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 () () 1()()() Hs Cs 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 Hs w t Fs w t Ds w t → → → 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. 111 222 22 11 12 333 22 2 () ( )( ) () ( )( ) () ()( ) () ( )( ) () () () () () 2 () () () H F FH D ct w it d ot w ct d dw dw it dt w st d et ot dt et ot otdt dt τττ τττ τ τττ ττ τττ ∞ ?∞ ∞ ?∞ ∞∞ ?∞ ?∞ ∞ ?∞ =? =? ? =? =? =? + ∫ ∫ ∫∫ ∫ 2 22 11 12 44 33 34 11 22 33 44 12 34 11 22 33 () ( ) ( )( ) ( ) ( )( ) () () () ()( )( ) () () () FH FH HFHF HFH ot dw dw it dw dw it dw dw dw dw it it dw dw dw d τ τττ ττ ττττ ττ ττ ττ ττ ττ ττ ττ ττ ττ ττ ∞∞ ∞∞ ?∞ ?∞ ?∞ ?∞ ∞∞∞∞ ?∞ ?∞ ?∞ ?∞ ∞∞ ?∞ ?∞ ???? ? ? ???? ???? = ∫∫ ∫∫ ∫∫∫∫ ∫∫ 44 1234 ()( ) Fii wRττττττ ∞∞ ?∞ ?∞ +?? ∫∫ 22 11 12 33 3 11 22 33 12 3 11 22 33 123 () () ( ) ( )( ) ( )( ) () () ()( )( ) () () () ( ) FH D HFD HFDis otdt d w d w it d w st dw dw dw it st dw dw dw R τ τττ ττ ττ τ ττ ττ ττ ττ τ ττ ττ τττττ ∞∞ ∞ ?∞ ?∞ ?∞ ∞∞∞ ?∞ ?∞ ?∞ ∞∞∞ ?∞ ?∞ ?∞ ???? =?? ???? ???? ? =+ ∫∫ ∫ ∫∫∫ ∫∫∫ We shall not require 2 ()dt in integral form. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 5 The problem now is to choose () H wt so as to minimize this 2 ()et , for which we use variational calculus. Let: 0 () () () H wt wt wtδ=+ where 0 ()wt is the optimum weighting function (to be determined) and ()wtδ is an arbitrary variation – arbitrary except that it must be physically realizable. Calculate the optimum 2 e and its first and second variations. 22 222 0 22 2 () 2 () () () ee e e eot otdtdt δδ=+ + =+ + The optimum 2 e ( 2 e for () 0wtδ = ): 2 010122303441234 2 10 1 2 2 3 3 1 2 3 () ( ) ( ) ( ) ( ) ( ) 2() () ()( )() FFi FDis et dw dw dw dw R dw dw dw R dt τ τττττττττττ ττ ττ ττ τττ ∞∞ ∞∞ ?∞ ?∞ ?∞ ?∞ ∞∞ ∞ ?∞ ?∞ ?∞ ? ?++ ∫∫∫∫ ∫∫∫ The first variation in 2 ()et is 2 11 22 303 44 1234 10 1 2 2 3 3 4 4 1 2 3 4 11 22 33 123 () ( ) ( ) ( ) ( ) ( ) () () () () ( ) 2()()()( ) FFi FDis et d w dw dw dw R dw dw d w dw R dw dw dw R δ τδ τ τ τ τ τ τ τ τ τ τ τ τ ττττδτττττττ τδ τ τ τ τ τ τ τ τ ∞∞∞∞ ?∞ ?∞ ?∞ ?∞ ∞∞∞∞ ?∞ ?∞ ?∞ ?∞ ∞∞ ?∞ ?∞ ?∞ =+? ++ ?+ ∫∫∫∫ ∫∫∫∫ ∫∫∫ In the second term, let: 13 24 31 42 τ τ τ τ τ τ τ τ ′= ′= ′= ′= and interchange the order of integration. 2 nd term 11 22 303 44 3412 () () () ()( ) FFi d w dw dw dw Rτ δτττ ττ ττττττ ∞∞∞∞ ?∞ ?∞ ?∞ ?∞ ′′ ′′ ′′′ ′′′′′′=+? ∫∫∫∫ 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 5 but since 3412 1234 ()() ii ii RRτ τττ ττττ′′′′ ′′′′+?? = +?? we see that the second term is exactly equal to the first term. Collecting these terms and separating out the common integral with respect to 1 τ gives 2 11 22 303 44 1234 22 33 123 () 2 ( ) ( ) ( ) ( ) ( ) () () ( ) FFi FDis et d w dw dw dw R dw dw R δ τδ τ τ τ τ τ τ τ τ τ τ τ ττ τττττ ∞∞∞∞ ?∞ ?∞ ?∞ ?∞ ∞∞ ?∞ ?∞ ? =+? ? ? ? ?+? ? ? ∫∫∫∫ ∫∫ The second variation of 2 ()et is 22 11 22 33 44 1234 () ( ) ( ) ( ) ( ) ( ) FFi et d w dw d w dw Rδ τδ τ τ τ τδ τ τ τ τ τ τ τ ∞∞∞∞ ?∞ ?∞ ?∞ ?∞ =+? ∫∫∫∫ By comparison with the expression for 2 ()ot , this is seen to be the mean squared output of the system () 2 22 output ( ) 0, non-zero inputetδ=> This second variation must be greater than zero, so the stationary point defined by the vanishing of the first variation is shown to be a minimum. In the expression for the first variation, 1 () 0wδ τ = for 1 0τ < by the requirement that the variation be physically realizable. But 1 ()wδ τ is arbitrary for 1 0τ ≥ , so we can be assured of the vanishing of 2 ()etδ only if the { } term vanishes almost everywhere for 1 0τ ≥ . The condition which defines the minimum in 2 ()et is then 22 303 44 1234 22 33 123 () () ()( ) () () ( ) 0 FFi FDis dw dw dw R dw dw R τ τττττττττ ττ τττττ ∞∞∞ ?∞ ?∞ ?∞ ∞∞ ?∞ ?∞ +?? ?+?= ∫∫∫ ∫∫ for all 1 τ , non-real-time. Using this condition in the expression for 2 0 ()et and remembering that 0 () 0wt= for 0t < gives the result 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 5 222 00 () () ()et dt ot=? which is convenient for the calculation of 2 0 ()et . Also since 222 00 () () ()ot dt et=?, this says the optimum mean squared output is always less than the mean squared desired output. Autocorrelation Functions We have arrived at an extended form of the Wiener-Kopf equation which defines the optimum linear system under the ground rules stated before. Recall that: () () () () () () () () ii ss sn ns nn is ss ns RRRRR RRR τ ττττ τττ =+++ =+ since isn=+. The free configuration problem is a specialization of the semi-free configuration. In this expression we would take () 1Fs= , or () () F wt tδ= . In that case we have 22 303 44 1234 22 3 3 123 03 133 3 133 1 () () ()( ) () () ( ) ()() ()()0 for 0 ii Dis ii D is dd dw d R ddwR wR d wR d ττ τ τ τδτ ττττ τδτ τ τ τ τ τ ττττ ττττ τ ∞∞ ∞ ?∞ ?∞ ?∞ ∞∞ ?∞ ?∞ ?∞ ?∞ +?? ?+?= ?? ?= ≥ ∫∫∫ ∫∫