8/6/2019 PI Control Best

http://slidepdf.com/reader/full/pi-control-best 1/7

Analytical design of a proportional-integral controller for constrained optimalregulatory control of inventory loop Joonho Shin a , Jongku Lee a , Seungyoung Park a , Kee-Kahb Koo b , Moonyong Lee c, Ã

a Corporate R&D, LG Chem, Moonji-dong, Yuseong-gu, Taejon 305-701, Republic of Koreab Department of Chemical and Biomolecular Engineering, Sogang University, Seoul 121-742, Republic of Koreac School of Chemical Engineering and Technology, Yeungnam University, Kyongsan, Kyongbuk 214-1, Republic of Korea

a r t i c l e i n f o

Article history:Received 31 October 2007Accepted 21 April 2008Available online 10 June 2008

Keywords:Inventory controlPI (proportional-integral) controller tuningOptimal controlConstraint controlLiquid level control

a b s t r a c t

An analytical design method for a simple proportional-integral (PI) controller is developed for theoptimal control of a constrained inventory loop. The proposed method explicitly deals with theimportant constraints in the inventory loop, such as the maximum allowable rate of change in themanipulated variable, the maximum allowable decay ratio and damping coefcient in the outputresponse, as well as minimizing the optimal control specication. The simple and explicit form of theresulting tuning rule is clearly advantageous to practitioners.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Inventory control loops are commonly encountered in theprocess industry. Typical examples include accumulator andbottom level control in a distillation column and the inventorycontrol of a tank ( Yang, Seborg, & Mellichamp, 1994 ). Inventorycontrol is extremely important for the successful operation of most chemical plants, because it is through the proper control of the ows and levels that the desired production rates andinventories are achieved ( Marlin, 1995 ).

Many studies have been conducted in an attempt to enhance thecontrol performance of the inventory loop. Cheung and Luyben (1979)studied the liquid level control system with P-only and PI feedbackcontrollers. They proposed a procedure with a design chart for thetuning of a PI controller in response to a step change of the inlet ow

rate. However, it is quite complicated to determine the tuningparameters of a PI controller. Proportional-lag control ( Luyben &Buckley, 1977 ) is a potentially good solution for liquid level controlsystems with feedforward compensation, but such feedforwardcontrol schemes ( Luyben & Buckley, 1977 ; Wu, Yu, & Cheung, 2001 )require an additional measurement which may be unavailable. Rivera,Morari, and Skogestad (1986) proposed the P-only controller using theinternal model control (IMC) principle for the critically dampedclosed-loop response of a liquid level control system. Buckley (1983)discussed several nonlinear PI controllers to provide fast control

action for large errors and slow action for small errors in the liquidlevel loops. MacDonald, McAvoy, and Tits (1986) proposed aninteresting method of deriving an averaging level control algorithmto minimize the maximum rate of change of the manipulated ow.

In practice, the operation of an inventory control system shouldbe located somewhere between the two extreme situations: therst, referred to as tight inventory control, is where the level is veryimportant but any variation in the manipulated ow is not of greatimportance; the second, referred to as averaging inventory control,occurs when some variation in the level is acceptable as long as thevalue remains within specied limits, but the manipulated owshould not experience rapid variations of a signicant magnitude.Thus, the control objective of inventory loops should considervariations not only in the controlled variable but also in themanipulated variable. Furthermore, inventory loops often have

several important constraints associated with both the controlledand manipulated variables. This feature of the inventory loop oftennecessitates an optimal control strategy with constraint handling.However, since most inventory loops make use of a simple PIcontroller, constrained optimal control is rarely implemented in theinventory loops. In this study, an analytical design method for PIcontrollers is developed for optimal regulatory control withexplicitly handling the major specications in the inventory loop.

2. Liquid level control dynamics

The liquid level control system presented in Fig. 1 is describedby the following differential equation with the nomenclature of

ARTICLE IN PRESS

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/conengprac

Control Engineering Practice

0967-0661/$ - see front matter & 2008 Elsevier Ltd. All rights reserved.doi: 10.1016/j.conengprac.2008.04.006

à Corresponding author. Tel.: +8253 8102512; fax: +8253 8113262.E-mail address: mynlee@yu.ac.kr (M. Lee).

