INSTRUMENTATION

Implement a constrained optimalcontrol in a conventional levelcontroller—Part 2Novel tuning method enables a conventional PI controller to explicitlyhandle the three important operational constraints of a liquid levelloop in an optimal manner as well as copes with a broad range of levelcontrol from tight to averaging control

M. LEE, Yeungnam University, Kyongsan, Korea; and J. SHIN and J. LEE, LG Chem.,Daejeon, Korea

Part 1 of this article outlined the methodology. This parewill discuss PI controller design for ughtest constraint control andprovide 5ome examples.

PI controller design for tightest constraint control.For a leve! loop being operaied near the constraint, it is necessaryto control the loop on the tightest possible constraint, the so-called "tightest constraint control":

• For a given//,^,j (or Q'^„^), the PI controüer for the tight-est constraint control with respect to Q',,,,, v (or ^WA) can bedesigned using either

7

gii )where Ç' = 0.404. In this case, the tightest available Q„„^ is1.3606 SCI or equivalently (,„,„ ^ 0.404.

• For a given (Q'««^-. „^x) set with yi,^ 7^, the PI control-ler for the tightest constraint control of Q, „,^ can he obtained bydesigning it on '(,"', j"'ff. The tightest available (¿,„^ is calculated*>y (¿,max= AQ,/(4'^ from Eq. 11 and the tightest available „„,becomes t,„,m = V"

• For a given (Q, max-' ^mjx) set, the PI controller that gives thetightest Q',, ,„ „. can be designed based on {t,„,i„, T''"//). The tightestavailable Q'„ „¡^.^ is calculated hy

• For a given (Q'„ ,^, ( ¿ ^ set, the PI controller that givesthe tightest //„,^^ can be obtained using (t,„i„, T"^//). The tightestavailable //„,^. is obtained as

SiC )

Illustrative examples . Consider a liquid level loop: A = 1^ ^„ = 2 m, Q,„^^ = 4 m^/min, Q'„ , ^ = 5 m^/min^. The

initial steady-state level is assumed to be at 50%. The expected

maximum AQ, is 1 m^/min, and w is chosen as 0.8.Seven examples are investigated to illustrate the corresponding

seven possible cases. The constraint sets and the resulting optimal

PI parameters are listed in Table 2. Figs. 5a and 5b show the

responses afforded by the resulting optimal PI controller. In the

simulation, a step change in the inflow is made at í - 1. As seen

^

65

6057,5

^*

f

1 1 1

^ ^ Example 1• - — Example 2

Example 3Example 4

0.0

32.3

21

p 0.5

0,0

0.5 1.0 1.5 Z.O 2,5Time, min

3.0 3,5 4.0

4 ——^====

0 -0.0 0.5 1.0 1.5 2.0 2,5

Time, min3,0 3.5 4.0

1.5 2.0 2.5Time, min

FlC. SAi Liquid level loop system responses for examples 1, 2, 3J and 4.

HYDROCARBON PROCESSING FEBRUARY 2010 81

INSTRUMENTATION

65

80^ 5 7 . 5•5 55

^ 50

45

7 " " " • ^

— 'I

— Example 5" Example 6- - Example 7

0.0 0.5

3 -

1-0 1-5 2.0 2.5 3.0 3.5 4-0Tifiie, min

s. 2 \\

— ' •'

0.0 0.5 1.0 1.5 2.0 2.5Time, min

3.0 3.5 4.0

1.50

-="1.15E 1.00

0,50

0.00

/

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Time, min

F I C . SB Liquid level loop system responses for examples 5, 6 and 7.

from [he figures, the level loop is optimally controlled and strictly

satisfies the given three constraints in every case.

It should be noted thar the PI parameters for exampleü 4, 6 and

7 also imply those for the tightest constraint control with respect

to Q^m^' Q'«,^' and ^ m ^ when the ( Q ' ^ , ^ , H„^), (Q^^^.,

H,,,,¿x) and (Q\ ,„^ , Off max) set are assumed ro be given as those of

examples 4, 6 and 7 in Table 2, respectively. The tightest Q„ ,„,^.,Q'ö w,UL' ä"J //„„tv specifications are calculated as 1.2476 mVniin,

4.7746 mVmin- and 0.2387 m, respectively. The responses of

examples 4, 6 and 7 in Figs. 5a and 5b clearly meet these tightest

constraints. HP

End of ser ies : Part 1, January 2010.

