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Unformatted text preview: EE363 Prof. S. Boyd EE363 homework 5 1. One-step ahead prediction of an autoregressive time series. We consider the following autoregressive (AR) system p t +1 = αp t + βp t- 1 + γp t- 2 + w t , y t = p t + v t . Here p is the (scalar) time series we are interested in, and y is the scalar measurement available to us. The process noise w is IID zero mean Gaussian, with variance 1. The sensor noise v is IID Gaussian with zero mean and variance 0 . 01. Our job is to estimate p t +1 , based on knowledge of y ,...,y t . We will use the parameter values α = 2 . 4 , β = − 2 . 17 , γ = 0 . 712 . (a) Find the steady state covariance matrix Σ x of the state x t = p t p t- 1 p t- 2 . (b) Run three simulations of the system, starting from statistical steady state. Plot p t for each of your three simulations. (c) Find the steady-state Kalman filter for the estimation problem, and simulate it with the three realizations found in part (b). Plot the one-step ahead prediction error for the three realizations. (d) Find the variance of the prediction error, i.e. , E (ˆ p t − p t ) 2 . Verify that this is consistent with the performance you observed in part (c). 2. Performance of Kalman filter when the system dynamics change. We consider the Gauss-Markov system x t +1 = Ax t + w t , y t = Cx t + v t , (1) with v and w are zero mean, with covariance matrices V > 0 and W ≥ 0, respectively. We’ll call this system the nominal system . We’ll consider another Gauss-Markov sysem, which we call the perturbed system : x t +1 = ( A + δA ) x t + w t , y t = Cx t + v t , (2) where δA ∈ R n × n . Here (for simplicity) C , V , and W are the same as for the nominal system; the only difference between the perturbed system and the nominal system is that the dynamics matrix is A + δA instead of A . 1 In this problem we examine what happens when you design a Kalman filter for the nominal system (1), and use it for the perturbed system (2). Let L denote the steady-state Kalman filter gain for the nominal system (1), i.e. , the steady-state Kalman filter for the nominal system is ˆ x t +1 = A ˆ x t + L ( y t − ˆ y t ) , ˆ y t = C ˆ x t . (3) (We’ll assume that ( C,A ) is observable and ( A,W ) is controllable, so the steady-state Kalman filter gain L exists, is unique, and A − LC is stable.) Now suppose we use the filter (3) on the perturbed system (2)....
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This document was uploaded on 10/05/2011.
- Winter '09