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A Turbo Fan Nonlinear Model and a Gain Scheduling Control Strategy Design

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... This static model is used to simulate the dynamic behavior of a pneumatic circuit composed of a turbo fan, an input valve, an output valve and the corresponding piping of a teaching and research installation that emulates a typical industrial gas
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  A TURBO FAN NONLINEAR MODEL AND A GAIN SCHEDULINGCONTROL STRATEGY DESIGN Leonardo AndradesElectronics DepartmentUniversidad T´ecnica Federico Santa Mar´ıaValpara´ıso, Chileemail: leonardo.andrades@ieee.orgManuel OlivaresElectronics DepartmentUniversidad T´ecnica Federico Santa Mar´ıaValpara´ıso, Chileemail: manuel.olivares@usm.cl ABSTRACT In this paper, a nonlinear static model of a variable speedturbo fan is obtained. This static model is used to sim-ulate the dynamic behavior of a pneumatic circuit com-posed of a turbo fan, an input valve, an output valve andthe corresponding piping of a teaching and research in-stallation that emulates a typical industrial gas extractionsystem. For this, the whole system is phenomenologicallymodeled, getting a model able to reproduce the  surge  phe-nomena present in this type of pneumatic circuits. As thedynamic system behavior is multivariable and highly non-linear, presenting a stable zone limited by the  surge line ,a single PI flow controller tuned using a first order linearmodel obtained from a step response at one specific stableoperating point shows the required closed loop responseonly for that operating point. To overcome that, a novelPI and GPC gain scheduling control strategies based ontwo experimental linear models are proposed and validatedby simulation. Both gain scheduling closed loop systemresponses, to set-point changes and input and output dis-turbances, are compared against each other and evaluated,showing a better performance than the usual single linearmodel tuned PI control strategy. KEY WORDS Modelling, Parametric Identification, Predictive Control,PID Control, Gain Scheduling 1 Introduction A turbo fan is a machine that converts the rotatory energyprovidedbyanACvariablespeedmotorintokineticgasen-ergy. The turbo fan operates in pneumatic piping circuits,supplyingcompressedairatpressurebetween 1 and 25[  psi ] and flow up to  60000[ m 3 h  ]  [1]. This machineis widely usedin industry for large gases mass flow transportation, par-ticularly in the copper processing industry at the smeltingstage, as shown in figure 1. Here, several coordinatedturbofan machines are needed for gas extraction from the con-tinuous Fusion Flash Furnace (HFF) and four batch PierceSmith Converters (CPSs) to the acid plant [2]. In particu-lar, forthe CPSs gas extractionit is neededto keepconstantboth, pressure and gas flow at the VTI-CPS turbo fan dis-charge, avoiding the  surge  phenomena [3] at different op-erating points required by the process, as the CPSs get intooperation asynchronously.The  surge  phenomena is a not desired flow and dis-charge pressure oscillation due to an input load that makesthe machine to accumulate gas mass elevating its outputpressure to an extent such that the flow is transitorily re-versed even though the turbo speed is constant. This phe-nomenacandamagethemachineandtheenvironment,thuscontrol strategies should avoid to make the turbo fan op-erates in the  surge  zone, regulating the flow and keepingtheoutputpressureconstantsimultaneously,actingoverthemotor