Fault Detection and Diagnosis of Distributed Parameter Systems Based on Sensor Networks and Artificial Intelligence

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1 Fault Detecton and Dagnoss of Dstrbuted Parameter Systems Based on Sensor Networks and Artfcal Intellgence CONSTANTIN VOLOSENCU Department of Automatcs and Appled Informatcs Poltehnca Unversty of Tmsoara Bd. V. Parvan nr. 2 Tmsoara ROMANIA constantn.volosencu@aut.upt.ro Abstract: - Ths paper presents some approaches on the new applcatons n fault estmaton, detecton and dagnoss emerged from three powerful concepts: theory of dstrbuted parameter systems, appled to large and complex physcal processes, artfcal ntellgence, wth ts tool adaptve-network-based fuzzy nference and the ntellgent wreless ad-hoc sensor networks. Sensor networks have large and successful applcatons n the real world. They may be placed n the areas of dstrbuted parameter systems, to be seen as a dstrbuted measurng sensor for the physcal varables. Usng sensor networks multvarable estmaton technques may be appled n dstrbuted parameter systems. Fault detecton and dagnoss n dstrbuted parameter systems become more easly and more performng usng these concepts. The paper presents some applcatons n fault detecton and dagnoss based on the adaptve-network-based fuzzy nference, allows treatment of large and complex systems wth many varables by learnng and extrapolaton. Key-Words: - Fault detecton and dagnoss, wreless sensor networks, non-lnear system dentfcaton, dstrbuted parameter systems, adaptve-network-based fuzzy nference, multvarable estmaton technques, auto-regresson, heat dstrbuton, partal dfferental equatons. 1 Introducton The supervson, fault detecton and fault dagnoss are mportant to mprove relablty, safety and effcency n mantenance of ndustral processes, seen lke lump parameter systems. In the last decades these methods were appled wth success n electrcal drves, power plants, arcrafts or chemcal plants. The classcal approaches n the feld of fault detecton and dagnoss are usng analytcal methods [1] of system dentfcaton based on lnear models as: parameter estmaton, state space observers and party equatons. For non-lnear systems dentfcaton usage of the artfcal ntellgence concepts as fuzzy logc, neural networks and the adaptve-network-based fuzzy nference [2] represents powerful tools n system dentfcaton. The dstrbuted parameter systems are n practce more complex processes, descrbed usng partal dfferental equatons, such as the propagaton of sound or heat, electrostatc phenomena, flud flows, elastcty. Processes consdered wth varables dstrbuted n space may be watched usng modern wreless ntellgent sensor networks [3, 4]. The commercal sensor networks have sensors for all knd of varables from physcal dstrbuted parameters systems as: temperature, pressure, radaton, lght ntensty, acceleraton and other. Some classcal methods are developed for dentfcaton of the general dstrbuted parameter system dentfcaton [5, 6]. Recent approaches n the above feld are reported n [7, 8]. The author has developed and publshed several theores related of usng multvarable estmaton technques based on artfcal ntellgence for the dentfcaton of dstrbuted parameter systems [9, 10, 11], n the new context of ntellgent sensor networks as a dstrbuted sensor. As a dstrbuted tool they may be used to measure tme varables n the complex dstrbuted parameter systems. In ths applcaton, wth a large feld of nterest n scence and engneerng, all the above topcs contrbute, convergng to the same objectve dentfcaton, detecton and dagnoss of fault n dstrbuted parameter systems. The paper presents a general theory, wth 2 estmaton algorthms and a general method for fault detecton and dagnoss based on those estmaton algorthms. ISSN: ISBN:

