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 (ie, coupling effect) and implement proper strategies to eliminate potential loop interactions (ie, 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
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 (ss) 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 ss 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
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 1, top (19), top (0) 1 1, top (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 1, top by Eq 5: 1 1 1 + (5) 1, top 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) 1, top 1, top 1, top 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, top 1, top 1, in τ1, tops, top, top, in τ, tops were (31) (3) em Eng Process Tec 3(1): 1037 (017) 3/10
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 G1txt Process Variable 41 40 39 38 Manipulated Variable 630 61 61 603 00 33 646 969 19 1615 1938 61 Time (min) Gain (K) 007867, 1st Time onstant (min) 1041, nd Time onstant (min) 41 Goodness of it: -Squared 0997, SSE 003101 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) 1, top 1, top 1, top (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, top 1, top 1, in τ1, tops 1, top 1, top 1 1 1, in 1 τ1, tops (37) (38) em Eng Process Tec 3(1): 1037 (017) 4/10
, 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 G G G 1 1 1 1, in, in 1, top ( τ s ) 1,, ( τ tops ), top 1 ( τ s ) 1, ( τ tops ) 1, top 1 ( τ s ) ( τ tops ) (45) (46) (47) (48) (49) G, top ( τ s ), ( τ tops ) (50) 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 Eqs55-58: f 1, in f + f 1 (55) 1, top + 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 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, top 1 1 1, top +, 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 615% in te manual mode, te controller output to te inlet control valve on te left side is canged from 615% to 630% and maintained at 630% until liquid levels reac new plateaus, ten dropped to 600% and maintained at 600% until liquid levels reac oter new plateaus inally, te controller output is increased to 615% 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 615% 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): 007867 004585 initial (1091s ) (1091s ) G 004007 00845 (91s ) (91s ) (63) 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 0996) Te final results are presented in igures 4-7 and te refined G matrix for te transfer functions are 007867 004585 ( 41s )(1041s ) ( 1041s )(61s ) G 004007 00845 (64) ( 41s )(141s ) ( 61s )(141s ) Te fine-tuned best-fit time constants are listed in Table 1 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 007867, K 1 004585, K 1 004007, and K 00845 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 138 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 10 If loop interaction were absent, we would expect te λ value to be exactly 10 If te second controller output were to increase te response of te process variable in te first loop, we would expect 0 <λ< 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 1041 61 141 em Eng Process Tec 3(1): 1037 (017) 6/10
Process Variable Manipulated Variable Model: Second Order Overdamped 41 40 39 38 630 61 61 603 Loop-Pro: Design Tools ile Name: Multi Tank G1 Gtxt 00 317 634 951 168 1585 190 19 536 Time (min) Gain (K) 004585, 1st Time onstant (min) 41, nd Time onstant (min) Goodness of it: -Squared 09967, SSE 00173 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 40 39 38 630 61 61 603 Loop-Pro: Design Tools ile Name: Multi Tank G G1txt 00 33 646 969 19 1615 1938 61 Time (min) Gain (K) 004007, 1st Time onstant (min) 41, nd Time onstant (min) 161 Goodness of it: -Squared 09965, SSE 0009501 igure 6 esult of data fitting to te exact second-order model for te disturbance model G 1 10 On te oter and, if λ value is very large or even negative, we 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 ( τ )( τ ) p 1 (68) PID tuning parameters can be determined by using te IM tuning metod for single-input-single-output (SISO) [3(b)][5]: K Proportional Gain 1 c (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 10 min Loop 1 K c 3 % /m τ I 18 min em Eng Process Tec 3(1): 1037 (017) 7/10
τ D 544 min Loop K c 3689 %/m τ I 40 min τ D 600 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 60 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 396 meters to 45 meters and back to 396 meters, wile maintaining te level setpoint of te tom tank on te rigt side (, sp ) at 39 meters Similar simulation is done by canging, sp from 39 meters to 45 meters and back to 39 meters wile maintaining 1,,sp at 396 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 60 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 (ie, 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 (ie,, ) Te decouplers used in tis simulation are D D G 1 1 G G 1 G 004585 1041 (61s ) 007867 41 (1041s ) ( s ) ( s ) 004007 41 (141s ) 00845 61 (141s ) ( 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 Manipulated Variable Process Variable Model: Second Order Overdamped 41 40 39 38 37 630 61 61 603 Loop-Pro: Design Tools ile Name: Multi Tank G1 Gtxt 00 317 634 951 168 1585 190 19 536 Time (min) Gain (K) 00845, 1st Time onstant (min), nd Time onstant (min) 161 Goodness of it: -Squared 09966, SSE 00446 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
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 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 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 ~ -004585/007867 and D ~ -004007/00845 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 em Eng Process Tec 3(1): 1037 (017) 9/10
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 EEENES 1 Jang LK Level ontrol by egulating ontrol Valve at te Bottom of A 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