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Formula & Calculator

PID Controller Output

Combines proportional, integral, and derivative actions to compute a controller's corrective output.

Written by the Vorzep Team

Control SystemsPID Control

PID Controller Output Calculator u(t) = Kp·e + Ki∫e dt + Kd·(de/dt)

u(t) = Kp·e + Ki∫e dt + Kd·(de/dt)
u(t) = controller output  ·  Kp, Ki, Kd = gains  ·  e = error, ∫e dt = integral of error, de/dt = derivative of error
⟹ Solve ⟷ u(t), Kp, Ki, Kd
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Please fix the errors above.
Solve for:
Presets:
Output u(t) —
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u(t): — Kp: — Ki: — Kd: — e: — ∫e dt: — de/dt: —
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Output Magnitude —
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u(t) = Kp·e + Ki·∫e dt + Kd·(de/dt)  ·  The PID controller output is the sum of proportional, integral, and derivative terms.
u(t) = K_p·e + K_i∫e dt + K_d·(de/dt)
PID Controller Output

Variables

SymbolQuantityUnit
u(t)Controller outputV (or variable)
K_pProportional gaindimensionless
eError signalV (or variable)
K_iIntegral gain1/s
∫e dtIntegral of errorV·s
K_dDerivative gains
de/dtDerivative of errorV/s

What it means

The PID (Proportional‑Integral‑Derivative) controller output is given by u(t) = K_p·e(t) + K_i·∫e(t)dt + K_d·(de/dt). The proportional term gives an output proportional to the current error; the integral term eliminates steady‑state error by summing past errors; the derivative term anticipates future error by responding to the rate of change. PID controllers are used in industrial automation, robotics, and process control. Tuning the gains (K_p, K_i, K_d) is essential for achieving desired performance (stability, overshoot, settling time). Example: In a temperature control system, the PID output adjusts the heater power. If the temperature is below setpoint, the P term increases power, the I term eliminates the residual error, and the D term dampens oscillations.

Worked example

PID Controller Output – Practical Example

Real‑World
Scenario: A temperature controller has Kp = 2, Ki = 0.5, Kd = 0.1. The error is 5 °C, integral of error is 10 °C·s, derivative is 2 °C/s. Calculate the PID output.
ParameterValue
e5
∫e10
de/dt2
Kp, Ki, Kd2, 0.5, 0.1
Formulau = Kp·e + Ki·∫e + Kd·(de/dt)
1P = 2×5 = 10
2I = 0.5×10 = 5
3D = 0.1×2 = 0.2
4u = 10 + 5 + 0.2 = 15.2
Final Design u = 15.2 ✓ PID output
Why: PID combines proportional, integral, and derivative actions to achieve fast, accurate, and stable control.

Common mistakes

Watch unit consistency and the assumptions behind the formula; misapplying it outside its valid conditions is the most frequent error.

Applications

PID controller output u(t) = K_p·e + K_i∫e dt + K_d·(de/dt) combines proportional, integral, and derivative actions to control a system. It is the most widespread control algorithm, used in industrial process control, robotics, and automation. Engineers tune the gains to achieve desired performance. Understanding PID control is essential for any control engineer.

  • Industrial process control (temperature, pressure, flow)
  • Robotics and motion control
  • Automotive cruise control and engine management
  • HVAC and building automation
  • Educational foundation of control theory