{"id":5456,"date":"2024-03-09T12:01:01","date_gmt":"2024-03-09T04:01:01","guid":{"rendered":""},"modified":"2024-03-09T12:01:01","modified_gmt":"2024-03-09T04:01:01","slug":"\u6377\u8054\u60ef\u5bfc\u7cfb\u7edf\u5b66\u4e606.4(\u5e73\u65b9\u6839\u6ee4\u6ce2 )","status":"publish","type":"post","link":"https:\/\/mushiming.com\/5456.html","title":{"rendered":"\u6377\u8054\u60ef\u5bfc\u7cfb\u7edf\u5b66\u4e606.4(\u5e73\u65b9\u6839\u6ee4\u6ce2 )"},"content":{"rendered":"

\n <\/path> \n<\/svg> <\/p>\n

\u5e73\u65b9\u6839\u6ee4\u6ce2<\/font><\/h3>\n

\u9488\u5bf9\u7ecf\u5178Kalman\u6ee4\u6ce2\u4e2d\uff0c\u72b6\u6001\u8bef\u5dee\u9635 P k P_k <\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u72b6\u6001\u4f30\u8ba1\u7684\u5e73\u65b9\uff0c\u53ef\u80fd\u5360\u6709\u66f4\u591a\u4f4d\u6570\uff0c\u6240\u4ee5\u4f7f\u7528\u72b6\u6001\u8bef\u5dee\u9635 P k P_k <\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u5e73\u65b9\u6839\u8fdb\u884c\u66f4\u65b0\uff0c\u51cf\u5c11\u6570\u503c\u4f4d\u6570\u548c\u8ba1\u7b97\u8bef\u5dee\uff08\u5728\u65e9\u671f\u8ba1\u7b97\u673a\u4e2d\u6bd4\u8f83\u6709\u6821\uff09\u3002<\/p>\n

Potter\u5e73\u65b9\u6839\u6ee4\u6ce2<\/font><\/h3>\n

\u5c06\u5747\u65b9\u8bef\u5dee\u9635 P k P_k <\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5206\u89e3\u4e3a\u4e0b\u4e09\u89d2\u9635\uff0c\u518d\u6c42\u5e73\u65b9\u6839
Cholesky\u5206\u89e3<\/font>
P : n \u9636 \u6b63 \u5b9a \u5bf9 \u79f0 \u77e9 \u9635 P:n\u9636\u6b63\u5b9a\u5bf9\u79f0\u77e9\u9635 <\/span><\/span>P<\/span><\/span>:<\/span><\/span><\/span><\/span>n<\/span>\u9636<\/span>\u6b63<\/span>\u5b9a<\/span>\u5bf9<\/span>\u79f0<\/span>\u77e9<\/span>\u9635<\/span><\/span><\/span><\/span><\/span>
\u603b\u53ef\u4ee5\u5f97\u5230\uff1a
P = [ P 11 P 12 . . . P 1 n P 21 P 22 . . . P 2 n . . . . . . . . . . . . P n 1 P n 2 . . . P n n ] , \u0394 = [ \u03b4 11 \u03b4 12 . . . \u03b4 1 n 0 \u03b4 21 . . . \u03b4 2 n . . . . . . . . . . . . 0 0 . . . \u03b4 n n ] P=\\left[\\begin{matrix} P_{11}&P_{12}&...&P_{1n}\\\\P_{21}&P_{22}&...&P_{2n}\\\\ ...&...&...&...\\\\P_{n1}&P_{n2}&...&P_{nn} \\end{matrix}\\right],\\Delta=\\left[\\begin{matrix} \\delta_{11}&\\delta_{12}&...&\\delta_{1n}\\\\0&\\delta_{21}&...&\\delta_{2n}\\\\ ...&...&...&...\\\\0&0&...&\\delta_{nn} \\end{matrix}\\right] <\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>P<\/span><\/span>n<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>P<\/span><\/span>n<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>2<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>P<\/span><\/span>n<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u0394<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>2<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>n<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
P = \u0394 \u0394 T = [ \u03b4 11 \u03b4 12 . . . \u03b4 1 n 0 \u03b4 21 . . . \u03b4 2 n . . . . . . . . . . . . 0 0 . . . \u03b4 n n ] [ \u03b4 11 \u03b4 12 . . . \u03b4 1 n 0 \u03b4 21 . . . \u03b4 2 n . . . . . . . . . . . . 0 0 . . . \u03b4 n n ] T P=\\Delta \\Delta^T =\\left[\\begin{matrix} \\delta_{11}&\\delta_{12}&...&\\delta_{1n}\\\\0&\\delta_{21}&...&\\delta_{2n}\\\\ ...&...&...&...\\\\0&0&...&\\delta_{nn} \\end{matrix}\\right]\\left[\\begin{matrix} \\delta_{11}&\\delta_{12}&...&\\delta_{1n}\\\\0&\\delta_{21}&...&\\delta_{2n}\\\\ ...&...&...&...\\\\0&0&...&\\delta_{nn} \\end{matrix}\\right]^T <\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span>\u0394<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>2<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>n<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>2<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>n<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u53ef\u4ee5\u5f97\u5230\uff1a
