Reconstructible system
Undergraduate
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
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 reconstructible and detectable, because it is observable.
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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 reconstructible and detectable, because the state is $0$ after one step, so the current state is known.
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$\dot{x} = x$, $y = x$
Undergraduate
$\dot{x} = x$, $y = x$ is reconstructible and detectable, because it is observable.
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 reconstructible, because the output does not show the position $x_1$.
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$\dot{x} = -x$, $y = 0$
Undergraduate
$\dot{x} = -x$, $y = 0$ 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.
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.