Positive systems
Positive systems[1][2] constitute a class of systems that has the important property that its state variables are never negative, given a positive initial state. These systems appear frequently in practical applications,[3][4] as these variables represent physical quantities, with positive sign (levels, heights, concentrations, etc.).
The fact that a system is positive has important implications in the control system design.[5] For instance, an asymptotically stable positive linear time-invariant system always admits a diagonal quadratic Lyapunov function, which makes these systems more numerical tractable in the context of Lyapunov analysis.[6]
It is also important to take this positivity into account for state observer design, as standard observers (for example Luenberger observers) might give illogical negative values.[7]
Conditions for positivity
A continuous-time linear system [math]\displaystyle{ \dot{x} = Ax }[/math] is positive if and only if A is a Metzler matrix.[1]
A discrete-time linear system [math]\displaystyle{ x(k+1) = A x(k) }[/math] is positive if and only if A is a nonnegative matrix.[1]
See also
References
- ↑ 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 (June 2006). "A positive systems model of TCP-like congestion control: asymptotic results". IEEE/ACM Transactions on Networking 14 (3): 616–629. doi:10.1109/TNET.2006.876178. http://mural.maynoothuniversity.ie/1764/1/HamiltonPositiveSystems.pdf. Retrieved 15 February 2023.
- ↑ Tadeo, Fernando; Rami, Mustapha Ait (July 2010). "Selection of Time-after-injection in Bone Scanning using Compartmental Observers". Proceedings of the World Congress on Engineering 1. https://www.iaeng.org/publication/WCE2010/WCE2010_pp656-661.pdf. Retrieved 15 February 2023.
- ↑ Hmamed, Abelaziz; Benzaouia, Abdellah; Rami, Mustapha Ait; Tadeo, Fernando (2008). "Memoryless Control to Drive States of Delayed Continuous-time Systems within the Nonnegative Orthant". IFAC Proceedings Volumes 41 (2): 3934–3939. doi:10.3182/20080706-5-KR-1001.00662. https://folk.ntnu.no/skoge/prost/proceedings/ifac2008/data/papers/3024.pdf. Retrieved 15 February 2023.
- ↑ Rantzer, Anders (2015). "Scalable control of positive systems" (in en). European Journal of Control 24: 72–80. doi:10.1016/j.ejcon.2015.04.004. https://linkinghub.elsevier.com/retrieve/pii/S094735801500059X.
- ↑ Ait Rami, M.; Helmke, U.; Tadeo, F. (June 2007). "Positive observation problem for linear time-delay positive systems". 2007 Mediterranean Conference on Control & Automation. pp. 1–6. doi:10.1109/MED.2007.4433692. ISBN 978-1-4244-1281-5. https://advantech.gr/med07/papers/T19-027-598.pdf. Retrieved 15 February 2023.
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