Digital Control Systems Waseem Gulsher BS (Evening) 9 Oct, 17 Laplace Transform Lecture – f Digital Control Systems Waseem Gulsher

Laplace Transform

Laplace Transform Laplace transform is an example of a class called integral transform. It takes a function f(t) of one variable t (refer to as time) into a function F(s) of another variable s (the complex frequency). The attraction of the Laplace transform is that it transforms differential equation in t (time) domain into algebraic equations in the s (frequency) domain.

Laplace Transform Solving differential equation in time domain therefore reduces to solving algebraic equations in s-domain. Having done the latter (last) for the desired unknowns, their values as functions of time may be found by taking inverse transforms. Another advantage of using the Laplace transform for solving differential equations is that initial conditions play an important role in the transformation process, so they are automatically incorporated into the solution.

Laplace Transform The Laplace transform is an ideal tool for solving initial-value problems such as those occurring in the investigations of electrical circuits and mechanical vibrations. In modeling the system by a differential equation, it has been assumed that both the input and output signals can vary any instant of time.

Definition and Notation We define Laplace transform of a function f(t) by the expression Where s is a complex variable and e-st is called the kernel of the transformation.

Definition and Notation It is usual to represent the Laplace transform of a function by corresponding capital letter, so we write that

Transforms of the Simple Functions

Example # 1 Determine the Laplace transform of the function f(t) = c where c is a constant. Solution: Using the definition

Example # 1

Example # 1

Example # 2 Determine the Laplace transform of the ramp function f(t) = t Solution: Using the definition

Example # 2 Following the same procedure as in previous example, limits exist provided that Re(s)>0, when Thus, provided that Re(s)>0, Giving us the Laplace transform pair

Example # 3 Determine the Laplace transform of the one-sided exponential function f(t) = ekt Solution:

Example # 3 If k is real, then, provided that σ=Re(s)>k, the limit exists, and is zero. If k is complex, say k=a+jb, then the limit will also exists, and will be zero, provided that σ>a (that is, Re(s) > Re(k)).

Example # 3 Under these conditions, we then have Giving us Laplace transform pair

Table of Laplace Transform

Table of Laplace Transform

Table of Laplace Transform

Assignment # 1 Solve all above examples with complete steps. Date of submission. 16 Oct, 17

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