查看“︁滾動時域估計”︁的源代码

'''滾動時域估計'''（Moving horizon estimation、MHE）是一種利用一連串量測的信號進行[[最优化]]的作法，量測的信號中包括{{link-en|統計雜訊|statistical noise|雜訊}}（隨機變異）以及其他的不準確性，根據這些信號產生未知參數或是變數的估計值。滾動時域估計和確定性的作法不同，滾動時域估計需要遞迴式的求解法，利用[[线性规划]]或[[非线性规划]]來找到對應的解<ref>{{cite journal|author=J.D. Hedengren |author2=R. Asgharzadeh Shishavan | author3=K.M. Powell | author4=T.F. Edgar|title=Nonlinear modeling, estimation and predictive control in APMonitor|journal=Computers & Chemical Engineering|year=2014|volume=70|issue=5|pages=133–148|doi=10.1016/j.compchemeng.2014.04.013}}</ref>。

若在一些可以簡化的條件下，滾動時域估計可以簡化成[[卡尔曼滤波]]<ref>{{Cite journal | author = Rao, C.V. |author2=Rawlings, J.B. |author3=Maynes, D.Q | journal = IEEE Transactions on Automatic Control | volume = 48 | issue = 2 | year = 2003 | title = Constrained State Estimation for Nonlinear Discrete-Time Systems: Stability and Moving Horizon Approximations | pages = 246–258 | doi=10.1109/tac.2002.808470}}</ref>。在針對{{le|擴展卡爾曼濾波器|Extended Kalman filter}}及滾動時域估計的評估中，發現滾動時域估計的性能有所提昇，唯一需要付出的代價是其計算成本<ref>{{Cite journal | author = Haseltine, E.J. | author2 = Rawlings, J.B. | journal = Ind. Eng. Chem. Res. | volume = 44 | issue = 8 | year = 2005 | title = Critical Evaluation of Extended Kalman Filtering and Moving-Horizon Estimation | url = http://pubs.acs.org/cgi-bin/abstract.cgi/iecred/2005/44/i08/abs/ie034308l.html | pages = 2451–2460 | doi = 10.1021/ie034308l | access-date = 2018-12-24 | archive-date = 2008-10-05 | archive-url = https://web.archive.org/web/20081005154309/http://pubs.acs.org/cgi-bin/abstract.cgi/iecred/2005/44/i08/abs/ie034308l.html | dead-url = no }}</ref>。因為滾動時域估計在計算上的成本較高，因此一般會應用在運算資源較充裕的系統，而且是反應較慢的系統。不過在文獻中已有不少加速的方法<ref name=Hashemian>{{Cite journal | author = Hashemian, N. |author2=Armaou, A. | journal = Proceedings of the American Control Conference | year = 2015 | title = Fast Moving Horizon Estimation of nonlinear processes via Carleman linearization | url = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7171854&tag=1 | pages = 3379–3385 | doi=10.1109/ACC.2015.7171854}}</ref><ref>{{Cite journal | author = Hashemian, N. | author2 = Armaou, A. | journal = AIChE Journal | year = 2016 | title = Simulation, model-reduction and state estimation of a two-component coagulation process | url = http://onlinelibrary.wiley.com/doi/10.1002/aic.15146/pdf | volume = 62 | pages = 1557–1567 | doi = 10.1002/aic.15146 | access-date = 2018-12-24 | archive-date = 2016-08-17 | archive-url = https://web.archive.org/web/20160817150422/http://onlinelibrary.wiley.com/doi/10.1002/aic.15146/pdf | dead-url = no }}</ref>。

