Laplace Transform enables one to get a very simple and elegant method of solving linear differential equation by transforming them into algebraic equations. It is well known that chemical processes are mathematically represented through a set of differential equations involving derivatives of process states. Analytical solution of such mathematical models in time domain is not only difficult but sometimes impossible without taking the help of numerical techniques. Laplace Transform comes as a good aid in this situation. For this reason, Laplace Transform has been included in the text of this “Process Control” course material though it is purely a mathematical subject.
Definition of Laplace Transform
Consider a function f(t). The Laplace transform of the function is represented by f(s) and defined by the following expression:
Hence, the Laplace Transform is a transformation of a function from the t -domain (time domain) to s -domain (Laplace domain) where both t and s are independent variables.
Properties of Laplace Transform
• The variable s is defined in the complex plane as S= a + jb where j = √-1.
• Laplace Transform of a function exists if the integral has a finite value, i.e. , it remains bounded; eg . if f(t) = eat, then f(s) exists only for s>a, as the integral becomes unbounded for s<a.
• Laplace Transform is a linear operation.
Laplace transform of a few basic functions
The Fig.1 shows a few basic functions which are frequently used in process control applications
Fig.1: Few basic functions which are frequently used in process control applications
Step function : See Fig. 1(a) for the schematic of a step function
Ramp function : See Fig. 1(b) for the schematic of a ramp function f(t)=at for t>0 where a is a constant
Exponential function : for t>0 where a is a constant
Sinusoidal function :
Delayed function :f(t - td) , i.e .f(t) is delayed by td seconds
Now, let us take , hence . At and at . Thus,
Pulse function : See Fig. 1(c) for the schematic of an unit pulse function. The area under the pulse is 1. The duration of pulse is T and hence it achieves maximum intensity of 1/T. Thus the
Function is defined by
It can also be defined as the “addition” of two step functions which are equal but with opposite intensity, however, the second function is delayed by T .
Hence, it is evident that f2(t) is equal to -f1(t) in intensity however it is delayed by time T.
Thus, . Since f1(t) is a step function of intensity 1/T, the following expression will hold.
Impulse function : See Fig. 1(d) for the schematic of an unit impulse function. This is analogous to a pulse function whose duration is shrinked to zero without losing the strength. Hence the area under the impulse remains 1. The function can be expressed as the following:
As the duration of the impulse tends to zero, its maximum intensity ideally tends to ∞. Mathematically it is termed as Dirac Delta function and is represented as . The following relation holds for unit impulse:
Thus the Laplace transform of the impulse function can be derived as the following:
L'Hospital's rule has been applied in the above derivation. Hence,
The following table presents the Laplace transforms of various functions.
Table 1: Laplace transforms of various functions (t>0)