MIT OpenCourseWare http://ocw.mit.edu 16.346 Astrodynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.• Estimation Error Vector:  = PHA−1 α• Covariance Matrix of Estimation Errors: E(T )= T T = PHA−1ααTA−1HTP T = PHA−1 ααT TT A−1H P = PHA−1AA−1H P = PHA−1HTP = PP−1P = P =(P−1)−1 Covariance Matrix =(Information Matrix)−1 A Matrix Identity (The Magic Lemma) Let X mn and Y nm be rectangular compatible matrices such that X mn Y nm and Y nm X mn • Lecture 22 The Covariance, Information & Estimator Matrices The Covariance Matrix #13.6 • Random Vector of Measurement Errors: α where αT =[α1 α2 ... α n] Assume measurement errors independent. First and Second Moments: E(α)= α = 0 and E(αiαj )= αiαj =0 (i = j) Variances: E(α2)= α2 = σ2 • i ii⎡ ⎤ ⎥⎥⎦ σ12 0 0··· 0 σ22 0···⎢⎢⎣ Variance Matrix: E(ααT )= ααT = A = . ... • . . . ... . . 0 0 σ2 ··· n are both meaningful. However, R = X Y is an m ×m matrix while S = Y X is an nmm mn nm nn nm mn ×n matrix. With this understanding, the following sequence of matrix operations leads to a remarkable and very useful identity: (Imm + XmnYnm)(Imm + XmnYnm)−1 = Imm Y nm(I mm + X mn Y nm)(I mm + X mn Y nm)−1 = Y nm (I nn + Y nm X mn)Y nm(I mm + X mn Y nm)−1 = Y nm (I nn + Y nm X mn)Y nm(I mm + X mn Y nm)−1X mn = Y nm X mn I nn +(I nn + Y nm X mn)Y nm(I mm + X mn Y nm)−1X mn = I nn + Y nm X mn (I + YX)−1[I +(I + YX)Y(I + XY)−1X]=(I + YX)−1(I + YX) (I + YX)−1 + Y(I + XY)−1X = I Hence: (I nn + Y nm X mn)−1 = I nn − Y nm(I mm + X mn Y nm)−1X mn To generalize: Let Y = A B and X = C−1 BT . Then nm nn nm mn mm mn (A−1 + BC−1BT )−1 = A − AB(C + BT AB)−1BT A 16.346 Astrodynamics Lecture 22� � Inverting the Information Matrix Using the Magic Lemma P −1 = P−1 + h(σ2)−1hTRecursive formulation: [= A−1 + BC−1BT ]• (A−1 + BC−1BT )−1 = A − AB(C + BT AB)−1BT A Using the magic lemma: P = P − Ph(σ2 + hT Ph)−1hT P• 1Define: a = σ2 + hT Ph and w = Ph so that • a P =(I − whT )P The Square Root of the P Matrix #13.7 The matrix W is the Square Root of a Positive Definite Matrix P if P = WWT 1P = P − PhhTP a � 1 � WW T = W I − WThhTW WT a � 1 � = W I − zzT WT a = W(I − βzzT)(I − βzzT)WT = W(I − 2βzzT + β2zzTzzT)WT = W[I − (2β − β2 z 2)zzT]WT But z 2 = hT WWT h = hT Ph = a − σ2 11Hence: 2β − β2 z 2 == β = a ⇒ a + √aσ2 W = W zzT z = WT hI − a + √aσ2 Properties of the Estimator Linear• • Unbiased: If measurements are exact ( α = 0) then δq�= δq = HT δr so that δ�r = PP−1 δr = δr • Reduces to deterministic case (δ�r = H−T δq�) if no redundant measurements. If H is square & non-singular, then P = H−T AH−1 and δ�r = H−T AH−1HA−1 δq�= H−T δq� 16.346 Astrodynamics Lecture 22� � � � � � � � � �� � � � Recursive Form of the Estimator #13.6 Define q�δ�r = Fδq� F = PHA −1 δq� = δδq�q�P =(I − whT)P H = � Hh � A = A 0 0T σ2 Then � �� � hF =(I − whT)P Hh A−1 0 =(I − whT)P HA−1 σ2 0T σ−2 = (I − whT)F aw − w(a − σ2) =[(I − whT)Fw ]σ2 σ2 δ�r  = F δq� = (I − whT)Fw δq�=(I − whT) δ�r + wδq�δq�= δ�r + w(δq�− hT δ�r)= δ�r + w(δq�− δq�) Since δq = hT δr, then δq�= hT δ�r is the best estimate of the new measurement. δ�r  = δ�r + w(δq�− δq�) δ�r = δ�r + w(δq�− δq�) z = WTh 1 1 w = Ph or w = Wz σ2 + z2σ2 + hTPh P =(I − whT)P zzT W = W I − a + √aσ2 Triangular Square Root ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 w1 0 0 w4 m1 m2 m3 WWT = ⎣ 0 w2 w3 ⎦ ⎣ 0 w2 w5 ⎦ = ⎣ m2 m4 m5 ⎦ w4 w5 w6 w1 w3 w6 m3 m5 m6 where w 2 1 = m1 w3 = m2 w6 = m3 w1 w1 w 2 2 = m1m4 − m2 2 m1 w 2 4 = det M m1w2 2 w5 = m5 − w3w6 w2 16.346 Astrodynamics Lecture