Observable system
High school
Consider a system whose starting state is unknown, but whose input and output you can see. The system is observable if you can figure out its starting state by watching its output (and knowing its input) for a while.
Undergraduate
Consider a system with state, input, and output. The system is observable if its initial state is uniquely determined by its input and output over some finite time interval.
Examples
Examples
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$\dot x_1 = x_2$, $\dot x_2 = u$, with position output $y = x_1$
Undergraduate
The double integrator with position output is observable because the velocity can be inferred from how the position changes.
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$\dot{x} = x$, $y = x$
Undergraduate
$\dot{x} = x$, $y = x$ is observable because the output is the state.
Non-examples
Things that may be mistaken for examples.
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$\dot x_1 = x_2$, $\dot x_2 = u$, with velocity output $y = x_2$
Undergraduate
The double integrator with velocity output is not observable, because the output does not show the position $x_1$.
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$x_{k+1} = 0 \cdot x_k$, $y_k = 0$
Undergraduate
$x_{k+1} = 0 \cdot x_k$, $y_k = 0$ is not observable, because the output carries no information about the initial state.
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$\dot{x} = -x$, $y = 0$
Undergraduate
$\dot{x} = -x$, $y = 0$ is not observable, because the output carries no information.
Properties
- For $\dot{x} = A x + B u$, $y = C x$ with $n$ states, observability holds exactly when \[\operatorname{rank} \begin{bmatrix} C \\ C A \\ \vdots \\ C A^{n-1} \end{bmatrix} = n.\]
- Observability is dual to controllability: $(A, C)$ is observable if and only if $(A^{\top}, C^{\top})$ is controllable.
Distinctions involving this term
- Observable vs. reconstructible vs. detectable: Observability, reconstructibility, and detectability describe how much the output of a system reveals about its state. An observable system’s initial state can be determined from its output, whereas a reconstructible system’s current state can be determined. A detectable system may have unobservable parts but those parts must that decay to zero so that an observer’s error still converges.