P i j = \u03b4 i j \u03b4 j j + \u03b4 i , j + 1 \u03b4 i , j + 1 + \u03b4 i , j + 2 \u03b4 i , j + 2 + . . . + \u03b4 i , j + n \u03b4 i , j + n = \u2211 k = j + 1 n \u03b4 i k \u03b4 j k + \u03b4 i j \u03b4 j j ( 1 \u2264 i \u2264 n , i \u2264 j \u2264 n ) P_{ij}=\\delta_{ij}\\delta_{jj}+\\delta_{i,j+1}\\delta_{i,j+1}+\\delta_{i,j+2}\\delta_{i,j+2}+...+\\delta_{i,j+n}\\delta_{i,j+n}=\\sum_{k=j+1}^n\\delta_{ik}\\delta_{jk}+\\delta_{ij}\\delta_{jj}(1\\leq i\\leq n,i\\leq j \\leq n) <\/span><\/span>P<\/span><\/span>i<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>j<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>,<\/span>j<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>,<\/span>j<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>,<\/span>j<\/span>+<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>,<\/span>j<\/span>+<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>,<\/span>j<\/span>+<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>,<\/span>j<\/span>+<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>k<\/span>=<\/span>j<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>j<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>j<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>1<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>i<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>n<\/span>,<\/span><\/span>i<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>j<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>
\u4ece\u800c\u5f97\u5230:
\u03b4 i j = { ( P i j \u2212 \u2211 i + j n \u03b4 i k \u03b4 j k ) \/ \u03b4 j j i < j P j j \u2212 \u2211 k = j + 1 n \u03b4 j k 2 i = j 0 i > j \\delta_{ij}=\\begin{cases} (P_{ij}-\\sum_{i+j}^{n}\\delta_{ik}\\delta_{jk})\/\\delta_{jj} &i<j\\\\ \\sqrt{P_{jj}-\\sum_{k=j+1}^{n}\\delta_{jk}^2}&i=j\\\\ 0&i>j\\\\ \\end{cases} <\/span><\/span>\u03b4<\/span><\/span>i<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>P<\/span><\/span>i<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u2211<\/span><\/span>i<\/span>+<\/span>j<\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>i<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>j<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\/<\/span>\u03b4<\/span><\/span>j<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>j<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u2211<\/span><\/span>k<\/span>=<\/span>j<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>j<\/span>k<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\n \n <\/path> \n <\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span><\/span><<\/span><\/span>j<\/span><\/span><\/span><\/span>i<\/span><\/span>=<\/span><\/span>j<\/span><\/span><\/span><\/span>i<\/span><\/span>><\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
QR\u5206\u89e3<\/font>
A m \u00d7 n : \u4e3a \u5217 \u6ee1 \u79e9 \u77e9 \u9635 \uff08 m \u2265 n \uff09 , r a n k ( A m \u00d7 n ) = n A_{m\u00d7n}:\u4e3a\u5217\u6ee1\u79e9\u77e9\u9635\uff08m\\geq n\uff09,rank(A_{m\u00d7n})=n <\/span><\/span>A<\/span><\/span>m<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>\u4e3a<\/span>\u5217<\/span>\u6ee1<\/span>\u79e9<\/span>\u77e9<\/span>\u9635<\/span>\uff08<\/span>m<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>n<\/span>\uff09<\/span>,<\/span><\/span>r<\/span>a<\/span>n<\/span>k<\/span>(<\/span>A<\/span><\/span>m<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>
\u90a3\u672b\u5fc5\u6709\uff1a
