Calculator · Dynamics & Control
Steady-State Error Calculator
Enter the open-loop transfer function L(s) of a unity negative feedback loop. The system type, the static error constants Kp, Kv and Ka and the resulting steady-state errors follow from the final-value theorem — and are reported only when the closed loop actually settles.
Input — loop transfer function L(s)
Result
| System type | Type 1 |
|---|---|
| Kp | ∞ |
| Kv | 5 |
| Ka | 0 |
| Closed loop settles | Yes |
| System type | Type 1 |
|---|---|
| Kp = lim s→0 L(s) | ∞ |
| Kv = lim s→0 sL(s) | 5 |
| Ka = lim s→0 s²L(s) | 0 |
| Unit step (Kp) | 0 |
|---|---|
| Unit ramp (Kv) | 0.2 |
| Unit parabolic (Ka) | ∞ |
The type is the number of integrators in L(s). Each integrator removes one order of tracking error: a Type 1 loop follows a step exactly but lags a ramp, a Type 2 loop follows a ramp exactly. Errors shown are e_ss = 1/(1 + Kp), 1/Kv and 1/Ka for unit inputs.
Explore further
Technical reference
Kp = lim s→0 L(s) · Kv = lim s→0 sL(s) · Ka = lim s→0 s²L(s)
e_step = 1/(1 + Kp) · e_ramp = 1/Kv · e_parabolic = 1/Ka
- L(s)
- open-loop (loop) transfer function under unity negative feedback
- Kp, Kv, Ka
- static position, velocity and acceleration error constants
- e_step, e_ramp, e_parabolic
- steady-state error to a unit step, ramp and parabolic reference
System type is the number of pure integrators (s = 0 poles) in L(s), read structurally from its s^k factors.
The final-value theorem — and so every error value above — applies only when the closed loop is asymptotically stable; otherwise no error is reported.
| System type | Step error | Ramp error | Parabolic error |
|---|---|---|---|
| Type 0 | 1/(1 + Kp), finite | Infinite | Infinite |
| Type 1 | 0 | 1/Kv, finite | Infinite |
| Type 2 | 0 | 0 | 1/Ka, finite |
Assumptions and limitations
- · Unity negative feedback with the error taken at the summing junction.
- · Common factors of the entered polynomials are not cancelled before the s^k integrator count is read.
- · Classical constants for step, ramp and parabolic references only — no disturbance or noise inputs.
- · Continuous-time systems only.