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FAULT DIAGNOSIS OF
A HEAT EXCHANGER SYSTEM USING
UNKNOWN INPUT OBSERVERS
Howard Hao-Yuan Chou
A thesis subrnitted in conformity with the requirements For the degree of Master of Applied Science
Graduate Department of Electrical and Cornputer Engineering University of Toronto
O Copyright by Howard Hao-Yuan Chou 2000
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ABSTRACT
Fault Diagnosis of A Heat Exchanger System Using Unknown Input Observers
Master of Applied Science, ZOO0
Howard Hao-Yuan Chou
Department of Electrical and Computer Engineering
University of Toronto
In this thesis, the fault diagnosis of a heat exchanger system. consisting of two heat exchangers
in series. is investigated. The system. modeled by ten fint-order nonlinear differential
equations. has three unknown disturbances and three output measurements. The p a l is to
determine the degradation levels in heat exchangen based on the sizes of the residuals generated
via unknown input observee. The residuals must be robust against disturbances and sensitive to
the degradation. It is proved that this is only achievable when the number of output
measurements is greater than that of disturbances. Therefore. either some of the disturbances
must be eliminated or more sensors must be installed. With one of the disturbances assumsd
constant, the degradation in the first heat exchanger cm be determined accurately with
reasonable precision. The addition of one more sensor results in a more precise diagnosis of the
first heat exchanger. The degradation in the second heat exchanger can be determined when a
second sensor is added. but the diagnosis is crude and only eighty percent accurate at b e a
ACKNOWLEDGEMENTS
1 would like to thank:
Professor Kwong for giving my directions and advice throughout rny thesis work.
Professor Wonham for his insightful feedback during the process of my research.
Dr. ChunHo Lam for providing usehl information on various approaches to tackle the problem.
and Dr. Maher Khalil for helping me to Ieam about turbomachinery and heat rxchangers.
TABLE OF CONTENTS
1. Fault Diagnosis And Prognostic Health Management
1.1 Introduction
1.2 Observer-based Approac hes
1.3 Parity Relation Approaches
i .l Puanieter Estimation hpproachcs
1.5 Fuzzy Logic Approaches 7
1.6 Neurai Network Approaches 9
1.7 Summary I I
2. The Target Heat Exchanger System For Fault Diagnosis Study 12
2.1 Background 12
2.2 Target Heat Exchanger System 13
2.3 Development of A Mathematical Mode1 15
2.4 Possible Faults of the System 20
3. Residual Generation Using Unknown Input Observers 21
3.1 introduction 2 1
3.2 Theory of Unknown Input Obsewers 22
3.3 Robust Residual Generation and Its Sensitivity to Faults 27
4. Robust Fault Diagnosis Of The Target System Using Unknown Input Observers 31
4.1 Preliminaries 3 1
4.2 Fault Modeling and Problrm Definition 35
4.3 Application o f the UIO-based Approach 36
4.3.1 Linearization 36
4.3.2 Violation of Existence Conditions 38
4.3.3 Physical interpretation and Proposed Solution 40
4.4 Residual Generation with Additional Sensors 12
4.4.1 installation of One Additional Sensor 42
4.4.2 Installation of Two Additional Sensors 5 1
4.5 Residual Generation with Fewer Disturbances 54
4.6 Fault Diagnostic Scheme and Simulation Results 55
4.6.1 Fault Diagnostic Simulation with TLi Assumed Constant
4.6.2 Fault Diagnostic Simulation with Th Measured
4.6.3 Fault Diagnostic Simulation with Tfi and T'fi2 Measured
4.7 Summary
5. Conclusion
S. 1 Discussion of Results
5.2 Future Research
Appendices
A Curve Fitting Using Experirnental Data of Heat Transfer Coefficients Versus Flow
Rates
B Matlab Files
C Residual Response to Degradation in Engine Oil Heat Exchanger
D Residual Response to Degradation in Sink Heat Exchanger
E Diagnostic Simulation Results
Reference
LIST OF TABLES
Table 2.1
Table 4.1
Table 4.2
Table 4.3
Table 4.4
Table 4.5
Table 4.6
Table 4.7
Table 4.8
Table 4.9
Table 4.10
Table 4.1 1
Table 4.12
Table A. I
Description of Parameten in the Mathematical Mode1 of the Heat Exchanger
S y stem
Numerical Values of Parameters
Modeling of Faults in the Target System
EEect of Degrd~ttion in Hrat Exchmgers on Residuals
Uncertainties in Residuals
Calibration of Degradation in Engine Oil Heat Exchanger with TLi Assumed
Constant
Diagnosis of Degradation in Engine Oil Heat Exchanger with TLI Assumed
Constant
Calibration of Degradation in Engine Oil Heat Exchanger with T f 2 Measured
Diagnosis of Degradation in Engine Oil Heat Exchanger with Tji: Measured
Calibration of Degradation in Sink Heat Exchanger with T f i and T f 2 Measured 63
Diagnosis of Degradation in Sink Heat Exchanger with T f i and Tfi? Measured 68
Calibration of Degradation in Sink Heat Exchanger with Smaller Variations
in Input and Disturbances 69
Diagnosis of Degradation in Sink Heat Exchanger with Smaller Variations
in Input and Disturbances 7 1
Heat Transkr Coefficients Versus FIow Rates 76
LIST OF FIGURES
Figure 1.1
Figure 1.2
Figure 1.3
Figure 1.4
Figure i.5
Figure 2.1
Figure 2.7
Figure 2.3
Figure 2.4
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.1 0
Figure A. 1
Figure A.2
Figure A.3
Figure A.4
Observer-based Fault Diagnosis
Residual Generation Using Parity Reiations
Fault Diagnosis Using Fuzzy Logic
Typical Architecture of A Neural Network
Input-output Relation of X Node
The Environmental Control System
The Target Heat Exchanger System
Cross-tlow Plate Heat Exchanger
Schematic Diagram of the Cross-tlow Heat Exchanger Model
Typical Noisy Measurement
Cornparison of Step Responses of the Differential Equation Model luid the
Honeywell Model
Dependence of HLIZICL on WL
Dependence of Hf?I3ICL on FVf
Settings of Input. Disturbances and Heat Exchanger Degradations for Fault
Diagnostic Simulation
Average Residual and Threshold with TLi Assurned Constant
Average Residual and Threshold with T f i Measured
Gaussian Distribution and the ThreshoIds
Average Residual and Thresholds with Tji and Tfi Measured
Settings of Simulation Conditions for Diagnosis oCSink Heat Exchanger
Hf = 3.83 ~ f ~ ~ . ' ' ' ' '
Hp = 1.1 3 w ~ ~ * ~ ~ ~ Hf2 = 1.41 w?."" HL = 1.77 WL*.'""
vii
Chapter 1
Fault Diagnosis And Prognostic Health Management
1.1 Introduction
Fault diagnosis is a method that can be applied to a physical system to identi@ component
failures within the system. The development and research in this area began in the 1970s.
Some of the early texts on this subject include Collacon ( 1977) and Himmelblau (1978). As
engineering systems become more and more sophisticated. the demand for higher safëty and
reliability of the systems is increasing. nierefore. fault diagnosis has received a lot of attention
and has become an important subject in modem control theory. Fault diagnosis of a system is
accomplished by combining information on controlled inputs and rneasured outputs. together
with the knowledge of the system to detemine the health of the system. It usually consists of
two phases: fauit detection and fault isolation. Fault detection refers to the decision whether
there is something wrong with the system. Fault isolation refers to the identification of the
location of the fault; for instance, which component has failed. The traditionai fault diagnosis
method mady deds with the detection and isolation of abrupt component failures. Recently
more attention has been focused on the diagnosis of incipient faults and gradua1 degradation of
die cornponents. If a fault is detected at its early stage, preventive measures can be taken before
the component fails.
Prognostic health management, abbreviated as PHM. is a new notion in the area of fault
diagnosis. The term PHM originates from medical sciences to represent the prediction of
possible developing diseases based on early syndromes. PHM for physical systems has two
tasks: 1. To detect and isolate component degradation before a failure occurs. 2. To predict the
remaining life of the components. The fint task is an extension to the conventional fault
diagnosis technique. Instead of identifying cornponent failures a f er they have occurred. PHM
determines the degree of component degradation. which may eventually lead to a component
failure. The second task requires a wear model based on historical performance data for each
component. Given the current degree of degradation and operating condition. the Wear model
cm be used to predict the remaining life of the component. The PHM approach is particularly
useful for cntical systems in which a component failure may be disastrous because it makes
possible the unscheduled maintenance to take place to prevent abrupt failures. The developrnent
of PHM for physical systems is still at an early stage and this term is not standardized in the
literature. Some other notions similar to PHM include condition monitoring. and health and
usage monitoring.
This thesis focuses on the identification of both abrupt f'lures and component degradation
in a physical system. which will be referred to together as Fault diagnosis in the rest of the thesis.
The remainder of this chapter is devoted to a discussion of the cumnt fauit diagnosis techniques
and their applicability to abrupt and graduai faults. These techniques inciude observer-based
approaches. parity relation approaches. parameter estimation approaches. h c y logic
approaches. and neural network approaches. Chapter 2 details the heat exchanger system
selected for fault diagnosis study and its model construction. Chapter 3 presents the theory of
unknown input observers and the framework of residual generation using unknown input
observee. Chapter 4 describes the application of the unknown input observer approach to fault
diagnosis of the target system. Chapter 5 concludes the thesis with a discussion of the results
and possible areas of fiiture research.
1.2 Observer-based Approaches
The observer-based fault diagnosis requires the knowledge of a mathematical mode1 and the
concept of residuals. An observer provides the estimates of the states of the system. which are
supposed to track the real states asymptotically under normal condition. The residuais are
formed as differences or a linear transformation of the differences between the measured and the
estimated outputs. In the case where the system is operating normally. the residuals should be
identically zero. If a fault occurs within the system, the residuals will deviate from zero. A
threshold on the nom of the residuals is selected. and then a fault is said to be detected when
this threshold is exceeded. The general structure of this approach is illustrated in Figure 1.1.
Inputs System
Residual Generation
1 Residuals
Residual Evaluation
-- Outputs (measured)
* Diagnosis
Figure 1.1 : Observer-based Fault Diagnosis
The mode1 of a linear system c m be described in the following state space representation:
where z ( t ) E \Rn is the state vector. ~ i ( r ) E 93' is the controlled input vector. and y(r) E !Rn' is the
measured output vector; A. B. C and D are known matrices with appropriate dimensions. Let f ( r ) be the state estimate. then the residuals are formed as r(r ) = Q( y(t ) - Cï(r ) - Dl@ )). where Q is an
user - defined matrix. which may be the identity matrix in the simplest case.
The possible faults of the system include actuator faults. sensor faults and component faults.
al1 of which cm be represented by an additive fault vectorflt) and the fault entry matrices R I
and Rz as in Equation ( 1.2).
The vector d(t) models the disturbances of the system. which rnay include modeling errors. plant
variations and unknown inputs. One important consideration in fault diagnosis is its robustness.
In the observer-based approac h. this translates into making the residuals ro bust against the
disturbances: that is. under normal condition. the residuals should remain close to zero in the
presence of disturbances. Therefore. the design objective is to make the residuals sensitive to
faults and insensitive to disturbances. Some methods to achieve robust observer-based fault
diagnosis include the unknown input observer and eigenstructure assignment as described in
Chen and Patton ( 1999).
Fault isolation is achieved by drsigning the residuals to be stmctural or directional.
Structural residuals are formed by a bank of observers. rach of which has a different subset of
vectors y(t) and u(r) as its inputs. such that the residuals generated with each observer are
sensitive to different groups of faults. A single fault or a pariicular group of faults can be
isolated by the patterns of structural residuals. Directional residuals utilize the design freedom
of the observer gain. which is chosen so that different faults cause the residuals to point to
different directions in the residual space. Both of these fault isolation schemes are discussed in
Chen and Patton ( 1999).
The robust fault diagnosis for nonlinear systems is an area of ongoing research. The design
of nonlinear unknown input observers is cornplicated and is currently restricted to particular
classes of nonlinear systems. Some of these designs can be found in Seliger and Frank (1 991)
and Alcorta Garcia and Frank (1997). Conventionally. residual evaluation invofves a binary
decision. which only determines whether the threshold is exceeded. This works well for abrupt
failure. but is not adequate to determine gradual faults. More information on gradual faults can
be obtained from the sizes of the residuals. This is further explored in Chapter 4.
1.3 Parity Relation Approaches
The parity relation approach is similar to the observer-based approach because it also uses
the mathematical model. the input vector id(!) and the output vectorfit) of the system to generate
residuals. The difference is that the residuals are formed directly without relying on the state
estimation (see Figure 1.2). This method of generating residuals is linear in nature and hence
cm only be applied to a linearized system.
- Outputs y(t) (measured)
Residuat Generation
(Parity Relation)
_+
1 Residual Evaluation
+ Diagnosis
Figure 1.3: Residual Generation Using Parity Relations
Consider a discrete-time linear system described in the input-output equation:
where z is the shifi operator, G(z) and Y(=) are polynomiai matrices. h(z) is a scalar polynomial.
and p(t) = ~ t ) ' d(r)T]T is a vector containing both faults and disturbances. Then the residual
generator takes the fom:
The design objective is to choose S(2) and Q(z) such that the residual vector r(r) has the
following properties: 1. It is sensitive to faults and robust against disturbances. 2. It exhibits
fixed directions in response to particular faults so that fault isolation can be achieved. The
detailed design procedures c m be found in Gertler and Monajemy ( 1995).
1.4 Parameter Estimation Approaches
A more intuitively direct method of fault diagnosis is tu estimate the Fault-sensitive physical
parametes of the system. This approach is best suited for gradua1 component Faults that c m be
represented by changes in physical parameten. If some physical parameter of a particular
component has changed from its normal value. there might be something wrong with that
component. The degree of degradation cm be inferred from the size of the change in the
parameter. The diagnosis of abrupt failures such as sensor faults and actuator faults is not easy
using parameter estimation since these drastic faults cannot be represented well by changes in
the parameters. Fault isolation c m be achieved by identifying the association of the physical
parameters with the components. The robustness depends on the parameter estimation methods:
however. the parameter estimates are usually quite sensitive to disturbances. A detailed
description of the parameter estimation methods can be found in the texts written by Koch
(1 999), Vas (1 993) and Walter (1 997).
For a linear system represented by Equation (1.1). the parameters in system matrix A can be
estimated by a recursive algorithm as described in Jonsson, Palsson and Sejling (1992). The
algorithm searches for parameter esthates that minimize an output error function. Another
approach is to estimate the coefficients of input-output transfer functions. This is simpler and
more feasible than trying to estimate the parameters in the matrix A. Its disadvantage is that
there might not be a one-to-one correspondence between the transfer function coefficients and
the physical parameters; in other words. the physical parameters may not be uniquely
determined from the estimated coefficients. A single-input-single-output system can be
described by the following transfer function:
In time domain. this cm be witten as y'"' = ylr8. where = [-y'"-" - y dm' * - O ir]
ander =[a,- , a, b, -.- b, 1. The transfer function coeffcient vector 0 is estimated. Application of the l e s t square algorithm to estimate the transfer function coefficients can be
found in Isermann ( 1 992) and P feufer ( 1 997).
Nonlinear p m e t e r estimation technique does exist but is computationally intensive. It
involves simulating the nonlinear system with the pararneter estimates recursively until the
minimum of an error function is reached. Bard (1974) and Stortelder (1998) provide good
references on the algorithms of nonlinear parameter estimation. The problems arise due to the
complexity of the nonlinear system. First of dl. the convergence criteria are not well defined.
Secondly. the global minimum is not easy to reach since the algorithm may get mick in a local
minimum. Application of the linear pararneter estimation to a linearized system may or may not
present a better solution because the nonlinear dynamics of the system may be excited by the
deviation h m the equilibrium. which wil1 introduce errors to the parameter estimates.
1.5 Fu- Logic Approaches
The application of fuzzy logic to fault diagnosis is a relatively new idea. An introductory
reference on fuzzy logic can be found in Nguyen (2000). A method of generating nnictural
residuals using fuPy process models is proposed by Ballé. Fischer. Fussel. Nelles and I se rmm
(1 998). The structural residuals are formed from subsystem models. which depend on different
sets of inputs. The residuals are sensitive to different faults and hence fault isolation can be
achieved. Each of the subsystem models is constntcted using f u q logic. The fuPy process
mode1 is nonlinear in nature; hence it is able to represent both linear and nonlinear dynarnic
systems. In general, a discrete-time single-input-singlesutputput nonlinear dynamic system can be
described by:
The output y(k) depends on the input il. measurable disturbances d, (i = 1.. .m) and the previous
outputs. The dead times of u and d, are denoted by s and s,: the dynarnic orden of if. d, and y are
denoted by nu. nd, and ny. According to TaKagi and Sugen (1985 j. ille fwz?; mijdcl for :bc
system in Equation (1 -6) can be constructed by a rule base with M rules of the following fom.
R,: IF zl is AJTl AND ~1 is ilJmr AND. ..AND 2, is ilJvnr
r where A,, is a funy set defined on the trajectory of zl. both z = [il zt...=,Jr and .r = [.ri x2....r,,]
contain subsets of the elements of ~ k ) and y,., is a p m e t e r to be determined. T The rules are developed with expert knowledge and the parameters. w = [ivi IV? ... IV,,] . are
identified off-Iine using expenmental data. Once the residuals are generated. they can be
evaluated using funy logic (see Figure 1.3). As implied by its name. the hzzy logic approach
produces qualitative diagnoses nther than quantitative mes. Because the diagnosis is
qualitative. it is not as precise as that generated by the previous approaches. but it is more
accurate for the same reason.
Inputs u(t) -7 Systern 1 1
Figure 1.3: Fault Diagnosis Using Fuay Logic
1.6 Neural Network Approaches
Artificial neural network has recently emerged as a strong tool in modeling of nonlinear
systems. The fundamental theory of neural networks c m be found in Anthony (1999) and De
Wilde (1997). The system is treated as a black box and no mathematical mode1 of the system is
required. The basic elements of the neural network are nodes and edges. The nodes are
comected by the edges unidirectionally. Each node c m have a number of fan-ins and km-outs:
hence a network structure is obtained. A simple neural network is illustrated in Figure 1.4.
