Fault Tolerant Control of Automotive Air Conditioning Systems using a GIMC Structure

0.91 MB·

by  Xu Zhang

1
Fault Tolerant Control of Automotive Air Conditioning Systems
using a GIMC Structure
Xu Zhang
Abstract—Although model-based fault tolerant control (FTC) models where the performance after fault occurrences was
hasbecomeprevalentinvariousengineeringfields,itsapplication unsatisfactory in transient [22].
to air-conditioning systems is limited due to the lack of control-
Although steady-state analysis is the main stream in FTC,
orientedmodelstocharacterizethephasechangeofrefrigerantin
studies have recently emerged on dynamic fault diagnosis
thevaporcompressioncycle.Theemergenceofmovingboundary
method (MBM) illuminates a promising way for FTC design. [23], in which transient data and models are used to identify
In this paper, we exploit a control-oriented nonlinear model faulty system behaviors. For example, a lumped parameter
7
1 comparable to MBM to design an FTC framework with a method has been applied to a vapor compression cycle with a
generalized internal model control (GIMC) approach. A fault
0 fixedorificedevicetodevelopanobserver-basedscheme[24].
detector and isolator (FDI) is developed to identify potential
2 Black box models obtained through data-driven techniques,
actuator and sensor faults. A fault compensator is employed to

n compensate these faults if detected. Comprehensive simulations such as ARX and ARMAX, were used to generate structured
a are carried out to evaluate the developed FTC framework with residuals to predict faults in an air handling unit [25]. In
J promising results. Plant variations are explicitly considered to [22], a comprehensive model was used to represent the vapor

2 enhance the gain-scheduled FTC developments.
compression cycle as opposed to an air handing unit and
1
Index Terms—Air Conditioning System, Fault Tolerant Con- a linearized model was used to explore the sensitivity of

trol, Generalized Internal Model Control, Moving Boundary each output to a variety of faults. However, no practical fault
]
Y Method, Robust Control diagnosis algorithm was implemented and tested.
S In summary, model-based FTC design for automotive A/C
. systems is far from satisfactory on three aspects. Firstly,
s I. INTRODUCTION
c the vapor compression cycle, a critical subsystem in the
[ Fault tolerant control (FTC) has attracted much attention A/C system for energy conversion, has not been comprehen-

1 from control engineers since it can accommodate sensor and sively studied; previous work focused on the air handling
v actuatorfaultstopreservedesiredclose-loopperformance[1]– unit. Secondly, lumped parameter modeling and data-driven
4 [5]. FTC schemes mainly fall into two categories: passive modeling approaches are not physics-based, rendering little
7 and active. Passive approaches aim to retain stability and understandingoftherelationshipsbetweenpossiblefaultsand
6
performance against all faults by a single controller. One corresponding symptoms. Thirdly, a gap exists between fault
3
0 the other hand, an active approach typically consists of a diagnosis and control design, from the fact that a seamlessly
. fault diagnosis stage followed by a controller reconfiguration. integrateddesignapproachcombiningbothfaultdiagnosisand
1
Althoughactiveapproachesrequiremorecomputationalpower control action has not been investigated yet. In this paper, we
0
7 during implementation, they typically yield less conservative aim to provide a benchmark for dynamic fault diagnosis and
1 results and better closed-loop performance when faults occur. control of vapor compression cycle in a unified framework,
v: Recently, great research attention has been given to the by merging the latest advances in both practitioner and theory
i automotive industry to achieve improved handling, comfort developments.
X
and safety, as well as route planning, road condition assess- A promising approach to develop control-oriented models
r ment, and pothole detection capability, fuel efficiency predic- for heat exchangers was proposed in [26]–[28] by exploiting
a
tion, etc. [6]–[13]. In control-related areas, various model- the Moving Boundary Method (MBM). The refrigerant is
based control and estimation approaches have been widely lumped based on its phase status, i.e., pure vapor, pure liquid
developed [14]–[20]. However, model-based FTC design for or mixed vapor and liquid. By exploiting mean void fraction
automotive Air Conditioning (A/C) system has rarely been and volumetric ratio of vapor over liquid, a set of differential
touched. The main reason of this stagnation is the lack of equations describing the mass, momentum, and energy bal-
a control-oriented model that can balance model accuracy ances of the phase change process was developed and solved.
andcomputationalcomplexitytocharacterizethethermo-fluid Thismodelisadvantageousinmodelingthetransientbehavior,
dynamics of the refrigerant [21]. During heat removal/release and it is more precise than existing modeling techniques. The
at the heat exchanger, refrigerant experiences a liquid/vapor use of a first-principle A/C model can significantly reduce the
phase change. This phase change is a complicated process in time required to develop and implement FTC algorithms.
