Diagnosis of process faults and sensor faults in a class of nonlinear uncertain systems

Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011, pp.2232Available online at www.jseepub.comDiagnosis of process faults and sensor faults ina class of nonlinear uncertain systemsXiaodong Zhang*, Qi Zhang, and Niharika SontiDepartment of Elcrical Engineering, Wright State University, Dayton, OH 45435, USAAbstract: This paper presents a fault diagnosis method for pro-Moreover, a faulty sensor output may also cause inaccuratecess faults and sensor faults in a class of nonlinear uncertaindiagnosticlprognostic results, resulting in unecessary re-systems. The faut detection and isolation architecture consistsplacement of system component or mission abortion. Oneof a faut detection estimator and a bank of adaptive isolation es-of the most difficult problems in sensor validation is that, intimators, each corresponding to a particular fault type. Adaptivemany practical systems, both the process components (i.e.thresholds ftor fault detection and isolation are presented. Faultdetectability cnditins characterizing the class of process faultsactuators and the controlled plant) and sensors are proneand sensor fauts that are detectable by the presented method areto faults. However, many nonlinear fault diagnostic meth-derived. A simulation example of robotic arm is used to ilustrateods deal with process faults and sensor faults separately,the efectiveness of the fauit diagnosis method.especially for nonlinear uncertain systems. Specifcally,Keywords: fault detection, fault isolatin, fault detecabilit, ro-when dealing with sensor validation, it is usually assumedbustness, sensor bias, process faults.that there are no process faults; when dealing with processfaults, it is often assumed that the sensors are“healthy". InDOI: 10.3969/ssn.10044132 2011.01.003the former case, a process fault may be misinterpreted as asensor fault; in the ltter case, a sensor fault may be misin-1. Introductionterpreted as a process fault. Both cases may potentiallyIn recent years, there was significant research activity inlead to a high false alarm rate and unnecessary mainte-the design and analysis of fault diagnosis and accommo-nance. Therefore, it would be desirable to consider sensordation schemes (see, for example, [1- 4]). Considerablefaults and process faults under one unified framework.research effort was devoted to the development of faultIn previous papers [10, 15-17], nonlinear fault diagno-diagnosis and accommodation schemes for nonlinear sys- sis methods were developed for process faults and sensortems in the framework of various kinds of assumptions andfaults, respectively. However, process faults and sensorfault scenarios (seior instance, [5- -10], and the refer-faults were considered separately. Specifically, for the pro-ences cited therein).cess fault diagnosis problem [10, 17], we assumed the sen-The overall fault diagnosis procedure can be dividedsors are healthy, and for the sensor fault diagnosis probleminto three steps: () fault detection is the process of deter- [15, 16], we assumned there are no process faults. In [18],mining whether a fault has occurred or not; (iI) fault iso-a unified fault diagnosis method for sensor faults and pro-lation deals with the issue of determining the location/type cess faults was developed for a class of nonlinear uncertainof fault; and (ii) fault estimation provides an estimate ofsystems in which the known system nonlinearity is repre-the magnitude or severity of the fault. In addition to faultsented as a function of known system signals (i.e. inputdiagnosis, there is also the problem of fault accommoda-and output variables). In this paper, we extend these pre-tion, which deals with control reconfiguration methods forvious results by considering