Institute of Advanced Control Technology 1Institute of Advanced Control Technology
IFAC PID 2018, May 9-11, 2018, Ghent, Belgium
New PID designs for sampling control and
batch process optimization
Tao Liu
Institute of Advanced Control Technology
Dalian University of Technology
Institute of Advanced Control Technology 2
Outline
3
1
2
4
Introduction
PID design for sampled control systems
Learning PID design for batch optimization
Conclusions & Outlook
Introduction
[1] Karl J. Åström, P.R. Kumar. Control: A perspective. Automatica, 2014, 50(1), 3-43.
[2] Tao Liu, Furong Gao. Industrial Process Identification and Control Design: Step-test and Relay-
Experiment-Based Methods. London UK: Springer, 2012.
Some facts of PID controllers
Advantages:
➢ Simple structure and low cost;
➢ Easily understood and commanded
by users;
➢ Most widely used and commercialized
in industrial applications.
Disadvantages [1, 2]:
Generally not optimal in control performance;
(often used as an inferior to demonstrate other advanced control designs)
Difficult to analyze the robust stability against system uncertainties.
(lowly valued for theoretical contribution in top control-relevant journals)
PID Controller: >1000 papers in 2017
PI Controller: >400 papers in 2017
Searching results by ISI web of Science
Steadily increasing over the past 30 years!
Trends: 1. More digital implementation in computer systems;
2. Performance improvement for batch/repetitive processes.
More applications
for various systems
Motivation of this talk
Relationship between the PID control and the internal model control (IMC)
C G+ + + y
d
e
i
doysp u
yG
-
-
Gd
od 1
The internal model control (IMC) structure
The unity feedback control structure
Equivalent relationship
1
CK
GC
IMC: applicable for stable processes
K G+ + + y
d
e
i
-
doysp u
Gd
od 1
Process model:( )
( )
dB zG z
A z
PID: applicable for stable, integrating,
and unstable processes
( )
( )
sB sG e
A s
Frequency domain:
PID design for sampled control systems
PID
where , , .0 0 1 0 2 0
Review of the IMC design in frequency domain
Step 1. Decompose the model into the minimum-phase (MP), non-MP, and all-pass parts.
p 0
1 2
( 1)
( 1)( 1)
sk s
G es s
For example:
p
mp
1 2( 1)( 1)
kG
s s
ap
sG e
nmp 0 1G s
Step 2. Specify the desired closed-loop transfer function for set-point tracking.
1 2 nmp apT F F G G
where and are two low-pass filters.2F1F
For a stable process described as above,
1 3
1
( 1)F
s
1 mpFG
mp nmp apG G G G
strictly properensure
2 nmpF Gensure2
0
1
1F
s
all-pass, i.e.
020
0
1
1
sse
s
0
3
0
1 1
( 1) 1
ssT e
s s
Desired closed-loop transfer function
single tuning parameter
No overshoot
Minimal ISE
where , , , .1 1a 2 1a 2 1b
Discrete-time domain IMC design
Step 1. Decompose the model into the minimum-phase (MP), non-MP, and all-pass parts.
p 1 2
1 2
( )( )
( )( )
dk z b z b
G zz a z a
For example:
p 1
mp
1 2
( )
( )( )
k z bG
z a z a
ap
dG z
nmp 2G z b
Step 2. Specify the desired closed-loop transfer function for set-point tracking.
1 2 nmp apT F F G G
where and are two low-pass filters.2F1F
For a stable process described as above,
2
c1 2
c
(1 )
( )F
z
1 mpFG
mp nmp apG G G G
strictly properensure
2 nmpF Gensure1
22 1
2 2
1
(1 )( )
bF
b z b
all-pass, i.e.
1
2 2
1
2 2
(1 )( )
(1 )( )
b z b
b z b
2 1
c 2 2
2 1
c 2 2
(1 ) (1 )( )
( ) (1 )( )
db z bT z
z b z b
Desired closed-loop transfer function
1 1b
Discrete-time domain IMC design
In case , it could provoke inter-sample ringing in the control signal !
For a stable process described as above,
Step 3. Determine the IMC controller.
IMC
( )( )
( )
T zC z
G z
2 1
1 2 c 2IMC 2 1
p 1 c 2 2
( )( )(1 ) (1 )( )
( )( ) (1 )( )
z a z a bC z
k z b z b z b
T GC
11 0b
0 50 100 150 200 250 300 350 400 450 5000
1
2
3
4
5
6
Time (sec)
Con
trol
Sig
nal
single tuning parameter c
Discrete-time domain IMC design
1 13
1
( )1
z bF z z
b
Solution: Introduce another filter to remove such a zero for implementation
RIMC 3 IMC( ) ( ) ( )C z F z C z
c
c
1( ) ( )dY z z R z
z
s -c ,
0, ,( )
1 .k d
k dy kT
k d
Step response in time domain
c c cr s c
1 1 c c
(1 )[1 ( )] lim
1 1
nk d
nk d k d
IAE y kT
cd
c
1( )T z
z
( )1
zR z
z
p1
1
p
( ) dK
G z zz z
p1
d s
1 c p
( )(1 )(1 )k d
KIAE y kT
z
For a stable process described by
Control performance assessment
Set-point tracking error
Disturbance rejection error
to a step changeQuantitative tuning
0lim ( )s
K s
1
CK
GC
1 2
3
p 0 0
( 1)( 1)
[( 1) ( 1) ( 1) ]s
s sK
k s s s e
( )M
K ss
21 (0)( ) [ (0) (0) ]
2!