Figure 1 A: Typical Architecture of A Neural Network
An activation fùnction is associated with each node to detemine the output according to the
input to that node. A weight is associated with each edge. which is multiplied with the output of
the preceding node in calculating the input of the next node. More precisely. the input to a node
is the sum of the product of the outputs from preceding node and their associated weights (see
Figure 1.5).
Figure 1.5: input-output Relation of A Node
Here a, is the output of node j. a, is the output of the preceding nodes. and w, is the weight
between node i and j. The input to node j is given by the sum:
A common activation tùnction is the sigrnoid function. which is of the foilowing form:
A neural network has to be trained before it c m be used to predict the system outputs.
Training is cmied out using experimental inputsutput data. The weights are assigned to
associate the input with the corresponding output. This is an optimization problem where a set
of weights is sought to minimize the output erron (defined as the differences between the
experimental outputs and the outputs of the neural network) for al1 input patterns. The neural
network finds its use in fault diagnosis in two ways. Firstly. it c m be used to evaluate the
residuals; that is, it can be trained to associate the residuai patterns to the fault conditions.
Secondly, the neural network cm be used to directly associate output measurements and input
commands to the fault conditions. An example of application of the neural network to fault
diagnosis can be found in Juuma and Parkkinen (1994). The strength of the neural network lies
in its architecture and its parallel processing capabilities. In fact, the performance of the neural
network depends heavily on the way the nodes and edges are htercomected. Some systems are
better modeled with certain architecture while some systems are better represented by another.
1.7 Summary
Among the above five approaches for fault diagnosis. the observer-based, parity relation and
parameter estimation approaches fall into the category of model-based fault diagnosis because
they al1 require a mathematical model of the target system. Most of the developrnent in these
three approaches deals with the linear systems. The nonlinear unknown input observen are only
applicable to systems of some particular forms. The nonlinear parity relation approach does not
exist yet while the nonlinear parameter estimation is complicated and hard to implement. Since
the majority of the real-world systems are nonlinear. robustness against plant mismatches. in
addition to disturbances, is an important issue. The parameter estimation approach has the
disadvantage of being relatively sensitive to disturbance compared wi-th the other two model-
based approaches. The observer-based and parity relation approaches are more suited to
diagnose additive faults while the parameter estimation approach c m easily identify the
parameter faults.
Although the fuzzy logic approach does not require a detailed model. some expert
knowledge about the system is needed in order to construct the nile base. On the other hand. the
neural network is a black box approach. which does not need any information about the system.
The advantage of not requiring a mathematical model cornes at the expense of off-Lne leaming
or training with large amount of expenmental data. Both the funy logic and neural network
approaches are able to handle nonlinear systems. There is no forma1 formulation of disturbance-
decoupling schemes for these two approaches. Presumably a complete set of experimental data
would include the scenarios of al1 possible effects of disturbances. and through training. the
fuzzy logic model and the neural network will be able to generate correct outputs in the presence
of disturbances. However. one can never obtain the complete experimental data that coven the
entire range of the system dynamics. Also. the selection of the appropriate architecture for the
neurai network is a dificult task even for experienced usen. The typical process of constructing
a neural network involves trials and errors of different architectures. This procedure is very
time-consuming as each neurai network has to be trained and tested before the best architecture
can be selected.
Chapter 2
The Target Heat Exchanger System
For Fault Diagnosis Study
2.1 Background
This thesis is a result of the research work done for the Joint University of Toronto and
Honeywell Prognostic Health Management Project. The ultimate goal of the project is to
develop PHM techniques for physical systems in an aircrafl. such as environmental control
system, electrical power generation and management system. landing system. secondary power
system, emergency power system. hydraulics. and engines. Traditionally component failures
are identified using conventional built-in-test techniques and the regular maintenance is
scheduled according to histoncd performance data. However. sometimes component failures
can occur between maintenances and the after-the-fact detection is unacceptable for some
critical systems. The application of P H M techniques would ailow for early detection of gradua1
faults and unscheduied maintenance to take place before zibmpt failures occur. This will
increase the reliability and deaease the life cycle cost of the system. This work is the fim step
towards developing PHM techniques for practical applications at Honeywell.
2.2 Target Heat Exchanger System
A heat exchanger system inside the environmental control system of a particular aircraft is
selected as the target system for PHM study. The objective is to investigate possible approaches
for diagnosing both abrupt and graduai faults associated with the system. A heat exchanger is a
device where heat transfer takes place between two Buid flows. The core of the heat exchanger.
usually made up of metal, separates the two fluids. Heat will tlow fiom the hot nuid to the cold
fluid through the metal core. The main components of the environmental control system include
a compressor and a cooling turbine joined by a rotating shaft. The bleed air From the engine is
compressed first to raise its temperature and pressure. Then. the hot air is passed through a heat
exchanger to be cooled d o m by a cooling liquid named PAO. The air is tùrther cooled down
when it expands in the cooling turbine and tums the shaft. which drives the compressor.
Finally, this cold air cm be used to cool the cabin and the avionics. A block diagram of the
environmental control system is illustrated in Figure 2.1.
Lube Oil PA0 From Gquid Engine
i Fuel F rom Target Heat Fuel To Fuel Tank - Exchanger System / Engine
Bleed Air
Lube Oil Fram Engine
Cold A
To Engine
L To Luinni-
Heat Hot Ai; Air Exchanger 1
PA0 Used for Cooling Usewhere
Figure 2.1 : The Environmental Control System
The task of the target heat exchanger system is to cool down the PA0 liquid and the
lubrication oil for the engine. It consists of two heat exchmgers, one bypass valve and three
temperature sensors, as s h o w in Figure 2.2. The he l acts as the cold fluid in both heat
exchangen. The PA0 liquid enters the sink heat exchanger to be cooled down before it is used
for cooling in the environmental control system. The flow rates of fuel. PA0 liquid and lube oil
are denoted by Wf, Wp and WL respectively. The bypass valve divides the fuel into two flows,
Wfi and Wf2, which join together before entering the engine oil heat exchanger. The inlet fuel
temperature, the inlet PA0 temperature. the outlet hie1 temperature and the outlet PA0
temperature for the sink heat exchanger are represented by Tfi. Tpi. Tfo and Tpo. The inlet fuel
temperature, the iniet oil temperature. the outlet fuel temperature and the outlet oil temperature
for the engine oil heat exchanger are represented by Te2. TLi. Tfoz and TLo.
The three temperature measurements are TI. T2 and Ti. It is desired to keep the temperature
T l below a certain threshold. If this threshold is exceeded. the bypass valve wiil be adjusted to
let more fuel through the sink to cool the PA0 liquid. The fuel is also used to cool the lube oil
in the engine oil heat exchanger. However. there is an upper limit on the fuel temperature T3 to
prevent the fuel from self-igniting. If this limit is rxceeded. the bypass valve will be adjusted to
let more fuel bypass the si&. Although the temperature limits on TI and T3 are two conflicting
requirernents. the latter has a higher priority due to safety reasons.
Lube Oil
Fuel
Valve
wr,
Figure 23: The Target Heat Exchanger System
Tp i WL TLi
I 1 *
Tfi2 OilHeat Exchanger
wf, fi Wf, m i n e
Bypass exch changer ~f~~
5 Sink Heat T ~ O
b
23 Development of A Mathematical Mode1
A mathematical model of the system cm be developed using differential equations. Both
the sink and engine oil heat exchangers are cross-flow plate heat exchangee, in which the flow
directions of the coId and hot Buids are perpendicular to each other and the tluids are separated
by a metd plate inside the heat exchanger (see Figure 2.3).
Hot Flow
Side View
Top View I I Metal 11 ! Plate
Cold Flow - 1
Figure 2.3: Cross-flow Plate Heat Exchanger
Heat exchangen can be rnodeled by direct lumping of the process. Jonsson and Palsson
(1991) describe this approach in detail. The heat exchangers are divided into smailer sections.
each of which is modeled from the first pnnciple of thermodynamics. The outlet temperatures
of each section become the inlet temperatures of the next one and these smdl models are
combined to give rise to the complete heat exchanger model. The cross-flow heat exchanger
can then be divided into n x m sections as show in Figure 3.4.
Thi Thi Thi
Tci , WC,
TCOIT mI2 Section 11 section 12 ' - . - . - .
Tci 1 1 1 ,,,,$ -7 Section 21 1 7
Section 1 m 4 r=
1 Tco,, J section nrn 1-.
Figure 2.4: Sc hematic Diagram of the Cross-flow Heat Exchanger Mode1 - - - - - - - - - - - - - - - - - - - - - -
The hot flow is equally divided into m tlows (Whi. Wh2. . .. Wh,), and the cold tlow is equally
divided into n flows (WC,. WC?. ... WC,). The inlet temperature of the hot fluid. Thi. is the hot
inlet temperature for the sections in the fint row: the inlet temperature of the cold tluid. Tci. is
the cold inlet temperature for the sections in the first column. The overall outlet temperature of
the hot fluid is the average of the hot outlet temperatures (Thoni, Thod. ... Thon,) from the
sections in the last row; the overall outlet temperature of the cold fluid is the average of the cold
outlet temperatures (Tco ,, Tc*,. . . . Tco,,) from the sections in the last column.
The following assumptions are made: 1. The heat transferred tu the surroundings is
negligible. 2. There is no heat conduction in the Buids themselves. 3. The temperature is
uniform in each section. 4. The specific heat capacities are constant for both fluids. With these
assumptions, differential equations (2.1) to (2.3) cm be written for each section. as in Jonsson
and Palsson ( 1 99 1 ).
b
Mc - Cc. Tco = WC #CC(TC~ - TCO) - Hc (TciSTco - Tm)
M ~ . c ~ - G = ~ h ( ~ ~ ~ + ~ ~ ~ - 3 - T ~ ) + H C ( TC^ + - 3 TCO - h)
where Ah, Mc and Mm are the masses of hot tluid. cold fluid and metal plate inside each
section, Ch, Cc and C m are the specific heat capacities of hot Auid. cold fluid and metal plate.
Tho. Tco and Tm are the hot outlet temperature. cold outlet temperature and the temperature of
the metal for each section. Thi and Tci are the inlet temperatures of hot fluid and cold fluid, Wh
and WC are the flow rates of hot tluid and cold tluid, Hh and Hc are the heat transfer coefficients
of hot fluid and cold fluid and they are functions of flow rates and the size of the section.
Equation (2.1) is the energy balance equation for the hot side. which says the total change in
energy of the hot fluid is a result of an increase in energy due to the hot inlet tluid minus the
heat loss through the metal plate. Equation (2.2) describes the energy balance for the cold side
following the same principle. Equation (2.3) states that the change in energy of the metal cornes
from the heat transferred from the hot side and the coid side.
The bypass valve is modeled with a gain and a first-order time constant. In frequency
domain. the transfer Function relating the tlows Wfi and Wf (see Figure 2.1 ) takes the following
fonn.
where kv is the valve gain. rv is the time constant, u is the valve command. Wf is the total Fuel
flow, Wh is the fuel let through the sink heat exchanger and it can be varied between the value
zero and Wf by adjusting the valve command 11. In time dom ai^ the differential equation is
given by :
The fuel that bypasses the sink heat exchanger is simply Wh = Wf - WJ.
The temperature sensor is rnodeled with a first-order lag between the measured value and the
actual value. For instance, Ti is the measurernent of Tpo, the outlet temperature of PA0 From
the sink heat exchanger, and the differential equation relating Ti and Tpo is:
where rsi is the tirne constant of the sensor.
'l'he model of the system in Figure 7.2 cm be consuucrrd by çumbininy ilie niuclels t j r Iieaî
exchangen, the bypass valve and sensos. For simpiicity. both the sink heat exchanger and the
engine oil heat exchanger are modeled with only one section (m = n = 1): hence each heat
exchanger model has three States. One c m always increase the order of the model by dividing
the heat exchanger into more sections. The overall model is described by the following
differential equations.
T Tpo q =-A+- 81 =,
Tm = Hp Tpo+ Hf rfo- Hp+Hf Tm+ 2- Mm-Cm 29 Mm. Ch
Hp Tpi+ Hf Tfi ~tfrn- Cm 2 Mm- Cm 2 Mm* Cm
where
The description of parameren in the abore equations is s u ~ î ~ z c d in Table 2.1.
1 TLo
lp- TLi
Unit " F O F
Type State Variable, measured State Variable. rneasured
Table 2.1 : Description of Parameten in the Mathematical Model of the Heat Exchanger System
Description Measurement of Tpo Measurement of TLo
O F
O F
O F
O F
O F
O F
O F
I brnis none "F O F
_ O F I bm/s Ibm/s Ibmh none s s s
State Variable, measured State Variable State Variable State Variable State Variable State Variable State Variable State Variable Controlled Input Non-controlled Input Non-controtled Input Non-controlled Input Non-conuolled Input Non-control led Input Non-controlled Input Physical Parameter Physical Pmmeter PhysicaI Parameter Physical Parameter
Phys ical parameter Physical Parameter Physical Parameter Physical Parameter Physicai Parameter Physical Parameter Phy sical Parameter Physical Parameter
Measurement of Tfo2 Outlet P A 0 temperature Out let fuel temperature from sink heat exchanger Metal plate rempenture of sink heat exchanger Outlet lube oil temperature Outlet fuel tempemure from engine oii heat exchanger Metal plate temperature of engine oil heat exchanger Fuel flow through sink heat exchanger Valve position command Inlet fuel temperature to sink heat exchanger Inlet P A 0 temperature Inlet oil temperature Total fuel tlow P A 0 flow Lube oil flow Gain of bypass valve Time constant of bypass valve Time constant of sensor # I Time constant of sensor #2
Physical Parameter Physical Parameter P hy sical Parameter Physical Parameter
Mass of metal plate in sink heat exchanger Mass of fuel in engine oil heat exchanger Mass of lube oil in engine oil heat exchanger Mass of metat plate in engine oiI heat exchanger Specific heat capacity of fuel Specific heat capacity of PA0 Specific heat capacity of lube oil Specific heat capacity of metal plate
Ibm Ibm Ibm 1 bm Btu/Ibmf°F Btu/Ibm/"F Btu/lbm/"F Btu/lbm/*F
Time constant of sensor #3 Heat transfer coefficient of fuel in sink heat exchanger Heat transtèr coeficient of PA0 in sink heat exchanger
s Btuis Bhds
Physical Parameter Physical Parameter Physical Parameter
Heat transfer coefficient of fuel in engine oil heat exchanger . Btu/s p.
Heat transfer coefficient oflube oil in engine oil heat exchanger 1 Btuh Mass of tiiel in sink heat exchanger Mass of P A 0 in sink heat exchanser
Ibm 1 bm
The model is nodinear because the state W' appears in products with other states in sorne of
the equations. The first five equations in (2.7) descnbe the dynamics of the engine oil heat
exchanger and the temperature sensors measuring its outlet temperatures while the rest of the
equations model the bypass valve, the sink heat exchanger and one temperature sensor. The
coupling between the sink heat exchanger and the engine oil heat exchanger results fiom the
term Th, the inlet fuel temperature of the engine oil heat exchanger. which is given by equation
(2.8). Equations (2.9) through (2.12) are obtained by cuve fitting using experimental data of
heat transfer coefficients versus tlow rates for both heat exchangers (see Appendix A). As
previousIy mentioned. the heat transfer coefficient depends on the tlow rate and the dimension
of the heat exchanger. Here Equations (2.9) and (2.10) relate Hf and Hp to their corresponding
flows with the dimension of the sink heat exchanger incorporated: Equations (2.1 1) and (2.12)
also incorporate the dimension of the engine oil heat exchanger.
2.4 Possible Faults of the System
Before developing fault diagnosis techniques for the target heat exchanger system. al1 the
faults one wishes to detect need to be specified. The faults under consideration c m be divided
into two categories. the abrupt faults and the graduai faults. The abrupt faults include Valve
Stuck Closed (VSC). Valve Stuck Open (VSO) and sensor faults. VSC occurs when the valve
lets al1 the fuel bypass the sink heat exchanger regardless of the valve command: this c m be
modeled by Ietting kv x ri = O. On the other hand. VSO rneans that al1 the fuel is let through the
sink heat exchanger and this is simulated by letting kv x ti = 1. Sensor failures are caused by
either open circuits or shon circuits. and as a result. the readings will stay unchanged at the
lowest or the highest values of the designed range. These faults cm be modeled by setting the
sensor states. Ti. T2 and T3. at the possible extreme values. The gradua1 faults are the
performance degradation of the heat exchangers. which results from the deposit and corrosion
on the metai plate and the inside walls of the heat exchangers. The etrèct is a loss of eEciency
of heat transfer inside the heat exchangers and this c m be represented by a decrease in the
magnitude of heat transfer coefficients. Therefore. the degradation in the sink heat exchanger is
modeled by decreasing the values of Hf and Hp: the degradation in the engine oil heat exchanger
is modeled by decreasing the values of Hf2 and HL.
Chapter 3
Residual Generation Using Unknown Input Observers
3.1 Introduction
ARer reviewing the various fault diagnostic approaches described in Chapter 1 and their
applicability to the target heat exchanger system. the Unknown Input Observer (abbreviated as
UIO hereafter) is chosen as the basis of the fault diagnostic scheme to be developed for the
target system. Under normal conditions. an U t 0 generates the state estimates that
asymptotically track the real states of a system in the presence of unknown inputs also referred
to as disturbances. The state estimates will deviate fiom the real states when a fault occurs
within the system. The residuals can be formed by taking the differences between the estimated
outputs and the measured outputs. The values of the residuals are non-zero only when there is a
fault in the system; thus, the residuals are robust against disturbances and sensitive to faults.
The robustness of the UIO approach is a crucial property because the inlet temperatures of
the heat exchmgers are unknown and hence are rnodeled as disturbances in the target system.
