A Survey of Active Fault Diagnosis Methods

10th IFAC Symposium on Fault Detection, 10th on Fault Detection, 10th IFAC IFAC Symposium Symposium Detection, Supervision and Safetyon forFault Technical Processes 10th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes Available online at www.sciencedirect.com Supervision and Safety for Technical Processes Warsaw, Poland, August 29-31, 2018 10th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Warsaw, Poland, Poland, August August 29-31, 29-31, 2018 2018 Processes Warsaw, Supervision and Safety Technical Warsaw, Poland, Augustfor 29-31, 2018 Processes Warsaw, Poland, August 29-31, 2018

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A A A A

Survey Survey Survey Survey

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Active Active Fault Fault Diagnosis Diagnosis Active Fault Diagnosis Methods Active Fault Diagnosis Methods Methods Methods ˇ Ivo Punˇ coch´ aˇ r, Jan Skach ˇ

ˇ Ivo c a rr,, Jan Ivo Punˇ Punˇ coch´ och´ aˇ ˇ Jan Skach Skach ˇ Ivo Punˇ coch´ aˇ r, Jan Skach ˇ Ivo Punˇ c och´ a ˇ r , Jan Skach European European Centre Centre of of Excellence Excellence --- NTIS, NTIS, Faculty Faculty of of Applied Applied Sciences, Sciences, European Centre of Excellence NTIS, Faculty of Sciences, European Centre ofBohemia, Excellence306 - NTIS, Faculty of Applied Applied Sciences, University of West 14 Pilsen, Czech Republic (e-mail: University of West Bohemia, 306 14 Pilsen, Czech Republic (e-mail: University of West Bohemia, 306 14 Pilsen, Czech Republic (e-mail: European Centre ofBohemia, Excellence306 [email protected]). NTIS, Faculty of Applied Sciences, University of West 14 Pilsen, Czech Republic (e-mail: [email protected], [email protected], [email protected]). [email protected], [email protected]). University of West Bohemia, 306 14 Pilsen, Czech Republic (e-mail: [email protected], [email protected]). [email protected], [email protected]). Abstract: During the last three decades, there has been growing interest in active fault Abstract: During During the the last last three three decades, decades, there there has has been been aaa growing growing interest interest in in active active fault fault Abstract: Abstract: During the last three decades, there has diagnosis (AFD) and several important results have been reported in the literature. This survey been a growing interest in active fault diagnosis (AFD) and several important results have been reported in the literature. This survey diagnosis (AFD) and several important results have been reported in the literature. This survey Abstract: During the last up-to-date three decades, there has been a growing intoactive faulta diagnosis (AFD) several important results have theinterest literature. survey aims to provide a compact overview of main research directions and propose aims to to provide provide aand compact up-to-date overview of been main reported research in directions and to toThis propose aims compact up-to-date overview of main research directions and propose diagnosis (AFD) a and several important results have been reported in the literature. This surveyaaa aims to provide a compact up-to-date overview of main research directions and to propose basic classification of AFD methods. The contributions are presented almost in the chronological basic to classification of AFD methods. methods. The contributions contributions are presented presenteddirections almost in in the the chronological chronological basic classification AFD The are almost aims provide a of compact up-to-date ofaremain todiffer propose a basic classification of AFD methods. Theoverview contributions are research presented almost inand the chronological order as they appeared in the literature and they divided into two groups that in the order as they appeared in the literature and they are divided into two groups that differ in in the the order as they appeared in the literature and they are divided into two groups that differ basic classification of AFD methods. The contributions are presented almost in the chronological order as uncertainties they appeared the literature and they are divided into two groups that differtrends, in the way the are The survey is concluded with the description of way the the uncertainties areinmodeled. modeled. The survey survey is concluded concluded with the description of main main trends, way are The is with the description of order as uncertainties they appeared the future literature and they are divided into two groups that differtrends, in the way the uncertainties areinmodeled. modeled. The survey isin concluded with the description of main main trends, some application results, and challenges the area of AFD. some application results, and future challenges in the area of AFD. some application results, and challenges the AFD. way uncertainties are modeled. survey isin with the description of main trends, somethe application results, and future futureThe challenges inconcluded the area area of of AFD. © 2018, IFAC (International Federation ofchallenges Automaticin Control) Hosting by Elsevier Ltd. All rights reserved. some application results, and future theinput area of AFD. Keywords: Active fault diagnosis, Survey, Auxiliary signal design, methods Keywords: Active Active fault fault diagnosis, diagnosis, Survey, Survey, Auxiliary Auxiliary input input signal signal design, design, Model-based Model-based methods methods Keywords: Keywords: Active fault diagnosis, Survey, Auxiliary input signal design, Model-based Model-based methods Keywords: Active fault diagnosis, Survey, Auxiliary input signal design, Model-based methods 1. INTRODUCTION signal to noise ratio of a fault or masking of aa fault by 1. INTRODUCTION INTRODUCTION signal to ratio of or of 1. signal to noise noise ratio mode of aa fault fault or masking masking of system. a fault fault by by aa current operating of the monitored In 1. INTRODUCTION signal to noise ratio of a fault or masking of a fault by current operating mode of the the