Central Chemical Engineering & Process Techniques

Cite this article: Jang LK (2017) Feedback Control for Liquid Level in a Gravity-Drained Multi-Tank System. Chem Eng Process Tech 3(1): 1037.

*Corresponding author

Larry K. Jang, Department of Chemical Engineering, California State University, Long Beach, CA 90815, USA, Email:

Submitted: 03 May 2017

Accepted: 17 July 2017

Published: 20 July 2017

ISSN: 2333-6633

Copyright© 2017 Jang

OPEN ACCESS

Keywords•Liquid level•Gravity-drained tank•Decoupling•MIMO•IMC

Review Article

Feedback Control for Liquid Level in a Gravity-Drained Multi-Tank SystemLarry K. Jang*Department of Chemical Engineering, California State University, USA

Abstract

Dynamic models for liquid level in a four-tank system are derived in this work by applying the principle of analogy to a single-tank case. In this system, there are two top tanks and two bottom tanks. Each of the two top tanks receives liquid from a feed stream, while discharging liquid to the two bottom tanks by gravity. Each of the two bottom tanks receives liquid from the two top tanks and discharges liquid by gravity from the bottom of the tank. The process models and the disturbance models for the levels of the two bottom tanks showing the effects of both feed streams are derived. Relative gain array (RGA) based on the results of simulation from Loop-Pro’s multi-tank process is used to predict the extent of loop interaction (or coupling). Feedback PID control parameters are obtained by using internal model control (IMC) tuning rule. The performance of the multiple-input/multiple-output (MIMO) feedback control system with and without decoupling strategy is compared and analyzed.

INTRODUCTION Liquid level control of all aspects remains one of the

most important case studies due to its widespread industrial applications. Mathematical models for the dynamic responses of liquid level are more easily perceived due to its simplicity in physical setup. In the literature, open-loop and closed-loop dynamic models as well as tuning rules are well developed for single-tank systems [1,2][3(a)]. In this paper, the process model for an open tank with liquid fed to the top and drained by gravity from the bottom via a hole or valve of fixed opening is reviewed (Figure 1). The transfer functions showing the effects of feed rate on the liquid level and the draining rate are derived. This system is then expanded to one that contains two top tanks and two bottom tanks. Each of the two top tanks receives liquid from one feed stream and discharges liquid to the two bottom tanks by gravity via two valves with fixed openings. Each of the two bottom tanks has two feed streams, one directly from the tank above, and the other from the other top tank. The liquid is then discharged from each of the two bottom tanks via one valve with fixed opening (Figure 2). It is of interest to find the effects of the two feed streams on the liquid levels in all four tanks as well as the draining rates of the six streams leaving the four tanks. In this work, dynamic models for the liquid levels in all four tanks are derived based on the principle of analogy to the single-tank case. Simulation data from a case study in Loop Pro (Control Station, Inc.) are used to generate process models and disturbance models for the system. When a multiple-input/multiple-output (MIMO) feedback control system is established to control the

liquid levels of the two bottom tanks, it is important to identify the extent of loop interactions (i.e., coupling effect) and implement proper strategies to eliminate potential loop interactions (i.e., decoupling). This paper will outline the procedure of tuning individual feedback controllers as well as improving the controller performance by implementing decoupling strategy.

Figure 1 Schematic diagram for a single open-tank, gravity-drained system. The tank has a cross-sectional area of A and a valve with fixed opening at the bottom.

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Figure 2 Schematic diagram of the feedback control loops for a four-tank system.

DYNAMIC MODELS

Transfer functions for an open tank with single inlet stream and single outlet stream

The analysis below is for a vertical open tank with constant cross-sectional area A (m2). Liquid is fed at a rate of fin (m3/s) to the top of the tank and drained by gravity at a rate of fout (m3/s) via a valve or a hole located at the bottom of the tank. The liquid draining rate is governed by liquid level [1][3(a)]

outf C h= (1)

Where C is discharge coefficient, a lumped parameter that includes the effects of gravitational acceleration, size and type of the valve, and valve stem position; and h is liquid level measured from the bottom of the tank. At the initial steady-state (s.s.) condition (denoted by overbar “ – “ ), the rate of accumulation of liquid hold-up in the tank can be described by the equation below:

0 in outdhA f fdt

= = − (2)

Assuming that at t = 0, the feed rate fin starts to deviate from

the initial s.s. value of inf , transient-state volumetric balance of liquid gives

in outdhA f fdt

= − (3)

