Steady-State Error Calculator
Find how far a unity-feedback system settles from its target — solve for the steady-state error, the error constant or the input size, or derive the constants straight from the system type and open-loop gain.
Sweeps the error constant around your value for your input and for twice the input magnitude. Error falls as the constant grows, but with diminishing returns, and it scales directly with how large the input is.
Left: the closed loop. The error E(s) is the input minus the fed-back output, and the final value theorem gives its limit as t → ∞. Right: a type-1 system tracking a ramp settles into a parallel line a fixed distance ess below the input.
| Type | Step A/s | Ramp A/s² | Parabola A/s³ |
|---|---|---|---|
| 0 | A/(1+Kp) | ∞ | ∞ |
| 1 | 0 | A/Kv | ∞ |
| 2 | 0 | 0 | A/Ka |
| 3 | 0 | 0 | 0 |
| Constant | Definition | Type 0 / 1 / 2 |
|---|---|---|
| Kp | lim G(s) | K / ∞ / ∞ |
| Kv | lim s·G(s) | 0 / K / ∞ |
| Ka | lim s²·G(s) | 0 / 0 / K |
Raise the loop gain
A larger K increases every error constant and shrinks ess, but too much gain hurts stability and overshoot.
Add an integrator
Each pole at the origin raises the system type by one, driving the error for that input class to zero. This is why PI control removes step offset.
Match input to system type
A system can track a step, ramp or parabola with finite error only if its type is high enough for that input.
What Is Steady-State Error?
Steady-state error is the difference between the desired input and the actual output of a control system after the transient response has died away. It measures accuracy: a thermostat that settles half a degree below its setpoint has a steady-state error of 0.5 degrees, however quickly it got there.
The Steady-State Error Formula
This is the final value theorem applied to the error signal. For a unity-feedback loop, E(s) = R(s) / (1 + G(s)), where G(s) is the open-loop transfer function. The theorem is only valid when the closed loop is stable, so check stability first. Plugging in standard inputs turns the limit into simple formulas using the error constants Kp, Kv and Ka.
How to Calculate Steady-State Error
Pick the reference input, choose the unknown, and enter the other two values. The calculator uses the matching error-constant formula and rearranges it as needed.
| Input | Error | Constant | Solve for constant · input |
|---|---|---|---|
| Step | ess = A / (1 + Kp) | Kp = lim G(s) | Kp = A/ess − 1 · A = ess·(1 + Kp) |
| Ramp | ess = A / Kv | Kv = lim s·G(s) | Kv = A/ess · A = ess·Kv |
| Parabola | ess = A / Ka | Ka = lim s²·G(s) | Ka = A/ess · A = ess·Ka |
Steady-State Error Formula Variables
| Symbol | Meaning |
|---|---|
| ess | Steady-state error as t → ∞ |
| A | Magnitude of the reference input (height, slope or acceleration) |
| Kp, Kv, Ka | Position, velocity and acceleration error constants of G(s) |
| N | System type: the number of integrators (poles at s = 0) in G(s) |
| K | Open-loop gain of G(s) written in time-constant form |
| R(s) | Laplace transform of the input: A/s, A/s² or A/s³ |
System Type and Error Constants
The system type N counts the pure integrators in G(s) and decides which inputs the loop can follow with finite error. A type-0 system settles with a fixed offset on a step but falls behind a ramp without limit. A type-1 system follows a step exactly and a ramp with a constant lag, and a type-2 system follows a ramp exactly. Turn on the derive option in Advanced mode to compute Kp, Kv and Ka from the system type and gain K directly.
Steady-State Error Calculation Example
ess = A / (1 + Kp) = 1 / 10 = 0.1 → the output settles 10% below the target
Ramp input, A = 1, type-1 system with K = 5 (e.g. G(s) = 10 / (s(s + 2)))
Kv = lim s·G(s) = 5, so ess = 1 / 5 = 0.2 → the output lags the ramp by 0.2 units
Frequently Asked Questions
The final value theorem only holds if s·E(s) has all its poles in the left half-plane. If the loop is unstable the error grows or oscillates forever, so the limit formula can produce a number that means nothing. Always confirm stability before trusting ess.
A type-0 system has Kv = 0, so ess = A/Kv is unbounded. The output cannot keep climbing at a constant rate without an integrator, so the gap to the ramp keeps growing with time.
With Kp ≥ 0, A/(1 + Kp) can never exceed A. An error bigger than the input would need a negative Kp, which does not correspond to a normal stable loop, so the calculator flags it as an input problem.
An integrator adds a pole at the origin, raising the system type by one. With the extra integrator the relevant error constant becomes infinite, so the error for that input class drops to zero, as in the type table above.
No. This tool covers the reference input for a unity-feedback loop. Load disturbances and sensor or feedback gains change the error expression, so for those cases derive E(s) for your specific block diagram and apply the final value theorem to it.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| e_ss | Steady-state error | V (or variable) |
| s | Laplace variable | 1/s |
| E(s) | Laplace transform of error signal | V·s (or variable·s) |
What it means
The steady‑state error e_ss is the final difference between the desired output and the actual output of a control system after transients have died out. It is found using the final value theorem: e_ss = lim_{s→0} s·E(s). It depends on the system type (number of integrators) and the nature of the input (step, ramp, parabola). A type‑0 system has finite error for a step input; type‑1 has zero error for a step, but finite for a ramp. Reducing steady‑state error often requires integral action in the controller. Example: For a type‑0 system with a step input, the steady‑state error is finite and given by 1/(1+K_p). If the system has an integrator (type‑1), the error for a step is zero. This is why PI controllers are used to eliminate steady‑state error.
Worked example
Steady‑State Error – Practical Example
Real‑World| Parameter | Value |
|---|---|
| System type | 0 |
| K | 10 |
| Input | Step |
| Formula | ess = 1/(1+K) |
Common mistakes
Watch unit consistency and the assumptions behind the formula; misapplying it outside its valid conditions is the most frequent error.Applications
Steady‑state error e_ss = lim_{s→0} s·E(s) is the final error remaining after transients have died out in a control system. It depends on the system type and input. Engineers use it to design controllers that reduce errors, to meet performance specifications, and to evaluate system accuracy. This formula is essential for precision control systems.
- Control system design for accurate tracking
- Compensator design to improve steady‑state behaviour
- System type classification and analysis
- Error analysis in positioning and regulation systems
- Educational understanding of steady‑state response
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