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The inverse Laplace transform

The inverse Laplace transform is described by Eq. (A.2). As already mentioned in section 2.2 in many cases a direct evaluation of the complex inverse integral is not necessary, as the most important elementary functions are given in Table 2.1. A complicated function, , not given in Table 2.1 must be decomposed into a sum of simple functions of that is

(2.15)

which have a known inverse Laplace transform:

(2.16)

For many problems in control the function is a ratio of polynomials in s, known as rational fraction, that is

(2.17)

where and are the numerator and the denominator, respectively.

If , then is divided by , where a polynomial in and a ratio of polynomials are obtained. The numerator of the fraction has a lower order than . E.g, if , then

(2.18)

whereby and and are constants.

A rational fraction given in Eq. (2.17) can be decomposed into more simple functions by application of partial fraction decomposition, as shown in Eq. (2.15). In order to perform this decomposition the denominator polynomial must be factorised into the form

(2.19)

For a denominator polynomial of -th order one obtains roots or zeros . The zeros of are also known as the poles of , since they define where is infinite. The partial fraction decomposition for different types of poles is shown in the following.

Case 1: has only single poles.

Here can be expanded into the form

(2.20)

where the residuals are real or complex constants. Using the table of correspondences one immediately can obtain the corresponding function of time

    for (2.21)

The values can be determined either by comparing the coefficients or by using the theorem of residuals from the theory of functions according to

(2.22)

for with .

Case 2: has multiple poles.

For multiple poles of each with multiplicity the corresponding partial fraction decomposition is

   with (2.23)

The back transformation of Eq. (2.23) into the time domain is

    for (2.24)

The real or complex coefficients for determined by the theorem of residuals are

(2.25)

This general relation also contains the case of single poles of . The poles may be real or complex.

Case 3: has also conjugate complex poles.

As both, the numerator and the denominator of the function are rational algebraic functions, complex factors always arise as conjugate complex pairs. If has a conjugate complex pair of poles , then for the function in the partial fraction decomposition of

   

Eq. (2.20) can be applied to give

(2.26)

where the residuals

   

are also a conjugate complex pair. Therefore, both fractions of can be combined, and one obtains

(2.27)

with the real coefficients

(2.28)

The determination of the coefficients and is performed again using the theorem of residuals by

(2.29)

As is complex, both sides of this equation are complex. Comparing the real and imaginary parts of both sides one gets two equations for the calculation of and . This procedure is demonstrated now using the following example.

Example 2.4.1   Find the inverse Laplace transform of

   

The partial fraction decomposition of is

   

where the function contains the conjugate pair of poles

   

In addition the third pole of is

   

For the coefficients and it follows from Eq. (2.29)

   

   

Comparing the real and imaginary parts on both sides one obtains

    and    

and from this finally .

Using Eq. (2.22) the residual is

   

The partial fraction decomposition of is thus

   

which can be rearranged in the form

   

such that correspondences given in Table 2.1 can be directly applied to find the inverse transformation. Using the correspondences 16, 15 and 6 of this table, it follows that

    for    

which can be rearranged as

    for    

The graphical representation of is shown in Figure 2.1a. Figure 2.1b shows the corresponding poles, marked by a x, for this in the complex  plane.

Figure 2.1: (a) Graph of the original function (function in the time domain) and (b) position of the poles of in the  plane

It can be seen from this example that the position of the poles and affects the shape of the graph of . In this case all poles of have negative real parts, therefore the graph of shows a damped behaviour, i.e. it decreases to zero for . If the real part of one pole be positive, then the graph of would be infinitely large for .

Since in control problems the original function always represents the time behaviour of a system variable, the behaviour of this system variable can be judged to a large extent by investigation of the positions of the poles of the corresponding mapped function . This will be further commented on in later sections.



Next: Solving linear differential equations Up: The Laplace transform Previous: Main theorems of the   Contents
Christian Schmid 2005-05-09