The requirement of robustness eliminates the choice of parameter estimation approaches since
the estimation is quite sensitive to disturbances. For the purpose of study, the parameter
estimation algorithm described in Isermann (1992) has been applied on the linearized mode1 of
the sink heat exchanger of the target system. The simulations of the system and the estimation
algorithm are run in ~ a t l a b ' . It is found that the parameter estimates do not agree with the real
parameters even though the output response of the linear system with the estimated parameters
matches qualitatively with the output response of the actual nonlinear system to a reasonable
degree. This is a result of the errors in linearization and it shows, indeed, a lack of robustness in
the estimation.
The fuPy logic approaches (Ballé. Fischer. Fussel. Nelles and Iserrnann. 1998) and the
neural network approaches (Juuma and Parkkinen. 1994) are not chosen due to two reasons.
Firstly, the target system can be modeled mathematically with a reasonable degree of accuracy
and it is agreed between University of Toronto and Honeywell that a model-based approach.
instead of a black-box approach. should be pursued. Secondly. sufficient input-output data
needed for off-line training is not readily available.
Although the parity relation approaches are similar to the observer-based approach. they are
designed to handle additive faults only. Thus. they are not suitable for diagnosing the target
system because the degradation of the heat exchanger in the target system is represented by
changes in parameten. On the other hand. the UIO approach c m be used to diagnose additive
faults as well as parametric faults.
3.2 Theory of Unknown Input ~bservers'
The theory of unknown input observen is applicable to a class of linear systems of the
following fom.
where ~ ( t ) E 'Rn is the state vector. i i ( t ) E 9' is the known input vector, d( t ) E 'Rq is the
unknown input vector and y ( [ ) E Sm is the measured output vector. A. B. C. E are known
matrices with appropriate dimensions. The rnatrix E is assurned to have full colurnn rank.
- - -
' A language of technical computing and simulation developed by The Math Works Inc. ' Section 3.2 in Chen and Patton (1999).
An observer i s defined as an unknown input observer for the system described in Equation
(3.1) if its state estimation error approaches zero asymptotically regardless of the presence of the
unknown inputs. The structure of a full-order UIO is given by:
where ~ ( 1 ) E gn i s the state vector of the UIO and f t r ) E !Rn is the estimated state vecior. F. T.
K. H are matrices to be designed to stabilize the UIO and to de-couple the unknown inputs.
Define the state estimation error as e(r) = .r(t) - .? ( I ) and let K = Ki + Kz. Then the hme
derivative of e(t) can be found using Equations (3.1) and (3.2).
è = ... - .3 = h + B u + E d - F z - T B i d - K Y - H y = AX + BU + Ed - FZ - TBu - K,y - K,y - HCrLr - HCBil - HCEd
= ( A - K A - K , C ) x + ( A - HC.4 - K , C ) ( S X ) + ( I - T - HC)Bid+(l -HC)Ed- FZ - K,y + ( A - HCA -K,C)(H' - Hy)
=(A-HCA-K,C)(X-.~)+(A-HCA-K,C)(.T-H~)+[(I-HC)-T]B~~+(~- HC)Ed- Fi
- [K, - ( A - HCA - K,C)H]y
= ( A - HC.1- K ,C)e - [F- (A- H c A - K , C ) ] Z - [ T - ( I - HC)]Bi<-(HC- I)Ed
- [K, - ( A - HCA - K,C)H]y
To achieve disturbance de-coupling. the following conditions must be met.
(HC- I )E=O T = I - H C F = A - H C A - K , C K, = FH K = K, + K,
When Equations (3.3) to (3.7) are satisfied. the dynamic enor equation is given by:
If the matrix F is stable (eigenvalues of F have negative real parts), e(t) will approach zero
asymptotically and the states estimates track the real states regardless of the values of the
unknown input d(i). Therefore. the design of the UIO involves solving Equations (3.3) to (3.7)
and stabilizing the matnx F. The necessary and sufficient conditions for this UIO to exist are
given in Theorem 3.1. Both the theorem and its proof can be found in Section 3.2 of Chen and
Patton (1 999).
Theorem 3.1 For a system described in Equation (3.1). the necessary and sufficient existence
conditions for an UIO described in Equation (3.2) are:
ci j runk(CE) = r.utik(E)
(ii) (C, A ,) is detectable, where
To prove Theorem 3.1, the following two Lernmas are introduced.
Lemma 3.1 Equation (3.3) is solvable if and only if rank(CE) = rank(0 and a special
solution is:
H = E[(C@~CEJ-'(CE)' (3.10)
Proof:
1. Assume Equation (3.3) has a solution H.
Then HCE = E or (CE)%' = E ~ : that is. E' belongs to the range space of (CE')'.
Hence, rank(~') I rank((~E)') or ronk(E) 5 mnk(CE).
However, rank(CE) < min (rank(C). rank(E) } < runk(E).
.: rank(CE) = rank(E)
11. Assume rank(CE) = rank(E).
Then CE is fi111 column rank since E is full colurnn rank.
Hence, a pseudo inverse of CE exists:
(CE)' = [(cE)*cEJ-'(cE)' (3.1 1)
Let H = E(CE)' = E[(cE)~cE~-'(cE)~ and substitute it into Equation (3.3).
Left Side = (HC - I)E = HCE - E = E[(cE)'cEJ~(cE)~cE - E = E - E = O = Right Side
:. H = E(cE)' is a solution to Equation (3.3)
The proof of Lemma 3.1 is complete.
L
Lcmma 3.2 Let C, = 1 CA 1, then the detectability of (C,. A) is equivalent to that of (C, A).
Proof:
1. If si EC is an unobservable mode of (CI 2).
Then there e'cists a vector a E Cn such that
Therefore, si is also an unobservable mode of (C. -4).
II. I f s z d is an unobservablemode of ( C A .
Then there exists a vector p E Cn such that
Therefore, s2 is also an unobservable mode of (Ci, A) .
Since (C, A ) and (Ci, A) have the same unobservable modes. their detectability is equivalent.
With Lemma 3.1 and Lemrna 3.2, Theorem 3.1 can be proved as follows.
Proof of Theorem 3.1 :
1. Necessity: Assume the UIO in Equation (3.2) exists for the system in Equation (3.1).
Then Equation (3.3) is solvable and. according to Lemma 3.1, condition (i) hoids mie.
The general solution of Equation (3.3) is given by:
where (CE)' is given in Equation (3.1 l), Ho E !Rn'" is an arbitrary matrix and I,,, E %"'" is
the identity matrix.
Substitute Equation (3.12) into Equation (3.5) and use Equation (3.9) to simpliQ the
expression for matnx F.
= A, - K'C'
where K. = [X, Ho ] and Cl = [A,] .
Since it is assumed that the UIO exists. F is stable and (C'. A ,) is detectable.
According to Lemrna 3.2. (C. '4 1 ) is also detcctable and condition (ii) holds true.
Therefore, conditions (i) and (ii) are necessary for the existence of the UIO.
11. Suficiency: Assume conditions (i) and (ii) hold true.
According to Lernma 3.1. Equation (3.3) is solvable and a special solution for H is given by
Equation (3.1 0).
Substitute Equation (3.10) into Equation (3.5) to obtain the following expression.
Since (C, A i ) is detectable. F can be stabilized by a proper choice of KI.
Once H and Ki are detemined, the remaining matrices for the UIO can be found using
Equations (3.4) to (3.7).
Therefore, conditions (i) and (ii) are suficient for the existence of the UIO.
The proof of Theorem 3.1 is complete.
Condition (i) in Theorem 3.1 implies that the number of independent rows in matrix C
cannot be less than the number of independent colurnns in matrix E; that is, the number of
independent measurernents must be equal to or greater than the number of disturbances to be de-
coupled for the UIO to exist. Condition (ii) depends on the structure of the system. specifically
the matrices A, C and E. A general design procedure for the UIO in Equation (3.2) is
summarized below.
i , Check the existence conditions ii.i Thecmrn 3.1.
2. Compute H using Equation (3.1 0).
3. Compute A i using Equation (3.9).
4. Find Ki to stabilize F = A 1 - KIC using techniques such as pole placement'.
5. Compute 7'. F and K using Equations (3.4) to (3.7).
An UIO successfully designed using this procedure will produce state estimates that
asymptotically track the real states of the system in the presence of unknown inputs.
3.3 Robust Residual Generation and Its Sensitivity to Faults
For a linear system described in Equation (3.1 ). if an U t 0 in Equation (3 2) exists. residuals
that are robust against disturbances c m be fonned as the differences between the measured
outputs and the estimated outputs.
r(r ) = y ( ( ) - C.?(r) Ce(() (3.13)
Up to now the discussion is only focused on making the residuals robust against the
disturbances. Nonetheless. the sensitivity of the residuals to faults is equally important for the
purpose of diagnosis. Only parametric faults are considered since these are the type of faults to
be diagnosed in the target heat exchanger system. A linear system with unknown inputs and
parametric faults can be represented as follows.
--
3 Section 73 in BeIanger ( 1995).
where ~ ( t ) E Ttn , ~ ( t ) E Sr , d( i ) E !KV , y ( [ ) E %"' and M E '93"" represents the change in
parameten in system matrix A. Therefore, the system in Equation (3.14) c m be considered as
the faulty version of the system in Equation (3.1).
Assume that the UIO in Equation (3.2) exists for the system in Equation (3.1). and the UIO
is used to estimate the states of the system in Equation (3.14). The time denvative of the
estimation error is given by:
é = x - x = ( A + M ) x + Bir + Ed- Fz- TBii- Ky - Hy
= ( A + bA)x + Bir + Ed - F(î - Hy) - TBlr - K , y - K,y - HC(A + M ) s - HCBU - HCEd
= FL>-(K? - F H ) y - ( T - ( I -HC))Bi l - (HC- I)Ed +(1 -HC)Mx
= Fr+(I - HC)MX
Due to the presence of the term ( I - HC)Mx in the dynamic error equation. c ( t ) will not
approach zero and the residuals. r = Ce . will also be different from zero. The trajectory of r(t)
is described by:
Equation (3.1 5) shows that the residuals generated via the UIO are robust against disturbances
and sensitive to parametric faults.
As previously discussed. the existence of the UIO requires the number of measured outputs
to be equal to or greater than the number of disturbances. However, it is found that when the
numbers of measurements and disturbances are equal. assuming that the UIO exists, the
residuals are insensitive to parametric faults. The formal description of this result is presented
in Theorem 3 2.
Theorem 3.2 Assume the parametric faults of the system in Equation (3.1) result in the system
in Equation (3.14). Given that the UIO in Equation (3.2) exists for the system in Equation (3.1)
and the U I 0 is used to estimate the states in Equation (3.14), if the nurnbers of measurements
and disturbances are equal, m = q, then the residuals will be insensitive to parametric faults; that
is, r(t) in Equation (3.15) goes to zero as t + a.
Proof of Theorem 3.2:
Let r( t ) = r, ( t ) + r,(t) , I
where r, ( t ) = C exp(Ft)e(O) and 3 ( t ) = IC exp(f;(t - r))( I - HC)bAx(r)dr. O
Because F is stable, r l ( t ) + O as t + 00 .
Next we show that r2(t) = O for al1 f.
- Fk '' - r 'X and substiiute it into r2([). txpand expi F jr - r j j into 1 + 1 k!
= ~ ~ ( t - r ) ~ This giveî 5 ( t ) = ](c - CHC + CC (1 - HC))Mx(r)dr
O k=l k!
Because rn = q. the matnx CE is a Full-rank square matrix and the expression for H in Equation
(3.1 0) is reduced to H = E(CE)" . Hence.
It is claimed that ak = O for al1 natural number k; therefore. r2(f) = O.
The claim is proved by induction.
1. Fork=l .
a, = CF(I - HC)
= C ( A - Hc4 - KIC)(I - HC) = C ( A - HCA)( I - HC) - CK, (C - CHC) =(C-CHC)A(I - HC) = O
II. Assume a, = O.
III. For k = j+ 1,
a J+l = C F ~ + [ ( I - HC) = C F J ( A - HCA - K,C)(I - HC)
= CFJ [ ( A - K A ) ( I - HC) - K, (C - CHC)]
= C F J ( A - HCA)(I - HC)
= C F J ( I - HC)A(I- HC)
= a , ( l - HC)
= O
Therefore, ak = O for al1 na- num bers k.
Since ri([) -t O as r + a and r&) = O for d l t . r(t) + O as t -t a.
The proof is complete.
It c m be seen that Equation (3.16) play the key role in the proof of Theorem 3.2. When the
nurnbee of measurements and disturbances are equal. Equation (3.16) holds true and the
residuals are insensitive to the parametric faults represented by M. Therefore. for a linear
system descri bed in Equation (3.14) with equal nurnbers of measurements and disturbances.
there are not enough degrees of design Freedom left to make the residuals sensitive to parametric
faults afier making them robust to disturbances. It is only when the number of measurements
exceeds the number of disturbances c m both objectives be achieved. In practice. this means
that the minimum number of senson required For the UIO-based fault diagnosis is always the
number of disturbances that need to be decoupled plus one.
Chapter 4
Robust Fault Diagnosis Of The Target System
Using Unknown Input Observers
The environmentai control system. which contains the target heat exchanger system. is
strongly nonlinear and its operation depends on the openting mode of the airplane. To sirnplifi
the problem, it is assumed that the airplane is in cmising mode. Although the target system is
sitting in a control loop, which regulates the valve command u to meet the temperature
requirements on Ti and T3 as described in Section 2.2. only open-loop operation of the system is
considered. The bypass valve is normdly fùily open, letting al1 the fuel through the sink heat
exchanger, but it rnay be commanded to any position. The valve gain kv is set to 0.1: hence the
valve command u can be varied fiom O (fully closed) to 10 (fuily open). The following
assumptions about the system are made with the help of the expertise fiom Honeywell.
Assumptions
1. The flow rates Wf, Wp and WL are constant inputs.
2. The inlet temperatures Tfi, Tpi and TLi are disturbances with known nominal values.
3. Each disturbance is allowed to vary within +lO°F Iiom the nominal value: that is.
where ATJ. ATpi and ATLi are the deviations of Tfi. Tpi and TLi from their nominal values.
4. The relative deviation between any two disturbances from their nominal values is within
5. Al1 temperature measurements are contaminated by white noise with a variance of 0.09.
Assurnption 4 accounts for the fact that the inlet temperatures tend to change in the same
direction. Assurnption 5 is based on the typical noisy measurement show in Figure 4.1. which
is plotted using the data provided by Honeywell.
Temperature Measurement wrth Noise 1 1 1 a
. . O 500 1000 1 500 2000 2500
ti me (s)
Figure 4.1 : Typical Noisy Measurement
From i = O to 1000 seconds, it cm be calculated that the measurements have a variance of 0.09.
Therefore, the measurements in the target system are assumed to have white noises with 0.09
variance. The spikes between i = 1000 to 1600 seconds are assurned to be caused by
disturbances of the system. For example, a spike in the measurement Ti c m be modeled by
introducing a spike in the disturbance Tpi.
Unit O F
O F
O F
Ibmfs
Name ?F Tpi TLi Wf Wp WL kv N
2s i SI
~3
Mf Mp Mm
Table 4. I : Numerical Values of Parameters
Type Disturbance Disturbance Disturbance Constant In~ut
-
brtrf;, hfL Mnz cf
. CP CL Cm
To validate the model given in Equation (2.7). cornparison is made between the step
Value 100 +, 10 136 _+ I O 266 + 10 0.833
Ibm/s Ibrn/s none s
Constant Input 1 1.67
Physical Parameter Physical Parameter Physical Parameter Physical Parameter Physical Pameter Physical P m e t e r
response of the differential equation model and the model developed by Honeywell. The
Constant Input Physicsil Parameter Physical Parameter
differential equation model is simulated in Matlab using the subroutine ODE45 The values of
2 O. I 6.5 5 5 5 4.15 2.69 8
lbm I bm I bm BIu/lbmI0F
-
. Physical Parameter 1 2.19
the constant inputs and physical parameters used in the simulation and subsequent analyses are
s s s Ibm Ibm Ibm
Physical Parameter Physical Parameter Physical Parameter
listed in Table 4.1. Al1 the initial temperatures are set to 1 15 OF. The responses of Tl and T3 to
1.55 10 0.52
Physical Parameter Physical Pararneter Physical Parameter
a step change in u from 10 to 5 at t = 50 s are plotted in Figure 4.2. The outlet PA0 temperature
Ti Uicreases after t = 50 s because 50% less fuel is used to cool down the PA0 liquid. For the
0.54 0.5 0.2
same reason, the outlet fuel temperature T3 drops after t = 50 S. The Honeywell model, written
Btu/lbdQF Btu/lbm/"F Btu/lbml°F
in ACSL', is nui in the ACSL Simulator under the same conditions and the responses plotted in
4 Advanced Continuous Simulation Language developed by MGA Sofhvare
Figure 4.2. The major difference between the two models is that the heat exchanger in the
Honeywell mode1 is represented by a steady-state eficiency model with a first order lag. From
the plots, smail discrepancies of less than 1 O F in steady state temperatures are observed.
Nonetheless, the responses match qualitatively and the differential equation model is approved
by Honeywell. It is detemined that the targei system has a dominant first order time constant of
13 seconds.
Step Response of T l : Differential Madel
............ C . . . . . . . . . . . . .........................
Step Repsonse of T3 : Differenb'al Model
I 1 ; 2300 1 50 1 O0 150 200
time (sec)
Step Response of T l . Honeywell Model Step Response of T3 Honeyweli Model - - - '-
1 j- j - -- , . -<
1 I
Figure 4.2: Cornparison of Step Responses of the Differential Equation Mode1 and the Honeywell Mode1
4.2 Fault Modeiing and Probkm Definition
As discussed in Section 2.4, the possible faults include abrupt faults and gradual faults,
which can be modeled by changing the corresponding system parameters. It is assumed that
only one abrupt fault can occur at a time while the degradation in both heat exchangers can
occur simultaneously, even in the presence of an abrupt fault. Table 4.2 lists al1 the faults to be
considered and the way the parameters are adjusted to mode1 them.
