Robust model-based fault detection and isolation for nonlinear processes using sliding modes

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust. Nonlinear Control 2012; 22:89–104Published online 23 October 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1807

Robust model-based fault detection and isolation for nonlinearprocesses using sliding modes

Iván Castillo1, Thomas F. Edgar1,*,† and Benito R. Fernández2

1Department of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712, USA2Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA

SUMMARY

This paper proposes a robust fault detection and isolation system for nonlinear processes that can be formu-lated as differential algebraic equations. For open-loop stable or closed-loop stabilized systems that operateunder strict nonlinear detectability conditions, a methodology to design a nonlinear state estimator basedon sliding mode theory was proposed. The extended observer can handle both parameter estimation andparameters with uncertainties. As a result, the state estimator is able to follow the faulty system, detectingfaults by examining changes in the controlled outputs with respect to setpoint and then probing variationsin the parameters estimated. Once the fault has occurred, the isolation mechanism uses the informationprovided from the state estimator, generated from recovery actions in the presence of a fault. These dif-ferences from normal operation trends can be derived through statistical analysis and then can be used toidentify faults. A steam generator system was used to validate this approach, where process faults wereconsidered. The proposed robust fault detection and isolation method shows significant advantages whenapplied to nonlinear model systems with parameter uncertainties or with complex nonlinearities. The com-plex nonlinearities can be simplified with algebraic nonlinear functions that have bounded uncertainties intheir parameters. Copyright © 2011 John Wiley & Sons, Ltd.

Received 30 December 2010; Revised 18 July 2011; Accepted 1 August 2011

KEY WORDS: robust observer; nonlinear fault detection and isolation (FDI); sliding modes; boundeduncertainties; parameter estimation; nonlinear systems; statistical analysis

1. INTRODUCTION

Complex systems are very demanding in their performance goals, particularly, in the presence offaults, such as sensor, actuator, or process faults. Increasing plant availability, avoiding undesirableshutdown actions in the process, and reducing the risk of safety hazards are examples of essen-tial goals that can be achieved through a fault tolerant control system (FTCS) [1, 2]. An importantclass of an FTCS is an active fault tolerant control system (AFTCS), in which by means of con-trol reconfiguration, the AFTCS responds actively in the event of a fault, guaranteeing stability andacceptable performance of the entire system. FTCS examples that are applied to nonlinear systemscan be found in [1–4], where switching control strategies are designed by using either hybrid controlor supervisory control methods. A fault detection and diagnosis system is a key component of anAFTCS [2,5]. Its main function is to gather as much information about a fault as possible in order todetermine the presence of a fault and the time it occurred in the system (fault detection), the natureand location of the fault (fault isolation), and the size of the fault (fault identification). This paper’scontribution is focused on the design of a robust fault detection and isolation (RFDI) system that

*Correspondence to: Thomas F. Edgar, Department of Chemical Engineering, The University of Texas at Austin, Austin,TX 78712 USA.

†E-mail: [email protected]

Copyright © 2011 John Wiley & Sons, Ltd.

90 I. CASTILLO, T. F. EDGAR AND B. R. FERNÁNDEZ

can be applied to nonlinear chemical processes by using a state estimator capable of dealing withboth bounded uncertainties and estimation of the model parameters.

There are multiple alternatives that can be used for solving the fault detection and isola-tion (FDI) problem, including model-based [6–8], data-driven [9, 10] or combined approaches[11]. A successful model-based fault isolation technique depends on a sufficiently accuratemodel of the system. Uncertainties in the model lead to false alarms or failures in detection.These issues make applying model-based techniques to nonlinear processes difficult. The major-ity of chemical processes have many uncertainties in their parameters due to their complexity.Research that addresses this problem consists of designing observers capable of parameterestimation [12]. However, restrictions surrounding the observability of these parameters limitthis approach. Another alternative is represented by the design of RFDI techniques that dealwith parameter uncertainty and are based on the sliding mode concepts, in which slidingobservers are formulated for restrictive nonlinear systems. Examples of these FDI techniquesbased on residual generation, defined as the difference between the measured signals and thevariables estimated, can be found in [13–15]. Also, linear sliding observers are used in [16]by transforming the nonlinear model. A disadvantage of FDI model-based techniques lies inthe inability of distinguishing or isolating faults when the complexity of the nonlinear modelincreases.

Data-driven approaches have the ability to handle a large number of measurements by com-pressing them into a few indexes, which enables operating conditions to be visualized in lowerdimensional plots. However, these techniques have limitations when they are applied to nonlinearsystems with significant plant variability.

