How do I create a transfer function of a high order

I am trying to write a high order transfer function in matlab that i need to approximate and reduce to a lower order transfer function. My question is, how do i write a transfer function that has multiplication in between in the denominator:

Looks like this:

G(s) = (14.14s^2 + 318.2s + 707) / (s^2 +20s+101)*(100*s+1)*(0.2*s^2 + 1.2*s+1)

Answer 1

[Sulaymon Eshkabilov]

It can be done this way:

s = tf('s');
G = (14.14*s^2 + 318.2*s + 707) / ((s^2 +20*s+101)*(100*s+1)*(0.2*s^2 + 1.2*s+1))

G =
 
                     14.14 s^2 + 318.2 s + 707
  ---------------------------------------------------------------
  20 s^5 + 520.2 s^4 + 4525 s^3 + 1.417e04 s^2 + 1.024e04 s + 101
 
Continuous-time transfer function.

% Simulate and get step response of this TF:
step(G)

Answer 2

[Sam Chak]

Hi @DAL

Here is an alternative approach that should produce the same result.

Note that num, p3, and den are vectors of polynomial coefficients.

num = [14.14 318.2 707]             % numerator

num = 1×3

14.1400 318.2000 707.0000

p3  = conv([1 20 101], [100 1]);    % obtain a 3rd-order polynomial via Convolution
den = conv(p3, [0.2 1.2 1])         % denominator

den = 1×6

1.0e+04 * 0.0020 0.0520 0.4525 1.4165 1.0241 0.0101

G   = tf(num, den)                  % transfer function

G =
 
                     14.14 s^2 + 318.2 s + 707
  ---------------------------------------------------------------
  20 s^5 + 520.2 s^4 + 4525 s^3 + 1.417e04 s^2 + 1.024e04 s + 101
 
Continuous-time transfer function.

To find a reduced-order approximation rsys of the LTI model G, balred() function can be used.

https://www.mathworks.com/help/control/ref/lti.balred.html

rsys = balred(G, 1)                 % Model order reduction

rsys =
 
  -0.06452 s + 0.06998
  --------------------
      s + 0.009997
 
Continuous-time transfer function.

subplot(2, 1, 1)
step(G, 1000)                       % step response of G behaves like a 1st-order system
subplot(2, 1, 2)
step(rsys, 1000)                    % step response of rsys