lecture18

lecture18 - 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 5 Lecture 18 Last time: Semi-free configuration design This is equivalent to: Note , ns enter the system at the same place. F is fixed. We design C (and perhaps ) B . We must stabilize F if it is given as unstable. () 1( ) ( ) ( ) Cs Hs CsFsBs = + so that having the optimum H , we determine C from ) ( ) ( ) HsFsBs = We do not collect H and F together because if F is non-minimum phase , we would not wish to define H by opt HF H F = This leads to an unstable mode which is not observable at the output – thus cannot be controlled by feeding back. Associate weighting functions with the given transfer functions. H F D w t Fs Ds
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 5 If () Fs is unstable , put a stabilizing feedback around it, later associate it with the rest of the system. Error Analysis We require the mean squared error. 11 1 22 2 1 2 33 3 2 ( )( ) ( ) ( ) ( ) ) 2 () () H F FH D ct w it d ot w d dw i t dt w st d et ot dt otdt ττ τ −∞ −∞ ∞∞ −∞ −∞ −∞ =− + ∫∫ 2 1 2 44 3 4 1 2 3 4 ( ) ) ) ( ) ( ) HFHF HFH o t i t i t i t i t d −∞ −∞ −∞ −∞ ∞∞ ∞ ∞ −∞ −∞ −∞ −∞ −∞ −∞ ⎡⎤ ⎢⎥ ⎣⎦ = ∫∫∫∫ 1 2 3 4 ()( ) Fi i wR −∞ −∞ +−− 1 2 3 1 2 3 1 2 3 () () ) ) ( )( ) () ( ) D HFD i s d w i t s t R −∞ −∞ −∞ ∞∞ ∞ −∞ −∞ −∞ −∞ −∞ −∞ =+ ∫∫∫ We shall not require 2 in integral form.
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16.322 Stochastic Estimation and Control, Fall 2004
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lecture18 - 16.322 Stochastic Estimation and Control, Fall...

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