2 2 Mathematcal Models A dstrbuted parameter system has a general mathematcal model n contnuous tme as a partal dfferental equaton, as an example for a parabolc case, as: θ θ θ f (,, 2,...) = 0 t ζ ζ 2 (1) wth varables θ(ζ, t), dependng on tme t and on space ζ, where ζ s x for one axs, (x, y) for two axs or (x, y, z) for three axs. In the practcal applcaton case studes lmts and ntal condtons of the equaton (1) are mposed: θ( 0, t) = θ ζ, t [ 0, T],θ(ζ, 0) = 0, ζ [ 0, l], (2) 0 θ( l, t) = θ, t [, T] Boundary condtons for the equaton (2) are: when the varable value the boundary s specfed we are speakng about Drchlet condtons and when the varable flux and transfer coeffcent are specfed there are Neumann condtons. A system wth fnte dfferences may be assocated to the equaton (1). For ths purpose the space S s dvded nto small peces of dmenson l p : l p = l / n (3) In each small pece S p, =1,,n of the space S the varable θ could be measured at each moment t k, usng a sensor from the sensor network, n a characterstc pont P (ζ ), of coordnate ζ. Let t be θ k the varable value n the pont P (ζ ) at the moment t k. It s a general known method to approxmate the dervatves of a varable wth small varatons. In the equaton wth partal dervatves there are dervatves of frst order, n tme, and dervatves of frst and second order n space. So, theoretcally, we may approxmate the varable dervatve n tme wth a small varaton n tme, wth the followng relaton: k+ 1 θ - θ = t t - t θ k+ 1 ζl 0 k k (4) The frst and the second dervatves n space may be approxmated wth small varatons n space to obtan the followng relatons: θ = x θ k - θ l p k 2 k k k -1 θ θ+ 1-2θ + _ 1, θ 2 = 2 x lp (5) We may consder the varable s measured as samples at equal tme ntervals wth the value: h = t k+1 - t k (6) called sample perod, n a samplng procedure, wth a dgtal equpment. Combnng the equatons (4, 5, 6) n the equaton (1) a system wth dfferences results: k k k+ 1 k+ 1 f (θ,θ,θ,θ ) 0 (7) -1-1 = Takng account of equaton (7) s obvous that two estmaton algorthms may be developed as follows. We may use several estmaton algorthms based on dscrete models of the partal dervatve equaton. Estmaton algorthm 1. It estmates the value of + 1 the varable θ k at the moment t k+1, measurng the k k k values of the varables θ , θ at the anteror moment t k : k k k ( ) k+ θ = f1 θ-1+ 1, θ 1 (8) Ths s a multvarable estmaton algorthm, based on the adjacent nodes [9]. Estmaton algorthm 2. It estmates the value of +1 the varable θ k at the moment t k+1, measurng the k k-1 k-2 k-3 values of the same varable θ, but at four anteror moments t k, t k-1, t k-2 and t k-3. k k-1 k-2 k 3 ( ) k+ 1 - θ = f 2 θ,θ (9) Ths s an autoregressve algorthm. Estmator model. The estmator s a non-lnear one, descrbed by the functon y=f(u 1, u 2, u 3, u 4 ), usng the adaptve-network-based fuzzy nference [2, 10]. Its general structure s presented n Fg. 1. It has four nputs u 1, u 2, u 3 and u 4 and one output y. The ANFIS procedure may use a hybrd learnng algorthm to dentfy the membershp functon parameters of sngle-output, Sugeno type fuzzy nference system. A combnaton of least-squares and backpropagaton gradent descent methods may be used for tranng membershp functon ISSN: ISBN:

3 parameters, modelng a gven set of nput/output data. Fg. 1. The estmator nput-output general structure In the nference method and may be mplemented wth product or mnmum, or wth maxmum or summaton, mplcaton wth product or mnmum and aggregaton wth maxmum or arthmetc meda. The frst layer s the nput layer. The second layer represents the nput membershp or fuzzfcaton layer. The neurons represent fuzzy sets used n the antecedents of fuzzy rules determne the membershp degree of the nput. The actvaton functon represents the membershp functons. The 3 rd layer represents the fuzzy rule base layer. Each neuron corresponds to a sngle fuzzy rule from the rule base. The nference s n ths case the sum-prod nference method, the conjuncton of the rule antecedents beng made wth product. The weghts of the 3 rd and 4 th layers are the normalzed degree of confdence of the correspondng fuzzy rules. These weghts are obtaned by tranng n the learnng process. The 4 th layer represents the output membershp functon. The actvaton functon s the output membershp functon. The 5 th layer represents the defuzzfcaton layer, wth sngle output, and the defuzzfcaton method may be the centre of gravty. 