Control Engineering Practice 16 (2008) 1391– 1397

8/6/2019 PI Control Best

http://slidepdf.com/reader/full/pi-control-best 2/7

8/6/2019 PI Control Best

http://slidepdf.com/reader/full/pi-control-best 3/7

as follows:

z ¼12 ffiffiffiffiffit I

t H r ¼12 ffiffiffiffiffiffiffiffiffit I K c

t V s (10)

3. Formulation of optimal regulatory control

Regulatory control in response to load changes is the majorconcern in inventory control systems. The PI controller for a liquidlevel loop is often required to have a smoother, less aggressive controlaction, even at the expense of less error minimization, which meansthat the performance measure of the control system must includeminimizing not only the error in the controlled variable but also therate of change of the manipulated variable. At the same time, thecontroller should be designed to meet all or some of the followingtypical specications or constraints in the level loop: (i) the rate of change of the outlet ow should be under a maximum allowablelimit, (ii) the decay ratio in the response should be under a maximumallowable limit to avoid a severe oscillatory response, and (iii) thedamping coefcient should also be less than a maximum allowable

limit in order to secure a suppression speed required.Based on the operational goal and the three constraints listedabove, the optimal design problem of the PI controller in theinventory loop can be dened as nding the controller parametersthat minimize the performance measure in (11-1), subject to theconstraints in (11-2)–(11-4)

min F ¼o Z 10

H ðt ÞD H

2

d t þ ð1 Ào ÞZ 10

Q 0oðt ÞQ 0o max

2

d t (11-1)

subject to

Q 0oðt Þp Q 0o max (11-2)

DRp DRmax (11-3)

zp zmax . (11-4)The rst and second terms in (11-1) describe the normalized

deviation of the liquid level and the normalized rate of change of the outlet ow, respectively.

3.1. Optimal solution for unconstrained case

Throughout this study, the regulatory problem to a step changein the inlet ow rate (i.e., Q i(s) ¼D Q i/s) is considered. Byperforming certain mathematical manipulations, the objectivefunction F given in (11-1) can be expressed in terms of t H and z asfollows (see Appendix A for details):

F ðt H ; zÞ ¼2oD Q i

AD H 2

t3H z

2

þ ð1 Ào ÞD Q i

Q 0o max 2

Â1

2 t H 1 þ

14z2 (12)

Thus, the unconstrained optimality conditions, i.e., the extre-mum in the absence of constraints, can be found by solving thefollowing equations simultaneously:

q Fq t H ¼6o

D Q i AD H

2

t 2H z

2À

1 Ào2 D Q i

Q 0o max 2

Â1

t 2H !1 þ

14z2 ¼0 (13)

qF

q z ¼4oD

Q i AD H 2

t 3H z À 1 Ào4 D

Q iQ 0o max 2

1t H z

3 ¼0 (14)

Multiplying (14) by t H /z gives

ðt H zÞ4¼

1 Ào16 o AD H

Q 0o max 2

(15)

The optimal t H for the unconstrained case can then be obtainedby substituting (15) into (13) and rearranging to give

t y2H

¼AD H

2Q 0o max ffiffiffiffiffiffiffiffiffi1 Ào

or (16)

where the superscript y denotes the optimum in the uncon-strained case. The optimal z can also be obtained as

zy¼1

ffiffiffi2p (17)

It is noted that the optimal value of the damping coefcientis equal to 1

ffiffi2p and is independent of the process dynamicsand weighting factor. The optimal tuning values of K yc and t I

y

for the unconstrained case can be simply calculated using (7)and (10).

The following simple relation can also be derived for theoptimal tuning of the unconstrained case:

K yc t yI ¼

2 t V . (18)

Note that product of K c y and t I

y is proportional to the hold-uptime of a level tank. Seki and Ogawa (1998) also came to the sameconclusion as that described by (17) and (18) for the optimalcontrol of a level loop.

Remark 1. For the optimal tuning, the product of K c y and

t I y should be kept constant by doubling the hold-up time,

regardless of the weighting factor. When one of the PI para-meters, K yc and t I

y, is changed during tuning, the other para-meter should also be adjusted so that the product remainsconstant.