monitoredof system. system. In asignal current operating mode of monitored In 1. INTRODUCTION to noise ratio of a fault or masking a fault by a current operating mode of the monitored system. In such situations, some faults can remain undetected for an The interest in fault diagnosis (FD) has grown substansuch situations, some faults can remain undetected for an The interest interest in in fault fault diagnosis diagnosis (FD) (FD) has has grown grown substansubstan- such situations, some faults can remain undetected for an The asuch current operating mode of the monitored system. In unacceptable long period of time. The interest in fault diagnosis (FD) has grown substantially over last decades due to increasing complexity of situations, some faults can remain undetected for an long period of time. tiallyinterest over last last decades due to to (FD) increasing complexity of unacceptable unacceptable long period of can time.remain undetected for an tially over decades due increasing complexity of such situations, some faults The in decades faultand diagnosis has grown substanengineering systems the urge to produce high quality unacceptable long period of time. tially over last due to increasing complexity of engineering systems and due the urge urge to produce producecomplexity high quality quality engineering systems and the to high tially overatlast decades to increasing of unacceptable long period of time. highefficient, quality products low costs while systems engineering systems and the keeping urge to the produce PFDr products at low costs while keeping the systems efficient, products at low costs while keeping the systems efficient, PFDr engineering systems and the urge to produce high quality PFDr environmentally friendly, and safe. Since a classical conproducts at low costs while keeping the systems efficient, PFDr environmentally friendly, and safe. Since a classical conenvironmentally friendly, and safe. Since a classical conproducts at low costs while keeping the systems efficient, RG environmentally friendly, and safe. Since a classical controller is usually able to keep satisfactory control perforISG/C S PFDr RG Decision troller is usually able to keep satisfactory control perforISG/C Input RG S troller iseven usually ablea tosystem keep satisfactory control perforISG/C S Output environmentally friendly, and satisfactory safe. Sincefrom acontrol classical conmance though deviates its nominal RG Decision S troller is usually able to keep perforISG/C Input Output Input Output mance even though a system deviates from its nominal mance even though system deviates from its nominal RG Decision troller iseven usually ableaa tosystem keeptosatisfactory control perforDM ISG/C Input Output Decision S behavior, FD techniques aim detect changes that might mance though deviates from its nominal DM behavior, FD techniques aim to detect changes that might DM Input Output Decision behavior, FD techniques aim to detect changes that might mance even though a system deviates from its nominal DM behavior, FD techniques aim to detect changes that might not be managed by controllers themselves. not be managed by controllers themselves. not be managed by controllers themselves. DM behavior, FD techniques aim tothemselves. detect changes that might not be managed by controllers Fig. 1. A block diagram of the PFD. The FD methods can be classified at the highest level not managed bycan controllers themselves. Fig. 1. 1. A A block block diagram diagram of of the the PFD. PFD. The be FD methods can be classified classified at the the highest highest level level Fig. The FD methods be at Fig. 1. A block diagram of the PFD. The FD methods can be classified at the highest level into two groups based on the interaction between a fault into two groups based on the interaction between a fault Fig. 1. A block diagram of the PFD. into two groups based on the interaction between a fault PFD architecture does not provide any to The FD and methods can on besystem. classified atfirst the highest level Since into two groups based the interaction between a fault detector a monitored The group includes Since PFD architecture does not provide any means means to Since PFD architecture does not provide means to detector and a monitored system. The first group includes detector and a monitored system. The first group includes Since PFD architecture does not provide any any means to address this issue, an active fault diagnosis (AFD) archiinto twoFD groups based on system. the that interaction between a fault group includes passive (PFD) methods are theoretically well detector and a monitored The first address this issue, an active fault diagnosis (AFD) archiaddress this issue, an active fault diagnosis (AFD) archipassive FD (PFD) methods that are theoretically well Since PFD architecture does not provide any means to passive FD (PFD) methods that are theoretically well tecture has been proposed in the literature. A key idea of detector and a monitored system. The first group includes address this issue, an active fault diagnosis (AFD) archideveloped firmly established the FD community. passive FDand (PFD) methods that in are theoretically well tecture has been proposed in the literature. A key idea of tecture has been proposed in the literature. A key idea of developed and firmly established in the FD community. address this issue, an active fault diagnosis (AFD) archideveloped and firmly established in the FD community. AFD is to use an auxiliary input signal that is injected passive FD (PFD) methods that are theoretically well tecture has been proposed in the literature. A key idea of developed and firmly established in the FD community. Their utility has been proven through many successful AFD is to use an auxiliary input signal that is injected AFD is to use an auxiliary input signal that is injected Their utility has been proven through many successful tecture has been proposed in the literature. A key idea of Their utility has been proven through many successful