A m \u00d7 n = Q m \u00d7 n R n \u00d7 n , Q m \u00d7 n T Q m \u00d7 n = I n \u00d7 n , R n \u00d7 n \u4e3a \u4e0a \u4e09 \u89d2 \u9635 \u6216 \u4e0b \u4e09 \u89d2 \u9635 A_{m\u00d7n}=Q_{m\u00d7n}R_{n\u00d7n},Q_{m\u00d7n}^TQ_{m\u00d7n}=I_{n\u00d7n},R_{n\u00d7n}\u4e3a\u4e0a\u4e09\u89d2\u9635\u6216\u4e0b\u4e09\u89d2\u9635 <\/span><\/span>A<\/span><\/span>m<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>m<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>Q<\/span><\/span>m<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>m<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>R<\/span><\/span>n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a<\/span>\u4e0a<\/span>\u4e09<\/span>\u89d2<\/span>\u9635<\/span>\u6216<\/span>\u4e0b<\/span>\u4e09<\/span>\u89d2<\/span>\u9635<\/span><\/span><\/span><\/span><\/span><\/span>
\u5747\u65b9\u8bef\u5dee\u7684\u91cf\u6d4b\u66f4\u65b0<\/font>
\u7cfb\u7edf\u72b6\u6001\u7a7a\u95f4<\/font>
X k : n \u7ef4 \u72b6 \u6001 \u5411 \u91cf X_k:n\u7ef4\u72b6\u6001\u5411\u91cf <\/span><\/span>X<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>n<\/span>\u7ef4<\/span>\u72b6<\/span>\u6001<\/span>\u5411<\/span>\u91cf<\/span><\/span><\/span><\/span><\/span>
Z k : m \u7ef4 \u6d4b \u91cf \u5411 \u91cf Z_k:m\u7ef4\u6d4b\u91cf\u5411\u91cf <\/span><\/span>Z<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>m<\/span>\u7ef4<\/span>\u6d4b<\/span>\u91cf<\/span>\u5411<\/span>\u91cf<\/span><\/span><\/span><\/span><\/span>
\u03a6 k \/ k \u2212 1 : \u5df2 \u77e5 \u7684 \u7cfb \u7edf \u7ed3 \u6784 \u53c2 \u6570 \\Phi_{k\/k-1}:\u5df2\u77e5\u7684\u7cfb\u7edf\u7ed3\u6784\u53c2\u6570 <\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>\u5df2<\/span>\u77e5<\/span>\u7684<\/span>\u7cfb<\/span>\u7edf<\/span>\u7ed3<\/span>\u6784<\/span>\u53c2<\/span>\u6570<\/span><\/span><\/span><\/span><\/span>
\u0393 k \/ k \u2212 1 : \u5df2 \u77e5 \u7684 \u7cfb \u7edf \u7ed3 \u6784 \u53c2 \u6570 \uff0c \u5206 \u522b \u4e3a n \u00d7 l \u9636 \u7cfb \u7edf \u5206 \u914d \u566a \u58f0 \\Gamma_{k\/k-1}:\u5df2\u77e5\u7684\u7cfb\u7edf\u7ed3\u6784\u53c2\u6570\uff0c\u5206\u522b\u4e3an\u00d7l\u9636\u7cfb\u7edf\u5206\u914d\u566a\u58f0 <\/span><\/span>\u0393<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>\u5df2<\/span>\u77e5<\/span>\u7684<\/span>\u7cfb<\/span>\u7edf<\/span>\u7ed3<\/span>\u6784<\/span>\u53c2<\/span>\u6570<\/span>\uff0c<\/span>\u5206<\/span>\u522b<\/span>\u4e3a<\/span>n<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>l<\/span>\u9636<\/span>\u7cfb<\/span>\u7edf<\/span>\u5206<\/span>\u914d<\/span>\u566a<\/span>\u58f0<\/span><\/span><\/span><\/span><\/span>
H k : \u5df2 \u77e5 \u7684 \u7cfb \u7edf \u7ed3 \u6784 \u53c2 \u6570 \uff0c \u5206 \u522b \u4e3a m \u00d7 n \u9636 \u6d4b \u91cf \u77e9 \u9635 H_k:\u5df2\u77e5\u7684\u7cfb\u7edf\u7ed3\u6784\u53c2\u6570\uff0c\u5206\u522b\u4e3am\u00d7n\u9636\u6d4b\u91cf\u77e9\u9635 <\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>\u5df2<\/span>\u77e5<\/span>\u7684<\/span>\u7cfb<\/span>\u7edf<\/span>\u7ed3<\/span>\u6784<\/span>\u53c2<\/span>\u6570<\/span>\uff0c<\/span>\u5206<\/span>\u522b<\/span>\u4e3a<\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span>\u9636<\/span>\u6d4b<\/span>\u91cf<\/span>\u77e9<\/span>\u9635<\/span><\/span><\/span><\/span><\/span>
V k : m \u7ef4 \u6d4b \u91cf \u566a \u58f0 \uff0c \u9ad8 \u65af \u767d \u566a \u58f0 \uff0c \u670d \u4ece \u6b63 \u592a \u5206 \u5e03 V_k:m\u7ef4\u6d4b\u91cf\u566a\u58f0\uff0c\u9ad8\u65af\u767d\u566a\u58f0\uff0c\u670d\u4ece\u6b63\u592a\u5206\u5e03 <\/span><\/span>V<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>m<\/span>\u7ef4<\/span>\u6d4b<\/span>\u91cf<\/span>\u566a<\/span>\u58f0<\/span>\uff0c<\/span>\u9ad8<\/span>\u65af<\/span>\u767d<\/span>\u566a<\/span>\u58f0<\/span>\uff0c<\/span>\u670d<\/span>\u4ece<\/span>\u6b63<\/span>\u592a<\/span>\u5206<\/span>\u5e03<\/span><\/span><\/span><\/span><\/span>
