Observability

 ( Classical Control Theory ) 

$N\; =\; \backslash bigcap\_\{k=0\}^\{n-1\}\; \backslash ker(CA^k)=\; \backslash ker\{\backslash mathcal\{O\}\}$

Since the system is observable if and only if $\backslash operatorname\{rank\}(\backslash mathcal\{O\})\; =\; n$, the system is observable if and only if $N$ is the zero subspace.

The following properties for the unobservable subspace are valid:

• $N\; \backslash subset\; Ke(C)$
• $A(N)\; \backslash subset\; N$
• $N=\; \backslash bigcup\; \backslash \{\; S\; \backslash subset\; R^n\; \backslash mid\; S\; \backslash subset\; Ke(C),\; A(S)\; \backslash subset\; N\; \backslash \}$

Detectability conditions are important in the context of Sensor Networks.

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

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

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

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

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 matrix
• $M(t\_0,t\_1)$ 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

: $M(t\_0,t\_1)\; =\; M(t\_0,t)\; +\; \backslash varphi(t,t\_0)^T\; M(t,t\_1)\backslash varphi(t,t\_0)$

Roger W. Brockett (1970). 9780471105855, John Wiley & Sons. ISBN 9780471105855

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}) &
     

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

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

= \begin{bmatrix}
     

Define the observation space $\backslash mathcal\{O\}\_s$ to be the space containing all repeated , 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, Kou, Elliot and Tarn, and Singh.

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

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

• MATLAB function for checking observability of a system
• Mathematica function for checking observability of a system