a different class of nonlinearcompensating for the effects of faults.systems, in which the known nonlinearity is modeled as aAn important area of fault diagnosis and accommoda-nonlinear function of the system input and state variablestion is sensor validation and fault-tolerant control[11-15].and satisfes a Lipschitz condition. With partially mea-A sensor fault may lead to poor regulation or tracking per-surable states and the Dresence of modeling uncertainty,formance, or even affect the stability of control systems. the de中国煤化工sis methods becomeManuseript received November 18, 2010.more|YHCNMHG*Coresponding author.uus paper, d lauit ueuecuon and isolation (FDI)Xiaodong Zhang et al: Diagnosis of process faults and sensor faults in a class of nonlinear uncertain systems23method is developed for a class of nonlinear Lipschitz sys-Remark 1 In addition to nonlinear uncertain systemstems with possibly nonlinear and unstructured modelingin the form of (1), the FDI design and analysis presenteduncertainty. Unstructured modeling uncertainty refers to in this paper are also applicable to more general nonlinearthe case where the modeling uncertainty function appearssystems that are transformable into (1) using a change ofpossibly in all state equations without being pre-multiplied coordinates (for instance, see [9, 16, 17]). It is worth not-by a dstribution matrix that satisfies certain conditions.ing that the nonlinear term p(z, u) is not considered in theThe FDI architecture consists of a fault detection estima-previous paper [18], which only deals with system nonlin-tor (FDE) and a bank of adaptive fault isolation estimators earity modeled as a function of measurable system input(FIEs), each corresponding to a particular fault type underand output variables. With P being a nonlinear function ofconsideration. Adaptive thresholds for FDI are presented.partially measurable state vector z, the design and analy-Fault detectability conditions characterizing the class ofsis of a unified fault diagnosis for process faults and sensorprocess faults and sensor faults that are detectable by thefaults are more challenging.presented method are derived, respectively.Assumption 1 The faults are assumed to be abrupt2. Problem formulation(sudden) ones. Therefore, βe() and By() in (1) take theform of a step function B() given byConsider a class of nonlinear multi-input multi-output(MIMO) dynamic systerms described byB(t-T)=0， t l),ρ: R"xRmH R',Vi :R' xoutput measurement, while all the remaining componentsRm→Rn-,4x:R'xRm→R',m:R"xR"xR+→are zero. Specifically, depending on the location of theR(n-), andn:R"x R"x R+→Rl are smooth vectorfault, the distribution vector F belongs to a class of l possi-fields. The constant matrix A11∈R(n-l)x(n-l) is Hur-ble vectors {F"',F...,F"}, where, for anyq= 1...,witz, A12, A21, A22 are constant matrices of appropriateonly the qth component of vector F9 is non-zero. Accord-dimensions, and C∈Rlxi is a nonsingular matrix.ingly, the scalar 09 is the unknown fault magnitude in theThe model described bygth sensor.z= A11z1 + A12z2 + ψr(y,u)The process fault function f(y, u) under considerationis modeled as a nonlinear function of input and output sig-in= A2121 + A22z2 + p(z, u) + 42(y,u)nals. It is assumed that there are N types of possible pro-y= Cz2zcess faults in the process fault class;, secifially, f(y,u)is the known nominal system model. The vector fields η1belongs to a finite set of functions given byand nη appearing in (1) represent the modeling uncertainty.Fe{f'(y,u), ... fN(,u)}(2)The change in the system dynamics as a result of a pro-cess fault is represented by the term Br(t - Tr)f(y,u) inEach fault function fp (p = 1....N is described by(1). Speifially, B-(t - Tr) denotes the time profile of aprocess fault which occurs at some unknown time Tr. andf(y,1) A [P)Fg(v,u), .... (?)”"