Fault Type
Abnipt Abrupt Abrupt Abru~t
Table 4.2: Modeling of FauIts in the Target System
Abmpt Valve stuck closed k v = O d
Fault
Gradua1 1 ~ & k heat exchanger degrades 10°/0 1 Hflfaulty) = 0.9 Hf; and Hp(faulty) = 0.9 Hp
The gradual degradation in the performance of the heat exchanger is calibrated by a percentage
\ Fault Modeling
Valve snick open Sensor fC 1 failure Sensor $2 failure Sensor #3 failure
decrease in its abilities to transfer heat. which is modeled by a percentage decrease of the
u = IO T, = 50 or Tl = 300 T; = 50 or Ti = 300 Tt = 50 or T: = 300
Hflfaulty) = 0.8 Hf: and Hdfaulty) = 0.8 Hp HAfaulty)=0.7Hf:and Hp(faulty)=0.7 Hp Hflfaulty) = 0.6 Hf: and HHfaulty) = 0.6 Hp Hf(fau1ty) = 0.9 Hfi, and HL(faulty ) = 0.9 HL Hfi(faulty) = 0.8 Hf,, and HL(fau1ty) = 0.8 HL Hh(faulty) = 0.7 Hfi, and HL(faulty) = 0.7 HL Hh(faulty) = 0.6 Hfi, and HL(fau1ty) = 0.6 HL
Gradua1 Gradua1 Gradua1 Gradua1 Gradua1 Gradual Gradual
normal values of the heat transfer coefficients. The percentage degradation of the sink heat
Sink heat exchanger degrades 20% Sink heatexchangerdegrades30% Sink heat exchanger degrades JO% Oil heat exchanger degrades 10% Oil heat exchanger degrades 20% Oil heat exchanger degndes 30% Oil heat exchanger degrades 40%
exchanger is represented by a percentage decrease in the values of Hf and Hp. as s h o w in Table
4.2. Similady. the percentage degradation of the engine oil heat exchanger is represented by a
percentage decrease in the vaiues of Hf? and HL. The mavimum degradation is assumed to be
40% in both heat exchangen. The calibration of the gradual faults is an important objective in
the fault diagnosis of the target system. The precise problem definition cm now be given.
Problem Definition For the heat exchanger system modeled by Equations (2.7) to (2.12),
under Assumptions 1 to 5. using the knowledge of the controlled input rt and measurements Ti,
Tt and T', it is desired to accomplish the following tasks:
1. To identify the performance degradation in the heat exchangers and estimate the percentage
degradation for each of them.
2. To identiQ the abrupt faults, assurning that only one abrupt fault can occur at a time.
Additional temperature senson may be installed in order to facilitate the tasks. but justifications
need to be provided.
The abrupt faults tend to induce a much more drastic change in the system behaviour than
the gradual faults would do. If both an abrupt fault and a gradual fault occur. very often the
change caused by the gradual fault will be masked by the change caused by the abrupt fault.
Therefore, when both the abrupt Fiuh aiid die grahal fault arc prcscnt, usudly only the ûbmpt
fault can be detected. Since more than one gradual fault can occur at the sarne time. the main
challenge of the first task is to isolate the gradual faults and to estimate the magnitude of each
fault.
4.3 Application of the UIO-based Approach
It is proposed to solve the fault diagnosis problem of the target system by using the UtO-
based approach described in Chapter 3. Since the U t 0 theory is developed for linear systems.
the nonlinear model in Equation (2.7) must be linearized before this method can be applied.
With the linearized system. an UIO cm be designed and used to observe the states of the
nonlinear system. Frorn the state estimates and the measurements. the residuals are generated.
The steady state values of the residuals should be averaged over a period of time to filter out the
noise. The average values of the residuals are evaluated to diagnose the faults. Most
importantly. the sizes of the residuals are used to calibrate the degree of degradation in the heat
exchanger.
The system is linearized around the steady state with a constant input ir = uo. Observe that
with u being a constant, the last state of the system. Wh, would also be constant because the
position of the bypass valve is fixed. Hence. this state is ignored in the linearized model and
substituted with the expression, WA = ÇYf . kv ri,. The states of the sensors, Ti, T2 and T3, have
no effect on the steady state values, so they are also ignored. These simplifications are justified
because only the steady state values of the residuals are utilized in the fault diagnosis.
Let x = [TLO Tfo2 Tmz Tpo T f o ~rnr be the linearized states.
be the disturbances and y = TLo be the outputs. [hl Let the valve command be a constant input ( u = uo).
The iinearized sysiarn is rlrscriberl by :
where
WL HL O
HI. O
.CfL 2ML - CL Mi , CL
HP 2 M m . Cm
Ml the other parameten are given by Equations (2.9) to (2.12) and Table 4.1.
4.3.2 Violation of Existence Conditions
It is found bat, with the system parameters having the values listed in Table 4.1, an UIO
fails to exist at dl the linearization points. Even if the UIO did exist, the residuais generated
would not be sensitive to the degradation in heat exchangers. The reason is that the number of
measurements is equal to the number of disturbances in the target system. and according to
Theorem 3.2, the residuals would be insensitive to parametric faults. However, for the purpose
of stuciy, rhe checking UT the necessary md sufficicm ionditions for the existence of an UIO is
demonstrated with the linearization point zio set to 5. which corresponds to the bypass valve
being half open.
With uo = 5,
Check the conditions in Theorem 3.1.
(i) rank(CE) = rank(E)
:. rank(CE) = ranR(E) = 3 and the first condition is satisfied.
(ii) (C, A ,) is detectable
Observe that rows 1,2 and 4 in d 1 are dl zeros. so the eigenvalues of A 1 contain 3 zeros.
Form the sub-matrix G by delrting rows 1.2.4 and columns 1.2.4 in A 1:
The eigenvalues of G are (7.6453. 4.2007. -5.9367). and they are equivdent to the
remaining eigenvalues o f 2 1.
For (C. A l ) to be detectable. the unstable eigenvalues of dl have to be observable. An
eigenvalue is observable if and only if it c m be moved by the state feedback. i l 1 - KIC. with
a proper choice of the gain matrix Ki. Using Matlab. a random gain matrix KI is generated
and the eigenvdues of (Al - KlC3 are calculated. It is verified that the three zero
eigenvalues can be moved and hence are observable. However. the other three eigenvalues.
given by the eigenvalues of G, remain the same for d l Ki, which means they are
unobservable. Since one of the unobservable eigenvalues is unstable. (C. dl ) is not
detectable and the second condition is not satistied.
The matrix A l is determined by Equation (3.9) with the objective of disturbance decoupling.
For the target system, the pair (C. A i ) ~ i r n s out to be undetectable aithough disturbance
decoupling is achieved. This is the case for al1 values of eo. The reason is that the unstable and
unobservable eigenvalue at 7.6453 is not dependent on the input u. T'herefore, A i will dways
have an unstable and unobservable eigenvalue regardless of the linearization point, and (C, A !)
will always be undetectable.
4.3.3 Physical Interpretation and Proposed Solution
It is worthwhile to further examine the target system to find out where this unstable and
unobservable pole cornes from. This pole corresponds to the entry in the fint row and first
column of the matrix G. which has the following expression:
uljeZl uljejl clj3 - - - - 7.6453
"i; "1
where the a,,'s and e,'s are the entnes in matrices .-l and E. which are given below.
(fj, = )= -1.8350 Mm. Ch
CVL ti,; = - - HL
= -0.24 1 M L Zhh!' . CL
e, , - Hf. /((7.&lrn,Cm) -- =-1.7 Hf,
These parameters are independent of the input ii: hence. this pole cannot be shifled by varying
the linearization point. The terni a: ir negatire. but the Iast two tcrms
the left side of Equation (4.2) are positive, which make the whole expression positive and result
in an unstable pole. Note that the signs of u33, ai^, 4 3 and az are fixed because they are the
products of some positive physical parameters. However, el3 and e3i/ezi can be made positive
by increasing WL and Wf respectively in an attempt to make the pole stable. The dependence of
HL HL on WL is described by Equation (2.12) and the plot of - versus WL is shown in Figure
2CL
HL 4.3. The dotted line represents the equation WL = - and el3 is positive for the part of the
3CL
Hf, curve that lies below the dotted line. Similar plot for - and Wf is shown in Figure 4.4 with 2Cf
the help of Equation (2.1 1).
Figure 4.3: Dependence of HLi2CL on WL
From Table 4.1, the values of Wf and WL are 0.833 lbrn/s and 2 Ibrn/s. With these
HL HI' values it cm be seen from the plots that WL c - and ~f -; hence. both e13 and e3 l/ezi
2CL 2Cf
are negative. Theoretically. it is possible to make the pole negative by increasing WL and WJ
In kt, it is found that with the oil flow WL increased fiom 2 Ibm/s to 3 Ibmls and 110 set to 5.
the eigenvalues of the matrix G become stable at (-22.21, -0.20. -5.94). This makes the pair
(C. cl , ) detectable and guarantees the existence of the UIO. The amount of oil flow is not
dependent on the target system itselC rather, it is determined by the operating condition of the
environmentai control system. Therefore, it c m be stated that the non-detectability of (C, A l ) is
not a fundamental property of the target system because (C, A i ) cm be made detecrable by
increasing the oii flow.
Figure 4.4: Dependence of HhI2CL on IVf
The practical issues involved in increasing the oil flow are not pursued further since the
residuais generated in this case would not be sensitive to parametric faults due to an equal
number of measurements and disturbances. Thus. to accomplish the diagnostic tasks, either
additional sensors need to be instalied or the number of the disturbances have to be reduced. [f
a temperature sensor is added or one of the disturbances is assumed to be a constant input. the
number of measurements will be exactly one greater than the number of disturbances. Only
then c m the UIO-generated residuals be sensitive to parametric faults and robust to disturbances
as well. Both of these solutions are investigated.
4.4 Residual Generation with Additional Sensors
4.4.1 Installation of One Additionai Sensor
The installation of one additional temperature sensor is considered k t . There are five
possible locations for the sensor: Tfo. Tfi? Tpi, TLi. and T h . An UIO is designed in each case
and simulations carried out in Matlab to veri& that the residuals are robust against the
disturbances and sensitive to degradation in heat exchangers. The best location for the sensor
installation is then determined based on a specific criterion described later. The linearization
and residual generation for the five cases are as follows.
Case 1 : Tfo Measured
The linearized system is described by:
where s. ci, A. E are the same as in Equation (4.1) and y =
Here four output measurements are available to decouple three disturbances and four residuals
can be generated. Two UIOs with different linearization points. one with no = 5 and the other
with uo = 9. are designed with the procedure outlined in Section 3.2. The existence conditions
for the UIO are verified at both linearization points. The nonlinear model in Equation (2.7) is
written in a Matlab file named "fau1ty.m". which is simulated with a constant input qua1 to the
linearization point and with the disturbances or faults injected. The disturbances and faults are
introduced into the model by changing the parameters in "fau1ty.m". They are constant rather
than time varying. The designed UIO is used to observe the States of the nonlinear faulry system
and to generate residuals. The noise on the measurements is not yet considered at the present
stage. The steady state residual values are recorded in Table 1.3. The Matlab file "U1O.m"
illustrates the design of an UIO in this case. and also the generation of residuals. The above-
mentioned Matlab files are included in Appendix B.
Case 2: Tfi Measured
The linearized system is described by:
where - y, A, C are the same as in Equation (4.1 ) and v = Tfi, d = [Tpi TLI]',
Since the disturbance Tfi is measured. it becomes a known input. Hence. only iwo disturbances
are left to be decoupled using the existing three rneasurements. Ii is verified that the existence
conditions for the UIO are satistied at u o = 5 and iio = 9. Three residuals are generated and the
simulations are done in the sarne marner as in Case 1. The steady state values of the residuais
are dso recorded in Table 4.3.
Case 3: Tpi Measured
The linearized system becomes:
where r y. A. C are the same as in Equation (4.1) and v = Tpi. d = [Tji 72ilT.
("- Hf: )p) hI/, 7 M f 4 f
In this case the disturbance Tpi is measured so it becomes a known input. Now three output
measurements are used to decouple two disturbances and generate three residuals. The
existence conditions for the UIO are verified for ico = 5 and if0 = 9. Similady, the simulations of
the residual response to disturbances and faults are carried out in Matlab. The steady state
values of the residuals are recorded in Table 4.3.
Case 4: TLi Measured
The linearized system becomes:
where x, y, A. C are the same as in Equation (4.1) and v = TLi. d = [Tfi ~ ~ i ] ~ .
It is found that the system in this case does not satisQ the second existence condition of the UIO
for a certain range of linearization points. Specifically. the pair (C. A !) is undetectable for il0
greater than 5. Therefore, this possible location for the additional sensor is elirninated.
Case 5: 7''' Measured
The linearized system is described by:
where x, d, A, E are the same as in Equation (4.1) and y = [ T p T f o TLo %Ir,
The UIO theory cannot be applied to this system because it has an extra term Rd(t) in the output
equation and does not beiong to die ciass or the sysienis: Jescribed in Equation (3.1). Ilowvcr.
with Tfi2 measured. an UIO can be constructed for the engine oil heat exchanger itself. The
resulting system of the engine oil heat exchanger is linear because the only nonlinear terni Tfiz is
now a known input. Ignoring the sensor dynarnics. the linear mode1 is given by:
where x = [TLo Tfoz ~ r n ~ j ' . v = Tfi?. d = TLi. -v = [TLo vo2] and
rvf Hfi --- Hf, 1. B=l-- Wf Hf, 1Mf2 2 1\& Cf !tv, - Cf kg, ZMf, .Cf
In this case, an UIO is designed for the engine oil heat exchanger. which has a known input Th, two measured outputs and one disturbance. It is expected. however. that the wo residuals
generated would not be sensitive to the degradation in the sink heat exchanger. It is verified that
the existence conditions for the UIO are satisfied. Because the system is linear, one UIO will be
able to handle al1 the input conditions. The residual response to disturbances and faults are
simulated for u = 5 and u = 9. The results are listed in Table 4.3.
Tfi
Tpi
Tfi2
u
5
Disturbance ATfi = 5
None None None None None
1
#DIV/O! 3,094 3.115. 3.115 3.1 15 3.1 15
]
9
5
9
5
9
5
9
Table 4.3: Effect of Degradation in Heat Exchmgers on Residuals
None Sink 10% Sink 40% Oi110% 011 40% Sink 40% Oil 10% None Sink 10% Sink 40% Oi110% 0i140% Sink 40%. Oil 10% None Sink 10% Sink40% Oi1 10°h Oi140% Sink 40%. 011 10% None Sink 10% Sink 40% Oi110% Oi140% Sink 40%, 011 10% None Sink 10% Sink40% Oi110% 0i140°h Sink 40%, Oil 10% None Sink 10% Sink40% Oi110% Oi140% Sink 40%, OiI 10% None Sink 10%
S i n W h Oi110% Oil40% Sink 40%. Oil10%
Degradation None Sink 10% Sink40% Oit 10% OiI 40% Sink 4O%, 011 10%
rdrz #DIV/O!
-1.772 -1.769 -1.769 -1 -769 -1.769
ATpi = 5 None None None None None
A l L i = 5 None None None None None
A Tpi = -5 None None None None None
ATfî = -5 None None None None None
ATLi = -5 None None None None None
aTf i = 10 None None None None None
aTLi = 10 None - -
- None None None None
O O
0.0031 -0.0661 -0.3243 -0.0648
O 0.0012 0.0066
-0.0671 -0.3296 -0.06 16
O 0.005
0.0256 -0.1249 -0.6128 -0. 1027
O -0.0958 -0.5056 4.1963
20.5971 3.7565
O -0.0871 -0.4434 2.8926
14.1983 2.5299
O O O
1.6606 8.1512 1.6866
O O O
1.5945 3.3986
1 1.6388
rt
O 0.0085 0.0445 -0.283
-1.3891 -0.243
r 4
O -0.1 165
-0.613 3.8999
19.1426 3.3482
rz O
0.0263 0.1386
-0.8816 4.3274 -0.7569
O' 0.0032 0.0164 -0.351
-1 .7226 -0.3443
O -0.0302 -0.1595 1.6317 8.0089 1.4979
O -0.0618 -0.31 41 1.5322 7.5204 1 .2606
O 9.0483 -0.2548 2.1136
10.3743 1.8921
O -0.0909 4.463 3.0209
14.8276 2.642
O O O
4.1129 20.1878 4.1772
O O O
3.9489 8.41 72 4.0588
r3
O -0.0466 4.2452 1.5593 7.6535 1.3387
O -0.0143 -0.074 1.5804 7.7571 1.5503
O -0.075
-0.3967 4.0594
19.9252 3.7265
O -0.1548 -0.7869 3.8374
' 18.8355 3.1 574
O 0.0323 0.1703 -1.413
-6.9358 -1.2649
O -0.0606 0.3095
-2.01 96 -9.913
-1 -7663
#DIVIO! 4.469 -4.512 -4.503 -4.503 4.503
#DlVIO! 2.483 2.487 2.488 2.488 2.488
#DIV/O! 2.505 2.505 2.505 2.505 2.505
#OIVIO! 4.669 -0.668 4.669 -0.669 -0.669
#DIVIO! 0.669
4.668 -0.669 4.669 4.669
0' -0.0353 -0.1833 3.9131
19.2076 3.8388
_
#DIVIO! #DIVIO!
5.290 5.310 5.31 2 5.31 3
#DIVIO! -25.167 -24.167 -24.317 -24.299 -24.31 7
#DIV/O! -12.360 -1 2.270 -12.267 -12.272 -1 2.275
#DIVIO! 0.504 0.504 0.504 0.504 0.504
#DIVIO! 1.044 1.044 1.044 1.044 1.044
#DIV/O ! #OIVIO! #D IV10 !
2.477 2.477 2.477
#OIV/O! #DIVIO! #DiV/O!
2.477 2.477 2.477
In Table 4.3, the disturbances are represented by deviations h m their nominal values. It
can be seen that the residuals are indeed robust to disturbances. Although more ùian one
residual is generated in each case. the ratios between any pairs of residuals under the same
linearization point are the same for al1 types of faults. as shown in the 1s t two columns of Table
4.3. Therefore, no additionai information can be deduced frorn the relative sizes of residuals. In
the first three cases where the UIO is constructed for the whole system, the degradations in sink
heat exchanger and engine oil heat exchanger result in residuals with opposite signs. However.
the residuais are much more sensitive to the degradation in engine oil heat exchanger in the
sense that the residuals induced by the degradation in engine oïl heat exchanger have a larger
magnitude than those caused by the same degree of degradation in sink heat exchanger. The
main reason is that the difference of inlet temperatures for the sink heat exchanger is much
smailer than that for the engine oil heat exchanger. and hence. the change in outlet temperatures
for the si& heat exchanger due to performance degradation is not as significant.
