TRANSFER_FUNCTION
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. It is defined as the Laplace transform of the output divided by the Laplace transform of the input, with zero initial conditions.
For a continuous-time polynomial representation, the transfer function is expressed as:
H(s) = \frac{num(s)}{den(s)} = \frac{b_m s^m + b_{m-1} s^{m-1} + \dots + b_1 s + b_0}{a_n s^n + a_{n-1} s^{n-1} + \dots + a_1 s + a_0}
The function accepts the coefficients of the numerator and denominator polynomials in descending order of powers of s (or z for discrete systems).
Excel Usage
=TRANSFER_FUNCTION(num, den)num(list[list], required): Polynomial coefficients of the numerator, in descending powers of s.den(list[list], required): Polynomial coefficients of the denominator, in descending powers of s.
Returns (str): String representation of the TransferFunction object.
Example 1: First order system 1/(s+1)
Inputs:
| num | den | |
|---|---|---|
| 1 | 1 | 1 |
Excel formula:
=TRANSFER_FUNCTION({1}, {1,1})Expected output:
"<TransferFunction>: sys[0]\nInputs (1): ['u[0]']\nOutputs (1): ['y[0]']\n\n 1\n -----\n s + 1"
Example 2: Second order system (s+2)/(s^2+3s+2)
Inputs:
| num | den | |||
|---|---|---|---|---|
| 1 | 2 | 1 | 3 | 2 |
Excel formula:
=TRANSFER_FUNCTION({1,2}, {1,3,2})Expected output:
"<TransferFunction>: sys[1]\nInputs (1): ['u[0]']\nOutputs (1): ['y[0]']\n\n s + 2\n -------------\n s^2 + 3 s + 2"
Example 3: Simple Integrator 1/s
Inputs:
| num | den | |
|---|---|---|
| 1 | 1 | 0 |
Excel formula:
=TRANSFER_FUNCTION({1}, {1,0})Expected output:
"<TransferFunction>: sys[2]\nInputs (1): ['u[0]']\nOutputs (1): ['y[0]']\n\n 1\n -\n s"
Example 4: Third order denominator
Inputs:
| num | den | ||||
|---|---|---|---|---|---|
| 2 | 1 | 1 | 4 | 5 | 2 |
Excel formula:
=TRANSFER_FUNCTION({2,1}, {1,4,5,2})Expected output:
"<TransferFunction>: sys[3]\nInputs (1): ['u[0]']\nOutputs (1): ['y[0]']\n\n 2 s + 1\n ---------------------\n s^3 + 4 s^2 + 5 s + 2"
Python Code
Show Code
import control as ct
import numpy as np
def transfer_function(num, den):
"""
Create a transfer function system from its numerator and denominator polynomial coefficients.
See: https://python-control.readthedocs.io/en/latest/generated/control.tf.html
This example function is provided as-is without any representation of accuracy.
Args:
num (list[list]): Polynomial coefficients of the numerator, in descending powers of s.
den (list[list]): Polynomial coefficients of the denominator, in descending powers of s.
Returns:
str: String representation of the TransferFunction object.
"""
try:
def flatten_to_1d(x):
# Extract first row or column
if not isinstance(x, list):
return [x]
flat = []
for row in x:
if isinstance(row, list):
for val in row:
if str(val) != "":
flat.append(float(val))
elif str(row) != "":
flat.append(float(row))
return flat
num_arr = flatten_to_1d(num)
den_arr = flatten_to_1d(den)
if not num_arr or not den_arr:
return "Error: Numerator and denominator coefficients must be provided"
if all(d == 0 for d in den_arr):
return "Error: Denominator cannot be zero"
sys = ct.tf(num_arr, den_arr)
return str(sys)
except Exception as e:
return f"Error: {str(e)}"