Unit-ramp Response Of First-order Systems

Unit-ramp response of first-order systems:

Fig: 1

Since the Laplace transform of the unit-ramp function is 1/s², we obtain the output of the system of Figure 1 as

Taking the inverse Laplace transform of Equation above, we have,

The error signal e(t) is then

Fig: 2 Unit-ramp response of the system

The unit-ramp input and the system output are shown in Figure 2. The error in following the unit-ramp input is equal to T for sufficiently large t. The smaller the time constant T, the smaller the steady-state error in following the ramp input.