無跡卡曼濾波器 (英文 :unscented Kalman filter ,UKF )係一種卡曼濾波器 ,攞嚟解決種問題係擴展卡曼濾波器 喺高度非線性系統 模型下表現嘸佳[ 1] 噉嘅情況;呢種卡曼濾波器使到一種叫做無跡變換 (UT)嘅確定性抽樣技巧,即喺平均值 周圍揀一組最細嘅樣本點(喊做sigma點)。然之後,啲sigma點識透過非線性函數傳播(propagated),從中形成到個新估計值畀平均值同埋協方差 。噉樣可以避免似喺EKF當中噉經由線性化 個潛在嘅非線性模型嚟傳播啲協方差。係噉呢種濾波器嘅具體效能取決於點樣計UT啲轉換統計量同埋使到邊一組採樣到嘅sigma點。需要注意嘅係,啲新嘅UKF始終可以以一致(consistent)嘅方法整出。[ 2] 對於某啲系統,噉樣得到嘅UKF更加準確噉估計得到真實嘅平均值同協方差,[ 3] 個論斷可以透過蒙地卡羅抽樣 或者啲後驗統計值個泰勒級數 展開嚟驗證到。另外,種技術免走明式噉計算Jacobi矩陣嘅要求:對於啲複雜函數來講,一嘸係啲函數嘸微分得所以冇辦法計得出Jacobi矩陣、就係計算本身好難無論係使分析法(導數 複雜)抑或係數值法(計算成本高)都好。
對於隨機向量
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{\displaystyle \mathbf {x} =(x_{1},\dots ,x_{L})}
,啲Sigma點係任何啲向量嘅集:
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{\displaystyle \{\mathbf {s} _{0},\dots ,\mathbf {s} _{N}\}={\bigl \{}{\begin{pmatrix}s_{0,1}&s_{0,2}&\ldots &s_{0,L}\end{pmatrix}},\dots ,{\begin{pmatrix}s_{N,1}&s_{N,2}&\ldots &s_{N,L}\end{pmatrix}}{\bigr \}}}
有:
一階權重
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{\displaystyle W_{0}^{a},\dots ,W_{N}^{a}}
滿足
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{\displaystyle \sum _{j=0}^{N}W_{j}^{a}=1}
對所有啲
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{\displaystyle i=1,\dots ,L}
有:
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{\displaystyle E[x_{i}]=\sum _{j=0}^{N}W_{j}^{a}s_{j,i}}
二階權重
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{\displaystyle W_{0}^{c},\dots ,W_{N}^{c}}
滿足
∑
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{\displaystyle \sum _{j=0}^{N}W_{j}^{c}=1}
對所有啲𠵿
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∈
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{\displaystyle (i,l)\in \{1,\dots ,L\}^{2}:E[x_{i}x_{l}]=\sum _{j=0}^{N}W_{j}^{c}s_{j,i}s_{j,l}}
UKF演算法入便一種揀啲sigma點同埋權重畀
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{\displaystyle \mathbf {x} _{k-1\mid k-1}}
嘅簡單方式係:
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{\displaystyle {\begin{aligned}\mathbf {s} _{0}&={\hat {\mathbf {x} }}_{k-1\mid k-1}\\-1&<W_{0}^{a}=W_{0}^{c}<1\\\mathbf {s} _{j}&={\hat {\mathbf {x} }}_{k-1\mid k-1}+{\sqrt {\frac {L}{1-W_{0}}}}\mathbf {A} _{j},\quad j=1,\dots ,L\\\mathbf {s} _{L+j}&={\hat {\mathbf {x} }}_{k-1\mid k-1}-{\sqrt {\frac {L}{1-W_{0}}}}\mathbf {A} _{j},\quad j=1,\dots ,L\\W_{j}^{a}&=W_{j}^{c}={\frac {1-W_{0}}{2L}},\quad j=1,\dots ,2L\end{aligned}}}
其中
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{\displaystyle {\hat {\mathbf {x} }}_{k-1\mid k-1}}
係
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{\displaystyle \mathbf {x} _{k-1\mid k-1}}
個平均估計。枚向量
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{\displaystyle \mathbf {A} _{j}}
係
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{\displaystyle \mathbf {A} }
個第j 棟,其中
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{\displaystyle \mathbf {A} }
有
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{\displaystyle \mathbf {P} _{k-1\mid k-1}=\mathbf {AA} ^{\textsf {T}}}
。矩陣
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{\displaystyle \mathbf {A} }
應使數值上高效又穩定嘅方法似Cholesky分解 嚟計算。平均值嘅權重
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{\displaystyle W_{0}}
可以任意噉揀。
第個流行嘅參數化(對上述普適化)方式係:
