__Laplace Transformation__

We shall first present a definition of the Laplace transformation and a brief discussion of the condition for the existence of the Laplace transform and then provide examples for illustrating the derivation of Laplace transforms of several common functions.

Let us define

f ( t ) = a function of time t such that f ( t ) = 0 for t < 0

s = a complex variable

L = an operational symbol indicating that the quantity that it prefixes is to be transformed by the Laplace integral

F(s) = Laplace transform of f( t )

Then the Laplace transform off ( t ) is given by

The reverse process of finding the time function f ( t ) from the Laplace transform F(s) is called the inverse Laplace transformation. The notation for the inverse Laplace transformation is L^{-1}, and the inverse Laplace transform can be found from F(s) by the following inversion integral:

where c, the abscissa of convergence, is a real constant and is chosen larger than the real parts of all singular points of F(s). Thus, the path of integration is parallel to the jw axis and is displaced by the amount c from it.This path of integration is to the right of all singular points.

Evaluating the inversion integral appears complicated. In practice, we seldom use this integral for finding f (t ). There are simpler methods for finding f ( t ) .

It is noted that in this book the time function f ( t ) is always assumed to be zero for negative time; that is,

__First Order Systems:__

__Fig: Block Diagram of first order systems__

__Fig: Exponential Response Curve:__

**Figure 5-3**

Unit-ramp response of the system shown in above figure.

**Figure 5-41**

Unit-impulse response of the system shown in above figure.

__SECOND-ORDER SYSTEMS:__

**Figure 5-5**

(a) Servo system;

(b) block diagram;

(c) simplified block diagram.

**Figure ***5-6 ***Second-order system.**

**Figure **5-7

Unit-step response curves of the system shown in above figure.

**Figure **5-8

Unit-step response curve showing t_{d, }t_{r}, t_{p}, M_{p }and t_{s}.

The time-domain specifications just given are quite important since most control systems are time-domain systems; that is, they must exhibit acceptable time responses. (This means that, the control system must be modified until the transient response is satisfactory.)

Note that not all these specifications necessarily apply to any given case. For example, for an over damped system, the terms peak time and maximum overshoot do not apply.

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