# Standard Test Signals of control systems

In this post we will see the different test signals of control systems. In my early post we discussed on **mathematical modelling of electrical systems.** Now the next step would be, to obtain its response, both transient response and steady state response for a specific input. Now question is the input can be a time varying function or it may be a *random signal. *Thus we need some standard test signals of control systems which strain the system very severely.

These standard input signals are

- an impulse,
- a step,
- a ramp and
- a parabolic input.

By using above standard test signals of control systems, analysis and design of control systems are carried out, defining certain performance measures for the system.

**Impulse Signal**

In bellow an impulse signal is shown in Fig

The impulse function is zero for all *t *not equal to 0 and it is infinity at *t *= 0. It rises to infinity at *t *= 0^{–} and comes back to zero at *t *= 0^{+} enclosing a finite area. If this area is A it is called as an impulse function of strength A. If A = 1 it is called a unit impulse function. Thus an impulse signal is denoted by *f(t) *= A (t).

**Step Signal**

In bellow a step signal is shown in Fig.

It is zero for *t *< 0 and suddenly rises to a value A at *t *= 0 and remains at this value for *t *> 0: It is denoted by *f(t) *= *Au (t). *If A = 1, it is called a *unit step function.*

**Ramp signal**

In bellow a ramp signal is shown in Fig

It is zero for *t *< 0 and uniformly increases with a slope equal to A. It is denoted by *f (t) *= *At.*

If the slope is unity, then it is called a *unit ramp signal.*

**Parabolic signal**

In bellow a parabolic signal is shown in Fig

A parabolic signal is denoted by *f (t) *= .If A is equal to unity then it is known as a *unit parabolic signal.*

Now see the interesting thing that the step function is obtained by integrating the impulse function from 0 to . A ramp function is obtained by integrating the step function and finally the parabolic function is obtained by integrating the ramp function. Similarly ramp function, step function and impulse function can be obtained by successive differentiations of the parabolic function.

Such a set of functions which are derived from one another are known as *singularity functions. *If the response of a linear system is known for anyone of these input signals, the response to any other signal, out of these singularity functions, can be obtained by either differentiation or integration of the known *response.*