Lecture Notes on PID Control

Jul 27, 2024

Notes on PID Control Introduction

Overview

  • PID controllers are the most popular control scheme in history.
  • This lecture aims to introduce core concepts surrounding PID control in a brief and high-level manner.

Feedback Control Architecture

  • Standard feedback control architecture includes:
    • Plant (G): Produces output.
    • Reference Signal (R): Desired output signal.
    • Error Signal (E): Difference between output and reference (E = R - y).
    • Controller: Computes control signal based on error to adjust the plant.

Simple Control Law

  • Proportional Control (P):
    • Simplest form of control involves multiplying the error by a gain (K):
      • Control signal (U) = K * E
    • This is referred to as proportional control (P) in PID.

PID Control Components

  1. Proportional Control (P)
    • Contribution to control:
      • Uᵖ = Kᵗ * E
  2. Integral Control (I)
    • Computed by integrating the error over time:
      • Uⁱ = Kᵢ * ∫E dt
  3. Derivative Control (D)
    • Based on the rate of change of error:
      • Uᵈ = Kᵈ * dE/dt

Block Diagram Representation

  • The PID controller combines these three control signals to compute the total control signal (U).

Characteristics of PID Components

Proportional Control (P)

  • Instantaneously responds to error.
  • Drawback: may result in steady state error.

Integral Control (I)

  • Eliminates steady state error.
  • Tends to overshoot due to the accumulation of error over time.

Derivative Control (D)

  • Responds to changes in error, providing damping.
  • Does not affect steady state conditions, as it outputs zero when steady state is reached.

Interaction of PID Components

  • At steady state:
    • The control action comes mainly from the integrator.
    • Zero error means the proportional (P) and derivative (D) components contribute nothing:
      • Uᵖ = 0 and Uᵈ = 0.

System Behavior and Physical Demonstration

  • Demonstrated using a physical system with a weight and springs to visualize the impact of P, I, and D terms.
  • Increase of proportional gain (Kᵗ) decreases rise time but increases overshoot.
  • Integral gain (Kᵢ) eliminates steady state error but can increase overshoot and settling time.
  • Derivative gain (Kᵈ) decreases overshoot, maintains settling time, and typically has no effect on steady-state conditions.

Tuning PID Parameters

  • KP (proportional gain):

    • Increases rise time (decreases), increases overshoot, very small effect on settling time, decreases stability.

  • KI (integral gain):

    • Increases rise time (decreases), increases overshoot, eliminates steady state error, adds potential for instability.

  • KD (derivative gain):

    • Little effect on rise time, decreases overshoot, may slightly decrease settling time, no effect on steady state error.
    • Can improve stability if kept small.

Conclusion

  • PID controllers balance proportional, integral, and derivative components to optimize control signal output.
  • Future lectures will further explore various aspects of PID control and tuning methods (e.g. root locus).

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