MK s M M s s
s
DC
I F
1( )
1
sK s k
s s
F D(0.01 0.1) C
I
D
(0)
1/ (0)
(0) / 2
k M
M
M
Step 4. Determine the equivalent controller in a unity feedback control structure.
IMC-based PID design in frequency domain
p 0
1 2
( 1)
( 1)( 1)
sk s
G es s
For
Step 5. Determine the PID controller by the Taylor approximation [1].
Since , let
[1] Lee, Y., Park, S., Lee, M., & Brosilow, C. PID controller tuning for desired closed-loop responses for
SI/SO systems. AIChE Journal, 44, 106-115, 1998.
Important advantage:
single tuning parameter
Alternatively, the delay term in the process model or the IMC controller may be rationally
approximated to derive a PID, e.g.,
IMC-based PID design in frequency domain
The first-order Taylor approximation [2]:
[2] Rivera, D. E., Morari, M., Skogestad, S. Internal model control. 4. PID controller design. Ind. Eng.
Chem. Res., 25, 252-265, 1986.
[3] Fruehauf P. S., Chien I.-L., Lauritsen M. D. Simplified IMC-PID tuning rules. ISA Transactions, 1994,
33, 43-59.
[4] Skogestad, S. Simple analytical rules for model reduction and PID controller tuning. Journal of
Process Control, 2003, 13, 291-309.
[5] Lee, J. Cho, W. Edgar, T. F. Simple analytic PID controller tuning rules revisited, Industrial &
Engineering Chemistry Research, 53(13), 5038-5047, 2014.
[6] Wang, Q. G., Yang, X. P. Single-loop controller design via IMC principles. Automatica, 37, 2041-
2048, 2001.
Pade approximation [3]:
1 se s
(1 ) / (1 )2 2
se s s
Model reduction [4, 5]:
2
2
21 21
1 1
( 1)( 1) ( ) 12
s
es s s
Frequency response fitting [6]:PID
1
min (j ) (j )m
i i
i
K K
cb(0.1, 1)i
‘Half rule’
1lim ( )z
K z
1
CK
GC
21 (1)( ) (1) (1)( 1) ( 1) ...
1 2!
MK z M M z z
z
DPID c D
I D
1 ( 1)( ) 1 (1 )
( 1)
zK z k
z z
(0.01, 0.1)
C
I
D
(1)
(1) / (1)
(1) / 2 (1)
k M
M M
M M
For sampling control implementation, the first-order differentiation is generally used to
discretize the above frequency domain design.
IMC-based PID design in discrete-time domain
A PID controller is determined by the Taylor approximation [1, 2].
owing to .
[1] Wang, D., Liu T.*, Sun X., Zhong Chongquan. Discrete-time domain two-degree-of-freedom control
for integrating and unstable processes with time delay. ISA Transactions, 63, 121-132, 2016.
[2] Cui J., Chen Y., Liu T.*. Discrete-time domain IMC-based PID control design for industrial processes
with time delay. The 35th Chinese Control Conference (CCC), Chengdu, China, 5946-5951, 2016.
( )( )
1
M zK z
z
For direct PID design in discrete-time domain, let the equivalent controller in a unity
feedback control structure be
single tuning parameter
IMC-based PID design in discrete-time domain
Closed-loop system robust stability analysis
m( ) ( ) 1T z z
p PID
p PID
( ) ( )( )
1 ( ) ( )
G z C zT z
G z C z
m m m( ) [ ( ) ( )] ( )z G z G z G z
T
+
+
T The structure
Small gain theorem
p
1( ) dk
G z zz
p dd
i d
p mdd
i d
( 1)11 (1 )
( 1) 1
( )( 1)11 1 (1 )
( 1)
d
d
k T zz K T
z T z z T
k zT zz K T
z T z z T
For a stable process described byNo analytical solution!
Magnitude/Phase margin, Maximal sensitivity index sM
IMC-based PID design in discrete-time domain
Disturbance rejection performance
Tuning guideline: The single adjustable parameter can be monotonically
increased or decreased in a range of to meet a good trade-off
between the closed-loop control performance and its robust stability.
0.20.4
0.60.8
1
10
15
20
25
300.5
0.6
0.7
0.8
0.9
1
c
d
DP
c (0.8, 1)
c
Disturbance response peak
0.0065904( 0.9222)( )
( 0.8669)( 0.9048)
dzG z z
z z
IMC-based PID design in discrete-time domain
[10] Panda R.C. Synthesis of PID tuning rule using the desired closed-loop response, Industrial &
Engineering Chemistry Research, 47(22), 8684-8692, 2008.
