Fault detection and isolation of smart actuators using bond graphs and external models

ARTICLE IN PRESS

Control Engineering Practice 13 (2005) 159–175

Fault detection and isolation of smart actuators using bond graphs and external models B. Ould Bouamama*, K. Medjaher, M. Bayart, A.K. Samantaray, B. Conrard Ecole Polytechnique Universitaire de Lille, LAGIS, UMR CNRS 8021, Cit!e Scientifique, 59655 Villeneuve d’Ascq Cedex, France Received 9 August 2002; accepted 3 March 2004

Abstract This paper deals with fault detection and isolation (FDI) of smart actuators combining the benefits of bond graph modelling with external models. An external model is a generic method which can be used to specify and verify the functional specifications of smart equipment. It uses the concept of services provided to the user and the organization of operating modes as a state graph with the logical conditions (depending on available services) required to move from one operating mode to another. One drawback of the external model is that it describes the system in terms of functions without taking into account the dynamics of the equipment. In addition, the availability of the services is not determined by FDI procedures. The integration of the bond graph methodology as a graphical and multi-disciplinary modelling tool quantifies the services by associating one or more bond graph elements to each service. Furthermore, the causal and structural properties of the bond graph methodology can be used to design FDI algorithms (i.e. the generation of fault indicators) to determine the availability of the services. This information is used to determine the transition conditions in the state graph. This technique has been applied to monitor a complex pneumatic servo-positioner. r 2004 Elsevier Ltd. All rights reserved. Keywords: Intelligent instrumentation; Bond graphs; Fault detection and isolation (FDI); Smart actuators; External models

1. Introduction Nowadays, the automation of large-scale complex industrial processes requires the broadening of the traditional scope, which has so far been limited to real-time control algorithms. The design of these systems, now, has to take into account additional functionality such as maintenance, management and supervision facilities. This evolution of the user’s need together with the technological advances made in the fields of the sensors, the actuators and the field bus communication, lead to the concept of distributed intelligent systems where smart sensors and actuators integrate with the information system of the automated process. The local processing powers of the smart sensors and actuators increase the quality of the processed information. This additional information extends the range of the possibilities of the global *Corresponding author. Tel.: +33-3-20-337139; fax: +33-3-20337189. E-mail address: [email protected] (B. Ould Bouamama). 0967-0661/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2004.03.003

system. In addition to the control possibilities, smart equipment creates a large number of data items which can be communicated to the various operators, who manage it all along its life cycle. Considerable work has been carried out to provide the functional, the behavioral, the object-based, the internal, and the external models of smart equipment (Robert, 1993; Albus & Proctor, 1996; Staroswiecki & Bayart, 1996; Maffezzoni, Ferrarini, & Carpanzano, 1998). The external model, which uses the concept of services offered to the users and organizations based on operating modes, leads to a generic model description, which can be used to specify and verify the functional specifications of smart instruments and hybrid systems (Bayart & Lemaire, 1997). Contrary to the black box models, which carry no information about the component’s behavior in different operating situations (whether normal or faulty), external models describe smart components from the point of view of the user who receives services and can use them in different operating modes. At any time, the smart actuator, as a system, carries out a coherent subset of missions. Each subset is called a User Operating Mode (USOM). The logical conditions

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to move from one USOM to another are described by a state graph termed ‘‘USOM graph management’’. This graph defines the logical sequences, of operating steps, in which some missions start and some others end. The achievement of the mission depends on the services provided by the components of the system. The lowlevel components are directly connected to the physical process, and they provide the elementary services, whereas higher-level services and components are realized by proper aggregation of the low-level components. Decisions to trigger alarms are taken according to the availability of the elementary system components. The drawback of the external model is that it describes the services and the USOMs in terms of the component’s functions, without taking into account the component’s physical, and dynamic behavior. This leads to some ambiguity. Furthermore, the availability of the resources is not determined by a fault detection and isolation (FDI) procedure. This is why, the bond graph methodology as a graphical modelling language is a convenient and useful complimentary tool for obtaining both the behavioral and the diagnostic models. Moreover, the causal properties of the bond graph methodology can help to design FDI algorithms, i.e. the generation of fault indicators (Ould Bouamama, Dauphin-Tanguy, Staroswiecki, & Amo-Bravo, 2000). In this way, by associating bond graph models, it becomes possible to obtain the behavioral knowledge of the actuators, and to improve their monitoring; and consequently, to ensure an improved safety standard. In this paper, the external model concepts are briefly presented. A short introduction to the bond graph modelling in the interest of actuator monitoring is given. Then the method to associate the bond graph description to the external model, in order to complete the equipment description, is explained. In Section 4, the use of the causal and the structural properties of bond graph method for FDI design is discussed. The application of the proposed method to a smart actuator that is composed of a control valve, a pneumatic servo-motor, and a positioner, followed by some simulation results, are presented in the final section.

ment of these goals is obtained as long as the actuators are able to correctly process the operating requests communicated to them from the higher levels. In that sense, any actuator fault or malfunction may have heavy consequences on the safety of the operators, or on the security and the productivity of the process itself. FDI techniques are developed in order to avoid such situations, as far as possible (Patton, Frank, & Clark, 1989). The actuators are installed mainly in harsh environments, which influence their predicted lifetime. Their malfunctions or failures cause long-term process disturbances, or even sometimes force the shut down of the installation. Moreover, such faults may influence the final product quality and lead to potential economic losses. For fault prevention or prediction, the real-time diagnosis of final control elements may be applied. Continuously or periodically performed, diagnosis of key actuators reduces the maintenance costs. The introduction of remote real-time diagnosis of actuators can reduce the periodical inspection costs. In such cases, the inspections and repairs of the actuators are undertaken only if necessary. A functional model of a general smart actuator is given in Fig. 1. 2.1. Bond graphs for smart actuator monitoring Actuators are complex and non-linear systems characterized by the coupling of different forms of energy (mechanical, electrical, pneumatic, etc.). They are used in processes where the environment is usually harsh and complex. This puts them at high risk to failures. The improvement of the actuator’s safety is essentially based on the FDI procedures. Different model-based methods for the FDI procedures have been developed, depending on the kind of knowledge used to describe the process (transfer function, state equation, structural model, etc.) (Patton et al., 1989; Isermann, 1994; Staroswiecki & Comtet-Varga, 2001). Monitorability analysis (ability to detect and to isolate the faults which may affect the

to Communicate

2. Smart actuators For the smart actuators, the primary functions are concerned with the power transformation (from an electrical, pneumatic, or hydraulic field into a mechanical one, for example) and their performances can be characterized by the shorter response time, better position, the peak speed, the precision, the stability, control decoupling, the power efficiency, and the dead band measurement (Staroswiecki & Bayart, 1996), etc. However, actuators are generally used in a larger system with higher-level production goals. The correct achieve-

to Input

Process

to Act

to Validate

to Manage

to Elaborate

to Manage data base

to Decide

Fig. 1. Functional structure of a smart actuator.