In the situation when both heat exchmgers have degraded. the degradation in sink heat
exchanger will be masked by the degradation in engine oil heat exchanger even if the
degradation in sink heat exchanger is much wone. This is demonstrated with the test runs
where 40% degradation in sink heat exchanger and 10% degradation in engine oil heat
exchanger are both injected. Therefore. the degradation in sink heat exchanger cannot be
detected in the presence of the degradation in engine oil heat exchanger. On the other hand. it is
possible to detect the degradation in engine oil heat exchanger even if the sink heat exchanger
has degraded senously. In the last case where the UIO is constructed only for the engine oii
heat exchanger. the residuals are insensitive to the degradation in sink heat exchanger as
expected. Therefore. it is concluded that the degradation in sink heat exchanger cannot be
diagnosed with the installation of only one additional sensor.
The objective now is to determine the optimal sensor location for diagnosing the degradation
in engine oil heat exchangr. Although the effect of the degradation in sink heat exchanger on
the residuals is small, it still introduces some uncertainties when the sizes of the residuals are
used to calibrate the degree of degradation in engine oil heat exchanger. AISO. it is found that
the values of the residuals will be affected by disturbances when a fault is present even though
the residuds are robust against disturbances when the system is fault-free. This is another cause
of variations in the residuais. These uncertainties in the residuais reflecting the degradation in
engine oil heat exchanger are demonstrated by simulations of the system in Equation (4.5). The
results are summarized in Table 3.4.
1 Tpi Measured
iOil 1 O?$ 4.Q801 2.0551 -! 374 O 504 -0.669 Oil 10% 4.0339 2.031 8 _ -1.3583 0.504 -0.669 Oil 10% 4.4743 2.2536 -1.5067 0.504 -0.669 0il10% 4.31 23 2.1 72 -1.4521 0.504 -0.669 - -
1
OiI 10% 4.359 2.1955 -1.4678 , 0.504 -0.669 Oil 10% 3.9178 1.9733 -1.3193 0.504 -0.669 Oil10% 3.9995 2.0144 -1.3468 0.504 -0.669 Oil 10% 4.2198 2.1254 -1.421 0.504 -0.669 Oit 10% 4.1734 2.1 02 -1.4053 0.504 -0.669 Oil 10% 3.9795 2.0025 -1.3388 0.503 -0.669 OiI10% 4.4171 2.2247 -1.4874 0.504 -0.669 Oil 10% 4,3931 2.2128 -1,4794 0.504 -0.669 Oil 1 O%, Sink 40% 3.4785 1.752 -1.1 71 3 0.504 -0.669 Oil20% 8.9451 4.5054 -3.0121 0.504 -0.669 Oil20% 9.537 4.8036 -3.21 15 0.504 -0.669 Oil20%, Sink 40% 7.9866 4.0226 -2.6893 0.504 -0.669 Oil30% 14.3608 7.2332 4.8358 0.504 -0.669 Oil30% 15.3129 7.7127 -5.1 564 0.504 -0.669 Oil 30%. Sink 40% 13.128 6.6123 -4.4207 0.504 -0.669 . - I
Oil40% 20.5971 10.3743 -6.9358 0.504 -0.669 Oil40% 21.9627 1 1 -0621 -7.3956 0.504 -0.669 Oil40%. Sink 40% 19.0485 9.5943 -6.4143 0.504 -0.669
Table -5.4: Uncenainties in Residuals
As c m be seen from Table 4.4. the values of the residuals corresponding to the same level of
degradation in engine oil heat exchanger can vary with disturbances and the degradation in
heat exchanger. The last two columns again show that the residuals always scale the same.
variations in the residuds are the most signifiant when the relative changes (IATfi - ATpil. 1, - ATLiI and (ATpi - ATLil) in the disturbances from their nominal values are the largest.
sink
The
ATfi
The
deviations occur in both directions. The maximum is reached when ATLi is 10. and the
minimum occurs when ATLi is -10 with 40 % degradation in sink heat exchanger. Another
source of uncertainties in the residuals is the input variation around the linearization point.
Although it is assumed that the input can be kept constant for a penod of time long enough for
the steady state to be reached, small variations are not uncornmon. Therefore, an upper bound
and a lower bound should be found for the residual at each degradation level of the engine oil
heat exchanger by taking into account the combined effect of disturbances. the degradation in
sink heat exchanger and input variations. The bounds of the residual measure the maximum
deviations from its normal value and hence serve as a criterion on which the effectiveness of
diagnosing the degradation in engine oil heat exchanger c m be judged.
Because al1 the residuals respond to the faults in the same way for each system. the residual
with the largest magnitude is used for fault diagnosis. For exarnple. the second residual r~ for
the engine oil heat exchanger described in Equation (4.7) has the largest magnitude in response
to faults: hence it is chosen as the reference on which fault diagnosis is based. The word
"residual" is used to refer to the residual with the largest magnitude for the rest of the thesis.
Simulations are carried out to determine the normal values. the upper bounds and the lower
bounds of the residual at different levels of degradation in engine oil heat exchanger for the
systems in Equations (4.3). (4.4). (4.5) and (4.7). The maximum variations in disturbances are
as defined in Assumptions 3 and 4. The maximum degradation of the sink heat exchanger is
assimed to be 40%. For the first three cases. residuals are generated using UIOs ai different
linexkation points with the input variation assumed to be within +l around each linearization
point. For the linear system in Equation (4.7). residuais are generated for die entire range of
input values using one UIO. The results are tabulated in Appendix C.
The difference between the upper and lower bounds of the residual is divided by the normal
value of the residual to give the ratio that compares the uncertainty in the residual to the
magnitude of the residual at each level of degndation. The smaller this ratio is. the more
precise the diagnosis will be. The average of these ratios under the same input condition is
calculated and a weighted surn of these averages is foound in each case. Because the system
normally operates with the valve command 1i being close to 10, larger weight is assigned to the
average ratio obtained at larger input value. This is done by multiplying each average ratio with
a weighting factor of the fom w = hlo -". where ii is the input value and h is set to 0.8. This is
the exponential forgetting factor. which increases with the value of 11 up to 1 . These weighted
sums are a rneasure of merit in the determination of the optimal location for the additional
sensor. Based on this cntenon. it can be concluded that a sensor installed to measure Tfiz would
produce the most precise diagnosis for the degradation in engine oil heat exchanger because the
weighted surn is the smallest in this case. This is expected since the UIO constructed in this
case is dedicated to the engine oil heat exchanger. It is reasonable that the residuals generated
are less sensitive to the degradation in sink heat exchanger and some disturbances. compared to
other cases.
4.4.2 tnstallation of Two Additional Sensors
In order !o diagnose the degradation in qink heat exchanger. residuals should be generated
with an UIO constructed for the sink heat exchanger itselE This way the residuals will not be
affected by the degradation in enginr oil heat exchanger. Since there are two disturbances (W. Tpi) and only one measurement Tl for the si& heat exchanger. two more temperature sensors
are needed to make the number of measurements lârger than that of disturbances. There are four
possible cases: 1. Measuring Tpi and ub. 2. Measuring Tpi and Tfi. 3. Measuring Tfo and T f i . 4.
Measuring Tfi and Tji?. The optimal configuration is found according to the sarne criterion
described in Section 44.1. The four linearized models of the sink heat exchanger are given
below.
Case 1 : Tpi and Tfi> Measured
The linearized system is given by:
X(C) = .-1x(i) + B v ( t ) + E d ( t )
y(r ) = C k ( 0
where .Y = [Tpo Tfo ~ m ] * v = Tpi. d = Tfi. y = [Tpo -1 r. and
O wf; --- H f ' 1 M f 21tlf-Cf
Case 2: Tpi and Tfi Measured
The linearized system is given by:
where .Y, A are the same as in Equation (4.8). and v = [ T f i ~ ~ i ] ~ . y = Tpo,
Case 3: Tfo and Tfi Measured
The linearized system is given by:
where .Y. y. A. C are the sarne as in Equation (4.8). and v = Tfi . d = Tpi.
Case 4: Tfi and T f 2 Measured
The linearîzed systern is given by:
where x, v, d, A, B, E are the same as in Equation (4.10). and y = [Tpo ?j'&lT,
The system in Equation (4.11) does not belong to the class of the systems in Equation (3.1) due
to the extra term fi([). However. the problem could be solved by replacing the output y([) by
y(t) = y ( [ ) - Dv(t) . This way the system becomes equivalent to the system in Equation (4.1 0)
and the UIO theory can be applied.
The UIOs are designed using the systems in Equations (4.8) to (4.10) for the entire range of
linearization points. The simulations of the UIOs and the Faulty nonlinear system are done in
Matlab. This time the normal values and the bounds of the residual at each degradation level of
the sink heat exchanger are found by taking into account the disturbances and an input variation
of f0.5. The results are tabulated in Appendix D. It can be seen that the ratios of the
uncertainties over the normal values of the residuals are much larger than those for the case of
the engine oil heat exchanger. This means the residuals for the sink heat exchanger are
relatively more sensitive to the variations in disturbances and input because the degradation in
sink heat exchanger has a smaller effect on the residuals. The weighted sum is calculated in the
same way and the smallest sum is obtained in Case 3 (Case 4) when Tfo and TTfi (Tfi and Th) are measured. Therefore. the most precise diagnosis of the degradation in sink heat exchanger
cm be achieved when the two sensors are placed to measure Tfo and Tfi. or Tfi and T h . Since
the optimal sensor location for the diagnosis of the engine oil heat exchanger is at Th. it makes
sense to install the second sensor to measure Tfi.
In conclusion. to diagnose the degradation in engine oil heat exchanger. one additional
sensor is needed and the best location is at T f 2 ; to diagnose the degradation in sink heat
exchanger in addition to engine oil heat exchanger. a second sensor is needed and the best
location is at Tfi.
4.5 Residual Generation with Fewer Disturbances
Another solution to making the number of measurements greater than the number of
disturbances is to make the assurnption that there are only two disturbances. With the expertise
from Honeywell, it was decided to mode1 the iniet oil temperature TLi as a constant input rather
than a disturbance because it has the least fluctuation among the three inlet temperatures. Now
the linearized system becomes:
where r y. A, C are the same as in Equation (4.1 ) and v = TLi. d = [Tfi ~~ i ] ' ;
The linearization point uo is set to 5. It can be verified that the existence conditions for the UIO
are met and this UIO is robust against the input variation. In other words. the residuals remain
zero when no fault is present for al1 possible input values although the UIO is designed for uo =
5. However, it is show by simulation that the degradation in sink heat exchanger cannot be
reliably diagnosed for the same reason as discussed in Section 4.4.1. Similarly. Matlab
simulations are carried out to calibrate the degradation in engine oil heat exchanger using the
residual. The results are summarised in Table 1.5. The residual is generated for different
degrees of degradation in engine oil heat exchanger. The lower and upper bounds of the
residual are found by considering the combined effect of the disturbances and the degradation in
sink heat exchanger. Although TLI is assumed to be a constant input, a variation of 52 O F
around the normal value is taken into account when determining the bounds for the residual.
Specifically, the upper bound is reached when ATfi = -10 O F , ATLi = -2 O F and the sink heat
exchanger has degraded 40%; the lower bound is reached when ATfi = 10 O F and ATLi = 2 OF.
The data in Table 4.5 is used in the diagnosis of the engine oil heat exchanger in the next
section.
, O Cil 2C% 10.7301 ? 1 .R7Y? 0.5843 O Oil30% 17.2273 18.6865 15.7682 O 011 40% 24.7089 26.5287 22.8895 2 Oil 5% 2.3259 3.0447 1.6241
2 Oil 30% 16.4113 17.6729 15.2661 2 Oil40% 23.5386 25.0743 22.1692 4 Oil5% 2.2489 2.9593 1 .5764
Lower Bound 1.6953
4 Oit 10% 4.6364 5.4332 3.91 8 4 Oil20% 9.8828 10.8692 9.0632 4 , Oil 30% 15.8665 17.0704 14.9312 4 OiI 40% 22.7571 24.2101 21.6881 6 Oil5% 2.1922 2.9004 1.5416 6 Oil 10% 4.51 97 5.312 3.8.46 1
h.
6 Oil20% 9.6337 10.6108 8.9092 6 Oil 30% 15.4674 16.6555 14.6853
r [Upper Bound 2.44161 3.1884
Input u O
- - - -
6 Oit 40% 22.1842 23.61 5 21.3355 8 Oil5% 2.1486 2.8569 1.5147 8 Oil 10% 4.4296 5.2222 3.7905
Fault Oil5%
- - - - - I
8 Oil40% 21.7423 23.1741 2 1 -0636 10 Oil5% 2.1 137 2.823 1 1.4934 10 Oil 10% 4.3577 5.1525 3.7465
Table 4.5: Calibration of Depdation in Engine Oil Heat Exchanger with TLi Assumed Constant
- - 10 O
4.6 Fault Diagnostic Scheme and Simulation Results
The general fault diagnostic scheme using the UIO-generated residual is as follows. Before
any diagnosis takes place, the residual is averaged over a penod of 60 seconds to filter out the
- . - - -
Oi140% VSO
21.3892 O
I
22.831 8 20.8463
noise in the measurements. This is done by checking the input il for a period of at least 120
seconds during which it is constant and averaging the corresponding residual over the last 60
seconds. The first 60 seconds allow the system to reach steady state after a change in input.
nie abrupt faults are diagnosed first. Al1 the temperature rneasurements are checked to see if
anyone of them is out of the range between 50 O F and 300 OF. If so. the failed sensor is
reported. Then the VSC and VSO faults are checked using the average residuai. If any one of
the abrupt faults is detected. the diagnosis stops. Otherwise, the degradation levels of the sink
heat exchanger and the engine oil heat exchanger are detemined using previously generated
tables relating the degradation levels to the sizes of the residual. Therefore. if the input is kept
constant, one diagnosis will be genented eevry 60 seconds.
4.6.1 Fault Diagnostic Simulation with TLi Assumed Constant
For the case where no additional sensors are installed and TLi is assumed to be constant,
Table 4.5 is the basis of the fault diagnostic scheme. The only gradua1 fault that cm be
diagnosed is the degradation in engine oil heat exchanger. I t can be seen tiom Table 4.5 that the
residual r is insensitive to the abrupt faults. VSO and VSC. Therefore. it is assurned that the
bypass valve is tàult-free. The UIO is designed for the system in Equation (4.12) lineaiized at
uo = 5. The sensors al1 have a sarnpling period of 1 second. A white noise with 0.09 variance is
added to each measmement. When 60 sarnples of the residual are averaged. the variance
decreases to 0.09/60 = 0.00 15. More samples cm be taken if it is desired to decrease the
variance even further as long as the input is constant. In fact. the constant input is defined using
the rounded value of ii because it is not practicaf to keep the input perfectly constant in the real
system.
For every average residual calculated. the corresponding input u is also avenged over that
60 seconds. Each pair of the average residual and the average input is used in determining the
degradation levels in the engine oil heat exchanger. The lower bound of the residual at 5%
degradation level is used as a threshold. below which the degradation in engine oil heat
exchanger will be diagnosed as less than 5%. Because this threshold varies with the input, the
data of the lower bounds at 5% degradation venus the input values is interpolated to find the
threshold at a particular average input. If the average residual is less than the threshold, one can
be sure that the degradation is less than 5%. On the other hand, if the average residual exceeds
this lower bound, a range of possible degradation is given as the diagnosis.
In the worst case the average residual might be very close to the upper bound or the lower
bound corresponding to some level of degradation. To be conservative, the minimum possible
degradation is found by assuming the upper bound is rqual to the average residual and
determining the level of degradation corresponding to this upper bound. The data of the upper
bounds venus the input values is interpolated to find the upper bound corresponding to the
average input value at each defined degradation level. Then the data of the degradation ievels
versus these upper bounds is interpolated to find the degradation level corresponding to the
upper bound. which is assumed to be equal to the average residual. Similady. the maximum
possible degradation cm be found by assuming the lower bound is equal to the average residual
and determining the level of degradation corresponding to this lower bound. For example. given
that the input is 6 and the average residual is 5. the range of possible degradation can bc
determined as follows. In Table 4.5. if Ir = 6 and the upper bound is 5. the corresponding level
of degradation is found to be 9.4% by interpolation. If u = 6 and the lower bound is 5. the
corresponding lrvel of degradation c m be interpolated to be 12.3%. Therefore. the possible
range of degradation lies between 9.4% and 12.3%.
The proposed fault diagnostic scheme is then tested by simulation of the system with faults
and disturbances injected. The residual generation is simulated using a Matlab a file named
D~U1OOas.m" and the fault diagnoser is run with another Matlab file named g*diagOas.m". Both
files cm be Found in Appendix B. The settings of input. disturbances and heat exchanger
degradations for the tault diagnostic simulation are iilustrated in Figure 4.5. The input 14 is a
step that models a change in the valve command From 9 to 7 at r = 400 S. The disturbances T$.
Tpi and TLI are modelled as a smooth pulse. a smooth step and a sinusoid respectively because
these are the typical forms of disturbances according to Honeywell. The degradation in both
heat exchangers is assumed to increase Iinearly with time from 0% to 15%. No abrupt fault is
introduced and the average residual is used to detemine the degradation in engine oil heat
exchanger. Figure 4.6 is a plot of the average residual and the threshold of 5% degradation. An
upward trend of the average residual is observed. which corresponds to the increase in the
percentage degradation of the engine oil heat exchanger. The threshold value depends on the
input; hence a small change in the level of the threshold can be observed at t = 400 S.
