Reachable system
High school
Consider a control system with an input that you choose. The system is reachable if you can steer it from the origin to every state you choose in a finite amount of time.
Undergraduate
Consider a system with state $x$ and input $u$, starting from the origin. The system is reachable if every state can be reached in finite time by some input.
Examples
Non-examples
Things that may be mistaken for examples.
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$\dot{x} = x + 0 \cdot u$
High school
$\dot{x} = x + 0 \cdot u$ is not a reachable system because the input has no effect.
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$x_{k+1} = 0 \cdot x_k + 0 \cdot u_k$
Undergraduate
$x_{k+1} = 0 \cdot x_k + 0 \cdot u_k$ is not reachable, because no nonzero state can be reached from $0$.
Properties
- For $\dot{x} = A x + B u$, reachability and controllability are equivalent.
- For $x_{k+1} = A x_k + B u_k$, reachability implies controllability. The converse holds when $A$ is invertible, but fails in general: if $A = 0$ and $B = 0$, every state reaches the origin in one step, but no other state can be reached.