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system) is based on the fault signatures deduced from the analytical redundancy relations (ARRs). System modelling is an important and difficult step in the generation of the ARRs. Because of the multi-domain energies involved in the actuators, the bond graph methodology as a multi-disciplinary and unified modelling language proves a convenient tool for the given purpose. The bond graph method, as a modelling tool, provides many possibilities. It allows both a causal and a behavioral system analysis. On the one hand, the bond graph can be refined by adding graphically more elements, like thermal losses, inertia, or storage effects, without having to start all over again. The bond graph methodology is widely used for modelling purposes, but only few works deal with the monitoring of complex systems using bond graph models. The bond graph modelling methodology allows for the generation of not only a behavioral model (Thoma, 1975; Karnopp, Margolis, & Rosenberg, 1990; Thoma & Ould Bouamama, 2000), but also it can be used for the structural analysis of the system. Owing to its causal properties, a bond graph model can greatly contribute to the design and the development of the process monitoring application. Bond graph models have been used to study the structural control properties (observability, controllability, etc.) (Sueur & Dauphin-Tanguy, 1991), monitorability analysis (Tagina, Cassar, Dauphin-Tanguy, & Staroswiecki, 1995), and they have also been used for qualitative reasoning (Linkens & Wang, 1995). Yet, on the other hand, the causal properties of the bond graph language enable the modeller to resolve the algorithmic level of modelling (e.g. singularities, invertibility, etc.) by assigning adequate derivative or integral causality in the formulation stage, even before the detailed equations have been derived. The innovative interest is the use of only one representation (the bond graph) for both the modelling, and for the monitoring of the system. One of these depends on the use of quantitative dynamic models, which leads to the determination of ARRs, and allows the real-time monitoring of the actuator. These ARRs are classically generated from the model using a method that will be briefly exposed later. Contrary to other classical model based methods (Staroswiecki, 1989), the ARRs can be directly and systematically determined from the bond graph model. However, in the industrial supervision tasks, human operators consider the running system in the terms of its functions. In order to give the operator, or the industrial user a clear view of the system’s functional organization, and to allow the user to estimate the system’s ability to achieve its goal, a functional modelling tool using an external model is used along with the bond graph model. How bond graph modelling can be used to complement external modelling is shown in the next section.

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3. External model in a bond graph sense Since bond graph theory has been presented extensively in the literature (Paynter, 1961; Karnopp et al., 1990; Thoma & Ould Bouamama, 2000), the essential goals of this paper will be to deal with those aspects of bond graphs in the sense of component model developed by Staroswiecki and Bayart (1996), and the use of bond graph modelling for smart actuator monitoring. From an external point of view, an automation system provides services to users. Users request services for specific purposes, for example, storage, energy transformation, etc. The external model describes the device from the point of view of the services, it is able to provide, to external entities (operators, other field instruments, computers, and so on). It introduces the notion of the services, the user operating modes, and the versions of the services. 3.1. Services A service is defined as a procedure whose execution results in the modification of at least one data item in the instrument’s database, and/or at least one signal on its output interface. Each service, Si ; is defined by a quintuplet of sets: consumed variables (Ci ), produced variables ðPi Þ; conditions of activation ðCai Þ; data processing ðdpi Þ; and resources ðRi Þ: Si ¼ /Ci ; Pi ; dpi ; Ri ; Cai S:

ð1Þ

A service is an operational entity corresponding to a physical object of the hardware architecture. The service has a hardware existence, which means; for a service to be achieved, it needs one or more resources of the process. Besides, a service possesses a functional identity, i.e., every service is associated with an activity. Any given service performs only one task at a time. The set of services S; which are offered by an equipment, is finite: S ¼ fsi j iA½1yng:

ð2Þ

In the sense of bond graph, the services correspond to some set of bond graph elements such as C for storage component (tank, boiler, etc.), TF and GY elements for transformation of energy from one domain to another, R for resistive element (pipe, valve, etc.), Se for effort source (pressure pump, voltage generator, etc.), and Sf for flow source (flow pump, current generator, etc.). *

The services offered by the sensors (to measure) are delivered by effort ðDeÞ and flow ðDf Þ detectors. The sensors can of course provide other services, such as keeping memory of minimum and maximum levels, providing alarms, etc. In this paper, only those services that provide a signal as a function of a measured variable are considered.

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162 *

*

*

*

The services provided by the equipment of sources of energy (mechanical motor, potential or kinetic energy of fluid and hydraulic pump, etc.) are represented by effort ðSeÞ or flow ðSf Þ sources. The services provided by the information system (controllers, positioners, etc.) are presented as block diagrams since it is assumed that these systems are not energetic, but only carry the signals. In the bond graph models, these systems are modelled by information bonds represented by full arrows. The consumed and produced variables are the power variables (the effort-flow pair). The request is represented by an information bond in the bond graph. The services offered by the functional role of the equipment (storage, transformation, transport, etc.) are represented by the bond graph elements R; C; and TF ; etc.

Thus, in a bond graph, the services as defined in the external model represent a set of elements necessary for the physical realization of the corresponding service. The execution of these services by a request can be controlled intentionally by an external entity (order coming from an operator or automaton) or by an internal one (from the process). The availability of each service offered by a component is modelled by a boolean variable associated with an internal or an external request. This boolean variable is provided by the decision procedure implemented in the FDI system. For example, when a sensor is faulty, the corresponding ‘‘measure’’ service will not be available. A hardware failure implies the unavailability of some services and could bring some missions to a halt. That is where the proposed modelling method allows the analysis of the consequences of a failure on the availability of the services offered by a system. 3.2. User Operating Mode

3.2.1. Versions of services According to the resource states decided by the supervision of the process, several versions, V ; of the services (nominal and degraded) may be designed. As an example, if the set of services (set of bond graph elements) is fS1 ; S2 ; S3 ; y; Sn g and fV1 ; V2 ; V3 ; y; Vn g is the corresponding set of versions, then the running operating mode is represented as MBG ¼ fS1 ðV1 Þ; S2 ðV2 Þ; y; Sn ðVn Þg:

ð4Þ

3.3. Operating mode management At any given time, the process (or the equipment) runs in a nominal operating mode represented by the corresponding bond graph model. Let MBG be the set of the bond graph models, and MBGj be the bond graph model corresponding to the jth operating mode, and n be the total number of the operating modes. MBG ¼ fMBG1 ; MBG2 ; y; MBGl ; y; MBGn g:

ð5Þ

Let b be the set of transitions: b ¼ fbij g;

ð6Þ

where bij indicates the required condition to move from MBGi to MBGj : The transition condition is represented by the boolean variable bij Af0; 1g: It is considered that all mode change conditions (noted Cm Þ are correctly defined, i.e., 8MBGi ði ¼ 1; nÞAMBG; ( j

As discussed previously, in order to avoid simultaneous attempt at performing incompatible services by the system (initialization and production services, for example), the set of services is organized into coherent subsets, called USOMs. Each USOM contains at least one service, and each service belongs to at least one USOM. At any given time, and according to the technical specifications, a system carries out a coherent subset of missions. Each of these subsets is called an operating mode. Each USOMi can be associated to a bond graph model noted MBGi ; where i is an index for the operating mode. USOMi -MBGi :

regime, starting mode, etc. An automaton, to change the USOMs, specifies the USOM (initial USOM, and the others) and the conditions to change from one USOM to another. From this description, a formal specification of the intelligent equipment can be obtained.