3 Sensor network capabltes A Crossbow sensor network was used n practce. It has the followng components: a starter kt, a MICA2 2,4 GHz wreless module, and an MTS320 sensor board. Ther nodes are 2 MICAz 2,4 GHz modules, wth 2 sensors MTS400, whch are measurng temperature, humdty, pressure, ambent lght ntensty; 1 MICAz 2,4 GHz wth 2 sensors MTS310 and 1 module MICAz 2,4 GHz workng as a central node when t s connected through the UB port. A gateway MIB520 for node programmng and a data acquston board MDA320 wth 8 analogue channels are provded. The network has the followng software: MoteVew for hstory sensor network montorzaton and real tme graphcs and MoteWorks for nod programmng n MesC language. The user nterface allows some facltes, as: admnstraton, searchng, connectons optons and so on. Ths modern wreless sensor network has multple measurng capabltes. So, t can measure temperature, humdty, lght ntensty or acceleraton on 2 axes. For these knd of physcal varables the mathematcal models are as follows. For temperature: 2 2 θ θ θ = a + Q( x, y, t) 2 2 t + x y (10) where Q s the tme varable source of heatng,postoned n space and θ s the temperature. For lght ntensty: I E( x) = 2 2, h + x Φ S E =, Φ = I = I. α (11) 2 S r where I s the lumnous ntensty of the lght source, at the dstance x and hgh h, as a measure of the source ntensty as seen by the eye, E s the lumnance at the specfc pont, defned as a rato, wth Φ representng the flux that strkes a tny area S, calculated consderng a sphercal surface of radus r, wth α representng the sold angle. For acceleraton: a dv dv x y dx dy =, ay =, vx =, v (12) y dt dt dt dt x = where the above notatons represents the acceleraton a x, a y, the speed v x, v y and the space x, y on two axs for an object of the mass m, under a force F. Some characterstcs measured for the sensor network are presented n Fg. 2. Fg. 2. Temperature am humdty transent characterstcs measured wth the sensor network ISSN: ISBN:

4 A sensor network s made by hhundred or thousands of ad-hoc tny sensor nodes spread across the space S. Sensor nodes collaborate among themselves, and the sensor network provdes nformaton anytme, by collectng, processng, analysng and dssemnatng temperature measured data. Sensor network s workng as a dstrbuted sensor. The constructve and functonal representaton of a sensor network s presented n Fg. 3. [ u u ] T u = (14) u n and f s the non-lnear estmaton functon of nonlnear regresson, n s the order of the regresson. By conventon all the components u 1 (t),,u n (t) of the multvarable tme seres u(t) are assumed to be zero mean. The functon f may be estmated n case that the tme seres u(t), u(t-1),, u(t-n) s known (recursve parameter estmaton), ether predct future value n case that the functon f and past values u(t-1),, u(t-n) are known (AR predcton). The method uses the tme seres of measured data provded by each sensor and reles on an (auto)- regressve multvarable predctor placed n base statons as t s presented n Fg. 4. Fg. 3. A sensor network wth moble access The sensor networks have dfferent structures, as the star networks (pont-to-pont), whch are networks n whch all sensors are transmttng drectly wth a central data collecton pont. New nodes automatcally are detected and ncorporated. The number and the place pont of the de sensor nodes may be dscussed accordng to the desred accuracy of estmaton [10, 11] usng dfferent dentfcaton methods. 4 Estmaton and Detecton Structure The present paper consders two multvarable estmaton models, one as regressve (8) and the second as an autoregressve (9), both based on nonlnear ANFIS estmator, whch can effcently approxmate the tme evoluton n space of the measured values provded by each and every sensor wthn the coverage area. An estmaton model descrbes the evoluton of a varable measured over the same sample perod as a non-lnear functon of past evolutons. Ths knd of systems evolves due to ts non-lnear memory", generatng nternal dynamcs. The estmaton model defnton s: y( t) = f ( u1 ( t),..., un ( t)) (13) where u(t) s a vector of the seres under nvestgaton (n our case s the seres of values measured by the sensors from the network): Fg. 4. Estmaton and detecton structure The prncple s the followng: the sensor nodes wll be dentfed by comparng ther output values θ(t) wth the