Optimization-based PI/PID control for SOPDT process

Summary on optimization-based PI/PID control

Best achievable IAE performance by PI/PID control of FOPDT process

Optimal rise-time vs, IAE in PI/PID control of SOPDT process

Optimal rise-time vs, IAE in PI/PID control of SOPDT process

FOPDT

SOPDT

According to the IMC theory, nominal loop transfer function of a control system that has an inverse-based controller will be of the following:

( )( ) ( )p

lp lp

G sG s F s

s

P

( ) ( ) ( )

where, ( ) serves as a loop filter in a control system,

and the ( ) represents the non-invertible part of G .

plp lp

lp

p

G sG s F s

sF s

G s

Loop transfer functions of IMC-PID Controllers

IMC-PID for FOPDT process

( )1

sp

p

k eG s

s

The resulting ( ) becomes (Chien and Fruehauf, 1990)lpG s

1

s

lp

eG

s

1 0 5

( )( 1)

s

lpf

e sG

s s

Loop transfer functions of IMC-PID Controllers

2( )

2 1

sp

p

k eG s

s s

the resulting loop transfer function becomes:

1s

lp

eG

s

FOPDT processes:

SOPDT processes:

(1 )( )

1

sp

p

k s eG s

s

( 2 )

(1 ) (1 )( )

1 (1 )

(1 ) (1 ) =

1 (1 )

(1 ) =

1

sp

p

sp

sp

k s e sG s

s s

k s e s

s s

k s e

s

* 1.38 ( 2 )IAE

(1 )( )

(1 )

so

loopk s e

G ss s

(1 )( )

(1 )

so

LP n

k s eG s

s s

• We should learn what happens to the Z-N tuned controllers?

• How inverse-based controllers are synthesized?

Inverse-based Controller Design

• Inverse-based synthesis approach is used– Target loop transfer function (LTF)

– This LTF has satisfactory control performance as well as reasonable stability robustness

• ko and a are selected to meet desired control specification

Defaulted value: ko=0.65 a=0.4 GM = 2.7 PM = 60 o

(1 )( ) ( ) ( )

(1 )

so

loop c pf

k a s eG s G s G s

s s

Loop transfer functions of Inverse-based Controllers

PI/PID Controllers Based on FOPDT Model

• A direct synthesis approach is used– PI controller

ko=0.5

• Controller parameters (actual PID)

– PID controller

ko=0.65 , a=0.4

( )s

oloop

k eG s

s

(1 )

( )(1 )

so

loopf

k a s eG s

s s

'

'

'

oc

p

R

D

kk

k

a

PID Controller Based on SOPDT Model

• Controller parameters (ideal PID)

( )s

oloop

k eG s

s

ko=0.5

(2 )

2

2

oc

p

R

D

kk

k

2( )

2 1

sp

p

k eG s

s s

0 0ˆ(1 ) (1 )s s

lp

k s e k s eG

s s

0 0ˆ andk k s s

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

^

p

ˆp

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

Am

m

Gain margin vs. phase margin at a=0.4

Phasemargin

Gainmargin

Auto-tune

• Autotuning via relay feedback: Astrom and Hagglund (1984)

Referred as autotune variation (ATV): Luyben (1987)

Main advantage: under closed-loop

4 2cu u

u

hk

A P

Apply Z-N or T-L tuning

MODEL-BASED CONTROLLERS DESIGN

• Reduced order models– FOPDT Monotonic step response

• For zero offset, PI or PID controller is considered

• Usage of PI or PID controller depend on:– The application occasions– The dynamic characteristics of given process

• Processes are classified into two groups for controller tuning

- Underdamped SOPDT

Oscillatory step response

( )1

sp

p

k eG s

s

2 2( )2 1

sp

p

k eG s

s s

Criterion for Classifying model order

• In general, processes with overdamped or slightly underdamped SOPDT dynamics can be identified with FOPDT models for controller tuning

Q: When an SOPDT process could be reduced to an

FOPDT parameterization?

A: Ku > 1

1 22 tan 11

; ; whereu

uu p cu

u u

KKK k k

10.707 1

2UK

PI/PID Controllers Based on FOPDT Model

• A direct synthesis approach is used– PI controller

ko=0.5

• Controller parameters (actual PID)

– PID controller

ko=0.65 , a=0.4

( )s

oloop

k eG s

s

(1 )

( )(1 )

so

loopf

k a s eG s

s s

'

'

'

oc

p

R

D

kk

k

a

In terms ofultimate data

(Ku = kp kcu)

2'

1 2

2'

1 2

'

1

tan 1

1

tan 1

o uc

p u

uR

u

u

Du

k Kk

k K

K

Ka

• PI controller

Defaulted value: ko=0.55 a=0.4

1( ) 1c c

R

G s ks

0.9

o Rc

p

R

kk

k

a

2

1 2

2 1 2

0.9 1

tan 1

10.9 1 tan 1

uoc

p u

R u uu

Kkk a

k K

K a K

In terms of

ultimate data

PID Controller Based on SOPDT Model

• Only PID controller is used for significant underdamped SOPDT dynamics, i.e.

• Controller parameters (ideal PID)

• The values of kp and need to be estimated in advance

( )s

oloop

k eG s

s

ko=0.5

(2 )

2

2

oc

p

R

D

kk

k

In terms ofultimate data

sin( )

sin( )

1 cos( )

sin( )

o u uc

p u

u uR

u

u uD

u u u

k Kk

k

K

K

K

1 2

DynamicProcess

FOPDT Model SOPDT Model

PIDController

Group I Group II

PIController

PIDController

( )s

oloop

k eG s

s

(1 )

( )(1 )

so

loopf

k a s eG s

s s

1 2 1 2

• Estimation of process gain kp

– Start the ATV test with a temporal disturbance to setpoint or process input

– Define

– and have

cycling responses

0

0

( ) ( )

( ) ( )

I t

I t

y t y d

u t u d

Iav

p Iav

yk

u

Iy Iu

• Estimation from is subject to error,

sometimes as high as 20%

• From Fourier series expansion

• Ultimate gain is computed exactly as:

4cuk h A

0

0

0

0

( )( )

( )

uu

uu

j tt Pt

p u j tt Pt

y t e dtG j

u t e dt

1

( )cup u

kG j

Estimation of Apparent Deadtime

• In an ATV test, two measured quantities are used to characterize the effect of the apparent deadtime

–

–

• For SOPDT process, this two quantities are

functions of and

A

A

pT

• Underdamped SOPDT processes

cos( 3) sin( 3)

sin( 3) cos( 3)

p

p

AX

T A

AY

T A

• Algorithm for estimation of apparent deadtime– Starting from a guessed value of – Calculate and , and feed them into networks to compute

and– Check if the eq. holds

– If not, increase the value

of until the above eq.

holds. At that time,

is the estimated apparent

deadtime

o X

Y

12 2

2tan

1u

uu

• In ATV test, it provides and which are functions of and

pA k h uP

Locate on this figure • Zone I: FOPDT parameterization • Zone II: SOPDT parameterization

1 2

1 2

,u pP A k h

• Initiate ATV test by a short period of manual disturbance and record y(t) and u(t) until constant cycling is attained

– Compute kp and kcu

– Estimate the apparent deadtime

– Classify the process by the location of

– If the process belongs to Group I, tune PI or PID controller based on FOPDT model parameterization

– If the process belongs to Group II, tune PID controller based on SOPDT model parameterization

( , )u pP A k h

• Examples

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Outp

ut

ProposedZ-N

0 10 20 30 40 50 600

0.5

1

1.5

Time

Outp

ut

ProposedZ-N

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Outp

ut

ProposedZ-N

Ex. 1 Ex. 2 Ex. 3

• Ideal PID controller with an extra filter

• The value of kp and need to be known in advance

(2 )

2

2

oc

p

R

D

kk

k

sin( )

sin( )

1 cos( )

sin( )

o u uc

p u

u uR

u

u uD

u u u

k Kk

k

K

K

K

'

1 1( ) 1

1c c DR f

G s k ss s

Optimal IAE Value for Set-point Tracking

– PI control

– PID control

• These optimal systems have reasonable stability robustness– PI control gain margin = 2.6 For unit step set-point change (H

uang and Jeng, 2002 )– phase margin = 55o

– PID control gain margin = 2.1, phase margin = 60o

1.0695s*PI

2.104 0.6023 for 5

2.104 for 5

eJ

1.5541s*PID

1.37 0.1134 for 3

1.37 for 3

eJ

• Control systems designed for optimal input disturbance response will give smaller gain margin and phase margin than those designed for optimal set-point response.

• The optimal IAE value occurs at a phase margin about 30o to 50o – trade-off between disturbance performance and phase margin is not

always needed

PI control PID control

Optimal System for Disturbance Rejection

Optimal IAE Value for Disturbance Rejection• The smaller the gain margin is (i.e. less robust), the lower the optimal IAE

value can achieve. – trade-off between disturbance performance and gain margin is

needed

• PI control

• PID control

d* 0.8931PI 0.8035 exp 0.0984m p mJ A k A

d* 1.0527PID 0.4681 exp 0.1071m p mJ A k A

PI control PID control

1.5 5mA Gain margin