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 is positive if and only if A is a Metzler matrix.[1]

A discrete-time linear system is positive if and only if A is a nonnegative matrix.[1]

See also

References
  1. 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 (June 2006). "A positive systems model of TCP-like congestion control: asymptotic results" (PDF). IEEE/ACM Transactions on Networking. 14 (3): 616–629. doi:10.1109/TNET.2006.876178. S2CID 14066559. Retrieved 15 February 2023.
  4. Tadeo, Fernando; Rami, Mustapha Ait (July 2010). "Selection of Time-after-injection in Bone Scanning using Compartmental Observers" (PDF). Proceedings of the World Congress on Engineering. 1. Retrieved 15 February 2023.
  5. Hmamed, Abelaziz; Benzaouia, Abdellah; Rami, Mustapha Ait; Tadeo, Fernando (2008). "Memoryless Control to Drive States of Delayed Continuous-time Systems within the Nonnegative Orthant" (PDF). IFAC Proceedings Volumes. 41 (2): 3934–3939. doi:10.3182/20080706-5-KR-1001.00662. Retrieved 15 February 2023.
  6. Rantzer, Anders (2015). "Scalable control of positive systems". European Journal of Control. 24: 72–80. arXiv:1203.0047. doi:10.1016/j.ejcon.2015.04.004. S2CID 31821230.
  7. Ait Rami, M.; Helmke, U.; Tadeo, F. (June 2007). "Positive observation problem for linear time-delay positive systems" (PDF). 2007 Mediterranean Conference on Control & Automation. pp. 1–6. doi:10.1109/MED.2007.4433692. ISBN 978-1-4244-1281-5. S2CID 15084715. Archived from the original (PDF) on 5 March 2016. Retrieved 15 February 2023.
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