Given the constraints of model-based and data-driven techniques, this paper proposes a combinedapproach in which the detection procedure is inspired by model-based ideas and the isolation mech-anism is based on data-driven techniques by means of statistical analysis. Sliding mode conceptsare applied to address the bounded uncertainties of some model parameters that cannot be esti-mated. They also simplify the complexity of the fault-free model with algebraic nonlinear functionsthat can have bounded uncertainty in its parameters. Traditionally, a model-based fault detectionmechanism is based on detecting abnormal changes between an observer, or state estimator esti-mates, and the measured signals from the plant. One disadvantage of this approach is that oncefaults are detected, the observer is not able to follow the behavior of the faulty system. When thenonlinear system has to operate in closed-loop control, the observer may be unnecessary, becausethe faults can be detected when the controlled outputs are unable to follow the setpoint trajec-tories. In the new approach proposed here, where the state estimator is able to follow the faultysystem, the mechanism of detection is based on detecting both the abnormal deviations betweenthe controlled outputs and the setpoint trajectories and the variations of the estimated parametersfrom their normal values. Additionally, a typical model-based isolation system is based on para-metric approaches [17] where parameters of the model are used to distinguish faults. Examplesof chemical applications that use nonlinear parametric fault isolation found in the literature arecontinuous stirred tank reactor [18], distillation columns [19], and polymerization reactors [20].The disadvantage of this method lies in observability restrictions and the incapability of distin-guishing faults with similar signatures. In the designed isolation approach, the fault information isextracted statistically using principal component analysis (PCA) [21] over the corrections that thenonlinear robust observer applies to the state variable estimates, which are provided by the nonlinearmodel.

This paper is divided into six sections. After a brief introduction, Section 2 presents the modelassumptions, which are the basis for the formulation of the FDI approach. Section 3 presents thedetails of the FDI system developed, in which an algorithm for the nonlinear state observer and thestatistical approach is presented. Next, in Section 4, a brief description of a steam generator systemfollowed by a brief analysis of the main characteristics of its dynamic model are illustrated. Com-parisons with both an extended Kalman filter (EKF) and extended Luenberger observer (ELO) arealso presented. Subsequently, the robust model-based technique is validated using a steam generatormodel with simulations during normal operation and under a process fault. Finally, the conclusionsof this paper are contained within Section 6.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 2012; 22:89–104DOI: 10.1002/rnc

NONLINEAR ROBUST FAULT DETECTION AND ISOLATION 91

2. MODELING ASSUMPTIONS

The proposed approach can be utilized for nonlinear systems formulated as a differential algebraicequation (DAE) of the following form:

dx

dtD f .x,u, � ,pd , q/ , (1)

q .x, � ,pd /D 0, (2)

y D h .x, � ,pd , q/ , (3)

where

� x 2 <n is the vector of state variables;� u 2 <m is the vector of system inputs;� y 2 <p is the vector of system outputs;� f .�/ is the nonlinear state equation function and is assumed to be sufficiently smooth for all t

on <n;� q.�/ is a nonlinear function that can be used in Equations (1) and (3);� h.�/ is the nonlinear output function and is assumed to be sufficiently smooth for all t on <n;

and� pd 2 <

r are the parameters that cannot be estimated, and these are defined as pd D pd C ıpd .Each of these parameters are assumed to have a nominal value pd with a bounded uncertaintydefined by jıpd j6 �.� � 2 <s are bounded parameters that can be estimated by the observer. The parameters are

assumed fixed (not time-variant) but unknown. The maximum limit of these parameters isdefined by �min < � < �max.� The solution for Equations (1)–(3) is assumed to exist for all time t . The first r time derivatives

of y exist where r is defined as the relative order of the output y. External disturbances are notconsidered in the model and are assumed to have zero mean.

As the observer executes parameter estimation, the state variables are augmented with the esti-mated parameters, � , and are defined as ex 2 <enDnCs D Œ xT �T �T . Thus, Equations (1)–(3)become Equations (4)–(6):

dexdtD ef .ex,u,pd , q/, (4)

q.ex,pd /D 0, (5)

y D h.ex,pd , q/, (6)

with ef .ex,u,pd , q/D�f .x,u, � ,pd , q/ Œ0 � � � 0�

�T. (7)

A simple discrete version of these equations can be written as follows:

exk Dexk�1C Ts �ef ı .exk�1,uk�1,pd , q/ , (8)

q .exk�1,pd /D 0, (9)

y D h.exk ,pd , q/, (10)

where Ts is the sample time and limTs!0ef ı D ef .