values y(t) predcted usng past/present values provded by the same sensors or adjacent sensors (adj). After ths ntalzaton, at every nstant tme t the estmated values are computed relyng only on past values θ A (t-1),, θ A (0) and both parameter estmaton and predcton are used as n the followng steps. Frst the parameters of the functon f are estmated usng tranng from measured values wth a tranng algorthm as backpropagaton for example. After that, the present values θ A ( t) measured by the sensor nodes may be compared wth ther estmated values y(t) by computng the errors: e A ( t) = θ A ( t) - y( t) (15) If these errors are hgher than the thresholds ε A at the sensor measurng pont a fault occurs. Here, based on a database contanng the known models, on a knowledge-based system we may see the case as a mult-agent system, whch can do crtcs, learnng and changes, takng decson based on node ISSN: ISBN:

5 analyss from network topology. Two parameters can nfluence the decson: the type of data measured by sensors and the computng lmtatons. Because both of them are a pror known an off-lne methodology s proposed. Realstc values are between 3 and 6.We are choosng 4 as n equatons (8) and (9). So, the method for fault detecton and dagnoss provded by ths paper may be synthessed as follows: The method recommended for fault detecton and dagnoss based on dentfcaton, sensor network and ANFIS. -Placng a sensor network n the feld of the dstrbuted parameter system. -Acqurng data, n tme, from the sensor nodes, for the system varables. -Usng measured data to determne an estmaton model based on ANFIS. -Usng measured data to estmate the future values of the system varables. -Imposng an error threshold for the system varables. -Comparng the measured data wth the estmated values. -If the determned error s greater then the threshold a default occurs. - Dagnosng the default, based on estmated data, determnng ts place n the sensor network and n the dstrbute parameter system feld. Wth the above condtons the equaton may be solved usng the fnte element method. The optmze mean meshes and nodes are presentng n Fg. 5. Fg. 5. The optmze meshes and nodes The temperature represented heght 3D over the surface analyzed s presented n Fg Case Study In ths paper a case study consstng n a heat dstrbuton flux through a plane square surface of dmensons l=1, wth Drchlet boundary condtons as constant temperature on three margns: h θ θ = r (16) wth r=0, and a Neuman boundary condton as a flux temperature from a source nk θ + qθ = g (17) where q s the heat transfer coeffcent q=0, g=0, h θ =1. The heat equaton, of a parabolc type, s: Fg. 6. The temperature over the plane In practce we are usng a reduced number of sensors, whch s equvalent to a number reduced of nodes and meshes, for example a sensor network wth only 13 nodes, placed lke n Fg. 7. θ ρc = ( k θ) + Q + hθ(θext _ θ) t (18) where ρ s the densty of the medum, C s the thermal (heat) capacty, k s the thermal conductvty, coeffcent of heat conducton, Q s the heat source, h θ s the convectve heat transfer coeffcent, θ ext s the external temperature. Relatve values are chosen for the equaton parameters: ρc=1, Q=10, k=1. Fg. 7. Sensor network poston n the feld ISSN: ISBN:

6 For ths case the soluton wth the fnte element method s represented n Fg. 8. The tme perod was 1 and the samplng perod was 0,01. In Fg. 10 the temperature for nodes 13 and 12 are the same, because they are on the same sotherm. We are chosen as an example the node 5 to be the node wth the estmated temperature, based on the frst recursve algorthm: k 1 = f (θ k,θ k,θ k,θ k ) (19) θ + And also for the node 5 we wll apply the second algorthm, auto-recursve: θ k k k-1 k-2 k- = f (θ,θ,θ,θ ) (20) Fg. 8. Soluton for 13 nodes The fuzzy nference system structure s presented n Fg. 11. The repartton of temperature on sotherms n plane s presented n Fg. 9. Fg. 11. FIS structure Fg. 9. Temperature n plane In the applcaton we are choosng the nodes 8, 13, 12 5 and 11 from the Fg. 7 to apply the estmaton method. The transent characterstcs of the temperature are presented n Fg. 10 for 101 samples. The comparson transent characterstcs for tranng and testng output data are presented n Fg. 12. Fg. 12. Comparson between tranng and testng output Fg.10. Temperature transent characterstcs The average testng error s 2, Number of tranng epochs s 3. For the second algorthm the tranng error was of 0,007, number of epochs 3 and the testng error 0,007. The FIS general structure s the same, but wth dfferent parameter values. ISSN: ISBN:

