emical Engineering & Process Tecniques eview Article eedback ontrol for Liquid Level in a Gravity-Drained Multi-Tank System Larry K. Jang* Department of emical Engineering, alifornia State University, USA Abstract Dynamic models for liquid level in a four-tank system are derived in tis work by applying te principle of analogy to a single-tank case. In tis system, tere are two top tanks and two tom tanks. Eac of te two top tanks receives liquid from a feed stream, wile discarging liquid to te two tom tanks by gravity. Eac of te two tom tanks receives liquid from te two top tanks and discarges liquid by gravity from te tom of te tank. Te process models and te disturbance models for te levels of te two tom tanks sowing te effects of feed streams are derived. elative gain array (GA) based on te results of simulation from Loop-Pro s multi-tank process is used to predict te extent of loop interaction (or coupling). eedback PID control parameters are obtained by using internal model control (IM) tuning rule. Te performance of te multiple-input/multiple-output (MIMO) feedback control system wit and witout decoupling strategy is compared and analyzed. *orresponding autor Larry K. Jang, Department of emical Engineering, alifornia State University, Long Beac, A 90815, USA, Submitted: 03 May 017 Accepted: 17 July 017 Publised: 0 July 017 ISSN: 333-6633 opyrigt 017 Jang OPEN AESS Keywords Liquid level Gravity-drained tank Decoupling MIMO IM INTODUTION Liquid level control of all aspects remains one of te most important case studies due to its widespread industrial applications. Matematical models for te dynamic responses of liquid level are more easily perceived due to its simplicity in pysical setup. In te literature, open-loop and closed-loop dynamic models as well as tuning rules are well developed for single-tank systems [1,][3(a)]. In tis paper, te process model for an open tank wit liquid fed to te top and drained by gravity from te tom via a ole or valve of fixed opening is reviewed (igure 1). Te transfer functions sowing te effects of feed rate on te liquid level and te draining rate are derived. Tis system is ten expanded to one tat contains two top tanks and two tom tanks. Eac of te two top tanks receives liquid from one feed stream and discarges liquid to te two tom tanks by gravity via two valves wit fixed openings. Eac of te two tom tanks as two feed streams, one directly from te tank above, and te oter from te oter top tank. Te liquid is ten discarged from eac of te two tom tanks via one valve wit fixed opening (igure ). It is of interest to find te effects of te two feed streams on te liquid levels in all four tanks as well as te draining rates of te six streams leaving te four tanks. In tis work, dynamic models for te liquid levels in all four tanks are derived based on te principle of analogy to te single-tank case. Simulation data from a case study in Loop Pro (ontrol Station, Inc.) are used to generate process models and disturbance models for te system. Wen a multiple-input/multiple-output (MIMO) feedback control system is establised to control te liquid levels of te two tom tanks, it is important to identify te extent of loop interactions (i.e., coupling effect) and implement proper strategies to eliminate potential loop interactions (i.e., decoupling). Tis paper will outline te procedure of tuning individual feedback controllers as well as improving te controller performance by implementing decoupling strategy. igure 1 Scematic diagram for a single open-tank, gravity-drained system. Te tank as a cross-sectional area of A and a valve wit fixed opening at te tom. ite tis article: Jang LK (017) eedback ontrol for Liquid Level in a Gravity-Drained Multi-Tank System. em Eng Process Tec 3(1): 1037.