Finally, Equations (8)–(10) will be used to develop the RFDI approach presented in Section 3.



3. FORMULATION OF THE ROBUST FAULT DETECTION AND ISOLATION SYSTEM

The FDI conclusions are based on the discrete version of the fault-free model formulated inEquations (1)–(3). In the event a fault occurs in the system, the error in the model predictions (inthis paper caused by process faults) is represented by ıf and affects the nonlinear state equation asfollows:

fsystem.x,u, � ,pd , q/D f .x,u, � ,pd , q/C ıf .x,u, � ,pd , q/, (11)

where ıf is assumed to be bounded and fsystem represents the system in which it is assumed thatf .�/ has the same structure.

The detection of a fault is performed by using both a nonlinear discrete observer that predicts thetrends of the measurable outputs,by, and a fault detector system that determines the time of the fault,tfault, once differences are found either between the setpoint trajectories and the process variables orin variations of the parameters estimated, � , from their normal operating values.

The isolation of the fault is carried out by analyzing the corrections of the state variables, x�k2 <en,

derived from the observer at each time step. Figure 1 illustrates how the robust observer, the faultdetector, and the isolation block components are connected. The fault contributions, f i

k, for each

i state variable, are obtained by applying PCA over the vector x�k

. Further detail regarding thesecomponents will be presented in the following subsections.

3.1. Robust observer formulation

The predictions of the state variablesbexk , at each time step k, can be obtained bybexk Dex�k C x�k �e�yk � , (12)

where the approximate values of the state vectorex�k

, determined by applying Equation (13), are in

terms of the a priori state estimatesbexk�1 and calculated by using Equation (8):

ex�k Dbexk�1C Ts �ef ı �bexk�1,uk�1,pd , q�

. (13)

These approximate values, ex�k

, are corrected by adding the term x�k

�e�yk

�that is in function of

the error e�yk ,

e�yk D ´k � y�k , (14)

where the vector ´k 2 <p corresponds to the measurements of the nonlinear system, and theestimates of the measurement vector y�

kare computed by applying Equation (15):

y�k D h�ex�k ,pd , q

�. (15)

In order to deal with bounded uncertainties in the parameters of the model, the discrete observeris based on the sliding observers theory [22], whereby it is demonstrated that the robust performance

Figure 1. Robust fault detection and isolation architecture.



of these type of observers is possible in the presence of parameter inaccuracies. The sliding observeris designed such that it drives the states to a particular surface, called the sliding surface S.t/ anddefined by Equation (16):

S.t/D°exk 2 <en W e�yk D 0± . (16)

Once this surface has reached the sliding motion, it is ensured that the states will remain close to thesurface. The sliding motion is generated by using a switching function ,vk 2 <p , defined by

vi DMi sign�yi � y

�i

�, i D 1, : : : ,p, (17)

where p is the number of measurable variables, and Mi is chosen in order to match all the boundeduncertainties that were defined in the model parameters.

The sliding observer used is based on the linear Utkin observer [22], which has been extended tononlinear systems. Furthermore, the observer proposed also allows parameter estimation wherebyan ELO in combination with the sliding motion guarantee a robust estimation. Equation (18) showshow the corrected vector x�

kis determined:

x�k DKke�ykC T �1c

"�Lkvk

�e�yk

�vk�e�yk

� #, (18)

where the matrix Kk 2 <enxp corresponds to the Luenberger gain, the matrix Lk 2 <.en�p/xp repre-sents the sliding gain, and the matrix Tc 2 <enxen is used to introduce a coordinate transformation. InSection 3.1.1, the calculation of these matrices will be presented in detail.

3.1.1. Robust observer algorithm. The prediction of the states can be achieved by using thefollowing algorithm at each time step:

Step 1: Approximate values of the state vectorex�k

are obtained from Equation (13).Step 2: Estimates of measurement vector y�

kare determined from Equation (15).