(w0]”(3) .f(y, u) is the nonlinear fault function. The change in thesystem dynamics as a result of a sensor bias fault is repre-whereOP (i = 1...,11 is an unknown parameter vectorsented by By(t - T)F0 in (1). Specifially,F∈R' is theassumed to belong to a known compact and convex set日?fault distribution vector describing the location of the sen- (i.e. 时∈θ? C R%), andgp: R' x RmH RU? isasor fault, θ∈R is the unknown magnitude of a constantknov中国煤化Ised in [10,19], thesensor bias, and the function By(t - Ty) characterizes theprocEand (3) caraterirestime profile of the fault with Ty being the unknown faultagenYHCNMHGrethevectorfeldg?occurrence time.represents the functional structure of the pth fault directly2Journal of Systems Engineering and Electronics Vol. 22, No.1, February 2011affecting the ith component of state vector z2, and the un-FDI schemes that completely decouple the fault from mod-known parameter vector 0? characterizes the fault magni-eling uncertainty.tude. The following assumptions are made throughout theAssumption 3 requires the boundedness of the state vari-paper.ables before and after the occurrence of a fault. Hence, itAssumption 2 The modeling uncertainties, repreis assumed that the feedback control system is capable ofsented by η and n2 in (1), are unstructured and unknownretaining the boundedness of the state variables even in thenonlinear functions of z, u, and t, but bounded by givenpresence of a fault. This is a technical assumption requiredfunctionals,ie. V(z,y,u)∈ZxY xU,Vt≥0for well-posedness since the FDI design that we considerdoes not influence the closed-loop dynamics and stability.|m(,u,t)|≤in(y,u,t)4)The problem of fault tolerant control design is beyond the|n2(z,u,t)|≤(2(,u,)scope of this paper. However, it is important to note thatwhere the bounding functions in(y, u, t) and 2(y, u,t) arethe proposed FDI design does not depend on the structureknown and uniformly bounded in Y x U x R+. Here,of the controller.Remark2 It is worth noting that the above formula-ZCR",UCR”,andYCR'arecompactsetsoftion considers both process faults and sensor faults underadmissible states, inputs, and outputs, respectively.Assumption 3 The state z remains bounded beforeone unifed framework. The unknown process fault mag-nitude is represented by P, forp = .....N, and theand after fault occurrence,i.e. z(t)∈Lo, Vt≥0.Assumption 4 The known nonlinear term p(z,u) inunknown sensor bias magnitude is represented by 09, for(1) satifies the following inequality: Vu C U,Vz,i∈Zq = ....,. respectively.|lp(z,u)- (z,u|≤γ|z-引5)3. Fault detection methodThe FDI architecture is based on a bank of N +1+1 nonlin-where r is a known constant.ear adaptive estimators, where N is the number of differentAssumption 2 characterizes the class of modeling uncer-nonlinear process faults in the fault class F defined in (3),tainty under consideration. The bound on the unstructuredand l is the number of sensors under monitoring. One ofmodeling uncertainty is needed in order to be able to dis-the nonlinear adaptive estimators is the FDE used for de-tinguish between the effects of faults and modeling uncer-tecting the occurrence of any faults, while the remainingtainty ([10, 20]). For instance, in the aircraft engine faultN + l nonlinear adaptive estimators are FIEs, which arediagnosis application considered in [21], the modeling un-activated for the purpose of fault isolation only after a faultcertainty is the deviation of the actual engine dynamicsis detected. In this section, we describe the fault detectionfrom a nominal engine model representing the dynamicsmethod.of a new engine, which results from normal engine com-Based on the system model given by (1), the FDE isponent degradation during its service life. Such normalchosen ascomponent degradation can be modeled by small changesin certain engine component health parameters (e.g.. effi-名= Alzi + Al2C-ly+ vh(y,u)ciency and flow capacity parameters of the fan, compres-2= A2121 + A22 + p(i,u) +sor, and turbine). Therefore, the bounding function onthe modeling uncertainty (ie.