Editing Observability (section)

=== Related concepts ===

==== Observability index ====
The ''observability index'' <math>v</math> of a linear time-invariant discrete system is the smallest natural number for which the following is satisfied: <math>\text{rank}{(\mathcal{O}_v)} = \text{rank}{(\mathcal{O}_{v+1})}</math>, where

:<math> \mathcal{O}_v=\begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{v-1} \end{bmatrix}.</math>

==== Unobservable subspace ====
The ''unobservable subspace'' <math>N</math> of the linear system is the kernel of the linear map <math>G</math> given by<ref name=":1">Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998</ref><blockquote><math> \begin{align}
G \colon \mathbb{R}^{n} &\rightarrow  \mathcal{C}(\mathbb{R};\mathbb{R}^n) \\
         x(0)  &\mapsto  C e^{A t} x(0)
\end{align} </math></blockquote>where <math>\mathcal{C}(\mathbb{R};\mathbb{R}^n)</math> is the set of continuous functions from <math>\mathbb{R}</math> to <math>\mathbb{R}^n </math>.  <math>N</math> can also be written as <ref name=":1" />

:<math> N = \bigcap_{k=0}^{n-1} \ker(CA^k)= \ker{\mathcal{O}} </math>

Since the system is observable if and only if  <math>\operatorname{rank}(\mathcal{O}) = n</math>,  the system is observable if and only if <math>N</math> is the zero subspace.

The following properties for the unobservable subspace are valid:<ref name=":1" />
*<math> N \subset Ke(C) </math>
*<math> A(N) \subset N  </math>
*<math> N= \bigcup \{ S \subset R^n \mid S \subset Ke(C), A(S) \subset N  \} </math>

==== Detectability ====
A slightly weaker notion than observability is ''detectability''. A system is detectable if all the unobservable states are stable.<ref>{{Cite web | url=http://www.ece.rutgers.edu/~gajic/psfiles/chap5traCO.pdf | title=Controllability and Observability | access-date=2024-05-19}}</ref>

Detectability conditions are important in the context of [[Sensor Networks|sensor networks]].<ref>{{Cite journal|last1=Li|first1=W.|last2=Wei|first2=G.|last3=Ho|first3=D. W. C.|last4=Ding|first4=D.|date=November 2018|title=A Weightedly Uniform Detectability for Sensor Networks|journal=IEEE Transactions on Neural Networks and Learning Systems|volume=29|issue=11|pages=5790–5796|doi=10.1109/TNNLS.2018.2817244|pmid=29993845|s2cid=51615852}}</ref><ref>{{Cite journal|last1=Li|first1=W.|last2=Wang|first2=Z.|last3=Ho|first3=D. W. C.|last4=Wei|first4=G.|date=2019|title=On Boundedness of Error Covariances for Kalman Consensus Filtering Problems|journal=IEEE Transactions on Automatic Control|volume=65|issue=6|pages=2654–2661|doi=10.1109/TAC.2019.2942826|s2cid=204196474}}</ref>