Steady-State Error Calculator

Find the steady-state error for step, ramp, and parabolic inputs from a system's type number and open-loop gain, using the Final Value Theorem.

📉 Steady-State Error Calculator
Open-loop gain (K)10
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0.1100
T₁ s
T₂ s
Applicable error
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System type
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Position constant Kp
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Velocity constant Kv
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Acceleration constant Ka
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Step error ess
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Ramp error ess
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Parabola error ess
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Step-by-step working

📉 What is Steady-State Error?

Steady-state error is the gap that remains between a control system's commanded input and its actual output after every transient wiggle has died away, as time goes to infinity. It is found by applying the Final Value Theorem to the closed-loop error signal, and for a standard unity-feedback (or equivalent) loop, the result depends only on two things about the open-loop transfer function: its system type number (how many pure integrators it contains) and its DC gain K, not on any of the transient dynamics that pole time constants control.

Engineers check steady-state error whenever a control loop must track a moving target, not just settle at a fixed setpoint. A CNC machine tool positioning system must reach a commanded position with essentially zero final error, requiring at least a Type 1 design. A radar or camera gimbal tracking a target moving at constant angular velocity needs a finite, small velocity error, again a Type 1 system, since a Type 0 system's ramp error grows without bound. A missile or robotic arm tracking an accelerating target needs a Type 2 system to keep even its parabolic-input error finite.

A common misconception is that a "better" (faster, more heavily damped) transient response automatically means a smaller steady-state error. In reality these are separate design axes entirely, a system can settle very quickly (good transient response, set by pole locations and time constants) while still carrying a large, nonzero steady-state error (set by the type number and gain), or vice versa. Increasing gain K reduces the steady-state error for inputs the system's type can already track, but it does nothing for inputs of a higher order, and it can also degrade stability margins.

This calculator takes an open-loop system's type number (0, 1, or 2), gain K, and pole time constants T1 and T2, and returns the position, velocity, and acceleration error constants (Kp, Kv, Ka), the steady-state error for step, ramp, and parabolic inputs, and a chart showing how the one input your system type can actually track improves as gain K increases.

📐 Formula

ess,step  =  1 / (1+Kp)     ess,ramp  =  1 / Kv     ess,parabola  =  1 / Ka
Open-loop transfer function: G(s)H(s) = K / [sN(1+sT1)(1+sT2)], where N is the system type number
Kp = lims→0 G(s)H(s), finite (=K) only for N=0, otherwise infinite
Kv = lims→0 sG(s)H(s), finite (=K) only for N=1, zero when N is less than 1, infinite when N is greater than 1
Ka = lims→0 s²G(s)H(s), finite (=K) only for N=2, zero when N is less than 2, infinite when N is greater than 2
Example: Type 1, K = 10 → Kv = 10, ess,ramp = 1/10 = 10%, while ess,step = 0% and ess,parabola is unbounded.

📖 How to Use This Calculator

Steps

1
Select the system type. Choose Type 0, Type 1, or Type 2 based on how many pure integrators (poles at the origin) the open-loop transfer function has.
2
Enter the open-loop gain and time constants. Type K, the open-loop DC gain, plus T1 and T2, the two pole time constants (they do not affect the error constants, only the transient response).
3
Read the error constants and steady-state error. See Kp, Kv, Ka and the steady-state error for step, ramp, and parabolic inputs, with the applicable one for your system type highlighted.

💡 Example Calculations

Example 1 — Type 0 System Tracking a Step Input

Type 0, K = 10, T1 = 0.5 s, T2 = 0.1 s

1
N = 0, so Kp = K = 10; Kv = 0; Ka = 0
2
ess,step = 1/(1+10) = 1/11 = 9.0909%
3
ess,ramp = 1/0, unbounded; ess,parabola = 1/0, unbounded
Applicable step error = 9.0909%, ramp and parabola errors are unbounded for a Type 0 system
Try this example →

Example 2 — Type 1 System Tracking a Ramp Input

Type 1, K = 10, T1 = 0.5 s, T2 = 0.1 s

1
N = 1, so Kp is infinite; Kv = K = 10; Ka = 0
2
ess,step = 1/(1+∞) = 0%
3
ess,ramp = 1/10 = 10%; ess,parabola = 1/0, unbounded
Applicable ramp error = 10%, step error is exactly zero, parabola error is unbounded
Try this example →