Step 3: The nonlinear state equation and output functions are linearized around the current estimatedstate,bexk�1, as in Equations (19) and (20):

AŒi ,j � D@ef i@exj

???bexk�1,uk�1A 2 <enxen, (19)

CŒi ,j � D@hi

@exj???bexk�1,uk�1

C 2 <pxen. (20)

Thus, the linear nominal system, at each time step, can be written as in Equation (21) wherebynonexternal disturbances are considered:

xk D Akxk�1,

yk D Ckxk ,(21)

where Ak and Ck correspond to the discretized matrices.Step 4: The gains that are used in the proposed nonlinear state estimator of Equation (12) are calcu-

lated in this step. These gains are calculated on the basis of the linearized model of Equation (21),which is obtained at each time step. The state estimator in the linear domain has the followingform:

bxk D Akbxk�1CKkeyk�1 C T �1c ��Lkvkvk

,

byk D Ckbxk ,(22)

where Kk and Lk correspond to the Luenberger and sliding gains, respectively. The switchingfunction vk is defined by Equation (28). The matrix Tc is obtained by Equation (23), and the



error eyk , which is the difference between the measurement vector yk and the estimates of theoutputsbyk , is determined by Equation (24).

Tc D�

null.Ck/T Ck�T

, (23)

eyk D yk �byk , (24)

exk D xk �bxk . (25)

Kk 2 <enxp is determined such that the eigenvalues of the error exk , given by Equation (26), have

strictly negative real parts. This error exk , defined as the difference between the true states xk andestimates bxk , is formulated in Equation (25). Thus, the first-order difference equation, given byEquation (26), is derived by using Equations (21), (22), (24), and (25).

exk D .Ak �KkCk/exk�1 . (26)

In order to obtain the sliding gain Lk , the matrix Ak of Equation (21) is transformed such thatthe outputs appear as components of the states. With the use of the following transformationTcx! �, the state estimator model of Equation (22) becomes� b�kbyk

D

�A11 A12A21 A22

� b�k�1byk�1C TcKkeyk C

��Lkvkvk

,

byk D � 0 Ip� � b�kbyk

, (27)

where each row of the discontinuous vector vk 2 <p is defined by

vi DMi sign.yi �byi /, i D 1, : : : ,p. (28)

Equation (29) shows the error between the transformed estimates and true states:�e�keyk

D

�A11 A12A21 A22

�e�k�1eyk�1

C TcKkeyk �

��Lkvkvk

. (29)

After some finite time when the states have reached the sliding surface, as defined by Equation(16), the errors eyk�1 and eyk approach 0. Equation (29) then reduces to

e�k D .A11 �LkA21/e�k�1 . (30)

The matrix Lk 2 <.en�p/xp is obtained such that the eigenvalues of Equation (30) have strictlynegative real parts.

Step 5: A posteriori estimate bexk , or a correction of the values ex�k

, is computed by utilizingEquations (12) and (18). Although this nonlinear observer equation is similar to the observermodel presented in Equation (22), the difference lies in the error of the outputs e�yk , which isdefined in Equation (14). The gains Kk and Lk are calculated by using Equations (26) and (30).Finally, the states that correspond to the parameters of the model are verified such that they areinside the limits defined previously in Section 2.

3.2. Fault detector system

The detection of a fault is performed by verifying differences in the controlled variables with set-point trajectories, followed by an examination of the changes in the parameter values that wereestimated from normal operation trends, as is shown in Figure 1. This procedure will be clarified inSection 5.



3.3. Statistical analysis

A statistical analysis is performed over the correction vector x�k

, defined in Equation (18). A statis-tical model of this vector is created using PCA. With the collection of data of the vector x�

kduring

normal process operation, the data matrix X 2 <Nxen is created, where N represents the number ofsamples anden represents the number of state variables. The main idea behind PCA is that instead ofusing the matrix X , it is possible to predict the variability in the data by building a low-dimensionalmodel or by using l principal components that are an orthogonal set of basis vectors. In other words,the covariance matrix S of the normalized matrix X, scaled to zero mean and to unit variance, canbe decomposed as follows:

S w 1

N � 1XTX D PƒP T C ePeƒeP T , (31)

where P 2 <enxl is the principal loadings, eP 2 <enx.en�l/ corresponds to the residual loadings, andthe diagonal matrices � and e� contain the principal and residual eigenvalues.

