Use this calculator to determine the output of a Proportional-Integral-Derivative (PID) controller for a single control loop iteration. Input the current system state, controller gains, and historical error values to see the calculated control signal.
Understanding PID Control
PID (Proportional-Integral-Derivative) controllers are widely used in industrial control systems to maintain a desired output from a process. They continuously calculate an "error" value as the difference between a desired setpoint (SP) and a measured process variable (PV). The controller then adjusts its output based on three terms: proportional, integral, and derivative, each responding to the error in a different way.
The Three Terms:
Proportional (P) Term: This term is directly proportional to the current error. A larger error results in a larger proportional response, providing a quick initial correction. The proportional gain (Kp) determines the strength of this response.
Integral (I) Term: This term accounts for past errors by summing them over time. It helps to eliminate steady-state errors (also known as offset) that the proportional term might leave. The integral gain (Ki) determines how quickly the controller responds to accumulated errors.
Derivative (D) Term: This term anticipates future errors by considering the rate of change of the current error. It helps to dampen oscillations, reduce overshoot, and improve system stability. The derivative gain (Kd) determines the strength of this anticipatory response.
How the Calculator Works:
This calculator computes the controller output for a single iteration based on the provided inputs. It simulates one step of a PID control loop:
Setpoint (SP): The target value you want the system to reach (e.g., 100°C, 50 PSI, 10 m/s).
Process Variable (PV): The current measured value of the system (e.g., 95°C, 48 PSI, 9.8 m/s).
Kp, Ki, Kd: The tuning parameters for each term. These values are crucial for optimal controller performance and are typically determined through tuning methods.
Time Step (dt): The time interval between control loop calculations (e.g., 0.1 seconds). This is important for the integral and derivative terms.
Previous Error (e_prev): The error from the immediately preceding time step (SP - PV_at_previous_step). This is used to calculate the rate of change of error for the derivative term.
Accumulated Integral Error (e_integral_current): The sum of (error * dt) from all previous time steps, up to the start of the current step. This represents the historical accumulation of error for the integral term.
The final PID output is the sum of these three terms, which then acts as the control signal to adjust the process (e.g., motor power, heater output, valve opening).
Example Calculation:
Let's use the default values provided in the calculator:
Setpoint (SP) = 100
Process Variable (PV) = 95
Proportional Gain (Kp) = 0.5
Integral Gain (Ki) = 0.1
Derivative Gain (Kd) = 0.2
Time Step (dt) = 0.1 seconds
Previous Error (e_prev) = 8
Accumulated Integral Error (e_integral_current) = 50