SME
Control Systems · Problem type
Ziegler–Nichols tuning
Use this for “A loop oscillates steadily at Kᵤ = 48 with period Tᵤ = 2.22 s. Find the Ziegler–Nichols PID gains.”
Chart
A steady oscillation at ωₙ = 2.83 rad/s: with ζ = 0 nothing takes energy out, so the swing never shrinks
This is an example. Type your own values to start a new problem.
Kₚ
=
0.6
×
Kᵤ
Tᵢ (s)
=
Tᵤ (s)
2
Td (s)
=
Tᵤ (s)
8
ωᵤ
Ultimate frequency
worked out (not typed) · rad/s
Kᵢ
Integral gain
calculated · s⁻¹
Kd
Derivative gain
calculated
Formulas
ωᵤ
=
2π
Tᵤ
2.8303
=
2π
2.22
Kₚ = 0.6 × Kᵤ
28.8 = 0.6 × 48
Tᵢ
=
Tᵤ
2
1.11
=
2.22
2
Td
=
Tᵤ
8
0.2775
=
2.22
8
Kᵢ
=
Kₚ
Tᵢ
25.9459
=
28.8
1.11
Kd = Kₚ × Td
7.992 = 28.8 × 0.2775
Assumptions
•
Under P control alone, raising the gain to the ultimate gain Kᵤ makes the loop oscillate steadily with period Tᵤ.
•
The classic PID rules: Kₚ = 0.6Kᵤ, integral time Tᵢ = Tᵤ ÷ 2, derivative time Td = Tᵤ ÷ 8.
•
The controller is Kₚ + Kᵢ ÷ s + Kd s, so Kᵢ = Kₚ ÷ Tᵢ and Kd = Kₚ × Td.
•
The rules aim at a quarter-decay response, so expect overshoot and fine-tune afterward.
Step-by-step
WE KNOW
Kᵤ = 48, Tᵤ = 2.22 s
FIND
ultimate frequency (ωᵤ), proportional gain (Kₚ), integral time (Tᵢ), derivative time (Td), integral gain (Kᵢ), derivative gain (Kd)
Step 1 · Find ultimate frequency (ωᵤ)
Use ωᵤ = 2π ÷ Tᵤ
One
full
oscillation
every
Tᵤ
seconds
is
2π
Tᵤ
radians
a
second.
ωᵤ
=
2π
2.22
ωᵤ = 2.8303 rad/s
Step 2 · Find proportional gain (Kₚ)
Use Kₚ = 0.6 × Kᵤ
Ziegler and Nichols set the PID gain at 0.6 of the ultimate gain.
Kₚ = 0.6 × 48
Kₚ = 28.8
Step 3 · Find integral time (Tᵢ)
Use Tᵢ = Tᵤ ÷ 2
The integral time is half the ultimate period.
Tᵢ
=
2.22
2
Tᵢ = 1.11 s
Step 4 · Find derivative time (Td)
Use Td = Tᵤ ÷ 8
The derivative time is an eighth of the ultimate period.
Td
=
2.22
8
Td = 0.2775 s
Step 5 · Find integral gain (Kᵢ)
Use Kᵢ = Kₚ ÷ Tᵢ
The
integral
term
Kₚ
Tᵢ s
has
gain
Kₚ
Tᵢ
.
Kᵢ
=
28.8
1.11
Kᵢ = 25.9459 s⁻¹
Step 6 · Find derivative gain (Kd)
Use Kd = Kₚ × Td
The derivative term Kₚ Td s has gain Kₚ × Td.
Kd = 28.8 × 0.2775
Kd = 7.992 s
Check
Put the numbers back in. Both sides should match.
2.8303
=
2π
2.22
✓
28.8 = 0.6 × 48 ✓
1.11
=
2.22
2
✓
0.2775
=
2.22
8
✓
25.9459
=
28.8
1.11
✓
7.992 = 28.8 × 0.2775 ✓
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