MEGR 3171  |  UNC Charlotte Mechatronics 2
Dr. Roger Tipton
Mechanical Engineering & Engineering Science
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Week 14 — Module 4: Feedback Control

PID Control Theory

Designing and tuning PID controllers using systematic methods, understanding the physical interpretation of each control action, and addressing practical implementation challenges.

Learning Objectives

1. PID Controller Structure

Continuous PID Controller
u(t) = K_p · e(t) + K_i · ∫e(t) dt + K_d · de(t)/dt Where e(t) = r(t) - y(t) (error = reference - output) Transfer function (ideal PID): C(s) = K_p + K_i/s + K_d·s = (K_d s² + K_p s + K_i) / s Parallel form parameters: K_p, K_i = K_p/T_i, K_d = K_p·T_d

Physical Interpretation of Each Action

Proportional (Kp)

Responds to current error. Reduces error but cannot eliminate steady-state error (offset) in Type 0 systems. Increasing Kp speeds response but increases overshoot and can destabilize.

Integral (Ki)

Responds to accumulated past error. Eliminates steady-state offset by adding an integrator (raises system type by 1). Can cause slow response and windup. Reduces phase margin.

Derivative (Kd)

Responds to rate of change of error (predictive). Reduces overshoot and improves transient response. Amplifies high-frequency noise — always use with a low-pass filter in practice.

2. Ziegler-Nichols Tuning

Method 1: Open-Loop Reaction Curve

Apply a step change to the plant input in open loop. Measure the S-shaped response curve. Fit a first-order plus dead-time (FOPDT) model: gain K, time constant T, dead time L.

Z-N Reaction Curve Formulas
Model: G(s) = K·e^(-Ls) / (Ts + 1) Extract from step response: K = Δy/Δu, L = apparent dead time, T = time constant Z-N Tuning: P only: K_p = T/(K·L) PI: K_p = 0.9T/(K·L), T_i = L/0.3 PID: K_p = 1.2T/(K·L), T_i = 2L, T_d = 0.5L

Method 2: Ultimate Sensitivity (Closed-Loop)

Increase Kp in proportional-only mode until sustained oscillations begin. Record the ultimate gain Ku and ultimate period Tu.

Z-N Ultimate Sensitivity Formulas
PID tuning: K_p = 0.6 · K_u T_i = T_u / 2 T_d = T_u / 8
Z-N Tuning Limitations Ziegler-Nichols rules are a starting point. They typically yield roughly 25% overshoot and are optimized for disturbance rejection rather than setpoint tracking. Manual fine-tuning is almost always required.

3. Anti-Windup and Derivative Filtering

Integrator Windup

When the actuator saturates (e.g., motor at max speed), the error persists and the integral term continues to grow (winds up). When the error finally reverses, the large integral must be "unwound" before the controller can respond, causing large overshoot. Anti-windup strategies:

Derivative Filtering

Ideal derivative D(s) = Kd·s has infinite gain at high frequency, amplifying sensor noise. A practical derivative uses a first-order filter: D(s) = Kd·s / (1 + s·Tf) where Tf = Td/N and N = 5 to 20.

Practice Problems

Problem 1 — Z-N Tuning A plant step response gives K = 2.5, T = 8 s, L = 1.5 s. Calculate PID gains using Ziegler-Nichols reaction curve rules.

K_p = 1.2T/(K·L) = 1.2×8/(2.5×1.5) = 9.6/3.75 = 2.56

T_i = 2L = 2×1.5 = 3.0 s → K_i = K_p/T_i = 2.56/3.0 = 0.853 s&sup-¹

T_d = 0.5L = 0.5×1.5 = 0.75 s → K_d = K_p×T_d = 2.56×0.75 = 1.92 s

K_p = 2.56, K_i = 0.853 s&sup-¹, K_d = 1.92 s
Problem 2 — Control Action Effects A temperature control system with P-only control has a steady-state error of 5°C. What change to the controller eliminates the steady-state error and what side effect must be managed?

Adding integral action (PI controller) eliminates steady-state offset by integrating the accumulated error until it reaches zero. The integral action raises the system type by 1, ensuring zero error for step reference inputs.

Add integral action (PI controller). Side effect: integrator windup must be managed with anti-windup logic; phase margin will decrease, potentially increasing overshoot.