0 2.5 5 7.5 10 12.5 150
0.2
0.4
0.6
0.8
1
1.2
Time(sec)
Out
put
Proposed
Ref.[10]
0 2.5 5 7.5 10 12.5 150
2
4
6
8
10
12
14
16
18
Time(sec)
Con
trol
Sig
nal
Proposed
Ref.[10]
0.251( )
1
sG s es
An illustrative example:
Sampling period:
250.0003947( 0.9868)( )
( 1)( 0.9608)
zG z z
z z
PID
1( ) 1.7049 1
105.2388( 1)
2.4956( 1)
0.4791
C zz
z
z
cIMC
c
( 0.99)(1 )( )
0.00995( )
zC z
z
s 0.02( )T s
Avoid differential kick!
Improved IMC-based PID design for disturbance rejection
For a slow process described by with a large time constant ,p
p 1
sk eG
s
i
(1 )y
G GCd
d
o
(1 )ˆ
yG GC
d
p
Load disturbance transfer function
The slow pole of or affects the disturbance rejection performance !G dG
Solution: Introduce an asymptotic constraint to remove the effect of slow dynamics.
1 1GC T (optimal by IMC)
pRIMC
1/lim (1 ) 0
sT
p2f
p
p
[1 ( 1) ]e
RIMC 2
f
( 1)
( 1)
ss eT
s
Modify the desired transfer function:
p
p
G( ) dk
z zz z
Discrete-time domain model: with a pole close to the unit circlep 1z
d
d
c 0 1d
c
(1 ) ( )( )
( )
n
n
zT z
z
d d
d
p c c
1
p c
0 1
( ) (1 )
( 1)(1 )
1
n n
n
z
z
p
d1/
lim (1 ) 0z z
T
d
1lim(1 ) 0z
T
Discrete-time domain:
(20 1)(2 1)
seG
s s
Improved IMC-based PID design for disturbance rejection
An illustrative example:
[1] Tao Liu, Furong Gao. New insight into internal model control filter design for load disturbance
rejection. IET Control Theory & Application, 4 (3), 448-460, 2010.
Comparison with the
standard IMC and
SIMC by Skogestad
(JPC, 2003)
‘long tail’
Slow pole:1
20s
Discrete-time PID design for integrating and unstable processes
Advantage: the set-point tracking is
decoupled from load disturbance
rejection.
Problem: The classical PID control structure could not suppress large overshoot for
set-point tracking ! There exists severe water-bed effect!
+
+r y+
e
u
-
-
c
uf
u
yr
Solution: A two-degree-of-freedom (2DOF) control structure
Set-point tracking: open-loop control
IMC design: r sT GC
Disturbance rejection: closed-loop
control using a PID controller,
1
fd 0 1 1
f
(1 )( ) ( )
( )
mm d
m mT z z z z
z
fC
The desired transfer function for set-point tracking is the same as the standard IMC design
The desired transfer function for
disturbance rejection :
Discrete-time PID design for integrating and unstable processes
The following asymptotic tracking constraints must be satisfied,
Integrating process:
dlim(1 ) 0z
T
uz
d1
lim (1 ) 0z
dT
dz
d1
lim(1 ) 0z
T
pz or (close to the unit circle)
p 0
I
p
( )( )
( 1)( )
dk z z
G z zz z z
p 0
U
u p
( )( )
( )( )
dk z z
G z zz z z z
Unstable process:
df
d
1
1
TC
T G
Derive the closed-loop controller:
Taylor approximation
PID
fd
f1
GCT
GC
Discrete-time PID design for integrating and unstable processes
Integrating process: 50.1
( )(5 1)
seG s
s s
s 2
c
(5 1)( )
0.1( 1)
s sC s
s
df
d
( )1( )
( ) 1 ( )
T sC s
G s T s
5
r 2
c
( )( 1)
seT s
s
252 1
d 4
f
1( )
( 1)
ss sT s e
s
1 f4 5 4 1
2 1 f5 25(0.2 1) 25e
0 20 40 60 80 100 1200
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Outp
ut
Proposed
Ref.[13]
Discretized ref.[13]
Ref.[25]
0 20 40 60 80 100 120-5
0
5
10
15
20
Time (sec)
Con
trol
Sig
nal
Proposed
Ref.[13]
Discretized ref.[13]
Ref.[25]
[1] Wang, D., Liu T.*, Sun X., Zhong Chongquan. Discrete-time domain two-degree-of-freedom control
for integrating and unstable processes with time delay. ISA Transactions, 63, 121-132, 2016.
[13] Liu T.*, Gao, F. Enhanced IMC design of load disturbance rejection for integrating and unstable
processes with slow dynamics. ISA Transactions, 50 (2), 239-248, 2011.
Frequency domain design:
Discrete-time PID design for integrating and unstable processes
Sampling period:
250.0003947( 0.9868)( )
( 1)( 0.9608)
zG z z
z z
2
cs 2
c
( 1)( 0.9608)(1 )( )
0.0007842( )
z zC z
z
0 20 40 60 80 100 1200
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Ou
tpu
t
Proposed
Ref.[13]
Discretized ref.[13]
Ref.[25]
0 20 40 60 80 100 120-5
0
5
10
15
20
Time (sec)
Con
trol
Sig
nal
Proposed
Ref.[13]
Discretized ref.[13]
Ref.[25]
[25] Torrico B.C., Cavalcante M.U., Braga A.P.S., Normey-Rico J.E., Albuquerque A.A.M. Simple tuning
rules for dead-time compensation of stable, integrative, and unstable first-order dead-time processes. Ind.