which mass and energy balance is difficult to characterize. As we mentioned, active FTC schemes rely on fault de-
Hence, previous studies were based on static or empirical tection to introduce fault-feedback compensation or control
reconfiguration based on identified fault information. For in-
Xu Zhang is with Department of Control Science and stance, an integrated control and fault detection design, based
Engineering, Harbin Institute of Technology, Harbin, 150080, China
on a four parameter controller, was proposed in [29]–[32],
[email protected]
whichincorporatesadditionaltwoDegreesofFreedom(DoFs)
2
for the purpose of fault diagnosis. An alternative implementa- 2) Two measurements are available for the controller and
tion ofthe integrated control,referred to as Generalized Inter- the FDI module, the evaporator pressure p and the
e
nal Model Control (GIMC), has been introduced in [33]–[36], superheat temperature SH .
which is able to overcome conflicts between performance and The air mass flow rates and temperatures on the evaporator
robustness in traditional feedback frameworks. With GIMC, a sidem˙ ,T andcondensersidem˙ ,T ,aremeasurablebut
ea eo ca co
high performance controller is active under normal conditions noncontrollabledisturbances.Theairtemperaturesleavingthe
and a robust controller will be activated when sensor/actuator heat exchangers T ,T and tube wall temperatures T ,T
eo co ew cs
faults or external disturbances are identified. are calculated using a control-oriented A/C model.
In this paper, MBM A/C modeling and FTC using the A controller is designed to track prescribed trajectories of
GIMC structure are integrated in a unified framework. The two output variables, namely the evaporator pressure p , and
e
contributions of this paper include the following. First, a first- the superheat temperature SH [37]. The reference values for
principlevaporcompressioncyclemodelisexploitedtomodel thetrackedvariablesarelabeledasp andSH ,respectively,
e,r r
thecomplexthermo-fluidprocess.Furthermore,afaulttolerant which are generated by higher level optimization algorithms
control scheme is developed by using the GIMC method. Fi- developed for improving energy conversion efficiency. Mean-
nally,again-schedulingcompensatorforfaultaccommodation while, the controller is expected to reject disturbances caused
is developed and simulations are presented to demonstrate the by the variation of air mass flow rate at the evaporator, m˙ ,
ea
efficacy of the proposed framework. whichismanuallysetbythedriverorintelligentlyregulatedby
The rest of the paper is organized as follows. The fun-
thecabincontrolunit.Atdifferentcoolingloads,thecoupling
damental theory of FTC with GIMC structure is detailed in
between inputs and outputs change significantly. Hence, the
Section II. Mathematical modeling of the A/C plant and gain-
controllerissupposedtobescheduledaccordingtothesystem
scheduled H controller design are introduced in Section III.
∞ operating condition.
Faultdetectionandisolationalgorithmforsensorandactuator
An FTC scheme is targeted to achieve stability and perfor-
faults is developed in Section IV and fault tolerant controller
mance, not only when all modules function normally, but also
isdesignedinSectionVwithapreliminarystudyonthegain-
in cases when there are faults in sensors, actuators, or other
scheduled GIMC structure. Conclusion remarks are made in
system components [5]. The faults of interest are one actuator
Section VI.
fault (compressor speed) and one sensor fault (pressure read-
ing). Both are assumed to enter the A/C system additively,
II. FAULTTOLERANTCONTROL(FTC)SCHEME
meaning that 1) the actual compressor speed N is the
cmp,act
In this section, a general FTC scheme for automotive
sum of the commanded compressor speed N sent by the
cmp
A/C systems is introduced with individual modules detailed.
controller and a faulty compressor speed f ; 2) the pressure
N
Integration of control action and reconfiguration mechanism
measurement p available to both the controller and FDI
e,msr
is realized through a method called the GIMC structure that
module is the actual pressure in the system p plus a faulty
e
can actively reconfigure the controller once faults occur.
reading f .
p
If the FDI module and reconfiguration mechanism are
A. Controller Reconfiguration
absent,theFTCschemeisconsideredpassiveasthecontroller
TheFTCschemeofanautomotiveA/Csystemisillustrated
is pre-determined in the design phase. Since it aims to be
in Figure 1. The scheme consists of four modules: A/C plant,
robust against a class of presumed faults, the fault-tolerant
gain-scheduledcontroller,FaultDetectionandIsolation(FDI),
capabilities are limited. In contrast, an FTC scheme with both
and reconfiguration mechanism.
the FDI module and reconfiguration mechanism is considered
AsillustratedinFigure1,abasicautomotiveA/Csystemis
active, because it reacts to faults actively by reconfiguring
composed of four primary components, evaporator, compres-
control actions so that the stability and performance of the
sor, condenser and expansion valve. The vapor compression
closed-loop system can be preserved. For a successful control
cycleremovesheatfromtheairflowingintothecabinthrough
reconfiguration,areal-timeFDImoduleisrequiredtoprovide
the evaporator, as the refrigerant evaporates from two-phase
precise information of the faulty components in the system.
