13 4 Scalar Kalman Filter Data Model To derive the Kalman filter we need the data model s n as n 1 u n State Equation x n s n w n Observation Equation Assumptions 1 u n is zero mean Gaussian White E u 2 n u2 2 w n is zero mean Gaussian White E w 2 n n2 3 The initial state is s 1 N s s2 4 u n w n and s 1 are all independent of each other Can vary with time To simplify the derivation let s 0 we ll account for this later 1 Goal and Two Properties Goal Recursively compute s n n E s n x 0 x 1 x n Notation X n is set of all observations x n is a single vector observation X n x 0 x 1 x n T Two Properties We Need 1 For the jointly Gaussian case the MMSE estimator of zero mean based on two uncorrelated data vectors x1 x2 is see p 350 of text E x1 x 2 E x1 E x 2 2 If 1 2 then the MSEE estimator is E x E 1 2 x E 1 x E 2 x a result of the linearity of E operator 2 Derivation of Scalar Kalman Filter Recall from Section 12 6 Innovation x n x n x n n 1 By MMSE Orthogonality Principle MMSE estimate of x n given X n 1 prediction E x n X n 1 0 x n is part of x n that is uncorrelated with the previous data x n Now note X n is equivalent to X n 1 Why Because we can get get X n from it as follows X n 1 X n 1 X n x n x n x n x n n 1 ak x k k 0 x n n 1 3 What have we done so far Have shown that X n X n 1 x n uncorrelated Have split current data set into 2 parts 1 Old data 2 Uncorrelated part of new data just the new facts x n s n n E s n X n E s n X n 1 Because of this So what Well can now exploit Property 1 s n n E s n X n 1 E s n x n s n n 1 prediction of s n based on past data Update based on innovation part of new data Now need to look more closely at each of these 4 Look at Prediction Term s n n 1 Use the Dynamical Model it is the key to prediction because it tells us how the state should progress from instant to instant s n n 1 E s n X n 1 E as n 1 u n X n 1 Now use Property 2 s n n 1 a E s n 1 X n 1 E u n X n 1 s n 1 n 1 By Definition E u n 0 By independence of u n X n 1 See bottom of p 433 in textbook s n n 1 as n 1 n 1 The Dynamical Model provides the update from estimate to prediction 5 Look at Update Term E s n x n Use the form for the Gaussian MMSE estimate E s n x n E s n x n x n 2 E x n x n x n x n n 1 k n So E s n x n k n x n x n n 1 s n n 1 w n n 1 by Prop 2 Prediction Shows Up Again Put these Results Together s n n s n n 1 k n x n s n n 1 as n 1 n 1 How to get the gain 0 Because w n is indep of x 0 x n 1 This is the Kalman Filter 6 Look at the Gain Term Need two properties A E s n x n s n n 1 E s n s n n 1 x n s n n 1 Aside x y x z y for any z y Linear combo of past data thus w innovation B E w n s n s n n 1 0 x n x n n 1 x n The innovation proof w n is the measurement noise and by assumption is indep of the dynamical driving noise u n and s 1 In other words w n is indep of everything dynamical So E w n s n 0 s n n 1 is based on past data which include w 0 w n 1 and since the measurement noise has indep samples we get s n n 1 w n 7 So we start with the gain as defined above Plug in for innovation E s n x n E s n x n s n n 1 k n 2 E x n E x n s n n 1 2 Use Prop A in num Use x n s n w n in denominator E s n s n n 1 x n s n n 1 E s n s n n 1 w n 2 E s n s n n 1 s n s n n 1 w n E s n s n n 1 w n 2 Use x n s n w n in numerator E s n s n n 1 2 E s n s n n 1 w n E s n s n n 1 2 M n n 1 2 n Expand 2 E s n s n n 1 w n 0 by Prop B MSE when s n is estimated by 1 step prediction 8 This gives a form for the gain k n M n n 1 n2 M n n 1 This balances the quality of the measured data against the predicted state In the Kalman filter the prediction acts like the prior information about the state at time n before we observe the data at time n 9 Look at the Prediction MSE Term But now we need to know how to find M n n 1 E as n 1 u n as n 1 n 1 E a s n 1 s n 1 n 1 u n M n n 1 E s n s n n 1 2 2 2 Use dynamical model exploit form for prediction Cross terms 0 Est Error at previous time M n n 1 a 2 M n 1 n 1 u2 Why are the cross terms zero Two parts 1 s n 1 depends on u 0 u n 1 s 1 which are indep of u n 2 s n 1 n 1 depends on s 0 w 0 s n 1 w n 1 which are indep of u n 10 Look at a Recursion for MSE Term M n n By def M n n E s n s n n 2 E s n s n n 1 k …