W k \u2212 1 : m \u7ef4 \u7cfb \u7edf \u566a \u58f0 \u5411 \u91cf \uff0c \u9ad8 \u65af \u767d \u566a \u58f0 \uff0c \u670d \u4ece \u6b63 \u592a \u5206 \u5e03 W_{k-1}:m\u7ef4\u7cfb\u7edf\u566a\u58f0\u5411\u91cf\uff0c\u9ad8\u65af\u767d\u566a\u58f0\uff0c\u670d\u4ece\u6b63\u592a\u5206\u5e03 <\/span><\/span>W<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>m<\/span>\u7ef4<\/span>\u7cfb<\/span>\u7edf<\/span>\u566a<\/span>\u58f0<\/span>\u5411<\/span>\u91cf<\/span>\uff0c<\/span>\u9ad8<\/span>\u65af<\/span>\u767d<\/span>\u566a<\/span>\u58f0<\/span>\uff0c<\/span>\u670d<\/span>\u4ece<\/span>\u6b63<\/span>\u592a<\/span>\u5206<\/span>\u5e03<\/span><\/span><\/span><\/span><\/span>
V k \u4e0e W k \u2212 1 \u4e92 \u4e0d \u76f8 \u5173 V_k\u4e0eW_{k-1}\u4e92\u4e0d\u76f8\u5173 <\/span><\/span>V<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e0e<\/span>W<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e92<\/span>\u4e0d<\/span>\u76f8<\/span>\u5173<\/span><\/span><\/span><\/span><\/span>
{ X k = \u03a6 k \/ k \u2212 1 X k \u2212 1 + \u0393 k \/ k \u2212 1 W k \u2212 1 Z k = H k X k + V k s t . { E [ W k ] = 0 , E [ W k W j T ] = Q k \u03b4 k j Q k \u2265 0 E [ V k ] = 0 , E [ V k V j T ] = R k \u03b4 k j , E [ W k V j T ] = 0 R \u2265 0 \\begin{cases} X_k=\\Phi_{k\/k-1}X_{k-1}+\\Gamma_{k\/k-1}W_{k-1}\\\\ Z_k=H_kX_k+V_k\\\\ \\end{cases} \\\\ st. \\\\ \\begin{cases} E[W_k]=0,E[W_kW_j^T]=Q_k\\delta_{kj} &Q_k \\geq 0\\\\ E[V_k]=0,E[V_kV_j^T]=R_k\\delta_{kj},E[W_kV_j^T]=0&R\\geq 0\\\\ \\end{cases} <\/span><\/span>{
\n <\/span><\/span><\/span>X<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>\u0393<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>W<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Z<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>V<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span>t<\/span>.<\/span><\/span><\/span><\/span><\/span>{
\n <\/span><\/span><\/span>E<\/span>[<\/span>W<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>=<\/span><\/span>0<\/span>,<\/span><\/span>E<\/span>[<\/span>W<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>W<\/span><\/span>j<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>=<\/span><\/span>Q<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>k<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span>[<\/span>V<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>=<\/span><\/span>0<\/span>,<\/span><\/span>E<\/span>[<\/span>V<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>j<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>=<\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>k<\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>E<\/span>[<\/span>W<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>j<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span>0<\/span><\/span><\/span><\/span>R<\/span><\/span>\u2265<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
kalman\u6ee4\u6ce2<\/font>
{ X ^ k \/ k \u2212 1 = \u03a6 k \/ k \u2212 1 X ^ k \u2212 1 \u72b6 \u6001 \u4e00 \u6b65 \u9884 \u6d4b P k \/ k \u2212 1 = \u03a6 k \/ k \u2212 1 P k \u2212 1 \u03a6 k \/ k \u2212 1 T + \u0393 k \u2212 1 Q k \u2212 1 \u0393 k \u2212 1 T \u72b6 \u6001 \u4e00 \u6b65 \u9884 \u6d4b \u5747 \u65b9 \u5dee \u9635 K k = P k \/ k \u2212 1 H k T ( H k P k \/ k \u2212 1 H k T \u2212 R k ) \u2212 1 \u6ee4 \u6ce2 \u589e \u76ca X ^ k = ( I \u2212 K k H k ) X ^ k \/ k \u2212 1 + K k Z k \u72b6 \u6001 \u4f30 \u8ba1 P k = ( I \u2212 K k H k ) P k \/ k \u2212 1 \u72b6 \u6001 \u4f30 \u8ba1 \u5747 \u65b9 \u8bef \u5dee \u9635 \\begin{cases} \\hat X_{k\/k-1}=\\Phi_{k\/k-1}\\hat X_{k-1}&\u72b6\u6001\u4e00\u6b65\u9884\u6d4b\\\\ P_{k\/k-1}=\\Phi_{k\/k-1}P_{k-1}\\Phi^T_{k\/k-1}+\\Gamma_{k-1}Q_{k-1}\\Gamma_{k-1}^T&\u72b6\u6001\u4e00\u6b65\u9884\u6d4b\u5747\u65b9\u5dee\u9635\\\\ K_k=P_{k\/k-1}H_k^T(H_kP_{k\/k-1}H_k^T-R_k)^{-1}&\u6ee4\u6ce2\u589e\u76ca\\\\ \\hat X_k=(I-K_kH_k)\\hat X_{k\/k-1}+K_kZ_k&\u72b6\u6001\u4f30\u8ba1\\\\ P_k=(I-K_kH_k)P_{k\/k-1}&\u72b6\u6001\u4f30\u8ba1\u5747\u65b9\u8bef\u5dee\u9635\\\\ \\end{cases} <\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>\u0393<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0393<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>K<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span>K<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>X<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>K<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Z<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span>K<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u72b6<\/span>\u6001<\/span>\u4e00<\/span>\u6b65<\/span>\u9884<\/span>\u6d4b<\/span><\/span><\/span><\/span>\u72b6<\/span>\u6001<\/span>\u4e00<\/span>\u6b65<\/span>\u9884<\/span>\u6d4b<\/span>\u5747<\/span>\u65b9<\/span>\u5dee<\/span>\u9635<\/span><\/span><\/span><\/span>\u6ee4<\/span>\u6ce2<\/span>\u589e<\/span>\u76ca<\/span><\/span><\/span><\/span>\u72b6<\/span>\u6001<\/span>\u4f30<\/span>\u8ba1<\/span><\/span><\/span><\/span>\u72b6<\/span>\u6001<\/span>\u4f30<\/span>\u8ba1<\/span>\u5747<\/span>\u65b9<\/span>\u8bef<\/span>\u5dee<\/span>\u9635<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u6ee4\u6ce2\u589e\u76ca\u5e26\u5165\u72b6\u6001\u4f30\u8ba1\u5747\u65b9\u8bef\u5dee\u9635<\/font>
P k = ( I \u2212 P k \/ k \u2212 1 H k T ( H k P k \/ k \u2212 1 H k T \u2212 R k ) \u2212 1 H k ) P k \/ k \u2212 1 P_k=(I-P_{k\/k-1}H_k^T(H_kP_{k\/k-1}H_k^T-R_k)^{-1}H_k)P_{k\/k-1} <\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u5047\u8bbe P k \u2212 1 , P k \/ k \u2212 1 , P k P_{k-1},P_{k\/k-1},P_k <\/span><\/span>P<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u5e73\u65b9\u6839\u5206\u522b\u4e3a \u0394 k \u2212 1 , \u0394 k \/ k \u2212 1 , \u0394 k \\Delta_{k-1},\\Delta_{k\/k-1},\\Delta_k <\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/font>
P k \u2212 1 = \u0394 k \u2212 1 \u0394 k \u2212 1 T , P k \/ k \u2212 1 = \u0394 k \/ k \u2212 1 \u0394 k \/ k \u2212 1 T , P k = \u0394 k \u0394 k T P_{k-1}=\\Delta_{k-1}\\Delta_{k-1}^T,P_{k\/k-1}=\\Delta_{k\/k-1}\\Delta_{k\/k-1}^T,P_{k}=\\Delta_{k}\\Delta_{k}^T <\/span><\/span>P<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u5e26\u5165 P k P_k <\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/font>
\u0394 k \u0394 k T = P k = \u0394 k \/ k \u2212 1 [ I \u2212 \u0394 k \/ k \u2212 1 T H k T ( H k \u0394 k \/ k \u2212 1 \u0394 k \/ k \u2212 1 T H k T + R k ) \u2212 1 H k \u0394 k \/ k \u2212 1 ] \u0394 k \/ k \u2212 1 T = \u0394 k \/ k \u2212 1 ( I \u2212 \u03c1 k \u2212 2 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 ) \u0394 k \/ k \u2212 1 T s t : \u03c1 k 2 = H k \u0394 k \/ k \u2212 1 \u0394 k \/ k \u2212 1 T H k T + R k \\Delta_{k}\\Delta_{k}^T=P_k=\\Delta_{k\/k-1}[I-\\Delta_{k\/k-1}^TH_k^T(H_k\\Delta_{k\/k-1}\\Delta_{k\/k-1}^TH_k^T+R_k)^{-1}H_k\\Delta_{k\/k-1}]\\Delta_{k\/k-1}^T \\\\ =\\Delta_{k\/k-1}(I-\\rho_{k}^{-2}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1})\\Delta_{k\/k-1}^T \\\\ st:\\\\ \\rho_k^2=H_k\\Delta_{k\/k-1}\\Delta_{k\/k-1}^TH_k^T+R_k <\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span>t<\/span><\/span>:<\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u6c42\uff1a P k P_k <\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u6839\u7b49\u4ef7\u4e8e I \u2212 \u03c1 k \u2212 2 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 I-\\rho_{k}^{-2}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1} <\/span><\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u6839<\/font>
