State-Space Representation
State-space models describe a system by first-order equations in a state vector, handling multivariable and nonlinear dynamics naturally.
The State Vector
A state-space model captures a system's dynamics through a set of first-order differential equations. The state x is a vector of variables that together summarize everything about the system's past needed to predict its future. For a linear time-invariant system the model is x' = A*x + B*u and y = C*x + D*u.
The four matrices
- A (state matrix): governs the internal dynamics; its eigenvalues are the system poles.
- B (input matrix): how the inputs u drive the state.
- C (output matrix): how the state maps to the measured outputs y.
- D (feedthrough matrix): direct pass-through from input to output, often zero.
Why it generalizes classical methods
A single transfer function relates one input to one output. State-space handles multiple inputs and outputs together in one matrix framework, and it exposes internal variables that a transfer function hides. It also extends naturally to time-varying and, in modified form, nonlinear systems, where transfer functions do not apply.
Relation to transfer functions
For an LTI system the transfer function is recovered as G(s) = C*(s*I - A)^(-1)*B + D. The poles are the eigenvalues of A. Different state-space models can share the same transfer function; the extra freedom is a change of state coordinates, which leaves input-output behavior unchanged.
Choosing states
States are often physical quantities such as position, velocity, temperature, or stored charge, chosen because they hold the system's energy or memory. For a mechanical system, position and velocity are the natural pair; for an electrical one, capacitor voltages and inductor currents. The number of states equals the system order.
State-space is the language of modern control. Controllability, observability, pole placement, the linear-quadratic regulator, the Kalman filter, and model-predictive control are all defined on these matrices. It is also the form solved numerically when simulating complex coupled systems such as the many magnetic and thermal states of a fusion device model.