Input u
I 1 Inter Fuel Temperature
1311 Heat Exchanger GegradaDon
Figure 4.5: Seninp of Input. Disturbances and Heat Exchanger Degradations for Fault Diagnostic Simulation
Solid Line Average Residual. Doned Line 5% Threshold 7 1 1
I
O 200 400 600 800 1000 1200 Dme (s)
Figure 4.6: Average Residual and Threshold with TLI Assumed Constant
The possible ranges of degradation in engine oil heat exchanger are determined and listed in
Table 4.6. The "Time" colurnn lists the end point of each 60-second period over which the
residual is averaged. Because degradation is a slow process. its variation should be negligible
during the 60 seconds. The actual degradation at the mid-point of each 60-second period is
taken as the accepted value. If this accepted value lies in the range determined. the diagnosis is
said to be correct. The uncertainty is calculated by dividing the difference between the
maximum and minimum degradation by the actual degradation. From Table 4.6, it cm be seen
that al1 the diagnoses are correct within a teasonable degree of precision and the uncertainty
decreases with increasing degradation in cngine oil heat exchanger. More simulations results
with different settings are attached in Appendix E and the consistency in the diagnosis of the
engine heat exchanger is verified.
Table 4.6: Diagnosis of Degradation in Engine Oil Heat Exchanger with TLi Assumed Constant
Time (s)
120 180 240 300 360 521 581 641 701 761 821 881 941
1001 1061 1121 1181
4.6.2 Fault Diagnostic Simulation with YiZ Measured
For the case where one more sensor is installed to measure T h , the residual is generated
using an UIO constnicted for the engine oil heat exchanger. Only the sensor failures and the
degradation in engine oil heat exchanger can be detected. The residual is insensitive to the
Average Residual
0.2691 0.7447 1.1979 1.6949 1.972
2.5683 2.7734 3.0244 3.4016 3.8313 4.3463 4.869
5.3718 5.7936 6.1389 6.4058 6.5613
valve failms and the degradation in sink heat exchanger because they affect the engine oil heat
Oïl Heat Exchanger Degradation (Oh)
1.13 1.88 2.63 3.38 4.13 6.14 6.89 7.64 8.39 9.14 9.89
10.64 11.39 12.14 12.89 13.64 14.39
U ncertainty
0.7230 0.6206 0.4578 0.421 1 0.3876 O. 3565 0.3294 0.2963 0.2754 0.2547 0.2389 0.2250 0.21 26 0.2085
Diagnosis Min %
O O O
2.98 3.47 4.46 4.82 5.31 6.1
6.99 8.07 9.17 10.2
11 11.7 12.2 12.5
Max % 5 5 5
5.42 6.03 7.27 7.72 8.27 9.09
10 11
12.1 13.1 13.9 14.6 15.1 15.5
exchanger through Th, which is now a known input decoupled From the residual. Therefore. it
is assumed that the bypass valve is normal. Table 4.7 provides the data needed io determine the
degradation levels of the engine oil heat exchanger fiom the residual. The upper bound is found
with ATLi = 10 O F and 40% degradation in sink heat exchanger; the Iower bound is found with
ATLi = -1 O OF.
Input u I
O
Table 4.7: Calibration of Degndation in Engine 0i1 Heat Exchanger with TA Measured
Upper Bound 2-3277
Fault Ir Oi15% 1 2.1955
I
Lower Bound 2.0633
22.2185 2.1373
O 1
L
Oi140% Oil 5%
4.4066 9.3932
1 1
23.5569 2.2736
Oil 10% Oil20%
20.88 2.0051
4.6878 9.991 5
4.1341 8.8123
The residual generation is simulated using a Matlab file named "UIO 1as.m". The time delay
of the measurement fiom the added sensor is also rnodelled. The fault diagnoser is written in
"diag1as.m". The same diagnostic scheme as in Section 4.6.1 is applied. The sensor tàilures
are checked first and then the average residual is used to find the possible range of degradation
in engine oil heat exchanger by interpolation using Table 4.7. For the purpose of comparison.
the same settings as in Figure 4.5 are used in the fault diagnostic simulation. The average
residual and the threshold of 5% degradation in engine oil heat exchanger are plotted in Figure
4.7. The diagnostic results are s h o w in Table 4.8. The diagnoses are correct in the sense that
the actual degradation always falls within the range detenined. The uncertainty in this case is
relatively consistent regardless of the levels of degradation. Also. it is iess han that in the
previous section by about 31%. Therefore. the installation of one more sensor results in a more
precise diagnosis of the degradation in engine oil heat exchanger. especially when the
degradation is low. in the presence of one more disturbance. This is confirmed by more
simulation results in Appendix E.
Sol~d Line Average Residual, Dotted Line 5% Threshold 7 - 1 1 1 1 1
tirne (s)
Figure 4.7: Average Residual and Threshold with Tfil Measured
Table 4.8: Diagnosis o f Degradation in Engine Oil Heat Exchanger with T/ii Measured
Time (s)
120 180 240
4.6.3 Fault Diagnostic Simulation with Tji and Tb2 Mtasured
When two sensors are installed to rneasure Tfi and TJz. the fault diagnosis of the target
Average Residual
0.4313 0.7164 0.963
system is done in two parts. The tirst part is to determine the degradation in engine oil heat
exchanger the same way as descnbed in Section 4.6.2. The second part deals with the sub-
Oil Heat Exchanger Degradation (%)
1.1 3 1.88 2.63
system involving the bypass valve and the sink heat exchanger. The fault diagnostic scheme of
the sub-system is based on Table 4.9. which is pre-determined with Matlab simulation. The
residuals are generated by UIOs constructed for the system described in Equation (4.1 1) at
Uncertainty Diagnosis
various linearization points. The second residual is selected as the reference residual r to be
Min % O O O
used in fault diagnosis because it has a larger magnitude. The upper bound is found with the
Max % 5, 5 5
real input being 0.5 higher than the linearization point (11 = 1c0 + 0.5) and the variation in
disturbance Tpi being 10 O F . The lower bound occurs when u = uo - 0.5 and ATpi = -10 O F .
The numben in the colurnn with the headings "(UB-r)I3" are calcdated by dividing the
difference between the upper bound and the normal value of the residual by 3. Similarly. the
nurnbers in the "(LB-r)/3" column are calculated by dividing the difference between the lower
bound and the normal value of the residual by 3. It can be seen that the upper and lower bounds
are not symmetric about the normal value of the residual. In dl but the case with ni* = 10, the
difference between the upper bound and the normal residual is bigger. The numbers in the 1st
column are the bigger vahes between the previous two columns, which are used as the standard
deviation of some Gaussian distribution centred at the normal value of the residuai. This is
M e r explained later in this section.
Table 4.9: Calibration of Degradation in Sink Heat Exchanger with TJ and Tfi? Measured
O O
Sink 5% O
O O
, O - 2
2 2 2 2 4 4 4 4 4 6 6 6 6 6 8
Sink 10% , O O O O
0.4179 0.6074 1.0412 1.5672 2.2182 0.4621 0.7531 1.4105 2.1923 3.1374 0.4928 0.8522 1.6574 2.6028 3.7283 0.5153
Bound O
Sink 20% Sink 30% Sink 40%
Br ik 5% Sink 10% Sink 20°h Sink 30% Sink 40% Sink 5% Sink IOo! Sink 20% Sink 30% Sink 40% Sink 5% Sink tOOh Sink 20°h Sink 30°h Sink 40°h Sink 5%
O O O O
-0.0707 -0.0054 0.1454 0.3309 0.5646 0.01 19 0.129
0.3953 0.7146 1.1047 0.0638 0.21 54 0.5562 0.9586 1.441 1 0.0999
8 8 8 8 10 10 10 10 10
, O O 2 2 4 4 6 6 8 8 10 10
O O O
0.1127 0.2357 0.5182.
0.863 1.2933 0.1858 0.3867 0.8418 1.3852 2.0454 0.2355 0.4888 1.0572 1.7264 2.5258- 0.2712
0.2756 0.668
1 .l263 1.6688 0.1264 0.3198 0.7494 1.2472 1.8309
O O
0.7992 -0.9376 0.6416
-2.4558 0.4321 -3.803 0.21 48
-5.0334 O
-6.1761
Bound O O
O O O
0.1017 0.1239 0.1743 0.2347 0.3083 0.0921 0.1221 0.1896 0.2690 0.3640 0.0858 0.121 1 0.2001 0.2921 0.4008 0.0814
Sink 10% Sink 20% Sink 30% Sink 40% Sink 5% Sink 10% Sink 20% Sink 30% Sink 40% VSO VSC VSO VSC VSO VSC VSO VSC VSO VSC VSO VSC
0.1207 0.2079 0.3088 0.4269 0.0382 0.0789 0.1 693 0.2739 0.3963
O O
O O O
0.061 1 0.0804 0.1243 0.1774 0.2429 0.0580 0.0859 0.1488 0.2235 0.3136 0.0572 0.091 1 0.1670 0.2559 0.3616 0.0571
0.5616 1.2091 1.9639 2.8549 0.2977 0.6155 1.3206 2.1363 3.0908
O O
1.299 -0.91 17
1.042 -1 -8998 0.7022
-2.8234 0.3476
-3.6852 O
-4.4945
O O O
0.1017 0.1239 0.1743 0.2347 0.3083 0.0921 0.1221 0.1896 0.2690 0.3640 0.0858 0.121 1 0.2001 0.2921 0.4008 0.0814
0.0953 0.1804 0.2792 0.3954 0.0571 0.0986 0.1904 0.2964 0.4200
O
0.9236 - 1.8328
2.8903 4.1356 0.4122 0.8523 1.8285 2.9579 4.2796
O O
4.8658 -0.851 2 5.0622
-0.9064 4.8922 -1 -1 55 4.6078
-1 -4795 4.2796
-1.8341
0.1207 0.2079 0.3088 0.4269 0.0571 0.0986 0.1904 0.2964 0.4200
Dev iation O
O O
From Table 4.9, it cm be seen that the abrupt fault VSO is indistinguishable from the
degradation in sink heat exchanger while the fault VSC c m be isolated. Therefore. it is assumed
that o d y one fault. VSC, can occur for the bypass valve. The degradation in sink heat
exchanger cannot be diagnosed the same way as the engine oil heat exchanger because the
variation in the residual is too big. Observe that the upper bound at a given degradation level is
larger than the normal value of the residual at the next higher degndation level. The overlap of
the uncertainties in residuals at ditferent degradation levels will result in diagnoses of wide
ranges of possible degradation if the previous method is employed. Hence. a new method called
multi-hypothesis testing is proposed.
The multi-hypothesis testing chooses one of the pre-defined hypotheses based on a reference
signal and gives the probability of a correct decision as described in Chapter 3 of Barkat ( 199 1 ).
The hypotheses in this case are detined as:
HI: 0 % 4 % degradation in sink heat exchanger Hz: 5% 10% degradation in sink heat exchanger H3: 10%-20% degradation in sink heat exchanger HA: 20%-30% degradation in sink heat exchanger H j: 300/~40% degradation in sink heat exchanger
The average residual F over 60 seconds is used as the reference signal to choose one of the
hypotheses with the following decision d e s :
Decide H i if Y <[hi Decide Hi if (hl < F < th2 Decide Hj if thZ < F < th3 Decide if th3 < Y < rlt4 Decide Hs if F > rh4
where th,, i = I . . . 4. is the ihreshold given by the normal residual value corresponding to 5%.
10%. 20% and 30% degradation in sink heat exchanger respectively. These thresholds Vary
with the linearization point lc.
Besides making the decision. it is also necessary to calculate the probability of a correct
decision. Some probabilistic assurnptions need to be given before this can be done. It is
assurned that the probability of the degradation in sink heat exchanger is equally distnbuted
between 0% and 40%. Therefore, the probability of each hypothesis occurring is:
For a given linearization point and a given degradation level in the sink heat exchanger. the
average residual is assurned to be a Gaussian random variable with the mean being the normal
residual value and the standard deviation being the corresponding number in the last column of
Table 4.9. In other words. the standard deviation for a specific linearization point and a specific
normal residual value (mean) is tound by interpolation. For example. if uo = 8 and r = 7. then
the standard deviation is interpolated to be 0.314 from Table 4.9. The probability density
function of the average residual is given by :
where m = r is the rnean and a i s the standard deviation.
Having specified the probability distribution for the average residual. the mathematical
meaning of the five hypotheses can be stated as follocvs.
If Hi is me. the mean of the Gaussian distribution for F should lie between O and thl.
If Hz is true. the mean of the Gaussian distribution for F should lie between thl and rh2.
If H3 is true. the mean of the Gaussian distribution for F should lie between th. and th3.
If E& is true. the mean of the Gaussian distribution for F should lie between th3 and th4.
If Hs is truc, the mean of the Gaussian distribution for F should lie between th4 and ihj.
The threshold th,, i = 1 . . . 5. is given by the normal residual value corresponding to 5%. 10%.
20%. 30% and 40% degradation in si& heat exchanger respectively. The probability of a
correct decision Pc is the sum of the products of the probability of each hypothesis occurring
and the probability that it is correct. Mathematically.
where P(0 < F < thilHi), P(thi < F < thzJH2). P(îh2 < F < rh3(H3), P(th3 < F < th$&) and P(th4
< F < ihslHj) are the conditional probabilities to be calculated. The probability l'(th, < F <
th,(Hk) represents the probability that the average residual lies in the range which defines Hr,
conditioned on Hk being true.
th,
Average Residual 7
Figure 4.8: Gaussian Distribution and the Thresholds
If the hypothesis Hi; is ûue. the mean of the Gaussian distribution for F should lie between
the thresholds th, and th,, as shown in Figure 4.8. This is because the mean represents the
normal value of the residual. which must lie between the thresholds if the decision is correct.
Shce there is equal chance for the mean to lie anywhere between the thresholds. P(th, < F <
th,(Hk) is calculated by averaging al1 the "shaded area" with the mean rn of the Gaussian cuve
going fiom th, to th,. Let p, be the shaded area below the Gaussian curve. Then.
Note that the probability pm is calculated by:
*
Y - m = 1 -tr-m'-d 2 s :
Q ( ~ ) = IG , which can be approximated by a complernentary error
function as shown in the file "Q.m" in Appendix B.
The residuai generation is simulated in Matlab with "UIO2as.m" and the Fault diagnostic
scheme is laid out in "diag2as.m". The probability of a correct decision is calculated using
"Pc.mV. These files cm be found in Appendix B. The simulation is run under the settings in
Figure 4.5. Unlike previous cases where only one UIO is needed. a bank of ten UlOs are
designed for the sink heat exchanger at different linearization points. ira = 1 ... 10. This is
because the degradation in sink heat exchanger has a relatively srnaIl effect on the residual while
the input variation is a major source of uncertainty. As a result. ten residuals are generated at
different lineaization points and the correct one is selected according to the curent input u. For
example if the input is kept constant at 8. the residual generated for the system linearized at uo =
8 is used for diagnosing the sink heat exchanger.
The sensor failures are checked fint, followed by the VSC fault. Then the degradation
levels in the sink heat exchanger are determined by multi-hypothesis testing using the average
residual. The thresholds at a particular input are found by interpolating the data of normal
residuals versus the inputs in Table 4.9. To caiculate the probability of a correct decision, the
standard deviations at that input are first found by interpolation. Then the standard deviation at
each mean (normal residual) is interpolated using previously obtained data of standard
deviations venus normal residuals. lnstead of computing Equation (4.13) by integration. the
integral is approximated by averaging the values of p, over a nurnber of points evenly
distributed between the thresholds. The diagnostic results are shown in Figure 4.9 and Table
4.1 O.
Figure 4.9: Average Residual and Thresholds w i h Tfi and Tfi? Measured
Table 4.10: Diagnosis of Degradation in Sink Heat Exchanger with Tfi and TJ2 Measured
Figure 4.9 shows an upward trend of the average residual corresponding to the linear
increase of the degradation in sink heat exchanger. The change in thresholds is caused by a step
change in the input at r = 400 s because the thresholds are dependent on the input. It c m be seen
fiom Table 4.10 that not al1 the diagnoses are correct although they are cmder than the
diagnoses for the engine oil heat exchanger. Speci fically , the diagnoses at t = 5 8 1. 88 1. 94 1.
and 1001 are wrong. The calculated probability of a correct decision is around 0.8. The actual
ratio of the correct decision in his run is i3ii 7 0.765. Tiie Ji ynosis cm bc made morc
precise by defining more hypotheses with smaller ranges. However. this will decrease the
probability of a correct decision and render the diagnoser unreliable. Results from more
simulation runs in Appendix E show that the probability of a correct decision increases from
0.75 to 0.83 when the input zr increases from 5 to 10.
[ u, IFault Ir IUpper Bound ILower Bound 1 O O O O
TabIe 4.1 1: Calibration of Degradation in Sink Heat Exchanger with Smaller Variations in Input and Disturbances
Sink 5% Sink 10% Sink 20% Sink 30%
0.2712 0.5616
0.4203 0.7709
8 8
O O O O
0.1493 0.3807
Sink 5% Sink 10%
O O O O
O O O O
The accuracy and the precision of the diagnosis for the sink heat exchanger can be improved
by confinhg the variations in input and disturbances to smaller ranges. This assumption is vaiid
if the environment is less varying. For the purpose of demonstration. the diagnostic simulation
is carried out assurning the input variation is within + 5% and the disturbances are constrained
by IATfi - ATpil 5 5 OF. As before, the UIO is constructed for the sink heat exchanger with Tfi
and Tfiz measured. The normal value and the bounds of the residuai are listed in Table 4.11.
The upper bound is found with the real input being 5% higher than the lineaization point (u =
1 . 0 5 ~ ~ ) and the variation in disturbance Tpi being 5 O F . The lower bound occurs when u =
0 . 9 5 ~ ~ and ATpi = -5 OF. In this case, the upper and lower bounds are closer together and the
overlap of the range of uncertainty is not as severe compared with the previous case. Therefore,
instead of using the multi-hypothesis testing method. the same diagnostic scheme used for the
engine oil heat exchanger is employed. The settings of the simulation conditions are s h o w in
Figure 4.10.
