state-space representation

State-space is the 'modern control' way of describing a system: instead of one input-output transfer function, you track a vector of internal state variables — the minimum set of numbers that, together with future inputs, completely determines the system's future. For a moving cart you might pick position and velocity; for a circuit, the capacitor voltages and inductor currents. The whole dynamics then collapse into two compact matrix equations: ẋ = Ax + Bu (how the state evolves and how input pushes it) and y = Cx + Du (how you observe the state as output).

Why bother when transfer functions already work? Because state-space scales and generalises. It handles multiple-input multiple-output (MIMO) systems naturally — a jetliner with many actuators and sensors — where transfer functions become an unwieldy matrix of ratios. It exposes the internal structure a transfer function hides, enabling powerful tools: pole placement that puts every closed-loop pole exactly where you want via state feedback u = −Kx, optimal LQR control, Kalman filtering, and the crisp concepts of controllability and observability. It also extends to time-varying and (with care) nonlinear systems. The four matrices A, B, C, D are the entire system, and almost all advanced and computer-based control lives in this language.