Which statement best describes controllability in a state-space system?

Which statement best describes controllability in a state-space system?

• A. The ability to observe all internal states from outputs
• B. The ability to drive the state to any value with an input
• C. The requirement of model order reduction
• D. The ability to maintain zero steady-state error with integral action

Answer: (B)

Controllability means you can move the system’s internal state to any desired value by choosing an appropriate input. For a state-space model ẋ = Ax + Bu, this means there exists an input u(t) that drives the state from any initial x0 to any target xf in finite time. When the system is controllable, you can place the closed-loop dynamics as you wish by state feedback. The mathematical test is that the controllability matrix [B AB A^2B ... A^{n-1}B] has full rank (equal to the number of states). If it isn’t full rank, some states can’t be reached no matter what input you apply. The other statements describe different ideas: observing all internal states from outputs relates to observability, not controllability; reducing model order is about simplifying the model, not about steering the state with input; and zero steady-state error with integral action concerns steady-state performance of a controller, not the ability to reach all states.

Controllability means you can move the system’s internal state to any desired value by choosing an appropriate input. For a state-space model ẋ = Ax + Bu, this means there exists an input u(t) that drives the state from any initial x0 to any target xf in finite time. When the system is controllable, you can place the closed-loop dynamics as you wish by state feedback. The mathematical test is that the controllability matrix [B AB A^2B ... A^{n-1}B] has full rank (equal to the number of states). If it isn’t full rank, some states can’t be reached no matter what input you apply.

The other statements describe different ideas: observing all internal states from outputs relates to observability, not controllability; reducing model order is about simplifying the model, not about steering the state with input; and zero steady-state error with integral action concerns steady-state performance of a controller, not the ability to reach all states.