Once the PCA model is computed, the objective is to determine the fault contributions for eachstate variable after a fault is detected by the mechanism illustrated in Section 3.2. PCA projectsthe information contained in x�

kinto two subspaces: the principal subspace and the residual sub-

space. Statistical indices [9,21,23] are used for projecting the data into these subspaces and are thenutilized for purposes of FDI. The most common statistical indices are (1) squared prediction errorwhose index is calculated in the residual subspace; (2) Hotelling’s T 2 whose index is calculatedin the principal subspace; and (3) combined index ' in which both indices T 2 and squared predic-tion error are combined. The latter statistical index ' will be used in this paper and is defined byEquation (32):

'k

�x�

T

k

�D x�

T

k

eCı2CD

�2

!x�k , (32)

where eC D PP T and D D P��1P T . The control limits ı2 and �2 are defined in Appendix A.Furthermore, once the statistical index is calculated, isolation is carried out by decomposing the

information provided by this index into each element of the vector x�k

that is associated with the statevariables of the system. The most utilized isolation technique is contribution plots [21]. However,here, the reconstruction-based contribution (RBC) [23] isolation technique will be utilized. RBCreconstructs the fault contribution along each variable i with associated direction �i 2 <en by usingthe following expression that is defined for the case of the ' index:

RBC'i D .�ifi /T

eCı2CD

�2

!.�ifi / , (33)

where fi is calculated in order to minimize the value of the statistical index '�x�k� �ifi

�, given by

Equation (32). The calculation of the derivative of this index '�x�k� �ifi

�with respect to fi and

equating to 0 results in the calculation of the fi as

fi D

�Ti

eCı2CD

�2

!�i

!�1�Ti

eCı2CD

�2

!x�k . (34)

Thus, with the use of the results of Equations (33) and (34), the fault contributions for eachvariable are calculated. The application of this technique will be presented in Section 5.

4. ANALYZING THE SLIDING CONTRIBUTIONS IN THE ESTIMATION

In this section, the motivating case study, which is a steam generator, will be briefly described withits main characteristics (derived from nonlinear complexity) that are very common in chemical pro-cesses. The proposed robust observer will also be compared with different nonlinear state estimatorswhereby the advantages of including the sliding modes will be highlighted.



4.1. Description of the nonlinear process and its main characteristics

The steam generator model is used in applications of fault diagnosis [1, 24, 25]. Figure 2 shows thepiping and instrumentation diagram of this system. This paper focuses on the dynamic of the boiler,which possesses the most severe nonlinearities. The boiler system has four inputs: (1) the flow offeed water, FAL Œkg/s�, which is proportional to the opening of the control valve V1; (2) the heatercontrol signal uTH; (3) the steam flow in the boiler output, FGV Œkg/s�, which is proportional to theopening of the control valve V2; and (4) the temperature of the feed water, TAL [°C]. Three measur-able outputs are considered: (1) the water volume of the boiler, L [l]; (2) the pressure in the boiler,PGV Œbar�; and (3) the temperature of the metal body of the boiler, TMG [°C]. The system’s modelcontains three state variables: (1) the mass of the water–steam mixture in the boiler, MGV Œkg�; (2)enthalpy in the boiler, HGV ŒJ�; and (3) the temperature of the metal body of the boiler, TMG [°C].Two proportional-integral controllers maintain the pressure and level in the boiler within a desiredrange. The pressure in the boiler is controlled by manipulating the thermal resistor, which has amaximum power of 55 kW, whereas the level in the boiler is controlled by operating the controlvalve V1, which transfers water from the storage tank. Additionally, the steam flow is controlled byusing control valve V2, which is operated in manual mode. The differential equations that representthe behavior of the boiler are

dMGV

dtD FAL �FGV, (35)

dHGV

dtD uTHPTHC cpeTALFALC V

dPGV

dt�FGVhV �KGM.TGV � TMG/, (36)

dTMG

dtD

1

CGM.KGM.TGV � TMG/�Kex.TMG � Tex//, (37)

y.t/D�.1�X/MGVvL PGV TMG

�T. (38)

where the parameters (V, cpe, PTH, KGM, Kex, CGM, and TGV) and the thermodynamical properties(hGV, vGV, hL, hV, vL, and vV) are defined in Appendix B. The polynomials of the thermodynamicalproperties were fitted using the data provided by the International Association for the Properties ofWater and Steam [26].

The outputs of the model are given by Equation (38). Also, Equations (39) and (40) are used tomodel the two-phase water–steam mixture. In using these equations, both the pressure PGV of theboiler and the steam quality X are obtained by solving the polynomials of Equations (41) and (42),respectively.

Figure 2. Piping and instrumentation diagram of the steam generator. This model is based on a pilot processoperating at the University of Lille (France).



hGV DHGV

MGVD hV.PGV/ �X C hL.PGV/ � .1�X/, (39)

vGV DV

MGVD vV.PGV/ �X C vL.PGV/ � .1�X/, (40)

.hGV � hL/.vV � vL/� .vGV � vL/.hV � hL/D 0, (41)

X D.hGV.PGV/� hL.PGV//

.hV.PGV/� hL.PGV//D.vGV.PGV/� vL.PGV//

.vV.PGV/� vL.PGV//.(42)

Three interesting characteristics based on the nonlinear model equations were found. These arethe following:

1. The uncertainty of some parameters of the model. For instance, in Equations (36) and (37),the parameters KGM and Kex are difficult to determine. As a result, errors in the prediction ofoutputs can lead to false alarms.