When Eq. 2 is subtracted from Eq. 3, a non-linear term is encountered and it can be linearized as

in outf f C h C h− = −

( )| h

d C h h hd h −≈ −

= ( )

2−

c h hh

(4)

A first-order transient-state equation in terms of deviation quantities can be obtained:

indH H RFdt

τ + = (5)

where

H h h= − (6)

in in inF f f= − (7)

2Re tan hR sis ceC

≡ = Resistance

2Re tan hR sis ceC

≡ = (8)

– τ = =first order time constant A R (9)

Equation 4 also yields the relationship between Fout and H:

= − =out out outHF f fR (10)

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Laplace transform of Eq. 5 with the initial condition H(t) = 0 at t = 0 yields the transfer function relating the liquid level H(s) to the liquid feed rate Fin(s) in the Laplace domain:

( ) ( ) 1 inRH s F ssτ

=+

(11)

Substituting Eq. 11 into Eq. 10 yields

( ) ( ) 1 ( ) 1out in

H sF s F s

R sτ= =

+ (12)

Transfer functions for a four-tank system

With the transfer functions derived for the single-tank case above, we may expand the system into one that contains four open tanks. In Figure 2, liquid is fed to the two top tanks via control valves at flow rates f1, in and f2, in, respectively. Liquid is discharged via two outlet streams from each of the two top tanks and then fed to the two bottom tanks as shown in Figure 2. The flow rate of liquid discharged from top tank j to bottom tank i is fij, where i = 1,2 and , j = 1,2. Finally, each of the two bottom tanks has one outlet stream with flow rates fi, bot(i = 1,2). The valves in all six outlet streams have fixed openings. The liquid level of the bottom left tank is monitored by level indicator and controller (LIC)#1. The feedback signal is sent to the top-left control valve in order to regulate the liquid feed rate to the top left tank, f1,in. Likewise, the liquid level of the bottom right tank is monitored by level indicator and controller (LIC)#2. The feedback signal is sent to the top-right control valve in order to regulate the liquid feed rate to the top right tank, f2,in.

Assuming that the gravity-drained rates fij from the two top tanks (index j) to the two bottom tanks (index i) are governed by the liquid levels of the two top tanks like the single tank case (Eq. 1), we may express fij as

11 11 1,topf C h= (13)

12 12 2,topf C h= (14)

21 21 1,topf C h= (15)

22 22 2,topf C h= (16)

Where Cij ‘s (i = 1,2; j = 1,2) are the discharge coefficients of the four valves below the two top tanks. Likewise, the gravity-drained rates from the two bottom tanks are governed by their liquid levels:

1, 1 1,out botf C h= (17)

2, 2 2,out botf C h= (18)

where Ci’s (i = 1,2) are the discharge coefficients of the two valves below the two bottom tanks. If the linearization procedure similar to Eq. 4 is employed, one may easily define the resistances of the six valves, four of which located below the two top tanks (Rij’s ) and the other two below the two bottom tanks (Ri,bot’s ):

1,11

11

2 topRCh

= (19)

2,12

12

2 topRCh

= (20)

1,21

21

2 topRCh

= (21)

2,22

22

2 topRCh

= (22)

1,1,

1

2 botbot

hRC

= (23)

2,2,

2

2 botbot

hRC

= (24)

The two valves with resistances R11 and R21 in the two streams leaving top tank no. 1 on the left side is analogous to the two resistors in parallel in an electric circuit. We may define their overall resistance R1, top by Eq. 25:

1, 11 21

1 1 1

topR R R= + (25)

Likewise, the overall resistance R2, top of the two valves in the two streams leaving the top tank no. 2 on the right side can be defined by Eq. 26:

2, 12 22

1 1 1

topR R R= + (26)

By analogy to Eq. 9, we may define the first-order time constants of the four tanks in the system:

1, 1, 1,top top topA Rτ = (27)

2, 2, 2,top top topA Rτ = (28)

1, 1, 1,bot bot botA Rτ = (29)

2, 2, 2,bot bot botA Rτ = (30)

where A’s are the cross-sectional areas of the four tanks. By analogy to Eq. 11, we my write the transfer functions relating liquid levels for the two top tanks to the two feed streams. For simplicity, the symbol “(s)” for the Laplace domain is omitted hereafter:

1,1, 1,

1, 1top

top intop

RH F

sτ=

+ (31)

2,2, 2,

2, 1top

top intop

RH F

sτ=

+ (32)

where

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Figure 3 Screen shot of simulation for response of liquid level to a doublet input by using Loop Pro algorithm.