AFD is to use an auxiliary input signal that is injected into the monitored system in order to improve the quality developed and firmly established in the many FD community. Their utility has been proven through successful applications (Chiang et al., 2001; Isermann, 2011). A block the system in order to quality into the monitored system in order to improve improve the quality applications (Chiang et al., al., 2001;through Isermann, 2011). A block block into AFD is monitored to use an idea auxiliary input signal that the ishas injected applications (Chiang et 2001; Isermann, 2011). A of decisions. The of the active approach been Their utility has been proven many successful into the monitored system in order to improve the quality diagram of the PFD is depicted in 1. applications (Chiang etarchitecture al., 2001; Isermann, 2011). AFig. block decisions. The idea of the active approach has been of decisions. The in idea of in the active approach hasquality been diagram of of the the PFDet architecture is depicted depicted in A Fig. 1. of into the employed monitored system order to improve the diagram PFD architecture is in Fig. 1. already other engineering fields such opapplications (Chiang al., 2001; Isermann, 2011). block of decisions. The in idea of the active approach hasas been The input to the monitored system (S) is generated by diagram of the PFD architecture is depicted in Fig. 1. already employed other engineering fields such as opalready employed in other engineering fields such as opThe input to the monitored system (S) is generated by of decisions. The idea of the active approach has been The input to the monitored system (S) is generated by timal experimental design (Atkinson et al., 2007), system diagram of to thethe PFD architecture is (ISG) depicted infeedback Fig. by 1. already employed in other engineering fields such as opThe input monitored system (S) is generated a feedforward input signal generator or a timal experimental design (Atkinson et al., 2007), system timal experimental design (Atkinson et al., 2007), system a feedforward input signal generator (ISG) or a feedback already employed in other engineering fields such as opa feedforward input signal generator (ISG) or a feedback timal experimental design (Atkinson et al., 2007), system identification (Zarrop, 1979), or dual control (Filatov and The input (C). to the monitored system (ISG) (S) is the generated by identification (Zarrop, 1979), or dual control (Filatov and To obtain a decision, both input and acontroller feedforward input signal generator or a feedback (Zarrop, 1979), or dual et control (Filatov and controller (C).input To obtain obtain agenerator decision, (ISG) both the the input and identification timal experimental design (Atkinson al., 2007), system controller (C). To a decision, both input and Unbehauen, 2004). acontroller feedforward signal or a feedback identification (Zarrop, 1979), or dual control (Filatov and output are processed by passive fault detector (PFDr) (C). To obtain decision, both the input and Unbehauen, Unbehauen, 2004). 2004). output are are(C). processed by aa aaapassive passive fault detector (PFDr) identification (Zarrop, 1979), or dual control (Filatov and output processed by fault detector (PFDr) controller To obtain decision, both the (RG) input and Unbehauen, 2004). that usually consists of residual generator and output are processed by a passive fault detector (PFDr) A block diagram of the AFD architecture is depicted in that usually consists of a residual generator (RG) and Unbehauen, 2004). that usually consists of a residual generator (RG) and A block diagram of the AFD is depicted in output are processed by passive fault detector (PFDr) A block diagram of the monitored AFD architecture architecture is is depicted in that usually consistsblock of aa (DM). residualSince generator (RG) and a decision making the ISG and C Fig. 2. The output of the system (S) processed A block diagram of AFD architecture is depicted in a decision making block (DM). Since the ISG and C a decision making block (DM). Since the ISG and C Fig. 2. The output of the monitored system (S) is processed that usuallymaking consists of a in residual generator (RG) and Fig. 2. The output of the monitored system (S) is processed assumed to be fixed the PFD architecture, the A block diagram of the AFD architecture is depicted in aare decision block (DM). Since the ISG and C in the active fault detector (AFDr) that consists of a residFig. 2. The output of the monitored system (S) is processed are assumed to be fixed in the PFD architecture, the are assumed to beisblock fixed in the PFD architecture, the the active fault detector (AFDr) that consists of a residaare decision making (DM). Since the ISG and C in in the active fault detector (AFDr) that consists of a residmonitored system excited only unintentionally by Fig. 2. The output of the monitored system (S) is processed assumed to be fixed in the PFD architecture, the ual generator (RG), aa decision making block and the active fault detector (AFDr) that consists of a residmonitored system is excited excited only unintentionally by the the in monitored system is only unintentionally by ual generator (RG), decision making block (DM), (DM), and are assumed to beunknown inexternal the PFD architecture, generator (RG), aa decision making block (DM), and known inputs and disturbances. in the active fault detector (AFDr) that consists of a residmonitored system is fixed excited only unintentionally by the the ual ual generator (RG), decision making block (DM), and an auxiliary input signal generator (ASG) that generates known inputs and unknown external disturbances. known inputs and unknown external disturbances. an auxiliary input signal generator (ASG) that generates monitored system is excitedexternal only unintentionally auxiliary input signal generator (ASG) that generates known inputs and unknown disturbances. by the an ual generatorinput (RG), a the decision making block (DM), and