18.2: A.2- Laplace Transform of a Ratio of Two Polynomials

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:

1. 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);
2. the degree of \(\operatorname{Den}(s)\) exceeds that of \(\text { Num(s) }\), \(0 \leq m<n\); and
3. 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

\[f(t)=L^{-1}\left[F_{n}(s)\right]=L^{-1}\left[\frac{\operatorname{Num}(s)}{\operatorname{Den}(s)}\right]=\sum_{k=1}^{n} \frac{\operatorname{Num}\left(p_{k}\right)}{\operatorname{Den}^{\prime}\left(p_{k}\right)} e^{p_{k} t}, t \geq 0\label{eqn:A.7} \]