正向系統

正向系統（Positive systems^[1]^[2]）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^[3]^[4]，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^[5]。例如漸近穩定的正向線性時不變系統可以使用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^[6]。

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^[7]。

正向的條件

連續時間線性系統 ${\dot {x}}=Ax$ 為正向系統，若且唯若A是梅茲勒矩陣（英語：Metzler matrix）^[1]。

離散時間線性系統 $x(k+1)=Ax(k)$ 為正向系統，若且唯若A是非負矩陣（英語：nonnegative matrix）^[1]。

參考資料

^ ^1.0 ^1.1 ^1.2 T. Kaczorek. Positive 1D and 2D Systems. Springer- Verlag, 2002
^ L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York, 2000
^ Shorten, Robert; Wirth, Fabian; Leith, Douglas. A positive systems model of TCP-like congestion control: asymptotic results (PDF). IEEE/ACM Transactions on Networking. June 2006, 14 (3): 616–629 [15 February 2023]. S2CID 14066559. doi:10.1109/TNET.2006.876178. （原始內容存檔 (PDF)於2024-04-22）.
^ Tadeo, Fernando; Rami, Mustapha Ait. Selection of Time-after-injection in Bone Scanning using Compartmental Observers (PDF). Proceedings of the World Congress on Engineering. July 2010, 1 [15 February 2023]. （原始內容存檔 (PDF)於2023-07-12）.
^ Hmamed, Abelaziz; Benzaouia, Abdellah; Rami, Mustapha Ait; Tadeo, Fernando. Memoryless Control to Drive States of Delayed Continuous-time Systems within the Nonnegative Orthant (PDF). IFAC Proceedings Volumes. 2008, 41 (2): 3934–3939 [15 February 2023]. doi:10.3182/20080706-5-KR-1001.00662. （原始內容存檔 (PDF)於2023-12-03）.
^ Rantzer, Anders. Scalable control of positive systems. European Journal of Control. 2015, 24: 72–80. S2CID 31821230. arXiv:1203.0047 . doi:10.1016/j.ejcon.2015.04.004 （英語）.
^ Ait Rami, M.; Helmke, U.; Tadeo, F. Positive observation problem for linear time-delay positive systems (PDF). 2007 Mediterranean Conference on Control & Automation. June 2007: 1–6 [15 February 2023]. ISBN 978-1-4244-1281-5. S2CID 15084715. doi:10.1109/MED.2007.4433692. （原始內容 (PDF)存檔於5 March 2016）.

[:0-1] 1.0 ^1.1 ^1.2 T. Kaczorek. Positive 1D and 2D Systems. Springer- Verlag, 2002

[2] L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York, 2000

[3] Shorten, Robert; Wirth, Fabian; Leith, Douglas. A positive systems model of TCP-like congestion control: asymptotic results (PDF). IEEE/ACM Transactions on Networking. June 2006, 14 (3): 616–629 [15 February 2023]. S2CID 14066559. doi:10.1109/TNET.2006.876178. （原始內容存檔 (PDF)於2024-04-22）.

[4] Tadeo, Fernando; Rami, Mustapha Ait. Selection of Time-after-injection in Bone Scanning using Compartmental Observers (PDF). Proceedings of the World Congress on Engineering. July 2010, 1 [15 February 2023]. （原始內容存檔 (PDF)於2023-07-12）.

[5] Hmamed, Abelaziz; Benzaouia, Abdellah; Rami, Mustapha Ait; Tadeo, Fernando. Memoryless Control to Drive States of Delayed Continuous-time Systems within the Nonnegative Orthant (PDF). IFAC Proceedings Volumes. 2008, 41 (2): 3934–3939 [15 February 2023]. doi:10.3182/20080706-5-KR-1001.00662. （原始內容存檔 (PDF)於2023-12-03）.

[6] Rantzer, Anders. Scalable control of positive systems. European Journal of Control. 2015, 24: 72–80. S2CID 31821230. arXiv:1203.0047 . doi:10.1016/j.ejcon.2015.04.004 （英語）.

[7] Ait Rami, M.; Helmke, U.; Tadeo, F. Positive observation problem for linear time-delay positive systems (PDF). 2007 Mediterranean Conference on Control & Automation. June 2007: 1–6 [15 February 2023]. ISBN 978-1-4244-1281-5. S2CID 15084715. doi:10.1109/MED.2007.4433692. （原始內容 (PDF)存檔於5 March 2016）.

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正向的條件

相關條目

參考資料