Type 0, 1, 2, 3 Systems

Topics: Feedback Control Systems

The type of the system refers to the number of poles at the origin $s=0$ in the system, also known as free integrators.

Type 0:

$$G(s) = K \frac{(s-z_1) ... (s-z_m)}{(s-p_1) ... (s-p_m)}$$

Type 1:

$$G(s) = K \frac{(s-z_1) ... (s-z_m)}{s(s-p_1) ... (s-p_m)}$$

Type 2:

$$G(s) = K \frac{(s-z_1) ... (s-z_m)}{s^2(s-p_1) ... (s-p_m)}$$

Type 3:

$$G(s) = K \frac{(s-z_1) ... (s-z_m)}{s^3(s-p_1) ... (s-p_m)}$$

The system type relates to the steady-state error of the system given certain types of inputs:
For zero steady-state error to a waveform with a Laplace transform $1/s^k$, the forward transfer function must have at least $k$ free integrators.

In other words,

• to have $e_{ss} = 0$ for a step input $1/s$ (constant position), the system must be at least Type 1
• to have $e_{ss} = 0$ for a ramp input $1/s^2$ (constant velocity), the system must be at least Type 2
• to have $e_{ss} = 0$ for a step parabolic input $1/s$ (constant acceleration), the system must be at least Type 3

The table below summarizes the steady-state error conditions

References

1. https://www.ocw.mit.edu/courses/2-004-dynamics-and-control-ii-spring-2008/519745ee31d586a1320098c606fc31aa_lecture_24.pdf