Example 3 — Type 2 System Tracking a Parabolic Input

Type 2, K = 10, T1 = 0.5 s, T2 = 0.1 s

1
N = 2, so Kp and Kv are infinite; Ka = K = 10
2
ess,step = 0%; ess,ramp = 0%
3
ess,parabola = 1/10 = 10%
Applicable parabola error = 10%, step and ramp errors are both exactly zero
Try this example →

❓ Frequently Asked Questions

What is steady-state error in a control system?+
Steady-state error is the difference between a system's commanded input and its actual output after all transient behavior has died out, as time goes to infinity. It is found using the Final Value Theorem applied to the closed-loop error signal, and its value depends only on the open-loop system's type number and DC gain, not on its transient dynamics.
What is the system type number and how do you find it?+
The type number N is the number of pure integrators, poles located exactly at s=0, in the open-loop transfer function G(s)H(s) = K / [s^N(1+sT1)(1+sT2)]. A system with no integrator is Type 0, one integrator is Type 1, two integrators is Type 2, and so on.
How do you calculate the position error constant Kp?+
Kp is the limit as s approaches 0 of the open-loop transfer function G(s)H(s). For G(s)H(s) = K/[s^N(1+sT1)(1+sT2)], Kp equals K for a Type 0 system and is infinite for Type 1 or higher, since the s^N term in the denominator forces the limit to blow up.
How do you calculate the velocity error constant Kv?+
Kv is the limit as s approaches 0 of s times G(s)H(s). For a Type 1 system this limit equals K exactly (the single s in the denominator cancels the extra s in the limit), it is 0 for Type 0 (too few integrators) and infinite for Type 2 or higher (too many).
How do you calculate the acceleration error constant Ka?+
Ka is the limit as s approaches 0 of s squared times G(s)H(s). For a Type 2 system this limit equals K exactly, it is 0 for Type 0 and Type 1 (too few integrators) and infinite for Type 3 or higher (too many).
What is the steady-state error formula for a step input?+
For a unit step input, the steady-state error is ess = 1/(1+Kp). A Type 0 system has a finite Kp and therefore a finite, nonzero step error. A Type 1 or higher system has Kp equal to infinity, making the step error exactly zero.
What is the steady-state error formula for a ramp input?+
For a unit ramp input, the steady-state error is ess = 1/Kv. A Type 0 system has Kv = 0, giving an infinite (unbounded, ever-growing) ramp error. A Type 1 system has a finite Kv and a finite ramp error. A Type 2 or higher system has Kv = infinity, giving zero ramp error.
Why don't the pole time constants T1 and T2 affect the steady-state error?+
Because the error constants are evaluated as s approaches 0, and the factors (1+sT1) and (1+sT2) both approach exactly 1 regardless of what T1 and T2 are, since s itself is approaching zero. T1 and T2 only shape the transient response (how the system gets to steady state), not the final steady-state value itself.
How can I reduce steady-state error without changing the system type?+
Increasing the open-loop gain K reduces the steady-state error for whichever input the system's type number can already track with a finite error, for example raising K on a Type 1 system reduces its ramp error (ess = 1/Kv = 1/K). It has zero effect on inputs of a higher order than the type number supports.
What is the difference between a Type 1 and Type 2 system?+
A Type 1 system (one open-loop integrator) tracks a step input with zero steady-state error and a ramp input with a finite, nonzero error, but cannot track an accelerating (parabolic) input at all. A Type 2 system (two open-loop integrators) tracks both step and ramp inputs with zero error and a parabolic input with a finite error.
Why is a higher system type not always better?+
Each added open-loop integrator (higher type number) reduces steady-state error for more demanding inputs, but it also adds 90 degrees of phase lag at every frequency, which erodes phase margin and pushes the closed-loop system closer to instability. Real designs balance steady-state accuracy against the stability margins checked by the Bode Plot and Nyquist calculators on this site.
Tags: Steady State Error Calculator System Type Number Calculator Position Velocity Acceleration Error Constant Calculator Final Value Theorem Calculator Control Systems Error Calculator Calculator Formula Free Online

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What is steady-state error in a control system?

Steady-state error is the difference between a system's commanded input and its actual output after all transient behavior has died out, as time goes to infinity. It is found using the Final Value Theorem applied to the closed-loop error signal, and its value depends only on the open-loop system's type number and DC gain, not on its transient dynamics.

