Introduction
Dynamical System
PID control
PID control - Simple tuning methods Ulf Holmberg
Experiment
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
Tank process
• Tank with level control • Two connected tanks • Pump for in-flow of water • Level measurements • Valve for out-flow (disturbance)
Experiment
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
Experiment
Open-loop system System u
Actuator
Control object
Sensor
y
• u control signal (pump voltage) • Actuator (tubes to pump, power amplifier , in-flow) • Control object (level in Tank with in- and out-flow) • Sensors (pressure sensors, tubes, elektronics) • y output (measurement of water level)
Introduction
Dynamical System
PID control
Experiment
Closed-loop system
r
+
e −
Controller
u
System
y
Give reference (set-point of water level in tank) r to controller in stead of pump voltage
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
Experiment
Time constant and stationary gain u(t)
y (t) Kp
Step
Step response
1 t
u
System
y
T
Example–Stove plate • u(t) power to stove plate • y (t) temperature on plate
• Time constant T • Stationary gain Kp
t
Introduction
Dynamical System
PID control
Experiment
Time constant and stationary gain u(t)
y (t) Kp ∆u
Step
Step response
∆u t
u
System
y
• Step response size proportional to step size • Start step from equilibrium
T
t
Introduction
Dynamical System
PID control
Experiment
Dead-time (time delay) u(t)
y (t) Step response
Kp
Step 1 t
u
System
y
t L T
Example – roll transport time
Dead-time (time delay, lag) L
Introduction
Dynamical System
PID control
Experiment
Step response model u(t)
y (t) Step response
Kp
Step 1 t
u
System
y
t A L T
Step response model from unit step: • T Time constant • Kp Stationary gain • L Dead-time • A see measure above
Introduction
Dynamical System
PID control
Experiment
Step response model u(t)
y (t) Step response
Kp ∆u
Step ∆u t
u
System
y
t A∆u L T
Step response model from experiment: • T Time constant • Kp Stationary gain • L Dead-time • A see figure
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
Experiment
Self-oscillation model
r
+
e −
Ku
u
System
Experiment • P-control (closed-loop system!) • Crank up gain to self-oscillation • Reference-step starts oscillation
Self-oscillation model • Ultimate gain Ku • Ultimate period Tu
y Tu
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
Experiment
PID structure r
+
e −
Controller
u
y
System
• r reference, set-point (SP) • y output, measured signal to be regulated • e = r − y control error
PID-controller: P e
I D
+
u
Introduction
Dynamical System
PID control
Experiment
P-controller
e(t)
u(t) Ke0 e0 t e
K
u
Control signal Proportional to control error
t
Introduction
Dynamical System
PID control
Experiment
P-controller with offset
e(t)
u(t)
Ke0
u0 e0
u0 Offset t e
K
+
u
Control signal Proportional to error plus ’offset’
Example: speed control of car u0 given gas at control-start
t
Introduction
Dynamical System
PID control
Experiment
I-part (integrator) e(t)
I-del
e0
e0 t
e
1 Ti
R
e
I-del
I-part • ∝ surface under e(t) so far • becomes as big as constant error e0 in time Ti • used to eliminate remaining error
Ti
t
Introduction
Dynamical System
PID control
Experiment
D-part (derivator) e(t)
D-del
t e
D-part • ∝ slope on e(t) • used to damp oscillations
Td de dt
D-del
t
Introduction
Dynamical System
PID control
PID-controller structure
Control signal = P + I + D 1 u(t) = K [e(t) + Ti
Z
t
e(s)ds + Td
Three parameters to tune K , Ti and Td
de(t) ] dt
Experiment
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
Ziegler-Nichols step response method
• Measure A and L from step response • Tp expected time constant for closed-loop system
