What is steady-state error?

Steady-state error is the difference between the desired reference input and the actual system output after all transients have died out. A well-designed system minimizes this error for the expected input type.

What does system type mean?

System type is the number of pure integrators (1/s poles) in the open-loop transfer function G(s). Type 0 has none, Type 1 has one, Type 2 has two. More integrators reduce steady-state error for higher-order inputs.

What is the open-loop gain K?

K is the DC gain of the open-loop transfer function after factoring out the integrators. For G(s) = K/s, K is the velocity constant directly. It scales the error constants Kp, Kv, and Ka.

Why is the error infinite for some combinations?

When system type is lower than required for the input (e.g., Type 0 tracking a ramp), the error grows without bound — the output cannot keep up with the reference. This is represented as infinity (∞).

Does this apply to non-unity feedback systems?

The standard error constants assume unity feedback (H(s)=1). For non-unity feedback, you need to convert the loop gain to an equivalent unity-feedback form first.

What is the parabola (acceleration) input?

A parabolic reference is r(t) = t²/2, corresponding to constant acceleration. Only Type 2 or higher systems can track it with finite error; all lower types diverge.

Steady State Error Calculator

Enter system type and open-loop gain to instantly compute steady-state error for step, ramp, and parabola inputs.

System Parameters

Unity-feedback, open-loop transfer function G(s).

System Type (N)

Type 0

Type 1

Type 2

1 integrator in G(s) — tracks step with zero error.

Open-loop Gain K

Must be a positive number. For G(s)=K/s, this is the velocity gain.

Reference Amplitude A

Amplitude of the reference signal (e.g. step height, ramp slope).

Error Constants

Kp (Position)

—

Kv (Velocity)

—

Ka (Accel.)

—

Steady-State Errors

Input Type	r(t)	e_ss
Step	A·u(t)	—
Ramp	A·t	—
Parabola	A·t²/2	—

System Type vs. Input Summary

Type	Step	Ramp	Parabola
0	A/(1+Kp)	∞	∞
1	0	A/Kv	∞
2	0	0	A/Ka

Summary

This calculator computes the steady-state error of a unity-feedback control system based on system type and open-loop gain. It derives the position error constant (Kp), velocity error constant (Kv), and acceleration error constant (Ka), then calculates the final steady-state error for step, ramp, and parabola reference inputs. Results follow standard control theory: a Type 0 system has finite error to a step, Type 1 tracks steps perfectly but has finite ramp error, and Type 2 tracks ramps perfectly but has finite parabola error.

How it works

Select the system type (0, 1, or 2) — the number of open-loop integrators.
Enter the open-loop position gain K (the DC gain of G(s) excluding integrators).
The tool computes Kp, Kv, and Ka from system type and K.
Steady-state errors for step, ramp, and parabola inputs are derived from the error constants.
Results update instantly; zero means perfect tracking, infinity means the output diverges from the input.

Use cases

Verify steady-state performance of a designed controller before simulation.
Determine the minimum system type needed to track a given reference input.
Compare error constants across different gain values during design.
Quickly classify an existing plant model by its steady-state behavior.
Teach or study control theory concepts for step/ramp/parabola inputs.
Check if integral action is needed to eliminate steady-state error.

Frequently Asked Questions

Last updated: 2026-06-18 · Reviewed by Nham Vu