ð3Þ

USOMs are given by a general classification of the operating modes of industrial devices: off-operation, configuration, manual or automatic mode, nominal

such that bij ACm and bij a0:

ð7Þ

The running operating mode can be represented as MBGj ¼ fsj1 ðVj1 Þ; sj2 ðVj2 Þ; y; sji ðVji Þ; y; sjm ðVjm Þg ¼ fSj g=sji ði ¼ 1; kÞ ¼ ffCji g; fRji g; fTFji g; fGYji g; fSeji g; fSfji g; fDeji g; fDfji gg;

ð8Þ

where m is the number of services (functions) offered by the smart actuator to accomplish the given operating mode associated to the MBGj ; and k is the number of bond graph elements necessary for the physical performance of the ith service. Each USOM (or bond graph model) MBGj contains at least one service, i.e., one bond graph element (otherwise, the physical process does not exist). In fact, the USOM’s organization can be described as a deterministic automaton.

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SUPERVISION

MBG1

FDI algorithms

u_ref

b13

163

b12 Information system

b41 MBG3

b34

b32

MBG2

Control system

b23

Ym = {De, Df }

Sensors De,Df

u

Actuators MSe, MS f

b24

Se, Sf

MBG4

Y

MSe, MSf

Process MBG

X

Bond graph model

Fig. 2. USOM graph in the bond graph sense.

Fig. 3. Bond graph representation of a supervised system.

Let G ¼ fMBG; b; Cm g be the USOM graph. It can be seen as a deterministic transition if 8MBGi AMBG; ( bij ACm ; ( at the most one transition with MBGi as origin mode and bij as label. MBG is the set of bond graph models. The sets of the USOMs and the transition conditions can be represented as shown in Fig. 2. The whole set of the available services and its set of associated versions are established for a given USOM by an internal function called ‘‘operating mode management’’. The request for a change of an operating mode must indicate the destination mode. However, in presence of faults, because of safety, some couples (origin, destination) may not be allowed. As an example, the request ‘‘automatic control mode’’ would be rejected as long as the service ‘‘measure’’ (provided by the sensors) is unavailable.

An information bond is shown as a full arrow (Fig. 3) on the bond graph and represents the transmitted signal by a sensor, or by the control algorithm block. 4.1.1. Formulation of the bond graph based FDI system A system, S; may be described by a set of constraints, F (which represents the system model); a set of variables, Z; and a set of parameters y: Each variable may be known, or unknown. S ¼ SðF ; Z; yÞ: 1. Constraints. The constraints, F ; can be seen as any relation which links the system variables and the parameters. It has to include information about the structure ðFJ Þ; the behavior ðFB Þ; the measurement ðFY Þ; the control system ðFC Þ; and the controlled sources ðFA Þ: F ¼ fFJ ; FB ; FY ; FC ; FA g:

4. FDI of smart actuators using bond graphs

*

4.1. Bond graph representation of a monitored system A monitored system can be represented as shown in Fig. 3. It consists mainly of two parts: a bond graph model and an information system. The bond graph model represents the energetic part of the system. It consists of the process plant and the set of actuators. The actuators are modelled by sources (of efforts and/or of flows). The sources can be simple (Se; Sf ), i.e., pump, heater, pressure supply, etc. or modulated ðMSe; MSf Þ; i.e., controlled by an external signal provided by the controllers or the user. The sensors and the control system (PID controllers, on–off controllers, etc.) form the information system. In the first system (energetic one), the power exchanged is represented by a half arrow (a bond), which labels a power variable (effort and flow). In the second system (information system), the exchanged power is negligible, and it is represented by an information bond similar to the block diagrams.

Structural equations FJ : They represent a set of conservation laws (of mass, energy, etc.) and/or equilibrium equations. They are deduced from the junction equations. The number of FJ equations is equal to the sum of the number of ‘‘0-junction’’, ‘‘1junction’’, and the ‘‘2-port’’ elements: TransFormers (TF) and GYrators (GY). FJ ¼ fFJ0 g,fFJ1 g,fFTF g,fFGY g;

*

where FJ ARnj ; and nj is the number of junctions. Behavioral equations FB : The physical laws expressing how the energy is transformed are mathematically described by the behavior model. In a bond graph model, they describe the physical phenomena which are represented in lumped-parameter bond graph elements (R; C; and I). These equations are called ‘‘constitutive laws’’. FB ¼ fFC g,fFI g,fFR g; ne

ð9Þ

where FB AR ; and ne is the total number of power bonds in the bond graph R; C; and I elements.

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Measurement model FY : It describes the measurements which are available. It expresses the way in which the sensors transform some state variables of the process into output signals which can be used for FDI and control purposes. In bond graph models, the sensors are represented as detectors of flow ðDf Þ and detectors of effort ðDeÞ: FY ¼ fFDe g,fFDf g;

*

where FY ARns ; and ns is the number of sensors. Control algorithm model FC : It describes the controller’s algorithm. The inputs are the set points and the sensor outputs. The output acts on the actuator which is represented as a modulated effort (MSe) or flow (MSf) source. FC ðu; uref ; Ym ; yreg Þ ¼ 0;

ð10Þ

where yreg is a set of controller parameters (gain, sampling time, etc.); u; uref ; and Ym represent, respectively, the controller output, the set point value, and the sensor outputs. FC ARnr and nr is the number of regulators. Unlike structural and behavioral equations, which use the power variables (effort, and flow) as input–output variables; measurement and control algorithms use the information signals as variables. *

Modulated sources model FA : It describes the modulated power sources (controlled pump, controlled current generator, etc.). The input signals, u; are provided by the controllers, and the outputs are the regulated variables, MSe and MSf : FA1 ðMSf ; uÞ ¼ 0;

FA2 ðMSe; uÞ ¼ 0;

where FA ARna ; and na is the number of the modulated sources. Thus, F ARnj þne þns þna þnr : 2. Variables. The set of constraints, F ; map to a set of variables, Z: known ðKÞ and unknown ðX Þ: Z ¼ X ,K:

ð11Þ

The unknown variables are the power variables (flow and effort) that label the bonds. The vector X containing all the power variables is X ðtÞ ¼ fe1 ðtÞ; f1 ðtÞg,fe2 ðtÞ; f2 ðtÞg?,fenc ðtÞ; fnc ðtÞg; ð12Þ where nc is the number of power bonds in the components (I; R; and C). Then, the dimension of the vector X is 2nc : X AR2nc : The set, K; of known variables contains the control variables, u; the variables whose values are measured by the sensors, Ym ; and the supervision parameters (such as uref ). K ¼ MSe,MSf ,Se,Sf ,Ym :

