Standard Test Inputs and Their Laplace Transforms

 

Standard Test Inputs and Their Laplace Transforms

In control systems, the following standard test signals are commonly used to analyze system response:

1. Step Input

• Definition: $u(t)=1$ for $t\ge 0$

• Laplace Transform:

$$U(s)=\frac{1}{s}$$

2. Ramp Input

• Definition: $r(t)=t$ for $t\ge 0$

• Laplace Transform:

$$R(s)=\frac{1}{s^2}$$

3. Parabolic Input

• Definition: $p(t)=\frac{t^2}{2}$ for $t\ge 0$

• Laplace Transform:

$$P(s)=\frac{1}{s^3}$$

4. Impulse Input

• Definition: $\delta (t)$ (Dirac delta function)

• Laplace Transform:

$$\mathrm{\Delta }(s)=1$$