speed and the output valve position, taking into ac-count energy efficiency issues [4] [5]. Also, the flow set-point must be set in accordance to the amount of operatingCPSs to send all their sulphur-rich output gases to the acidplant process, avoiding any gas leakage to the environmentas they are hot and extremely toxic.This paper is focussed to the development of a vari-able speed turbo fan nonlinear model and a pneumatic cir-cuit dynamic simulation that emulates the industrial CPSsgas extraction process at the Angloamerican Chagres cop-persmelterlocatedinChile. Thisallowstodesignandeval-uate two different output gas flow control strategies to keepa specified dynamic performance independent of the oper-ating point required by the gas extraction process, actingon the turbo fan speed. The output pressure is changedacting on the output valve position, as output disturbances,and the batch CPSs operation is simulated by changing theinput valve position, as input disturbances.The paper is organized as follows. The second sec-tion shows a turbo fan nonlinear characteristic curve fittingprocedure from experimental data and a pneumatic circuitdynamicmodelingusingfirst principlesofmass andenergyconservation,ideal gas law and quadratic valve model. Thesimulated turbo fan operation in stable and unstable zonesare shown in section three. In the fourth section the gasflow closed loop system is simulated using a standard PIcontroller, tuned from a single experimental linear model.In the fifth section, the proposed gain scheduling [6] PIand GPC control strategies based on two experimental lin-ear models are developedand evaluated, showing improve-ments with respect to the single model tuned PI controller.Finally, in section six some results discussion and futurework concludes the paper.  Proceedings of the IASTED International Conference  February 14 - 16, 2011 Innsbruck, Austria Modelling, Identification, and Control (MIC 2011)  DOI: 10.2316/P.2011.718-078  39  Figure 1. Gas extraction process at Chagres smelterFigure 2. Turbo fan lab. rig 2 Turbo fan and piping modeling The CPSs gas extraction process is emulated by means of the Thermofluids lab. rig equipment shown in figure 2,which main components characteristics are listed in table1 [8]. At system components description level, the staticturbo fan model operating in the stable zone is developedand then the piping dynamic modeling is presented.Typical fan characteristic curve shows a quadraticstatic relationship between the pressure difference  ∆ P   andthe gas flow  q  , parameterized by the motor speed  n  [5].This data can be summarizedin the proposedmathematicalmodel (1), where A ,  B  and  C   are positive constants ∆ P   =  − A ( q   −  Bn ) 2 + Cn 2 (1)Bytheotherhand,fanflowchangesandpressuredifferencechanges depends on motor speed changes as indicated in(2) and (3) known as fan laws [7], relationships which areTable 1. Lab. rig equipment Equipment  Tech. data Turbo fun  1000[ m 3 h  ] Demag, Sez 2B  ∆ P   = 1 , 6[ ata ] AC drive motor  380[ V  ] ,  90[ A ] 3 φ  Siemens  50[ kW  ] P.F   : 0 , 872930[ rpm ]50[ Hz ] Freq. variator 3 φ  55[ kW  ] Leroy Somer, UMV 301satisfied by model (1) q  n 2 q  n 1 =  n 2 n 1 (2) ∆ P  n 2 ∆ P  n 1 =  n 2 n 1  2 (3)  Proof:  Let  q  ni  and  ∆ P  ni  be the gas flow and the pressuredifference at motor speed  n i , then using (1) we get ∆ P  n 1  =  − A ( q  n 1  − Bn 1 ) 2 + Cn 21  (4) ∆ P  n 2  =  − A ( q  n 2  − Bn 2 ) 2 + Cn 22  (5)Replacing (3) into (5),  n 2 n 1  2 ∆ P  n 1  =  − A ( q  n 2  − Bn 2 ) 2 + Cn 22  (6)and then replacing (2) into (6) finally, ∆ P  n 1  =  − A ( q  n 1  − Bn 1 ) 2 +  Cn 21  (7)whichdemonstratesthatproposition(1)satisfy thefanlaws(2) and (3).   