ð13Þ

MSe and MSf represent, respectively, a modulated effort source variable and a modulated flow source variable (controlled pump, controlled heater, etc.). They depend on the output, u; of the control system: MSe ¼ FC1 ðuÞ;

MSf ¼ FC2 ðuÞ:

ð14Þ

Output signal, u; depends on the control algorithm function, FC ; the set point value; and the sensor outputs. u is supposed to be known and calculated by the control algorithm, FC (Eq. (10)). Se and Sf ; represent, respectively, a constant effort source and a constant flow source (weight, flow pump, voltage generator, etc.). Ym is the sensor ðDe; Df Þ output vector of dimension ns ; Ym ARns : Finally, KARl ; where l ¼ ns þ na ; and na is the number of sources (including modulated and simple sources). From a technological point of view, na is equal to the number of actuators. Thus, the dimension of vector Z is l þ 2nc : ZARðlþ2nc Þ : 3. Parameters. yARp is the vector of parameters. In a bond graph model, it is associated with the characteristics of R; C; and I elements, i.e., the flow coefficient, the capacity (which can be variable), etc. 4.2. Generation of fault indicators 4.2.1. Structural analysis Let s be a binary relation between F and Z: sðfi ; zj Þ ¼ 1 means that the constraint fi AF applies to the variable zj AZ (s ¼ 0 otherwise). The structure leads to a bipartite graph whose binary incidence matrix represents the links between the known and the unknown variables, and the constraints. It is known that any finite-dimensional bipartite graph can be canonically decomposed into three subgraphs (which represent sub-systems), namely, the under ðZ > F Þ; the just ðZ ¼ F Þ; and the over ðZoF Þ constrained sub-systems. It has been shown by Declerck (1991), that the over-constrained sub-system is the monitorable part of the overall system; since, it is the only one to exhibit some redundancy, which can be expressed by ARRs. Indeed, the model of an overconstrained sub-system contains more constraints, F ; than unknowns, X ; so that some compatibility conditions must hold for this system to have solutions. The ARRs are just these compatibility conditions, which are expressed by the existence of relationships between the known variables (Staroswiecki, Coquempot, & Cassar, 1990; Patton et al., 1989). Thus, an ARR is a relationship between a set of known variables of the form f ðKÞ ¼ 0; where K is the set of known variables.

ð15Þ

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From the bond graph modelling view point, the ARR will be written under the following form: f ðDe; Df ; Se; Sf ; MSe; MSf ; ym Þ ¼ 0;

ð16Þ

where ym is a set of known parameters. A residual, r; is formed from each ARR: r  f ðKÞ ¼ 0:

ð17Þ

The problem of finding the ARRs is equivalent to the problem of finding the over-determined sub-systems in the structure of the system. An over-determined subsystem is characterized by a complete matching with respect to X only, so that the unknown variables can be computed in different ways by using a subset of the constraints and the known variables K: The ARRs, which are used for the residual computation, can be derived from the monitorable system using several approaches. The first one decomposes the generation process into two stages: the computation of the solution with regard to the unknown variables, followed by substitution of this solution to the remaining relations. The second approach aims to eliminate the set of unknown variables in a single operation. In the linear case (which is rare in real systems), the projection operation is used for this purpose, and that leads to the parity space techniques (Chow & Willsky, 1984). The structural approach, which is a method based on the resolution of the equations of the model, does not provide all the solutions for monitoring. Indeed, if the example of a linear system is taken: if the full rank hypothesis is not verified even when the system is just determined structurally, it is possible by linear combination to eliminate the unknown variables and to generate the expression of ARRs. The elimination method (called the theory of elimination working on polynomial rings) (Cox, Little, & O’Shea, 1992; Guernez, Petitot, Cassar, & Staroswiecki, 1997) gives the same results as the resolution method would do, plus the relations that the structural analysis does not detect. This method does not solve, but eliminates the unknown variables. The drawback of the elimination method is that the set of constraints used must be given in the form of a polynomial function (which is seldom true in the real systems). 4.2.2. Generation of ARRs from a bond graph model In the cited model based FDI methods, the model is given under complicated equations. Sometimes, physical meanings for unknown variables, and the location of sensors are not clearly shown in the process model. The constraint equations are not deduced in a systematic way. It is not trivial in the real systems to write the model under a form x’ ¼ f ðx; u; yÞ: Using a bond graph model allows to deal with the enormous amount of equations describing the dynamic behavior of different phenomena which occur in the system. Due to its

165

graphical representation, it allows (independent of the physical nature of the studied system), to display the exchange of power in a system, including storage and transformation, and to show explicitly the sensor placement in the real process. Furthermore, the ARRs can be deduced directly from bond graph models using the systematic methodologies (Ould Bouamama, Samantaray, Staroswiecki, & Dauphin-Tanguy, 2003; Tagina, 1995). A method to generate ARRs from a bond graph model using covering causal paths is proposed by Tagina (1995). The goal is to study all the causal paths relating the considered junction to the sources and the sensors. In this paper, this method is extended, to a closed-loop system, with systematic treatment of the equations. 4.2.3. Algorithm for generation of ARRs The bond graph model of the monitored process is generated by using preferred derivative causality. The integral causality is recommended for engineering simulation in order to avoid the numerical problems arising out of differentiation. However, the derivative causality is more suitable in ARR expression to avoid influence of the initial conditions. For this task, the detectors are dualized: the effort detectors are transformed into flow detectors and vice versa (Ould Bouamama et al., 2003). When a detector cannot be dualized, a material redundancy (two sensors measure the same variable, or the measurement from the sensor can be obtained as a function of the measurements from other sensors, etc.) in the considered junction can be detected. Globally the algorithm works as follows: 1. Choose a junction. 2. Find the corresponding ARR by writing the structural equation, FJ ; of the given junction. The main idea is to associate two indicators, knv and unv; with each constraint relation (constitutive equation). knv is the number of known variables and unv is the number of unknown variables. knv þ unv ¼ nvar ;

ð18Þ

where nvar ¼ 2nc þ ns is the total number of variables. nc is the number of plant items in a bond graph sense, i.e., number of elements C; I; R; TF ; and GY : The goal is to decrease unv and increase knv so that the ARR can be obtained when unv becomes zero, or alternately stated, when nvar ¼ knv:

ð19Þ

The ARR is generated, then, by the algorithm when unv ¼ 0: The unknown variables in symbolic format are directly deduced from the bond graph model using the covering causal path rules. It is important to recall that a causal path is a sequence of links which have the same causal orientation. Depending on the causality, the followed

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variable is either the effort or the flow. To change the variable, one can pass through a GY passive element, or return through an active element (I; C; or R). 3. Consider the next junction. 4. If the second ARR is independent of the others derived already, then keep it, else pass onto another junction. 5. Repeat the step (2) until all the junctions are considered, and the distinct signatures are obtained. 4.3. Detection and isolation procedure Faults can occur in the process itself; or in the controllers, sensors and actuators (e.g. a change in the parameters of some component). A residual will be coherent with the model of the system if it is null or less than a chosen threshold e: The coherence of each residual is tested by using a direct comparison between its value and a threshold e: This test, applied on the set of residuals, ri ; leads to a binary coherence vector, C ¼ ½c1 c2 ? cn : Each component, ci ; of C is obtained using the following rule: ci ¼ 1;

if jri j > e; otherwise; ci ¼ 0:

ð20Þ

The structure of a residual is a binary vector, V ; that expresses which measures influence the residual’s value.


