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Inverse Laplace transform of rational function

0

$begingroup$

Is it true that the inverse Laplace transform of a proper rational function (where all the coefficients are real and positive),
begin{equation}
s mapsto frac{a_1 , s^{m-1} + a_2 , s^{m-2} + dotsb + a_{m}}{s^m + b_1 s^{m-1} + dotsb + b_{m}},
end{equation}
can be expressed as $langle b, e^{A t} b rangle$, for a matrix $A in mathbb R^{n times n}$ and a vector $b in mathbb R^n$ and $n geq m$?

I would say that it is and that the inverse can be calculated by partial fraction decomposition, inverse Laplace transform of the partial fractions, and using an ansatz of $A$ in normal Jordan form, but would like confirmation or a reference.

laplace-transform

share|cite|improve this question

edited Mar 17 at 17:22

Roberto Rastapopoulos

asked Jan 12 at 20:07

Roberto RastapopoulosRoberto Rastapopoulos

950425

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  We can always fill up $A$ and $b$ with as many zeros as we wish.
  $endgroup$
  – A.Γ.
  Jan 12 at 20:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, but I don't see your point?
  $endgroup$
  – Roberto Rastapopoulos
  Jan 12 at 20:25
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  We can make $n$ as large as we want. Isn't it the question?
  $endgroup$
  – A.Γ.
  Jan 12 at 20:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The question is: Are there $A, b, n$ (with $n$ possibly larger than $m$), such that the Laplace transform of $<b, e^{-At} b>$ is the rational function given?
  $endgroup$
  – Roberto Rastapopoulos
  Jan 12 at 20:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  How do you think to write say $e^{-t}-e^{-2t}$ as $langle b,e^{At}brangle$? At $t=0$ it gives $|b|=0$. Do I miss something here?
  $endgroup$
  – A.Γ.
  Jan 12 at 21:23
  

  
  
  
  

 | 
show 1 more comment

0

$begingroup$

Is it true that the inverse Laplace transform of a proper rational function (where all the coefficients are real and positive),
begin{equation}
s mapsto frac{a_1 , s^{m-1} + a_2 , s^{m-2} + dotsb + a_{m}}{s^m + b_1 s^{m-1} + dotsb + b_{m}},
end{equation}
can be expressed as $langle b, e^{A t} b rangle$, for a matrix $A in mathbb R^{n times n}$ and a vector $b in mathbb R^n$ and $n geq m$?

I would say that it is and that the inverse can be calculated by partial fraction decomposition, inverse Laplace transform of the partial fractions, and using an ansatz of $A$ in normal Jordan form, but would like confirmation or a reference.

laplace-transform

share|cite|improve this question

edited Mar 17 at 17:22

Roberto Rastapopoulos

asked Jan 12 at 20:07

Roberto RastapopoulosRoberto Rastapopoulos

950425

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  We can always fill up $A$ and $b$ with as many zeros as we wish.
  $endgroup$
  – A.Γ.
  Jan 12 at 20:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, but I don't see your point?
  $endgroup$
  – Roberto Rastapopoulos
  Jan 12 at 20:25
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  We can make $n$ as large as we want. Isn't it the question?
  $endgroup$
  – A.Γ.
  Jan 12 at 20:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The question is: Are there $A, b, n$ (with $n$ possibly larger than $m$), such that the Laplace transform of $<b, e^{-At} b>$ is the rational function given?
  $endgroup$
  – Roberto Rastapopoulos
  Jan 12 at 20:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  How do you think to write say $e^{-t}-e^{-2t}$ as $langle b,e^{At}brangle$? At $t=0$ it gives $|b|=0$. Do I miss something here?
  $endgroup$
  – A.Γ.
  Jan 12 at 21:23
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Is it true that the inverse Laplace transform of a proper rational function (where all the coefficients are real and positive),
begin{equation}
s mapsto frac{a_1 , s^{m-1} + a_2 , s^{m-2} + dotsb + a_{m}}{s^m + b_1 s^{m-1} + dotsb + b_{m}},
end{equation}
can be expressed as $langle b, e^{A t} b rangle$, for a matrix $A in mathbb R^{n times n}$ and a vector $b in mathbb R^n$ and $n geq m$?

I would say that it is and that the inverse can be calculated by partial fraction decomposition, inverse Laplace transform of the partial fractions, and using an ansatz of $A$ in normal Jordan form, but would like confirmation or a reference.

laplace-transform

share|cite|improve this question

edited Mar 17 at 17:22

Roberto Rastapopoulos

asked Jan 12 at 20:07

Roberto RastapopoulosRoberto Rastapopoulos

950425

$endgroup$

share|cite|improve this question

edited Mar 17 at 17:22

Roberto Rastapopoulos

asked Jan 12 at 20:07

Roberto RastapopoulosRoberto Rastapopoulos

950425

share|cite|improve this question

edited Mar 17 at 17:22

Roberto Rastapopoulos

asked Jan 12 at 20:07

Roberto RastapopoulosRoberto Rastapopoulos

950425

asked Jan 12 at 20:07

Roberto RastapopoulosRoberto Rastapopoulos

950425

asked Jan 12 at 20:07

Roberto RastapopoulosRoberto Rastapopoulos

950425