Chapter 3: PID Tuning Methods

Why Tuning Matters

Poorly tuned PID controllers can cause:

• Slow response
• Excessive overshoot
• Oscillations
• Instability

Ziegler-Nichols Method

Step Response Method

1. Apply step input to open-loop system
2. Measure delay time (L) and time constant (T)
3. Calculate gains:

Controller	Kp	Ti	Td
P	T/L	-	-
PI	0.9T/L	L/0.3	-
PID	1.2T/L	2L	0.5L

Ultimate Gain Method

1. Set Ki = 0, Kd = 0
2. Increase Kp until sustained oscillation
3. Record Ku (ultimate gain) and Tu (period)

Controller	Kp	Ti	Td
P	0.5Ku	-	-
PI	0.45Ku	Tu/1.2	-
PID	0.6Ku	Tu/2	Tu/8

def ziegler_nichols_ultimate(Ku, Tu, controller_type='PID'):
    if controller_type == 'P':
        return {'Kp': 0.5 * Ku}
    elif controller_type == 'PI':
        return {'Kp': 0.45 * Ku, 'Ki': 0.45 * Ku / (Tu/1.2)}
    elif controller_type == 'PID':
        Kp = 0.6 * Ku
        Ti = Tu / 2
        Td = Tu / 8
        return {'Kp': Kp, 'Ki': Kp/Ti, 'Kd': Kp*Td}

Cohen-Coon Method

Better for systems with significant dead time.

def cohen_coon(K, T, L):
    r = L / T
    Kp = (1.35/K) * (T/L + 0.185)
    Ti = 2.5 * L * (T + 0.185*L) / (T + 0.611*L)
    Td = 0.37 * L * T / (T + 0.185*L)
    Ki = Kp / Ti
    Kd = Kp * Td
    return {'Kp': Kp, 'Ki': Ki, 'Kd': Kd}

Manual Tuning Guidelines

1. Start with Kp only, increase until oscillation
2. Add Ki to eliminate steady-state error
3. Add Kd to reduce overshoot
4. Fine-tune iteratively

Next: Chapter 4 - State-Space Representation!