== 簡介 ==
滾動時域估計一般會用在[[动力系统]]中，估計一些有量測或是無法量測的狀態。會透過滾動時域估計來調整模型的初始狀態以及參數，讓估計結果接近量測結果。滾動時域估計是以在有限時間區間內，對程序模型及量測的最佳化為基礎。在時間{{math|t}}時，針對當前程序狀態進行取樣，再針對包括過去在內，較短的時間區間 <math>[t-T,t]</math>計算可以最小化策略（會使用數值的最小化）。滾動時域估計會用即時運算（透過[[歐拉-拉格朗日方程]]）來找到在時間<math>t</math>之間可以讓目標函數最小化的策略。但只有估計策略中的最後一步會用到，之後再針對滾動後的時域重新對程序數據取樣，再進行計算，得到新的狀態路徑以及估測參數。因為估計的時間區間會一直往前移動，因此此法會稱為'''滾動時域估計'''。此作法不一定是最佳的，但在實務上和[[卡尔曼滤波]]及其他估計策略比較，有不錯的結果。

===原理===
滾動時域估計是多變數的估計演算法，會用到
* 程序的內在動態模型
* 過去量測值的歷史
* 在估計時間區間內的最佳化費用函數
來計算最佳的狀態及參數

[[File:Moving horizon estimation scheme.png|thumb|滾動時域估計的架構<ref name=Hashemian />]]

其最佳化估計函數為

<math>J=\sum_{i=1}^N w_{y} (x_i-y_i)^2 + \sum_{i=1}^N w_{\hat{x}} (x_i-\hat{x_i})^2 + \sum_{i=1}^N w_{p_i} {\Delta p_i}^2</math>

並且沒有違反狀態或是參數的限制條件（例如上下限）

其中

<math>x_i</math> = 第''i''個模型估計變數（例如估計溫度）

<math>y_i</math> = 第''i''個量測變數（例如實測估計溫度）

<math>p_i</math> = 第''i''個估計參數（例如熱傳係數）

<math>w_{y}</math> = 加權係數，反應量測值<math>y_i</math>的相對重要性

<math>w_{\hat{x_i}}</math> = 加權係數，反應之前模型預測<math>\hat{x_i}</math>的相對重要性

<math>w_{p_i}</math> = 加權係數，避免<math>p_i</math>的大幅變化

滾動時域估計使用滾動的時間區間。在每一次取樣時，時間區間會往前前進一個時間間隔，會分析量測的輸出信號以及最近的輸出信號，來估測目前時間區間的狀態。

==應用==
* MATLAB、Python及Simulink都已有滾動時域估計的程式碼<ref>{{Cite web |url=http://apmonitor.com/do/index.php/Main/MovingHorizonEstimation |title=Python, MATLAB, and Simulink CSTR Example |accessdate=2018-12-24 |archive-date=2015-04-11 |archive-url=https://web.archive.org/web/20150411013442/http://apmonitor.com/do/index.php/Main/MovingHorizonEstimation |dead-url=no }}</ref> 
* 監控工業製程的污染<ref>{{Cite journal | author = Spivey, B. | author2 = Hedengren, J. D. | author3 = Edgar, T. F. | journal = Industrial & Engineering Chemistry Research | volume = 49 | issue = 17 | year = 2010 | title = Constrained Nonlinear Estimation for Industrial Process Fouling | url = http://pubs.acs.org/doi/abs/10.1021/ie9018116 | pages = 7824–7831 | doi = 10.1021/ie9018116 | access-date = 2018-12-24 | archive-date = 2019-04-10 | archive-url = https://web.archive.org/web/20190410151404/https://pubs.acs.org/doi/abs/10.1021/ie9018116 | dead-url = no }}</ref>
* 石油及天然氣產業<ref>{{Cite book | author = Hedengren, J.D. | year = 2012 | title = Advanced Process Monitoring | editors = Kevin C. Furman, Jin-Hwa Song, Amr El-Bakry | url = http://prism.groups.et.byu.net/uploads/Members/hedengren_apm2012.pdf | series = Springer’s International Series in Operations Research and Management Science | access-date = 2018-12-24 | archive-url = https://web.archive.org/web/20160304081817/http://prism.groups.et.byu.net/uploads/Members/hedengren_apm2012.pdf | archive-date = 2016-03-04 | dead-url = yes }}</ref>
* 聚合物製造<ref>{{cite journal | last=Ramlal | first=J. | title=Moving Horizon Estimation for an Industrial Gas Phase Polymerization Reactor | journal=IFAC Symposium on Nonlinear Control Systems Design (NOLCOS) | year=2007 | url=http://apmonitor.com/Documents/mhe.pdf | deadurl=yes | archiveurl=https://web.archive.org/web/20090920070424/http://apmonitor.com/Documents/mhe.pdf | archivedate=2009-09-20 | df= }}</ref>
* 無人航空系統<ref>{{cite journal | last=Sun | first=L. | title=Optimal Trajectory Generation using Model Predictive Control for Aerially Towed Cable Systems | journal=Journal of Guidance, Control, and Dynamics | year=2013 | url=http://apm.byu.edu/prism/uploads/Members/sun_2013.pdf | access-date=2018-12-24 | archive-date=2014-03-03 | archive-url=https://web.archive.org/web/20140303205912/http://apm.byu.edu/prism/uploads/Members/sun_2013.pdf | dead-url=no }}</ref><ref>{{cite journal | last=Sun | first=L. | title=Parameter Estimation for Towed Cable Systems Using Moving Horizon Estimation | journal=IEEE Transactions on Aerospace and Electronic Systems | year=2015 | url=http://apm.byu.edu/prism/uploads/Members/SunCastagnoHedengren14.pdf | access-date=2018-12-24 | archive-date=2015-06-10 | archive-url=https://web.archive.org/web/20150610224510/http://apm.byu.edu/prism/uploads/Members/SunCastagnoHedengren14.pdf | dead-url=no }}</ref>