Eng. Chem. Res., 52, 11646-11654, 2013.
2 4250 1 2 f
d 4
f
( )(1 )( )
( )
z zT z z
z
s 0.2( )T s
1 2
f
42
1d
4 3 4
p f p f p p f
2 4 2
f p
( ) 4(1 ) (1 ) ( 1)(1 )
(1 ) ( 1)
d d d
4 2
f 0 1 2 p
f 4 2 4 25
p 0 f 0 1 2 f
(1 ) ( )( 1)( )( )
(1 )[( ) ( )(1 ) ]
z z z zC z
k z z z z z z
Perturbation: the process gain and time
constant are actually 20% larger
Discrete-time PID design for long time delay processes
[1] Tao Liu*, Pedro García, Yueling Chen, Xuhui Ren, Pedro Albertos, Ricardo Sanz. New Predictor and
2DOF Control Scheme for Industrial Processes with Long Time Delay. IEEE Transactions on Industrial
Electronics, 65 (5), 4247-4256, 2018.
[2] Yueling Chen, Tao Liu*, Pedro García, Pedro Albertos. Analytical design of a generalized predictor-
based control scheme for low-order integrating and unstable systems with long time delay. IET Control
Theory & Application, 10(8), 884-893, 2016.
Predictor based control structure
Predictor
Delay-free output prediction
PIDSet-point filter
Discrete-time PID design for long time delay processes
The nominal system transfer function
The desired closed-loop transfer function (free of time delay)
pp
f
( ) ( )( ) ( ) ( )
ˆ1 ( ) ( )
dK z G zy z K z z r z
K z G z
pp
1
( )1 ( ) ( ) ( )
ˆ1 ( ) ( )
dG zK z F z z w z
K z G z
d
( ) ( )
1 (
( )( )
( ) ) ( )
K z G z
K z
u zT
w z zz
G
Closed-loop transfer function for disturbance rejection
d
d
cd
0c
(1 )( )
( )
n li
ini
T z zz
0
1l
i
i
d
d p
( ) 1( )
1 ( ) ( )
T zK z
T z G z
Taylor approximation
PID
27.5
( )52.5 1
seG s
s
550.009479( )
0.9905G z z
z
Sampling period: s 0.5( )T s
Example:
0 100 200 300 400 5000
0.2
0.4
0.6
0.8
1
1.2
Time (s)
Ou
tpu
t
Proposed
Ref.[16]
Ref.[24]
Ref.[19]
0 100 200 300 400 5000
0.5
1
1.5
2
2.5
3
3.5
Time (s)
Co
ntr
ol
Sig
nal
Proposed
Ref.[16]
Ref.[24]
Ref.[19]
[16] Kirtania, K.M. Choudhury, A.A.S. A novel dead time compensator for stable processes with long dead
times. J. Process Control, 22(3), 612-625, 2012.
[24] Zhang, W., Rieber, J.M., Gu, D. Optimal dead-time compensator design for stable and integrating
processes with time delay. J. Process Control, 18(5), 449-457, 2008.
[19] Normey-Rico J.E., Camacho E. F. Unified approach for robust dead-time compensator design. J
Process Control, 19: 38-47, 2009.
Discrete-time PID design for long time delay processes
2
c 1 0(1 ) ( )( )
0.009479( 1)
zK z
z
PI controller:
2 2
c ff 2 2
c 0 1 f
( ) (1 )( )
(1 ) ( )( )
z zK z
z z
2 2 2
1 p c c p c( ) (1 ) / ( 1)(1 )z z
0 11
100.250.0004529( )
(760.40 1)
sG s es s
Sampling period:s 3( )T s
A temperature control system of a 4-liter jacketed
reactor for pharmaceutical crystallization
Discrete-time PID design for long time delay processes
4 2
f 0 1 2
6
(1 ) ( )( )
4.9557 10 ( 1)( 0.9899)
z zK z
z z z
4 3
f sf 4 2 3
f 0 1 2 s
( ) (1 )( )
(1 ) ( )( )
z zK z
z z z
1 f 24 / (1 ) 0 1 21
0 200 400 600 800 1000 1200 140020
25
30
35
40
45
50
55
60
Time (s)
Tem
pera
ture
Process
Model
Pt100 Jacketed reactor
PLC Heater Step response identification:
6342.6765 10 ( 0.9989)
( )( 1)( 0.9961)
zG z z
z z
Integrating type
process model:
4
f2 2 4 2
f f
(0.9961 ) 4 1.
(0.9961 1) (1 ) (1 0.9961)(1 ) (0.9961 1)
Set-point filter:
Closed-loop controller:
Compared with the filtered PID
method [21] based on delayed
output feedback, more than 30
minutes are saved for
recovering the solution
temperature to the operation
zone of (45±0.1)oC against the
load disturbance.
Input constraint:
regulate the heating power
Heat up the aqueous solution
from the room temperature
(25oC) to 45oC.
Load disturbance arises from
feeding 200(ml) solvent of
distilled water.