Jang (017) igure Scematic diagram of te feedback control loops for a four-tank system. DYNAMI MODELS Transfer functions for an open tank wit single inlet stream and single outlet stream Te analysis below is for a vertical open tank wit constant cross-sectional area A (m ). Liquid is fed at a rate of f in (m 3 /s) to te top of te tank and drained by gravity at a rate of f out (m 3 /s) via a valve or a ole located at te tom of te tank. Te liquid draining rate is governed by liquid level [1][3(a)] fout (1) Were is discarge coefficient, a lumped parameter tat includes te effects of gravitational acceleration, size and type of te valve, and valve stem position; and is liquid level measured from te tom of te tank. At te initial steady-state (s.s.) condition (denoted by overbar ), te rate of accumulation of liquid old-up in te tank can be described by te equation below: d A 0 fin f out dt () Assuming tat at t 0, te feed rate f in starts to deviate from te initial s.s. value of liquid gives d A f f f in, transient-state volumetric balance of in out dt (3) Wen Eq. is subtracted from Eq. 3, a non-linear term is encountered and it can be linearized as f f in out d c ( ) ( ) d A first-order transient-state equation in terms of deviation quantities can be obtained: d τ + in (5) dt were (4) (6) in fin fin (7) esistance (8) τ first order time constant A (9) Equation 4 also yields te relationsip between out and : out fout fout (10) em Eng Process Tec 3(1): 1037 (017) /10
Jang (017) Laplace transform of Eq. 5 wit te initial condition (t) 0 at t 0 yields te transfer function relating te liquid level (s) to te liquid feed rate in (s) in te Laplace domain: s s τ s in ( ) ( ) Substituting Eq. into Eq. 10 yields ( ) 1 s out ( s) in ( s) τ s Transfer functions for a four-tank system () (1) Wit te transfer functions derived for te single-tank case above, we may expand te system into one tat contains four open tanks. In igure, liquid is fed to te two top tanks via control valves at flow rates f 1, in and f, in, respectively. Liquid is discarged via two outlet streams from eac of te two top tanks and ten fed to te two tom tanks as sown in igure. Te flow rate of liquid discarged from top tank j to tom tank i is f ij, were i 1, and, j 1,. inally, eac of te two tom tanks as one outlet stream wit flow rates f i, (i 1,). Te valves in all six outlet streams ave fixed openings. Te liquid level of te tom left tank is monitored by level indicator and controller (LI)#1. Te feedback signal is sent to te top-left control valve in order to regulate te liquid feed rate to te top left tank, f 1,in. Likewise, te liquid level of te tom rigt tank is monitored by level indicator and controller (LI)#. Te feedback signal is sent to te top-rigt control valve in order to regulate te liquid feed rate to te top rigt tank, f,in. Assuming tat te gravity-drained rates f ij from te two top tanks (index j) to te two tom tanks (index i) are governed by te liquid levels of te two top tanks like te single tank case (Eq. 1), we may express f ij as f 1,top (13) f1 1,top (14) f1 1 1,top (15) f,top (16) Were ij s (i 1,; j 1,) are te discarge coefficients of te four valves below te two top tanks. Likewise, te gravitydrained rates from te two tom tanks are governed by teir liquid levels: f (17) 1, out 1 f (18), out, were i s (i 1,) are te discarge coefficients of te two valves below te two tom tanks. If te linearization procedure similar to Eq. 4 is employed, one may easily define te resistances of te six valves, four of wic located below te two top tanks ( ij s ) and te oter two below te two tom tanks ( i, s ): 1 1, (19), top (0) 1 (1) 1, top () 1, (3) 1, (4) Te two valves wit resistances and 1 in te two streams leaving top tank no. 1 on te left side is analogous to te two resistors in parallel in an electric circuit. We may define teir overall resistance by Eq. 5: 1 1 1 + (5) 1 Likewise, te overall resistance, top of te two valves in te two streams leaving te top tank no. on te rigt side can be defined by Eq. 6: 1 1 1 + (6), top 1 By analogy to Eq. 9, we may define te first-order time constants of te four tanks in te system: τ A (7) τ A (8), top, top, top τ A (9) τ A (30),,, were A s are te cross-sectional areas of te four tanks. By analogy to Eq., we my write te transfer functions relating liquid levels for te two top tanks to te two feed streams. or simplicity, te symbol (s) for te Laplace domain is omitted ereafter: 1, in τs, top, top, in τ, tops were (31) (3) em Eng Process Tec 3(1): 1037 (017) 3/10