2. The stability of the system. The first state equation, given by Equation (35), is unstable inopen loop, forcing the system to be analyzed under closed-loop control. This violates causalityassumptions in system identification [27, 28].

3. A noninvertible characteristic of the pressure trajectory because of the modeling of the water–steam mixture, where the thermodynamic properties are used in Equations (39) and (40).Figure 3 shows a typical trajectory of the boiler pressure, PGV, which is obtained by solvingthe polynomial given in Equation (41). Based on this pressure trajectory, that is, the functionof the state variables MGV and HGV, a noninvertible characteristic can be found. For instance,if the pressure of the boiler is 7.5 bar (grey plane shown in Figure 3), then the pressure valuecan be obtained from three different values in the state variables MGV and HGV. This factmakes it very difficult to derive a nonlinear function that approximates the trajectory of thepressure in terms of the state variables. Therefore, it is a formidable task to analyze the prop-erties of the model, such as observability and controllability, while also using the model as astate estimator.

To deal with the first characteristic, two alternatives are given: (1) verify if Kgm and Kex can beestimated by applying an observability analysis; or (2) if these parameters are not observable, thenbounded uncertainties can be defined for them.

PGV D b1C b2MGVC b3HGV. (43)

Figure 3. Pressure trajectory, PGV, versus mass of the steam–water mixture, MGV, and enthalpy, HGV.



To handle the noninvertible characteristic, instead of using Equation (41), which is a high-orderpolynomial, an explicit function that solves for the pressure PGV can be defined, as shown inEquation (43). Notice that this function corresponds to the function q, defined in Equation (9),that could be in function of the state variables and parameters that cannot be estimated. In Equa-tion (43), the parameters b are defined as b D Œ 11.96 �0.23 2.78e�7 �T with the followingbounded uncertainties: jıb1 6 4j, jıb2 6 0.1j, and

ˇ̌ıb3 6 1e�07

ˇ̌. The bounded uncertainties were

included after determining that these parameters cannot be estimated.In Section 4.2, the performance of different nonlinear state estimators is compared using the

simplified model that includes Equation (43).

4.2. Nonlinear state estimator comparisons

The performance of the proposed nonlinear robust observer is compared with an ELO and an EKF.For these three state estimators, parameter estimation is performed. Figure 4 shows the pressure andvolume trajectories of the boiler, respectively, where random noise has been included in solving themodel formulated in Equations (35)–(42). The pressure is operated at 7, 10, and 8 bar. The watervolume is operated at 149, 153, and 147 l, respectively. These controlled variables, pressure andwater volume of the boiler, successfully follow the setpoint trajectories in normal operation.

To summarize, three steps are used to define the robust observer:

Step 1: The complex nonlinear dynamics, as in the case of the noninvertible characteristic, illus-trated in Section 4.1, can be simplified so that they can be formulated like the model defined inEquations (8)–(10). For the steam generator, the function that solves for the pressure PGV, givenby Equation (41), is replaced by Equation (43).

Step 2: An observability analysis is performed in order to define the parameters of the nonlinearmodel that can be estimated with their associated maximum limits. In this example, Kgm andKex were the parameters selected. The estimation of these parameters will be obtained over thefollowing limits:

06Kex 6 10, (44)

06Kgm 6 2100. (45)

Step 3: The parameters that cannot be estimated but could lead to false alarms are defined withbounded uncertainties. The value of Mi of Equation (17) is also determined on the basis of themagnitude of these uncertainties. For the case study, the parameters of Equation (43) were definedas the parameters with associated uncertainties, and their M1..r values are M1..r D 0.25.

Figure 4. Pressure and level trajectories of the boiler simulated at different operating points.



