正向系统

正向系统（Positive systems^[1]^[2]）是指一种系统，在给定正的初始值下，其状态变数不会是负的。在一些实务应用中常会用到正向系统^[3]^[4]，因为这些变数表示实际的物理量，而这些物理量（如浓度、水位、高度等）不会为负值。

正向系统在控制系统的设计中相当重要^[5]。例如渐近稳定的正向线性时不变系统可以使用对角二次的李亚普诺夫函数，这在李亚普诺夫分析上的数值追踪性更好^[6]。

在状态观测器设计中也需考虑系统是否是正向系统，因为一般的状态观测器（例如龙贝格观测器）可能会出现不合理的负值^[7]。

正向的条件

连续时间线性系统 ${\dot {x}}=Ax$ 为正向系统，当且仅当A是梅兹勒矩阵（英语：Metzler matrix）^[1]。

离散时间线性系统 $x(k+1)=Ax(k)$ 为正向系统，当且仅当A是非负矩阵（英语：nonnegative matrix）^[1]。

参考资料

^ ^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. （原始内容存档 (PDF)于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]. （原始内容存档 (PDF)于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. （原始内容存档 (PDF)于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 （英语）.
^ 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. （原始内容 (PDF)存档于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. （原始内容存档 (PDF)于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]. （原始内容存档 (PDF)于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. （原始内容存档 (PDF)于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 （英语）.

[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. （原始内容 (PDF)存档于5 March 2016）.

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正向的条件

相关条目

参考资料