Hw5 - EE363 Prof S Boyd EE363 homework 5 1 One-step ahead prediction of an autoregressive time series We consider the following autoregressive(AR

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)....
View Full Document

This document was uploaded on 10/05/2011.

Page1 / 5

Hw5 - EE363 Prof S Boyd EE363 homework 5 1 One-step ahead prediction of an autoregressive time series We consider the following autoregressive(AR

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online