SME
Process Dynamics & Control
Controller tuning
Use this for “A loop cycles steadily at Kᵤ = 8 with a period of 4 min. Find the Ziegler–Nichols PID settings.”
Table
Controller
Kc
τI
(min)
τD
(min)
P
4
—
—
PI
3.6
3.333
—
PID
4.8
2
0.5
Ultimate gain Kᵤ = 8
Ultimate period Pᵤ = 4 min
PID integral gain KI = 2.4 min⁻¹
This is an example. Type your own values to start a new problem.
Kc
=
0.6
×
Kᵤ
τI (min)
=
Pᵤ (min)
÷
2
τD (min)
=
Pᵤ (min)
÷
8
KI
PID integral gain
calculated
Kc,P
P-only gain
worked out (not typed)
Kc,PI
PI gain
worked out (not typed)
τI,PI
PI integral time
worked out (not typed)
Formulas
Kc = 0.6 × Kᵤ
4.8 = 0.6 × 8
τI
=
Pᵤ
2
2
=
4
2
τD
=
Pᵤ
8
0.5
=
4
8
KI
=
Kc
τI
2.4
=
4.8
2
Kc,P = 0.5 × Kᵤ
4 = 0.5 × 8
Kc,PI = 0.45 × Kᵤ
3.6 = 0.45 × 8
τI,PI
=
Pᵤ
1.2
3.333
=
4
1.2
Assumptions
•
Kᵤ and Pᵤ come from a closed-loop test: P control raised until the loop cycles steadily.
•
Ziegler–Nichols settings aim for a quarter decay ratio, so they are aggressive; detune them for a gentler loop.
•
The P row takes 0.5Kᵤ; the PI row takes 0.45Kᵤ with τI = Pᵤ ÷ 1.2.
Step-by-step
WE KNOW
Kᵤ = 8, Pᵤ = 4 min
FIND
PID gain (Kc), PID integral time (τI), PID derivative time (τD), PID integral gain (KI), P-only gain (Kc,P), PI gain (Kc,PI), PI integral time (τI,PI)
Step 1 · Find PID gain (Kc)
Use Kc = 0.6 × Kᵤ
PID carries 0.6 of the gain that just made the loop cycle; derivative lead lets it keep more.
Kc = 0.6 × 8
Kc = 4.8
Step 2 · Find PID integral time (τI)
Use τI = Pᵤ ÷ 2
The integral time is half the period the loop cycled at: slower cycling, slower reset.
τI
=
4
2
τI = 2 min
Step 3 · Find PID derivative time (τD)
Use τD = Pᵤ ÷ 8
The derivative time is an eighth of the period, a quarter of the integral time.
τD
=
4
8
τD = 0.5 min
Step 4 · Find PID integral gain (KI)
Use KI = Kc ÷ τI
The
controller
Kc(1
+
1
τI s
+
τD
s)
has
integral
gain
Kc
τI
,
the
reset
rate.
KI
=
4.8
2
KI = 2.4 min⁻¹
Step 5 · Find P-only gain (Kc,P)
Use Kc,P = 0.5 × Kᵤ
P alone takes half the ultimate gain, a gain margin of 2.
Kc,P = 0.5 × 8
Kc,P = 4
Step 6 · Find PI gain (Kc,PI)
Use Kc,PI = 0.45 × Kᵤ
Integral action adds phase lag, so PI backs the gain off below the P row.
Kc,PI = 0.45 × 8
Kc,PI = 3.6
Step 7 · Find PI integral time (τI,PI)
Use τI,PI = Pᵤ ÷ 1.2
Without
derivative
lead,
PI
resets
more
slowly
than
PID:
Pᵤ
1.2
,
not
Pᵤ
2
.
τI,PI
=
4
1.2
τI,PI = 3.333 min
Check
Put the numbers back in. Both sides should match.
4.8 = 0.6 × 8 ✓
2
=
4
2
✓
0.5
=
4
8
✓
2.4
=
4.8
2
✓
4 = 0.5 × 8 ✓
3.6 = 0.45 × 8 ✓
3.333
=
4
1.2
✓
Topic check
See what you know: 5 questions on this topic
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