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{\displaystyle {\begin{aligned}\mathbf {s} _{0}&={\hat {\mathbf {x} }}_{k-1\mid k-1}\\W_{0}^{a}&={\frac {\alpha ^{2}\kappa -L}{\alpha ^{2}\kappa }}\\W_{0}^{c}&=W_{0}^{a}+1-\alpha ^{2}+\beta \\\mathbf {s} _{j}&={\hat {\mathbf {x} }}_{k-1\mid k-1}+\alpha {\sqrt {\kappa }}\mathbf {A} _{j},\quad j=1,\dots ,L\\\mathbf {s} _{L+j}&={\hat {\mathbf {x} }}_{k-1\mid k-1}-\alpha {\sqrt {\kappa }}\mathbf {A} _{j},\quad j=1,\dots ,L\\W_{j}^{a}&=W_{j}^{c}={\frac {1}{2\alpha ^{2}\kappa }},\quad j=1,\dots ,2L.\end{aligned}}}
α
{\displaystyle \alpha }
同
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{\displaystyle \kappa }
啲sigma點嘅擴散。
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{\displaystyle \beta }
同
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{\displaystyle x}
嘅分佈有關。
啱嘅值取決於手頭係咩樣嘅問題,而典型建議係揀
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{\displaystyle \alpha =10^{-3}}
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{\displaystyle \kappa =1}
同埋
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{\displaystyle \beta =2}
。之不過,大啲嘅值畀
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{\displaystyle \alpha }
(似
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{\displaystyle \alpha =1}
)可能捕捉返個分佈嘅擴散同埋啲可能有嘅非線性會好啲。[ 4] 若果個
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{\displaystyle x}
真實分佈係高斯分佈 ,係噉揀
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{\displaystyle \beta =2}
係最啱嘅。[ 5]
同EKF一樣,UKF預測可以獨立於UKF更新,可以佮線性(或者EKF)更新一齊用或者反過嚟。攞有估計值
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{\displaystyle {\hat {\mathbf {x} }}_{k-1\mid k-1}}
、
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{\displaystyle \mathbf {P} _{k-1\mid k-1}}
畀平均值同協方差嘅話,可以攞到
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{\displaystyle N=2L+1}
粒sigma點,似上節所講。啲sigma點係透過轉換函數f (transition function)傳播嘅:
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{\displaystyle \mathbf {x} _{j}=f\left(\mathbf {s} _{j}\right)\quad j=0,\dots ,2L}
.
着傳播過嘅 sigma 點又着加權嚟產生啲預測嘅平均值同埋協方差。
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{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}_{k\mid k-1}&=\sum _{j=0}^{2L}W_{j}^{a}\mathbf {x} _{j}\\\mathbf {P} _{k\mid k-1}&=\sum _{j=0}^{2L}W_{j}^{c}\left(\mathbf {x} _{j}-{\hat {\mathbf {x} }}_{k\mid k-1}\right)\left(\mathbf {x} _{j}-{\hat {\mathbf {x} }}_{k\mid k-1}\right)^{\textsf {T}}+\mathbf {Q} _{k}\end{aligned}}}
其中
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{\displaystyle W_{j}^{a}}
同埋
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{\displaystyle W_{j}^{c}}
分別係一階同埋二階權重喺啲原始sigma點上。矩陣
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{\displaystyle \mathbf {Q} _{k}}
係轉換噪音
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{\displaystyle \mathbf {w} _{k}}
嘅協方差。
畀有預測估計
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{\displaystyle {\hat {\mathbf {x} }}_{k\mid k-1}}
同
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{\displaystyle \mathbf {P} _{k\mid k-1}}
,計得出新一組
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{\displaystyle N=2L+1}
粒sigma點
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{\displaystyle \mathbf {s} _{0},\dots ,\mathbf {s} _{2L}}
有相應一階權重
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{\displaystyle W_{0}^{a},\dots W_{2L}^{a}}
二階權重
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{\displaystyle W_{0}^{c},\dots ,W_{2L}^{c}}
嘅。 [ 6] 啲sigma點通過
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{\displaystyle h}
轉變:
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{\displaystyle \mathbf {z} _{j}=h(\mathbf {s} _{j}),\,\,j=0,1,\dots ,2L}
.
