Predictors Prediction is difficult especially of the future. Attributed to Niels Henrik David Bohr 1885-1962 CHAPTER FOCUS Estimation Problem This is the problem of estimating the state of a linear stochastic system by using measurements that are linear functions of the state. We suppose that stochastic systems can be represented by the types of plant and measurement models for continuous and discrete time shown as Equations in Table with dimensions of the vector and matrix quantities as shown m Table . The symbols Sfk and ft .v stand for the Kronecker delta function and the Dirac delta function actually a generalized function respectively. TABLE Linear Plant and Measurement Models Model Continuous Time Discrete Time Equation Number Plant x t F t x t w t xk xk-1 wk-1 Measurement z t H t x t v t zk nkxk vk Plant noise E w t 0 E w t wT s 5 t - s Q t E wk 0 E wkwf A k - i Qk Observation noise E Mt 0 E v t vT s 5 t - s R t E vk 0 E vkvf A k - i Rk 114 CHAPTER FOCUS 115 TABLE Dimensions of Vectors and Matrices in Linear Model Symbol Dimensions Symbol Dimensions x w n x 1 O Q n x n z v f. x 1 H l x n R I X I A S scalar The measurement and plant noise vk and wk are assumed to be zero-mean Gaussian processes and the in iti a I value x0 is a Gaussian variate with known mean x0 and known covariance matrix Po. Although the noise sequences wk and vk are assumed to be uncorrelated the derivation m Section will n nrwe tins restriction and modify the estimator equations accordingly. The objective will be to find an estimate of the n state vector xk represented by xk a linear function of the measurements z - . zk that minimizes the weighted mcan-squared error E xk - xk T xi - xk 4-6 .
