查看“︁正向系統”︁的源代码

'''正向系統'''（Positive systems<ref name=":0">T. Kaczorek. Positive 1D and 2D Systems. Springer-
Verlag, 2002</ref><ref>L. Farina and S. Rinaldi, Positive Linear Systems; Theory and
Applications, J. Wiley, New York, 2000</ref>）是指一種系統，在給定正的初始值下，其[[狀態變數]]不會是負的。在一些實務應用中常會用到正向系統<ref>{{cite journal |last1=Shorten |first1=Robert |last2=Wirth |first2=Fabian |last3=Leith |first3=Douglas |title=A positive systems model of TCP-like congestion control: asymptotic results |journal=IEEE/ACM Transactions on Networking |date=June 2006 |volume=14 |issue=3 |pages=616–629 |doi=10.1109/TNET.2006.876178 |s2cid=14066559 |url=http://mural.maynoothuniversity.ie/1764/1/HamiltonPositiveSystems.pdf |access-date=15 February 2023 |archive-date=2024-04-22 |archive-url=https://web.archive.org/web/20240422030344/https://mural.maynoothuniversity.ie/1764/1/HamiltonPositiveSystems.pdf |dead-url=no }}</ref><ref>{{cite journal |last1=Tadeo |first1=Fernando |last2=Rami |first2=Mustapha Ait |title=Selection of Time-after-injection in Bone Scanning using Compartmental Observers |journal=Proceedings of the World Congress on Engineering |date=July 2010 |volume=1 |url=https://www.iaeng.org/publication/WCE2010/WCE2010_pp656-661.pdf |access-date=15 February 2023 |archive-date=2023-07-12 |archive-url=https://web.archive.org/web/20230712212628/https://www.iaeng.org/publication/WCE2010/WCE2010_pp656-661.pdf |dead-url=no }}</ref>，因為這些變數表示實際的物理量，而這些物理量（如[[濃度]]、[[水位]]、[[高度]]等）不會為負值。

正向系統在[[控制系統]]的設計中相當重要<ref>{{cite journal |last1=Hmamed |first1=Abelaziz |last2=Benzaouia |first2=Abdellah |last3=Rami |first3=Mustapha Ait |last4=Tadeo |first4=Fernando |title=Memoryless Control to Drive States of Delayed Continuous-time Systems within the Nonnegative Orthant |journal=IFAC Proceedings Volumes |date=2008 |volume=41 |issue=2 |pages=3934–3939 |doi=10.3182/20080706-5-KR-1001.00662 |url=https://folk.ntnu.no/skoge/prost/proceedings/ifac2008/data/papers/3024.pdf |access-date=15 February 2023 |archive-date=2023-12-03 |archive-url=https://web.archive.org/web/20231203062654/https://folk.ntnu.no/skoge/prost/proceedings/ifac2008/data/papers/3024.pdf |dead-url=no }}</ref>。例如[[李雅普诺夫稳定性|漸近穩定]]的正向[[线性时不变系统理论|线性时不变系统]]可以使用[[對角矩陣|對角]]二次的[[李亞普諾夫函數]]，這在李亞普諾夫分析上的數值追蹤性更好<ref>{{Cite journal|last=Rantzer|first=Anders|date=2015|title=Scalable control of positive systems|url=https://linkinghub.elsevier.com/retrieve/pii/S094735801500059X|journal=European Journal of Control|language=en|volume=24|pages=72–80|doi=10.1016/j.ejcon.2015.04.004|arxiv=1203.0047|s2cid=31821230}}</ref>。

在[[狀態觀測器]]設計中也需考慮系統是否是正向系統，因為一般的狀態觀測器（例如龍貝格觀測器）可能會出現不合理的負值<ref>{{cite book |last1=Ait Rami |first1=M. |last2=Helmke |first2=U. |last3=Tadeo |first3=F. |title=2007 Mediterranean Conference on Control & Automation |chapter=Positive observation problem for linear time-delay positive systems |date=June 2007 |pages=1–6 |doi=10.1109/MED.2007.4433692 |isbn=978-1-4244-1281-5 |s2cid=15084715 |chapter-url=https://advantech.gr/med07/papers/T19-027-598.pdf |access-date=15 February 2023|archive-url=https://web.archive.org/web/20160305062015/https://advantech.gr/med07/papers/T19-027-598.pdf|archive-date=5 March 2016}}</ref>。

== 正向的條件 ==
連續時間線性系統<math>\dot{x} = Ax</math>為正向系統，若且唯若A是{{le|梅茲勒矩陣|Metzler matrix}}<ref name=":0" />。

離散時間線性系統<math>x(k+1) = A x(k)</math>為正向系統，若且唯若A是{{le|非負矩陣|nonnegative matrix}}<ref name=":0" />。

==相關條目==
*[[正回饋]]

==參考資料==
{{Reflist}}

[[Category:控制理論]]
[[Category:系統理論]]