Positive systems

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,{{cite journal |last1=Shorten |first1=Robert |last2=Wirth |first2=Fabian |last3=Leith |first3=Douglas |title=A positive systems model of TCP-like congestion control: asymptotic results |journal=IEEE/ACM Transactions on Networking |date=June 2006 |volume=14 |issue=3 |pages=616–629 |doi=10.1109/TNET.2006.876178 |s2cid=14066559 |url=http://mural.maynoothuniversity.ie/1764/1/HamiltonPositiveSystems.pdf |access-date=15 February 2023}}{{cite journal |last1=Tadeo |first1=Fernando |last2=Rami |first2=Mustapha Ait |title=Selection of Time-after-injection in Bone Scanning using Compartmental Observers |journal=Proceedings of the World Congress on Engineering |date=July 2010 |volume=1 |url=https://www.iaeng.org/publication/WCE2010/WCE2010_pp656-661.pdf |access-date=15 February 2023}} 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.{{cite journal |last1=Hmamed |first1=Abelaziz |last2=Benzaouia |first2=Abdellah |last3=Rami |first3=Mustapha Ait |last4=Tadeo |first4=Fernando |title=Memoryless Control to Drive States of Delayed Continuous-time Systems within the Nonnegative Orthant |journal=IFAC Proceedings Volumes |date=2008 |volume=41 |issue=2 |pages=3934–3939 |doi=10.3182/20080706-5-KR-1001.00662 |url=https://folk.ntnu.no/skoge/prost/proceedings/ifac2008/data/papers/3024.pdf |access-date=15 February 2023}} 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.{{Cite journal|last=Rantzer|first=Anders|date=2015|title=Scalable control of positive systems|url=https://linkinghub.elsevier.com/retrieve/pii/S094735801500059X|journal=European Journal of Control|language=en|volume=24|pages=72–80|doi=10.1016/j.ejcon.2015.04.004|arxiv=1203.0047|s2cid=31821230}}

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.{{cite book |last1=Ait Rami |first1=M. |last2=Helmke |first2=U. |last3=Tadeo |first3=F. |title=2007 Mediterranean Conference on Control & Automation |chapter=Positive observation problem for linear time-delay positive systems |date=June 2007 |pages=1–6 |doi=10.1109/MED.2007.4433692 |isbn=978-1-4244-1281-5 |s2cid=15084715 |chapter-url=https://advantech.gr/med07/papers/T19-027-598.pdf |access-date=15 February 2023|archive-url=https://web.archive.org/web/20160305062015/https://advantech.gr/med07/papers/T19-027-598.pdf|archive-date=5 March 2016}}