Editing Observability (section)

== Nonlinear systems ==
Given the system <math>\dot{x} = f(x) + \sum_{j=1}^mg_j(x)u_j </math>, <math>y_i = h_i(x), i \in p</math>. Where <math>x \in \mathbb{R}^n</math> the state vector, <math>u \in \mathbb{R}^m</math> the input vector and <math>y \in \mathbb{R}^p</math> the output vector.  <math>f,g,h</math> are to be smooth vector fields.

Define the observation space <math>\mathcal{O}_s</math> to be the space containing all repeated [[Lie derivative]]s, then the system is observable in <math>x_0</math> if and only if <math>\dim(d\mathcal{O}_s(x_0)) = n</math>, where

:<math>d\mathcal{O}_s(x_0) = \operatorname{span}(dh_1(x_0), \ldots , dh_p(x_0), dL_{v_i}L_{v_{i-1}}, \ldots , L_{v_1}h_j(x_0)),\ j\in p, k=1,2,\ldots.</math><ref>Lecture notes for Nonlinear Systems Theory by prof. dr. D.Jeltsema, [[Jacquelien Scherpen|prof dr. J.M.A.Scherpen]] and prof dr. A.J.van der Schaft.</ref>

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,<ref>{{cite journal|doi=10.1016/0022-247X(71)90241-1|title=On the observability of nonlinear systems: I |year=1971 |last1=Griffith |first1=E. W. |last2=Kumar |first2=K. S. P. |journal=Journal of Mathematical Analysis and Applications |volume=35 |pages=135–147 |doi-access= }}</ref> Kou, Elliot and Tarn,<ref>{{cite journal|doi=10.1016/S0019-9958(73)90508-1|title=Observability of nonlinear systems |year=1973 |last1=Kou |first1=Shauying R. |last2=Elliott |first2=David L. |last3=Tarn |first3=Tzyh Jong |journal=Information and Control |volume=22 |pages=89–99 |doi-access=free }}</ref> and Singh.<ref>{{cite journal|doi=10.1080/00207727508941856|title=Observability in non-linear systems with immeasurable inputs |year=1975 |last1=Singh |first1=Sahjendra N. |journal=International Journal of Systems Science |volume=6 |issue=8 |pages=723–732 }}</ref>

There also exist an observability criteria for nonlinear time-varying systems.<ref>{{Cite journal |last=Martinelli |first=Agostino |date=2022 |title=Extension of the Observability Rank Condition to Time-Varying Nonlinear Systems |url=https://ieeexplore.ieee.org/document/9790057 |journal=IEEE Transactions on Automatic Control |volume=67 |issue=9 |pages=5002–5008 |doi=10.1109/TAC.2022.3180771 |s2cid=251957578 |issn=0018-9286}}</ref>