ACKNOWLEDGMENT

This research was supported by Yeungnam Umwrsity research gruits in 2008.

NOMENCLATURE

Roman leiiersDrum croüs-sectional area, m-Level deviation from setpoinr, mLtvei transmitter span, mMaximum alJowabIt level deviation from setpoinr, mProportional gain (dimensional}, m-/minProportional gain (dimensionless)Inñow change, m^/minOutflow change, m VminMaximum diowable change in outflow, m^/minMaximum achievable outUow through the uutlet valve, m'/minExpecred maximum -step change in inflow, mVminRate of change in outflow, dQ'„(i)/d/. mVmin^Maximum allowable rare of change in outflow. mVmin'Maximum achie -aWc rate of change in ourflow, m Vmin"

T A B L E 2 . Constraint specifications and PI parametersfor the iBustrative examples

Example Case Specification PI parameter

1

2

3

4

5

6

7

A

B

C

D

E

F

G

V OIIUX

4.0

2.3

5.0

4.0

5.0

6.0

3.0

''max

1.0

1.0

0.15

0.15

0.3

0.15

0.5

•í 0 máx

1.3

1.3

1.3

1.3

1.15

1.15

1.15

0.6325,

1.15,

2.3601,

2.0.

1.8766,

2.3873,

1.5.

1.5811

0.46

0.6796

0.3635

0.9111

0.7162

1.1398

Drum surge volume, A¡íH¡-,,,,, m^

Greek lenersDamping factorIntegra! time constant, minApparent holdup time, W/A^ or ly^K' minHoldup time, AAH,j,^,,l (¿„„„^, minWeighting factor of objecdve function, 0 < d x 1Objective function ot perfotmance measure for optimal controlI itgrangian multiplierSlack variable

SuperscriptsTangent poiniExtreme point of tbe objective function

< 'f Optimum point on the con.straint

nwpectivcly

IT. VU, vl Optimum point oti [be vertex formed by X,, — and

y y = f{Q, respectively

Time, min

LITERATURE CITEDBuckley, P (1983), "Recent advanct-s in averaging le\'el control," Productivitythrough CoHtTolTn-bnotogy. 18-21, Houston.Cheung, T. and Luyben. W. ( 1979), "Liquid-level control in single tanks andcascades ot tanks with P-only and PI feedback controllers," ¡nd. Eng. Chem.Fund., I. 15-21.Luyben, W. and Buckley, P.S. (1977), "A proportional-lag controller,"Instrum. Tecb., 24. 65-68.MacDonald. K.. McAvoy. T. and Tits, A. (1986). "Optimal averaging levelcontrol," AlCbEJ., 32. 75-86.Marlin, T. E. (1995). Process Control: Designing processes and ControlSystem-'i for Dynamic Performance, McGraw-Hill Inc.. New York, USA.Rivera, D.E., Morari, M. and Skogesrad, S. (1986). "Internal model con-trol. 4. PID controller design," Ind. Eng. Chcm- Process Des. Dev., 25,252-265.Seki, H. and Ogawa. M. (1998). Japan Patent # 2811041.St. Clair, D.W. ( 1993). Controller tuning and conctol loop performance:PIDwithout math, 2nd ed.. Straight-Line Control Co Inc.. Newark, Dciawart-.Wu. K., Yu, C. and Cbeung. Y. (2001 ). "A two degree of freedom level con-trol," J. of Prooss Control, 11.311-319.Shin, J., Lee, J., Park, S.. Koo, K. and Lee, M., "Analvtical design of aproportional-integral controller for constrained optimal regulatory control ofinventory loop," Com- Ktig. Prac.. Vol. 16. pp. 1391-1397, 2008,Lee, M. and Shin, J., '"Coasirain«! optimal control of liquid level loop usinga conventional proportional-integral controller," will appear in Chem. Eng.Comm., 2009.Dennis, J.B. (1959), .Marhemadcal Programs and Elcaron Networks, jhonWiiey & Sons, New York.