3.8

3.9

4.0

4.1

60.3

61.2

62.1

63.0

0.0 32.3 64.6 96.9 129.2 161.5 193.8 226.1

Loop-Pro: Design ToolsModel: Second Order Overdamped File Name: Multi Tank G11 G21.txt

Goodness of Fit: R-Squared = 0.9972, SSE = 0.03101Gain (K) = 0.07867, 1st Time Constant (min) = 10.41, 2nd Time Constant (min) = 11.41

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Figure 4 Result of data fitting to the exact second-order model for the process model G11.

1, 1, 1,in in inF f f= − (33)

2, 2, 2,in in inF f f= − (34)

1,1, 1, toptop topH h h= − (35)

2,2, 2, toptop topH h h= − (36)

Since the flow rates fij leaving the bottom of the two top tanks are governed by the liquid level in the top tank j and resistance Rij, one may derive Fij analogous to Eq. 12:

1,

1, 1111 1,

11 1, 1

top

topin

top

RH R

F FR sτ

= =

+ (37)

1,

1, 2121 1,

21 1, 1

top

topin

top

RH R

F FR sτ

= =

+ (38)

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2,

2, 1212 2,

12 2, 1

top

topin

top

RH R

F FR sτ

= =

+ (39)

2,

2, 2222 2,

22 2, 1

top

topin

top

RH R

F FR sτ

= =

+ (40)

where

11 11 11F f f= − (41)

12 12 12F f f= − (42)

21 21 21F f f= − (43)

22 22 22F f f= − (44)

For the two bottom tanks, each tank receives two inlet streams. The combined flow rate of the two inlet streams to bottom tank no. 1 on the left side is (f11 + f12). Likewise, the combined flow rate of the two inlet streams to bottom tank no. 2 on the right side is (f21 + f22). By analogy to Eq. 11, one can derive the following transfer functions for the liquid levels of the two bottom tanks by substituting Eqs. 37-44 into Eqs. 45 and 46:

( )

1,1, 1,

1,11 12

1,

11 1, 12 2,

1

botbot bot

bot

bot

in in

H h hR

F Fs

G F G Fτ

= −

= ++

= +

(45)

( )

2,2, 2,

2,21 22

2,

21 1, 22 2,

1τ

= −

= ++

= +

botbot bot

bot

bot

in in

H h hR

F Fs

G F G F

(46)

where

( )

1,

1, 1111

1, 1,( 1) 1τ τ

=

+ +

top

bot

bot top

RR R

Gs s

(47)

( )

2,

1, 1212

1, 2,( 1) 1τ τ

=

+ +

top

bot

bot top

RR R

Gs s

(48)

( )

1,

2, 2121

2, 1,( 1) 1τ τ

=

+ +

top

bot

bot top

RR R

Gs s

(49)

( )

2,

2, 2222

2, 2,( 1) 1τ τ

=

+ +

top

bot

bot top

RR R

Gs s

(50)

The resultant transfer functions can be expressed as linear combinations showing the effects of the liquid feed rates to the two top tanks on the liquid levels in the two bottom tanks. According to the control schematic diagram (Figure 2), the liquid levels of the two bottom tanks h1,bot and h2,bot would be controlled by regulating f1,in and f2,in, respectively. The transfer functions G11 and G22 are then considered the process models showing effects of F1,in on H1,bot and F2,in on and H2,bot, respectively. On the other hand, the transfer function G12 and G21 are considered the disturbance models showing the effects of F2,in on H1,bot and F1,in on H2,bot, respectively. Again, by analogy to Eq. 12,

1,1,

1,

botout

bot

HF

R= (51)

2,2,

2,

botout

bot

HF

R= (52)

Where

1, 1, 1,= −out out outF f f (53)

2, 2, 2,= −out out outF f f (54)

Effect of initial steady state on model parameters

The initial steady-state condition of the four-tank system

depends on the feed rates 1,inf and 2,inf and the discharge

coefficients of the six valves below the four tanks. Since the sum of the two outlet flow rates equals to the inlet flow rate for each

of the two top tanks at steady state, one may calculate the steady-

state liquid levels 1,toph and 2,toph by Eqs. 13-16 and Eqs.55-58:

=

1, 11 21

1, 1,11 21

in

top top

f f f

C h C h

= +

+ (55)

