Getting Started with PID Tuning

Introduction

PID (Proportional-Integral-Derivative) controllers are the workhorse of industrial process control. Despite decades of advances in advanced control, ~90% of industrial control loops still use some form of PID. This tutorial walks through three classic tuning methods using our interactive control_utils toolkit.

Prerequisites

• Basic understanding of first-order systems and time constants
• Familiarity with transfer function concepts (optional but helpful)
• Optionally: the PID Tuner web tool open in another tab to follow along visually

The Process Model: FOLPD

Before tuning any controller, we need a model of the process. The First-Order Lag Plus Deadtime (FOLPD) model is the industry standard:

$$G(s) = \frac{K \cdot e^{-Ls}}{Ts + 1}$$

Parameter	Meaning	How to Identify
$K$	Process gain	Steady-state output change per unit input change
$T$	Time constant	Time to reach ~63% of final value after delay
$L$	Dead time	Transport delay before response begins

From a step response plot:

from control_utils import FOLPD

# t and y are measured step response arrays
plant = FOLPD.from_step_response(t, y, method='tangent')
print(plant)  # G(s) = 2.0 * exp(-1.0s) / (5.0s + 1)

Method 1: Ziegler-Nichols Reaction Curve

The classic open-loop Z-N method uses the process reaction curve parameters $K$, $T$, and $L$ directly:

$$K_p = 1.2 \frac{T}{K \cdot L}, \quad K_i = \frac{K_p}{2L}, \quad K_d = K_p \cdot \frac{L}{2}$$

from control_utils import PIDGains, auto_tune_pid

gains = plant.ziegler_nichols_tuning('PID')
print(gains)  # PIDGains(kp=3.00, ki=1.50, kd=1.50)

Characteristics

• Fast response, ~25% overshoot typical
• Aggressive — good for setpoint tracking
• Can be oscillatory if $L/T$ is large (dead time dominant)

Method 2: Cohen-Coon Tuning

Cohen-Coon improves on Z-N for processes with significant dead time by accounting for the controllability ratio $r = L/T$:

$$K_p = \frac{1}{K}\left(1.33r + 0.167\right), \quad K_i = \frac{K_p}{2.67L + 0.5T}, \quad K_d = K_p \cdot \frac{L}{2}$$

gains = plant.coon_cohen_tuning('PID')
print(gains)

Characteristics

• Reduced overshoot compared to Z-N
• Better disturbance rejection
• More robust for $r > 0.5$ processes

Method 3: Internal Model Control (IMC)

IMC provides explicit robustness-performance trade-off through the closed-loop time constant $\tau_c$:

$$K_p = \frac{T + L/2}{K \cdot \tau_c}, \quad K_i = \frac{K_p}{T + L/2}, \quad K_d = \frac{T \cdot L}{2(T + L/2)}$$

gains = plant.imc_tuning(tau_c=1.0)  # conservative
print(gains)

Choosing $\tau_c$

$\tau_c$	Behavior	Best For
$\tau_c = L$	Aggressive	Fast servo tracking, low noise
$\tau_c = 2L$	Balanced	General purpose
$\tau_c = 5L$	Conservative	Noisy measurements, model uncertainty

Visual Comparison

Try each method in the PID Tuner with these typical values:

Method	Kp	Ki	Kd
Ziegler-Nichols	3.00	1.50	1.50
Cohen-Coon	2.63	0.75	1.32
IMC ($\tau_c=1$)	2.75	0.55	0.45

Observe: Z-N gives fastest rise but most overshoot. IMC is smoothest. Cohen-Coon balances both.

Summary Table

Method	Complexity	Overshoot	Robustness	Best Use Case
Z-N	Low	High	Low	Quick tuning, approximate
Cohen-Coon	Low	Medium	Medium	Dead-time dominant
IMC	Medium	Low	High	Robust performance, tunable

Next Steps

1. Try the interactive PID Tuner with your own process parameters
2. Read Process Identification to learn how to derive $K$, $T$, and $L$ from real plant data
3. Explore IMC Robustness Analysis for advanced tuning

References

1. Åström, K.J., & Hägglund, T. (2006). Advanced PID Control. ISA.
2. Seborg, D.E., Edgar, T.F., Mellichamp, D.A., & Doyle, F.J. (2011). Process Dynamics and Control (3rd ed.). Wiley.
3. Rivera, D.E., Morari, M., & Skogestad, S. (1986). Internal Model Control. Ind. Eng. Chem. Process Des. Dev., 25(1), 252–265.