excitation signal to system. It is assumed that the an auxiliary signal generator (ASG) that generates Since modeling errors and other uncertainties are inexcitation signal to system. It is assumed that the known inputs anderrors unknown disturbances.are an excitation signalsignal to the thegenerator system. It is assumed that the Since modeling modeling errors and external other uncertainties uncertainties are inin- an an auxiliarysystem input (ASG) that generates Since and other monitored either has unused inputs that can be an excitation signal to the system. It is assumed that the evitable in practice, a PFDr has to take them into account. Since modeling errors and other uncertainties are inmonitored system either has unused inputs that can be monitored system either has unused inputs that can be evitable in practice, a PFDr has to take them into account. an excitation signal to the system. It is assumed that the evitable in practice, a PFDr has to take them into account. utilized for the purpose of AFD or the auxiliary input Since modeling errors andhas other uncertainties are inmonitored system either has unused inputs that can be evitable in practice, a PFDr to take them into account. It is intuitively clear that even if the optimal PFDr is utilized for the purpose of AFD or the auxiliary input utilized for the purpose of AFD or the auxiliary input It is intuitively clear that even if the optimal PFDr is has unused inputs that can be It is intuitively clear that eventoiftake thequality optimal PFDr is monitored utilized forsystem theto purpose of AFD or the auxiliary input signal is added aa either control signal generated by an indepenevitable infor practice, aFD PFDr has them into account. designed a given problem, the of decisions It is intuitively clear that even if the optimal PFDr is signal is added to control signal generated by an indepensignal is added aboth control generated by an independesigned for aa given given FD problem, the quality of decisions decisions utilized for theto purpose ofsignal AFD or the auxiliary input designed for FD problem, the quality of dent controller. In cases it should be ensured that the It is intuitively clear that even if the optimal PFDr is signal is added to a control signal generated by an indepencan be mediocre when the input-output data carries insufdesigned for a given FD problem, the quality of decisions controller. In cases it should be ensured that the dent controller. Inaboth both cases itdrive should bemonitored ensured that the can be be mediocre mediocre whenFD theproblem, input-output data carries carries insuf- dent signal is added to control signal generated by an indepencan when the input-output data insufauxiliary input signal does not the system designed for a given the quality ofmonitored decisions dent controller. In both cases itdrive should bemonitored ensured that the can be mediocre when the input-output data carries insufficient information about a fault present in the auxiliary input signal does not the system auxiliary input signal does not drive the monitored system ficient information about a fault present in the monitored dent controller. In both cases it should be ensured that the ficient information about a fault present in the monitored auxiliary input signal does not drive the monitored system out of desired control performance specifications. can be information mediocre when theainput-output data carries insufficient about fault present in the monitored system. Lack of information could be caused by a small out of desired control performance specifications. out of desired control performance specifications. system. Lack of information could be caused by a small auxiliary input signal does not drive the monitored system system. Lack of information could be caused by a small out of desired control performance specifications. ficient information about a fault present in the by monitored system. Lack of information could be caused a small  This work The devoted to grown was of supported by the Ministry of Education, and out desired control specifications.  The ofliterature literature devotedperformance to AFD AFD has has grown considerably considerably  This system. Lack information could be caused byYouth a small The literature devoted to AFD has grown considerably work was supported by the Ministry of Education, Youth and This work was supported by the Ministry of Education, Youth and  The literature devoted to AFD has grown 1990. considerably since the first works were published around Besides Sports of thewas Czech Republic, project no. of LO1506 and Youth the Czech This work supported by the Ministry Education, and since the first works were published around 1990. Besides Sports of the Czech Republic, project no. LO1506 and the Czech since the first works were published around 1990. Besides Sports of the Czech Republic, project no. LO1506 and the Czech  The literature devoted topublished AFD hasaround grownavailable considerably since the first works were 1990. Besides dozens of papers, there are also few books that Science Foundation, project no.the GA-18-08531S. This work was supported by Ministry of Education, Youth and Sports of the Czech Republic, project no. LO1506 and the Czech dozens of papers, there are also few books available that Science Foundation, project no. GA-18-08531S. dozens of papers, there are also few books available that Science Foundation, project no. GA-18-08531S. since the first works were published around 1990. Besides Sports of the Czech project Republic, no. LO1506 and the Czech dozens of papers, there are also few books available that Science Foundation, no. project GA-18-08531S. dozens of papers, there are also few books available that Science Foundation, project no. GA-18-08531S. 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2018 IFAC 1091 Copyright © 2018 1091 Copyright © under 2018 IFAC IFAC 1091Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 1091 10.1016/j.ifacol.2018.09.726 Copyright © 2018 IFAC 1091

