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. http://eprints.nuim.ie/1764/1/HamiltonPositiveSystems.pdf
  4. http://www.iaeng.org/publication/WCE2010/WCE2010_pp656-661.pdf
  5. http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2008/data/papers/3024.pdf
  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.
  7. http://advantech.gr/med07/papers/T19-027-598.pdf
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