steady-state error (Ess) for different input signals (unit step, ramp, parabolic)

 To calculate steady-state error (Ess) for different input signals (unit step, ramp, parabolic) with systems of type 0, 1, and 2, we use the error constants:

• Position constant: $K_p={\lim }_{s\to 0}G(s)$

• Velocity constant: $K_v={\lim }_{s\to 0}sG(s)$

• Acceleration constant: $K_a={\lim }_{s\to 0}{s}^{2}G(s)$

where $G(s)$ is the open-loop transfer function.

General Formulas

• Step input:

$$e_{ss}=\frac{1}{1+K_p}$$

• Ramp input:

$$e_{ss}=\frac{1}{K_v}$$

• Parabolic input:

$$e_{ss}=\frac{1}{K_a}$$

Interpretation

• Type 0 system: Can only track step inputs with finite error. Ramp and parabolic inputs → infinite error.

• Type 1 system: Perfectly tracks step inputs (zero error), has finite error for ramp, but fails for parabolic.

• Type 2 system: Perfectly tracks step and ramp inputs, has finite error for parabolic.

Example

Suppose $G(s)=\frac{K}{s(s+2)}$ (Type 1 system):

• Step input → $e_{ss}=0$

• Ramp input → $e_{ss}=\frac{1}{K_v}=\frac{1}{K\mathrm{/}2}=\frac{2}{K}$

• Parabolic input → $e_{ss}=\mathrm{\infty }$