A transfer function model is a way of expressing linear time-invariant (LTI) ODE systems; these have constant coefficients and are linear in terms of the derivative terms. For single input and single output (SISO), a LTI ODE has the form
To express this more compactly, we first define
Then, the ODE is
Rearranging so that is isolated, we have
where is called the transfer function, a ratio of polynomials in .
Now, extending to multiple inputs and outputs , we have multiple differential equations. Each equation expresses a single in terms of , so we can write a transfer function for each . Then, we can write the entire system in matrix form
where is the term in for , and the entire matrix is called the transfer function matrix .
To convert a 🇺🇸 State Space Model into transfer function form, we can rewrite the state equation as
Plugging it into the output equation, we get
giving us
Observe that would be in the denominator (due to the inverse), so the poles of (roots of denominator) are a subset of the eigenvalues of ; it's a subset since some terms may cancel with the numerator.
Steady state gain is a measure of the system at equilibrium (assuming 🎱 Stability), with constant input . Borrowing the form from 🇺🇸 State Space Model, we can derive the equilibrium state with , so . Then, the output converges to
From this equation, we can see that output reaches a value proportional to the constant input , which is the steady state gain
Note that this is similar to the transfer function , and is in fact equivalent with :