==相關條目==
* {{link-en|Alpha beta濾波器|Alpha beta filter}}
* [[数据同化]]
* {{link-en|集合卡爾曼濾波器|Ensemble Kalman filter}}
* {{le|擴展卡爾曼濾波器|Extended Kalman filter}}
* {{link-en|不變擴展卡爾曼濾波器|Invariant extended Kalman filter}}
* {{link-en|快速卡爾曼濾波|Fast Kalman filter}}
* [[濾波問題]]
* {{link-en|核自适应滤波器|Kernel adaptive filter}}
* {{link-en|非線性濾波器|Non-linear filter}}
* [[粒子濾波器]]
* {{link-en|預測子-修正子法|Predictor corrector}}
* {{link-en|遞迴最小平方法|Recursive least squares}}
* [[施密特–卡尔曼滤波器]]
* [[滑動模式控制]]
* [[维纳滤波]]

==參考資料==
{{Reflist}}

==延伸閱讀==
{{Refbegin}}

* {{Cite book
 | first=James B.  |last= Rawlings
 | year = 2009
 | title = Model Predictive Control: Theory and Design
 | url=https://archive.org/details/modelpredictivec0000rawl  | publisher = [Nob Hill Publishing, LLC | location=Madison, WI
 | series = Mathematics in Science and Engineering
 | pages=576
 | isbn=978-0-9759377-0-9
}}

{{Refend}}

==外部連結==
* [http://apmonitor.com/wiki/index.php/Main/Estimation MHE Tutorial in Simulink and MATLAB] {{Wayback|url=http://apmonitor.com/wiki/index.php/Main/Estimation |date=20130606202608 }}
* [https://www.cds.caltech.edu/~murray/wiki/NCS:_Moving_Horizon_Estimation MHE lecture material] {{Wayback|url=https://www.cds.caltech.edu/~murray/wiki/NCS:_Moving_Horizon_Estimation |date=20181224121911 }}
* [http://apmonitor.com/do/index.php/Main/MovingHorizonEstimation Online Course:] {{Wayback|url=http://apmonitor.com/do/index.php/Main/MovingHorizonEstimation |date=20150411013442 }} MHE in Simulink, MATLAB and Python

[[Category:控制理论]]
[[Category:非線性濾波器]]
[[Category:线性滤波器]]
[[Category:信號估測]]