Controllability

A system is controllable if the input can drive its state from any starting point to any target in finite time.

Can the Input Reach Every State

Controllability asks a structural question about a system: using only the available inputs, can we steer the internal state from any initial value to any desired value in a finite time? If some part of the state cannot be influenced by the input no matter how it is applied, the system is not fully controllable.

The controllability matrix

For a linear system x' = A*x + B*u of order n, form the controllability matrix by stacking the columns of B, A*B, A^2*B, up to A^(n-1)*B side by side. The system is controllable if and only if this matrix has full rank n. This is a finite algebraic test, needing no simulation.

What loss of controllability means

• An uncontrollable mode evolves according to the plant's own dynamics regardless of input.
• If that mode is stable, it merely cannot be commanded; the system may still be usable (stabilizable).
• If that mode is unstable and uncontrollable, no feedback can stabilize the system: a fatal design flaw.
• Uncontrollability often signals a symmetry, a redundant state, or an actuator placed where it cannot affect a variable.

Stabilizability

A weaker but often sufficient property is stabilizability: all uncontrollable modes are already stable. A stabilizable system can be made stable by feedback even if not every mode can be independently commanded. Many real designs aim for stabilizability rather than full controllability.

Practical role

Controllability is a prerequisite for arbitrary pole placement: only a controllable system's poles can be moved to any chosen locations by state feedback. It also guides actuator placement. In a fusion device model, checking controllability tells the designer whether the available coil currents can actually command every plasma shape and position degree of freedom, or whether an actuator must be added.

Controllability is a yes-or-no structural property; how easily a controllable system can be steered, its degree of controllability, is measured separately by the controllability Gramian.