I \u2212 \u03c1 k \u2212 2 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 = ( I \u2212 r k \u2212 1 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 ) ( I + r k \u2212 1 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 ) I-\\rho_{k}^{-2}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1}=(I-r_k^{-1}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1})(I+r_k^{-1}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1}) <\/span><\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>r<\/span><\/span>k<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>(<\/span>I<\/span><\/span>+<\/span><\/span><\/span><\/span>r<\/span><\/span>k<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>
\u5229\u7528\u5f85\u5b9a\u7cfb\u6570\u6cd5\u6c42\u5f97\uff1a
r k = \u03c1 k ( \u03c1 k \u00b1 R k ) = 2 \u03c1 k 2 \u00b1 4 \u03c1 k 4 \u2212 4 \u03c1 k 2 ( \u03c1 k 2 \u2212 R k ) 2 r_k=\\rho_k(\\rho_k \\pm\\sqrt{R_k})=\\frac{2\\rho_k^2\\pm\\sqrt{4\\rho_k^4-4\\rho_k^2(\\rho_k^2-R_k)}}{2} <\/span><\/span>r<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00b1<\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\n \n <\/path> \n <\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00b1<\/span><\/span><\/span>4<\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>4<\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>
\n \n <\/path> \n <\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u0394 k \u0394 k T = \u0394 k \/ k \u2212 1 ( I \u2212 \u03c1 k \u2212 2 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 ) \u0394 k \/ k \u2212 1 T = ( \u0394 k \/ k \u2212 1 ( I \u2212 r k \u2212 1 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 ) ) ( \u0394 k \/ k \u2212 1 ( I \u2212 r k \u2212 1 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 ) ) T \\Delta_{k}\\Delta_{k}^T=\\Delta_{k\/k-1}(I-\\rho_{k}^{-2}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1})\\Delta_{k\/k-1}^T\\\\ =(\\Delta_{k\/k-1}(I-r_{k}^{-1}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1}))(\\Delta_{k\/k-1}(I-r_{k}^{-1}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1}))^T <\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>r<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>)<\/span>(<\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>r<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u0394 k = \u0394 k \/ k \u2212 1 ( I \u2212 r k \u2212 1 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 ) ( \u0394 k \u8ba1 \u7b97 \u4e4b \u540e \u4e00 \u822c \u4e0d \u4e3a \u4e09 \u89d2 \u9635 ) \\Delta _k=\\Delta_{k\/k-1}(I-r_{k}^{-1}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1})(\\Delta _k \u8ba1\u7b97\u4e4b\u540e\u4e00\u822c\u4e0d\u4e3a\u4e09\u89d2\u9635) <\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>r<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>(<\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8ba1<\/span>\u7b97<\/span>\u4e4b<\/span>\u540e<\/span>\u4e00<\/span>\u822c<\/span>\u4e0d<\/span>\u4e3a<\/span>\u4e09<\/span>\u89d2<\/span>\u9635<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/font>
\u5747\u65b9\u8bef\u5dee\u7684\u65f6\u95f4\u66f4\u65b0<\/font>
P k \u2212 1 = \u0394 k \u2212 1 \u0394 k \u2212 1 T , P k \/ k \u2212 1 = \u0394 k \/ k \u2212 1 \u0394 k \/ k \u2212 1 T , P k = \u0394 k \u0394 k T P_{k-1}=\\Delta_{k-1}\\Delta_{k-1}^T,P_{k\/k-1}=\\Delta_{k\/k-1}\\Delta_{k\/k-1}^T,P_{k}=\\Delta_{k}\\Delta_{k}^T <\/span><\/span>P<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u5e26\u5165 P k \/ k \u2212 1 = \u03a6 k \/ k \u2212 1 P k \u2212 1 \u03a6 k \/ k \u2212 1 T + \u0393 k \u2212 1 Q k \u2212 1 \u0393 k \u2212 1 T P_{k\/k-1}=\\Phi_{k\/k-1}P_{k-1}\\Phi^T_{k\/k-1}+\\Gamma_{k-1}Q_{k-1}\\Gamma_{k-1}^T <\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u0393<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0393<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/font>