7 The estmated output for the second algorthm s presented n Fg. 13. Fg. 13. The estmated output for the second algorthm Comparng the two algorthms the frst one had a better testng error. If a default appears at the sensor 5 an error occurs n estmaton, lke n Fg. 14. parameter systems, usng the measured values provde by the sensor and the estmated values computed by the ANFIS estmator, calculatng an error and detectng the fault based on a decson taken after a threshold comparson. Estmatons methods may be appled n the case of dscovery of malcous nodes n wreless sensor networks. A case study for the both algorthms s presented for heat transfer n plane. A comparson between the two algorthms s made. Good approxmatons were obtaned. Developng of the algorthms and the method are taken n consderaton n the future, n other applcatons, consderng all the capabltes of the sensor nodes to measure physcal varables. Ths approach allows treatment of large and complex systems wth many varables by learnng and extrapolaton. Acknowledgement: Ths work was developed n the frame of PNII- IDEI-PCE-ID CNCSIS - UEFISCSU grant. Fg. 14. Error at the ffth node for a fault n the network Detecton of ths error s equvalent to a default at sensor 5, from other pont of vew n the place of the senor 5 n the space of the dstrbuted parameter systems and n heat flow around the sensor 5. 6 Concluson The paper presents two algorthms and a method for fault detecton and dagnoss of dstrbuted parameter systems, wth the adaptve network based fuzzy nference systems and the ntellgent wreless sensor networks. The sensor network s seen as a dstrbuted sensor. The algorthms are one based on regresson usng the values provded by the adjacent nodes of the sensor network and the second s an autoregressve one based on the values from anteror tme moments of the same node. The method descrbed the way how to use all these concepts for fault detecton and dagnoss n dstrbuted References: [1] R. Isermann, Supervson, fault-detecton and fault dagnoss methods. An ntroducton. Control Eng. Practce, Vol. 5, No. 5, 1997, p [2] J. S. Roger Jang, ANFIS: Adaptve Network Based Fuzzy Inference Systems, IEEE Trans. on Systems, Man, and Cybernetcs, Vol. 23, No. 03, p , May [3] I. F. Akyldz, W. Su, Y. Sankarasubramanam, E. Cayrc, Wreless Sensor Networks: A Survey. Computer Networks, 38(4), March, [4] M. Tubashat, S. Madra, Sensor networks: an overvew, IEEE Potental, Apr. 2003, Vol. 22, Issue 2, p [5] C.S. Kubrulsky, M.R. de S. Vncente, Dstrbuted parameter system dentfcaton. A survey, Internatonal Journal of Control, Volume 26, Issue 4 Oct. 1977, p [6] D. Ucnsk, Optmal Measurement Methods for Dstrbuted Parameter System Identfcaton, CRC Press, [7] H. Wang, P. Chen, Fault Dagnoss for a Rollng Bearng used n a Recprocatng Machne by Adaptve Flterng Technque and Fuzzy Neural Network, WSEAS Transactons on Systems, Issue 1, Volume 7, January 2008, p ISSN: ISBN:

8 [8] S. Postalcoglu, K. Erkan, E. D. Bolat, Intellgent sensor fault detecton and dentfcaton for temperature control, Proc. of the 11th WSEAS Int. Conf. on Computers, Greece, 2007, p [9] C. Volosencu, Identfcaton of Dstrbuted Parameter Systems, Based on Sensor Networks and Artfcal Intellgence, WSEAS Transactons on Systems, Issue 6, Vol. 7, June 2008, p [10] C. Volosencu, D. I. Curac, A Comparatve Study for Identfcaton of Dstrbuted Parameter Systems usng Sensor Networks and Adaptve-Network-Based Fuzzy Inference, Proc. of the 11th WSEAS Int. Conf. on Automatc Control, Modelng and Smulaton, ACMOS 09, Istanbul, 2009, p [11] C. Volosencu, Identfcaton of Dstrbuted Parameter Systems Based on Sensor Networks and Multvarable Estmaton Technques, Proc. of the 9 th WSEAS Int. Conf. on Smulaton, Modelng and Optmzaton, SMO 09, Budapest, Hungary, 2009, p ISSN: ISBN: