Positive systems

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title: "Positive systems" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["control-theory", "systems-theory"] topic_path: "general/control-theory" source: "https://en.wikipedia.org/wiki/Positive_systems" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

Positive systemsT. Kaczorek. Positive 1D and 2D Systems. Springer- Verlag, 2002L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York, 2000 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, 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. 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.

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.

Conditions for positivity

A continuous-time linear system \dot{x} = Ax is positive if and only if A is a Metzler matrix.

A discrete-time linear system x(k+1) = A x(k) is positive if and only if A is a nonnegative matrix.

References

References

  1. (June 2006). "A positive systems model of TCP-like congestion control: asymptotic results". IEEE/ACM Transactions on Networking.
  2. (July 2010). "Selection of Time-after-injection in Bone Scanning using Compartmental Observers". Proceedings of the World Congress on Engineering.
  3. (2008). "Memoryless Control to Drive States of Delayed Continuous-time Systems within the Nonnegative Orthant". IFAC Proceedings Volumes.
  4. Rantzer, Anders. (2015). "Scalable control of positive systems". European Journal of Control.
  5. (June 2007). "2007 Mediterranean Conference on Control & Automation".

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