Dead-Time Compensation
(纯滞后补偿)
Lei Xie
Institute of Industrial Control,
Zhejiang University, Hangzhou, P. R. China
Contents





Introduction
Smith Predictor for Dead-Time
Compensation
Improved Smith Predictor
Simulation Examples
Summary
Problem Discussion
(1) For the controlled processes, configure your
Simulink model & compare their results.
(2) Can you provide some compensation approaches for
processes with variable & notable dead-time ?
D (s)
Process
R (s)
+ _
Process
Models:
Gc(s)
Kpgp (s)
2.0  2s
G p (s) 
e ;
4s  1
e
 s
+
+
2.0 8s
G p ( s) 
e .
4s  1
Y (s)
Conventional PID Control Systems
D (s)
Process
R (s)
+ _
Process
Models:
Gc(s)
G p (s) 
Kpgp (s)
2.0  2s
e ;
4s  1
e
 s
G p ( s) 
+
+
Y (s)
2.0 8s
e .
4s  1
Question：Use Ziegler-Nichols or Lambda tuning method to obtain
PID parameters and compare their values.
Please see the SimuLink model …SISODelayPlant / PIDLoop.mdl
Simulation Example #1
Output of Transmitter
78
2.0  2s
G p (s) 
e ;
4s  1
76
74
72
For PID Controller,
%
Z-N tuning: Kc = 1.2,
Ti = 4 min, Td = 1 min
70
68
66
64
Lambda tuning:
Kc = 0.83, Ti = 4 min ,
Td = 1 min
62
setpoint
Ziegler-Nichols Tuning
Lambda Tuning
60
58
0
20
40
60
80
100
120
Time, min
140
160
180
200
Simulation Example #2
Output of Transmitter
80
2.0 8 s
G p (s) 
e ;
4s  1
78
76
74
For PID Controller,
72
Z-N tuning: Kc = 0.3,
Ti = 16 min, Td = 4 min
Lambda tuning:
%
70
68
66
64
set point
Z-N tuning
Lambda tuning, Td = 1 min
Lambda tuning, Td = 4 min
62
Kc = 0.2, Ti = 4 min ,
Td = 1 min
60
58
0
20
40
60
80
100
120
Time, min
140
160
180
200
Smith’s Idea (1957)
D (s)
R (s)
+ _
Process
+
Gc(s)
+
e  s
Kpgp (s)
Y (s)
D (s)
+
R (s)
+ _
Gc(s)
+
Process
Kpgp (s)
e
 s
Y (s)
Basic Smith Predictor
D (s)
R (s)
+ _
U(s)
Gc(s)
Process
+
+
Kpgp (s)
+
+
 m s
_
+
Y 1(s)
e  s
e
Km gm (s)
Y (s)
Y 2(s)
Smith Predictor
Smith Predictor #2
D (s)
U (s)
R (s)
Gc(s)
+ _
+
K m g m (s )
+
+
K p g p ( s )e
Y (s)
 p s
K m g m ( s )e  m s
_
+
+
Please see the SimuLink model …SISODelayPlant / PID_Smith.mdl
Results of Basic Smith Predictor
with an Accurate Model
Output of Transmitter
80
Gm ( s)  G p ( s)
78
2.0 8 s

e ;
4s  1
76
74
72
Simple PID:
%
70
Kc = 0.2, Ti = 4 min ,
Td = 1 min
68
66
64
PID + Smith:
Kc = 2, Ti = 4 min ,
Td = 1 min
set point
PID with Smith compensator
Simple PID
62
60
58
0
20
40
60
80
100
120
Time, min
140
160
180
200
Results of Basic Smith Predictor
with an Inaccurate Model
Output of Transmitter
2.0 8 s
Gm ( s ) 
e ;
4s  1
2.0 6 s
G p (s) 
e
4s  1
85
80
75
%
Simple PID:
Kc = 0.2, Ti = 4 min ,
Td = 1 min
PID + Smith:
Kc = 2, Ti = 4 min ,
Td = 1 min
70
65
set point
PID + Smith
Simple PID
60
55
0
20
40
60
80
100
120
Time, min
140
160
180
200
Improved Smith Predictor
D (s)
R (s)
U (s)
Gc(s)
+ _
+
+
+
K p g p ( s )e
K m g m ( s )e  m s
K m g m (s )
Y (s)
 p s
_
+
+
Gf(s)
1
Prediction Error Filter：G f (s) 
Tf s 1
Results of Improved Smith
Predictor with an Inaccurate Model
Output of Transmitter
80
2.0 8 s
e ;
4s  1
2.0 6 s
G p (s) 
e ;
4s  1
1
G f (s) 
4s  1
Gm ( s ) 
78
76
74
72
%
70
68
66
PID + Smith:
Kc = 2, Ti = 4 min ,
Td = 1 min
64
set point
PID + Smith with Gm =Gp
PID + Smith with Gm <> Gp
Simple PID
62
60
58
0
20
40
60
80
100
120
Time, min
140
160
180
200
Summary



The principle of Smith predictor for deadtime compensation
Improved Smith predictor for a controlled
process with an inaccurate model
Comparison of the Simple PID and the PID
with a Smith predictor
Next Topic: Coupling of Multivariable
Systems and Decoupling






Concept of Relative Gains
Calculation of Relative Gain Matrix
Rule of CVs and MVs Pairing
Linear Decoupler from Block Diagrams
Nonlinear Decoupler from Basic Principles
Application Examples
Problem Discussion
for Next Topic
For the two-input-two-input
controlled system, design
your control schemes.
Suppose that
F1SP
F1m
FC
01
F2SP
F2m
FC
02
FT
01
C1 F1  C2 F2
F  F1  F2 , C 
;
F1  F2
FT
02
F1, C1
Fm 0.5 C
Am
1
e 5 s

, 
,

;
F s  1 C 4s  1 C 2s  1
Initial states:
F10  75, F20  25, C1  60%, C2  40%.
F2, C2
u2
u1
ASP
AC
11
FC
03
Am
FSP
Fm
AT
11
FT
03
C
F