Jang (017) igure 3 Screen sot of simulation for response of liquid level to a doublet input by using Loop Pro algoritm. Model: Second Order Overdamped Loop-Pro: Design Tools ile Name: Multi Tank G G1.txt Process Variable 4.1 4.0 3.9 3.8 Manipulated Variable 63.0 6.1 61. 60.3 0.0 3.3 64.6 96.9 19. 161.5 193.8 6.1 Time (min) Gain (K) 0.07867, 1st Time onstant (min) 10.41, nd Time onstant (min).41 Goodness of it: -Squared 0.997, SSE 0.03101 igure 4 esult of data fitting to te exact second-order model for te process model G. f f (33) 1, in 1, in 1, in f f (34), in, in, in (35) (36), top, top, top Since te flow rates f ij leaving te tom of te two top tanks are governed by te liquid level in te top tank j and resistance ij, one may derive ij analogous to Eq. 1: 1, in τs 1 1 1, in 1 τs (37) (38) em Eng Process Tec 3(1): 1037 (017) 4/10
Jang (017), top, top 1 1, in 1 τ, tops, top, top, in τ, tops were (39) (40) f f (41) 1 f1 f 1 (4) 1 f1 f 1 (43) f f (44) or te two tom tanks, eac tank receives two inlet streams. Te combined flow rate of te two inlet streams to tom tank no. 1 on te left side is (f + f 1 ). Likewise, te combined flow rate of te two inlet streams to tom tank no. on te rigt side is (f 1 + f ). By analogy to Eq., one can derive te following transfer functions for te liquid levels of te two tom tanks by substituting Eqs. 37-44 into Eqs. 45 and 46: 1, ( ) + 1 τ1, s G + G 1, in 1, in,,, ( ), 1 + τ, s G + G were 1 1, in, in tops ) (45) (46) G (47) S ) 1,, top 1 G 1 (48) S ), tops ), 1 G 1 (49), S ) 1, tops ), top, G (50), S ), tops ) Te resultant transfer functions can be expressed as linear combinations sowing te effects of te liquid feed rates to te two top tanks on te liquid levels in te two tom tanks. According to te control scematic diagram (igure ), te liquid levels of te two tom tanks 1, and, would be controlled by regulating f 1,in and f,in, respectively. Te transfer functions G and G are ten considered te process models sowing effects of 1,in on 1, and,in on and,, respectively. On te oter and, te transfer function G 1 and G 1 are considered te disturbance models sowing te effects of,in on 1, and 1,in on,, respectively. Again, by analogy to Eq. 1, 1, out (51) 1,,, out (5), Were f f (53) 1, out 1, out 1, out f f (54), out, out, out Effect of initial steady state on model parameters Te initial steady-state condition of te four-tank system depends on te feed rates f 1,in and f,in and te discarge coefficients of te six valves below te four tanks. Since te sum of te two outlet flow rates equals to te inlet flow rate for eac of te two top tanks at steady state, one may calculate te steadystate liquid levels 1,top and,top by Eqs. 13-16 and Eqs.55-58: f 1, in f + f 1 (55) + 1, top 1 f f + f, in 1 1 top + Or, f 1, in 1, top + 1,, top (56) (57) f, in, top (58) 1 + Once 1,top and,top are calculated, one may calculate te four discarge flow rates ( 1, ; 1, ) f ij i j from te two top tanks at steady state according to Eqs. 13-16. In turn, one may furter calculate te steady-state liquid levels of te two tom em Eng Process Tec 3(1): 1037 (017) 5/10
tanks due to entral te fact tat te sum of te two inlet flow rates equals to te outlet flow rate for eac of te two tom tanks: f + f f 1 1, out f 1 + f f 1, out Or,, + 1, top 1 1 +, top (59) (60) (61) (6) It is evident tat te initial steady-state condition is affected by te flow rates of te two feed streams. In turn, te resistances of all six valves below te four tanks and te gains and time constants of te process and disturbance models in Eqs. 47-50 are affected as well. Tis is typical of any process units wose dynamic models contain non-linear terms. SIMULATION O OPEN LOOP ESPONSES Simulation is done by using te multi-tank case of Loop Pro (ontrol Station, Inc.). A snapsot of te simulation procedure is sown in igure 3. Te two constant pumping rates D1 and D from te two tom tanks are set at zero. Wile