observability

Consider a physical system modeled in state-space representation. A system is said to be observable if, for every possible evolution of state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by sensors). 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.

For time-invariant linear systems in the state space representation, there are convenient tests to check whether a system is observable. Consider a SISO system with $n$ state variables (see state space for details about MIMO systems) given by

: $\backslash dot\{\backslash mathbf\{x\}\}(t)\; =\; \backslash mathbf\{A\}\; \backslash mathbf\{x\}(t)\; +\; \backslash mathbf\{B\}\; \backslash mathbf\{u\}(t)$

: $\backslash mathbf\{y\}(t)\; =\; \backslash mathbf\{C\}\; \backslash mathbf\{x\}(t)\; +\; \backslash mathbf\{D\}\; \backslash mathbf\{u\}(t)$

If and only if the column rank of the observability matrix, defined as

:$\backslash mathcal\{O\}=\backslash begin\{bmatrix\}\; C\; \backslash \backslash \; CA\; \backslash \backslash \; CA^2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; CA^\{n-1\}\; \backslash end\{bmatrix\}$

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

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

:$\backslash mathcal\{O\}\_v=\backslash begin\{bmatrix\}\; C\; \backslash \backslash \; CA\; \backslash \backslash \; CA^2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; CA^\{v-1\}\; \backslash end\{bmatrix\}.$

The unobservable subspace $N$ of the linear system is the kernel of the linear map $G$ given bySontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998

$\backslash begin\{align\}$

G \colon \mathbb{R}^{n} &\rightarrow \mathcal{C}(\mathbb{R};\mathbb{R}^n) \\

x(0) &\mapsto C e^{A t} x(0)

\end{align}

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

Detectability conditions are important in the context of sensor networks.{{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}}{{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}}

Consider the continuous linear time-variant system

: $\backslash dot\{\backslash mathbf\{x\}\}(t)\; =\; A(t)\; \backslash mathbf\{x\}(t)\; +\; B(t)\; \backslash mathbf\{u\}(t)\; \backslash ,$

: $\backslash mathbf\{y\}(t)\; =\; C(t)\; \backslash mathbf\{x\}(t).\; \backslash ,$

Suppose that the matrices $A$, $B$ and $C$ are given as well as inputs and outputs $u$ and $y$ for all $t\; \backslash in\; [t\_0,t\_1];$ then it is possible to determine $x(t\_0)$ to within an additive constant vector which lies in the null space of $M(t\_0,t\_1)$ defined by

: $M(t\_0,t\_1)\; =\; \backslash int\_\{t\_0\}^\{t\_1\}\; \backslash varphi(t,t\_0)^\{T\}C(t)^\{T\}C(t)\backslash varphi(t,t\_0)\; \backslash ,\; dt$

where $\backslash varphi$ is the state-transition matrix.

It is possible to determine a unique $x(t\_0)$ if $M(t\_0,t\_1)$ is nonsingular. In fact, it is not possible to distinguish the initial state for $x\_1$ from that of $x\_2$ if $x\_1\; -\; x\_2$ is in the null space of $M(t\_0,t\_1)$.

Note that the matrix $M$ defined as above has the following properties:

• $M(t\_0,t\_1)$ is symmetric
• $M(t\_0,t\_1)$ is positive semidefinite for $t\_1\; \backslash geq\; t\_0$
• $M(t\_0,t\_1)$ satisfies the linear matrix differential equation

:: $\backslash frac\{d\}\{dt\}M(t,t\_1)\; =\; -A(t)^\{T\}M(t,t\_1)-M(t,t\_1)A(t)-C(t)^\{T\}C(t),\; \backslash ;\; M(t\_1,t\_1)\; =\; 0$

• $M(t\_0,t\_1)$ satisfies the equation

:: $M(t\_0,t\_1)\; =\; M(t\_0,t)\; +\; \backslash varphi(t,t\_0)^T\; M(t,t\_1)\backslash varphi(t,t\_0)${{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}

The system is observable in $[t\_0,t\_1]$ if and only if there exists an interval $[t\_0,t\_1]$ in $\backslash mathbb\{R\}$ such that the matrix $M(t\_0,t\_1)$ is nonsingular.

If $A(t),\; C(t)$ are analytic, then the system is observable in the interval [$t\_0$,$t\_1$] if there exists $\backslash bar\{t\}\; \backslash in\; [t\_0,t\_1]$ and a positive integer k such thatEduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.

: $\backslash operatorname\{rank\}\; \backslash begin\{bmatrix\}$

& N_0(\bar{t}) & \\

& N_1(\bar{t}) & \\

& \vdots & \\

& N_{k}(\bar{t}) &

\end{bmatrix} = n,

where $N\_0(t):=C(t)$ and $N\_i(t)$ is defined recursively as

: $N\_\{i+1\}(t)\; :=\; N\_i(t)A(t)\; +\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}\; t\}N\_i(t),\backslash \; i\; =\; 0,\; \backslash ldots,\; k-1$

Consider a system varying analytically in $(-\backslash infty,\backslash infty)$ and matrices

$A(t)\; =\; \backslash begin\{bmatrix\}$

t & 1 & 0\\

0 & t^{3} & 0\\

0 & 0 & t^{2}

\end{bmatrix},\, C(t) = \begin{bmatrix}

1 & 0 & 1

\end{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} , and since this matrix has rank = 3, the system is observable on every nontrivial interval of $\backslash mathbb\{R\}$.

Given the system $\backslash dot\{x\}\; =\; f(x)\; +\; \backslash sum\_\{j=1\}^mg\_j(x)u\_j$, $y\_i\; =\; h\_i(x),\; i\; \backslash in\; p$. Where $x\; \backslash in\; \backslash mathbb\{R\}^n$ the state vector, $u\; \backslash in\; \backslash mathbb\{R\}^m$ the input vector and $y\; \backslash in\; \backslash mathbb\{R\}^p$ the output vector. $f,g,h$ are to be smooth vector fields.

Define the observation space $\backslash mathcal\{O\}\_s$ to be the space containing all repeated Lie derivatives, then the system is observable in $x\_0$ if and only if $\backslash dim(d\backslash mathcal\{O\}\_s(x\_0))\; =\; n$, where

:$d\backslash mathcal\{O\}\_s(x\_0)\; =\; \backslash operatorname\{span\}(dh\_1(x\_0),\; \backslash ldots\; ,\; dh\_p(x\_0),\; dL\_\{v\_i\}L\_\{v\_\{i-1\}\},\; \backslash ldots\; ,\; L\_\{v\_1\}h\_j(x\_0)),\backslash \; j\backslash in\; p,\; k=1,2,\backslash ldots.$Lecture notes for Nonlinear Systems Theory by prof. dr. D.Jeltsema, prof dr. J.M.A.Scherpen and prof dr. A.J.van der Schaft.

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,{{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= }} Kou, Elliot and Tarn,{{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 }} and Singh.{{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 }}

There also exist an observability criteria for nonlinear time-varying systems.{{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}}

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 $\backslash mathbb\{R\}^n$.{{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 }}{{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 }} Just as observability criteria are used to predict the behavior of Kalman filters or other observers in the dynamic system case, observability criteria for sets in $\backslash mathbb\{R\}^n$ are used to predict the behavior of 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.

• Controllability
• Hautus lemma
• Identifiability
• State observer
• State space (controls)

{{reflist}}

• {{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