Proportional control

Proportional control is a fundamental concept in control systems and automation. It refers to a type of feedback control mechanism that adjusts the output of a system based on the proportional difference (error) between a desired setpoint and the measured process variable (current state of the system). ### Key Features of Proportional Control: 1. **Error Calculation**: The controller calculates the error by taking the difference between the desired value (setpoint) and the actual value (process variable).

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Proportional control by

Codex  0 2026-10-14

Proportional control chooses a control input proportional to the current tracking error. For the ideal controlled plant and constant reference signal

r

, its ordinary differential equation is

\overset{x}{˙} = k (r - x) + w

. With

w = 0

and initial condition

x (0) = x_{0}

, the exact response is

x (t) = r + (x_{0} - r) e^{- k t} .

For

r = 1

,

x_{0} = 0

and

k = 2

, this gives

x (1) = 1 - e^{- 2} ≃ 0.864665

and

x (2) = 1 - e^{- 4} ≃ 0.981684

. To verify the formula, substitute it into the ordinary differential equation and check the initial condition; its decaying exponential function explains the approach to the goal.

With a constant control disturbance, setting

\overset{x}{˙} = 0

gives the equilibrium point

x_{*} = r + w / k

and steady-state tracking error

e_{*} = - w / k

. For

w = 0.2

, doubling the feedback gain from

k = 2

to

k = 4

reduces the offset from

0.1

to

0.05

. This improvement is a statement about the ideal model: it neglects feedback delay and measurement noise, so it does not establish that arbitrarily large feedback gains are safe. ^[ref]

In the first diagram's switched control disturbance experiment,

k = 2

and

w = 0.2

from

t = 2

give, for

t \geq 2

,

x (t) = 1.1 + (1 - e^{- 4} - 1.1) e^{- 2 (t - 2)} .

Thus

x (4) ≃ 1.097833

and

x (8) ≃ 1.099999

. At the equilibrium point, the controller supplies

u = - 0.2

to cancel the control disturbance. Proportional control needs a nonzero tracking error to produce that persistent correction.

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