18.2: A.2- Laplace Transform of a Ratio of Two Polynomials
- Page ID
- 7741
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Suppose that we have a Laplace transform as the ratio of two polynomials, from Equation 2.2.2:
\[F_{n}(s) \equiv \frac{\operatorname{Num}(s)}{\operatorname{Den}(s)}=\frac{b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m+1}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n+1}}=\frac{b_{1}\left(s-z_{1}\right)\left(s-z_{2}\right) \cdots\left(s-z_{m}\right)}{a_{1}\left(s-p_{1}\right)\left(s-p_{2}\right) \cdots\left(s-p_{n}\right)}\label{eqn:A.1} \]
The results derived in this section are based upon three assumptions:
- the roots \(p_{k}\) of \(\operatorname{Den}(s)\), which are the poles of \(F_{n}(s)\), are not repeated (such roots are called simple poles);
- the degree of \(\operatorname{Den}(s)\) exceeds that of \(\text { Num(s) }\), \(0 \leq m<n\); and
- none of the zeros of Equation \(\ref{eqn:A.1}\) equals any of the poles. Under these circumstances, we can expand transform Equation \(\ref{eqn:A.1}\) into partial fractions, from Equation 2.3.3:
\[F_{n}(s)=\sum_{k=1}^{n} \frac{C_{k}}{s-p_{k}}\label{eqn:A.2} \]
In Equation \(\ref{eqn:A.2}\) the residues are given by Equation 2.3.6 as
\[C_{k}=\left[\left(s-p_{k}\right) F_{n}(s)\right]_{s=p_{k}}=\left[\left(s-p_{k}\right) \frac{\operatorname{Num}(s)}{\operatorname{Den}(s)}\right]_{s=p_{k}}, k=1,2, \ldots, n\label{eqn:A.3} \]
Let us examine what might be considered the “total denominator” of Equation \(\ref{eqn:A.3}\):
\[D_{k}=\lim _{s \rightarrow p_{k}}\left[\frac{\operatorname{Den}(s)}{\left(s-p_{k}\right)}\right]\label{eqn:A.4} \]
Observe from \(\operatorname{Den}(s)\) in Equation \(\ref{eqn:A.1}\) that in \(D_{k}\) Equation \(\ref{eqn:A.4}\) has the indeterminate form 0/0. Since we assume that all zeros of \(F_{n}(s)\) are different from the poles, \(\operatorname{Num}\left(p_{k}\right)\) in Equation \(\ref{eqn:A.3}\) is non-zero and finite. Therefore, \(D_{k}\) must also be non-zero and finite, and we can use l’Hopital’s rule to cast Equation \(\ref{eqn:A.4}\) into a different form:
\[D_{k}=\lim _{s \rightarrow p_{k}}\left[\frac{\operatorname{Den}(s)}{\left(s-p_{k}\right)}\right]=\lim _{s \rightarrow p_{k}}\left[\frac{\frac{d}{d s} \operatorname{Den}(s)}{\frac{d}{d s}\left(s-p_{k}\right)}\right] \equiv\left[\frac{d}{d s} \operatorname{Den}(s)\right]_{s=p_{k}}\label{eqn:A.5} \]
Thus (Hildebrand, 1962, p. 548), residue Equation \(\ref{eqn:A.3}\) can be expressed alternatively as
\[C_{k}=\left[\left(s-p_{k}\right) \frac{\operatorname{Num}(s)}{\operatorname{Den}(s)}\right]_{s=p_{k}}=\left[\frac{\operatorname{Num}(s)}{\frac{d}{d s} \operatorname{Den}(s)}\right]_{s=p_{k}} \equiv \frac{\operatorname{Num}\left(p_{k}\right)}{\operatorname{Den}^{\prime}\left(p_{k}\right)}\label{eqn:A.6} \]
Finally (Meirovitch, 1967, p. 532), by substituting Equation \(\ref{eqn:A.6}\) back into Equation \(\ref{eqn:A.2}\) and then taking the inverse Laplace transform of each term in the summation, we find
\[f(t)=L^{-1}\left[F_{n}(s)\right]=L^{-1}\left[\frac{\operatorname{Num}(s)}{\operatorname{Den}(s)}\right]=\sum_{k=1}^{n} \frac{\operatorname{Num}\left(p_{k}\right)}{\operatorname{Den}^{\prime}\left(p_{k}\right)} e^{p_{k} t}, t \geq 0\label{eqn:A.7} \]