Q k \u2212 1 1 \/ 2 ( Q k \u2212 1 1 \/ 2 ) T = Q k \u2212 1 ( c h o l e s k y \u5206 \u89e3 \u6cd5 \u6c42 \u5f97 ) Q_{k-1}^{1\/2}(Q_{k-1}^{1\/2})^T=Q_{k-1}(cholesky \u5206\u89e3\u6cd5\u6c42\u5f97) <\/span><\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>c<\/span>h<\/span>o<\/span>l<\/span>e<\/span>s<\/span>k<\/span>y<\/span>\u5206<\/span>\u89e3<\/span>\u6cd5<\/span>\u6c42<\/span>\u5f97<\/span>)<\/span><\/span><\/span><\/span><\/span>
P k \/ k \u2212 1 = \u0394 k \/ k \u2212 1 \u0394 k \/ k \u2212 1 T = \u03a6 k \/ k \u2212 1 \u0394 k \u2212 1 \u0394 k \u2212 1 T \u03a6 k \/ k \u2212 1 T + \u0393 k \u2212 1 Q k \u2212 1 1 \/ 2 ( Q k \u2212 1 1 \/ 2 ) T \u0393 k \u2212 1 T = [ \u03a6 k \/ k \u2212 1 \u0394 k \u2212 1 \u0393 k \u2212 1 Q k \u2212 1 1 \/ 2 ] [ \u0394 k \u2212 1 T \u03a6 k \/ k \u2212 1 T ( Q k \u2212 1 1 \/ 2 ) T \u0393 k \u2212 1 T ] P_{k\/k-1}=\\Delta_{k\/k-1}\\Delta_{k\/k-1}^T=\\Phi_{k\/k-1}\\Delta_{k-1}\\Delta_{k-1}^T\\Phi^T_{k\/k-1}+\\Gamma_{k-1}Q^{1\/2}_{k-1}(Q^{1\/2}_{k-1})^T\\Gamma_{k-1}^T \\\\ =\\left[\\begin{matrix} \\Phi_{k\/k-1}\\Delta_{k-1}&\\Gamma_{k-1}Q_{k-1}^{1\/2}\\\\ \\end{matrix}\\right] \\left[\\begin{matrix} \\Delta_{k-1}^T\\Phi_{k\/k-1}^T\\\\(Q_{k-1}^{1\/2})^T\\Gamma_{k-1}^T\\\\ \\end{matrix}\\right] <\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u0393<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0393<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0393<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0393<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u4ee4 A ( n + l ) \u00d7 n = [ \u0394 k \u2212 1 T \u03a6 k \/ k \u2212 1 T ( Q k \u2212 1 1 \/ 2 ) T \u0393 k \u2212 1 T ] A_{(n+l)\u00d7n}=\\left[\\begin{matrix} \\Delta_{k-1}^T\\Phi_{k\/k-1}^T\\\\(Q_{k-1}^{1\/2})^T\\Gamma_{k-1}^T\\\\ \\end{matrix}\\right] <\/span><\/span>A<\/span><\/span>(<\/span>n<\/span>+<\/span>l<\/span>)<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a6<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>Q<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0393<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,\u5bf9 A ( n + l ) \u00d7 n A_{(n+l)\u00d7n} <\/span><\/span>A<\/span><\/span>(<\/span>n<\/span>+<\/span>l<\/span>)<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8fdb\u884cUD\u5206\u89e3<\/font>
A ( n + l ) \u00d7 n = Q \u203e ( n + l ) \u00d7 n R \u203e n \u00d7 n A_{(n+l)\u00d7n}=\\overline Q_{(n+l)\u00d7n}\\overline R_{n\u00d7n} <\/span><\/span>A<\/span><\/span>(<\/span>n<\/span>+<\/span>l<\/span>)<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>n<\/span>+<\/span>l<\/span>)<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
A ( n + l ) \u00d7 n T A ( n + l ) \u00d7 n = ( Q \u203e ( n + l ) \u00d7 n R \u203e n \u00d7 n ) T ( Q \u203e ( n + l ) \u00d7 n R \u203e n \u00d7 n ) = R n \u00d7 n T R n \u00d7 n = \u0394 k \/ k \u2212 1 \u0394 k \/ k \u2212 1 T A_{(n+l)\u00d7n}^TA_{(n+l)\u00d7n}=(\\overline Q_{(n+l)\u00d7n}\\overline R_{n\u00d7n})^T(\\overline Q_{(n+l)\u00d7n}\\overline R_{n\u00d7n})=R_{n\u00d7n}^TR_{n\u00d7n}=\\Delta_{k\/k-1}\\Delta_{k\/k-1}^T <\/span><\/span>A<\/span><\/span>(<\/span>n<\/span>+<\/span>l<\/span>)<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>(<\/span>n<\/span>+<\/span>l<\/span>)<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>n<\/span>+<\/span>l<\/span>)<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>n<\/span>+<\/span>l<\/span>)<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>R<\/span><\/span>n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\u0394 k \/ k \u2212 1 T \\Delta_{k\/k-1}^T <\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a A ( n + l ) \u00d7 n A_{(n+l)\u00d7n} <\/span><\/span>A<\/span><\/span>(<\/span>n<\/span>+<\/span>l<\/span>)<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8fdb\u884cUD\u5206\u89e3\u4e4b\u540e\u7684\uff0c\u4e0a\u4e09\u89d2\u6216\u4e0b\u4e09\u89d2\u9635
\u5e73\u65b9\u6839\u6ee4\u6ce2\u6d41\u7a0b<\/font>