(TP) status into superheated (SH) status, and rejects heat
In the FDI module, both commanded control inputs and
to the air flowing through the condenser, as the refrigerant
measurement outputs from the A/C plant are synthesized
condenses from superheated (SH) status into sub-cooled (SC)
to generate residuals which are signals essential for fault
status through two-phase (TP) status. The enthalpy, mass flow
detection, isolation and estimation.
rate and pressure, are exchanged via the four components.
Basically, the two heat exchangers set the pressures of the
B. GIMC Structure
system, while the compressor and expansion valve determine
the mass flow rates at the inlet and outlet of the evaporator The general FTC scheme in Figure 1 relies on an FDI
and condenser. algorithm, followed by a fault accommodation into the nom-
From a system-level perspective, the interfaces of the A/C inal controller. The FDI module is expected to detect and
plant to the rest of the scheme are: isolatetheoccurrenceofafaultintheclosed-loopsystem,and
1) ThecontrollersendscommandstotheA/Cplantthrough provide an appropriate compensation signal to the controller
the two controllable inputs, the compressor speed N inordertomaintaintheclosed-loopperformance.Thegeneral
c
and the valve position α, FTC scheme is realized through an integrated FDI design and
3
Residual Fault
Filter FDI
Reconfiguration
A/C System
pe,r Nc pe
SHr   G𝑯a∞in CSocnhterdoullleerd m mvt,h3t SC TP SH mcro t,h2t TSH
ea V Condenser sse Nc eo
Tea  ET rpm Tew
m Evaporator oC T
Tca mvt,h4t TP SH mct,h1t Tco
ca cw
Fig.1. InterconnectionsofPlant,Controller,FDIandReconfiguration.
reconfiguration mechanism referred to as the GIMC structure derived as:
[33], as shown in Figure 2, in which design methods for
P =M˜−1N˜, P =M˜−1N˜ , P =M˜−1N˜ , (3)
nominal conditions [36] are summarized below for reader’s uy dy d fy f
convenience. where M˜,N˜,N˜ ,N˜ ∈RH .
d f ∞
Now suppose a nominal controller K that stabilizes the
d f nominal plant P , and provides a desired closed-loop per-
uy
formance in terms of robustness, transient, and steady state
P P responses.Thecontrollercanberepresentedbyaleftcoprime
dy fy
factorization,
Nominal Controller
ref e U~ q V~1 u Puy y K =V˜−1U˜ (4)
The accommodation scheme adopted in this paper is moti-
Q N~ vated by a new implementation of the Youla parametrization
referred to as GIMC [33]. In this configuration, it allows the
r H fe M~ systemtoperformFDIandfaultaccommodationintheunified
structure, where these two processes can be carried out by
Fault Accommodation
selectingtwodesignparametersQ,H ∈RH .Consequently,
∞
the residual r is generated by selecting the detection/isolation
filter H and the accommodation signal q is generated by the
Fig.2. GeneralizedInternalModelControlStructure(Adaptedfrom[33]).
compensatorQ,usingthefilteredsignalf withthefollowing
e
criteria [36].
ConsideralinearsystemP(s)affectedbydisturbancesd∈
Rr and possible faults f ∈Rf described by • H(s):thefaultdetection/isolationfiltermustdiminishthe
effect of the disturbances or uncertainty into the residual
(cid:26)
x˙ =Ax+Bu+F f +E d, signal, and maximize the effect of the faults.
1 1 (1)
y =Cx+Du+F2f +E2d, • Q(s): the robustification controller must provide robust-
ness into the closed-loop system in order to maintain
where x ∈ Rn represents the vector of states, u ∈ Rm is
acceptable performance against faults.
the vector of inputs, and y ∈ Rp represents the vector of
The GIMC structure functions as follows: r = 0 if there
outputs. The nominal system is considered to be controllable
is no model uncertainties, external disturbances or faults and
and observable. The system response y can be analyzed in a
then the control system will be solely controlled by the
transfer matrix form as follows:
high performance controller K = V˜−1U˜. On the other
0
y =P u(s)+P f(s)+P d(s). (2) hand, the robustification controller Q will only be active
uy fy dy
when r (cid:54)= 0, i.e., there are either model uncertainties or
A left coprime factorization for each transfer matrix can be external disturbances or sensor/actuator faults. The advantage
4
of the GIMC structure is that if there is no uncertainty, the isentropic enthalpy difference. The first control input is the
controller will perform as well as a nominal controller does; compressor rotation speed N in the unit of rpm.
c
if uncertainty exists, the controller implementation should in The mass flow rate through the expansion valve is mod-
principleperformnoworsethanthestandardrobustcontroller eled by the orifice flow equation, approximated by assuming
implementation does as to robustness and performance. constant fluid density:
(cid:112)
m˙ =C A 2ρ (p −p ), (7)
III. A/CPLANT v d,v v 3 3 4
Realization of the general FTC scheme using the GIMC where Av is the valve curtain area and Cv is the discharge
structure needs left coprime factorization of the A/C plant coefficient.Theoutletenthalpyistypicallyfoundbyassuming
model in order to design the detection/isolation filter H and anidealthrottlingprocess,henceh4 =h3.Thesecondcontrol
the compensator Q. Here, the A/C plant described using input is the valve position α in percentage, determining the
MBMlanguageisutilizedtogenerateacontrol-orientedmodel effective flow area of the valve.