Controller P PI PID
K 1/A 0.9/A 1.2/A
Ti
Td
3L 2L
L/2
Tp 4L 5.7L 3.4L
Experiment
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
Experiment
Ziegler-Nichols self-oscillation method
• Measure ultimate gain Ku and period Tu from experiment • Tp expected time constant for closed-loop system
Controller P PI PID
K Ti Td Tp 0.5Ku Tu 0.4Ku 0.8Tu 1.4Tu 0.6Ku 0.5Tu 0.125Tu 0.85Tu
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
Experiment
Step response Step response model for the left tank 10
5.4
9
5.3
8
5.2
Estimated from figure:
5.1
7
• A∆u = 0.1, ∆u = 0.5
5 6
⇒ A = 1/5
4.9 5 4.8
• L=3
4 4.7 3
4.6
2
4.5 0
100
200
300
20
30
40
50
60
Introduction
Dynamical System
PID control
Experiment
P-control of the left tank
Valve opens/closes r, y 3
Ziegler-Nichols P-control
2.5
• K = 1/A = 5 0
20
40
60
80
100
120
140
160
180
• Stationary error after valve
u
opening even if offset is used
4
2
0 0
20
40
60
80
100
120
140
160
180
Introduction
Dynamical System
PID control
Experiment
PI-control of the left tank
Valve opens/closes r, y
Ziegler-Nichols PI-control
3
• K = 0.9/A = 4.5
2.5 0
50
100
150
200
u
4
• Ti = 3L = 9 • No stationary error after valve
disturbance 2
0 0
50
100
150
200
Introduction
Dynamical System
PID control
Experiment
PID-control of the left tank
Valve opens/closes
Ziegler-Nichols PID-control
r, y 3.2
• K = 1.2/A = 6
3
• Ti = 2L = 6
2.8 2.6
• Td = L/2 = 1.5
2.4 0
50
100
150
200
• No stationary error after valve
u
disturbance • Fast and damped
5
• Noisier control signal 0
50
100
150
200
Introduction
Dynamical System
PID control
Experiment
Self-oscillation experiment Self-oscillation model for the left tank r (bˆ¶rvˆ⁄rde) och y (ˆ⁄rvˆ⁄rde)
Ultimate gain and period
3 2
• Ku = 15 (from tuning)
1
• Tu = 10 (from figure)
0 0
20
40
60
80
100
Z-N PID-tuning:
u (styrsignal) 10
K = 0.6Ku = 8 Ti = 0.5Tu = 5 Td = 0.125Tu = 1.25
5
0 0
20
40
60
80
100
120
Introduction
Dynamical System
PID control
Experiment
Ziegler-Nichols PID from self-oscillation model t, y
5
0 0
50
100
150
50
100
150
200
250
u 10
5
0 0
200
250
Introduction
Dynamical System
PID control
PID control Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID control PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks
Experiment
Introduction
Dynamical System
PID control
Experiment
Step response Step response model for the right tank 4.7 8
Roughly estimated (bad precision!):
4.6 7
4.5
6
• A∆u ≈ 0.2, ∆u = 0.5
4.4
⇒ A = 2/5
4.3
5
• L ≈ 10
4.2 4
From larger figure: A∆u = 0.15, L = 8
4.1 3
4
2
3.9 0
100
200
300
400
30
40
50
60
70
Introduction
Dynamical System
PID control
Experiment
PI-control of the right tank
Valve opens/closes r, y
Ziegler-Nichols PI-control
3
• K = 0.9/A = 2.25
2 0
100
200
300
u
Disturbance eliminated in 60s (Compare Tp = 5.7L = 57s)
3
2
1
0 0
• Ti = 3L = 30
100
200
300
Introduction
Dynamical System
PID control
Experiment
PID-control of the right tank (step response tuning)
Valve opens/closes r, y 6
Ziegler-Nichols PID-control
5
• K = 1.2/A = 3
4
• Ti = 2L = 20
3 0
100
200
300
u 10
• Td = L/2 = 5
Disturbance eliminated i 40s (Compare Tp = 3.4L = 34s)
5
0 0
100
200
300
Introduction
Dynamical System
PID control
Experiment
Self-oscillation experiment Self-oscillation model for the right tank y 4.9
Ultimate gain and period
4.8
• Ku = 10 (from tuning) • Tu = 25 (from figure)
4.7 0
20
40
60
80
100
u
Z-N PID-tuning:
3
K = 0.6Ku = 6 Ti = 0.5Tu ≈ 13 Td = 0.125Tu ≈ 3
2
1
0 0
20
40
60
80
100
Introduction
Dynamical System
PID control
Experiment
PID-control of the right tank (self-oscillation tuning) r, y 5
Valve opens/closes 4
3 0
50
100
150
200
250
50
100
150
200
250
u 10
5
0 0