{ a k = \u0394 k \/ k \u2212 1 T H k T \u03c1 k 2 = H k \u0394 k \/ k \u2212 1 \u0394 k \/ k \u2212 1 T H k T + R k = a k T a k + R k \u0394 k = \u0394 k \/ k \u2212 1 ( I \u2212 r k \u2212 1 \u0394 k \/ k \u2212 1 T H k T H k \u0394 k \/ k \u2212 1 ) = \u0394 k \/ k \u2212 1 ( I \u2212 r k \u2212 1 a k a k T ) K k = \u0394 k \/ k \u2212 1 \u0394 k \/ k \u2212 1 T H k T ( H k \u0394 k \/ k \u2212 1 \u0394 k \/ k \u2212 1 T H k T + R k ) \u2212 1 = \u0394 k \/ k \u2212 1 a k \u03c1 k \u2212 2 ( \u6216 \u0394 k \u0394 k T H k T R k \u2212 1 ) \\begin{cases} a_k=\\Delta_{k\/k-1}^TH_k^T&\\\\ \\rho_k^2=H_k\\Delta_{k\/k-1}\\Delta_{k\/k-1}^TH_k^T+R_k=a_k^Ta_k+R_k\\\\ \\Delta _k=\\Delta_{k\/k-1}(I-r_{k}^{-1}\\Delta_{k\/k-1}^TH_k^TH_k\\Delta_{k\/k-1})=\\Delta_{k\/k-1}(I-r^{-1}_ka_ka_k^T)\\\\ K_k=\\Delta_{k\/k-1}\\Delta_{k\/k-1}^TH_k^T(H_k\\Delta_{k\/k-1}\\Delta_{k\/k-1}^TH_k^T+R_k)^{-1}=\\Delta_{k\/k-1}a_k\\rho_k^{-2}(\u6216\\Delta_{k}\\Delta_{k}^TH_k^TR_k^{-1})\\\\ \\end{cases} <\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23aa<\/span><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>a<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span>r<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>I<\/span><\/span>\u2212<\/span><\/span>r<\/span><\/span>k<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>K<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>k<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u6216<\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\"\u6377\u8054\u60ef\u5bfc\u7cfb\u7edf\u5b66\u4e606.4(\u5e73\u65b9\u6839\u6ee4\u6ce2<\/p>\n

\u5f53\u91cf\u6d4b\u4e3a\u77e2\u91cf\u65f6<\/font>
\u4e0a\u8ff0\u60c5\u51b5\u9488\u5bf9\u91cf\u6d4b\u4e3a\u6807\u91cf\u65f6\u7684\u60c5\u51b5\uff0c\u5f53\u91cf\u6d4b\u4e3a\u77e2\u91cf\u65f6\uff1a<\/p>\n

    \n
  1. \u4f7f\u7528\u5e8f\u8d2f\u6ee4\u6ce2\u5c06\u77e2\u91cf\u8f6c\u5316\u4e3a\u6807\u91cf<\/li>\n
  2. \u76f4\u63a5\u8fdb\u884c\u5411\u91cf\u91cf\u6d4b\u5e73\u65b9\u6839\u6ee4\u6ce2
    \u6b64\u65f6\uff1a\u5747\u65b9\u8bef\u5dee\u9635\u53ca\u5176\u5bf9\u5e94\u5e73\u65b9\u6839\u6ee4\u6ce2\u516c\u5f0f\u4e3a\uff1a
    P k = P k \/ k \u2212 1 \u2212 P k \/ k \u2212 1 H k T ( H k P k \/ k \u2212 1 H k T + R k ) \u2212 1 H k P k \/ k \u2212 1 \u0394 k = \u0394 k \/ k \u2212 1 [ I \u2212 \u0394 k \/ k \u2212 1 T H k T ( p k p k T + R k 1 \/ 2 P k T ) \u2212 1 H k \u0394 k \/ k \u2212 1 ] s t : p k p k T = H k P k \/ k \u2212 1 H k T + R k = [ H k \u0394 k \/ k \u2212 1 R k 1 \/ 2 ] [ \u0394 k \/ k \u2212 1 T H k T ( R k 1 \/ 2 ) T ] \u2212 > P k T ( Q R \u540c \u4e0a Q R \u5206 \u89e3 \u5f97 \u5230 ) P_k=P_{k\/k-1}-P_{k\/k-1}H_k^T(H_kP_{k\/k-1}H_k^T+R_k)^{-1}H_kP_{k\/k-1} \\\\ \\Delta_k=\\Delta_{k\/k-1}[I-\\Delta_{k\/k-1}^TH_k^T(p_kp_k^T+R_k^{1\/2}P_k^T)^{-1}H_k\\Delta_{k\/k-1}]\\\\ st:\\\\ p_kp_k^T=H_kP_{k\/k-1}H_k^T+R_k=\\left[\\begin{matrix} H_{k}\\Delta_{k\/k-1}&R_{k}^{1\/2}\\\\ \\end{matrix}\\right] \\left[\\begin{matrix} \\Delta_{k\/k-1}^TH_{k}^T\\\\ (R_{k}^{1\/2})^T\\\\ \\end{matrix}\\right]->P_k^T(QR\u540c\u4e0aQR\u5206\u89e3\u5f97\u5230) <\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>p<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>s<\/span>t<\/span><\/span>:<\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>\u0394<\/span><\/span>k<\/span>\/<\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>1<\/span>\/<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>><\/span><\/span><\/span><\/span>P<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>Q<\/span>R<\/span>\u540c<\/span>\u4e0a<\/span>Q<\/span>R<\/span>\u5206<\/span>\u89e3<\/span>\u5f97<\/span>\u5230<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"\u6377\u8054\u60ef\u5bfc\u7cfb\u7edf\u5b66\u4e606.4(\u5e73\u65b9\u6839\u6ee4\u6ce2 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