Formula & Calculator
Steady-State Error
Determines the residual error of a control system's output as time approaches infinity.
Type 0: ess = 1/(1+K) (step), ∞ (ramp/parabolic)
Type 1: ess = 0 (step), 1/K (ramp), ∞ (parabolic)
Type 2: ess = 0 (step/ramp), 1/K (parabolic)
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