永and n2) can be obtained(2(y,u)+ L(y- i)by using the knowledge of possible normal degradation ofj=Cizthese health parameters during a number of fights underthe worst case scenario. Additionally, it is worth notingwhere 21,名2, and y denote the estimated state and outputthat the unstructured modeling uncertainty under consid-vectors, respectively, L∈Rlxi is a design gain matrix,eration is more general than the types of uncertainties thatandi会[(《)T (C-1y)TIT. The initial conditions arehave been considered for fault diagnosis of Lipschiz non-z(O) = 0and z2(0) = C-1y(0). Let云 兰z1-幻.linear systems in literature, which often assumes the ab-and z台z2 - 2 denote the state estimation errors, andsence of modeling uncertainty [13] or structured models ofy兰y- y denote the output estimation error. Then, beforemodeling uncertainty (e.g.. [9, 11]). In the case of struc-fault ocurrence (i.e. fort < min(Tx, Ty), we havetured modeling uncertainty, certain rank conditions are of-(7ten assumed to be satisfied by the uncertainty distribution中国煤化工matrix to achieve robustness. On the other band, the uti-lization of structured uncertainty with aditional assump-MHCNMH+(⑧)tions on the distribution matrix may allow the design ofj=Ci.(9Xiaodong Zhang et al: Diagnosis of process faults and sensor faults in a class of nonlinear uncertain systemswhere A22台A22 - LC. Note that, since C is nonsingular,The fault detection time Td is defined as the first timewe can always choose L to make A22 Hurwitz.instant such that |ij;(Td)| > vj;(Ta), for some Td≥By using (4) and (7), and applying the triangle inequal-min(Tz,Ty) and some j∈.... thatis,Td 台ity, we obtaininf U {t≥0:!s(t)l> vf(t)}.|&()| < kowoe" -0ot+ko e-)o(t-r)n(y,u,r)dr (10)Remark 3 It is worth noting that the adaptive thresh-where keo and 入0 are positive constants that satisfyold vj(t) given by (14) has obvious advantages over a fixed|en||≤koe-hot (since A11 is Hurwitz, such con-threshold. Moreover, the adaptive threshold uj(t) can bestants ko and入o always exist [221), and wo is a (pos-easily implemented using linear fltering techniques [10].sibly conservative) constant bound for |z1 (0)|, such that4. Fault detectability analysis|i(0)| = |z(0)|≤wo (note云1(0) = 0).Based on (8), the state estimation error之2(t) is given byAs is well known in the fault diagnosis literature, thereis an inherent tradeoff between robustness and fault sensi-2(t)=| eAa(t-+)[Az1z(r) + p(z,u) -tivity. In this section, we investigate the fault detectabilityproperty of the fault detection method, which characterizesp(z,u) + n2(z,u, r)]dr(11)the class of detectable sensor faults and process faults.Note that before fault occurrence, we have4.1 Sensor fault detectability conditionz(7)-z(r)=zn(r)-(r)、1 =「云()](12)In this section, we derive the fault detectability condition。z(r)- C-'y(r)]=0of sensor faults. Specifcally, the following theorem char-Denote C; as the jth row vector of matrix C, forj =acterizes the class of sensor faults that are detectable by the1....1. By using (4), (5), (9), (10), (11), and (12), it canproposed method.be shown that each component of the output estimation er-Theorem 1 Consider the fault detection method de-ror, i.e. j(t)台Cz2(t), satisfies the following inequalityfined by the FDE (6) for residual generation and the adap-tive threshold (14) for residual evaluation. If there exist(:;(0| ≤。