Editing Observability

{{Short description|In control theory, visible state of a system}}
{{for-multi|the concept in quantum mechanics|Observable|the concept in software|Observability (software)}}
'''Observability''' is a measure of how well internal states of a [[system]] can be inferred from knowledge of its external outputs.
In [[control theory]], the observability and [[controllability]] of a linear system are mathematical [[duality (mathematics)|duals]].

The concept of observability was introduced by the Hungarian-American engineer [[Rudolf E. Kálmán]] for linear dynamic systems.<ref>{{cite journal|doi=10.1016/S1474-6670(17)70094-8|title=On the general theory of control systems |year=1960 |last1=Kalman |first1=R.E. |journal=IFAC Proceedings Volumes |volume=1 |pages=491–502 |doi-access= }}</ref><ref>{{cite journal|doi=10.1137/0301010|title=Mathematical Description of Linear Dynamical Systems |year=1963 |last1=Kalman |first1=R. E. |journal=Journal of the Society for Industrial and Applied Mathematics, Series A: Control |volume=1 |issue=2 |pages=152–192 }}</ref> A dynamical system designed to estimate the state of a system from measurements of the outputs is called a ''[[state observer]]'' for that system, such as [[Kalman filter]]s.

==Definition==
Consider a physical system modeled in [[state-space representation]]. A system is said to be '''observable''' if, for every possible evolution of [[State space representation#Linear systems|state and control vectors]], the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by [[sensor]]s). In other words, one can determine the behavior of the entire system from the system's outputs. On the other hand, if the system is not observable, there are state trajectories that are not distinguishable by only measuring the outputs.

== Linear time-invariant systems ==
For [[LTI system theory|time-invariant linear system]]s in the state space representation, there are convenient tests to check whether a system is observable. Consider a [[Single-Input and Single-Output|SISO]] system with <math>n</math> state variables (see [[state space (controls)|state space]] for details about [[MIMO]] systems) given by

: <math>\dot{\mathbf{x}}(t) = \mathbf{A} \mathbf{x}(t) + \mathbf{B} \mathbf{u}(t)</math>
: <math>\mathbf{y}(t) = \mathbf{C} \mathbf{x}(t) + \mathbf{D} \mathbf{u}(t)</math>

=== Observability matrix ===
If and only if the column [[Rank (linear algebra)|rank]] of the ''observability matrix'', defined as

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

is equal to <math>n</math>, then the system is observable. The rationale for this test is that if <math>n</math> columns are linearly independent, then each of the <math>n</math> state variables is viewable through linear combinations of the output variables <math>y</math>.

=== 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>

== Linear time-varying systems ==

Consider the [[continuous function|continuous]] [[linear]] [[time-variant system]]

: <math>\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t) \, </math>
: <math>\mathbf{y}(t) = C(t) \mathbf{x}(t). \, </math>

Suppose that the matrices <math>A</math>, <math>B</math>  and <math>C</math> are given as well as inputs and outputs <math>u</math> and <math>y</math> for all <math>t \in [t_0,t_1];</math> then it is possible to determine <math>x(t_0)</math> to within an additive constant vector which lies in the [[null space]] of <math>M(t_0,t_1)</math> defined by
: <math>M(t_0,t_1) = \int_{t_0}^{t_1} \varphi(t,t_0)^{T}C(t)^{T}C(t)\varphi(t,t_0) \, dt</math>
where <math>\varphi</math> is the [[state-transition matrix]].

It is possible to determine a unique <math>x(t_0)</math> if <math>M(t_0,t_1)</math> is [[Algebraic curve#Singularities|nonsingular]].  In fact, it is not possible to distinguish the initial state for <math>x_1</math> from that of <math>x_2</math> if <math>x_1 - x_2</math> is in the null space of <math>M(t_0,t_1)</math>.

