__Theory__

**Systems**

Systems are specified as zero-order, first-order, and second-order depending on the order (highest derivative term) of the differential equation. The following sections describe the system models, real life examples and equations of responses for the test inputs.

**Second-Order System**

The highest order term in the differential equation of the second-order systems is second derivative. The second-order systems follow the equation.

After rearranging the equation in the standard form, the equation for the second-order system is

Where is called undamped natural frequency, is called damping ratio, and

is the static sensitivity of the system.

** Fig .1 **

The transfer function of the second-order system is

An example of a second-order measurement system is a mass-spring-damper assembly shown in above Fig. 1

**1. Test Inputs**

The following test inputs are normally used for testing the dynamics of a system. For characterizing system in time domain, the test inputs used are impulse, step, and ramp. A swept frequency sine wave is used to characterize system in frequency domain.

**1.1 Impulse Input**

The unit impulse function is defined as

for and is zero elsewhere.

### ** Unit impule input**

The Laplace transform of unit impulse is given by

**1.2 Step Input**

The unit step function *u*(*t*) is defined as

The Laplace transform of unit impulse is given by

Unit step input

Unit step response

**1.3 Ramp Input**

The unit ramp function *r*(*t*) is defined as

Unit ramp input

The Laplace transform of unit ramp is given by

where .