that not only provides the M˜ and N˜ matrices for design The mass and energy balance equations for the two-phase
purpose,butalsoservesasanonlinearsimulatorforalgorithm region of the evaporator are given directly in Equations 8
validation. and 9, respectively, where pe is the evaporator pressure; ζ1
In the MBM modeling framework, the compressor and the is the two-phase region normalized tube length; he,SH is the
valvearemodeledasstaticcomponents.Thedynamicsrelated enthalpy of the refrigerant at the tube exit; m˙ is the mass
to the heat and mass transfer inside the heat exchangers are flow rate; Q˙ is the heat transfer rate; ρ denotes the density.
described using the MBM method [26], [28], where Reynolds In Equations 8 and 9, the left hands represent the variation
transporttheoremdescribingthemassandenergyconservation of independent states of the refrigerant, and the right hands
for transient one-dimensional flow is applied to each phase the exchanger of mass and energy at the inlet and outlet of
region of the condenser and evaporator with boundary con- individual phase region, as well as the heat transfer along the
ditions and refrigerant properties specified. After derivations wall of corresponding region. The terms multiplying the state
detailed in [38] and not included here for brevity, the final variations depend on the refrigerant inherent thermodynamic
mathematicalequationsdescribingsystemdynamicsareinthe properties, hence are state-dependent. The mass and energy
descriptor form, balancesforthesub-cooled,two-phaseandsuperheatedregion
of the condenser are not shown here for brevity.
dx
Z(x,f ) =f(x,f ,u,v,f ),
a dt a N (5)
y =g(x,f ,f ), B. Coprime factorization of Plant Model
a p
where f and f are, respectively, the compressor and the The A/C model was calibrated using the data collected
N p
pressure fault as aforementioned. The input vector u in- during the tests when vehicle/engine speeds are maintained
cludes the compressor rotation speed and expansion valve at nominal steady state, and verified with reference to the
opening percentage, i.e., u = (cid:2)N α(cid:3)T. The boundary SC03 Air Conditioning Cycle in which vehicle speed trace
c
for this regulatory driving cycle is shown in Figure 3(a). The
conditions are the variables describing the air side of the
calibrationprocessrequiresapplyingmultiplierscorrectingthe
heat exchangers, and could be treated as unknown distur-
(cid:2) (cid:3)T valuesoftheheattransfercoefficientspredictedbytheempiri-
bances, v = m˙ T . The state vector x , de-
ea ea,in e
calcorrelationsfoundintheliterature.Figure3alsocompares
scribing the evaporator status, includes 5 variables, x =
e
(cid:2) (cid:3)T the outputs of the model with the corresponding experimental
ζ p h T T . Finally, the output vector y
e1 e e2 e1w e2w
data. During the SC03 test, the compressor speed (related to
includes the evaporator pressure and superheat temperature,
(cid:2) (cid:3)T the engine speed) changes considerably, causing significant
y = p SH , which are algebraic functions of states,
e variations in the refrigerant flow rate that affect the pressure
g(x). The Z matrix and f vector are complex expressions of
dynamics in the heat exchangers. This is particularly evident
refrigerant properties, heat transfer coefficients and geometric
by observing the fluctuations of the condenser pressure, as
parameters [38].
shown in Figure 3(b). From the Root Mean Square Error
(RMSE) between calculated pressures and measured pressure,
A. Mathematical Model thecondenserpressureerroriswithin6%ofitsaveragevalue,
The compressor and expansion valve are the two main and the evaporator pressure error is around 8%. Therefore,
actuators regulating the pressure difference and enthalpy dis- the model appears quite accurate in capturing the pressure
tribution in the A/C loop. In the compressor, the mass flow dynamics at the condenser and evaporator.
rate m˙ and outlet enthalpy h are, respectively,defined as: The nonlinear A/C model is linearized at three operating
c 2
conditions, corresponding to low, medium and high cooling
m˙ =η V ρ ω ,
c v d 1 c loads, whose controlled inputs and steady state refrigerant
h −h (6)
h = 2s 1 +h , status change with respect to the inlet air temperature of the
2 η 1
s evaporator, as summarized in Table I.
where V is the compressor displacement; ρ ,h are the Note that as seen from Figure 2, the coprime factors M˜
d 1 1
refrigerant density and enthalpy at the compressor inlet, re- andN˜,insteadoftheplantmodelP ,areofinterestbecause
uy
spectively; ω is the compressor speed and h −h is the they are parts of the fault accommodation block.