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Variable finite time interval – Since several FD methods employ sequential statistical tests that allow a decision to be postponed until a sufficient information is accumulated, the auxiliary input signal is designed over the corresponding variable length time interval. Infinite time interval – The auxiliary input signal is continuously fed into the monitored system to facilitate FD. The issue with selecting the start and length of the testing interval is avoided at the cost of a potential decrease in control performance.

AFDr Output

ISG

Input

S

RG DM

Decision

ASG Fig. 2. A block diagram of the AFD. deal with particular approaches to AFD (Zhang, 1989; Campbell and Nikoukhah, 2004). Although many survey papers focused on FD methods can be found, they mostly consider the PFD methods and only a couple of recent ones also include a brief part devoted to AFD (Gao et al., 2015; Severson et al., 2015). To the best of our knowledge, no comprehensive up-to-date survey of the AFD methods has been published yet. Therefore, the goal of this paper is to provide an overview of the main research directions in AFD and categorize AFD methods according to their distinctive features. The paper is organized as follows. Section 2 proposes some classification features and provides a high level overview of this survey. The deterministic methods are reviewed in Section 3. The overview of the probabilistic methods is presented in Section 4. Final comments are provided in Section 5. 2. CLASSIFICATION FEATURES Although various classification features can be devised, this survey focuses only on the most distinctive features that allow a clear overview of AFD methods to be provided. The selected classification features cover the model, time horizon, and design goal. The classification of the AFD methods is governed by the first feature that is considered to be the most important in this survey.

Finally, the auxiliary input signal can be designed with the following two distinctive aims. Model discrimination – It is assumed that whatever the fault status of the monitored system is, it remains the same over the whole testing interval. Thus the aim of the auxiliary input signal is to facilitate the discrimination between fault-free model and individual faulty models. Model change detection – The fault status of the monitored system can change during the testing interval. The auxiliary input signal is designed to accentuate a potential change in the fault status. It makes this problem more challenging to solve compared to the model discrimination problem. 3. DETERMINISTIC METHODS The deterministic methods assume that the disturbances acting on the monitored system can be modeled as normbounded deterministic signals. To simplify the classification and keep together methods that are historically connected, this section also includes hybrid methods that make some probabilistic assumptions. 3.1 Integrated controller and detector design

The origins of active approach to FD for deterministic systems are tied to algebraic aspect of linear control staThe most distinctive classification feature concerns the bility (Nett, 1986). The early works (Nett et al., 1988; description of the uncertainties that arise from noises, Jacobson and Nett, 1991) introduced a four parameter initial condition, and modeling errors. The following two controller and analyzed inherent trade-offs of the integroups are considered. grated controller and detector design. A block diagram Deterministic – The uncertainties are assumed to be of the monitored system represented by a trasfer function described as norm bounded signals with known upper matrix S and the four parameter controller represented bounds on norms. The main difference being the type by transfer function matrices C11 , C12 , C21 , and C22 are of the norm that is used and whether the signals are depicted in Fig. 3. The transfer function matrices C21 and C22 represent the controller whereas a residual generator bounded point-wise or over a certain time horizon. Probabilistic – The uncertainties are modelled as stochas- is given by C11 and C12 . The block diagram also specifies tic processes. Similarly to the optimal stochastic control, relationships between the reference signal ry , input to the three basic information processing strategies can be em- system u, output of the system y, and residual signal r. ployed (Bar-Shalom and Tse, 1974). ry u C21 Another important feature is the length of the time interval for which the auxiliary input signal is designed. The following three possibilities occur in AFD. C11 C22 S Fixed finite time interval – The auxiliary input signal is injected into the monitored system during a time interval of a chosen fixed length. The start of such testing interval can be scheduled in advance or triggered by an event. The decisions about faults can be provided either at each time step of the testing interval or only one decision is made during the testing interval.

r

C12

y

Fig. 3. The block diagram of the four parameter controller. The four parameter controller was reformulated within the standard robust control framework (Tyler and Morari,

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1994; Stoustrup et al., 1997). This step not only provided computational means of robust control to be used for integrated design but it also allowed the relationship between the residual generator and controller to be analyzed more easily. It was concluded that the residual generator and controller can be designed independently for the nominal model because a separation principle holds in this case. If an uncertain model is considered, the integrated design is required to achieve the optimal solution. Since it is better to use different norms for measuring control and detection performance, mixed H2 and H∞ design of a modified four parameter controller was presented in Khosrowjerdi et al. (2004). The separation principle was shown to hold for this modified four parameter controller in nominal case as well. Although the integrated design balanced the control and detection performance, the explicit generation of an auxiliary input signal that improves the quality of FD was not considered. The auxiliary input signal was artificially introduced into the modified four parameter controller to facilitate detection of parametric faults (Niemann and Poulsen, 2005; Niemann, 2006b) as depicted in Fig. 4. ˜ N ˜ represent the left The transfer function matrices M, ˜ U ˜ are coprime factors of the monitored system and V, left coprime factors of a nominal controller that has as additional input an auxiliary input signal η. The key idea is to employ Youla-Jabr-Bongiorno-Kuˇcera (YJBK) parametrization and design the auxiliary input signal that manifests itself in the residual signal when a parametric fault occurs while degradation of control performance is acceptable for fault-free model. This idea was extended to include parametric uncertainties (Niemann, 2006a), more than two models (Niemann et al., 2007) and MIMO models (Niemann and Poulsen, 2014). The integrated design was also considered in the context of fault-tolerant control in Niemann (2012). Besides injecting the auxiliary input signal, an idea of modifying the nominal controller to make the residual signal more sensitive to parametric faults or even temporarily destabilize the monitored system was examined (Stoustrup and Niemann, 2010). Although the primary objective was to design the auxiliary input signal with distinctive signature in the residual signal, its evaluation under stochastic assumption was also considered in Poulsen and Niemann (2008). Finally, a survey of the integrated design schemes can be found in Ding (2009).

u

y

S η

˜ −1 V ˜ N

˜ U -

˜ M r

Fig. 4. The block diagram of a modified four parameter controller with the auxiliary input signal.