3.2. Optimal solution for constrained case

When the constraints given in (11-2)–(11-4) are considered inthe controller design, the global optimum can be located either onthe extreme point of the objective function or on the constraintboundary. Note that an extreme point denotes an extremum in theabsence of constraints. To deal with the constrained cases, all of the constraints have to be expressed in terms of the independentvariables, t H and z. The rate of change of the outlet ow rate for astep disturbance can be determined by using the properties of theLaplace transform of the derivative of the outlet ow, which isgiven by

LdQ oðt Þ

d t ¼sQ oðsÞ ÀQ oðt Þt ¼0 (19)

where Q o(t )t ¼ 0

¼0.

Throughout this study, it is assumed that the controller isdesigned to give a response with zX 0.5 in order to avoid a severeoscillatory response. In this case, since the largest value of the rateof change occurs at t ¼0, by applying the initial value theorem to(6), it is easily determined that

dQ oðt Þd t t ¼0 ¼

D Q it H

(20)

Therefore, the constraint given in (11-2) can be expressed interms of t H :

t H XD Q i

Q 0o maxt H min (21)

Furthermore, using the relation between the decay ratio and the

damping coefcient in the second-order process, the constraintimposed by (11-3) can be converted to the following inequality

ARTICLE IN PRESS

J. Shin et al. / Control Engineering Practice 16 (2008) 1391–1397 1393

8/6/2019 PI Control Best

http://slidepdf.com/reader/full/pi-control-best 4/7

condition in terms of z:

zX1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ ð2p = ln DRmax Þ2q

zmin (22)

Note that since all of the constraints are linear, the solutionregion is convex. The feasibility of the solution region should berstly checked for a given constraint set, which can be easily donefrom (11-4) and (22).

Once the constraints applied are conrmed as giving a feasiblesolution region, the next step is to determine the global optimalcondition. The global optimum can exist either on the extremepoint of the objective function or on the boundary of a constraint.The local optimum value of z on the boundary of t H ¼t Hmin can becalculated from (14) by replacing t H with t Hmin :

zü1

ffiffiffi2p 1

t H min 1 Ào4o

1=4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi AD H Q 0o maxs ¼

1

ffiffiffi2p t yH

t H min(23)

The local optimum value of t H on the boundary of z ¼zmin canbe found from (13) by replacing z with zmin :

t ÃH ¼ 4z2min þ 112 z4

min !1=4

1 Ào4o 1=4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi AD H

Q 0o maxs ¼ 4z2min þ 112 z4

min !1=4

t yH

(24)

Similarly, the local optimum value of t H on the boundary of z ¼zmax can be obtained as

t ÃÃH ¼4z2

max þ 112 z4

max !1=4

1 Ào4o

1=4 ffiffiffiffiffiffiffi AD H Q 0o maxs ¼

4z2max þ 1

12 z4max !

1=4

t yH

(25)

It is clear from (23) that the local optimum z* is always less than1

ffiffi2p (or zy) when t H y o t Hmin . Furthermore, since g (z) ¼(4 z2 +1/

12 z4 )1/4 is a monotonically decreasing function in z with g ð1= ffiffiffi2p Þ ¼1, t H * is less than t H

y for zy o zmin and t H ** is greaterthan t H

y for zy 4 zmax . These relationships indicate that thecontours of the objective function are left skewed in the zÀt H

plane.Based on the contour characteristics, the condition and

location of the global optimums of ( z,t H ) for every possiblecase are found as listed in Table 1 . Fig. 2 shows ve possiblecases of a global optimum with the contours of the objec-tive function and the constraints imposed by (11-2)–(11-4). Oncethe global optimum ( zopt ,t H

opt ) is obtained, the correspondingoptimal PI parameters can be simply calculated from (7), (8),and (10) as

K opt C ¼ ðD H Þ A

Q o max t opt H

(26)

ARTICLE IN PRESS

Table 1Global optimums of ( x,t H ) for the constrained case

Case Condition Location Solution

A (t H yX t H min and zmax X zyX zmin ) At the extremum ( zy t H

y) (zy t H y)

B (t H yo t H min and zmax X z*X zmin ) or ( zy4 zmax and t H **o t H min ) On the constraint t H ¼t H min (z*,t H min )

