What is state-space representation and what are its advantages?

State-space represents systems using first-order differential equations: dx/dt = Ax + Bu (state equation), y = Cx + Du (output equation). State variables capture internal system dynamics. Advantages over transfer functions include: handles multiple inputs/outputs (MIMO) naturally, reveals all system modes (including those canceled in TF), facilitates modern control design (optimal control, observers), and directly simulates on computers. State matrix A eigenvalues equal transfer function poles. Controllability and observability are key state-space properties.