4.3 Finding the inverse laplace transform  (Page 2/3)

Since the poles are ${\alpha }_1=-1$ and ${\alpha }_2=-2$ are distinct, we have the expansion

Using [link] then gives:

and

Therefore, we get:

The inverse Laplace transform of $X\left(s\right)$ can be found by looking up the inverse transform of each of the terms in the right-hand side of [link] giving

Repeated Poles:

Let's consider the case when each pole is repeated,

where $p_1+p_2+\cdots +p_r=p$ . In this case the partial fraction expansion goes like this:

We'll look at two methods. In the first method, the coefficients can be found using the following formula

where $i=1,2,...,r$ , $k=0,1,...,p_i-1$ and

Note that the computation of $A_{i,p_i}$ does not require any differentiation, since $k=0$ .

Example 3.4 Find the inverse Laplace transform of

Here we have a single repeated pole at $s=-2$ . The expansion is therefore given by

Using [link] , we begin with $k=0$ which corresponds to

Next, we set $k=1$ in [link]

The partial fraction expansion is then given by

Therefore,

In the second method, the coefficients $A_{i,p_i},i=1,...,r$ can be found via the cover up method. The remaining coefficients, $A_{k,p_i},i=1,...,r,k=1,...,p_i-1$ can be found by substituting values of $s$ that are not equal to one of the poles in [link] . This leads to a system of linear equations which can be used to solve for the remaining coefficients. This method is generally preferable if the order of each repeated pole as well as the number of poles is sufficiently small so that the number of unknown coefficients is at most two for hand calculations.

Example 3.5 Find the inverse Laplace transform of:

Using the cover-up method we can find $A_{1,3}$ as follows

So we are left with

Setting $s=0$ in [link] leads to

and setting $s=-2$ in [link] gives

These choices of $s$ were used to simplify the linear equations to the greatest extent possible. The solution to [link] and [link] is easily found to be $A_{1,1}=0$ and $A_{1,2}=1$ . The partial fraction expansion is given by

Using the corresponding Laplace transform pairs leads to

Distinct and repeated poles:

If a Laplace transform contains both distinct and repeated poles, then we would combine the expansions in [link] and [link] . Perhaps the easiest way to indicate this is by way of an example:

Example 3.6 Find the inverse Laplace transform of

The coefficients corresponding to the distinct poles can be found using [link] :

The coefficient $A_{3,2}$ corresponding to the double pole at $s=-5$ can be found using [link] with $k=0$ :

The remaining coefficient, $A_{3,1}$ can be found using [link] with $k=1$ :