16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Often this sensitivity matrix can only be evaluated by simulation. If a small is introduced and the Or noted(i=1, 2), the ratios are finite-difference OI measures of the -s, which are the values comprising the jth column of s. thus 6 perturbed trajectories must be calculated and the end state differenced with the nominal end state to define S. Each perturbed trajectory defines one column of s e.= Or△r Linearized analysis Given the dynamic 文=f(x) F=1 v=a+g Consider the errors as a small perturbation around the nominal x(t) trajectory x=i,+ox=f( ≈f(xn)+x 文n=f(x) where 3= ax(1)=Φ(o2,1)6x() d(Lo, 1) 6()=dox(0)=F()x() =F()Φ(0,0)6x(t0) Therefore, for arbitrary Sx(to) (2,1) =F()Φ(to,0 Φ(,)=16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Often this sensitivity matrix can only be evaluated by simulation. If a small δ xij δ rsi is introduced and the δ r noted (i =1,2), the ratios are finite-difference si δ xij ∂rs measures of the i , which are the values comprising the jth column of S. Thus 6 ∂xij perturbed trajectories must be calculated and the end state differenced with the nominal end state to define S. Each perturbed trajectory defines one column of S. es = [S]2 6× e1 ∂r ∆r = si ≈ si Sij ∂ ∆xij xij Linearized analysis Given the dynamics x& = f x( ) r v & = v a & = + g Consider the errors as a small perturbation around the nominal x() t trajectory. x x & = & + δ x& = f x ( + δ x) n n df ≈ f ( ) x + δ x n dx x& = f ( ) x n n δ δ x F x , where Fij = ∂fi & = ∂x j () 0 xt , ( 0 ∂ =Φ(t t )δ x t ) δ &( ) = dΦ( , 0 x t ( ()δ x t t t ) δ x t0 ) = F t ( ) dt () 0 Ft , ( = Φ 0 (t t )δ x t ) Therefore, for arbitrary δ x( ) t0 dΦ( , ( ) 0 , t t ) = Φ(t t ) 0 F t dt Φ( , t t ) = I 00 Page 6 of 6