I O 200 400 600 800 1000 1200
inlet PA0 Temperature
I
Sink He& Exchanger Degradanon
nme (5)
011 Heat Exchanger Degradaoon
Figure 4.10: Settings of Simulation Conditions for Diagnosis of Sink Heat Exchanger
Again, a threshold is set at the 5% degradation, beyond which the possible range of degradation
in sink heat exchanger is detennined. The results are shown in Table 4.12. It can be seen that
al1 the diagnoses are correct and the precision has improved from the multi-hypothesis testing
results. This trade-off between the tolerance of variations and the effectiveness of the diagnosis
is expected. It is up to the user to find the balance between thern.
llirne (s) IAverage ISink Heat Exchanger [ Diagnosis 1 1 1 Residual 1 Degradation (%) 1 Min % 1 Max %l
Table 12: Diagnosis of Degndation in Sink Heat Exchanger with Smaller Variations in lnpu t and Disturbances
4.7 Summary
With the existing system configuration. the diagnosis for the degradation in heat exchangen
is not possible. With one of the disturbances. TLi. assumed constant. the degradation in engine
oil heat exchanger cm be determined accurately to a reasonable degree of precision. With an
additional sensor installed to measure Th, the degradation in engine oil heat exchanger can be
detemined with greater precision in the presence of al1 three disturbances. but the degradation
in sink heat exchanger still cannot be diagnosed. With two additional sensors installed to
measure Tfi and Tf2, the degradation in si& heat exchanger and the VSC fault can be
detennined, in addition to the degradation in engine oil heat exchanger. The diagnostic result of
the engine oil heat exchanger is exactly the same as the case with Tfi2 measured only. However,
the diagnosis of the sink heat exchanger is crude and only 80% correct at best. The accuracy
and the precision of the diagnosis for the sink heat exchanger cm be improved by reducing the
ailowable variations in input and disturbances.
Chapter 5
Conclusion
5.1 Discussion of Results
In this thesis. the application of the UlOs-based approach to the fault diagnosis of the target
heat exchanger system is investigated. nie challenge is to generate residuals that are both
robust against disturbances and sensitive to parametric Faults using the UlOs. It is proved that
this is only achievable when the number of independent output measurements is greater than the
number of dimirbances. The degradation in the heat exchanger is modeled as the parametric
fault and the degree of degradation is related to the size of the residual. The main idea is to first
calibrate off-line the levels of degradation in heat exchanger by the size of the residual. and then
use the data to diagnose the heat exchanger. The calibration is done using Matlab simulation in
the thesis while in practice this information can be extracted from real performance data. The
diagnosis of the heat exchanger is given as a possible range of degradation according to the size
of the residual.
The degradation in engine oil heat exchanger can be determined accurately with reasonable
precision if one of the disturbances is assumed to be constant. The addition of one more sensor
would result in a more precise diagnosis of the degradation in engine oii heat exchanger. The
degradation in sink heat exchanger can only be determined when at least two more sensors are
added. However, the diagnosis is not precise because the eRect of the degradation in sink heat
exchanger on the residual is relatively small cornpared with the effect of the disturbances and
input variations. This is due to the smail difference in the inlet temperatures of the sink heat
exchanger. For the same reason, the degradation in sink heat exchanger will not affect the
performance of the target system seriously under the curent operating condition. Therefore. it
is recornmended to install just one sensor to diagnose the engine oil heat exchanger if the
existing senson cannot perform the diagnosis reliably. Besides decreasing the allowable
variations in input and disturbances. another way to increase the effectiveness of the UIO
approach in diagnosing the sink heat exchanger is to change the operating condition of the target
system to achieve a greater difference in its iniet temperatures. Only then will it become
meaningful to install a second sensor to diagnose the sink heat exchanger.
In assessing the research contributions of this work. it is felt that two items can be
highlighted. The first contribution of this thesis is Theorem 3.2. The implication of Theorem
3.2 is that robust fault diagnosis requires the number of independent measurements to be at least
one larger than the nurnber of disturbances. If there are not anough measurernents to decouple
the disturbances. robust fault diagnosis cannot be done no matter how good the mode1 is. This
theorem is the basis of the whole analysis on making the residuals sensitive to parametnc faults
by reducing the number of disturbances or installing additional senson for the target system.
The second contribution is the method of determining the percentage degradation in heat
exchangen based on the size of the residual. Associating the size of the residual with the degree
of gradua1 faults is a new idea in the residual-based fault diagnosis. Although it may be
surprising that the residuals generated respond to al1 the faults in the same way. this can be
accounted for by the fact that the number of rneasurernents is exactly one larger than the nurnber
of disturbances. The measurements are just enough to decouple the disturbances and to detect
faults, but not sufficient to isolate the faults. if more measurements are available. it is possible
to generate directional residuals pointing to different directions corresponding to different faults
so that fault isolation is achieved.
5.2 Future Research
The valve failures in the target system are not effectively diagnosed using the UIO-based
approach. The reason is that the valve mainly affects the outlet temperatures of the sink heat
exchanger, and it is only when both Tfi and Tfiz are measured can its effect be reflected in the
residual. but then the problem becomes how to isolate this effect from the degradation in sink
heat exchanger. More research needs to be donr on that issue.
Since the steady state value of the residual is used in the fault diagnosis of the target system,
the fauIts that only affect the transient response of the system cannot be detected. Such faults
include the increase in time constants OF a sensor due to deposits and the increase in time
constant of the bypass valve. These faults cause delays in the system response. Although they
do not affect the steady state of the system. they may eventually develop into abrupt fadts such
as sensor failures and valve failures. The early detection of these types of faults c m prevent the
abrupt faults from occming. Hence it is worthwhile to M e r investigate this subject.
The detection and isolation of component degradation is only the first task of PHM. The
second task is to predict the remaining life of the components. This cran be done by developing
a Wear rnodel for each component based on performance data. R. J. Hansen. D. L. Hall and S.
K. Kurtz ( 1995) descnbed the use of Wear rnodels in machinery prognostics. The Wear model
for the heat exchanger should be able to estimate the degree of degradation given the
information about initial degradation. the length of operation and the openting conditions. If
the mouimum tolerable degradation is defined. the remaining life of the heat exchanger cm be
predicted under specific operating conditions. More research needs to be done to construct Wear
models for the components in the target system.
Appendix A
Curve Fitting Using Experimental Data of
Heat Transfer Coefficients Versus Flow Rates
The mathematical relationship between the heat transfer coefficient and the tlow rate is
obtained by fining the experirnental data in Table A. 1 to the equation H = n FP. where H is the
heat transfer coefficient. FV is the tlow rate. cr and b are the parameters to be determined.
Take the natural logarithm of both sides to get h(H) = In(a) - b ln W).
Form the error term. tc = In(f7) - cr - b ln(Wj. where a = ln(a).
Calculate the sum of squares of the error terms associated with al1 data points. E = Te'.
Find u and b that minimize E by solving the tollowing system of rquations.
The resulting four relations are plotted in Figures A.1 through A.4 dong with the experimental
data points. As can be seen from the graphs. the rxperimental data is represented by the
equations to a reasonable degree of accuracy.
Table A, 1 : Heat Transkr CoeffTcients Versus Flow Rates
Hf Vs Wf, for Sink Heat Exchanger
. - .
Figure A. I : Hf = 3.83 wf,04"
Hp Vs Wp for Sink Heat Exchanger
Figure A2: Hp = 1.13 w ~ ~ . ~
Hf2 Vs M for Engine Oil Heat Exchanger
Figure A.3: Hf: = I .JI wPW
HL Vs WL for Engine Oil Heat Exchanger
Figure A.4: HL = 1.77 WL'.'"
Appendix B
Matlab Files
function [ra,th,ti = d i a q 0 z s ;
3 Diagnostic scneme f c r r e s i d u â l q s n e r s t e d : ~ ç L n g U1OOàs.n ? R e t u r x the ave rage r o s i d u a l r?, 5 - D q r z d z t F o n T h r e s h c l d ch and trme t
4 I n p u t v a l u e s uI = [ O 2 4 6 8 101;
i Levels of degradation d = [ O 5 10 20 30 401;
à Residual Generarion using VIO [ r , y i , y2, y31 = UIOOas (Tl ;
3 Checking f o r "constant" input f o r i = Z:length(T),
if round(u(i) ) == round(u ii-1) 1 count = count+1;
e i s e coun t = 1;
end
% C a l c u l a t i n g a v e r a g e r e s i d u a l and a v e r a g e i n p u t if c o u n t == 120
j = jtl; ra(j) = m e a n ( r ( i - 5 9 : i ) ) ; u a ( j ) = m e a n ( u ( i - 5 9 : i ) 1 ; t ( j ) = i; c o u n t = coun t -60 ;
e n d end
f o r i = l : j ,
t (1); t i m e = t(i) t h ( i ) = i n t e r p l ( u I r L B 5 , u a i i ) ~ ;
% Checking f o r senscr f a i l u r e s i f y l ( t ( i ) ) < - 4 9 t y l ( t ( i )
f a u l t = ' S e n s o r #IV e l s e i f y2(t(i) ) < - 4 9 1 y 2 i t ! i j j>299
f a u l t = ' S e n s o r # 2 ' e l s e i f y 3 i t (i) ) < - 4 9 ! y 3 ( t (F) ) >209
f a u l t = ' S e n s o r t 3 '
3 Deter rn in ing r ange of d e g r a d a t i o n i n o i l neat e x c h a n g e r e l s e
UBI (1 ) = 0 ; UBI(2) = i n t e r p l ( u I , U B 5 , u & ( i l ) ; UBI ( 3 ) = i n t e r p l (uI, UBlO,ua!i j 1 ; U B I ( 4 ) = i n t e r p l i u I f 8 B 2 0 , u z ( i ) ) ; UBI ( 5 ) = i n t e r p l (uI,UB30,~a(i) ) ; UBI(6 ) = i n t e r p f ( u I , U B 4 0 , u a ( i ) ) ;
L B I ( 1 ) = O; L B I ( 2 ) = i n t e r p 1 ( u I , L 9 5 , u a ( F ~ I ; LBI ( 3 ) = i n t e r p l ( u 1 , LBi0, ua (i)); LBI ( 4 ) = i n t e r p l ( u I r L B 2 0 , u a ( i ) 1 ; LEI ( 5 ) = i n t e r p l ( u I I l B 3 0 , u a i i ) ) ; LBI ( 6 ) = i n t e r p l (uI,LB40, u a ( F ) ) ; min = i n t e r p l (UBI, d , r a (i! ) max = i n t e r p l (LEI, a, r a (ij j
end e n d
p l o t (t, r a , t, t h , ' : ' 1 ; x l a b e l ( ' t i m e ( s ) ' ) ; y l a b e l ( ' d egF ' ) ; t i ; l e ( ' S o l i d Line: Average R e s i d u a l , D o t t e d L ine : 5% T h r e s h o l d t ) ;
Diag 1 as.m
function [ra, th, t 1 = diaglas;
% Diagnostic scheme for residual generated using U1Olas.m % Returns the average reçidual ra, 5% Degradation Threshold th and tirne t T = [0:1:1200]'; u = 8+tanh (400-T) ;
B Upper and lower bounds at different input va lues ana degradation Levels UB5=[2.3277 2.2736 2.2038 2.1583 2.1257 2.1009 2.09051; .*cl + u 0 ~ 2 - i : , 7332 4 .5U72 G . 5 : 3 O i*!?? !. 2-25 4 . 1 3 1 4 ?.7! ! : UB20=[10.2294 9.9915 9.6345 9 . 4 8 4 â 9.3413 9.2321 9.i864j; UB30=[16.4238 16.0422 1 5 . 5 4 4 3 15 .2283 14.998 14 .823 1 4 . 7 5 1 ; UB40=[23.5569 23.0092 22.3023 21.8419 21.5116 21.2606 21.15621;
3 Input values u I = [O 1 3 5 7 9 101;
4 Levels of degradation d = [ O 5 1 0 2' 30 4 O j ;
3 Residual Generation using YI0 [ r , y l , y2,Tfi2] = UIOlas (T! ;
3 Checking for "constant" input for i = 2:length(Tl,
if round(u(i1 ) == roundluii-ll l count = coun:+l;
else count = 1;
end
3 Calculating average r e s i d u a i and average i n p u t if coun t == 120
j = j + l ;
r a ( j ) = mean(r(i-59:i) ! ; ua(j) = mean(u(i-59:i) 1 ; t ( j ) = i; count = count-60;
end end
for i = I : j ,
t (il ; time = t(i)
th(i) = interpl (ur,~a5,u(t (i)) ) ) ;
i f y1 ( t ( i ) ) < - 4 9 l y l ( t ( i l 1 >299 f a u l t = 'Sensor # 2 '
elseif y2(t(i))<-491y2{t(i) ) > 2 9 9 f a u l t = 'Sensor $ 3 '
elseif Tfi2(t(i))<-491Tfi2{t(i) ) > 2 9 9 fault = 'Sensûr # 4 '
% Checking if the thresnold is exceeded e lse i f ra(i) < th(i)
fault = 'Deqradation in O i l 3X < 5 . '
,% Determining range of degr~dsticn in oil h e a t exchanger else
UBI(L) = 0; U B I (2) = interpl (UT, U 0 5 , Z P (il 1 ; U B I ( 3 ) = i n i e r p i ( u I , L J B l O , ü à [i) ! ; UBI(4) = interpliuI,UB2O,us(i!); UBI (5) = interpl (CI, VB30, ua (F)) ; UaI (6) = interpl (u1, üE40, u à ( i l ! ;
plot(ttraI t, tht ' : ' ) ; xlabel('tirne is) ' ! ; ylabel('degF'); title('So1id Line: Average Residuzl, 3o t reÙ Line: 5 i Thresnold'j;
function [ra,th,sdJ = diaq2âs;
S Diagnostic scherne for res iàua i generated using U102as.m 1 Returns the average residuai ra , degradation tnreshola th, Y standard deviation to the l e f t sdL, ûnd stanaard deviation to the riqht sdU
# Degradation thresholds at different input value r5=[O 0.1127 0.1858 0.2355 0.2712 0.29771; r10=[O 0 . 2 3 5 7 0 . 3 8 6 7 0.4888 0 . 5 6 1 6 0 . 6 i 5 5 1 ; r20=[0 0.5182 0.8418 1.0572 1.2091 1.32061; r30=[0 0.863 1.3852 1.7264 1.9639 2.13631; r40=[0 1.2933 2 . 0 4 5 4 2 . 5 2 5 8 2 . 8 5 4 9 3.09091;
% Standard deviation sd5=[0 0.1017 0 .0921 0 .0858 0 .0814 0 . 0 5 7 1 j ; sdlO=[O 0 .1239 0.1221 0.1211 0 ,1207 0.99061; sd20=[0 0 .1743 0.1896 0.2001 0.2079 C.19C4j; s d 3 0 = [ 0 0.2347 0,2690 0 .2921 0.3088 0.25641; sd40=[0 0.3083 0.3640 0.4008 0 .4269 0.42001;
2 Checking for "constant" input for i = 2:length(T),
if round(uii)) == round(u[F-1): count = count+i;
else count = 1;
end
i f count == i20 . . 4 = - * ( .