82 FEBRUARY 2010 HYDROCARBON PROCESSING

INSTRUMENTATION

Appendix. Derivation of performance measure andinequality constraints given in Eqs. I ia-d

Derivation of the performance measure i^ given hy Eq.lia'»

For a step change of magnitude AQ, in the inflow. Hit) andQoit) are obtained from Eqs. 4 and 5, respectively, as follows:

for

2 r,r 2 rrr,e- - r f '

r — rfor

(Al)

CA2)

where r, and ri are the roots ofthe characteristic equarlon

Therefore, the performance measure for optimal control is;

diAH

/S/IIUI

AAH

d-ft»

! '2

2

_2 \_+ r, 2

J 1' 1 . ^2

•l ' 2 J

r,' r/

2 2 r,+n

= 2(0AAH

2T,1 +

SfUII

1

Q'^ ^ Ci' TOI« I

CA3)

Derivation ofthe constraint given by £q. l i dThe outflow response to a step change in the inflow is obtained

from the inverse Laplace transform of Eq. 5 as follows:

AQ,, exp2rH

for ¿ < 1 where 0 —2TH

- 1 +2T H )

exp2r,

l+exp

for ^ > 1 where

-f

2r Í(A4)

The peak of Q (i) can be found from dQp(f)/di = 0 in termsC:

(A5)

= l + exp2 tan .V

for 0 < ^ < 1 where x =

= l+exp(—2) for ^ =

, 2 tanh ' x= l + exp

for f > 1 where x (A6}

Therefore, the constraint in Eq. 1 Id tan be expressed as:

Q , _ (A7)

Derivation of the constraint given by Eq. l i b "The rate of change of the outflow. Q',,(/). is obtained by dif-

ferentiating Eq. A4. The peak of Q „(i) can be found from dQ'„(f)/d/ in terms of £ and T//;

AQ.Q / - ÍA8)

where

hiO- exp3tan X —

for 0 < C < 0.5 where x - ^ J '

= 1 for ¿;> 0.5 (A9)

Therefore, the constraint given by Eq. l i b can be expressed as:

Q ' ™ . (Aio)

Derivation ofthe constraint given by Eq.The analytical solution of the level responses can be obtained

from the inverse Laplace transform of Eq. 4 as;

sin(0/)A <f> 2rH

for ¿f < 1 where 0 =

exp

1

A

_AQi 1^ A 0

exp2r.

sinh(0i)

sinh(0f)

for ¿ > 1 where 0 = (All)

Tbe peak of Hit) can be found in terms of £ and T^;

(A12)

84 FEBRUARY 2010 HYDROCARBON PROCESSING

INSTRUMENTATION

where g{Ç,) is;

tan X tan XI exptan X

for O < ¿ < 1 where .v

- 2 e x p ( - l ) forC=l

for 0 < ¿ < 1 where x —

C= 2

tanh X-1

(A: —tanh jc)exp

whaex =

tanh X

(AI 5)

> ] wíiere x —f

(A13)

Finally, the constraint given by Eq. 1 lc can be expressed as:

' ^MÍ^O < ^™, (A14)A "

Furthermore, g'iO > which denotes dg{Q/dl, is given by:

,jgMa^^r^^^r M o o n y o n g Lee is a professor m the school of chemical engi-

•5 » • reenng and technology at Yeungnam University, Korea. He holds a• ^ B iiS degree m chemical engineering from Seoul National University,

M j ^r,d MS and PhD degrees in chemical engineering from KAIST- *- ^ uf Lee worked in the refining and petrochemical plants of SK

; : , ; 10 years as a design and control specialist. Sincejoining the university¡n iy94, his areas of speciaiization have included modeling, design and chemicalprocess control.

Joonho Shin is a process systems engineer in corporate R&D atLG Chem Ltd., Korea. He holds a 8S degree in chemical engineer-ing from Korea University, and MS and PhD degrees in chemicalengineering from KAIST. Dr. Shin began his professional career asa design and control specialist at SK Engineering & Construction in

1997. His industrial experience has focused on modeling, optimization and controlof chemical and petrochemical industrial plants since 2002.

Jong K u Lee is a vice president of LG Chem Research Park,LG Chem, Ltd., Korea. He holds a BS degree from Seoul NationalUniversity, and MS and PhD degrees m chemical engineering fromKAIST. SincejoiningLGChem, Ltd in 1994, Dr. Leehasbeen work-ing on numerous process modeling and optimization related proj-

ects. Currently he is in charge of the process modeling & solution group and doingresearch in the areas of energy savings and sustainability in chemical plants.

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