The function computes the coprime factors of the stabilizable and detectable state space realization .

For each proper real rational transfer matrix , there exist eight transfer matrices in satisfying the equations:

and

The pairs and constitute the right and left coprime factors of the transfer matrix , respectively. They are said to be normalized when each of the transfer matrices , is norm preserving. The second equation is known as the Bezout identity. The state space formulae for normalized factors, and their certificates of coprimeness are given as follows.

Let be a stabilizable and detectable realization and choose , , and as:

Let and be the unique stabilizing solutions for each of the following Algebraic Riccati equations (AREs), respectively:

Define the state feedback and observer gains and as:

Then

satisfy the Bezout identity and the pairs and are normalized according to the definition above.

As output the function provides the structures and with the realizations of respectively:

- , , , ,
- , , , ,