Library / Control engineering / CTL-01
Control engineering
Proportional feedback on a first-order plant
Closed-loop time constant when a first-order plant b/(s + a) is controlled by a pure proportional gain with negative unity feedback.
- Ref
- CTL-01
- Rev
- v1
- Field
- Control engineering
- Result
- τcl [s]
The plant is b/(s + a), so on its own it has a time constant 1/a. Proportional feedback with gain Kc moves its pole from −a to −(a + Kcb), which makes the loop faster by the factor (a + Kcb)/a. The closed-loop DC gain is Kcb/(a + Kcb), which is always below 1, so with this controller the output never quite reaches a constant reference.
Closed-loop time constant when a first-order plant b/(s + a) is controlled by a pure proportional gain with negative unity feedback. A larger gain speeds the response up, but it leaves a steady-state error.
Calculator
- Closed-loop pole, −(a + Kcb)−5 1/s
- Closed-loop DC gain, Kcb / (a + Kcb)0.6
- Steady-state error to a unit step, 1 − DC gain0.4
- Open-loop time constant, 1/a0.5 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.
- At the defaults the loop has a time constant of 0.2 s, against 0.5 s for the plant alone: feedback made it 2.5 times faster. The price is that the output reaches only 0.6 of a unit step, a steady error of 40 %.
- Doubling the gain to 6 shortens the time constant to 0.125 s and cuts the steady error to 25 %, but it never reaches zero. Removing the error entirely needs integral action.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Kc Proportional controller gain | 0.2 to 20 | 0.04545 | 0.4545 | 0.4091 | +10% | −5.66 % |
| b Plant gain numerator b (plant is b/(s + a)) | 0.2 to 5 1/s | 0.05882 | 0.3846 | 0.3258 | +10% | −5.66 % |
| a Plant pole a | 0.2 to 5 1/s | 0.125 | 0.3125 | 0.1875 | +10% | −3.85 % |
Sweep of Proportional controller gain
| Proportional controller gain | Closed-loop time constant (s) |
|---|---|
| 0.2 | 0.4545 |
| 2.675 | 0.2139 |
| 5.15 | 0.1399 |
| 7.625 | 0.1039 |
| 10.1 | 0.08264 |
| 12.57 | 0.06861 |
| 15.05 | 0.05865 |
| 17.52 | 0.05122 |
| 20 | 0.04545 |
Sweep one variable
Variables and how each one contributes
Closed-loop time constant
What the equation gives you. Everything else feeds it.
Plant gain numerator b (plant is b/(s + a))
The numerator of the plant b/(s + a), in 1/s. The plant's own DC gain is b/a. Raising b strengthens the loop in the same way as raising Kc, because only the product Kcb enters the closed-loop pole. Raising it by 10 % (to 1.1) moves the result by −5.66 %, from 0.2 to 0.1887 s.
Plant pole a
The pole of the plant, in 1/s: left alone the plant settles as e−at. A larger a means a faster plant. A larger a makes the loop faster, but the effect is smaller than the change in a itself because feedback already contributes part of the pole. Raising it by 10 % (to 2.2) moves the result by −3.85 %, from 0.2 to 0.1923 s.
Proportional controller gain
The proportional controller gain: the control signal is Kc times the error. It is dimensionless when the plant gain b/a is dimensionless. A higher gain pushes the pole further left and shortens the time constant, but the benefit fades as Kcb grows larger than a. Raising it by 10 % (to 3.3) moves the result by −5.66 %, from 0.2 to 0.1887 s.
Worked example with the values above
- Closed-loop pole: a + Kcb = 2 + 3 × 1 = 5 1/s, so the pole is at s = −5.
- Time constant: τcl = 1 / 5 = 0.2 s.
- Final output for a unit step: Kcb / (a + Kcb) = 3 / 5 = 0.6. The time constant is 0.2 s.
Assumptions and limits
- A linear first-order plant with no time delay, no actuator limit and an ideal sensor.
- Negative unity feedback with a purely proportional controller (no integral or derivative action).
Common mistakes
- Treating the loop as accurate because it is fast. With proportional action alone the output stops short of the reference by 1/(1 + Kcb/a) of the step.
- Raising Kc without limit. Real actuators saturate, sensors add noise and unmodelled lag makes the loop unstable at high gain, so the first-order picture stops holding.
Statistics and uncertainty
The time constant is 1/(a + Kcb), so a relative error in the sum maps into τcl with the opposite sign. At the defaults the plant pole supplies 40 % of the sum and feedback supplies the rest. The higher the gain, the less the response depends on uncertainty in a, and the more it depends on Kcb, the quantity a designer controls.
Where it is used
- Estimating how fast a speed, temperature or level loop responds after a gain change.
- Showing why integral action is added: it removes the finite steady-state error this formula leaves.
Maths you need
- Poles and time constants: a pole at −σ gives a decay e^(−σt), time constant 1/σ
- Negative feedback: closed loop T = L / (1 + L)