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3.2 Guaranteed fault diagnosis Another group of deterministic methods focuses on designing an auxiliary input signal that can achieve guaranteed FD under the assumption of bounded disturbances. Besides a few exceptions, the early works used the multiple model framework with only two potential linear models, namely a fault-free model and a faulty model. A proper auxiliary input signal is defined as a signal that enable the discrimination between those two models by making the sets of all possible outputs under the fault-free and faulty model to be disjunctive as depicted in Figure 5. The optimal auxiliary input signal u is found off-line as the proper auxiliary input signal of a minimum norm, and subsequently it is injected into the monitored system in open loop as shown in Figure 6a.

a)

b)

Fig. 5. Sets of possible outputs under fault-free model (black ellipse) and faulty model (gray ellipse) for zero (a) and optimal auxiliary input signal (b). The problem of optimal auxiliary input signal design over finite-time interval for models with polyhedral additive disturbances was introduced in Nikoukhah (1998) and extended later on in Scola et al. (2003). A computationally more tractable design was devised under the assumption that the additive disturbances can be modeled as energy bounded signals (Nikoukhah et al., 2000a). The computational issues related to transformation of the dynamical problem into a large static problem were addressed in Nikoukhah et al. (2002), where the auxiliary input signal was obtained as the solution to a boundary value problem through Riccati type equations. The approach was also extended to infinite-time interval formulation in Nikoukhah et al. (2000b). Since perfect knowledge of fault-free and faulty model is seldom in practice, models with multiplicative disturbances were adopted in Nikoukhah et al. (2001). Contrary to additive disturbances, a proper auxiliary input signal may not even exist for large multiplicative disturbances. A further extensions for this kind of models included use of an a priori information about the initial state and consideration of an additional input signal that is known in advance (Nikoukhah and Campbell, 2005b, 2006). Although the original multiple model setup allowed only abrupt faults to be considered, it was also extended to incipient faults (Nikoukhah et al., 2010). The multiplicative disturbances were considered together with incipient faults in Nikoukhah and Campbell (2008) and computationally modest infinite-time horizon solution was proposed as an approximate solution to a long finite-time horizon. A similar approach was investigated in Ashari et al. (2011) for discrete-time systems with a priori information about initial state. The design of the auxiliary input signal in

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the case of two concurrent incipient faults was dealt with in Fair and Campbell (2009). Sampled-based systems were considered together with multiplicative disturbances for two competing models in Nikoukhah and Campbell (2005a). The setup with more than two models was addressed using a general purpose optimization solver in Campbell et al. (2002). Nonlinear models were also considered in Andjelkovic et al. (2008) where a linearization approach and direct approach were devised. Since the optimal auxiliary input signal can be quite complex, and thus difficult to apply to a real system, an idea of designing a piece-wise constant auxiliary input signal was elaborated in Choe et al. (2009) and the direct approach was further considered in Andjelkovic and Campbell (2011). Most of the above mentioned results can be found in a self-contained book (Campbell and Nikoukhah, 2004).

G

u

S

y

G

w

u

S

y

F a)

All above mentioned methods apply the auxiliary input signal in open loop. Nevertheless, new measurements contain information about the system state that could be beneficial for AFD. The moving horizon technique was applied for discrimination between several models in Raimondo et al. (2013a). The primary optimization criterion was the length of the auxiliary input signal and its norm was used as a secondary design criterion. The use of moving horizon optimization was employed together with constrained zonotopes in Raimondo et al. (2016b). The another closed loop design of auxiliary input signal for model falsification was introduced in Tabatabaeipour (2015). Since the deterministic and probabilistic approaches are in some sense complementary, a hybrid approach that benefits from both approaches was proposed in Scott et al. (2013). It assumes the uniform distribution over zonotopes. The minimum norm auxiliary input signal is designed to ensure guaranteed FD in desired number of step and keeping the probability of correct decision at an earlier time step above a prescribed limit. An extension of this approach to include arbitrary probability distribution over zonotopes is presented in Marseglia et al. (2014).

b)