=

2, 12 22

2, 2,12 22

in

top top

f f f

C h C h

= +

+ (56)

Or,2

1,1,

11 21

intop

fh

C C

= + (57)

2

2,2,

12 22

intop

fh

C C

= + (58)

Once 1,toph and 2,toph are calculated, one may calculate the

four discharge flow rates ( )1,2; 1,2ij i jf = = from the two top tanks at steady state according to Eqs. 13-16. In turn, one may further calculate the steady-state liquid levels of the two bottom

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tanks due to the fact that the sum of the two inlet flow rates equals to the outlet flow rate for each of the two bottom tanks:

11 12 1,

1,1

out

bot

f f

C

f

h

+ =

= (59)

21 22 2,

2,2

out

bot

f f

C

f

h

+ =

= (60)

Or 2

1, 2,11 121,

1

top topbot

ChC hhC

+ =

(61)

2

1, 2,21 222,

2

top topbot

ChC hhC

+ =

(62)

It is evident that the initial steady-state condition is affected by the flow rates of the two feed streams. In turn, the resistances of all six valves below the four tanks and the gains and time constants of the process and disturbance models in Eqs. 47-50 are affected as well. This is typical of any process units whose dynamic models contain non-linear terms.

SIMULATION FOR OPEN LOOP RESPONSESSimulation is done by using the multi-tank case of Loop Pro

(Control Station, Inc.). A snapshot of the simulation procedure is shown in Figure 3. The two constant pumping rates D1 and D2 from the two bottom tanks are set at zero. While the controller output to the inlet control valve on the right side is maintained at 61.5% in the manual mode, the controller output to the inlet control valve on the left side is changed from 61.5% to 63.0% and maintained at 63.0% until both liquid levels reach new plateaus, then dropped to 60.0% and maintained at 60.0% until both liquid levels reach other new plateaus. Finally, the controller output is increased to 61.5% until the initial steady state is reached. This pattern of input is called doublet input, a revised step or pulse input. Similar procedure is done by changing the controller output to the control valve on the right side while maintaining the controller output to the control valve on the left side at 61.5%.

The response data is collected and the overdamped second order model without dead time is selected when using Design Tools of Loop Pro to find the best-fit transfer functions. Loop Pro gives the initial results for the critically-damped case (with identical time constants for each second-order fit):

2 2

2 2

0.07867 0.04585(10.91 1) (10.91 1)

0.04007 0.08452(11.91 1) (11.91 1)

initial s sG

s s

+ + = + +

(63)

Since the model developed in this work suggests that there may be four distinct time constants, the time constants obtained in the initial fit are artificially fine-tuned while ensuring reasonably good fit (with goodness of fit at R2 greater than 0.996). The final results are presented in Figures 4-7 and the refined G matrix for the transfer functions are

( ) ( )

( ) ( )

0.07867 0.0458511.41 1 (10.41 1) 10.41 1 (11.61 1)

0.04007 0.0845211.41 1 (12.41 1) 11.61 1 (12.41 1)

s s s sG

s s s s

+ + + + = + + + +

(64)

The fine-tuned best-fit time constants are listed in Table 1.

In this simulation, it is assumed that the controller output to the control valves is proportional to the flow rate, which is a reasonable assumption if one uses control valves with linear trims and the feed streams have constant source pressures.

PREDICTION OF THE EXTENT OF LOOP INTERACTION

The transfer functions Gij (i, j = 1, 2) in Equation 64 have very close time constants in the denominators. Therefore, one may simply use the gains to analyze the extent of loop interaction:

11 12

21 22

K KK

K K

=

(65)

where K11 = 0.07867, K12 = 0.04585, K21 = 0.04007, and K22 = 0.08452. One may then calculate the parameter λ in the relative gain array (RGA) [3(c)][4(a)]:

11 12

21 22

11

RGAλ λ λ λλ λ λ λ

− = = −

(66)

where

12 21

11 22

1 1.3821 K K

K K

λ = =−

(67)

Note that the parameter λ means the ratio of the process gain for the bottom tank on the left side when both loops are open to that when the first loop is open while the second loop is closed. The fact that the parameter λ being greater than unity indicates that the controller output to the second loop (in order to maintain the level in the bottom tank on the right side) acts to reduce the response of the level in the bottom tank on the left side. Therefore, the parameter λ is a useful indicator for the extent of loop interaction. In this example, the extent of loop interaction is not severe because the value of λ is just somewhat above 1.0. If loop interaction were absent, we would expect the λ value to be exactly 1.0. If the second controller output were to increase the response of the process variable in the first loop, we would expect 0 <λ<

Table 1: Summary of the time constants for the simulation results of the four-tank system from Loop-Pro.