Figure 5 shows the estimate of the pressure PGV at the same operating points illustrated inFigure 4. The blue (thick) line represents the real trajectory of the pressure, which is obtained bysolving Equations (35)–(42). The red (thin) line has an oscillatory trajectory around the operationpoints of pressure that represent the sliding motion of the observer. The green (dashed) line corre-sponds to the estimated trajectory of pressure, which is a filtered version of the oscillatory red line.The error of estimation calculated was 1.53% and was computed by the mean absolute percentageerror (MAPE), defined as

MAPED

ˇ̌y.t/� y�

k

ˇ̌y.t/

100% , (46)

where y.t/ is given by Equation (38) and y�k

is determined by Equation (15).Figure 6 shows the estimated trajectories of parameters Kex and Kgm. The black (thick) lines

are the estimated parameters, which result from filtering the oscillatory trajectory, given by the red(thin) line trajectories. Because of the sliding motion, the oscillation presented is within the limitsdefined previously in Equations (44) and (45) for these parameters.

The estimated pressure trajectories by using an ELO and an EKF are illustrated in Figure 7. Theblue (thick) line corresponds to the real trajectory of the pressure, and the red (dashed) lines repre-sent the estimates of each estimator. Notice that the obtained MAPE error of estimation for the case

Figure 5. Estimate of the pressure trajectory with the use of the robust nonlinear observer. MAPE, meanabsolute percentage error.

(a) (b)

Figure 6. Parameter estimation results with the use of robust nonlinear observer: (a) estimate of thecoefficient Kex and (b) estimate of the coefficient Kgm.



(a) (b)

Figure 7. Estimate of the pressure trajectory with the use of (a) extended Luenberger observer and (b)extended Kalman filter. MAPE, mean absolute percentage error.

(a) (b)

Figure 8. Parameter estimation of the coefficient Kex: (a) estimate with the use of extended Luenbergerobserver and (b) estimate with the use of extended Kalman filter.

of the ELO is 0.93%, as shown in Figure 7(a), which is better than the error obtained with the use ofthe robust observer. The MAPE error for the EKF, as shown in Figure 7(b), is 5%. For this case, thetrajectory of the pressure estimated has a larger error than the case estimated in the proposed robustobserver (shown in Figure 5), stemming from the approximation of the pressure function defined inEquation (43). However, because of the uncertainties associated in the parameters of Equation (43),these estimators violate the assumptions formulated for parameter estimation, which are definedin Equations (8)–(10). In Figure 8, the estimated trajectories of the parameter Kex, for ELO andEKF, are not constant. Therefore, these parameter estimations cannot be used to detect faults. Thisfact is a fundamental advantage of including the sliding mode contributions in an observer whenuncertainties are presented in the model.

Finally, Figure 9 shows the pressure estimates of an EKF, red (dashed) line. The performance ofthis estimator is evaluated under the gas leak case, where further detail can be found in Section 5.Before the fault occurs, the estimated trajectory has a similar trend to the one shown in Figure 7(b).Once the fault has occurred, the estimates of the EKF are no longer valid.

5. FAULT DETECTION AND ISOLATION RESULTS

Two faults are introduced to validate the proposed approach. First, a gas leak is introduced in thesystem at t D 1, 000 s. Figure 10(a) shows the estimated trajectories. As a result of the fault, thepressure value, blue (thick) line, falls at t D 1, 000 s and separates from the setpoint trajectory. The



Figure 9. Estimates of the pressure PGV in the presence of a gas leak with the use of an extended Kalmanfilter. The extended Kalman filter trace is truncated to show a relevant scale.

(a) (b)

Figure 10. Gas leak fault: (a) pressure trajectory estimates and (b) estimate of the parameter Kex. MAPE,mean absolute percentage error.

observer is able to follow the pressure trajectory, green (dashed) line, with a MAPE error equal to2.17%. A similar situation happens when a liquid leak (second fault) is added into the system att D 600 s. For this fault, the level trajectory, blue (thick) line, separates from the setpoint trajectory.The observer follows the level trajectory, green (dashed) line, with a MAPE error equal to 0.44%.

Figure 10(b) shows the coefficient Kex trajectory, black (thick) line, in the presence of a gas leakfault. In this case, changes in the value of this coefficient from the nominal value have occurred,and the fault is detected as a result. The fault is first detected from the deviation of the pressurevalues to the setpoint trajectory and then verified once changes in the parameter value have beendetected. The fault is detected at t D 1, 026 s. For the case of the liquid leak, Figure 11(b) shows thecoefficient Kex estimates, and the fault is detected at t D 651 s.

Figures 12(a) and 12(b) illustrate the results of isolation for the gas leak and liquid leak, respec-tively, whereby the fault contributions, calculated by using Equation (33) for each state variable,are plotted. Although the contribution of each fault is similar, the faults can be distinguished on thebasis of the magnitude of the contributions associated for each state variable.

