Detectable system
Undergraduate (linear systems)
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.
Graduate (nonlinear systems)
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
Examples
- $\dot x_1 = x_2$, $\dot x_2 = u$, with position output $y = x_1$ Undergraduate
- $x_{k+1} = 0 \cdot x_k$, $y_k = 0$ Undergraduate
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$\dot{x} = -x$, $y = 0$
Undergraduate
$\dot{x} = -x$, $y = 0$ is detectable, because the unknown state decays to $0$ anyway.
- $\dot{x} = x$, $y = x$ Undergraduate
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 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.
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.