te controller output to te inlet control valve on te rigt side is maintained at 61.5% in te manual mode, te controller output to te inlet control valve on te left side is canged from 61.5% to 63.0% and maintained at 63.0% until liquid levels reac new plateaus, ten dropped to 60.0% and maintained at 60.0% until liquid levels reac oter new plateaus. inally, te controller output is increased to 61.5% until te initial steady state is reaced. Tis pattern of input is called doublet input, a revised step or pulse input. Similar procedure is done by canging te controller output to te control valve on te rigt side wile maintaining te controller output to te control valve on te left side at 61.5%. Te response data is collected and te overdamped second order model witout dead time is selected wen using Design Tools of Loop Pro to find te best-fit transfer functions. Loop Pro gives te initial results for te critically-damped case (wit identical time constants for eac second-order fit): 0.07867 0.04585 initial (10.91s ) (10.91s ) G 0.04007 0.0845 (.91s ) (.91s ) (63) Jang (017) Since te model developed in tis work suggests tat tere may be four distinct time constants, te time constants obtained in te initial fit are artificially fine-tuned wile ensuring reasonably good fit (wit goodness of fit at greater tan 0.996). Te final results are presented in igures 4-7 and te refined G matrix for te transfer functions are 0.07867 0.04585 (.41s )(10.41s ) ( 10.41s )(.61s ) G 0.04007 0.0845 (.41s )(1.41s ) (.61s )(1.41s ) Te fine-tuned best-fit time constants are listed in Table 1. (64) In tis simulation, it is assumed tat te controller output to te control valves is proportional to te flow rate, wic is a reasonable assumption if one uses control valves wit linear trims and te feed streams ave constant source pressures. PEDITION O TE EXTENT O LOOP INTEATION Te transfer functions G ij (i, j 1, ) in Equation 64 ave very close time constants in te denominators. Terefore, one may simply use te gains to analyze te extent of loop interaction: K K K 1 K1 K (65) were K 0.07867, K 1 0.04585, K 1 0.04007, and K 0.0845. One may ten calculate te parameter λ in te relative gain array (GA) [3(c)][4(a)]: GA were λ λ λ 1 λ λ λ 1 λ λ 1 1 1 λ 1.38 1 1 1 K K K K (66) (67) Note tat te parameter λ means te ratio of te process gain for te tom tank on te left side wen loops are open to tat wen te first loop is open wile te second loop is closed. Te fact tat te parameter λ being greater tan unity indicates tat te controller output to te second loop (in order to maintain te level in te tom tank on te rigt side) acts to reduce te response of te level in te tom tank on te left side. Terefore, te parameter λ is a useful indicator for te extent of loop interaction. In tis example, te extent of loop interaction is not severe because te value of λ is just somewat above 1.0. If loop interaction were absent, we would expect te λ value to be exactly 1.0. If te second controller output were to increase te response of te process variable in te first loop, we would expect 0 <λ< 1.0. On te oter and, if λ value is very large or even negative, we Table 1: Summary of te time constants for te simulation results of te four-tank system from Loop-Pro. τ 1,top (min) τ 1, (min) τ,top (min) τ, (min).41 10.41.61 1.41 em Eng Process Tec 3(1): 1037 (017) 6/10
Jang (017) Process Variable Manipulated Variable Model: Second Order Overdamped 4.1 4.0 3.9 3.8 63.0 6.1 61. 60.3 Loop-Pro: Design Tools ile Name: Multi Tank G1 G.txt 0.0 31.7 63.4 95.1 16.8 158.5 190. 1.9 53.6 Time (min) Gain (K) 0.04585, 1st Time onstant (min).41, nd Time onstant (min). Goodness of it: -Squared 0.9967, SSE 0.0173 igure 5 esult of data fitting to te exact second-order model for te disturbance model G 1. Process Variable Manipulated Variable Model: Second Order Overdamped 4.0 3.9 3.8 63.0 6.1 61. 60.3 Loop-Pro: Design Tools ile Name: Multi Tank G G1.txt 0.0 3.3 64.6 96.9 19. 