Transfer function to state-space
Convert a transfer function to a state-space realisation online. This tool produces the A, B, C, and D matrices in controllable canonical form from H(s) = N(s) / D(s), for x′ = Ax + Bu, y = Cx + Du.
Controllable canonical form
The denominator coefficients form the bottom row of the companion A matrix, B is a unit vector, and C carries the numerator. A non-zero D appears only when the numerator and denominator have the same degree (direct feedthrough).
Frequently asked questions
- What is a state-space model?
- A state-space model represents a system with first-order matrix equations x′ = Ax + Bu and y = Cx + Du, where A, B, C, and D are matrices. It is equivalent to a transfer function but extends naturally to multi-input, multi-output and modern control design.
- How do you convert a transfer function to state-space?
- This tool uses the controllable canonical form: the denominator coefficients build the companion A matrix, B is a unit input vector, C carries the numerator coefficients, and D is the direct feedthrough — non-zero only when the numerator and denominator have equal degree.
- What is controllable canonical form?
- Controllable canonical form is a standard state-space realisation built directly from the transfer-function coefficients, in which the system is guaranteed to be controllable. It is convenient for pole placement and controller design.
- Is this converter free?
- Yes, it runs entirely in your browser at no cost.