係噉轉變過嘅啲點嘅觀測平均值同埋協方差就噉計出:
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{\displaystyle {\begin{aligned}{\hat {\mathbf {z} }}&=\sum _{j=0}^{2L}W_{j}^{a}\mathbf {z} _{j}\\[6pt]{\hat {\mathbf {S} }}_{k}&=\sum _{j=0}^{2L}W_{j}^{c}(\mathbf {z} _{j}-{\hat {\mathbf {z} }})(\mathbf {z} _{j}-{\hat {\mathbf {z} }})^{\textsf {T}}+\mathbf {R} _{k}\end{aligned}}}
其中
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{\displaystyle \mathbf {R} _{k}}
係觀測噪音
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{\displaystyle \mathbf {v} _{k}}
嘅協方差矩陣。另外仲要到交叉協方差(cross covariance )矩陣:
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{\displaystyle {\begin{aligned}\mathbf {C_{sz}} &=\sum _{j=0}^{2L}W_{j}^{c}(\mathbf {s} _{j}-{\hat {\mathbf {x} }}_{k|k-1})(\mathbf {z} _{j}-{\hat {\mathbf {z} }})^{\textsf {T}}\end{aligned}}}
其中
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{\displaystyle \mathbf {s} _{j}}
係攞
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{\displaystyle {\hat {\mathbf {x} }}_{k\mid k-1}}
、
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{\displaystyle \mathbf {P} _{k\mid k-1}}
整出嘅啲未轉變過嘅sigma點。
個卡曼增益即係:
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{\displaystyle {\begin{aligned}\mathbf {K} _{k}=\mathbf {C_{sz}} {\hat {\mathbf {S} }}_{k}^{-1}.\end{aligned}}}
更新後嘅估計值畀平均值同埋協方差即係:
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{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}_{k\mid k}&={\hat {\mathbf {x} }}_{k|k-1}+\mathbf {K} _{k}(\mathbf {z} _{k}-{\hat {\mathbf {z} }})\\\mathbf {P} _{k\mid k}&=\mathbf {P} _{k\mid k-1}-\mathbf {K} _{k}{\hat {\mathbf {S} }}_{k}\mathbf {K} _{k}^{\textsf {T}}.\end{aligned}}}
↑ Julier, Simon J.; Uhlmann, Jeffrey K. (1997). "New extension of the Kalman filter to nonlinear systems" (PDF) . 出自 Kadar, Ivan (編). Signal Processing, Sensor Fusion, and Target Recognition VI . Proceedings of SPIE.第3卷. pp. 182–193. Bibcode :1997SPIE.3068..182J . CiteSeerX 10.1.1.5.2891 . doi :10.1117/12.280797 . S2CID 7937456 . 喺2008-05-03 搵到 .
↑ Menegaz, H. M. T.; Ishihara, J. Y.; Borges, G. A.; Vargas, A. N. (October 2015). "A Systematization of the Unscented Kalman Filter Theory". IEEE Transactions on Automatic Control . 60 (10): 2583–2598. doi :10.1109/tac.2015.2404511 . hdl :20.500.11824/251 . ISSN 0018-9286 . S2CID 12606055 .
↑ Gustafsson, Fredrik; Hendeby, Gustaf (2012). "Some Relations Between Extended and Unscented Kalman Filters" . IEEE Transactions on Signal Processing . 60 (2): 545–555. Bibcode :2012ITSP...60..545G . doi :10.1109/tsp.2011.2172431 . S2CID 17876531 .
↑ Bitzer, S. (2016). "The UKF exposed: How it works, when it works and when it's better to sample" . doi :10.5281/zenodo.44386 .
↑ Wan, E.A.; Van Der Merwe, R. (2000). "The unscented Kalman filter for nonlinear estimation" (PDF) . Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No.00EX373) . p. 153. CiteSeerX 10.1.1.361.9373 . doi :10.1109/ASSPCC.2000.882463 . ISBN 978-0-7803-5800-3 . S2CID 13992571 . 原著 (PDF) 喺2012年3月3號歸檔. 喺2021年7月5號搵到 .
↑ Sarkka, Simo (September 2007). "On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems". IEEE Transactions on Automatic Control . 52 (9): 1631–1641. doi :10.1109/TAC.2007.904453 .