Plotting an Inverse Laplace Function

I have an expression:

pretty(vsol)
(s + 100000) 147573952589676412928   14757395258967641292800000
---------------------------------- - --------------------------
                #1                               #1

       2
     (s  + 100000 s + 10000000000) 2213609288845146                       3                          2
   - ---------------------------------------------- + ((1106804644422573 s  + 110680464442257300000 s
                           #1

   + 18446744073709550646400000 s + 737869762948382064640000000000) 6)/(s #1)

where

                           3                          2
   #1 == 5534023222112865 s  + 700976274800962912928 s  + 106991115627515394524800000 s

      + 5165088340638674452480000000000

Where when I do an inverse laplace transformation it turned into this:

      pretty(ans)
    /   3            \
    |  ---           |
    |  \   exp(t #2) |
    |  /   --------- | 22136092888451460000000000
    |  ---     #1    |
6   \ k = 1          /
- - ---------------------------------------------
7                         7

   -

  /   3                                                                                  \
  |  ---                                                                                 |
  |  \                                     exp(#2 t) #2                                  |
  |  /   ------------------------------------------------------------------------------- | 73786976294838187072
  |  ---                                                 2                               |
  \ k = 11401952549601925825856 #2 + 16602069666338595 #2  + 106991115627515394524800000 /
  -------------------------------------------------------------------------------------------------------------
                                                        7

     /   3                \
     |  ---             2 |
     |  \   exp(t #2) #2  |
     |  /   ------------- | 2213609288845146
     |  ---       #1      |
     \ k = 1              /
   - ---------------------------------------
                        7

where

                             2
   #1 == 16602069666338595 #2  + 1401952549601925825856 #2 + 106991115627515394524800000

             /                             2
             |  3   700976274800962912928 z    21398223125503078904960000 z
   #2 == root| z  + ------------------------ + ----------------------------
             \          5534023222112865             1106804644422573

                                              \
        1033017668127734890496000000000       |
      + -------------------------------, z, k |
                1106804644422573              /

How do I plot this? Where did the z come from?

[Torsten]

Your code to determine vsol ?

[David Goodmanson]

Hello Feliciana,

z is just a dummy variable to express a cubic polynomial, which has three roots. The index k denotes which root, those roots are used in the solution, at which point z is left behind.

Answer 1

[Star Strider]

If you have the Control System Toolbox —

% (s + 100000) 147573952589676412928   14757395258967641292800000
% ---------------------------------- - --------------------------
%                 #1                               #1
%        2
%      (s  + 100000 s + 10000000000) 2213609288845146                       3                          2
%    - ---------------------------------------------- + ((1106804644422573 s  + 110680464442257300000 s
%                            #1
%    + 18446744073709550646400000 s + 737869762948382064640000000000) 6)/(s #1)
% where
%                            3                          2
%    #1 == 5534023222112865 s  + 700976274800962912928 s  + 106991115627515394524800000 s
%       + 5165088340638674452480000000000
      
s = tf('s');
num(1) = ((s + 100000) * 147573952589676412928 - 14757395258967641292800000) - (14757395258967641292800000) - ((s^2  + 100000 * s + 10000000000) * 2213609288845146);
num(2) = ((1106804644422573 * s^3  + 110680464442257300000 * s^2 + 18446744073709550646400000 * s + 737869762948382064640000000000) * 6);
den(1) = 5534023222112865 * s^3  + 700976274800962912928 * s^2  + 106991115627515394524800000 * s + 5165088340638674452480000000000;
den(2) = s * den(1);

H = num(1)/den(1) + num(2)/den(2)

H =
 
     2.45e31 s^6 + 6.37e36 s^5 + 1.296e42 s^4 + 1.622e47 s^3 + 1.405e52 s^2 + 8.548e56 s + 2.287e61
  ----------------------------------------------------------------------------------------------------
  3.063e31 s^7 + 7.758e36 s^6 + 1.676e42 s^5 + 2.072e47 s^4 + 1.869e52 s^3 + 1.105e57 s^2 + 2.668e61 s
 
Continuous-time transfer function.

figure
bp = bodeplot(H);
setoptions(bp, 'FreqUnits','Hz')
grid

figure
impulseplot(H)

figure
stepplot(H)

details = stepinfo(H)

Warning: Simulation did not reach steady state. Please specify YFINAL if this system is stable and eventually settles.

details = struct with fields:
         RiseTime: NaN
    TransientTime: NaN
     SettlingTime: NaN
      SettlingMin: NaN
      SettlingMax: NaN
        Overshoot: NaN
       Undershoot: NaN
             Peak: Inf
         PeakTime: Inf

This assumes that I correctly coded your results. It would be appropriate for you to check that to be certain.

.

Answer 2

[Sam Chak]

Not exactly an answer, but rather an attempt to recreate the 'answer' that was displayed in your Command Window.

It appears that this is a 13th-order Transfer Function.

%% Visualizing the 13th-order Transfer Function
s     = tf('s');
den   = 5534023222112865*s^3 + 700976274800962912928*s^2 + 106991115627515394524800000*s + 5165088340638674452480000000000;
term1 = ((s + 100000)*147573952589676412928)/den;
term2 = (14757395258967641292800000)/den;
term3 = ((s^2 + 100000*s + 10000000000)*2213609288845146)/den;
term4 = ((1106804644422573*s^3 + 110680464442257300000*s^2 + 18446744073709550646400000*s + 737869762948382064640000000000)*6)/(s*den);
G     = term1 - term2 - term3 + term4

G =
 
                                                                                                                                                            
  7.503e62 s^12 + 3.852e68 s^11 + 1.327e74 s^10 + 3.172e79 s^9 + 5.903e84 s^8 + 8.704e89 s^7 + 1.033e95 s^6 + 9.892e99 s^5 + 7.507e104 s^4 + 4.401e109 s^3  
                                                                                                                                                            
                                                                                                                     + 1.873e114 s^2 + 5.011e118 s + 6.1e122
                                                                                                                                                            
  -----------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                             
  9.379e62 s^13 + 4.752e68 s^12 + 1.628e74 s^11 + 3.869e79 s^10 + 7.167e84 s^9 + 1.052e90 s^8 + 1.243e95 s^7 + 1.186e100 s^6 + 8.966e104 s^5 + 5.236e109 s^4 
                                                                                                                                                             
                                                                                                               + 2.219e114 s^3 + 5.897e118 s^2 + 7.117e122 s 
                                                                                                                                                             
 
Continuous-time transfer function.

%% Describe the transfer function symbolically
syms  s
den   = 5534023222112865*s^3 + 700976274800962912928*s^2 + 106991115627515394524800000*s + 5165088340638674452480000000000;
term1 = ((s + 100000)*147573952589676412928)/den;
term2 = (14757395258967641292800000)/den;
term3 = ((s^2 + 100000*s + 10000000000)*2213609288845146)/den;
term4 = ((1106804644422573*s^3 + 110680464442257300000*s^2 + 18446744073709550646400000*s + 737869762948382064640000000000)*6)/(s*den);
vsol  = term1 - term2 - term3 + term4;

%% Convert the transfer function back to time-domain function
ft    = ilaplace(vsol)

ft =