τ1,top (min) τ1,bot (min) τ2,top (min) τ2,bot (min)

11.41 10.41 11.61 12.41

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3.8

3.9

4.0

4.1

60.3

61.2

62.1

63.0

0.0 31.7 63.4 95.1 126.8 158.5 190.2 221.9 253.6

Loop-Pro: Design ToolsModel: Second Order Overdamped File Name: Multi Tank G12 G22.txt

Goodness of Fit: R-Squared = 0.9967, SSE = 0.01273Gain (K) = 0.04585, 1st Time Constant (min) = 11.41, 2nd Time Constant (min) = 11.11

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cess

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Time (min)

Figure 5 Result of data fitting to the exact second-order model for the disturbance model G21.

3.8

3.9

4.0

60.3

61.2

62.1

63.0

0.0 32.3 64.6 96.9 129.2 161.5 193.8 226.1

Loop-Pro: Design ToolsModel: Second Order Overdamped File Name: Multi Tank G11 G21.txt

Goodness of Fit: R-Squared = 0.9965, SSE = 0.009501Gain (K) = 0.04007, 1st Time Constant (min) = 11.41, 2nd Time Constant (min) = 12.61

Pro

cess

Var

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Time (min)

Figure 6 Result of data fitting to the exact second-order model for the disturbance model G12.

1.0. On the other hand, if λ value is very large or even negative, we may conclude that the extent of loop interaction to be severe and/or the process variables and the manipulated variables may be paired incorrectly [4(a)]. When severe loop interaction exists, we may need to implement strategies such as decoupling or revising manipulated variable/process variable pairs [3(d)][4(b)].

IMC TUNING PARAMETERSIf the internal model control (IMC) tuning method is used,

the PID (proportional-integral-derivative) tuning parameters for both loops can be calculated by using process model parameters (from G11 and G22) with the expected closed-loop time constant τc as the adjustable parameter. For a general exact second-order process model

( )( )1 21 1p

p

KG

τ τ=

+ + (68)

PID tuning parameters can be determined by using the IMC tuning method for single-input-single-output (SISO) [3(b)][5]:

1 21 cp c

K Proportional GainK

τ ττ+

≡ = (69)

1 2 I Integral Timeτ τ τ≡ = + (70)

1 2

1 2

D Derivative Time τ τττ τ

≡ =+

(71)

Therefore, by using the model parameters from Eq. 64, PID tuning parameters may be calculated by Eqs. 69-71with the expected closed-loop time constant for both loops chosen arbitrarily at τc = 12.0 min.

Loop 1

Kc = 23.11 % /m

τ I = 21.82 min.

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τD = 5.44 min.

Loop 2

Kc = 23.689 %/m

τ I = 24.02 min.

τD = 6.00 min.

Since the extent of loop interaction is mild in this case, tuning rule based on SISO may yield satisfactory tuning parameters for multiple-input-multiple-output (MIMO) systems. However, in the presence of significant loop interactions, the tuning parameters based on the SISO must be detuned to suit MIMO cases. The procedures of detuning control parameters are recommended in the literature [6][7][3(e)][4(c)][8].

FEEDBACK CONTROL WITH AND WITHOUT DECOUPLING

The control block diagram for the feedback control of single loops showing the effect of loop interactions in a MIMO system is given in Figure 8. With the above PID tuning parameters entered to the PID controllers for both loops and the data sample time chosen at 6.0 seconds, the performance of the control system in tracking level setpoints in both loops are shown in Figure 9. Level setpoint for the bottom tank on the left side (h1,bot,sp) is changed from 3.96 meters to 4.5 meters and back to 3.96 meters, while maintaining the level setpoint of the bottom tank on the right side (h2, bot, sp) at 3.92 meters. Similar simulation is done by changing h2, bot, sp from 3.92 meters to 4.5 meters and back to 3.92 meters while maintaining h1,bot,sp at 3.96 meters (Figure 9). It appears that the PID controller implemented according to the procedure developed in this work provides satisfactory performance of setpoint tracking for both liquid levels. However, while the level for the left bottom tank (h1) is responding to a change in h1, setpoint, the level for the right bottom tank (h2) deviates from h2, setpoint due to the interference from Loop 1. The reverse is also true. The simulation results suggest that the loop interactions cannot be eliminated effectively by two individual PID feedback loops.