C (zy4 zmin and t H *X t H min ) On the constraint z ¼zmin (zmin ,t H *)D (zy4 zmax and t H **X t H min ) or ( t H

yo t H min and z*4 zmax ) On the constraint z ¼zmax (zmax ,t H **)E (t H

yo t H min and z*o zmin ) or ( zyo zmin and t H *o t H min ) On the vertex by t H ¼t H min and z ¼zmin (zmin ,t H min )

ζ = ζmin

ζ = ζminζ = ζmin

ζ = ζmin

ζ = ζmax

ζ = ζmax

ζ = ζmax

τH = τHmin

τH = τHmin

τH = τHmin

τH = τHmin

τH=

τHmin

optimum

optimumoptimum

optimum

optimum

ζ

ζζ

ζ ζ

τ H

τ H

τ H

τ H

τ H

CASE A

CASE B CASE C

CASE ECASE D

Fig. 2. Possible cases of a global optimum location: the shaded region denotes a feasible region.

J. Shin et al. / Control Engineering Practice 16 (2008) 1391–1397 1394

8/6/2019 PI Control Best

http://slidepdf.com/reader/full/pi-control-best 5/7

t opt I ¼4ðz

opt Þ2 t opt

H (27)

4. Illustrative examples

Consider a liquid level system as follows: the liquid level of atank with a cross-section area of 1 m 2 and a working volume( AD H ) of 2 m 3 is controlled by a PI controller. The maximum outletow ( Q omax ) is 4m 3 /min. The initial steady-state level is 50% andthe nominal ow rates of the inlet and outlet are both 1 m 3 /min.The maximum expected change in the inlet ow ( D Q i) is 1 m 3 /min.The maximum allowable rate of change of the outlet ow ( Q 0omax )is 1.5 m 3 /min 2 .

Example 1. Optimal tuning: unconstrained case.

Suppose that there is no hard control specication. Optimaltuning can then be calculated based on (16) and (17). For example,if o ¼0.5, t yH ¼1= ffiffiffi3p and zy¼1= ffiffiffi2p are obtained. The optimal PIparameters are calculated using (26) and (27) as K c ¼ ffiffiffi3p =2 andt I ¼2=

ffiffiffi3p min.

The responses in the level and outlet ow rate for the proposedoptimal tuning with various weighting factors are shown in Fig. 3.In the simulation, a step change of 1 m 3 /min in the inlet ow rateis introduced at 5 min and sequentially the level set-pointundergoes a 25% step increase at 20 min. As seen in the gure,the lower the weighting factor, o , the more slowly the outlet owchanges, but the higher the peak level becomes and the moresluggishly the level is controlled. As o is increased, a smaller peakcan be obtained at the cost of a higher rate of change in the outletow rate. In this manner, a clear tradeoff can be achieved betweenthe tightness in the level control and the smoothness in the outletow change with only the single tuning parameter, o .

To conrm the advantage of the proposed method, the closedloop performance provided by the proposed PI controller iscompared with that afforded by the IMC-PI tuning method ( Riveraet al., 1986 ). The weighting factor o is set to 0.5 in the proposed

method. In order to provide a fair comparison, the closed looptime constant in the IMC-PI method is adjusted so that bothcontrollers yield the same maximum peak level. As shown inFig. 4, the PI controller using the proposed method gives a smallermaximum rate of change of the outlet ow, as well as a fastersettling time in the level response. The value of the performancemeasure in (11-1) was also evaluated for each method, and the

proposed PI controller gave a smaller value of 0.2988 than thatgiven by the IMC-PI tuning method of 0.3580.

Example 2. Optimal tuning: constrained case.