c 2s 1
5
(cid:18) (cid:19)
ρ −ρ dζ 1 ∂ρ dp 1 ∂ρ dγ¯
e,TP g 1 + e,TP e ·ζ + e,TP e ·ζ
ρ dt ρ ∂p dt 1 ρ ∂γ¯ dt 1
e,TP e,TP e e,TP e
(cid:18) (cid:19)
m˙ m˙ ρ (h −h )dζ ∂h 1 dp ∂h dγ¯
= ρ vV − ρ 12V g eρ,TP g dt1 + ∂ep,TP − ρ dte ·ζ1+ ∂eγ¯,TP dte ·ζ1 (8)
e,TP e e,TP e e,TP e e,TP e
m˙ m˙ Q˙
= v (h −h )− 12 (h −h )+ TP ,
ρ V 4 e,TP ρ V g e,TP ρ V
e,TP e e,TP e e,TP e
(cid:18) (cid:19)
ρ −ρ dζ 1 ∂ρ dp 1 ∂ρ dh
− e,SH g 1 + e,SH e ·(1−ζ )+ e,SH e,SH ·(1−ζ )
ρ dt ρ ∂p dt 1 ρ ∂h dt 1
e,SH e,SH e e,SH e,SH
m˙ m˙ ρ (h −h )dζ 1 dp dh
= ρ 12V − ρ cV − g gρ e,SH dt1 + ρ dte ·(1−ζ1)− de,tSH ·(1−ζ1) (9)
e,SH e e,SH e e,SH e,TP
m˙ m˙ Q˙
= 12 (h −h )− c (h −h )+ SH ,
ρ V g e,SH ρ V 1 e,SH ρ V
e,SH e e,SH e e,SH e
TABLEI
60 A/COPERATINGPOINTS.
h]50
d [mp40 Q˙a Ta 0C Nc (rpm) α (%) Pe (kpa)
e Low 25 450 25 302.2
pe30
e S Med 30 1000 40 251.2
hicl20 High 40 2500 55 204.6
e
V10
0
0 100 200 300 400 500 600
Time [s]
transfer function between the disturbance ω and controlled
(a) VehicleSpeedProfile
output z, (cid:107)T (cid:107) , is as small as possible [43], [44].
ωz ∞
InordertofittheH synthesisframework,theperformance
∞
1600
MBM criteria z and unknown disturbances ω should be clarified for
1400 Data automotive A/C systems. Mathematically, the six elements in
a] the vector of weighted performance criterion z are selected as
e [kP1200
Pressur1000 z =(cid:2)epe eSH Ncmp α pe SH(cid:3)T , (10)
800
wheree =p −p ande =SH −SH areerrorsonthe
pe e,r e SH r
6000 100 200 300 400 500 600 outputevaporatorpressureandsuperheattemperature.Ncmp is
Time [s] the compressor rotation speed and α is the valve opening per-
(b) CondenserPressure centage. pe and SH are, respectively, the evaporator pressure
and superheat temperature.
700 The reference evaporator pressure p and superheat tem-
r
MBM
600 Data perature SHr are time-varying and regarded as additional
disturbances besides the unknown disturbances m˙ , as well
a]500 ea
P
Pressure [k340000 aass:the noises.wTh=ere[∆fomr˙e, t,hpe di,sStuHrba,nnce,nve]cTto.r ω is defi(n1e1d)
ea e,r r 1 2
200
100 The original A/C model in state-space form is augmented
0 100 200 300 400 500 600
Time [s] with the output vector and disturbance vector defined, and a
(c) EvaporatorPressure H controller is obtained by solving Linear Matrix Inequal-
∞
ities (LMIs) associated with the augmented system according
Fig.3. VerificationofMBMA/CModelfortheSC03drivingcycle.
tomethodsprovidedin[43],[44].Theabovedesignprocedure
isdetailedin previouswork[38],where simulationresultsare
providedtosupportthevalidityofthecontrollerduringoutput
C. Coprime factorization of H Controller
∞ tracking and disturbance rejection.
H control and estimation problems have been extensively Note that as seen from Figure 2, the coprime factors U˜ and
∞
studied in literature [39]–[42], and are considered to be very V˜, instead of the controller model K, are of interest because
promising for the automotive industry. In this work, a general the fault accommodation signal q is added in between U˜ and
H synthesis is to find a controller K such that the closed- V˜. Therefore, the system matrices of the coprime factors are
∞
loop system is asymptotically stable and the H norm of the provided below,
∞
6
IV. FAULTDETECTIONANDISOLATION 50 6
foaocprttRiivrmeeassitizeiddaututhiaaoellsncg,ogeamennnepderearnatvistoaealndtiodarabrteQyeddtiehnusesitgihnFnegDeGdItIhiMmneCotnhdoesuntlrfleuriancmaeturaeerrwe.rMoeTrqBkhuMeiorffiedlHAte/∞trCos Actuator Residual12340000 Sensor Residual 12345 fraeusildtual
0
model. 0
−1
0 200 400 600 800 1000 0 200 400 600 800 1000
Time [sec] Time [sec]
A. H optimization
∞
Fig.4. FaultIsolationwithoutDisturbance.