V ðiÞ ¼ 1

if the residual contains the ith measure;

V ðiÞ ¼ 0

otherwise: ð21Þ

The set of ARRs generates a binary matrix. The rows of this matrix are called failure signatures. A ‘‘1’’ entry in the ith row and the jth column of the matrix indicates that the residual, rj ; is sensitive to the ith fault (the fault can thus be detected). The aim of the isolation step is to provide a list of elements which are faulty. In the case of single failure hypothesis, all the information contained in the signatures and in the coherence vector can be exploited. The detection procedure compares the coherence vector with all the signatures and determines which of the signatures are the closest. The Hamming distance is used to calculate the isolation ability of a set of ARRs. Each of the component’s failure signature is compared with the others: the larger the distance between the signatures, the better is the isolation index. The designed FDI system is associated with detection and isolation quality levels; evaluated by the false alarms and missed detection probabilities, the detection delays, the possibilities to find out which faults really occurred among the different possible ones, etc. (Basseville, Benveniste, Moustakides, & Rougee, 1987).

5. Application to a smart pneumatic valve 5.1. Description of the system In order to show the practicality of the advocated approach, the methodology is applied to a smart actuator (Fig. 4), which represents a pneumatic valve chosen as the DAMADICS benchmark (Bartys & De Las Heras, 2002). The actuator consists mainly of a control valve, a servo-motor, and a positioner with the pressure supply system. The control valve (5), which can be a disk, ball, or plug, corrects the flow in the pipe (6) to maintain a desired flow set point. The driving force of the fluid is the difference between the upstream pressure ðSe3 :P1 Þ and the downstream pressure ðSe4 :P2 Þ: These pressures are measured by the sensors De3 and De2 : The movement of the valve is accomplished by a servo-motor which consists of a pneumatic chamber (1), a spring (3), a diaphragm (2), and a stem (4). This system is a compressible (air) fluid powered device, in which the fluid stored in (1) acts upon the flexible diaphragm and the spring to provide linear motion of the servo-motor stem. The positioner belongs to the group of microprocessor based smart devices. It is applied to eliminate the control-valve stem mis-positioning produced by the external or internal sources such as friction, pressure unbalance, hydrodynamic forces, etc. However, the communication facilities are limited only to two current (4–20 mA) signalling loops. One loop provides a set point value, Cv ref ; and the second one corresponds to the actual valve stem position. The algorithm is based on PID controller algorithms given by Bartys and De Las Heras (2002). The positioner output, Cv; acts on the Air pressure supply

Se2 :Pat

Cv

Rve

Se1 :Pal

.

m Rvs

Ps

De1:Psm

1

Ps

Cv_ref

2

∫

3 6

4 x

De3:P1m

5

xm

Positioner

. Df1 : xm

De2:P2m

Se3 :P1

. Df 2 : mm

Se4 :P2

Fig. 4. Scheme of the actuator.

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air pressure supply system in order to deliver the desired pressure, Ps : The control signal, Cv; is defined by the valve stem position, xm ; the controlled fluid mass flow, m ’ m ; the pressure in the pneumatic chamber, Psm ; and the reference value, Cv ref : Values for xm ; m ’ m ; Psm ; and Cv ref are provided respectively by the sensors Df1 ; Df2 ; De1 ; and the user. The air mass flow delivered by the air supply system depends on the state of the supply ðRvs Þ; and the exhaust ðRve Þ orifices. According to a control signal delivered by the positioner, Cv ref ; the pneumatic chamber (1) is exposed to supply pressure, Se1 :Pal ; or is exposed to exhaust (zero reference pressure), Se2 :Pat : Faults can occur in the process itself, or in the sensors and actuators; for example, a change in the parameters of some component. The FDI specifications are intended to list those faults which have to be considered. In total, 19 faults fFlt1yFlt19g are distinguished (Koj, 1998) in the assembly consisting of: the control valve, the pneumatic servo-motor, and the positioner. These faults are classified into following four groups: Control valve faults fFlt1yFlt7g; pneumatic servo-motor faults fFlt8yFlt11g; positioner faults fFlt12yFlt14g; and general/external faults fFlt15yFlt19g: 5.2. External bond graph models of the smart actuator 5.2.1. Bond graph model The bond graph model in normal operating regime is given in Fig. 5.

167

Two forms of energy are modelled here: hydraulic and mechanical. The thermal energy is not modelled, since an isothermal case is assumed. The power variables used for the mechanical energy are the force–velocity ðF ; xÞ ’ pair. The hydraulic (or pneumatic) energy is modelled using pressure–mass flow ðP; mÞ ’ power variables in a pseudo-bond graph form and pressure–volume flow ’ power variables are used in a true bond graph ðP; VÞ form. True bond graphs have the physical power as the product of effort and flow. More precisely, it is the instantaneous power transferred along a bond or entering a port. Such bond graphs are well suited for mechanical, hydraulic, and electrical systems. Pseudobond graphs are used in those cases, where the product of effort and flow is not power. True bond graphs are more convenient for modelling incompressible fluid flow. In fact, the selection of power variables in process engineering plants, including thermo-fluid, biological, and chemical engineering; is not trivial, and depends on the type of the process. More details are available in Thoma and Ould Bouamama (2000) and Ould Bouamama (2003). However, in pseudo bond graphs, the rules for causalities and signs remain valid for the entire graph. The inlet air mass flow, m ’ 16 ; from the supply system to the pneumatic chamber is given by the constitutive equation of modulated restrictors, R:Cvs and, R:Cve: These restrictors are modulated by the positioner output signal, Cv: When a fluid is moving through a restrictor, a

Fig. 5. Bond graph model of the smart actuator in normal operating mode.