0 50 100 150 200 2500
20
40
60
80
100
Time (min)
Con
trol
sig
nal
Proposed
Ref.[18]
Ref.[21]
[18] Normey-Rico J.E, Camacho E.F. Unified approach for robust dead-time compensator design. J.
Process Control, 19, 38-47, 2009.
[21] Jin, Q.B., Liu, Q. Analytical IMC-PID design in terms of performance/robustness tradeoff for
integrating processes: From 2-Dof to 1-Dof. J. Process Control, 24(3), 22-32, 2014.
0 50 100 150 200 25025
30
35
40
434445464748
Time (min)
Tem
per
atu
re r
esp
on
se
Proposed
Ref.[18]
Ref.[21]
100 110 120 130 140 15043
43.5
44
44.5
45
45.5
46
46.5
47
0 100u
Discrete-time PID design for long time delay processes
Our recent publications on discrete-time domain PID design
✓ Tao Liu*, Pedro García, Yueling Chen, Xuhui Ren, Pedro Albertos, Ricardo Sanz.
New Predictor and 2DOF Control Scheme for Industrial Processes with Long Time
Delay. IEEE Transactions on Industrial Electronics, 2018, 65 (5), 4247-4256.
✓ Yueling Chen, Tao Liu*, Pedro García, Pedro Albertos. Analytical design of a
generalized predictor-based control scheme for low-order integrating and unstable
systems with long time delay. IET Control Theory & Application, 2016, 10(8), 884-893.
✓ Dong Wang, Tao Liu*, Ximing Sun, Chongquan Zhong. Discrete-time domain two-
degree-of-freedom control for integrating and unstable processes with time delay. ISA
Transactions, 2016, 63, 121-132.
✓ Tao Liu*, Furong Gao. Enhanced IMC design of load disturbance rejection for
integrating and unstable processes with slow dynamics. ISA Transactions, 2011, 50 (2),
239-248.
✓ Tao Liu, Furong Gao*. New insight into internal model control filter design for load
disturbance rejection. IET Control Theory & Application, 2010, 4 (3), 448-460.
✓ Jiyao Cui, Yueling Chen, Tao Liu*. Discrete-time domain IMC-based PID control
design for industrial processes with time delay. The 35th Chinese Control Conference
(CCC), Chengdu, China, July 27-29, 2016, 5946-5951.
✓ Hongyu Tian, Xudong Sun, Dong Wang, Tao Liu*. Predictor based two-degree-of-
freedom control design for industrial stable processes with long input delay. The 35th
Chinese Control Conference (CCC), Chengdu, China, July 27-29, 2016, 4348-4353.
PID design for batch process optimization
Features: 1. Repetitive operation for production;
2. Historical cycle information for progressively improving system performance;
3. Time or batch varying uncertainties.
Industrial
batch
processes,
e.g.,
injection
molding
machines,
chemical
reactors
PID design for batch process optimization
IMC-based iterative learning
control (ILC) for perfect tracking
using historical cycle information
Tao Liu, Furong Gao, Youqing Wang. IMC-based iterative learning control for batch processes with
uncertain time delay. Journal of Process Control, 20 (2), 173-180, 2010.
Advantage: batch control optimization
G
+-
+
-C
Gd
D
1kU
G mo
kYMemory
kV+
k-1U
mse
mse-
+
k-1Y
k-1Y
kY
Y+R CU
+
--
PID design for batch process optimization
Type I: PID-based
ILC system
Wang, Y., Gao, F., Doyle F.III., Survey on iterative learning control, repetitive control, and run-to-run
control. J. Process Control, 19(10), 1589-1600, 2009.
Type II: ILC-based
PID control system
Set-point
Set-point
(modify the input
error for batch
operation)
(controller retuning
for batch operation)
PID design for batch process optimization
Indirect-type ILC based on the PI control structure
Youqing Wang, Tao Liu, Zhong Zhao. Advanced PI control with simple learning set-point design:
Application on batch processes and robust stability analysis. Chemical Engineering Science, 71(1), 153-
165, 2012.
Advantage: No need to modify the closed-loop PI controller for ILC design, i.e.,
The PI controller and ILC updating law can be separately designed.
PID design for batch process optimization
Indirect-type ILC design based on the PI control structure and current output error
PlantMEMORY
MEMORY-1z
MEMORY
1L
2L
3L
IK
pK
rY
y(t,k)u(t,k)
e(t,k)e(t+1,k-1)
sy (t, k)
sy (t, k 1)
se (t, k)
se (t, k)
se (t, k 1) sδ e (t 1,k)
Shoulin Hao, Tao Liu*, Qing-Guo Wang. PI based indirect-type iterative learning control for batch processes
with time-varying uncertainties A 2D FM model based approach. J. Process Control, submitted, 2018.