Cea4(1|a()|+)) 1(,2) -some time instantTa > Ty and some j∈{...,.. suchothat the sensor bias θ satisfiesp(,u)|+ |n2(z,u,7)|dr≤1旧1> 2(Tx)(15)σ(Tc)where-(i(u,)dro10el,[eAx(t-)(A214I1A12C-1F-withLF)dr+F;(AA12C-1F|+|C-1).x(t)会kowoe-2ot +ko | e-0(-)n(y,u,r)dr (13)2(1-e-)(-T)where kj and入j are positive constants that satisfy入|C;e422|≤kje-Xyt (since A22 is Hurwitz, such constantsthen the sensor fault will be detected at timet = Td, i.e.kj and入j always exist [22]). Therefore, we have the fol-lowing result.|j;(Ta)| > v;(Ta). .Fault detection decision scheme The decision on theProof In the presence of a sensor fault (i.e. fort ≥occurrence of a fault (detection) is made when the modu-Ty), based on (1) and (6), the dynamics of the state estima-lus of at least one component of the output estimation errortion erors兹兰z1-之andz2台z2 - 2 satisfy(i.e. j() exceeds its corresponding threshold uj(t) given之= A11zr- A12C lByF0+n(,u,t) (17byv;(t)兰k; e()(-()(IAa1| + r)x(r)+中国煤化工-p(i,1)+MHCNMHG0(18)(n(y,u, r)|dr(14)j=Cz2+ ByFθ(19)26Joural of Systems Engineering and Electronics Vol. 22, No. 1, February 2011Letz兰之一A1A12C-1ByFO (note that A11 is in.Based on the above inequality and (24), we haveverible since it is Hurwitz). Then, from (17), we have去= A11z + n(z,u,t)eax2(-)(AzrAt} A12C-1B,F -Based on the definition of z, we obtainLlLyF)dr + BAvE;|0)-云=云+ AI' Al2C-1By,FO =kη°-1)(-)|A4T+Ar2C-1B.F0| +5(t) + AI1 A1n2C-1ByFO(21)where|C-1B.F0|dr-5(t)会| ell(l()nl(a,r)dr +elutz(0) (22kj [ e-()(IAal|1 + r)()| +By using (18) and (19), fort > Ty, each componentn(y,u,T)|drof the output estimation error (i.e. j(t) = y;(t) - j(t),j= ...1) is given byBased on (13) and (22), we obtain |5(t)|≤x(t). Addi-tionally, by using (14) and the property of the step function访= Cjz + ByF;0=βy(t - Ty), the above inequality can be rewritten asC; | eAxa(t- r)[Az1行+ p(z,u)- p(i,u)+例|≥eAa(t-r)(A21A1A12C-1F-n2(,u,) - LByFA]dr + ByF;0(23)By substituting (21) into (23), we obtainLF)dr +E|1-k;y|8| _ e-1)(-)((1C-1|+Jr,j=C;exz(-)(A21AI Al2C-1B,F0-|4A12C 1F| )dr -vy(t)(26)Based on (26), it can be easily seen that if there existsLByFO)dr +C;| (ax()([(z,u) - (,w1)+some time Td > Ty such that condition (15) is satisfed,then we can conclude that |i(Td)| > ry;(Ta), i.e. the faultA21(r) + n2(z, r)]dr + ByFjOis detected at time t= T.According to Theorem 1, if the sensor bias fault is suchBy applying the triangular inequality, we havethat its magnitude 0 satisfies (15) for some Td > Ty, thenthe fault will be detected at Ta. Hence, Theorem 2 char-|回|≥C, (an-(4n41AnC-BAF-acterizes (in non-closed form) the class of sensor faultsthat are detectable by the robust nonlinear fault detectionLByF)dr + ByF}| -method.4.2 Process fault detectability conditionk; e-y(t-)| l(,1)- p(;,u)| + .In this section, we derive the fault detecability conditionfor process fault f(y, u). Specifically, the following the-Ill|()|) + n(,u,r)|dr(24) orem characterizes the class of process faults that are de-tectable by the proposed method.where the constants kj and λj are defined in (13). NoteTheorem 2 Consider the fault detection method de-that in the presence of a sensor fault, we havefined by the FDE (6) for residual generation and the adap-tive threshold (14) for residual evaluation. If there existz-z=21-元(25) some time instantTd > Tr and somej∈{.... such,z2-C-1]=[-0-_C-1ByFθthat the process fault function f(y, u) satisfiesTherefore, by using (5), (21), and (25), we have中国煤化工≥2vy;(Ta) (27)lo(a,u) - p(i,w)≤rli|+|C tB,Fe]≤then.YHCN M H Gd at timet=Ta.i.e.η(5()|. + IA4I A2C 1A,F01+ 1C-1B。