ilaplace and laplace answer

I am just learning how to use the Laplace/Inverse functions in Matlab and was hoping to get some understanding of the final answer below. When working out 1st and second order functions I can ascertain the answer as it does not contain symsum or rootof, or r4 etc. In any case I am unsure of what symsum and Rootof translates to for a final answer? Thanks

>> syms t s
>> F=5*(s+3)/((s^5+4*s^4-3*s^2+7*s+10))

F =

(5*s + 15)/(s^5 + 4*s^4 - 3*s^2 + 7*s + 10)

>> pretty(ans)
          5 s + 15
---------------------------
 5      4      2
s  + 4 s  - 3 s  + 7 s + 10

>> ilaplace(F)

ans =

15*symsum(exp(r4*t)/(5*r4^4 + 16*r4^3 - 6*r4 + 7), r4 in RootOf(s4^5 + 4*s4^4 - 3*s4^2 + 7*s4 + 10, s4)) + 5*symsum((r4*exp(r4*t))/(5*r4^4 + 16*r4^3 - 6*r4 + 7), r4 in RootOf(s4^5 + 4*s4^4 - 3*s4^2 + 7*s4 + 10, s4))

>>

Answer 1

[Star Strider]

If you have R2015a, you can invert it the old-fashion way with partfrac. Do a pratial fraction expansion and then invert it:

syms t s
Fsv=5*(s+3)/((s^5+4*s^4-3*s^2+7*s+10));
Fpf = partfrac(Fsv, s, 'FactorMode','real');
Ft = ilaplace(Fpf, s, t);
Ftv = vpa(Ft,5)

producing:

Ftv =

- exp(t*(0.97907 + 0.92185i))*(0.29419 + 0.30855i) - exp(t*(0.97907 - 0.92185i))*(0.29419 - 0.30855i) + exp(t*(- 1.1315 + 0.46501i))*(0.30547 - 0.6614i) + exp(t*(- 1.1315 - 0.46501i))*(0.30547 + 0.6614i) - 0.022557*exp(-3.6952*t)

I used vpa to make it easy to read. MATLAB maintains full precision.

Added —

This can be simplified further with rewrite and simplify:

Ftt = rewrite(Ftv, 'sincos');
Ftt = simplify(Ftt, 'Steps',10, 'IgnoreAnalyticConstraints',1);
Ftt = vpa(Ftt,5)

producing a much neater result:

Ftt =

1.3228*exp(-1.1315*t)*sin(0.46501*t) - 0.022557*exp(-3.6952*t) + 0.61094*exp(-1.1315*t)*cos(0.46501*t)

Answer 2

[Duc Huy Le]

Thanks, i have used tutorial but but results return a virtual:

syms t s

M=(40000/s+240)*(exp(-0.0936*s)-exp(-1.008*s));

N=(32000/s+240)*(exp(-0.2808*s)-exp(-0.288*s));

Q1=2000*s^2+2400*s+360000;

Q2=-120*s+34000;

Q3=120*s-34000;

Q4=2100*s^2+4062*s+604100;

F=((M+N)*Q4+(1.35*N-1.25*M)*Q2)/(Q1*Q4+Q2*Q3);

Fpf = partfrac(F, s, 'FactorMode','real');

Ft = ilaplace(Fpf, s, t);

Ftv = vpa(Ft,5);

Ftt = rewrite(Ftv, 'sincos');

Ftt = simplify(Ftt, 'Steps',10, 'IgnoreAnalyticConstraints',1);

Ftt = vpa(Ftt,5)