The isolation filter H (l×p transfer matrix) is designed to
l
isolatethefaultvectorf anddecoupletheperturbationd.The
150 30
trade-off between these two objectives is also formulated as 100 20 fault
an optiHmImi∈zRaintHio∞n(cid:107)p(cid:2)ro0blemT: (cid:3)−HI(cid:2) N˜d N˜f (cid:3)(cid:107)∞, (12) Actuator Residual −55000 Sensor Residual−11000 residual
−100 −20
where T ∈ RH is a diagonal transfer matrix to be deter-
∞
mined according to the frequency response of N˜ , in order to −1500 200 400 600 800 1000 −300 200 400 600 800 1000
f Time [sec] Time [sec]
achieve the isolation and decoupling objectives.
The above optimization problem can be solved by a Linear Fig.5. FaultIsolationwithDisturbance.
Fractional Transformation (LFT):
HIm∈RinH∞(cid:107)Fl(GHI,HI)(cid:107)∞, (13) itsrannosdtumceert.aCnudrreonntehathrderwmaarel ccoonufiplgeu,raitsioanb,lei.e.t,ooinseolparteesstuwroe
where G stands for the generalized plant associated to the
HI independent faults simultaneously. In order to ensure the
LFT transformation given by
robustness of the isolation filter, it is required to increase
(cid:18) 0 T −I (cid:19) the observability of the A/C system. Herein, an additional
G (s)= . (14)
HI N˜ N˜ 0 thermocouple is placed on the wall of the section of the heat
d f
exchanger encompassing vapor refrigerant. In other words, an
B. Performance Evaluation additionalstate,thetemperatureofthewallinsuperheatphase
region, is available. Using the same weighting function as
The fault isolation filter generates independent residuals
before, a new isolation filter H is calculated again as an H
corresponding to either an actuator fault or a sensor fault. I ∞
optimization problem stated in Equation 13.
Based on Equation 13, the performance of a fault isolation
Next, the robustness of the new filter H to external
filterisdeterminedbytheselectedweightingfunction.Wenext I
disturbance is checked with stepwise variation of boundary
design the fault isolation filter H by selecting the following
I conditions on air side of the evaporator as drawn in Figure 6.
weighting function
A manoeuver is performed that takes the A/C system slightly
(cid:20) (cid:21)
1 0 1 away from the equilibrium design point. The manoeuver
T(s)= · . (15)
0 1 10s+1 performedtakestheA/Csystemevaporatorpressureup5kpa
and superheat temperature down 2.5oC. Due to the existence
The fault isolation filter H calculated by solving an H
I ∞
of external disturbance, the tracking performance is slightly
optimization problem stated in Equation 13 is implemented
deteriorated as the controller tries to compensate the distur-
in the nonlinear closed-loop simulation, where the plant is
bance during transient. For the same fault setting, however, it
given by the nonlinear differential equations of the MBM
is seen that the filter residuals are able to detect and isolate
A/Cmodel.Thisenablesthetestformodeldiscrepancieswith
the fault signature with a high degree of accuracy, without
respect to the linear plants used in the filter design process.
significant variation introduced by the external disturbance.
The actuator fault of 40 rpm is added into the compressor
rotation speed at 400 second, and the sensor fault of 5oC is Hence, it can be concluded that detection and robustness
capability of the filter is guaranteed under current choice of
added into the pressure signal at 700 second. When external
the weight function and computation scheme.
disturbanceisnotconsidered,thetworesidualscorresponding
to actuator fault and sensor fault respectively, are presented
in Figure 4. It is clear that both residuals respond to their V. FAULTACCOMMODATION
respectivefault,andaredecoupledfromtheotherfault.When The GIMC structure is an active FTC scheme since the
external disturbance is added as a stepwise signal, however, compensator Q will be activated by the residual signals
the two residuals corresponding to actuator fault and sensor generated by the FDI module. From analysis of the influence
faultrespectively,asdepictedinFigure5,areverysensitiveto of a fault at different locations of the closed-loop system,
the variation of ambient condition on the air side of the heat the problem of designing a compensator Q is reduced to an
exchangers. optimizationproblemthatminimizestheinfluenceofafaultat
Sensitivity of the designed isolation filter to external distur- either input or output of the plant. The fault accommodation
bances is due to the fact that the fault observability condition capabilityofthecompensatorQisdemonstratedbycomparing
7
optimization strategy:
Pressure [kPa]2222455650500 200 400 600 raec8fte0ur0aelnce10 00 Temperature [C]112205050 200 400 600 rae8cft0eu0raelnc1e0 00 Q∈m+RiHn(cid:20)∞−(cid:107)α(cid:20)dSαodPS0uoyPV˜dy−1−(cid:21)αQf(cid:2)S0iNK˜PfyN˜(cid:21) (cid:3)(cid:107) . (20)
Compressor Speed [rpm]Actuator Residual11112468240000000000000 220000 44TT00iimm00ee [[ss66ee00cc00]] 88frae00us00ildtua11l00 0000 Valve Opening [%]Sensor Residual234000024600 220000fraeus44ilTTdt00iiumm00aeel [[ss66ee00cc00]] 880000 1100 0000 TFwGrhhaQeecr(taeisob)Gno=avQleTroerpapαntriesdm−fsSoeiα0ozrnQPmfatstS∈dmiaoyRitthViin˜Honen−∞p−g1r((cid:107)eoαLnFbfFellSeTr(0amiG)Kl:iQzcPe,adfQnyp)bd(cid:107)lae∞−nst,α−oglfdαivSvefeodSnPi∞bubV˜yyy−V˜a1−L1in(2e1a,r)
Time [sec] Time [sec] N˜ N˜ 0
d f
(22)
Fig.6. FaultIsolationwithDisturbanceafterAdditionalSensorInstalled. and α ,α ∈ [0,1] are two weighting factors to balance the
d f
tradeoff between perturbations and faults reduction.