40  For the specific case of this installation, the turbofan model constants  A ,  B  and  C   are listed in table 2,which were obtained from experimental data set at  n 2  =1500[ rpm ] , and validated with experimental data sets at n 1  = 1000[ rpm ]  and  n 3  = 2000[ rpm ] . That is, three( ∆ P  , q) experimentaldata pair sets at  n 1 ,  n 2  and n 3  motorspeeds were obtained varying the discharge valve positiongetting the stars, white squared dots and crosses shown infigure3. There, the dash-dotted,dashedand solid lines rep-resent the output model data at n 1 ,  n 2  and n 3  motor speedsrespectively, which match the corresponding experimentaldata points, validating the turbo fan model (1).In addition to the turbo fan, the lab. rig system repre-sented by the scheme on figure 4 has two mass and energyaccumulationzones, whose equations (8)-(28) are obtainedfrom first principles [9], and are listed using variables andconstant definitions given in table 2. Piping volume v 1  and v 2  of each zone and dry air density  1 . 2[ kg/m 3 ]  were con-sideredto calculate the initial air masses m 1 (0)  and m 2 (0) .To get the initial energy  E  1 (0)  and  E  2 (0) , equations (8)and (18) were used, considering a temperature of   293[ K  ] . 00.20.40.60.8100.511.522.5x 10 4   X: 0.33Y: 1.06e+004       ∆    P   [   P  a   ]  q 2  [kg/s] X: 0.22Y: 4738X: 0.44Y: 1.895e+004 Modeled at 1500 [rpm]Measured data 1500 [rpm]Modeled at 2000 [rpm]Measured data 2000 [rpm]Modeled at 1000 [rpm]Measured data 1000 [rpm]beta=60º Figure 3. Experimental data and nonlinear turbo fan modelFigure 4. Lab. rig system scheme T  1  =  E  1 c · m 1 (8) E  1  =  E  1 (0)+    t 0 ( w i 1  − w 011  − w 021  − w 031 ) dt  (9) w i 1  =  P  1  ·  q  1 ρ 1 + q  1  · c · T  atm  (10) w 011  =  r ( T  1  − T  atm )  (11) w 021  =  q  2  · c · T  1  (12) w 031  =  P  1  ·  q  2 ρ 1 (13) ρ 1  =  m 1 v 1 (14) P  1  =  ρ 1  ·  kµ  · T  1  (15) m 1  =  m 1 (0) +    t 0 ( q  1  − q  2 ) dt  (16) q  1  =  α · z  · sign ( P  atm  − P  1 )   | P  atm  − P  1 | (17) T  2  =  E  2 c · m 2 (18) E  2  =  E  2 (0)+    t 0 ( w i 2  − w 012  − w 022  − w 032 ) dt  (19) w i 2  =  P  2  ·  q  2 ρ 2 + q  2  · c · T  1  (20) w 012  =  k ( T  2  − T  atm )  (21) w 022  =  q  3  · c · T  2  (22) w 032  =  P  2  ·  q  3 ρ 2 (23) ρ 2  =  m 2 v 2 (24) P  2  =  ρ 2  ·  kµ  · T  2  (25) m 2  =  m 2 (0) +    t 0 ( q  2  − q  3 ) dt  (26) q  3  =  β   · z  · sign ( P  2  − P  atm )   | P  2  − P  atm | (27) ∆ P   =  P  2  − P  1  (28) 3 Lab. rig system simulation To simulate the stable and unstable operation zones, a fullturbo fan model is needed. The stable operation zone,where the gas flow  q   can be reduced by closing the outputvalve, getting an increase in the pressure difference  ∆ P  , ismodeledbyequation(1). Butthis increasein  ∆ P   is limitedby the point where the accumulation of gas molecules ei-ther expands the turbo fan volume or run away in the oppo-site direction, getting an instantaneous  ∆ P   reduction thatin response get an increase in the gas flow  q  , staying in the surge  oscillating condition [10] [11], as shown in figure 5.This unstable operating zone can be physically understoodas the zone where it is not possible to reduce the flow  q  41  Table 2. System variables and constants definition Variable  Description Value  p atm  Atmospheric pressure  101325[ Pa ] A  Turbo fan parameter  3 . 