PID Controller Theory

Introduction

The PID (Proportional-Integral-Derivative) controller is the most widely used feedback controller in process industries. It calculates an error value as the difference between a measured process variable and a desired setpoint, then applies a correction based on proportional, integral, and derivative terms.

PID Control Algorithm

The standard PID controller output is:

u(t) = K_p[e(t) + (1/T_i)∫e(τ)dτ + T_d(de/dt)]

Or in terms of separate gains:

u(t) = K_pe + K_i∫e·dt + K_d(de/dt)

Proportional (P)

Produces output proportional to current error. Higher K_p means faster response but can cause oscillation. Cannot eliminate steady-state error alone.

Integral (I)

Accumulates past errors to eliminate steady-state offset. Too much integral action causes overshoot and oscillation. K_i = K_p/T_i

Derivative (D)

Predicts future error based on rate of change. Improves stability and reduces overshoot. Sensitive to noise. K_d = K_p×T_d

FOPDT Process Model

Many processes can be approximated by a First Order Plus Dead Time (FOPDT) model:

G(s) = K × e^-θs / (τs + 1)

K = Process gain (output change / input change)
τ = Time constant (63.2% of final value)
θ = Dead time (delay before response begins)

The ratio θ/τ is critical for controllability. Processes with θ/τ > 1 are difficult to control.

Tuning Methods

Ziegler-Nichols (Open-Loop)

Classic method based on step response. Aggressive tuning with ~25% overshoot.

Controller	K_p	T_i	T_d
P	τ/(Kθ)	-	-
PI	0.9τ/(Kθ)	3.33θ	-
PID	1.2τ/(Kθ)	2θ	0.5θ

Cohen-Coon

Better for processes with larger dead time. Less aggressive than Z-N.

Controller	K_p	T_i	T_d
PI	(τ/Kθ)(0.9+θ/12τ)	θ(30+3θ/τ)/(9+20θ/τ)	-
PID	(τ/Kθ)(4/3+θ/4τ)	θ(32+6θ/τ)/(13+8θ/τ)	4θ/(11+2θ/τ)

IMC (Internal Model Control)

Model-based tuning with adjustable closed-loop time constant λ (aggressiveness).

Controller	K_p	T_i	T_d
PI	τ/(K(λ+θ))	τ	-
PID	(τ+θ/2)/(K(λ+θ))	τ+θ/2	τθ/(2τ+θ)

Recommended: λ = max(0.25τ, 0.2θ) for moderate response

Tyreus-Luyben

Conservative tuning for robust performance. Low overshoot.

Controller	K_p	T_i	T_d
PI	0.31τ/(Kθ)	2.2τ	-
PID	0.45τ/(Kθ)	2.2τ	τ/6.3

Performance Metrics

Rise Time:Time to go from 10% to 90% of final value

Settling Time:Time to stay within ±2% of final value

Overshoot:Peak value above setpoint as percentage

IAE:Integral of Absolute Error - overall performance measure

References

Seborg, D.E. "Process Dynamics and Control"
Åström, K.J. "PID Controllers: Theory, Design, and Tuning"
Marlin, T.E. "Process Control"