# Unit-ramp Response Of First-order Systems

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**Unit-ramp response of first-order systems:**

*Fig: 1 *

Since the Laplace transform of the unit-ramp function is 1/s^{2}, 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.