State-Space Representation

$\dot{x} = Ax + Bu \qquad y = Cx + Du$

$x$ is the state vector (e.g. position and velocity). The eigenvalues of $A$ are the system poles. Transfer function: $G(s) = C(sI - A)^{-1}B + D$.

• Controllability — can the input drive the system to any state? Check rank of $[B\ AB\ \dots\ A^{n-1}B]$.
• Observability — can the states be determined from outputs? Check rank of $[C;\ CA;\ \dots]$.

State feedback $u = -Kx$ places poles anywhere if the system is controllable (pole placement, LQR); an observer (or Kalman filter) estimates unmeasured states. State-space is the basis of modern control for robots, aircraft and motor drives.