Memory
PID
ILC scheme
PID design for batch process optimization
Indirect-type ILC based on the PID control structure plus feedforward control
Tao Liu, Xue Z. Wang, Junghui Chen. Robust PID based indirect-type iterative learning control for batch
processes with time-varying uncertainties. Journal of Process Control, 24 (12), 95-106, 2014.
m m
p
( 1, ) [ ( , )] ( , ) [ ( , )] ( , ) ( , )
( , ) ( , ), 0 ;
(0, ) (0), =1,2, .
x t k A A t k x t k B B t k u t k t k
y t k Cx t k t T
x k x k
:P
time index:t
:k batch index
batch periodp :T
Robust PID tuning for indirect-type ILC
A PID control law in discrete-time domain
PID P I D( ) ( ) ( ) [ ( 1) ( )]u t k e t k e t k e t e t
State-space closed-loop PID system description
r( , ) Y ( ) ( , )e t k t y t k ( ) ( 1)e t e t
[ ( ) ( 2) 2 ( 1)] / 2e t e t e t
m ( )B B B t
m ( )A A A t
D m P Iˆ ˆ ˆ( 1) ( ) I[ ( I ) ]
( )( ) ( 1)I
( )( )
( 1)
x t x tA B k C A k C Bkt
e t e tC
x ty t C
e t
0
0
1
I D m Iˆ (I )k k CB k
1
D D m Dˆ (I )k k CB k
1
P D m P Iˆ (I ) ( )k k CB k k where
approximate
Robust PID tuning for indirect-type ILC
The H infinity control objective for closed-loop system robust stability
PID2 2( ) ( )e t t
where denotes the robust performance level.PID
Theorem 1: The PID control system is guaranteed robustly stable if there exist
matrices , , , and positive scalars , , such that22 0P 11 0P
12P 2R1R 21
1 A1 A1 2 B1 B1 g
A2 B2
PID
PID
1
2
*
* * I0
* * * I
* * * * I
* * * * * I
T T
T T T T
P D
P PH C P P
0 0 0
0
0 0 0
0 0
0
g [I ]TD 0where [I ]H 0 A1 1[ , ]T TA 0 A2 2 11 2 12[ , ]A P A P
B1 1[ , ]T TB 0
B2 2 1 2 2[ , ]B R B R
11 12
22*
P PP
P
m 11 m 1 m 12 m 2
11 12 12 22
T
A P B R A P B R
CP P CP P
Robust PID tuning for indirect-type ILC
Correspondingly, the PID controller is determined by
1
D m P I 1 2ˆ ˆ ˆ[ ( I ) ] [ ]k C A k C k R R P
where is user specified for implementation. If , it is a PI controller.
I I D mˆ (I )k k k CB
P P D m Iˆ (I )k k k CB k
DkD 0k
PID( ), ( )Min
A t B t
An optimal program for tuning PID to accommodate for the uncertainty bounds,
Guideline: A smaller value of leads to faster output response with a more
aggressive control action, and vice versa.PID
Tao Liu, Xue Z. Wang, Junghui Chen. Robust PID based indirect-type iterative learning control for batch
processes with time-varying uncertainties. Journal of Process Control, 24 (12), 95-106, 2014.
Robust PI tuning for indirect-type ILC
Another robust tuning of PI controller by assigning the closed-loop system poles
to a prescribed circular region, centered at with radius
and , i.e., ( ) ( , )A D r
while the closed-loop transfer function satisfies
( , )D r
Theorem 2: The PI control system is guaranteed robustly D-stable if there exist
matrices , , , , , and positive scalar , such that3 0P 1 0P
T
2 2P P2R1R
T T
1 2
1 2 T T
1 PI
3
1
1
2
ˆ ˆ0 0
ˆ ˆ* 0 0
* * 0 0 00
* * * 0 0
* * * * 0
* * * * *
PC PF
I D D
I
P
I
1 1 | | where 1 1
2 1( 1) T 2 2 2 T
1ˆ ˆ ˆ ˆ( )PA AP r P EE
T T
2ˆ ˆ ˆPA EE
1ˆ ˆ( ) ( )H z C zI A D
T
3ˆ ˆP EE
T T T
T 1 A 1 B
T T T T
2 A 2 B
ˆ PF R FPF
P F R F
| | 1r ( ,0) r
PI|| ( ) ||H z
1 1 2 2
T
1 2 2 3
ˆAP BR AP BR
APCP P CP P
1 2
2 3
P PP
P P
Robust PI tuning for indirect-type ILC
Correspondingly, the PI controller is determined by
1
P I I 1 2( )K K C K R R P
To optimize the robust H infinity control performance, the PI controller can be
determined by solving the following optimization program,
( ), I
( )PMin
A t B t
Guideline: A smaller value of leads to faster output response with a more
aggressive control action, and vice versa.PI
Shoulin Hao, Tao Liu*, Qing-Guo Wang. PI based indirect-type iterative learning control for batch processes
with time-varying uncertainties A 2D FM model based approach. J. Process Control, submitted, 2018.
I I
T T 1
P P I
ˆ
ˆ ˆ( )
K K
K K C CC K
1
1 2 P Iˆ ˆ( ) ( )R R P K K
PI based set-point learning design for indirect-type ILC
s s 1 2 s 3 s( , ) ( , 1) ( 1, 1) ( 1, ) ( 1, )y t k y t k L e t k L e t k L e t k
Based on the PI or PID closed-loop, a learning set-point command is designed as
where is the set-point command in the previous cycle, s ( , )y t k
s s s( 1, ) ( 1, ) ( 1, 1)e t k e t k e t k s s s( , ) ( 1, ) ( , )e t k e t k e t k
Like PI
Memory
PID
ILC scheme
PI based set-point learning design for indirect-type ILC
Feedforward controllers are used to adjust the process input
Note: The setpoint tracking errors at the current moment, one-step ahead
moment, and the error integral in the current cycle are used to construct the
feedforward control.