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non-linear relationship exists relating the mass flow rate, m; ’ to the pressure drop, DP; across the restriction and the boolean variable, b1 : The compressibility effect and the air mass storage in the pneumatic chamber are modelled by the two-port C-field ðCc Þ: This multiport allows to calculate the pressure, which depends on the mass and the volume of air.  n P1 V2 ¼ P1 V1n ¼ P2 V2n ) P2 V1 which leads to 1 ’ ðm P’ ¼ ’  rVÞ; ½ðrV Þ=ðnPÞ

ð22Þ

where n is the polytropic index, r is the fluid density, and V is the volume of the collapsible chamber (which is variable). This C-field has, consequently, one ðP; mÞ ’ ’ port. This implies that the total port and one ðP; VÞ energy depends on m and V : The bond graph of this element represents a sophisticated version of models of compressibility effects in air servomechanism, in which the volume is variable. In the bond graph model (Fig. 5), the bond number 14 models the storage of mass, and the bond number 13 models the storage of volume. The pressure is the same in both these bonds, ðP13 ¼ P14 Þ: The constitutive relation for the Cc field is given by #    " R m K Kra P14 ’ dt R ; ð23Þ ¼ V’ dt P13 K Kra where ra is the air density and K ¼ 1=½ðra V Þ=ðnPÞ: In integral causality, the effort variable (pressure) is calculated by the conservation rules at the ‘‘03 ’’ junction and the Cc field (Eq. (24)). The corresponding block diagram (for n ¼ 1; or an isothermal case) is given

in Fig. 6. Z ne14 ðf14  ra f13 Þ dt þ e14 ð0Þ: e14 ¼ ra V

ð24Þ

The transformation of the pneumatic energy into mechanical energy leading to the servo-motor’s stem displacement is modelled by the transformer, TF ; whose modulus, Ae; is the cross-sectional area of the diaphragm. The dynamics of the servo-motor’s stem is modelled by the conservation of energy represented by the ‘‘13 -junction’’ linked to the R; C; and I elements. I:M; Kv; Ks; Kd; Se5 :  Mg; Fvc; and Fp account respectively for the inertia due to the mass of the stem, the force due to the friction, the force due to the stiffness of the spring, the force due to the elasticity of the diaphragm, the gravity, the vena contracta force (which depends on the pressure of the controlled fluid, diameter of the plug, etc.), and the active force. The flow through the control valve is modelled by the modulated Relement, R:Rvn: The resistance value of R:Rvn is the coefficient of hydraulic losses which depends on the stem position, x: 5.3. Missions and versions Based on the objectives fixed by the technical specifications for the actuator, a non-exhaustive list of missions and associated services, i.e., bond graph elements, has been defined (Table 1). 5.4. Operating mode management of the smart actuator As described previously, the operating mode management, given in Fig. 7, allows the operator to be informed about the conditions under which the transition from

Fig. 6. Bond graph and block diagram of the Cc -field.

ARTICLE IN PRESS B. Ould Bouamama et al. / Control Engineering Practice 13 (2005) 159–175 Table 1 List of missions and services for the smart actuator No. 1 2 3 4 5 6 7 8 9 10 11 12 13

Missions

BG elements

To supply the pneumatic chamber with air pressure To store the air in the pneumatic chamber To transform a pneumatic energy into mechanical one To provide a motion of servo-motor stem To control the servo-motor stem position To control the flow of the fluid To provide a fluid in the pipe To measure the pressure in the pneumatic chamber To measure the inlet pressure in the pipe To measure the outlet pressure in the pipe To measure the servo-motor stem position To measure the control valve flow To maintain equipment

Se1 ; Cv; R:Cvs; R:Cve; Se2 C:Cc TF TF ; C:1=Kd; TF; C:1=Ks; R:Kv Df1 ; Positioner Df2 ; Positioner Se3 ; R:Rvn De1 De3 De2 Df1 Df2 —

Nominal regime MBG1 1,2,3,4,5,6,7,8,9,10,11,12 b01 b13

b10 b12

Manual mode MBG3 1,2,3,4,7,8,9,10,11,12

b30

Stop mode MBG0 13

b20 b23

Cavitation mode MBG2 1,2,3,4,5,8,9,10,11,12

Fig. 7. Operating mode management of the smart actuator.

one USOM to another can be performed. In the current approach, and as a complement to the external modelling methodology, each USOM is represented by a bond graph model. It helps the operators to understand the basic physical principles, to visualize the physical phenomena of the process, and to quantify each effect. Furthermore, the bond graph model is used to generate the alarm decisions, and consequently the availability or unavailability of the services (sensor, actuator, or process faults). The smart actuator corresponds to the following nonexhaustive list of bond graph models. MBG0, stop operating mode: In this mode, the actuator carries out the mission no. 13. Since the process is at rest (no power), the bond graph model is of no interest in this mode.

169

MBG1, nominal regime: The missions carried out in the nominal regime are nos. 1–12. The bond graph model corresponding to this regime has been given in Fig. 5. MBG2, manual operating mode: In this mode, the smart actuator carries out mission nos. 1–4 and 7–12. MBG3, cavitation operating mode: Cavitation occurs if the fluid outlet pressure drops below the vapor pressure of the liquid. As the vapor cavities collapse, noise is generated and damage can occur. Cavitation damage produces a rough, pitted, cinder-like surface. The corresponding bond graph model is given in Fig. 8. Both flashing and cavitation limit the flow of the liquid through the valve. During flashing, and cavitation, bubbles begin to form in the flow stream when the pressure drops below the vapor pressure of the liquid. The bubble formation at the vena contracta restricts the amount of liquid that can be forced through the valve. This is commonly known as the choked-flow condition. The flow no longer increases with decreases in downstream pressure. Choked-flow can result in severe damage to the valve; and desired flow requirements may not be reached, which impedes the process. The difference of pressure at the choked flow condition is a function of the flow geometry of the control valve. The experimentally determined coefficient denoted, Km; is used to define the point of choked flow condition (Fig. 9). The valve recovery coefficient, Km; and critical pressure ratio of the liquid, rc ; can be determined from the tables and curves given by the valve manufacturer (Issakovitch, Loguinov, & Popadko, 1983). The main missions carried out in this mode are nos. 1–5 and 8–12. Such organization increases the operational safety of the system and forbids the access to those USOMs for which the meaningful services cannot be run properly. Consider, for instance, the current USOM: MBG1 and assume that service 2 (to store the air in the pneumatic chamber modelled by the two port Cc -field) is not available because of a leakage. Thus, according to Fig. 7, the transition to USOM: MBG3 or MBG2 will be rejected, since service 2 belongs to both these operating modes. Only the transition to MBG0 is allowed. The alarm generation in the smart actuator’s FDI system is developed in the next section.

5.5. Self-monitoring of the smart actuator 5.5.1. Variables The vector, X ; containing all the power variables is deduced from the bond graph model (Fig. 5). X ¼ ½e17 f17 e20 f20 e13 f13 e14 f14 e8 f8 e9 f9 e10 f10 e11 f11 e4 f4 :

ð25Þ

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170

Fig. 8. Bond graph model in cavitation mode.

