How do I find the impulse response function

How can I find the impulse response functiin from a transfer function. I do not wish to find the plot or the response values but rather the function or expression. I can manually enter the TF as a symbolic expression and determine the inverse Laplace transform, but the resulting expression is more complex (using exploring) then the plot of the 'impulse' function suggests.

[Jacques Hebert]

I had left out the specific transfer function i am working with to find a more general result. if you are interested, the specific function i am working with is

0.0001163 s^2 + 0.07919 s + 1.612
-----------------------------------------------------------------------
0.0001112 s^5 + 0.003739 s^4 + 0.05894 s^3 + 0.4631 s^2 + 0.7542 s + 1.612

Continuous-time transfer function.

the decomposition i determined with 'residue' function

b=[0.0001163 0.07919 1.612];
a=[0.0001112 0.003739 0.5894 0.4631 0.6542 1.612];
[r,p,k]=residue(b,a);

is

r =
   0.0005 + 0.0007i
   0.0005 - 0.0007i
  -0.2701 - 0.3806i
  -0.2701 + 0.3806i
   0.5392 + 0.0000i
p =
 -16.4209 +70.7384i
 -16.4209 -70.7384i
   0.3089 + 1.3667i
   0.3089 - 1.3667i
  -1.4001 + 0.0000i
k =
     []

using this result the inverse laplace transform

ilaplace(((0.0005 + 0.0007i)/(s-(-16.4209 +70.7384i)))+((0.0005 - 0.0007i)/(s-(-16.4209 -70.7384i)))+((-0.2701 - 0.3806i)/(s-(0.3089 + 1.3667i)))+((-0.2701 + 0.3806i)/(s-(0.3089 - 1.3667i)))+((0.5392 + 0.0000i)/(s-(-1.4001 + 0.0000i))))

gives the result

(337*exp(-(14001*t)/10000))/625 - exp(t*(3089/10000 - 13667i/10000))*(2701/10000 - 1903i/5000) - exp(t*(3089/10000 + 13667i/10000))*(2701/10000 + 1903i/5000) + exp(t*(- 4622072445068011/281474976710656 - 88423i/1250))*(1/2000 - 7i/10000) + exp(t*(- 4622072445068011/281474976710656 + 88423i/1250))*(1/2000 + 7i/10000)

as i stated previously, the plot of this result is similar

but markedly different (more exagerated) than the plot of 'impulse' of the original transfer function.

i am unsure where i am making a mistake, but i dont believe i have correctly found the impulse response function of the given transfer function

[Jacques Hebert]

If possible, I am having a similar, small problem finding the symbolic expression for the step response with a matching plot.

plotting the step response directly

calculating the step response from the previously determined impulse response

int(ilaplace((((0.0987 + 0.0000i)/(s-(-15.0950 + 0.0000i)))+(( 0.1672 + 0.0208i)/(s-(-8.5626 +12.1924i)))+(( 0.1672 - 0.0208i)/(s-(-8.5626 -12.1924i)))+(( -0.2165 - 1.1635i)/(s-(-0.7019 + 1.9580i)))+(( -0.2165 + 1.1635i)/(s-(-0.7019 - 1.9580i))))

this plot is similar, except centered around 0 rather than 1 for some reason. can you help me determine where ive gone wrong?

Answer 1

[David Goodmanson]

Hi Jaques,

I believe the problem is that you have a couple of small errors in the 'a' vector (they don't agree with the original transfer function denominator), which make a big difference in the result. If you make those corrections, then

b=[0.0001163 0.07919 1.612];
a=[0.0001112 0.003739 0.05894 0.4631 0.7542 1.612];
[r,p,k]=residue(b,a)

 r =
   0.0987 + 0.0000i
   0.1672 + 0.0208i
   0.1672 - 0.0208i
  -0.2165 - 1.1635i
  -0.2165 + 1.1635i

 p = 
 -15.0950 + 0.0000i
  -8.5626 +12.1924i
  -8.5626 -12.1924i
  -0.7019 + 1.9580i
  -0.7019 - 1.9580i

Now all the poles have negative real part, which is good.

At this point rather than more symbolism I just went for a numerical result:

t = 0:.001:10;
[tt rr] = meshgrid(t,r);
[tt pp] = meshgrid(t,p);

f = rr.*exp(tt.*pp);
plot(t,sum(f))
grid on

and the resulting plot agrees nicely with the second one of yours.

[Jacques Hebert]

thank you for pointing out my typo.

this does yield a result with negative real poles. but i am ultimately looking for a symbolic expression of the impulse response, rather than numeric result.

continuing with my previous method

this too results in an agreeable plot.

[David Goodmanson]

cool! it works