161.5 193.8 6.1 Time (min) Gain (K) 0.04007, 1st Time onstant (min).41, nd Time onstant (min) 1.61 Goodness of it: -Squared 0.9965, SSE 0.009501 igure 6 esult of data fitting to te exact second-order model for te disturbance tuning model metod G 1 for. single-input-single-output (SISO) [3(b)][5]: may conclude tat te extent of loop interaction to be severe and/ or te process variables and te manipulated variables may be paired incorrectly [4(a)]. Wen severe loop interaction exists, we may need to implement strategies suc as decoupling or revising manipulated variable/process variable pairs [3(d)][4(b)]. IM TUNING PAAMETES If te internal model control (IM) tuning metod is used, te PID (proportional-integral-derivative) tuning parameters for loops can be calculated by using process model parameters (from G and G ) wit te expected closed-loop time constant τ c as te adjustable parameter. or a general exact second-order process model G p K ) ) p 1 (68) PID tuning parameters can be determined by using te IM K 1 c Proportional Gain (69) K p τ c 1 τ + τ τ I Integral Time τ1 + τ (70) τ1τ τ D Derivative Time τ + τ (71) 1 Terefore, by using te model parameters from Eq. 64, PID tuning parameters may be calculated by Eqs. 69-71wit te expected closed-loop time constant for loops cosen arbitrarily at τ c 1.0 min. Loop 1 K c 3. % /m τ I 1.8 min. τ D 5.44 min. Loop em Eng Process Tec 3(1): 1037 (017) 7/10
Jang (017) K c 3.689 %/m τ I 4.0 min. τ D 6.00 min. Since te extent of loop interaction is mild in tis case, tuning rule based on SISO may yield satisfactory tuning parameters for multiple-input-multiple-output (MIMO) systems. owever, in te presence of significant loop interactions, te tuning parameters based on te SISO must be detuned to suit MIMO cases. Te procedures of detuning control parameters are recommended in te literature [6][7][3(e)][4(c)][8]. EEDBAK ONTOL WIT AND WITOUT DEOUPLING Te control block diagram for te feedback control of single loops sowing te effect of loop interactions in a MIMO system is given in igure 8. Wit te above PID tuning parameters entered to te PID controllers for loops and te data sample time cosen at 6.0 seconds, te performance of te control system in tracking level setpoints in loops are sown in igure 9. Level setpoint for te tom tank on te left side ( 1,,sp ) is canged from 3.96 meters to 4.5 meters and back to 3.96 meters, wile maintaining te level setpoint of te tom tank on te rigt side (,, sp ) at 3.9 meters. Similar simulation is done by canging,, sp from 3.9 meters to 4.5 meters and back to 3.9 meters wile maintaining 1,,sp at 3.96 meters (igure 9). It appears tat te PID controller implemented according to te procedure developed in tis work provides satisfactory performance of setpoint tracking for liquid levels. owever, wile te level for te left tom tank ( 1 ) is responding to a cange in 1, setpoint, te level for te rigt tom tank ( ) deviates from, setpoint due to te interference from Loop 1. Te reverse is also true. Te simulation results suggest tat te loop interactions cannot be eliminated effectively by two individual PID feedback loops. Te control block diagram for te feedback control of single loops using two-way decoupling strategy to eliminate or minimize loop interactions is sown in igure 10. By coosing PID wit Decoupler for loops in te Loop Pro s multi-tank case study, te same PID control parameters above are entered and data sample time is maintained at 6.0 seconds. Te decoupler D 1 in igure 10 is essentially a feed forward controller tat would reject te disturbance (or interference) from te controller of Loop on process variable of Loop 1 (i.e., 1, ). Likewise, te decoupler D in igure 10 is a feed forward controller tat would reject te disturbance (or interference) from te controller of Loop 1 on process variable of Loop (i.e.,, ). Te decouplers used in tis simulation are D D G 1 1 G G 1 G 0.04585 10.41 (.61s ) 0.07867.41 (10.41s ) ( s ) ( s ) 0.04007.41 (1.41s ) 0.0845.61 (1.41s ) ( s ) ( s ) (7) (73) Similar setpoint-tracking simulations are done as in te case witout decoupling; te results are sown in igure. By comparing (igure 9 and ), it appears tat te controllers move more aggressively if te decoupling strategy is implemented. Tis is obvious due to te additional feedforward action from te decouplers. One may observe a striking contrast in te response of liquid levels. In te case witout decoupling (igure 9), wen 1, and, are responding to teir respective setpoint canges, te level of, and 1,, respectively, are disturbed somewat from teir original setpoints. owever, suc disturbances are almost fully eliminated wen te decoupling strategy is implemented (igure ). Wen 1, is responding to step canges in its setpoint,, pretty muc stays very near its setpoint value. Te reverse is also true. Evidently, te control strategy developed in tis work not only successfully identifies model and model parameters, but also develops an effective Manipulated Variable Process Variable Model: Second Order Overdamped 4.1 4.0 3.9 3.8 3.7 63.0 6.1 61. 