Obtain a transfer function form a 2nd order D.E. using the Lapalce Transforms

Hello,

(Using MATLAB) Is it possible to obtain a transfer function H(s) from a 2nd order D.E. using the Laplace Transfroms?

The D.E. is; d^2y(t)/dt^2 + 7.6*dy(t)/dt + 4.2*y(t) = x(t)

Thanks!

Answer 1

[Star Strider]

It is, however it takes some effort and a bit of manual intervention in the end:

% d^2y(t)/dt^2 + 7.6*dy(t)/dt + 4.2*y(t) = x(t)
syms  s t x(t) y(t) X(s) Y(s) 
assume(X(s) ~= 0)
DE = diff(y,2) + 7.6*diff(y,1) + 4.2*y == x;
LDE = laplace(DE,t,s);
LDE = subs(LDE, {laplace(y, t, s), subs(diff(y(t), t), t, 0), laplace(x(t), t, s), y(0)},{Y(s), 0, X(s), 0})
LDETF = simplify( LDE, 'Steps',250)
LDETF = subs(LDETF,{X,Y},{1,1})
LDETF = ((5*s + 3)*(s + 7))/5

s = tf('s');
H = ((5*s + 3)*(s + 7))/5                               % Copy ‘LDETF’ Result From Command Window & Paste Here

bode(H)

[Joshua Scicluna]

Thanks! the script works perfectly.

one small question, can you explain the following line of code; Thanks again!

LDETF = simplify( LDE, 'Steps',250) %What is 'Steps', 250
 

[Star Strider]

As always, my pleasure!

The simplify function simplifies the argument expression. The 'Steps' name-value pair tells it to keep iterating until it cannot simplify it further, the limit here being 250 simplification steps.

[Joshua Scicluna]

Thanks!!

[Star Strider]

As always, my pleasure!