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lecture24 - 16.322 Stochastic Estimation and Control, Fall...

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 7 Lecture 24 Last time: () () () () () () TT d Xt AtXt XtAt BtNtBt dt =+ + If the system reaches a statistical steady state , the covariance matrix would be constant. The system would have to be invariant , 0 d X t AX XA BNB dt + = and the steady state covariance matrix can be solved for from this algebraic equation. Note that: X t is symmetric. If [ ] , A B is controllable and 0 () 0 > , > for all t . If, in addition, [ ] , AC is observable, then Cov ( ) 0 yt ⎡⎤ > ⎣⎦ for all . t Kalman Filter Our text treats two forms of the Kalman filter: Discrete time filter Based on the model o System: 1( ) kk x kx k w φ += + o Measurements: k zH x kv Continuous time filter o System: x Fx Gw & o Measurements: xv The most common practical situation is neither of these. In the aerospace area, and many other application areas as well, we are interested in dynamic systems that evolve continuously in time , so we must at least start with a continuous system model. The system may be driven by a known command input and may be subject to random disturbances. If the given disturbances cannot be well approximated as white, then a shaping filter must be added to shape the given disturbance spectrum from white noise.
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 7 The augmented system is modeled as () x At x Gtu Btn =++ & where nt is an unbiased white noise with correlation function () ( ) T Entn Nt t τ δτ ⎡⎤ =− ⎣⎦ We allow the system matrices to be time varying. The most common reason for dealing with time varying linear dynamics is because we have linearized nonlinear dynamics about some trajectory. But although the system dynamics are continuous in time, we do not process measurements continuously. The Kalman filter involves significant computations, and those computations are universally executed in a digital computer . Since a digital computer works on a cyclic basis, the measurements are processed at discrete points in time .
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lecture24 - 16.322 Stochastic Estimation and Control, Fall...

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