What is the system type number and how do you find it?

The type number N is the number of pure integrators, poles located exactly at s=0, in the open-loop transfer function G(s)H(s) = K / [s^N(1+sT1)(1+sT2)]. A system with no integrator is Type 0, one integrator is Type 1, two integrators is Type 2, and so on.

How do you calculate the position error constant Kp?

Kp is the limit as s approaches 0 of the open-loop transfer function G(s)H(s). For G(s)H(s) = K/[s^N(1+sT1)(1+sT2)], Kp equals K for a Type 0 system and is infinite for Type 1 or higher, since the s^N term in the denominator forces the limit to blow up.

How do you calculate the velocity error constant Kv?

Kv is the limit as s approaches 0 of s times G(s)H(s). For a Type 1 system this limit equals K exactly (the single s in the denominator cancels the extra s in the limit), it is 0 for Type 0 (too few integrators) and infinite for Type 2 or higher (too many).

How do you calculate the acceleration error constant Ka?

Ka is the limit as s approaches 0 of s squared times G(s)H(s). For a Type 2 system this limit equals K exactly, it is 0 for Type 0 and Type 1 (too few integrators) and infinite for Type 3 or higher (too many).

What is the steady-state error formula for a step input?

For a unit step input, the steady-state error is ess = 1/(1+Kp). A Type 0 system has a finite Kp and therefore a finite, nonzero step error. A Type 1 or higher system has Kp equal to infinity, making the step error exactly zero.

What is the steady-state error formula for a ramp input?

For a unit ramp input, the steady-state error is ess = 1/Kv. A Type 0 system has Kv = 0, giving an infinite (unbounded, ever-growing) ramp error. A Type 1 system has a finite Kv and a finite ramp error. A Type 2 or higher system has Kv = infinity, giving zero ramp error.

Why don't the pole time constants T1 and T2 affect the steady-state error?

Because the error constants are evaluated as s approaches 0, and the factors (1+sT1) and (1+sT2) both approach exactly 1 regardless of what T1 and T2 are, since s itself is approaching zero. T1 and T2 only shape the transient response (how the system gets to steady state), not the final steady-state value itself.

How can I reduce steady-state error without changing the system type?

Increasing the open-loop gain K reduces the steady-state error for whichever input the system's type number can already track with a finite error, for example raising K on a Type 1 system reduces its ramp error (ess = 1/Kv = 1/K). It has zero effect on inputs of a higher order than the type number supports.

What is the difference between a Type 1 and Type 2 system?

A Type 1 system (one open-loop integrator) tracks a step input with zero steady-state error and a ramp input with a finite, nonzero error, but cannot track an accelerating (parabolic) input at all. A Type 2 system (two open-loop integrators) tracks both step and ramp inputs with zero error and a parabolic input with a finite error.

Why is a higher system type not always better?

Each added open-loop integrator (higher type number) reduces steady-state error for more demanding inputs, but it also adds 90 degrees of phase lag at every frequency, which erodes phase margin and pushes the closed-loop system closer to instability. Real designs balance steady-state accuracy against the stability margins checked by the Bode Plot and Nyquist calculators on this site.

📌 Quick Tips

💡The system type number equals the number of pure integrators (poles at s=0) in the open-loop transfer function G(s)H(s), not the order of the denominator polynomial as a whole.
💡Pole time constants T1 and T2 change how fast the response settles, but they drop out entirely of the steady-state error constants Kp, Kv, and Ka, only the type number and the DC gain K matter for steady-state accuracy.
💡Raising the open-loop gain K reduces the steady-state error for the input the system's type number can track (a finite error), but it does nothing for inputs of a higher order, those still produce infinite steady-state error regardless of how large K is.
💡A Type 0 system tracks a step input with some finite error but cannot track a ramp at all (error grows without bound). A Type 1 system tracks a step with zero error and a ramp with finite error, but not a parabola. A Type 2 system adds finite parabola tracking on top of that.
💡This calculator's K, T1, and T2 use the same open-loop convention as the Bode Plot Gain and Phase Margin Calculator and the Nyquist Stability Criterion Calculator on this site, so the same gain and time constants can be checked for stability margins there and for steady-state accuracy here.