60.3 Loop-Pro: Design Tools ile Name: Multi Tank G1 G.txt 0.0 31.7 63.4 95.1 16.8 158.5 190. 1.9 53.6 Time (min) Gain (K) 0.0845, 1st Time onstant (min)., nd Time onstant (min) 1.61 Goodness of it: -Squared 0.9966, SSE 0.0446 igure 7 esult of data fitting to te exact second-order model for process model G. em Eng Process Tec 3(1): 1037 (017) 8/10
Jang (017) igure 8 Block diagram for x MIMO feedback control sceme witout using te decoupling strategy. igure 9 Setpoint tracking for te two tom tanks using PID settings for feedback controllers witout using decoupling strategy. decoupling strategy to eliminate loop interactions. If one examines Eqs. 7 and 73, it is evident tat te time constants involved in tis system are very close to eac oter. Terefore, one may ignore te dynamic part of te decouplers and simply use static decouplers D 1 ~ -0.04585/0.07867 and D ~ -0.04007/0.0845. Te results are very similar to tose in igure and not demonstrated ere. ONLUSION Te transfer functions for a four-tank system illustrated in tis work can be derived by using te principle of analogy to te single-tank case, wit resistances of te six valves below te four tanks and first-order time constants of te four tanks clearly defined. Te final results sow tat te transfer functions of te liquid levels of te two tom tanks are linear combinations of te effects of te two feed streams to te system. Simulation results for te four-tank system in Loop Pro sow tat te dynamic responses of process variables to te canges in controller outputs fit te expected overdamped second-order beaviors. Wit te process models and disturbance models clearly developed and model parameters obtained, one may identify te extent of loop interactions using relative gain array. Model-based controller tuning metod suc as IM provides adequate PID tuning parameters for te two feedback controllers. owever, te system encountered in tis work exibits certain degree of loop interaction by using two individual PID feedback controllers. Wit decoupling strategy applied to loops, loop interactions are almost eliminated entirely. EEENES 1. Jang LK. Level ontrol by egulating ontrol Valve at te Bottom of A em Eng Process Tec 3(1): 1037 (017) 9/10
Jang (017) igure 10 Block diagram for x MIMO feedback control sceme wit decoupling strategy. igure Setpoint tracking for te two tom tanks using PID settings for feedback controllers wit decoupling strategy implemented. Te legends are te same as tose of igure 9. Gravity-drained Tank. em Eng Educ. 016; 50: 45-50.. Lee M, Sin J. onstrained Optimal ontrol of Liquid Level Loop Using a onventional Proportional-Integral ontroller, em. Eng. ommun. 009; 196: 79 745. 3. iggs JB, Karim MN. emical and Bio-Process ontrol, 4t edition. 016; 18 183. 4. Smit A, orripio AB. Principles and Practices of Automatic Process ontrol, 3 edition. oboken, NJ: Wiley. 005; 441-44. 5. Luyben WL. Simple metod for tuning SISO controllers in multivariable systems. Ind Eng em Process Des Dev. 1986; 654 660. 6. 6. Malwatkar GM, Kandekar AA, Asutkar VG, Wagmare LM. Design of entralized PI/PID ontroller: Interaction Measure Approac, in 008 IEEE egion 10 and te Tird international onference on Industrial and Information Systems. 008; 1 6. 7. Lengare MJ, ile, Wagmare LM. Design of decentralized controllers for MIMO processes, omput Electr Eng. 01; 140 147. 8. Gatzke EP, Meadows ES, Wang, Doyle J. Model based control of a four-tank system. omput em Eng. 000; 1503 1509. ite tis article Jang LK (017) eedback ontrol for Liquid Level in a Gravity-Drained Multi-Tank System. em Eng Process Tec 3(1): 1037. em Eng Process Tec 3(1): 1037 (017) 10/10