The control block diagram for the feedback control of single loops using two-way decoupling strategy to eliminate or minimize loop interactions is shown in Figure 10. By choosing “PID with Decoupler” for both loops in the Loop Pro’s multi-tank case study, the same PID control parameters above are entered and data sample time is maintained at 6.0 seconds. The decoupler D1 in Figure 10 is essentially a feed forward controller that would reject the disturbance (or interference) from the controller of Loop 2 on process variable of Loop 1 (i.e., h1,bot). Likewise, the decoupler D2 in Figure 10 is a feed forward controller that would reject the disturbance (or interference) from the controller of Loop 1 on process variable of Loop 2 (i.e., h2,bot). The decouplers used in this simulation are

( )

( )

121

11

0.0458510.41 1 (11.61 1)

0.0786711.41 1 (10.41 1)

s sGDG

s s

−+ +−

= =

+ +

(72)

( )

( )

212

22

0.0400711.41 1 (12.41 1)

0.0845211.61 1 (12.41 1)

s sGDG

s s

−+ +−

= =

+ +

(73)

Similar setpoint-tracking simulations are done as in the case without decoupling; the results are shown in Figure 11. By comparing (Figure 9 and 11), it appears that the controllers move more aggressively if the decoupling strategy is implemented. This is obvious due to the additional feedforward action from the decouplers. One may observe a striking contrast in the response of both liquid levels. In the case without decoupling (Figure 9), when h1,bot and h2,bot are responding to their respective setpoint changes, the level of h2,bot and h1,bot , respectively, are disturbed somewhat from their original setpoints. However, such disturbances are almost fully eliminated when the decoupling strategy is implemented (Figure 11). When h1,bot is responding to step changes in its setpoint, h2,bot pretty much stays very near

3.7

3.8

3.9

4.0

4.1

60.3

61.2

62.1

63.0

0.0 31.7 63.4 95.1 126.8 158.5 190.2 221.9 253.6

Loop-Pro: Design ToolsModel: Second Order Overdamped File Name: Multi Tank G12 G22.txt

Goodness of Fit: R-Squared = 0.9966, SSE = 0.04246Gain (K) = 0.08452, 1st Time Constant (min) = 11.11, 2nd Time Constant (min) = 12.61

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Time (min)

Figure 7 Result of data fitting to the exact second-order model for process model G22.

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Figure 8 Block diagram for 2x2 MIMO feedback control scheme without using the decoupling strategy.

Figure 9 Setpoint tracking for the two bottom tanks using PID settings for both feedback controllers without using decoupling strategy.

its setpoint value. The reverse is also true. Evidently, the control strategy developed in this work not only successfully identifies model and model parameters, but also develops an effective decoupling strategy to eliminate loop interactions.

If one examines Eqs. 72 and 73, it is evident that the time constants involved in this system are very close to each other. Therefore, one may ignore the dynamic part of the decouplers and simply use static decouplers D1 ~ -0.04585/0.07867 and D2 ~ -0.04007/0.08452. The results are very similar to those in Figure 11 and not demonstrated here.

CONCLUSIONThe transfer functions for a four-tank system illustrated

in this work can be derived by using the principle of analogy to the single-tank case, with resistances of the six valves

below the four tanks and first-order time constants of the four tanks clearly defined. The final results show that the transfer functions of the liquid levels of the two bottom tanks are linear combinations of the effects of the two feed streams to the system. Simulation results for the four-tank system in Loop Pro show that the dynamic responses of process variables to the changes in controller outputs fit the expected overdamped second-order behaviors. With the process models and disturbance models clearly developed and model parameters obtained, one may identify the extent of loop interactions using relative gain array. Model-based controller tuning method such as IMC provides adequate PID tuning parameters for the two feedback controllers. However, the system encountered in this work exhibits certain degree of loop interaction by using two individual PID feedback controllers. With decoupling strategy applied to both loops, loop interactions are almost eliminated entirely.

Central

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Figure 10 Block diagram for 2x2 MIMO feedback control scheme with decoupling strategy.

Figure 11 Setpoint tracking for the two bottom tanks using PID settings for both feedback controllers with decoupling strategy implemented. The legends are the same as those of Figure 9.

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Jang LK (2017) Feedback Control for Liquid Level in a Gravity-Drained Multi-Tank System. Chem Eng Process Tech 3(1): 1037.

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