4. PROBABILISTIC METHODS

Fig. 6. The block diagram of open loop (a) and feedback auxiliary input signal (b): signal generator (G), monitored system (S), feedback (F). The question whether the affine feedback depicted in Figure 6b can improve the guaranteed AFD was initially addressed in Ashari et al. (2012b). The affine feedback F is assumed to be given and since the auxiliary input signal u depends on disturbances in this feedback setup, the test signal w is designed to minimize the norm of u for the worst-case disturbance and fault-free model. It was proven that in such a case the norm of the auxiliary input signal u cannot be reduced uniformly for all possible disturbances compared to the open loop auxiliary input signal. Nevertheless, it was shown in Ashari et al. (2012a) that a suitably chosen affine feedback F can be beneficial if an additive quadratic criterion that involves the auxiliary input signal and the system state is considered. A computationally feasible solution to feedback design with quadratic criterion was elaborated in Ashari et al. (2009). With the advances in mathematical programming solvers, the assumption of point-wise bounded disturbances has received increased attention in the last few years. An open loop auxiliary input signal design for several switching linear models was considered in Scott et al. (2014); Marseglia and Raimondo (2017) with the assumption that the disturbances and initial state belong to zonotopes. The design is translated to a mixed integer quadratic program that can be solved by available software tools. The proposed method was used in the context of faulttolerant control (Raimondo et al., 2013b) and further extended to distributed environment (Raimondo et al., 2016a). The design of a constant or periodic auxiliary input signal was considered for continuous-time systems with zonotopic uncertainties in (Blanchini et al., 2017). The proposed approach relied on finding auxiliary input signal that makes limiting sets of observer-based residual signal disjoint for fault-free and all faulty models.

The probabilistic methods assume that uncertainties can be modeled by random variables with known probability density functions. The methods can broadly be divided into those that use statistical tests without a priori information and methods that fit the Bayesian framework. 4.1 Statistical test If the monitored system is described by a stochastic model, the residual signal is a stochastic process with properties that are supposed to change when a fault occurs. Several FD methods assume that the fault-free model is correct and the change to a faulty model is detected using a sequential statistical test that is performed over the residual signal. The length of the testing interval is not known in advance and depends on the samples of the residual signal because the sequential statistical tests gather samples until a reliable decision can be made. A design of the auxiliary input signal that minimizes average detection delay of the sequential probability ratio test for discriminating between several ARMAX models was introduced in (Zhang, 1989). Later, minimization of the average detection delay while keeping the mean time between false alarms above a chosen threshold was considered in (Kerestecio˘glu and Zarrop, 1991). It was shown that the open loop optimal auxiliary input signal should concentrate all power at a single frequency. A summary can be found in book (Kerestecio˘glu, 1993). The design of closed loop auxiliary input signal using linear feedback for CUSUM test is discussed in (Kerestecio˘glu and Zarrop, 1994). A rather restrictive assumption that the parameters of the faulty model are known completely was partially removed in (Kerestecio˘glu and C ¸ etin, 1997). In particular, the direction of the parameter change is known but its magnitude is not. Further relaxation that allows for direction to also unknown was presented in (Kerestecio˘ glu and C ¸ etin, 2004). A summary of these ideas can be found in the book (Patton et al., 2000).

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4.2 Bayesian approach Bayesian approach provides a systematic way to update prior information with a measured data to obtain posterior information. The predictive probability density functions of the output conditioned by different models play crucial role because they are inherently used as likelihood functions to generate decisions. Therefore, the underlying idea is to use an auxiliary input signal to minimize the overlapping of these predictive probability density functions as depicted in Fig. 7 for the fault-free and faulty model. The overlapping between the pdfs can be measured in various ways that are connected to penalization of incorrect decisions. Note that if all random variable have bounded support, the guaranteed AFD would be possible. However, most contributions assume unbounded supports and thus there is always a nonzero probability of making an incorrect decision. output pdf

An open loop design of auxiliary input signal for model discrimination where detection criterion is minimized while the control criterion is keep below a specified threshold ˇ and vice versa was introduced in Sirok´ y et al. (2011). This idea was extended to nonlinear system that allow for state ˇ feedback linearization (Sirok´ y et al., 2012). A summary of this approach to active fault detection and control was provided in Punˇcoch´aˇr et al. (2015a). Another closed loop auxiliary input signal generator for discrimination between two models was considered in Kim et al. (2013), where the parameters of an affine feedback auxiliary input signal generator were found by solving a linear matrix inequality problem. Polynomial chaos theory was employed in Mesbah et al. (2014) to deal with uncertainty propagation and the problem of discrimination between models was solved. An extension to polynomial models was presented in Streif et al. (2014). Perfect information model y System

zero auxiliary signal

output output pdf

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Estimator

optimal auxiliary signal

ζ

Active fault detector Decision generator d Input signal u generator

Fig. 8. The block diagram of AFD with the perfect state information model.

output

Fig. 7. Probability density functions of output under fault-free and faulty model separated by the optimal auxiliary input signal. Inspired Bayes risk used in pattern recognition (Duda et al., 2000) the open loop auxiliary input signal is designed in Blackmore and Williams (2005) to minimize an overall risk of model selection error. Since the solution is impossible to be found in closed form, the Bhattacharyya bound is used in the case of two competing models. The extension to arbitrary number of models was considered in Blackmore and Williams (2006) and multiple model changes over a finite-time horizon was investigated in Blackmore et al. (2008). Lately, the Hellinger distance closely related to the Bhattacharyya bound was utilized (Mesbah et al., 2014). The three basic information processing strategies (IPSs) for the auxiliary input signal design were investigated ˇ in Simandl and Herejt (2003). Similarly to Kerestecio˘glu (1993) it was assumed that the detector is given and the aim was to improve the terminal decision. A general problem of AFD and control was formulated as a functional optimization problem with a general criterion over a finiteˇ time horizon in Simandl et al. (2005a). The key tool is use of dynamic programming and it approximate versions. An approximate solution based on rolling horizon was ˇ considered in Simandl et al. (2005b). Several special cases and their particular approximate solutions were elaborated ˇ in Simandl and Punˇcoch´ aˇr (2009). The problem of auxiliary input design for model discrimination was considered in (Punˇcoch´ aˇr et al., 2009). An compact overview of special ˇ cases was provided in Simandl et al. (2011).

