pid-controller

PID Controller Implementation

Overview

A PID (Proportional-Integral-Derivative) controller is a feedback control mechanism used in industrial control systems. It continuously calculates an error value and applies a correction based on proportional, integral, and derivative terms.

Control Law

output = Kp * error + Ki * integral(error) + Kd * derivative(error)

Where:

• error = setpoint - measured_value
• Kp = proportional gain (reacts to current error)
• Ki = integral gain (reacts to accumulated error)
• Kd = derivative gain (reacts to rate of change)

Discrete-Time Implementation

class PIDController:
    def __init__(self, kp, ki, kd, output_min=None, output_max=None):
        self.kp = kp
        self.ki = ki
        self.kd = kd
        self.output_min = output_min
        self.output_max = output_max
        self.integral = 0.0
        self.prev_error = 0.0

    def reset(self):
        """Clear controller state."""
        self.integral = 0.0
        self.prev_error = 0.0

    def compute(self, error, dt):
        """Compute control output given error and timestep."""
        # Proportional term
        p_term = self.kp * error

        # Integral term
        self.integral += error * dt
        i_term = self.ki * self.integral

        # Derivative term
        derivative = (error - self.prev_error) / dt if dt > 0 else 0.0
        d_term = self.kd * derivative
        self.prev_error = error

        # Total output
        output = p_term + i_term + d_term

        # Output clamping (optional)
        if self.output_min is not None:
            output = max(output, self.output_min)
        if self.output_max is not None:
            output = min(output, self.output_max)

        return output

Anti-Windup

Integral windup occurs when output saturates but integral keeps accumulating. Solutions:

1. Clamping: Limit integral term magnitude
2. Conditional Integration: Only integrate when not saturated
3. Back-calculation: Reduce integral when output is clamped

Tuning Guidelines

Manual Tuning:

1. Set Ki = Kd = 0
2. Increase Kp until acceptable response speed
3. Add Ki to eliminate steady-state error
4. Add Kd to reduce overshoot

Effect of Each Gain:

• Higher Kp -> faster response, more overshoot
• Higher Ki -> eliminates steady-state error, can cause oscillation
• Higher Kd -> reduces overshoot, sensitive to noise