When the control specications given by (11-2)–(11-4) have tobe strictly satised, the optimal tuning values can be obtained bycategorizing the global optimum case from Table 1 . Suppose thatthe weighting factor is set to o ¼0.8 for the liquid level systemabove. As an illustrative example, consider cases I, II, and III,where the maximum allowable decay ratios are 0.0005, 0.1, and0.1 and the maximum allowable damping coefcients are 1.0, 1.0,and 0.4, respectively, while the maximum allowable rate of change of the outlet ow is 1.5 m 3 /min 2 in all three cases. From(21), t H min ¼0 :6667 is obtained. The minimum allowable damp-ing coefcients corresponding to the maximum decay ratios,calculated using (22), are 0.7708 for case I and 0.3441 for cases IIand III. Therefore, cases I, II, and III correspond to cases E, B, and Din Table 1 and Fig. 2, respectively. The optimal PI parameters forthe three cases are K c ¼0.75, 0.75, and 0.5697 and t I ¼1.584, 1.0,and 0.562, respectively.

Fig. 5 compares the responses for the level and rate of changeof the outlet ow for the three cases. In the simulation, a stepchange of 1 m 3 /min in the inlet ow rate is introduced at 1 min.The global optimum is located at the vertex point by the twoconstraints z ¼zmin and t H ¼t H min for case I, on the constraintt H ¼t H min for case II, and on the constraint z ¼zmax for case III. Asseen in the gure, all of the responses strictly satisfy the givencontrol specications.

ARTICLE IN PRESS

100

90

80

70

60

50

40

L e v e

l ( % )

1.4

1.2

1

0.8

0.6

0.4

0.2

0 O u

t l e t f l o w r a

t e ( m

3 / m i n )

0 5 10 15 20 25 30 35Time (min)

0 5 10 15 20 25 30 35

Time (min)

w = 0.2w = 0.5w = 0.7w = 0.9

w = 0.2w = 0.5w = 0.7w = 0.9

Fig. 3. Level and outlet ow responses using the proposed PI controller (unconstrained case).

J. Shin et al. / Control Engineering Practice 16 (2008) 1391–1397 1395

8/6/2019 PI Control Best

http://slidepdf.com/reader/full/pi-control-best 6/7

Example 3. Robustness against modeling error in dead time andarea.

Level processes generally do not have a dead time except forcertain rare occasions such as level loops in which the controlvalve has a dead zone. Therefore, the dead time in level processescannot be too large and has to be maintained at almost 10% of thehold-up time ( Wu et al., 2001 ).

Fig. 6 shows the results assuming that the level process inExample 1 has a dead time of 20% of the hold-up time. Theweighting factor is set to 0.5. The issue of robustness is also

studied by examining 7 20% errors in the cross-sectional area andthe results are presented in Fig. 6. The responses shown in the

gure indicate that the realistic uncertainties in the dead time andarea have little effect on the control performance.

5. Conclusions

An analytical design method for the optimal control of a liquidlevel loop is developed by solving a constrained optimizationproblem. One of the main drawbacks of simple PID controllers isthat they cannot handle the constraints explicitly. The proposeddesign method explicitly deals with the important constraints in

the inventory loop, as well as minimizing the optimal controlspecication. Several examples were presented to illustrate the

ARTICLE IN PRESS

80

70

60

50

400 5 10 15 20 25

Time (min)

0 5 10 15 20 25Time (min)

1.5

1

0.5

0

-0.5

d Q o

/ d t ( m

3 / m i n 2 ) proposed

IMC-PI

proposedIMC-PI

L e v e

l ( % )

Fig. 4. Comparison of responses by the proposed method and the IMC-PI tuning method (unconstrained case, w ¼0.5).

80

70

60

50

40

0 5 10 15Time (min)

0 5 10 15Time (min)

2

1.5

1

0.5

0

-0.5

d Q o

/ d t ( m

3 / m i n 2 )

case Icase IIcase III

case Icase IIcase III

L e v e

l ( % )

Fig. 5. Level and rate of change of outlet ow responses using the proposed PI controller (constrained case): case I ( DRmax ¼0.0005, zmax ¼1.0), case II ( DRmax ¼0.1,zmax ¼1.0), case III ( DRmax ¼0.1, zmax ¼0.4).

J. Shin et al. / Control Engineering Practice 16 (2008) 1391–1397 1396

8/6/2019 PI Control Best

http://slidepdf.com/reader/full/pi-control-best 7/7

effectiveness of the proposed design method. The results alsoshowed that the proposed method gives satisfactory responses,not only in the nominal condition but also under uncertainties inthe dead time and cross-sectional area.

Acknowledgment

This research was supported by the Yeungnam Universityresearch grants in 2007.