*

FJ0 : Junction ‘‘0’’ linked to R; C; or I elements. Structural equation for the junction ‘‘01 ’’ is given by

F1 : f14  f17 þ f20 ¼ 0: *

ð27Þ

FJ1 : Equations for the ‘‘1’’ junctions ‘‘11 ’’, ‘‘12 ’’, ‘‘13 ’’, and ‘‘14 ’’ are given as

F2 : e17  Se1 þ e14 ¼ 0; F3 : e20  e14 þ Se2 ¼ 0; pd 2 ðSe3  Se4 Þ 4Km  e8  e9  e10  e11 þ e12 þ Se5 ¼ 0; F5 :  e4 þ Se3  Se4 ¼ 0:

F4 : 

*

Fig. 9. Determination of the coefficient Km:

ð28Þ

Transformers: TF : Ae; and MTF

F6 : e12 ¼ Aee13 ¼ Aee14 ; F7 : f13 ¼ Aef12 ; In this case, the number of components, nc ; is equal to 9. The set, K; of known variables is K ¼ ½Se1 Se2 Se3 Se4 De1 De2 De3 Df1 Df2 Cv:

ð26Þ

5.5.2. Constraints F ¼ fFJ ; FB ; FY ; FC ; FA g: (1) Structural equations, FJ : FJ ¼ fFJ0 g,fFJ1 g, fFTF g,fFGY g:

pd 2 F8 : e7 ¼  ðSe3  Se4 Þ: 4Km

ð29Þ

(2) Behavioral equations, FB : pffiffiffiffiffiffiffiffi F9 : R : RCVs -f17 ¼ jCvj je17 jb1 ; pffiffiffiffiffiffiffiffi F10 : R : RCVe -f20 ¼ jCvj je20 jb%1 ; Z 1 ðf14  ra f13 Þ dt; F11 : C : Cc-e14 ¼ ½ðra V Þ=ðne14 Þ

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Z 1 e8 dt; F12 : I : M-f8 ¼ M F13 : R : Kv-e9 ¼ Kvf9 ; Z F14 : C : 1=Ks-e10 ¼ Ks f10 dt; Z F15 : C : 1=Kd-e11 ¼ Kd f11 dt; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Se3  Se4 F16 : R : Rvn-f4 ¼ 100KvðxÞ : rf

f17 is calculated by covering the causal path 16–17: pffiffiffiffiffiffiffiffi f17 ¼ jCvj je17 jb1 : ð36Þ e17 is unknown, and is derived from ‘‘11 ’’ junction’s constitutive equation, e17  Se1 þ e14 ¼ 0; with e14 ¼ Psm : Thus, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f17 ¼ jCvj jSe1  Psm jb1 : ð37Þ ð30Þ

(3) Measurement equations, FY : Z F17 : Df1 -xm ¼ f8 dt ) x’ m ¼ f8 ; F18 : Df2 -m ’ m ¼ f4 ; F19 : De1 -Psm ¼ e14 ; F20 : De3 -P1m ¼ Se3 ; F21 : De4 -P2m ¼ Se4 :

171

ð31Þ

(4) P-controller algorithm model FC :

Now the second variable, f17 ; is known. Then knv ¼ knv þ 1 ¼ 2; and unv ¼ unv  1 ¼ 1: f20 is obtained from the constitutive relation Eq. (30):F10 and ‘‘12 ’’ junction’s equation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f20 ¼ jCvj jPsm  Se2 jb%1 : ð38Þ The third variable, f20 ; is known. Then: knv ¼ knv þ 1 ¼ 3; and unv ¼ unv  1 ¼ 0: Finally, the first residual, r1 ; can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P’ sm ra ðAexm þ V0 Þ r1 ¼ þ ra Aex’ m  jCvj jSe1  Psm jb1 nPsm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  jCvj jPsm  Se2 j b%1 ¼ r1 ðDe1 ; Se1 ; Se2 ; Df1 ; Cc f Þ;

F22 : Positioner: Cv ¼ Kp ðCv ref  12 þ ð3=pÞ

ð39Þ

arcsinððxm =0:0381Þ  ð12ÞÞ:
1 if Cv > 0; F23 : Logic block: b1 ¼ 0 otherwise; and b%1 ¼ 1  b1 : ð32Þ

where Cc f represents the leakage in the pneumatic chamber. Indeed, if the conservation law modeled by ‘‘01 ’’ junction does not hold, there may be a leakage. The density of air, ra ; can be calculated as R R m f14 dt þ mð0Þ ’ dt þ mð0Þ : ð40Þ ¼ ra ¼ V V

5.5.3. Generation of ARRs For the junction ‘‘01 ’’, the conservation relation is f14  f16 þ f15 ¼ 0: Covering causal paths 16–17, and 15–20, one obtains:

In integral causality, the residual can be easily calculated from Z ra V e14 ¼ ðf14  ra f13 Þ dt þ e14 ð0Þ: ð41Þ ne14

f14  f17 þ f20 ¼ 0:

In Eq. (41), unknown variables are e14 ; e14 ð0Þ; f13 ; f14 ; V ; and ra : By covering different causal paths, and using measurement equations, a residual can be written as:  Z 2  ram V f14m  r1 ¼ Psm   f13m dt  Psm ð0Þ; ð42Þ nPsm ram

ð33Þ

The number of variables nvar ¼ 3 (f14 ; f15 ; and f16 ). Initially, the number of unknown variables (f14 ; f17 ; and f20 ), unv ¼ 3: The number of known variables knv ¼ 0 ) nvar ¼ unv þ knv ¼ 3: f14 is given by the constitutive equation of Cc element (Eq. (30):F11). In derivative causality, e’14 ra ðAexm þ V0 Þ f14 ¼ þ ra f13 : ð34Þ ne14 e14 and f13 are unknown, they are determined from the measurement equation (Eq. (31):F19) and TF equation (Eq. (29):F7). f13 ¼ Aex’ m ;

e14 ¼ Psm :

Finally, the first variable f14 is expressed in the terms of all known variables as P’ sm ra ðAexm þ V0 Þ f14 ¼ þ ra Aex’ m : ð35Þ nPsm Thus, knv ¼ knv þ 1 ¼ 1; and unv ¼ unv  1 ¼ 2:

where

R

f14m dt þ mð0Þ ; Aexm þ V0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ jCvj jSe1  Psm jb1  jCvj jPsm  Se2 jb%1 ;

e14m ¼ Psm ; f14m and

V ¼ Aexm þ V0 ;

ram ¼

f13 ¼ Aex’ m :

The residuals are used in derivative form because it is considered that the initial conditions are unknown. However, this leads to some numerical problems; especially, if the measurements are noisy. In integral causality, residual evaluation needs values for the initial conditions, but the residuals are not sensitive to the measurement noise, so their numerical evaluation is more accurate.

