Steady-State Error & System Type
Error constants and the steady-state error to step, ramp and parabola inputs.
How many integrators do you have?
For a unity-feedback loop, the steady-state error to a given input depends only on the system type — the number of integrators (poles at the origin) in the open-loop transfer function — through three static error constants:
and the errors follow: e_step = 1/(1+K_p), e_ramp = 1/K_v,
e_parab = 1/K_a. The pattern is the whole lesson: each integrator kills
one more error. A Type 0 system has a finite step error and can't follow a ramp at all;
a Type 1 system tracks a step perfectly but lags a ramp by 1/K_v; a Type 2 system
nails both step and ramp and only errs on an accelerating (parabola) input. This is exactly why
the integral term in a PID controller exists — it adds an
integrator, raising the system type and driving the steady-state step error to zero.