F0||i;(Td)|> uj)(Td).Xiaodong Zhang et al: Diagnosis of process faults and sensor faults in a class of nonlinear uncertain systems27Proof In the presence of a process fault, based onBy substituting (14) into the above inequality, we obtain(1) and (6), the dynamics of the state estimation errors之≌z1-z1 andz2兰z2-名satisfies|,()|≥| Ceac(t-)paf(,u)dr |-y;() (33)玄= A1云+ nm(a,u,t)(28)Based on the property of the step function Bx, it follows初= A22 + A21云1 + p(z,u)- p(i,u)+that, if there exists some time Ta > Tr, such that condi-n2(z,u,t) + Baf(y,u)(29) tion (27) is satisfied, then it can concluded that |i(T)| >y=C(z- 2)= Ci(30)vj(Td), i.e. the process fault is detected at timet= Td. OAccording to Theorem 2, if the process fault f(y, u) sat-Based on (29) and (30), we can see that each compo-isfies (27) for some time Ta > Tr, then the fault will benent of the output estimation error, j(t) = yj(t) -的(t)detected at T. Hence, Theorem 2 characterizes (in non-(i = 1....). is given byclosed form) the class of process faults that are detectableby the robust nonlinear fault detection technique. Notingi(t)=C;etxa(t- -)[2z1云(r) + p(z,u)- p(;,w) +that the inegral on the lef.- hand side of (27) representsthe filtered fault function, in qualitative terms, the fault de-tectability theorem states that if the magnitude of the fil-n2(z,u, r)dr +"CeAa(t- r)Bzf(v,u)drtered fault function on the time interval [Tx Ta] becomessufficiently large, then the fault can be detected. The re-By applying the triangular inequality, we obtainsult also shows that if a fault function f(y, u) changes signover time then it may be difficult (or impossible) to detect.|i()|≥Cex(-(-}+f(,u)dr，5. Fault isolation method| Ceaxm(-()[2z()+ (z,1) - pi,u)dr-Now, assume that a fault is detected at some time Td; aC~Jcordingly, att = Ta the FIEs are activated. Each FIE cor-responds to one potential fault type. Specifically, among. Cjetxa(t -)n(z,r)drthe N + l FIEs employed in the fault isolation scheme, NFIEs are designed based on the functional structure of theNote that in the presence of a process fault, we haveprocess faults defined in (2) and (3), and the remaining lFIEs are designed based on the functional structure of po-2-名1-[吾(32)tential sensor faults..z2-C-1y5.1 FIEs for process faultsBy using (5), (3I), and (32), we haveThe following N FIEs are designed based on the functionalstructure of process faults: forp= 1....Nlij(t)|≥Cetxa(-r)Brf(y,u)dr -望= Au望+ Al2C- ly + y)n(y,u)|J君= A212P + A22站+ p(z,u)+ 2(y,u)+DP(y-捫)+ j(,u,)+ s?Pip(34)k; | e((-()(|A21| + r)z(r)dr加p = A22S2P + GP(y,u)Note that (28) is in the same form as (7). Thus, based onjP =C沼(10) and (13), we have |运(t)|≤x(t). Therefore, we ob-tainwhere望，望，and p denote the estimated state and out-put vectors, respectively,LP∈Rlx6 is a design gain ma-|is(t)|≥trix (for the simplicity of presentation and without loss ofgenerality, we letLP = L),?P台|(部)T (C-1y)TT,k;| e-(-)iz(u,u, r)dr-中国煤化工()Tr(u,u), and: of the fault param-eterHC N M H Glation estimaor. Itk, ((1[1|+ ]x(r)dris noted that according to (3), the fault approximation28Jourmal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011model fp is linear in the adjustable weights印. Conse-θ9 C R (in order to guarantee the stability of the learn-quently, the gradient matrix GP≌8fP(y, u, 0P)/80D = = ing algorithm in the presence of modeling uncertainty)diag(g{)T .... (gR)T] does not depend on oP. The initial[15,22], and r9∈R is the leamning rate.conditions are动(Td)= 0,遇(Td)= 0, QP(Td)=0.Remark 4 The parameter estimates印and 09 mayThe adaptation in the isolation estimators arises due toalso provide some fault information. However, it is im-the unknown parameter vector OP会[().... (P)T]T.portant to stress that it cannot be guaranteed that for theThe adaptive law for adjusting 的is derived using thematched FIE, the parameter estimate OP or 09 will con-Lyapunov synthesis approach (see, for example, [22, 23]).verge to the true values, unless we assume persistency ofSpecifcally, the leaming algorithm is chosen asexcitation [22, 23], which in general is a restrictive condi-印= Por{rnr"CTp}(35)tion in practical applications (here we do not assume thepersistency of excitation).where P(t)兰y(t) - p(t) denotes the output estimationRemark 5 The stability and learning capability of theerror of the sth estimator, r > 0 is a symmetric, positive-FIEs described by (34) and (36) have been established indefinite leamning rate matrix, and Per is the projection op-[16, 17]. The analytical results guarantee the boundednesserator restricting 0P to the corresponding known set θP (inof the variables in the bank of FIEs. Moreover, it has beenorder to guarantee the stability of the learing algorithm inshown that the ability of the isolation estimators to learnthe fault function is limited by the modeling uncertaintiesthe presence of modeling uncertainty [22, 23]).