16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde but since R,(t'+t4--T2=R,(G+r,) 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 t, gives ∫dn-()drya()R(+2-x) The second variation of e(is 8e(0'=dt, w(r)dr wr(r2) dt, w(t,) dr, wA(T)R,(r, +T2-r, -t By comparison with the expression for o(t), this is seen to be the mean squared output of the system output (output)=8e(0)>0, non-zero input 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, Ow(T,)=0 for t, <0 by the requirement that the variation be physically realizable. But Sw(t,) is arbitrary for t, 20,so we can be assured of the vanishing of &e(0 only if the( term vanishes almost everywhere for t, 20. The condition which defines the minimum in e(t) is then jdr,wr(z2)dr, wo(t,) dr,wE(TA)R, (4+t2-t3-z4) dr]wF(r2) dr, wp(r3)R, (T,+T2-T3)=0 for all t. non-real-time Using this condition in the expression for e(0 and remembering that wo(1)=0 for t<o gives the result16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 5 but since 3412 1234 ( )( ) R R ii ii τ ′′′′ ′′′′ + −− = + −− 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 1 1 2 2 30 3 4 4 1 2 3 4 2 2 3 3 123 () 2 ( ) ( ) ( ) ( ) ( ) () () ( ) F F ii F D is et d w d w d w d w R dw dw R δ τδ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τττ ∞ ∞ ∞∞ −∞ −∞ −∞ −∞ ∞ ∞ −∞ −∞ ⎧ = +−− ⎨ ⎩ ⎫ − +− ⎬ ⎭ ∫ ∫ ∫∫ ∫ ∫ The second variation of 2 e t( ) is 2 2 1 1 2 2 3 3 4 4 1234 () ( ) ( ) ( ) ( ) ( ) F F ii δ et d w d w d w d w R τδ τ τ τ τδ τ τ τ τ τ τ τ ∞∞ ∞ ∞ −∞ −∞ −∞ −∞ = +−− ∫∫∫∫ By comparison with the expression for 2 o t( ) , this is seen to be the mean squared output of the system ( )2 2 2 output ( ) 0, non-zero input = > δ e t 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 δw() 0 τ = for 1 τ < 0 by the requirement that the variation be physically realizable. But 1 δw( ) τ is arbitrary for 1 τ ≥ 0 , so we can be assured of the vanishing of 2 δ e t( ) only if the { } term vanishes almost everywhere for 1 τ ≥ 0 . The condition which defines the minimum in 2 e t( ) is then 2 2 30 3 4 4 1 2 3 4 2 2 3 3 123 () () () ( ) () () ( ) 0 F F ii F D is dw dw dw R dw dw R τ τ τ τ τ τ ττττ τ τ τ τ τττ ∞ ∞∞ −∞ −∞ −∞ ∞ ∞ −∞ −∞ +−− − +− = ∫ ∫∫ ∫ ∫ for all 1 τ , non-real-time. Using this condition in the expression for 2 0 e t( ) and remembering that 0 w t() 0 = for t < 0 gives the result