PID 1 s 2 s 3 s( , ) ( , ) ( , ) ( 1, ) ( 1, )u t k u t k F e t k F e t k F e t k
Memory
PID
ILC scheme
PI based set-point learning design for indirect-type ILC
where
s s
w
s s
s
s
( 1, ) ( , )
( , ) ( 1, )( )
( , ) ( 1, )
( 1, ) ( 1, 1)
( , )
( 1, )( , )
( 1, )
( 1, 1)
x t k x t k
e t k e t kD t
e t k e t k
e t k e t k
x t k
e t kt k G
e t k
e t k
[ ]G 0 0 0 I
w [ ]T TD C 0 0I
p i d 1 p i d 1 2 2 d
2
2
p i d 1 p i d 1 2 2 d
p i d 1 3 3 i p i d 1 1
3 1
3 1
p i d 1 3 3 i
( ) [( ) ]
( ) [( ) ]
[( ) ] ( )
[( ) ]
A B k k k F C B k k k F L F k
C L
C L
CA CB k k k F C CB k k k F L F k
B k k k F L F k B k k k F L
L L
L L
CB k k k F L F k
I
I p i d 1 1( )CB k k k F L
( , ) ( 1, 1)t k e t k
Two-dimensional (2D) system description of the indirect ILC scheme
PI based set-point learning design for indirect-type ILC
2D Roesser’s system stability [1]:
Robust stability condition [1]:
The control objectives for robust tracking from batch to batch
[1] Rogers, E., Galkowski, K. & Owens, D. H. Control Systems Theory and Applications for Linear
Repetitive Processes, Berlin: Springer, 2007.
1 p 22 21
BP ILC ILC2 20 0
( ( , 1) ( , 1) ) 0
N T N
t k
J t k t k
11 11 12 12
21 21 22 22
1 2
( 1, ) ( , )( , )
( , 1) ( , )
( , )( , )
( , )
, =0,1,2, .
h h
v v
h
v
A A A Ax i j x i ji j
A A A Ax i j x i j
x i jy i j C C
x i j
i j
0TA PA P 11 11 12 12
21 21 22 22
A A A AA
A A A A
1 2{ , }P diag P P
PI based set-point learning design for indirect-type ILC
Define
Lyapunov-Krasovskii function used for analyzing 2D asymptotic stability
The objective function of robust batch operation for minimization
s
s
( , )
( , ) ( 1, )
( 1, )
h
x t k
x t k e t k
e t k
( , ) ( 1, )vx t k e t k
( 1, ) ( , )
( , ) ( , 1)
h h
Q Qv v
x t k x t kV V V
x t k x t k
1 p 1 p2 22 21
BP ILC ILC2 20 0 0 0
( ( , 1) ( , 1) ) 0
N T N TN N
t k t k
J t k t k V V
s s s
s s s
(0,0) (0,1) (1,0) 0;
(0,0) (0,1) (1,0) 0;
(0,0) (0,1) (1,0) 0;
(0,0) (0,1) (1,0) 0.
x x x
e e e
e e e
e e e
1 p 2
0 0
0
N T N
t k
V
PI based set-point learning design for indirect-type ILC
Theorem 3: The 2D control system is guaranteed robustly stable with a H infinity
control performance level, , if there exist , , , ,
matrices , , , , , and positive scalars , , such that2 0Q ILC
2F 21
1 A1 A1 2 B1 B1 w
A2 B2
ILC
ILC
1
2
*
* * I0
* * * I
* * * * I
* * * * * I
T T
T T T
Q D
Q QG P P
0 0 0
0
0 0 0
0 0
0
g [I ]TD 0where [I ]H 01 2 3 4{ , , , }Q diag Q Q Q Q
A1 1 1[ , , , ]T T T TA A C 0 0 A2 2[ , , , ]A 0 0 0 B1 1 1[ , , , ]T T T TB B C 0 0
B2 2 p i d 1
2 p i d 1 2 2 d
2 p i d 1 3 3 i
2 p i d 1 1
( ) ,
ˆ ˆ [( ) ],
ˆ ˆ [( ) ]
ˆ ( )
B k k k F C
B k k k F L F k
B k k k F L F k
B k k k F L
m 1 m p i d 1 1 m p i d 1 2 m 2 m d 2
1 2
1 2
m 1 m p i d 1 1 m p i d 1 2 m 2 m d 2
m p i d 1 3 m 3 m i 3 m p i d 1 1
3
ˆ ˆ( ) ( )
ˆ
ˆ
ˆ ˆ( ) ( )
ˆ ˆ ˆ( ) ( )
ˆ ˆ
A Q B k k k F CQ B k k k F L B F B k Q
CQ L
CQ L
CA Q CB k k k F CQ CB k k k F L CB F CB k Q
B k k k F L B F B k Q B k k k F L
L
1
3 3 1
m p i d 1 3 m 3 m i 3 4 m p i d 1 1
ˆ ˆ
ˆ ˆ ˆ( ) ( )
L
Q L L
CB k k k F L CB F CB k Q Q CB k k k F L
1 0Q 3 0Q 4 0Q
3F 3L1L 2L
PI based set-point learning design for indirect-type ILC
Correspondingly, the PI type ILC controller is determined by
1
1 1 4
1
2 2 2
1
3 3 3
ˆ
ˆ
ˆ
L L Q
L L Q
L L Q
To optimize the set-point tracking performance, the PI type ILC controller can
be determined by solving the following optimization program,
( ), ILC
( )Min
A t B t
Guideline: A smaller value of leads to faster output response with a more
aggressive control action, and vice versa.ILC
Shoulin Hao, Tao Liu*, Qing-Guo Wang. PI based indirect-type iterative learning control for batch processes
with time-varying uncertainties A 2D FM model based approach. J. Process Control, submitted, 2018.