J J - '
. - . . i Calcüiating average resiansl 3: SpeCiriZ LnFuc va lue r a ( j ) = mean(r(i-5S:i,round~uii~ 1 1 ; ; ü a ( j ) = mean(u(i-59:i) !; t ( j ) = i; count = counc-60;
cnd end
f o r F = 1: j ,
5 Check ing f o r s e n s o r faiiures if yl{t(i) )<-49lyl(t(i? 1 >239
fault = ' S e n s o r *IV elseif Tfi(t(i) )<-491Tfi(t (i) )>299
fault = 'Çensor # S i elseif Tfi2 (t ( i ) ) < - 4 9 1 Tf F2 ( ï (i) ) Q 9 9
fault = 'Sensor # 4 '
i Interpolating thresnolàs and s ïanc iard aeviations else
th(1, i) = 0; tn(2,i) = interpl(uI,r5,uaii!) ; th(3, i) = interpl Iul, rl0, us (i) ) ; th(4, i) = i n t e r p l (u1 , r 2 0 , ua (i) ) ;
th(5,i) = interpl(uIfr30,ua(i)); th(6, i) = interpl (u1, r40 , ua (i) ) ;
sd(1) = 0; sd(2j = interpl(uItsd5,ua(i)); sd(3) = interpl(uI,sdl0,uâ(i~l; s d ( 4 ) = interpl(uI,sd20,ua(i)); ç d ( 5 ) = interpl(uItsd30,ua(iH ; sd(6) = interpl (uI, sd40, u a (il ) ;
3 Multihypothesis ïesïinq if ra(i) < th(2,F)
fault = 'Degradation in Sink :X < S i ' . . - eLseLL L a ; i ; >t>,;z,i; & : z ; : ; ::k,:2, il
fault = 'Degradâtion Ln SFnk HX Between 5 % - l o i ' e l se i f ra(i)>th(3, il & ra(i) < t i i ( 4 , il
f a u l ï = 'Degradation i . Çink HX b e ~ w e e n 1 Ù % - 2 0 5 ' elseif ra(L)>th(4,i) & r~[Fl<ih(5,ij
f a u l t = 'Degradation in Sink HX betwetn 8e tween 2 0 3 - 3 0 i t olse r a ( i ) >ih(5, i)
fault = 'Degradâticn in SFnk SX > 303' end
end end
function xp = f a u l i y (t, x: ;
1 Nonlinear mode1 of the f a u l t y sys tem
% Disturbances T f i = 100+ (tanh( (t-400) /10C) ftanh ( (900-tj 1 1 0 0 ) ) *3; T p i = 126+(tanh( (500-t) /100) -1) -2; TLi = 266+cos (t/i00) '2;
kv = 0.1; tv = o. 5; tsl = 5; ts2 = 5; t s 3 = 5;
rnp = 2.69; rnf = 4 . 1 5 ; mm = 8; m E 2 = 2.19; mL, = 1.55; mm2 = 10;
% F a u l t y h e a t t r a n s f e r c o e f f i c i e n t s H f = 3 . 8 3 * x ( 1 0 ) " 0 . 4 7 4 ~ ; - t / 8 0 0 0 + 1 ) ; - - -+-,T- * n a c- j - I - - / ~ n n n & - . iip ; L. I J rry u . - i u a , L I " V V k - , , Hf2 = 1 .41'WfAû. J64'(-t/SCOO*Ii ; HL = i.77*WLA0.424* ( - c / 8 0 0 0 t l ! ;
3 input u = 8 + t a n h ( 4 0 0 - t ) ;
S teady .m
f u n c t i o n [xO] = s t e a à y ( u 0 )
% Calcuiate t h e steady s t à t e s of Che syscem 5 [xO] = steady(u0) r e t u r n s t h e steaay s t a t e xO given i n p u t v a l u e u0
global u i ;
T = [ 0 : 0 . 1 : 2 0 0 ] ; ui = uOtones ( s i t e (Tl 1 ; xO = [238.81 215 .53 2 1 5 . 5 3 2 3 8 . 0 1 210.51 1 1 5 . 2 9 115.29 122.30 113.90 C.8331';
% S i m u l a t i o n of t h e nonlinear normai system [ t , x ] = o d e 4 5 i t s y s t e m ' , T , x 0 ) ;
f o r i = 1:10 x0 ( i l = s u m ( x ( 1 9 0 2 : 2 0 0 1 , i) ) /100;
end
function P = Pc(th,sd);
3 Calculate the probability of a correcc decision u s i n g 3 d a t a of thresholds ana stândard d e v i a t i o n s
; + L # - I \ . n n i ,&L 1 7 ) , L i& l - I . W . U r . L.. , d , ,
G = Q ( ( t h ( 2 ) -r) . / i n ç e r p I i c n , s d , r f -Q( ( t h i 3 ) -r) . / i n t e r p l ( t h , s d , r j ) ; P2 = mean ( G ) ;
r = t h ( 3 ) : 0 . 0 1 : t h ! 4 ) ; G = Q( i ~ n ( 3 ) -r) . / i n t î r p l (Ch,sd, r! - Q ( i i h I . 1 ; - . / i n t - r p l (th,sd, r ) ) ; P 3 = mean (G) ;
function y = Q ( x )
f u n c t i o n xp = s y s t e m ( t , x ) ;
Y Nonlinear mode1 of the normal s y s t e m
global ui
W f = 0.333; Wp = 1 . 6 7 ; WL = 2; T f i = 100; T p i = 126; T L i = 266; T f i 2 = Tfi+x(lO) / W f t ( ~ ( 8 ) -Tfii ;
kv = 0.1; tv = 6.5; ts l = 5; ts2 = 5; ts3 = 5;
?i Additional sensor to measure Tfo % UIO constructed for che t a r g e t sysïern
Y Linearization point uo = 5 ;
% Calculate the steady state ï c be usea as i n i t i a l condition [xO] = steady(u0) ;
% Constant flows Wf = 0.833; W f l = Wf*u0/10; Fip = 1.67; WL = 2;
% Nominal disturbance T f i = 100; Tpi = 126; T L i = 256;
% Specific Heat Capacities c p = 0 . 5 4 ; C f = 0 .52 ; Cm = 0.2 ; CL = 0 . 5 ;
1 # a - - - - - J b ' t C 1 3 3 L . J
rnp = 2.69; rnf = 4 . 1 5 ; mrt = 8; m f 2 = 2 . 1 9 ; mi, = 1.55; mm2 = 10;
$ Heat Trans fe r Coef f lcis?.rs Hf = 3.93*Wf1"0.474; Hp = 1.13 'WpA0.463; Hf2 = 1.41+WfA0.484; HL = L.77*WLA0.424;
i Sysrem Katrices n - .? - ;-WL/:L-~!L/~/EL/CL G ZL. /mL. /CL ! .: C r
O -Wf/mf2-Ef2/2/mf2/cf sf'imft/Cf a ; W f / m f Z - k i f 2 ~ 2 / n E S / C f ) *Wfl/Wf O , H L / Z / r r u d / C m Hf2/2/mmZ/Cm - i % L - ' . i f i ) immî/Cm 0 iiif2/2/;nm2/Cm) * W f l / ' n l f O,
t Weighting Matrices Q = d i a g ( [ l I L I I 111; R = diag([l i I i l ) ;
% Disturbances dl = Tfifones(length(T) ,1) ; d2 = Tpitones (length (T) ,1) ;
3 Simulation of nonlinear sysEem witn faults [t,x] = ode45 ( ' 5aulty1 ,T,xO) ;
3 Design of UIO CE = C l = ; H = Et i n v (CE' *CE) -CE ' ; Tk = Q - H'C; A l = Tk'A;
Çtablizinq (Al-KitC) using LQR c ies ion sub- rou t ine fKl,S,Ci] = lqr(AI',C',Q,3); K1 = KI'; F = Al - Kl'C; K2 = Fc3; K = KI 7 K2;
3 Residual Generation rL = yl - Xe(:, 4 ) ; r 2 = y2 - xe(:,5j; r3 = y3 - xe(:,l); r4 = y4 - xe(:,S);
subplot (2, 2, 1) p l o t ( T I u) g r i d title ('ut) ;
subplot ( 2 , 2 , 2 ) plot (Tt r l ) g r i d title ( ' r l t 1 ;
subplot (2,2,3) plot (T, r2) grid title('r2'); xlabel ( ' t (sec) ' ) ;
s u b p l o t (2,2,4) plot (T, r 3 ) g r i d t i t l e ( ' r3' 1 ; x l abe l ( ' t (sec) ' ) ;
S Calculate steady s t a t e values of residuals r n e a n ( r l ( 1 0 0 : 1 5 0 ) 1 mean ( r 2 ( 1 0 0 : 1 5 0 ) ) r n e a n ( r 3 ( 1 0 0 : 1 5 0 ) mean(r4 ( 1 0 0 : 1 5 0 ) )
Eunct ion [r, y l , y2 , y31 = U ï O Q a s (TI ;
3 3 sensors(TL,T2,T3), 2 aisturYanzes:Tfi,Tpi) 3 UT0 c o n s c r u c t e d for c h 2 ïarqet se{scen
% L i n e a r i z a c i o n point uû = 5;
5 Calcu lace tne steady siste [xO = steady (u0 } ;
Wf = 0 .933 ; W f l = Wf'u0/10; Wp = 1 .67 ; WL = 2;
% Nominal QisrurDance T f i = 100; Tpi = 126; TLi = 2 6 6 ;
3 Specific Heat C â p a c i t i e s Cp = 0.54; C f = 0 . 5 2 ; Cm = 0 .2 ; C L = 0.5;
C Masses rnp = 2.69; mf = 4.15; mm = 8; rnf2 = 2 . 1 9 ; mL, = 1 . 5 5 ; mm2 = 10;
S Heat T r a n s f e r Coefficients H f = 3 . 8 3 * W f l A 0 , 4 7 4 ; Hp = 1.13*WpA0.463; Hf2 = 1 .41 'WfA0.464; HL = 1.77*WLA0.424;
% System Matrices A = [-WL/mL-HL/2/mL/CL O HL/mL/CL O O 0,
O -Wf/mf2-Hf2/2/mfl/Cf Hf2/mf2/Cf O (Wf/mf2-Ef2/2/mf2/Cf)fWfl/Wf 0, HL/S/mm2/Cm Hf2/2/mrn2/Cm -(HL+Hf2)/mm2/Cm O (Hf2/2/m2/Cm)*Wfl/Wf 0, O O O -Wp/mp-Hp/2/mp/Cp O Hp/mp/Cp, O O O O - W f l / r n f - H f / 2 / r n f / C f H f / r n f / C f , O 0 O Hp/2/mm/Cm Hf/2/mm/Cm -(Hp+Hf)/mm/Cm];
B = [WL/mL-HL/2/mL/CL O HL/2/mm2/Cm G G O ] ' ;
D = ( O O O]';
? Keighting matrices Q = à i a g ( [ l I L I l L I ! ; R = diog([l I I l ) ;
.: Disturbances dl = TfiWones (length!T! , I) ; d2 = Tpi'ones (lengtn (T) ,1) ;
3 Simulation of nonlicesr sysrern wiik f a u l r s [ t , x ] = ode45 ( ' fau1ty1,T,x~);
j Whits noises with variânce 0.a9 vl = 0.3*randn(lenqth(T),I); v2 = 013*randn(length(TI,I); v3 = 0.3*randn(lengthiT?,I);
3 Adding noise r o measürements
y; : ;; :, 6) + vl; :,2) + v2;
y3 = x ( : , l ) + v3;
i Design of UIO CE = C'F; 3 = Et inv (CE' 'CE) 'CE' ; Tk = Q - HtC; Al = TktA;
3 Stablizing (Ai-Kl*C) usinq LQR d e s i g n s u b - r o u t i n e [Kl,S,Ei] = Lqr(AI1,C',Q,R); KI = KI' ; F = A l - KI'C; K2 = FfH; K = KI + K2;
% TLi is assumed to be â known input v = TLi'ones (length (T) , L) ; DD = zeros(6,l);
% Simulation of UIO [xe,zj=Isim(F, [Tk*B K] ,a, [DD HI, iv y1 y2 y31 ,T! ;
% Residual generztion rl = yl - xe(:,4); r2 = y2 - xei:,l); r3 = y3 - x e ( : , 2 ) ;
3 Residual to be usea f o r f a u i : d iaqr-osis r = - r 2 ;
f unc r ion
One sensor inscalled to messure T f i 2 I U I O conscructea f o r oil h e a t exchange r us ing Tfi2,T2,T3
i Calculais the steady s c a c e ï o b~ u s e i f o r initail concition uo = 5 ; [ x O ] = srear ly !uO! ;
i ModelLing t h e d e l a y of tne scided senso r [ T f i 2 , q ] = lsim (1, [ 5 11 , T f i Z , T I ;
CE = C'Z; Fi = Zcinv (CE' *CC) *CEt ; Tk = Q - H'C; Al = Tk%;
f u n c t i o n f r, yl, Tf i 2 , T f i 1 = ü I 0 2 a s (T, u0 1 ;
% Two sensors instâlled to measure T f i 2 a n d Tfi % LJIO constructed for sink heat exchanger u s i n g T f i , T f i S , T l
Wf = 0.833; W f l = Wf*u0/10; Wp = i.67; WL = 2 ; T f i = 100+(tanh( (T-400) /10O) +tanhi (900-Tl / I O O ) ) *3 ; T p i = 126; T L i = 266;
kv = 0.1; t v = 6.5; + S I = fi; ts2 = 5; ts3 = 5;
Cp = 0 . 5 4 ; C f = 0 . 5 2 ; Cm = 0.2; CL, = 0 . 5 ; mp = 2 . 6 9 ; rnf = 4 . 1 5 ; mm = 8 ; mf2 = 2 , 1 ? ; mL = L.55; mm2 = IO;
c = [I. O O , O I O ] ;
8 Modeling the sensor d e i a y s T f i 2 = Tfi+x(:,lO)/Wf.~(x(:,8)-Tfi); [ T f i 2 , q ] = Isirn(1, [ 5 11 , T f i 2 , T ) ; [ T f i , q ] = l s i m ( 1 , [ 5 11 , T f i , T ) ;
3 Adding noise to the measuremenïs y1 = x ( : , 6 ) i vl; T f i 2 = T f i 2 + v2; T f i = T f i + v3;
CE = C'F; H = E+Fnv (CE' 'CE) 'CE' ; Tk = Q - H'C; A l = Tk'A;
Appendix C
Residual Response to Degradation in Engine Oil Heat Exchanger
UIO for M o l e System - Tfo measured
I I I I I I I 1 1 1 1 Weighted Sum 1 0.5369
UIO for Whole System - Tfi measured
UIO for Whole System - Tpi measured, u , = 5
UIO for Engine Oil Heat Exchanger - Tfi, measured, u, = 5
Weighted Sum 0.307
Appendix D
Residual Response to Degradation in Sink Heat Exchanger
UIO for Sink Heat Exchanger - Tpi, Tfo measured
I I I I 1
I 1 1 1 1 Welahted Sum 1 7.94891
UIO for Sink Heat Exchanger - Tfo, Tfi measured I T f i , Tfiz measured
UIO for Sink Heat Exchanger - Tpi, Tfi measured
Weighted Sum
(UB-L6)lr . 7 1.4578
32.0765 18.8182 12.2141 1 1 -0000 5.2259 3.3058 2.3512 4.8527 2.5385 1.7687 1.3871 2.9878 1.7270 1.3082 1.1 008 2.1 746 1.3742 1 .IO87 0.9774
Weighted Sum
Lower Bound -0.1604 -0.1494 -0.1 358 -0.1 186 -0.1009 -0.0631 -0.01 76
2.3184
Average
33.641 7
5.4707
2.6367
1.7810
1.4087
8.5654
u o 1
b
1 1 1 3 3 3
Residual r 0.0083 0.01 83 0.0308 0.0467 0.0431 0.0943
O. 1 56
Degradation Sink10% Sink 20% Sink 30% Sink 40% SinkfO% Sink 20% Sink 30%
Upper Bound 0.4327 0.4376 0.4438 0.4518 0.3732 0.4297 0.4981
3 5 5 5 5 7 7 7 7 9 9 9 9
0.5828 0.3684 û.432 0.641 5 0.8209 O. 3857 0.5813 0.8109 1.0843 0.41 32 0.6773 0.9846 1 3464
0.0378 -0.0499 Û.0 185 0.0994 O. 1963
-0.0051 0.0941
. 0.2099 0.347
0.0346 O. 1632 0.31 23 0.487
Sink 40% Sink10% Sink 20% Sink 30% Sink 40% Sink10% Sink 20% Sink 30% Sink 40% Sink 10% Sink 20% Sink 30% Sink 40%
0.23181 0.0862 0- 167
0.3065 0.4503 O. 1308 0.2821 0.4594 0.6698 0.1 741 0.3741 0.6064 0.8793
Appendix E
Diagnostic Simulation Results
R u 1
Settings of Input, Disturbances and Degradations
inlet Fuel Tamperanire
tnlet 011 Temperature 166 1
l
011 Heat Exchanger Degradanon
"O 200 300 600 800 1000 1200 orne (SI
Diagnostic Results for the Engine Oil Heat Exchanger without Additional Sensors
Diagnostic Results for the Engine Oil Heat Exchanger with One More Sensor Measuring Tfi?
Time (s)
120 180 240 300 360 420 480 540
Average Residual
2.5294 2.8064 2.9636 3.3219 3.4895 3.7437 4.0241,
Uncertainty
_ O. 1357 0.1 360 0.1 378 0.1421 0.1445 O. 1442 0.1440 0,1416 O. 1426 O. 1395 O. 1 302 0.1333 O. 1 362 O. 1388 O. 1333 O. 1283 O. 1236 O. 1263 O. 1288
Oil Heat Exchanger Degradation (%)
5.75 6.25 6.75 7.25 7.75 8.25 8 75
Time (s)
120 180 240 300 360 420 -
480 540 600 660 720 780 840 900 960
1020 1080 1140 1200
4.1921 9.25
Oit Heat Exchanger Degradation (%)
5.75 6.25 6.75 7.25 7.75 8.25 8.75 9.25 9.75
10.25 10.75 11.25 1 1.75 12.25 12.75 13.25 13.75 14.25 14.75
Average Residual
2.2509 2.4436 2.6646 2.9349- 3.2136 3.3908 3.6346 3.779
3.9963 4.2981 4.5266 4.8154 5.0383 5.2291 5.3983 5.4233 5.5462 5.7444 6.037
Uncertainty
0.481 7 0.4640 0.4385 0.41 24 0.3884 0.3685 0.3486
Diagnosis
Diagnosis Min %
5.2 5.63 6.1 1 6.71 7.32 7.7
8.23 8.54 9.01 9.67 10.2 10.7 11.2 11.5 11.9 1 12.2 12.6 13.2
0.3254
Min % 4.32 4.79 5.08
_ 5.82 6.17 6.68 7.25 7.59
Max % 5.98 6.48 7.04 7.74 8.44 8.89 9.49 9.85 10.4 11.1 11.6 12.2 12.8 13.2 13.6 13.6 13.9 14.4 15.1
Max % 7.09 7.69 8.04 8.81 9.18 9.72 10.3 10.6
Diagnostic Results for the Sink Heat Exchanger with Two More Sensors Measuring Tfi and Tfi2
Time (s)
120 180 240.
i 3001 0.3161 i 0.21781 0.45251 0.90071 1.60511 17.25 i 5-101 0.76151
Sink Heat Exchanger Degradation (%)
15.751 16.25 16.75
Average Residual
0.6917 0.6312 0.4632
Diagnosis (%)
10-20 10-20 10-20
Pc
0.7545 0.7564 0.759
Thres hold 30%
1.5627 1.5739 1.5915
5% 0.2117 0.2133 0.2158
10% 0.4398 0.4432 0.4484
20% 0.9539 0.9609 0.972
Run 2
Settings of Input, Disturbances and Degradations
Si flk Heaf Excftanqer oegradatlon
lnler Fuel Temperature
160 [ O 200 40G 500 800 1000 1200
CM Hear Excnanger Oegradaaon
Diagnostic
Diagnostic Results for the Engine Oil Heat Exchanger with One More Sensor Measuring Th
Diagnostic Results for the Sink Heat Exchanger with Two More Sensors Measuring Tfi and Tfi?
Tirne (s)
120 180 240 300 360 420 480 540 764 824 884 944
1004 1064 t 124
i 1184
Average Residual
4.1915 4.3595 4.4186 4.5024 4.531
Oil Heat Exchanger Degradation (%)
10.38 10.63 10.88 11.13 11 -38
4.6552' 4.7145 4.8005 5.3205 5.3377 5.4937 5.5027 5.726
5.7879 5.93
6.0459
Uncertainty
, 0.1417 0.1412 0.1471 O. 1438 0.1319
Diagnosis
1 1.63 1 1.88 12.13 13.06 13.31 13.56 13.81 14.06 14.31 14.56 14.81
1
O. 1376 0.1432 O. 1402 O. 1532 O. 1503 0.1 549 O. 1521 0.1494 O. 1538 0.1511 O. 1486
Min % 9.63
10 10.1 70.3 10.4 10.6 10.7 f 0.9 1 2 12.1 12.4 12.4 12.9
13 13.3 13.6
Max% 17.1 11.5 11.7 11.9 11.9 12.2 12.4 12.6 14.1 14.1 14.5 14.5
15 15.2 15.5 15.8
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