The two H optimization schemes given in Equations 17
∞
simulationresultsofapassiveFTCschemeandanactiveFTC
and 20 are derived by attenuating the influences of faults on
scheme. Moreover, the variation of the compensator Q over
outputs and inputs, respectively. The solutions to the above
the plant operating point is investigated as a preliminary step
two problems using the GIMC structure usually generate a
towards the gain-scheduled GIMC structure. compensator Q in high order. Thus it is necessary to perform
acontrollerorderreduction.Oneapproachisthestandardway,
A. Theoretical Background such as balanced truncation appealing to analysis of Hankel
norm of every system state. Another approach is to design a
The following lemma originally presented in [35] charac-
specific compensator for every studied fault by replacing the
terizesthedynamicbehaviorofthecontrolinputuandoutput
transfer functions N˜ and P with their corresponding parts.
y of the closed-loop system. f fy
Lemma 1. In the GIMC configuration considering additive
faults, the resulting closed-loop characteristics for the control B. Actuator Fault Accommodation
signal u and output y are given by When an actuator fault occurs, an active FTC scheme is
u(s)=S Kr(s)−S˜(V)−1(U˜M˜−1+Q) supposed to ensure the system outputs unchanged, enabling
i i
thesysteminputstomaintaintheoriginalvaluesifthesteady-
×[N˜ d(s)+N˜ f(s)],
d f state input-output mapping relationships are fixed. Since the
(16)
y(s)=T r(s)+S M˜−1(I−N˜V˜−1Q) sum of the commanded inputs and the faulty input signals are
o o
×[N˜ d(s)+N˜ f(s)], unchanged, the commanded inputs by the controller are mod-
d f
ulated automatically to compensate the faulty signals entering
where the input sensitivity is Si = (I + KPuy)−1, output thesysteminputs.Hence,theH∞ optimizationschemesgiven
sensitivity is So =(I+PuyK)−1 and complementary output in Equation 17 are suitable for actuator fault accommodation.
sensitivity is To =I−So =(I+PuyK)−1PuyK. After block replacing, the optimization problem for actuator
If one desires to attenuate both faults and perturbations at compensator design is defined as:
the output y, the following optimization scheme is suggested:
min (cid:107)S M˜−1(I−N˜V˜−1Q)Q(cid:2) α N˜ α N˜ (cid:3)(cid:107) . min (cid:107)S M˜−1(I−N˜V˜−1Q )(cid:104) α N˜ α N˜act (cid:105)(cid:107) .
Q∈RH∞ o d d f f ∞ Qact∈RH∞ o act d d f f ∞
(17) (23)
The above optimization problem can be solved by a Linear ThedesignedcompensatorQ isimplementedintheFTC
act
Fractional Transformation (LFT): schemeshowninFigure1,wheretheplantisreplacedwiththe
MBM A/C model. The simulation results of actuator accom-
min (cid:107)F (G ,Q)(cid:107) , (18)
l Q ∞ modation are depicted in Figure 7, with the same manoeuver
Q∈RH∞
strategy as before and one actuator fault of 40 rpm added at
where G is given by
Q 500 rpm.
(cid:18) α S P α S KP −S P V˜−1 (cid:19) The solid lines represent the inputs and the outputs of the
G (s)= d o dy f o fy o uy .
Q α N˜ α N˜ 0 A/C system with a passive FTC scheme, which refers to
d d f f
(19) an output tracking controller designed using H synthesis
∞
If one wants to minimize the fault effects on the control withoutfaultaccommodationfunction.Whenanactuatorfault
signal while reducing the perturbation contribution at the out- occurs,apassiveFTCschemehassomeextentofcapabilityof
put, the compensator Q should be designed by the following fault compensation. In the top figures, the compressor speed
8
is reduced by 20 rpm to ensure that the evaporator pressure to calculate the commanded inputs. In the top figures, the
does not deviate from the reference value. measured evaporator pressure is maintained, while the actual
The dashed lines represent the inputs and the outputs of evaporatorpressureisenforcedtotrackanotherreferencevalue
the A/C system with an active FTC scheme using GIMC deviatingfromthenominalvaluebytheamplitudeofthefaulty
structure. Still from the top figures, the compressor speed is output,5kPa.Hence,thecompressorspeedboostsup50rpm
reduced by 40 rpm by the actuator fault compensator Q , inordertodrivetheactualevaporatorpressuretothedeviated
act
which is the exact amplitude of the actuator fault added. reference value.