63  ·  10 4 [ Pa · s 2 kg 2  ] B  Turbo fan parameter  5 . 18  ·  10 − 5 [  kgs · rpm ] C   Turbo fan parameter 0.0057  [  Parpm 2 ] n  Drive motor speed  [ rpm ] α  Input valve position [0..1] β   Output valve position [0..1] z  Valves coefficient  0 , 002384[  kgs ·√  Pa ] c  Specific air heat coefficient  1012[  J    kg · K  ) ] k  Boltzmann constant  1 , 38  ·  10 − 23 [  J K  ] µ  Average air molecular mass  4 , 81  ·  10 − 26[ kg ] v 1  Zone 1 volume  0 . 46[ m 3 ] v 2  Zone 2 volume  0 . 78[ m 3 ] E  1 (0)  Initial zone 1 energy  16 , 4  ·  10 4 [ J  ] E  2 (0)  Initial zone 2 energy  27 , 9  ·  10 4 [ J  ] m 1 (0)  Initial zone 1 mass  0 . 0672[ kg ] m 2 (0)  Initial zone 2 mass  0 . 094428[ kg ] P  1  Zone 1 pressure [Pa] P  2  Zone 2 pressure [Pa] T  1  Zone 1 temperature [K] T  2  Zone 2 temperature [K] E  1  Zone 1 energy [J] E  2  Zone 2 energy [J] w i 1  Zone 1 input power  [ J s ] w i 2  Zone 2 input power  [ J s ] w 0 x 1  Zone 1 output power ’x’  [ J s ] w 0 x 2  Zone 2 output power ’x’  [ J s ] r  Power transmission coefficient 5 [  J s · K  ] q  1  Zone 1 input gas flow  [ kgs  ] q  2  Zone 2 input gas flow  [ kgs  ] q  3  Zone 2 output gas flow  [ kgs  ] m 1  Zone 1 accumulated mass  [ kg ] m 2  Zone 2 accumulated mass  [ kg ] ρ 1  Zone 1 air density  [  kgm 3 ] ρ 2  Zone 2 air density  [  kgm 3 ] simultaneously reducing the pressure difference  ∆ P  , thatis the zone where  d ∆ P dq  ≥  0  (see figure 3). Thus equations(29)and (30) are proposedas the full turbofan model to in-clude this dynamic behavior. In equation (29) the slip rate r e  = 0  when the air is driven by the fan blades, showingstable operation, and  r e  = 1  when the air is slipping fromthem, causing the surge phenomena. Equation (30) sets thenext slip rate r ep  according to  ∆ P   hysteresis levels, that is,the next state of the slip rate  r e  for the next time step. q  2  =   Cn 2 − ∆ P A  +  Bn r e  = 0 −   ∆ P A  − Bn r e  = 1 (29) Figure 5. Surge phenomena r ep  =  0 0 ≤ ∆ P   ≤ δ  1 r e  δ  1  <  ∆ P < Cn 2 − δ  2 1  Cn 2 − δ  2  ≤ ∆ P  (30) Inequation(30) δ  1  and δ  2  set the effectivehysteresislevels,from which flow inversion is due, that in practice dependon the motor speed  n . For simulation purposes  δ  1  =  δ  2  =0 . 03  · C   · n 2 [Pa] are chosen. 3.1 Stable open loop operation The equilibrium point is stable when it is outside the surgezone, that is when  q > Bn  and  ∆ P < Cn 2 , or equiv-alently where  d ∆ P    dq  <  0 . This is validated by simulationapplying a  20%  step increase in motor speed, starting atthe equilibrium point  q   = 0 . 22[ kgs  ] ,  ∆ P   = 4738[ Pa ] , n  = 1000[ rpm ] , obtaining the gas flow response shownin figure 6. 98991001011020,220,264   q    3    [   k  g   /  s   ] t[s] 98991001011024000500060007000       ∆    P   [   P  a   ] t[s] Figure 6. Open loop system response to motor speed step 42  99.5100100.5101−0.500.5   q   [   k  g   /  s   ] t[s]   q 2 (t)q 3 (t)99.5100100.5101020004000      ∆     P     [     P   a     ] t[s]   ∆  P (t) Figure 7. Surge due to an input valve disturbance 3.2 Unstable open loop operation The lab. rig system of figure 4 can present the surge phe-nomena by either input valve or output valve disturbances.In practice, input valve disturbances are srcinated by theamount of operating CPSs, as they not always operate si-multaneously because of their batch operation. Outputvalve disturbances are usually due to operator maneuversor control loop operation intended