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.
observable ⊂ detectable, and for continuous-time linear systems, observable = reconstructible.
Definitions
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.
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
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$\dot x_1 = x_2$, $\dot x_2 = u$, with position output $y = x_1$
The double integrator with position output is observable because the velocity can be inferred from how the position changes.
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$\dot{x} = x$, $y = x$
$\dot{x} = x$, $y = x$ is observable because the output is the state.
Non-examples
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$\dot x_1 = x_2$, $\dot x_2 = u$, with velocity output $y = x_2$
The double integrator with velocity output is not observable, because the output does not show the position $x_1$.
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$x_{k+1} = 0 \cdot x_k$, $y_k = 0$
$x_{k+1} = 0 \cdot x_k$, $y_k = 0$ is not observable, because the output carries no information about the initial state.
-
$\dot{x} = -x$, $y = 0$
$\dot{x} = -x$, $y = 0$ 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.
Consider a system with state, input, and output, observed up to the present time. The system is reconstructible if its current state is uniquely determined by its past input and output over some finite time interval.
Examples
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$\dot x_1 = x_2$, $\dot x_2 = u$, with position output $y = x_1$
The double integrator with position output is reconstructible and detectable, because it is observable.
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$x_{k+1} = 0 \cdot x_k$, $y_k = 0$
$x_{k+1} = 0 \cdot x_k$, $y_k = 0$, is reconstructible and detectable, because the state is $0$ after one step, so the current state is known.
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$\dot{x} = x$, $y = x$
$\dot{x} = x$, $y = x$ is reconstructible and detectable, because it is observable.
Non-examples
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$\dot x_1 = x_2$, $\dot x_2 = u$, with velocity output $y = x_2$
The double integrator with velocity output is not reconstructible, because the output does not show the position $x_1$.
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$\dot{x} = -x$, $y = 0$
$\dot{x} = -x$, $y = 0$ is not reconstructible, because the output carries no information.
Properties
- For continuous-time linear systems, reconstructibility and observability are equivalent.
- For $x_{k+1} = A x_k + B u_k$, $y_k = C x_k$, observability implies reconstructibility. The converse fails when $A$ is singular: if $A = 0$, the current state is always $0$ after one step, even if nothing can be learned about the initial state.
Consider a linear system $\dot{x} = A x + B u$ with output $y = C x$, and its unobservable modes: the parts of the state that do not affect the output. The system is detectable if every unobservable mode is asymptotically stable.
Consider a system $\dot{x} = f(x)$ with output $y = h(x)$. The system is (zero-state) detectable if every solution such that $y(t) = 0$ for all $t \geq 0$ converges to the origin.
Examples
- $\dot x_1 = x_2$, $\dot x_2 = u$, with position output $y = x_1$
- $x_{k+1} = 0 \cdot x_k$, $y_k = 0$
-
$\dot{x} = -x$, $y = 0$
$\dot{x} = -x$, $y = 0$ is detectable, because the unknown state decays to $0$ anyway.
- $\dot{x} = x$, $y = x$
Non-examples
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$\dot x_1 = x_2$, $\dot x_2 = u$, with velocity output $y = x_2$
The double integrator with velocity output is not detectable, because the position’s mode (with eigenvalue $0$) does not decay.
Properties
- A linear system is detectable if and only if there is a matrix $L$ such that $A - L C$ is Hurwitz, so a Luenberger observer can estimate the state with an error that converges to zero.
- Detectability is dual to stabilizability, and every observable linear system is detectable.
Examples
| Example | observable | reconstructible | detectable | Why |
|---|---|---|---|---|
| $\dot{x} = x$, $y = x$ | ✓ | ✓ | ✓ | |
| $\dot x_1 = x_2$, $\dot x_2 = u$, with position output $y = x_1$ | ✓ | ✓ | ✓ | |
| $\dot x_1 = x_2$, $\dot x_2 = u$, with velocity output $y = x_2$ | ✗ | ✗ | ✗ | |
| $\dot{x} = -x$, $y = 0$ | ✗ | ✗ | ✓ | |
| $x_{k+1} = 0 \cdot x_k$, $y_k = 0$ | ✗ | ✓ | ✓ |
Duality
Each notion is dual to one on the controllability page. With $(A, C)$ paired with $(A^{\top}, C^{\top})$:
- observability is dual to reachability,
- reconstructibility is dual to controllability (to the origin), and
- detectability is dual to stabilizability.
For continuous-time linear systems, the first two pairs coincide, which is why observability is usually described as dual to controllability.