Library / Control engineering / CTL-08
Control engineering
Steady-state error to a ramp input
Constant tracking error of a stable unity-feedback loop following a reference that rises at a steady rate.
- Ref
- CTL-08
- Rev
- v1
- Field
- Control engineering
- Result
- ess [units]
For a stable unity-feedback loop following a reference that rises at a constant rate, the output trails behind by a fixed amount: the rate divided by the velocity error constant Kvel = lims→0 sL(s). A loop with no integrator has Kvel = 0 and its error grows without limit; a loop with one integrator has a finite, constant error; a loop with two integrators follows the ramp with zero error.
Constant tracking error of a stable unity-feedback loop following a reference that rises at a steady rate. Kvel is the limit of s L(s) as s tends to zero, so the loop needs at least one integrator for the error to be finite.
Calculator
- Following lag, 1/Kvel0.1 s
- Kvel needed to halve the error, 2Kvel20 1/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The following error equals the reference rate times a fixed lag of 1/Kvel = 0.1 s. A target that moves at 2 units/s is therefore always 0.2 units behind where it is aimed.
- To halve the error without changing the target, Kvel must double to 20. A higher loop gain is limited by stability, which is why feedforward of the reference rate is often added to cut the lag.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (units) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Kvel Velocity error constant, lim sL(s) | 1 to 100 1/s | 0.02 | 2 | 1.98 | +10% | −9.09 % |
| rrate Rate of rise of the reference | 0.5 to 10 units/s | 0.05 | 1 | 0.95 | +10% | +10 % |
Sweep of Velocity error constant, lim sL(s)
| Velocity error constant, lim sL(s) (1/s) | Steady-state ramp error (units) |
|---|---|
| 1 | 2 |
| 13.38 | 0.1495 |
| 25.75 | 0.07767 |
| 38.13 | 0.05246 |
| 50.5 | 0.0396 |
| 62.88 | 0.03181 |
| 75.25 | 0.02658 |
| 87.63 | 0.02282 |
| 100 | 0.02 |
Sweep one variable
Variables and how each one contributes
Steady-state ramp error
What the equation gives you. Everything else feeds it.
Rate of rise of the reference
The rate at which the reference rises, in output units per second. Directly proportional: a faster ramp leaves a proportionally bigger lag. Raising it by 10 % (to 2.2) moves the result by +10 %, from 0.2 to 0.22 units.
Velocity error constant, lim sL(s)
The velocity error constant, in 1/s: the limit of sL(s) as s tends to zero. It is the loop gain of a type-1 loop. Inversely proportional: doubling the loop gain halves the following error. Raising it by 10 % (to 11) moves the result by −9.09 %, from 0.2 to 0.1818 units.
Worked example with the values above
- Rate of rise of the reference: 2 units per s.
- Error: ess = rate / Kvel = 2 / 10.
- ess = 0.2 units, equivalent to a time lag of 0.1 s.
Assumptions and limits
- A stable unity-feedback loop with at least one integrator, so that Kvel is finite and non-zero.
- A reference that rises at a constant rate for long enough for the transient to die out.
Common mistakes
- Using it with a type-0 loop (no integrator). Kvel = 0 and the error grows without bound.
- Mixing up Kvel, which includes the plant, with the derivative gain Kd of a PID controller.
Statistics and uncertainty
The error is inversely proportional to Kvel, so a 10 % error in the measured loop gain gives about a 10 % error in the predicted lag, in the opposite direction. Kvel is found from the low-frequency slope of the open-loop gain, so it is easy to overestimate when the Bode plot is read at frequencies where the loop gain has already started to roll off.
Where it is used
- Sizing a tracking servo so that its following error stays within a limit.
- Explaining why a moving target is always tracked with a lag and how feedforward removes it.
Maths you need
- System type: the number of integrators in the loop
- Final value theorem: steady error from the limit of sE(s)