to keep the output pres-sure constant. Those industrial disturbances, are emulatedvarying valve positions  α  and  β  , respectively, measuredin degrees where  0 o corresponds to a fully closed throttlevalve and  90 ◦  to a fully open valve.Thesimulatedunstableresponsetoan inputvalvedis-turbance is shown in figure 7. Starting at an stable equilib-rium point with the output valve at  60 ◦ , the input valve isclosed from  90 o to  10 o , showing the surge phenomena. Bythe other hand, in figure 8 the simulated unstable responseto an output valve disturbance is shown. In this case thesurge phenomenais obtained closing the output valve from 60 o to  30 o , from an stable equilibrium point with the inputvalve fully open. 4 PI controller design and simulation One of the most used experimental method to design PIcontroller parameters consists on approximate a stableopenloop step input system response anduse tuningproce-dures based on that experimentalresponse. In this case, theoutput gas flow PI control loop acting on the motor speedis tuned using the same stable open loop step responseshown in figure 6. Starting at the stable equilibrium point q   = 0 . 22 [kg/s],  ∆ P   = 4738 [Pa] and  n  = 1000 [rpm], theopen loop  20%  step in motor speed  n  increase responseis approximated by a first order plant, where plant param-eters gain  K   and time constant  τ   p  are fitted using leastsquares [12]. As the system is nonlinear, a similar pro-cedure is applied to get a first order model at the stableequilibrium point q   = 0 . 4352 [kg/s],  ∆ P   = 29456 [Pa] and n  = 2500 [rpm]. The first order models  G 1 ( s )  and  G 2 ( s ) 9999.5100100.5101−0,500,220,5   q   [   k  g   /  s   ] t[s]   9999.5100100.5101020004000      ∆     P     [     P   a     ] t[s]   q 2 (t)q 3 (t) ∆  P (t) Figure 8. Surge due to an output valve disturbanceare obtained respectively, in (31). G 1 ( s ) = 0 . 002315 s  + 10 . 52 G 2 ( s ) = 0 . 0009022 s  + 4 . 101  (31) Then  K  c  and  K  i  parameters of a PI controller (32) aredesigned for each plant by means of the pole assignmentmethod [13], specifying an underdumped closed loop be-havior ( ξ   = 0 . 7 ) and a faster response than the open loopplant ( τ  lc  = 0 . 8 τ   p ), getting  PI  1  and  PI  2  controllers (33) u ( t ) =  K  c · e ( t ) +  K  i    t 0 ( e ( t )) dt  (32) PI  1 ( s ) =  K  c 1  +  K  i 1 s  = 6815 . 18 + 149649 . 55 sPI  2 ( s ) =  K  c 2  +  K  i 2 s  = 6765 . 12 + 52939 . 13 s  (33) Thus, using just onePI controllerfor the whole stable oper-ation zoneis notenoughto getthe specified closed loopbe-havior. Thisis showninfigure9, wherea poorperformanceto different closed loop set-point step responses is obtainedwhen using only  PI  1 , with higher overshot in both actu-ation and output signals. More over, when the set-pointis too far of the equilibrium point for which the controllerparameters were designed a transitory surge phenomena isobtained,which is finally stabilized. To overcomethis non-linear behavior a nonlinear control strategy is needed, thatallows to keep the specified closed loop behavior in thewhole stable zone operation of the turbo fan. 5 Gain scheduling control design As the lab. rig is a nonlinear two input two output mul-tivariable system, being  β   and  n  the control inputs,  q   and ∆ P   the system outputs, and  α  the input disturbance, theproposed control strategy is as follows: For stable opera-tion zone, gas flow q   is regulated acting on the motor speed 43
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