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Short notes

Steady State Error Short Notes

Set up a unity-feedback position control system with a Type 1 open-loop transfer function and command it to follow a ramp input — the output shaft never quite catches up, trailing behind by a constant angle. That constant angular error is the steady-state ramp error e_ss = R/K_v, where K_v is the velocity error constant and R is the ramp slope. Knowing the system type instantly tells you which input will produce zero error, finite error, or infinite error — that classification is the entire framework of this topic.

EEE, ECE

How it works

Steady-state error for a unity-feedback system with open-loop G(s): e_ss = lim(s→0) s·E(s) = R(s)/(1+G(s)). For step input R/s: e_ss = R/(1+K_p) where K_p = lim(s→0) G(s). For ramp R/s²: e_ss = R/K_v where K_v = lim(s→0) s·G(s). For parabolic R/s³: e_ss = R/K_a where K_a = lim(s→0) s²·G(s). System type is the number of open-loop poles at s = 0. A Type 0 system has finite step error, infinite ramp error. A Type 1 system has zero step error, finite ramp error, infinite parabolic error.

Key points to remember

Type 0 system: K_p = finite, K_v = 0, K_a = 0. Type 1: K_p = ∞ (zero step error), K_v = finite, K_a = 0. Type 2: K_p = ∞, K_v = ∞, K_a = finite. For G(s) = 10/[s(s+2)], Type 1, K_v = lim(s→0) s·10/[s(s+2)] = 10/2 = 5; ramp error = 1/5 = 0.2 for unit ramp. Adding an integrator to the forward path increases system type by 1 but may reduce stability. Final value theorem (lim(s→0) s·Y(s)) is used to compute steady-state output; error constants quantify the deviation from the desired value.

Exam tip

The examiner always asks you to find steady-state error for a given G(s) and all three standard inputs — identify the system type first, then compute only the relevant error constant, and clearly state which inputs give zero vs infinite error.

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