1
2 2 2
1
3 3 3
ˆ
ˆ
F F Q
F F Q
The feedforeward controller is determined by
PI based indirect-type ILC for batch injection molding
Shi, J., Gao, F., Wu, T.-J. Integrated design and structure analysis of robust iterative learning control system
based on a two-dimensional model. Ind. Eng. Chem. Res., 44, 8095-8105, 2005.
The nozzle pressure response to the hydraulic valve input was modeled [1] by
1 2
1 2
1.239( 5%) 0.9282( 5%)( , 1) ( , 1) ( , 1)
1 1.607( 5%) 0.6086( 5%)
z zy t k u t k t k
z z
injection molding machine
PI based indirect-type ILC for batch process opimtization
Robustly tuned PI controllers:
1.607 1 1.239 1( 1, 1) ( ) ( , 1) ( ) ( , 1) ( , 1)
0.6086 0 0.9282 0
( , 1) 1, 0 ( , 1)
x t k A x t k B u t k t k
y t k x t k
0.0804 ( ) 0 1 0 ( ) 0 0.0804 0( )
0.0304 ( ) 0 0 1 0 ( ) 0.0304 0
t tA t
t t
0.062 ( ) 1 0 ( ) 0 0.062( )
0.0464 ( ) 0 1 0 ( ) 0.0464
t tB t
t t
( ) 1t
ILC controllers:
p 1.2889k i 0.0336k
1 [ 1.3, 0.1]F
1 0.1776L 2 0L 3 0.029L
3 0.0097F 2 0F
m m p i d 1[ ( ) ] 1A B k k k F C
Equivalently, the process model is rewritten in a state-space form
time-varying
uncertainties
PI based indirect-type ILC for batch process opimtization
r
p
200, 0 100;
200+5( 100), 100 120;
300, 120 200.
t
Y t t
t T
Desired output profile
Case 2:
repetitive
disturbance
Case 1:
time-invariant
uncertainties
PI based indirect-type ILC for batch process opimtization
Plot of the output error for batch operation
p
p
1
ATE( )
( , ) /
T
t
k
e t k T
[1] Tao Liu, Xue Z. Wang, Junghui Chen. Robust PID based indirect-type iterative learning control for
batch processes with time-varying uncertainties. Journal of Process Control, 24 (12), 95-106, 2014.
[2] Y. Wang, Tao Liu, Z. Zhao. Advanced PI control with simple learning set-point design: Application on
batch processes and robust stability analysis. Chemical Engineering Science, 71 (1), 153-165, 2012.
[3] Shi, J., Gao, F., Wu, T.-J. Integrated design and structure analysis of robust iterative learning control
system based on a two-dimensional model. Ind. Eng. Chem. Res., 44, 8095-8105, 2005.
( , 1)
sin( ( ))
t k
t k
Case 3:
Time-varying
uncertainties
( ) [0,2 ]k
( ) 0.1t
Institute of Advanced Control Technology 50
Main Results:
➢ Analytical PID design in discrete-time domain for sampled control systems
➢ 2DOF control structure based PID design for improving disturbance
rejection
➢ Predictor-based PID design for long time delay systems
➢ Robust PID tuning methods with respect to the system uncertainty bounds
➢ PI based indirect type ILC design for batch process optimization
Conclusions & Outlook
Outlook:
➢ Data-driven PID tuning for sampled control systems
➢ Fractional-order PID design & PID scheduling for nonlinear systems
➢ PID +Memory for learning/intelligent control of industrial batch processes,
repetitive systems, and robots etc.
Institute of Advanced Control Technology 51
Thanks to my collaborators and students for PID design
Acknowledgement
Furong GaoPedro Albertos Youqing WangPedro García Wojciech Paszke
My group photo
in 2017
Institute of Advanced Control Technology 52
Institute of Advanced Control Technology
Research interests:
➢ Advanced control system design & system optimization;
➢ On-line monitoring, control design and control optimization
of chemical production batch process;
➢ Real-time model predictive control and optimization of
crystallization & drying processes;
➢ Design of in-situ measurement and control devices;
➢ PBM and CFD modeling of crystallization processes.
Thanks for your attention & comments!
Institute of Advanced Control Technology
Lab website: http://act.dlut.edu.cn/