Appendix A. Derivation of the objective function U in (12)

From (5) with Q i(s) ¼D Q i/s and H set (s) ¼0, the liquid levelresponse is obtained as

H ðt Þ ¼D Q i A

e r 1 t Àe r 2 t

r 1 Àr 2 for r 1 a r 2 (A1)

where r 1 and r 2 are the roots of the characteristic equation s2 +(1/t H )s+(1/ t I t H ) ¼0.

Thus,

r 1 r 2 ¼1

t I t H (A2)

r 1 þ r 2 ¼ À1

t H (A3)

r 1 Àr 2 ¼1t H ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 À

4 t H

t I s ¼1

t H ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiz2À1

z2s (A4)

Therefore, the integral square error of the normalized liquidlevel in (11-1) becomes

o Z 10

H ðt ÞD H

2

d t ¼oD Q i

AD H 2 1

r 1 Àr 2 2 2

r 1 þ r 2 À12

1r 1 þ

1r 2

¼2oD Q i

AD H

2

t 3H z

2 (A5)

Now, consider the integral square error of the normalized rate of change of the outlet ow rate in (11-1).

From (6), with Q i(s) ¼D Q i/s and H set (s) ¼0, the Laplace trans-form of the outlet ow rate is

Q oðsÞ ¼t I s þ 1

t H t I s2 þ t I s þ 1 D Q is

(A6)

Therefore, using the property of the Laplace transform for aderivative, i.e., Q 0o(s) L[dQ o(t )/d t ] ¼sQ o(s), the rate of change of the outlet ow rate is obtained by

Q 0oðt

Þ ¼D Q i

r 22 e r 2 t Àr 21 e r 1 t

r 1 Àr 2 for r 1 a r 2 (A7)

Therefore, the integral square error of the normalized rate of change of the outlet ow rate becomes

ð1 Ào ÞZ 10

Q 0oðt ÞQ 0o max

2

d t ¼ ð1 Ào ÞD Q i

Q 0o max 2 1

r 1 Àr 2 2

 Àr 322 À

r 312 þ

2r 21 r 22r 1 þ r 2

¼ ð1 Ào ÞD Q i

Q 0o max 2 1

r 1 Àr 2 2

À12

 ðr 1 þ r 2Þ3À3r 1 r 2ðr 1 þ r 2Þ À

4r 21 r 22r 1 þ r 2

¼ ð1 Ào ÞD Q

iQ 0o max 2 1

2 t H 1 þ1

4z2 (A8)

References

Buckley, P. (1983). Recent advances in averaging level control. In Productivitythrough control technology (pp. 18–21), Houston.

Cheung, T., & Luyben, W. (1979). Liquid-level control in single tanks and cascadesof tanks with P-only and PI feedback controllers. Industrial and Engineering Chemistry Fundamentals , 18(1), 15–21.

Luyben, W., & Buckley, P. S. (1977). A proportional-lag controller. InstrumentationTechnology , 24 (12), 65–68.

MacDonald, K., McAvoy, T., & Tits, A. (1986). Optimal averaging level control. AIChE Journal , 32 , 75–86.

Marlin, T. E. (1995). Process control . Mcgraw-Hill 581.Rivera, D. E., Morari, M., & Skogestad, S. (1986). Internal model control, 4: PID

controller design. Industrial & Engineering Chemistry Process Design andDevelopment , 25 (1), 252–265.

Seki, H., & Ogawa, M. (1998). Japan Patent # 2811041.Wu, K., Yu, C., & Cheung, Y. (2001). A two degree of freedom level control. Journal of

Process Control , 11, 311–319.Yang, D. R., Seborg, D. E., & Mellichamp, D. A. (1994). The inuence of inventory

control dynamics on distillation composition control. Control Engineering Practice , 2(6), 27–32.

ARTICLE IN PRESS

80

75

70

65

60

55

50

45

400 5 10 15

Time (min)

L e v e

l ( %

)

Nominal+20% Error in Dead Time+20% Error in Area

–20% Error in Area

Fig. 6. Effect of modeling error in dead time and area on the control performance.

J. Shin et al. / Control Engineering Practice 16 (2008) 1391–1397 1397