Note that the matrix <math>M</math> defined as above has the following properties:
* <math>M(t_0,t_1)</math> is [[symmetric matrix|symmetric]]
* <math>M(t_0,t_1)</math> is [[positive semidefinite matrix|positive semidefinite]] for <math>t_1 \geq t_0</math>
* <math>M(t_0,t_1)</math> satisfies the linear [[matrix differential equation]]
:: <math>\frac{d}{dt}M(t,t_1) = -A(t)^{T}M(t,t_1)-M(t,t_1)A(t)-C(t)^{T}C(t), \; M(t_1,t_1) = 0</math>
* <math>M(t_0,t_1)</math> satisfies the equation
:: <math>M(t_0,t_1) = M(t_0,t) + \varphi(t,t_0)^T M(t,t_1)\varphi(t,t_0)</math><ref>{{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}</ref>

=== Observability matrix generalization ===
The system is observable in <math>[t_0,t_1]</math> if and only if there exists an interval <math>[t_0,t_1]</math> in <math>\mathbb{R}</math> such that the matrix <math>M(t_0,t_1)</math> is nonsingular.

If <math>A(t), C(t)</math> are analytic, then the system is observable in the interval [<math>t_0</math>,<math>t_1</math>] if there exists <math>\bar{t} \in [t_0,t_1]</math> and a positive integer ''k'' such that<ref name=":0">Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.</ref>

: <math> \operatorname{rank} \begin{bmatrix}
 & N_0(\bar{t}) & \\ 
 & N_1(\bar{t}) & \\ 
 & \vdots & \\ 
 & N_{k}(\bar{t}) & 
\end{bmatrix} = n, </math>

where <math>N_0(t):=C(t)</math> and <math>N_i(t)</math> is defined recursively as

: <math>N_{i+1}(t) := N_i(t)A(t) + \frac{\mathrm{d}}{\mathrm{d} t}N_i(t),\  i = 0, \ldots, k-1 </math>

==== Example ====

Consider a system varying analytically  in <math> (-\infty,\infty) </math> and matrices<blockquote><math>A(t) = \begin{bmatrix}
t & 1 & 0\\ 
0 & t^{3} & 0\\ 
0 & 0 & t^{2} 
\end{bmatrix},\, C(t) = \begin{bmatrix}
1 & 0 & 1
\end{bmatrix}.</math> </blockquote>Then <math> \begin{bmatrix}
N_0(0) \\
N_1(0) \\
N_2(0) 
\end{bmatrix}
 = \begin{bmatrix}
1 & 0 & 1 \\ 
0 & 1 & 0 \\ 
1& 0 & 0 
\end{bmatrix}</math> , and since this matrix has rank = 3, the system is observable on every nontrivial interval of  <math>\mathbb{R}</math>.

== 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>

== Static systems and general topological spaces ==

Observability may also be characterized for steady state systems (systems typically defined in terms of algebraic equations and inequalities), or more generally, for sets in <math>\mathbb{R}^n</math>.<ref>{{cite journal|doi=10.1016/0009-2509(81)85004-X|url=https://gregstanleyandassociates.com/CES-1981a-ObservabilityRedundancy.pdf|title=Observability and redundancy in process data estimation |year=1981 |last1=Stanley |first1=G. M. |last2=Mah |first2=R. S. H. |journal=Chemical Engineering Science |volume=36 |issue=2 |pages=259–272 |bibcode=1981ChEnS..36..259S }}</ref><ref>{{cite journal|doi=10.1016/0009-2509(81)80034-6|url=https://gregstanleyandassociates.com/CES-1981b-ObservabilityRedundancyProcessNetworks.pdf|title=Observability and redundancy classification in process networks |year=1981 |last1=Stanley |first1=G.M. |last2=Mah |first2=R.S.H. |journal=Chemical Engineering Science |volume=36 |issue=12 |pages=1941–1954 }}</ref> Just as observability criteria are used to predict the behavior of [[Kalman filter]]s or other observers in the dynamic system case,  observability criteria for sets in <math>\mathbb{R}^n</math> are used to predict the behavior of [[data validation and reconciliation|data reconciliation]] and other static estimators. In the nonlinear case,  observability can be characterized for individual variables, and also for local estimator behavior rather than just global behavior.

== See also ==
* [[Controllability]]
* [[Hautus lemma]]
* [[Structural identifiability|Identifiability]]
* [[State observer]]
* [[State space (controls)]]

==References==
{{reflist}}

==External links==
*{{planetmath reference|urlname=Observability|title=Observability}}
* [http://www.mathworks.com/help/toolbox/control/ref/obsv.html MATLAB function for checking observability of a system]
* [http://reference.wolfram.com/mathematica/ref/ObservableModelQ.html Mathematica function for checking observability of a system]

[[Category:Classical control theory]]

[[fr:Représentation d'état#Observabilité et détectabilité]]