The output evaporator, after minor transient, returns to the The dashed lines represent the inputs and the outputs of
reference value without any deviation. The simulation results the A/C system with an active FTC scheme using the GIMC
are natural outcomes of the design process, since the GIMC structure. Still from the top figures, the compressor speed
structure activate a compensator to tolerate the faulty input. only boosts up 10 rpm after the sensor fault compensator
Q is activate. Since the actual evaporator output is almost
sen
unchanged, the measured evaporator pressure starts to deviate
256 1510
Pressure [kPa]222222555555012345 paactsivseiv eF TFCTC Compressor Speed [rpm]11111444456789000000 paactsivseiv eF TFCTC icmrfseroosumaumcdlphtdesthbenadeesrt.aentteAoenrdmalpt,thiuenotrrhafauolelgrrohmeaufcattethcnrioevecmenetcoetFehtsaTavolnCafsltduehsneeecssbhiopgyeranms4fspaeikruvoPlhetcaaeos5snaseak,f.ltPsreTiernaahcfdeaeiyusstlihtnmayeocuhGtoliuaIfetMutviploeluCdnyt
0 200 400 600 800 1000 0 200 400 600 800 1000
Time [sec] Time [sec] structure activate a compensator to ensure the actual output is
50 almost unchanged and the measured output changing by the
6
Actuator Residual12340000 fraeusildtual Sensor Residual 024 fraeusildtual amplit2u60de of the faulty signa l. 1550
0 passive FTC m] passive FTC
Fig.7. F0 ault20A0cco4T0mi0mem [soe6c0d]0atio8n00of10A00ctuato−r20 Faul2t0.0 4T0i0me [se6c0]0 800 1000 Pressure [kPa]222255552468 active FTC mpressor Speed [rp1500 active FTC
Co
250 1450
0 200 400 600 800 1000 0 200 400 600 800 1000
Time [sec] Time [sec]
50
C. Sensor Fault Accommodation fault 6 fault
dsyessWtierehmdenotouatpenusestsunrsteoortmhfeaauisnlyttsatioencmcthuiernsp,ouratisgniunanaclcthivvaaenlugeFesdT,Cpernosvacbihdleienmdgeththaiest Actuator Residual12340000 residual Sensor Residual 024 residual
0
the steady-state input-output mapping is preserved. Since 0 200 400 600 800 1000 −20 200 400 600 800 1000
Time [sec] Time [sec]
actual system outputs are unchanged, the sum of the actual
outputs and the faulty output signals deviate from the original
Fig.8. FaultAccommodationofSensorFault.
measured outputs. However, the controller disregards the de-
viation, and uses the original measured outputs as before. In
other words, the sensor fault does not affect the closed-loop
system performance significantly. Hence, the H optimiza- D. Plant Variation
∞
tion schemes given in Equation 20 are suitable for actuator
The variation of the compensator Q and Q over the
act sen
fault accommodation. After block replacing, the optimization
operatingpoint,orcoolingloadforA/Csystem,isexaminedto
problem for sensor compensator design is defined as
exploit the possibility of gain-scheduled GIMC structure. The
(cid:20) α S P 0 (cid:21) nonlinear MBM A/C model is linearized at three operating
min (cid:107) d o dy
0 −α S KPsen conditions, corresponding to low, medium and high cooling
Qsen∈RH∞ f i fy
(cid:20) −α S P V˜−1 (cid:21) (cid:104) (cid:105) (24) loads. The cooling load is regulated by changing the inlet air
+ d o uy Q N˜ N˜sen (cid:107) . temperature of the evaporator. For consistency, the superheat
−α S V˜−1 d f ∞
f i temperature is kept around 20oC by cooperation of the com-
The designed compensator Q is implemented in the pressorspeedN andtheexpansionvalvepositionα;however,
sen c
FTC scheme shown in Figure 1, where the plant is replaced the evaporator pressure P is allowed to vary according to
e
with the MBM A/C model. The simulation results of sensor the cooling load as a gain scheduling parameter. Specifically,
accommodation are depicted in Figure 8, with the same the boundary conditions, controlled inputs and steady-state
manoeuver strategy as before and one sensor fault of 5 kPa refrigeration status are summarized in Table below. For every
added at 500 sec. operating point, the actuator and sensor compensators are
The solid lines represent the inputs and the outputs of the designed following the active FTC scheme using the GIMC
A/C system with a passive FTC scheme. For a sensor fault, a scheme. From Figures 9 and 10, it is not difficulty to see that
passive fault tolerant controller uses the faulty measurements the GIMC can also be made to be adaptive to working point.
9
Bode Diagram
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