ilaplace() function is giving wrong results

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David 2020-9-3

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编辑： David Goodmanson 2020-9-7

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HI,

I am using ilaplace() function to transform a function with this aspect

The temporal form of that function is a linear combination of a constant term and two exponentials.

I am particularly interested in noting that f(t) is a linear combination of a,b, and c. If I run this code

syms s t a b c d e f;
Fs = (a*s^2+b*s+c)/(s*(d*s^2+e*s+f));
ft = ilaplace(Fs)

The result is

Which IMHO is NOT a linear combination of a, b, and c. Why am I getting this? The result seems wrong to me.

Thanks in advanced.

EDIT:

Found the solution. Just use this function before applyting inverse Laplace tansform to F.

partfrac(F,'FactorMode','complex')

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Walter Roberson 2020-9-3

Maple agrees with MATLAB as to what the inverse laplace transform is.

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Answer 1

David Goodmanson 2020-9-3

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https://ww2.mathworks.cn/matlabcentral/answers/588577-ilaplace-function-is-giving-wrong-results#answer_489322

编辑：David Goodmanson 2020-9-7

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Hi David,

plugging in

cosh(z) = ((exp(z) + exp(-z))/2
sinh(z) = ((exp(z) - exp(-z))/2

results in a constant and two exponentials, as you said.

Note that in the expression for #1, the d in the denominator belongs in the argument for the square root. It does not belong under the entire expression. The fraction bar is too long, and it looks quite likely that the expression for ft has similar problems.

If you want to find the coefficients of the exponentials with a minimum of fuss, the following works.

<< the original question has been modified and includes the form shown here below >>

Find the roots of (d*s^2 + e*s + f), and reverse their signs to obtain r1,r2. Then

F(s) = (a*s^2+b*s+c) / (d*s*(s+r1)*(s+r2))

The solution has the form A0 + A1* exp(-r1*t) + A2*exp(-r2*t).

For A0, evaluate F(s) at s = 0, neglecting s in the denominator.

For A1, evaluate F(s) at s = -r1, neglecting (s+r1) in the denominator.

For A2, evaluate F(s) at s = -r2, neglecting (s+r2) in the denominator.

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David 2020-9-4

编辑：David 2020-9-4

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Hi David,

Thanks for taking the time. The bigger problem is the following. I use ilaplace() in order to obtain a symbolic temporal function that is evaluated at a certain value of the time. As a result, I should have an equation with a number of variables (say a, b, and c in our example).

I repeat that with a number of expressions in order to obtain a system of linear equations. Well, MATLAB says the system is not that linear...

Error using mupadengine/feval_internal
Unable to convert to matrix form because the system does not seem to be linear.
Error in sym/equationsToMatrix (line 61)
T = eng.feval_internal('symobj::equationsToMatrix',eqns,vars);
Error in HVCon (line 142)
[A,B] = equationsToMatrix(eqn, [VCflyi VCparBoti VCparTopi VCflyf VCparBotf VCparTopf Vo]);

I was debugging when I found that odd symbolic expression with cosh and sinh. The result should be a linear combination of coefficients a, b, and c from

Fs = (a*s^2+b*s+c)/(s*(d*s^2+e*s+f));

But the obtained expression doesn't seem linear!

In conclusion, as a far as I see, ilaplace() is transforming an 's' expression which is linearly dependent of coefficients a, b, and c into a 't' expression which is non-linearly dependent of those coefficients.

Thanks again.

David 2020-9-4

编辑：David 2020-9-4

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Thanks for your time Walter.

After implenting your suggestion, I have this code

syms s t a b c d e f;
Fs = (a*s^2+b*s+c)/(s*(d*s^2+e*s+f));
ft = ilaplace(Fs);
cosh2exp = @(z) (exp(z) + exp(-z))/2;
ft = mapSymType(ft,'cosh',@(f) cosh2exp(children(f)));
sinh2exp = @(z) (exp(z) - exp(-z))/2;
ft = mapSymType(ft,'sinh',@(f) sinh2exp(children(f)))

The result is

Again, I have to make sure that f(t) is a linear combination of a, b, and c. In the expression I obtain, there is an odd fraction that led me think that the expression was not a linear combination of a, b, and c but it you carefully look at the whole expression I think that the denominator is canceled and everythin is OK.

(b*f-c*e)/(a*f-c*d)

However, in my real problem I have a bunch of laplace expressions which are inverted and then evaluted at a certain value of time. Thus, I have a system of equations in which coefficients a, b, and c contains the variables. When I try to solve the problem (it should be a system of 29 linear equations and 29 variables), I get this error

Error using mupadengine/feval_internal
Unable to convert to matrix form because the system does not seem to be linear.
Error in sym/equationsToMatrix (line 61)
T = eng.feval_internal('symobj::equationsToMatrix',eqns,vars);

I think this is because my simbolic expressions contain numbers and symbolic variables. This somehow is preventing MATLAB from correctly simplify the expression as I noted before.

David 2020-9-4

Exponential terms are gone once the function is evaluated at a certain value of time. The only variables are inside coefficients a, b, and c, which are not inside any exponential.

David 2020-9-4

编辑：David 2020-9-4

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I wrote a MWE of the whole problem. If I enter the laplace functions with the second order polynomial in the denominator, the non-linear problem arise.

syms s t a b c;
Fs1 = (100*a*s^2+98493*b)/(s*(784937*s^2+890384*s+8374));
Fs2 = (45450*a*s+983*c)/(s*(784337*s^2+8384*s-34));
Fs3 = (545*b*s+98563*c)/(s*(4337*s^2+834*s+345454));
ft1 = vpa(ilaplace(Fs1));
ft2 = vpa(ilaplace(Fs2));
ft3 = vpa(ilaplace(Fs3));
eqn1 = c == subs(ft1, 10);
eqn2 = b == subs(ft2, 10);
eqn3 = a == subs(ft3, 10);
[A, B] = equationsToMatrix([eqn1, eqn2, eqn3]);

However, if I replace the second order polynomial with its two roots, everything works fine.

syms s t a b c;
Fs1 = (100*a*s^2+98493*b)/(s*(s-1.1249)*(s-0.0095));
Fs2 = (45450*a*s+983*c)/(s*(s-0.0138)*(s+0.0031));
Fs3 = (545*b*s+98563*c)/(s*(s-9.0215)*(s+8.8282));
ft1 = vpa(ilaplace(Fs1));
ft2 = vpa(ilaplace(Fs2));
ft3 = vpa(ilaplace(Fs3));
eqn1 = c == subs(ft1, 10);
eqn2 = b == subs(ft2, 10);
eqn3 = a == subs(ft3, 10);
[A, B] = equationsToMatrix([eqn1, eqn2, eqn3]);