6. CONCLUSIONS

An RFDI system, utilizing a fault-free model, was designed. The detection mechanism proposedinvolves a robust state estimator that is based on an ELO combined with the theory of sliding



(a) (b)

Figure 11. Liquid leak fault: (a) level trajectory estimates and (b) estimate of the parameter Kex. MAPE,mean absolute percentage error.

(a) (b)

Figure 12. Isolation results in the case of process faults: (a) contribution plot for the gas leak and (b)contribution plot for the liquid leak.

modes. The advantage of including sliding modes in the presence of parameter uncertainties hasbeen demonstrated by comparing its performance with an EKF and an ELO. Consequently, thisrobust estimator is capable of dealing with parameters that can have associated bounded uncer-tainties as well as parameter estimation in the presence of noise. These capabilities provide theadvantage of avoiding false alarms in addition to allowing the simplification of nonlinear com-plex dynamics into explicit functions that can be modeled as DAEs. Therefore, this simplificationfacilitates the analysis of the main properties of the system such as observability, stability, andcontrollability. These two advantages are important contributions for a FTCS. Faults are detectedby verifying changes in the parameter values and deviations of the controlled variables from thesetpoint trajectories.

In regard to the isolation mechanism proposed, faults can be reconstructed determining the con-tributions over the state variables and parameters estimated. A statistical analysis is performed fromthe correction vector of the robust state estimator, which minimizes the error between the output esti-mates and the measurable variables of the system. This correction vector is calculated by linearizingthe nonlinear model at the current operating point that facilitates the use of statistical methods thatare based on linear models. This isolation mechanism provides further information to distinguishfaults with similar characteristics, as in the case of process faults.

This approach was successfully validated by using a steam generator system where two processfaults, a gas leak and a liquid leak, were considered. These results serve as important evidence toextend the robust FDI formulation to other nonlinear applications that can be defined as DAEs.



Future work will focus on analyzing how to distinguish the effect of external disturbances from thefaults when these are not zero mean.

ACKNOWLEDGEMENTS

The authors want to thank the Process Science & Technology Center (PSTC) and the Roberto RoccaFoundation for their support on this project.

APPENDIX A: CONTROL LIMITS FOR THE PRINCIPAL COMPONENT ANALYSIS MODEL

The control limits that are utilized in Equations (32)–(34) are defined as follows [21]:

ı2 D�2

�1�2˛

�21�2

�with .1� ˛/� 100% confidence level,

where �1 DPeniDlC1 i and �2 D

PeniDlC1

2i with i equal to the i th eigenvalue of the covariance

matrix S , defined by Equation (31).

�2 D �2˛.l/ with .1� ˛/� 100% confidence level.

APPENDIX B: PARAMETERS AND THE THERMODYNAMICAL PROPERTIES OF THESTEAM GENERATOR

KGM is the heat exchange coefficient from the water–steam mixture to the metal body of the boiler,KGM DKgmFVG.Kex is the heat exchange coefficient from the metal body of the boiler to the environment.hGV [J/kg] is the specific enthalpy of the water–steam mixture.vGV [m3/kg] is the specific volume of the water–steam mixture.The geometric volume of the boiler, V, is 0.165 m3.The specific heat of the feedwater flow, cpe , is 4200 J/kg °C.The heating power, PTH, is 55 kW or kJ/s.The average heat capacity of the metal, CGM [J/kg °C], is

CGM DMmetalcmetal D 102 kg� 498 J/kg °C.

The temperature of the boiler, TGV [°C], is

TGV D� 8.569e�3P 4GVC 0.336P 3GV � 4.805P 2GV

C 34.357PGVC 65.533.

The specific enthalpy of liquid, hL [kJ/kg], is

hL D� 3.578e�2P 4GVC 1.402P 3GV � 20.077P 2GV

C 144.7PGVC 273.96.

The specific enthalpy of steam, hV [kJ/kg], is

hV D� 4.068e�4P 6GVC 2.205e�2P 5GV � 0.472P 4GV

C 5.094P 3GV � 29.552P 2GVC 96.438PGVC 2599.3.

The specific volume of liquid, vL [m3/kg], is

vL D�3.591e�7P 2GVC 1.246e�5PGVC 1.039e�3.

The specific volume of steam, vV [m3/kg], is

vV D� 3.290e�5P 5GVC 1.606e�3P 4GV � 3.008e�2P 3GV

C 0.271P 2GV � 1.211PGVC 2.489.



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Robust model-based fault detection and isolation for nonlinear processes using sliding modes

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