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 simulate the dynamic behavior of a pneumatic circuit composed of a turbo fan, an input valve, an output valve andthe corresponding piping of a teaching and research installation that emulates a typical industrial gas extractionsystem. For this, the whole system is phenomenologicallymodeled, getting a model able to reproduce the
surge
phenomena present in this type of pneumatic circuits. As thedynamic system behavior is multivariable and highly nonlinear, 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 setpoint changes and input and output disturbances, 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 energyprovidedbyanACvariablespeedmotorintokineticgasenergy. 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, particularly in the copper processing industry at the smeltingstage, as shown in ﬁgure 1. Here, several coordinatedturbofan machines are needed for gas extraction from the continuous Fusion Flash Furnace (HFF) and four batch PierceSmith Converters (CPSs) to the acid plant [2]. In particular, forthe CPSs gas extractionit is neededto keepconstantboth, pressure and gas ﬂow at the VTICPS turbo fan discharge, avoiding the
surge
phenomena [3] at different operating points required by the process, as the CPSs get intooperation asynchronously.The
surge
phenomena is a not desired ﬂow and discharge 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 reversed even though the turbo speed is constant. This phenomenacandamagethemachineandtheenvironment,thuscontrol strategies should avoid to make the turbo fan operates in the
surge
zone, regulating the ﬂow and keepingtheoutputpressureconstantsimultaneously,actingoverthemotor speed and the output valve position, taking into account energy efﬁciency issues [4] [5]. Also, the ﬂow setpoint must be set in accordance to the amount of operatingCPSs to send all their sulphurrich 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 variable speed turbo fan nonlinear model and a pneumatic circuit dynamic simulation that emulates the industrial CPSsgas extraction process at the Angloamerican Chagres coppersmelterlocatedinChile. Thisallowstodesignandevaluate two different output gas ﬂow control strategies to keepa speciﬁed dynamic performance independent of the operating 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 section 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 linear models are developedand evaluated, showing improvements 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.2011.718078
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 dashdotted,dashedand solid lines represent 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 represented 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 consideredto 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 either expands the turbo fan volume or run away in the opposite 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 include 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 equivalently 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 phenomena 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 simultaneously because of their batch operation. Outputvalve disturbances are usually due to operator maneuversor control loop operation intended to keep the output pressure 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 inputvalvedisturbance is shown in ﬁgure 7. Starting at an stable equilibrium 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 tuningprocedures 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 parameters gain
K
and time constant
τ
p
are ﬁtted using leastsquares [12]. As the system is nonlinear, a similar procedure 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 behavior (
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 operation zoneis notenoughto getthe speciﬁed closed loopbehavior. Thisis showninﬁgure9, wherea poorperformanceto different closed loop setpoint step responses is obtainedwhen using only
PI
1
, with higher overshot in both actuation and output signals. More over, when the setpointis too far of the equilibrium point for which the controllerparameters were designed a transitory surge phenomena isobtained,which is ﬁnally stabilized. To overcomethis nonlinear 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 multivariable 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 operation zone, gas ﬂow
q
is regulated acting on the motor speed
43