Ftt =

0.10385*heaviside(1.0*t - 0.0936) - 0.096154*heaviside(1.0*t - 0.288) + 0.096154*heaviside(1.0*t - 0.2808) - 0.10385*heaviside(1.0*t - 1.008) + sin(t*(17.006 - 0.95036i))*heaviside(1.0*t - 0.288)*(0.006097 + 0.0015722i) - sin(t*(17.006 - 0.95036i))*heaviside(1.0*t - 0.2808)*(0.0062008 + 0.00081025i) - sin(t*(17.006 - 0.95036i))*heaviside(1.0*t - 1.008)*(0.018259 - 0.000076573i) - cos(t*(13.31 - 0.61679i))*heaviside(1.0*t - 0.0936)*(0.01834 + 0.044105i) - sin(t*(13.31 - 0.61679i))*heaviside(1.0*t - 0.0936)*(0.044105 - 0.01834i) - cos(t*(13.31 - 0.61679i))*heaviside(1.0*t - 0.288)*(0.050133 + 0.03844i) + cos(t*(13.31 - 0.61679i))*heaviside(1.0*t - 0.2808)*(0.053344 + 0.033319i) + cos(t*(13.31 - 0.61679i))*heaviside(1.0*t - 1.008)*(0.059611 + 0.059122i) - sin(t*(13.31 - 0.61679i))*heaviside(1.0*t - 0.288)*(0.03844 - 0.050133i) + sin(t*(13.31 - 0.61679i))*heaviside(1.0*t - 0.2808)*(0.033319 - 0.053344i) + sin(t*(13.31 - 0.61679i))*heaviside(1.0*t - 1.008)*(0.059122 - 0.059611i) - heaviside(1.0*t - 0.0936)*cos(t*(13.31 + 0.61679i))*(0.01834 - 0.044105i) + heaviside(1.0*t - 0.0936)*cos(t*(17.006 + 0.95036i))*(0.001173 + 0.0075669i) - sin(t*(13.31 + 0.61679i))*heaviside(1.0*t - 0.0936)*(0.044105 + 0.01834i) + cos(t*(17.006 - 0.95036i))*heaviside(1.0*t - 0.0936)*(0.001173 - 0.0075669i) - sin(t*(17.006 + 0.95036i))*heaviside(1.0*t - 0.0936)*(0.0075669 - 0.001173i) - heaviside(1.0*t - 0.288)*cos(t*(13.31 + 0.61679i))*(0.050133 - 0.03844i) + heaviside(1.0*t - 0.2808)*cos(t*(13.31 + 0.61679i))*(0.053344 - 0.033319i) - heaviside(1.0*t - 0.288)*cos(t*(17.006 + 0.95036i))*(0.0015722 + 0.006097i) + heaviside(1.0*t - 0.2808)*cos(t*(17.006 + 0.95036i))*(0.00081025 + 0.0062008i) + heaviside(1.0*t - 1.008)*cos(t*(13.31 + 0.61679i))*(0.059611 - 0.059122i) - sin(t*(17.006 - 0.95036i))*heaviside(1.0*t - 0.0936)*(0.0075669 + 0.001173i) - heaviside(1.0*t - 1.008)*cos(t*(17.006 + 0.95036i))*(0.000076573 - 0.018259i) - sin(t*(13.31 + 0.61679i))*heaviside(1.0*t - 0.288)*(0.03844 + 0.050133i) - cos(t*(17.006 - 0.95036i))*heaviside(1.0*t - 0.288)*(0.0015722 - 0.006097i) + sin(t*(13.31 + 0.61679i))*heaviside(1.0*t - 0.2808)*(0.033319 + 0.053344i) + cos(t*(17.006 - 0.95036i))*heaviside(1.0*t - 0.2808)*(0.00081025 - 0.0062008i) + sin(t*(17.006 + 0.95036i))*heaviside(1.0*t - 0.288)*(0.006097 - 0.0015722i) - sin(t*(17.006 + 0.95036i))*heaviside(1.0*t - 0.2808)*(0.0062008 - 0.00081025i) + sin(t*(13.31 + 0.61679i))*heaviside(1.0*t - 1.008)*(0.059122 + 0.059611i) - cos(t*(17.006 - 0.95036i))*heaviside(1.0*t - 1.008)*(0.000076573 + 0.018259i) - sin(t*(17.006 + 0.95036i))*heaviside(1.0*t - 1.008)*(0.018259 + 0.000076573i)

I want to draw a graph of the function returns the final. Please help me. Thanks!