# Types¶

linear_system_like

Any one of the following:

• gain : float

A system that multiplies its input by a constant gain.

$F(s) = \texttt{gain}$
• (num : array_like, den : array_like)

A transfer function with numerator polynomial num and denominator polynomial den.

$F(s) = \frac{\sum_{i=0}^{\texttt{len(num)-1}} \texttt{num[-i-1]} s^i} {\sum_{i=0}^{\texttt{len(den)-1}} \texttt{den[-i-1]} s^i}$
• (zeros : array_like, poles : array_like, gain : float)

A transfer function with numerator roots zeros, denominator roots poles, and scalar gain.

$F(s) = \texttt{gain} \ \frac{\prod_{i=0}^{\texttt{len(zeros)-1}} (s - \texttt{zeros[i]})} {\prod_{i=0}^{\texttt{len(poles)-1}} (s - \texttt{poles[i]})}$
• (A : array_like, B : array_like, C : array_like, D : array_like)

A state-space model described by four 2–dimensional matrices (A, B, C, D).

$\begin{split}\dot{{\bf x}}(t) &= A{\bf x}(t) + B{\bf u}(t) \\ {\bf y}(t) &= C{\bf x}(t) + D{\bf u}(t)\end{split}$

This has the transfer function $$F(s) = C (sI - A)^{-1} B + D$$.

• An instance of LinearSystem.

• An instance of nengo.LinearFilter.

Note: The above equations are for the continuous time-domain. For the discrete time-domain, replace $$s \rightarrow z$$ and $$\dot{{\bf x}} \rightarrow {\bf x}[k+1]$$.