η1 and 72, and the fault parameter estimaton eror.5.2 FIEs for sensor faults5.3 Fault isolation decision schemeAnalogously, the following l FIEs correspond to sensorNow let us consider the pth process fault, where p =faults: forq≈N + ....N +l1.,..,N, and the qth sensor fault, where q = N +望= A112 + A12C-+(y- F909)+ .1....N + l, under a unified framework. Then, we haveN+l faults in the augmented fault class. More seifiall,ψ1(y,u) + 729的9fors= 1...N +l, fault sis a process faultif1≤8≤N,望= A2129 + A22望+ p(幻,1) + (,12) +and fault 8 is a sensor faultifN+1≤8≤N+l.L(y-i們)+ s2θ(36)The fault isolation decision scheme is based on the fol-lowing intuitive principle: fors = 1...N + l, if fault的= A11829- A12C-1F98 occurs at time To and is detected at time Ta, then aset of adaptive theshold functions {4号()j = ....望=A22-L9P9can be designed for the sth isolation estimator such thatg=C盟+ F9的qthe jth component of its output estimation error satisfieswhere望,器, and 9 denote the estimated state and output|q号,(t)|≤4(t), for allt> T. Consequently, foreach s = ....N + I, such a set of adaptive thresh-vecors, retevey,ae {(3)T (C-'(u- F00yT}',olds {4(t),j = .... can be associated with theL9∈Rlxl is a design gain matrix (for the simplicity ofoutput estimation of the sth isolation estimator. In thepressentation, here we let L9 = L), and的is the esti-fault isolation procedure, if for each isolation estimatormate of the sensor bias magnitude provided by the qth iso-r∈{1...N + l}\{s}, there exists somej∈.,..lation estimator. The initial conditions are望(Ta) = 0,such that the jth component of its output estimation errorsatisfes |eE,(t)| > 45(t) for some finite timet > Td, then望(Ta) =0, 82q(Ta) = 0,and 2(Ta)=0.the possibility of the occurrence of fault r can be excluded.the unknown sensor bias magnitude 0. The adaptive lawBased on this intuitive idea, the following fault isolationfor adjusting 09 is derived using the Lyapunov synthesisdecision scheme is devised.approach (see, for example, [22,23). Secifcally, theFault isolation decision scheme If for each r∈{...N. + 1}\{s}, there exist some finite timet" > Tdlearming algorithm is chosen asand somej ∈.... such that|q,(") > [(t"),0=Peo {r(CQ29+F0"0}(37) then the ocurrence of fault 8 is concluded.t) play a key rolewhere 9(t)台y9(t) - 9(t) denotes the output estima- in th中国煤化工The ise of adaprtion error, the projection operator P restricts the parame- tive:m1HC N M H Gi has been rigorouslyter estimate 09 to a predefined compact and convex region invesugatea n [16, 1小. specincally, for1≤8≤NXiaodong Zhang et al: Diagnosis of process faults and sensor faults in a class of noalinear uncertain systems2S(i.e. FIEs corresponding to process faut), the following01threshold function is chosen「x1Jx2 |+i3x3|4号(t)=k。e((-)[I\2al|+ r)*(r)+‘JrLics」T4」. Jnm的]dr + (CjQn)Txe() + kyuxe-d)ft-Tz)-mghand forN +1≤8≤N +l (i.e. FIEs corresponding to. sinc1sensor faults), the threshold function is given byug(t)=kj|_ e(()(IA]||+ r)ζ∈° + |29|k*)+Jm“1100n+r|C-1P01x°|dr +1C;Q2+ FjI\r' +kxu2e-)j(t-Ta)) 0 10|2where0001ξ*(t)≌ko-0(t-)i(y, u,r)dr+kowre -o(t-Ta)Note that the FDI method presented in the previous pa-Tdper [18] is not applicable here because the state 工1 is notThe positive constants W1 and的are (possibly conser-assumed to be measurable. By using a linear change of co-vative) bounds for unknown initial conditions |z{(Td)| andordinatesz= [2,Z]T = TxwithT=[-50, 0, 0, 0;|z2(Td)|, respectively. Addionally, r*(t) is a suitable-1,-1,00; 001 0; 000 1], the state space model in thefunction satisfying |0° -的(t)|≤r*(t), chosen based onnew coordinate system isgeometric properties of the known compact parameter set5日, since the estimate θ belongs to θ°.