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Unformatted text preview: EE363 Prof. S. Boyd EE363 homework 5 1. Onestep 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 steadystate Kalman filter for the estimation problem, and simulate it with the three realizations found in part (b). Plot the onestep 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 GaussMarkov 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 GaussMarkov 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 steadystate Kalman filter gain for the nominal system (1), i.e. , the steadystate 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 steadystate 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

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