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172

It is easy to see from covering causal path analysis that the next residual deduced from the ‘‘11 ’’ junction relationship has the same signature as r1 : Indeed, for this junction (equation given by Eq. (28):F2 relation, e17  Se1 þ e14 ¼ 0), the unknown variables are e17 and e14 : Their determination uses the same causal paths as those used for r1 : Because this residual has the same fault signature as r1 (i.e., it is sensitive to the same faults), its analysis is abandoned and another junction is considered. The variables involved in Eq. (28):F4 (structural equation of junction ‘‘13 ’’), are e8 ¼ M f’8 ¼ M x. m ; e9 ¼ Kvf9 ¼ Kvx’ m ; Z e10 ¼ Ks f9 dt ¼ Ksxm þ e10 ð0Þ; Z e11 ¼ Kd f9 dt ¼ Kdxm þ e11 ð0Þ; e12 ¼ Psm Ae: Thus the following Eq. (28):F4:

residual

is

obtained

from

pd 2 ðSe3  Se4 Þ  Kvx’ m 4Km  Ksxm þ Psm Ae  Kdxm  M x. m

r2 ¼ Se5 

¼ r2 ðDf1 ; Se3 ; Se4 ; De1 ; TF ; KsÞ;

ð43Þ

where Ks is the stiffness of the spring. Change of this parameter from its nominal value leads to spring failure. Using integral causality, the relation for r2 can be written as Z  2 1 pd  ðSe3  Se4 Þ þ Mg þ Kvx’ m r2 ¼ x’ m þ M 4Km  þ Ksxm þ Kdxm  Psm Ae dt  x’ m ð0Þ: ð44Þ From ‘‘14 ’’ junction, the third residual is obtained as  2 rf Df2 Se3 þ Se4 r3 ¼ 10; 000 Kvðxm Þ ¼ r3 ðDf2 ; Se3 ; Se4 ; Df1 ; Cavit; RvnÞ;

ð45Þ

where Cavit represents a cavitation failure, and the fault which may affect the valve (such as blockage) is denoted as Rvn: From ‘‘03 ’’ and ‘‘04 ’’ junctions, these simple residuals are obtained: r4 ¼ Se3  De3 ; r5 ¼ Se4  De2 : ð46Þ *

Monitoring of the positioner

Let Cvm be the measured value of the positioner output. Thus the following residual can be obtained: 

x  3 m 1 1 r6 ¼ Cvm  Kp Cv ref  2 þ arcsin  p 0:0381 2 ¼ r6 ðDf1 ; pos f Þ;

ð47Þ

where pos f is a positioner failure, Kp is the gain of the controller, and xm is the valve position measured by the sensor Df1 : 5.5.4. Decision procedure The signature of a fault is the subset of the residuals which are influenced by the fault. For the system having six residual relations, the resulting signature matrix is given in Table 2. The 15 row vectors contain at least one non-zero element, and thus all the faults can be detected. Based on the binary vector of each component given in Table 2, all faults can be detected and isolated except the fault which may affect Se1 ; Se2 ; and Cc; or Cavit; and Rvn: Faults in these components are not isolable. Indeed, the components Se1 ; Se2 and Cc have the same fault signature, V ¼ ½1 0 0 0 0 0: Similarly, faults Cavit and Rvn have the same fault signature, V ¼ ½0 0 1 0 0 0: This signature of faults is used to generate alarms for the operating mode management of the smart actuator, given in Fig. 7. For example, if the service ‘‘To store the air in the pneumatic chamber’’, associated with the bond graph two-port element Cc (Table 1), is not available because of a pneumatic chamber leakage, only the residual r1 will be sensitive. Thus, the request ‘‘nominal regime mode’’ should be rejected as long as this service (provided by the decision procedure) is not available. Thus, according to Fig. 7, the transition to USOM: MBG3 or MBG2 will be rejected since service-2 is an element of both these operating modes; so only the transition to MBG0 will be allowed. 5.5.5. Simulation results To analyze the sensitivity of the residuals to the different faults, a number of simulation studies have been performed (chamber leakage, diaphragm perforation, positioner fault, sensor faults, etc.). Two of them are shown here in Figs. 10 and 11.

Table 2 Fault signatures of the monitored system

Se1 Se2 Se3 Se4 De1 De2 De3 Df1 Df2 Cc f pos f TF Ks Cavit Rvn

r1

r2

r3

r4

r5

r6

1 1 0 0 1 0 0 1 0 1 0 1 0 0 0

0 0 1 1 1 0 0 1 0 0 0 1 1 0 0

0 0 1 1 0 0 0 1 1 0 0 0 0 1 1

0 0 1 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 1 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 1 0 0 0 0

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In normal operation, the residuals should be zero in the ideal case (no modeling error, no uncertainties, no measurement noise). However, this is not practical, since modeling errors and noises are always present. A simple moving average over a 10 s time window has been used to filter out the effect of the noises in the residuals. Finally, taking into account the relative values of the residuals with respect to the corresponding signals shows that the precision is quite good.

173

The decision procedure, which has been applied is rather simple; the residuals have been compared with a threshold equal to three times the standard deviation, under the normal operation hypothesis. If this value exceeds the threshold, it is considered that a fault has occurred. Two kinds of faults are simulated: sensor faults and plant faults. It is assumed that the sensor signals are noisy; a white noise is thus added on each measurement (2% of each output measurement). The

Fig. 10. Residuals response to servo-motor’s diaphragm perforation.

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Fig. 11. The inlet pressure sensor fault.

fault signatures for the servo-motor’s diaphragm perforation and for the inlet pressure sensor fault ðDe3 Þ are given as examples. These faults are simulated from 4 s up to 10 s: The servo-motor’s diaphragm perforation is detected by the residuals r1 and r2 (Fig. 10); whereas the inlet pressure sensor fault ðDe3 Þ is detected only by the residual r4 (Fig. 11). As shown by the simulation results (r1 behavior), the introduced process fault (which can be considered as a disturbance) is automatically compensated by the valve position control system.

6. Conclusion The smart actuator, as an intelligent equipment, carries out a coherent subset of missions, called USOMs. The logical conditions to move from one USOM to another depends on the services provided by the system’s components. All low-level components are directly connected to the physical process and they

provide elementary services. In the presence of faults, the alarm decision is taken according to the availability of the elementary system components. In the real industrial supervision tasks, human operators consider the running system in the terms of its functions. Thus, the external model, which describes an intelligent equipment from the users’ point of view, is used. However, the drawback of this functional modelling is that it describes the services and USOMs in terms of the functions without taking into account its dynamics. Furthermore, the availability of the resources is not physically identified by using the FDI procedures. This is why, in this paper, the bond graph methodology as a graphical modelling language is used. A methodology based on the causal and structural properties of the bond graph models is developed to design FDI algorithms (i.e., the generation of fault indicators), which can provide the list of faulty (not available) components (services) to the USOM. The proposed methodology, incorporating both the external and bond

ARTICLE IN PRESS B. Ould Bouamama et al. / Control Engineering Practice 13 (2005) 159–175

graph models, is applied to a currently acting device (a control valve with a pneumatic servo-motor) in an industrial application.

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