new pid designs for sampling control and batch process ... · imc-based pid design in discrete-time...

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


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


In case , it could provoke inter-sample ringing in the control signal !


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


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:


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



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


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


[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 :


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


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]


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:


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]


Ref.[25]

0 20 40 60 80 100 120-5

0

5

10

15

20

Time (sec)

Con

trol

Sig

nal

Proposed

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


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.


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


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


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


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

+

--


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)


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.


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


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


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


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


aggressive control action, and vice versa.PI



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


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


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


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


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


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


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


aggressive control action, and vice versa.ILC



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


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


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


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.


Thanks to my collaborators and students for PID design

Acknowledgement

Furong GaoPedro Albertos Youqing WangPedro García Wojciech Paszke

My group photo

in 2017



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!


Lab website: http://act.dlut.edu.cn/

new pid designs for sampling control and batch process ... · imc-based pid design in discrete-time...

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