The optimal input signal generator for finite-time horizon is generally a time-varying system that is difficult to design and implement. Therefore, infinite-time horizon was of great interest because a simpler asymptotic solution can ˇ be found (Punˇcoch´aˇr and Simandl, 2014). The first step was problem reformulation that resulted in a perfect state information problem. The block diagram in Fig. 8 shows the monitored system complemented with an estimator that provides informative statistics ζ. These statistics are subsequently used to generate decision d and auxiliary input signal u. Nonlinear models with directly observable continuous part of the state was assumed and the auxiliary input signal generator was designed over infinite-time horizon using approximate value iteration algorithm. The algorithm of the generalized policy iteration was employed to reduce the computational demands of auxiliary input signal generator in Punˇcoch´aˇr et al. (2015b). The extension to linear models with noisy measurements of the continuous part of the state was done using informative statistics in Punˇcoch´aˇr et al. (2015). The machine learning techniques were used to train the auxiliary input ˇ signal generator on simulated data in Skach et al. (2016). This approach was extended later on to nonlinear models ˇ in Skach et al. (2017). 5. CONCLUSION Although the presented survey is not by any means exhaustive due to space limitations, it clearly shows that the number of contributions in the area of AFD has increased significantly in recent years. Despite all the theoretical advances in AFD, applications to real-world systems are still limited in the literature. It is clearly due to higher

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theoretical complexity and increased computational demands of AFD. Nonetheless, several academic studies have already demonstrated potential benefits of AFD. In Zhang (1989), the AFD in a stock tank and a pre-concentrator of an anhydrous caustic soda plant is applied to decrease average detection delay. The open loop auxiliary input signal for improving detection of typical aircraft faults was designed in Blackmore and Williams (2006). The AFD of a stuck air mixing damper of an air handling unit was ˇ considered in Sirok´ y et al. (2012). A simple AFD within a comprehensive methodology for robust fault detection and fault tolerant control of wind turbines can be found in Casau et al. (2015). The survey also helped to identify the main trends and future challenges in the area of active fault diagnosis. There is a strong emphasis on formulating the active fault diagnosis as a problem that can be solved using ready-to-use numerical solvers. In the case of probabilistic approaches the similar trend can be observed together with use of advanced machine learning techniques. Finally, the distributive active fault detection for large scale system with only few initial results for guaranteed fault diagnosis certainly poses a challenging problem for future research. REFERENCES Andjelkovic, I. and Campbell, S.L. (2011). Direct Optimization Determination of Auxiliary Test Signals for Linear Problems with Model Uncertainty. In Proceedings of the IEEE Conference on Decision and Control, 909– 914. Orlando, FL, USA. Andjelkovic, I., Sweetingham, K., and Campbell, S.L. (2008). Active Fault Detection in Nonlinear Systems Using Auxiliary Signals. In Proceedings of 2008 American Control Conference, 2142–2147. Seattle, WA, USA. Ashari, A.E., Nikoukhah, R., and Campbell, S.L. (2009). Asymptotic Behavior and Solution Approximation of Active Robust Fault Detection for Closed-Loop Systems. In Proceedings of Joint 48th Conference on Decision and Control and 28th Chinese Control Conference, 1026–1031. Shanghai, China. Ashari, A.E., Nikoukhah, R., and Campbell, S.L. (2011). Auxiliary signal design for robust active fault detection of linear discrete-time systems. Automatica, 47(9), 1887–1895. Ashari, A.E., Nikoukhah, R., and Campbell, S.L. (2012a). Active Robust Fault Detection in Closed-Loop Systems: Quadratic Optimization Approach. IEEE Transactions on Automatic Control, 57(10), 2532–2544. Ashari, A.E., Nikoukhah, R., and Campbell, S.L. (2012b). Effects of feedback on active fault detection. Automatica, 48(5), 866–872. Atkinson, A.C., Donev, A.N., and Tobias, R.D. (2007). Optimum Experimental Designs, with SAS. Oxford University Press, New York, NY, USA. Bar-Shalom, Y. and Tse, E. (1974). Dual Effects, Certainty Equivalence and Separation in Stochastic Control. IEEE Transactions on Automatic Control, 19(5), 494–500. Blackmore, L., Rajamanoharan, S., and Williams, B.C. (2008). Active Estimation for Jump Markov Linear Systems. IEEE Transactions on Automatic Control, 53(10), 2223–2236.

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