6线性最优滤波小结Xk+1=+1/k +Kk+1(y+1-H+1+1/k)(7)= (I - Kk+1Hk+1)Xk+1/k + Kk+1Yk+1比较(6)与(7)，可知Kk+1 = Pk+1H+1Rk+1(8)(9)Pk+1 = (I - Kk+1Hk+1)Pk+1/k(7)、(8)、(9)是常见的卡尔曼滤波公式。XUTUPROF.YUAN-LICAI
XJTU PROF. YUAN-LI CAI 线性最优滤波小结 6 𝑥̂𝑘+1 = 𝑥̂𝑘+1|𝑘 + 𝐾𝑘+1(𝑦𝑘+1 − 𝐻𝑘+1𝑥̂𝑘+1|𝑘) = (𝐼 − 𝐾𝑘+1𝐻𝑘+1 )𝑥̂𝑘+1|𝑘 + 𝐾𝑘+1𝑦𝑘+1 (7) 比较(6)与(7)，可知 𝐾𝑘+1 = 𝑃𝑘+1𝐻𝑘+1 𝑇 𝑅𝑘+1 −1 (8) 𝑃𝑘+1 = (𝐼 − 𝐾𝑘+1𝐻𝑘+1 )𝑃𝑘+1|𝑘 (9) (7)、(8)、(9)是常见的卡尔曼滤波公式
线性最优滤波小结2.最小二乘与KF在k+1时刻，我们有(10)Xk+1/k=Xk+1+Ek+1Ek+1 ~ (0, Pk+1|k)(11)Vk+1 ~ (0, Rk+1)Yk+1=Hk+1Xk+1+Vk+1将(10)也视为量测方程之一，于是XUTUPROF.YUAN-LI CAI
XJTU PROF. YUAN-LI CAI 线性最优滤波小结 7 2. 最小二乘与 KF 在𝑘 + 1时刻，我们有 𝑥̂𝑘+1|𝑘 = 𝑥𝑘+1 + 𝜖𝑘+1 , 𝜖𝑘+1 ∼ (0, 𝑃𝑘+1|𝑘) (10) 𝑦𝑘+1 = 𝐻𝑘+1𝑥𝑘+1 + 𝑣𝑘+1 , 𝑣𝑘+1 ∼ (0, 𝑅𝑘+1 ) (11) 将(10)也视为量测方程之一，于是
8线性最优滤波小结Ek+1HM7[Pk+1]k00Pk+1/kEVVTdef W-W0Rk+1Rk+]0由加权最小二乘估计可得Xk+1/kXk+1 = Xk+1/k+1 = (HTWH)-1HTИYk+1Pk+1 = (HTWH)-1XUTUPROF.YUAN-LICAI
XJTU PROF. YUAN-LI CAI 线性最优滤波小结 8 ℋ = [ 𝐼𝑛×𝑛 𝐻𝑘+1 ], 𝒱 = [ 𝜖𝑘+1 𝑣𝑘+1 ] 𝐸𝒱𝒱 𝑇 = [ 𝑃𝑘+1|𝑘 0 0 𝑅𝑘+1 ] ≝ 𝒲−1 , ⇒ 𝒲 = [ 𝑃𝑘+1|𝑘 −1 0 0 𝑅𝑘+1 −1 ] 由加权最小二乘估计可得 𝑥̂𝑘+1 = 𝑥̂𝑘+1|𝑘+1 = (ℋ𝑇𝒲ℋ) −1ℋ𝑇𝒲 [ 𝑥̂𝑘+1|𝑘 𝑦𝑘+1 ] 𝑃𝑘+1 = (ℋ𝑇𝒲ℋ) −1
9线性最优滤波小结其中-10Pk+1|k[Pk+1]/kHk+1Rk+1]HTW =[InxnHRk+10InxnHTWH =[Pk+1]k H+1R+1]P+1/k + H+1Rk+1Hk+1Hk+1因此XUTUPROF.YUAN-LI CAI
XJTU PROF. YUAN-LI CAI 线性最优滤波小结 9 其中 ℋ𝑇𝒲 = [𝐼𝑛×𝑛 𝐻𝑘+1 𝑇 ][ 𝑃𝑘+1|𝑘 −1 0 0 𝑅𝑘+1 −1 ] = [𝑃𝑘+1|𝑘 −1 𝐻𝑘+1𝑅𝑘+1 −1 ] ℋ𝑇𝒲ℋ = [𝑃𝑘+1|𝑘 −1 𝐻𝑘+1 𝑇 𝑅𝑘+1 −1 ][ 𝐼𝑛×𝑛 𝐻𝑘+1 ] = 𝑃𝑘+1|𝑘 −1 + 𝐻𝑘+1 𝑇 𝑅𝑘+1 −1 𝐻𝑘+1 因此
10线性最优滤波小结(12)Xk+1 =Pk+1(Pk+1)kXk+1/k + Hk+1Rk+1yk+1)其他讨论同上小节，不再赘述。XJTUPROF.YUAN-LICAI
XJTU PROF. YUAN-LI CAI 线性最优滤波小结 10 𝑥̂𝑘+1 = 𝑃𝑘+1 (𝑃𝑘+1|𝑘 −1 𝑥̂𝑘+1|𝑘 + 𝐻𝑘+1𝑅𝑘+1 −1 𝑦𝑘+1 ) (12) 其他讨论同上小节，不再赘述