--0.0270.89 -0.44Remark 6 The adaptive threshold functions giveni2=|above can be easily implemented using linear filtering- -0.04-2techniques. Additionally, note that since the effect of the(possibly conservative) bounds山1 and w2 decreases ex-.m1pontially (i.e. they are multiplied by e-2o(t- -Ta) and-4.36 sin(z1/50)e-Ar(t-1a), respectively), the performance of the fault iso-n2lation algorithm will not be significantly affected.6. Simulation results「-100y=010|2Consider a single-link robotic arm with a revolute elasticjoint, whose motion equations are given byNote that the modeling uncertainty terms n1 and n haveJign + Fiqn + k<(q1 - q2) + mghsing1 = 0been added to the model. Io this simulation example, themodeling uncertainty is assumed to be up to 5% inaccu-Jmg2+ Fmg2-k(q1 -q2)= khruracy in the amplifer gain ky, which gives η = 0 andwhere 91 and 92 are the angular positions of the link and 7 = 0.05kx[|u(t)|/Jm. Clearly, the above model is inthe motor, respectively. The robot parameters (in SI units)the form of (1). The nonlinear term -4.36 sin(z1/50) hasare:k=2,Fm=1,Fi=0.5,Jm=1,J=4.5, m =4, a Lipschitz constant of 0.087.g= 9.8, andl = 0.5,h = 1. The initial conditions ofWe consider the fllowing three types of faults, includ-the plant are chosen to be q1(0) = 92(0) = 0, and the in- ing an actuator fault and two sensor faults:put to the system is given by u = 2sin(t/2). By choosing●A simple multiplicative actuator fault by ltting u =出1 =q1,T2 = 91,I3= 92,T4 = g2, and by assuming i + 0中国煤化工ol input in the non-the motor position, motor velocity, and the sum of link po- fault qCNMH(Frameter characteriz-sition and link velocity are measured, the above model can ing theShat thecase01 =0be rewritten in state space form asrepresents the normal operation condition (no fault), while3Jourmal of Systems Engineering and Electronics Vol. 22, No.1, February 201102 = -1 corresponds to the complete failure of the ac~generated by FIE #2 and FIE #3 exceed their correspond-tuator. Therefore, the actuator fault can be described bying thresholds, respectively. Note that for each of these twof}兰[0,0, 0'g'(u)T, where g^(u) = kryu()/Jm andFIEs, only selected residual component is shown, since this0∈[-1,0]is sufficient to exclude the possibility of occurrence of the●A bias in the sensor measuring yn represented bycorresponding fault type based on the isolation logic de-F=F2=[0, 1, 0]T and 02∈[0,0.2].scribed above. Additionally, as shown in Fig. 2, all three●A bias in the sensor measuring y3 represented byresidual components generated by the FIE #1 always re-F=F3=[0, 0, 1]T and 03∈{0,0.2].main below their thresholds. Thus, the occurrence of anThe objective is to detect the occurrence of any faults.5rand further determine the specifc fault type.Based on the presented FDI scheme, an FDE and three.00.5FIEs are constructed. We choose the gain matrix L =. [-2.39, -0.44, 0; 0, 1.7, 1.0; 0, -2.0, -0.9], so that the6810poles of matrix A22 are located at -1.5, -1.7, and -1.9,Time/srespectively. Consequently, the related constants are cho-: Residual (y); --: Thresbhold (n)sentobeko= k1 = k2=kg= 1,λ= 1,λ1 = 1.2,λ2= 1.4, and\s = 1.9. The leamning rate of the adaptivealgorithm for fault parameter estimation of the FIEs is setto 0.1.2Fig. 1 and Fig. 2 show the simulation results when an: Reeidual (h);Threshold (I)actuator fault with0l = -0.5 occurs atT: = 5 s. Specif-.5厂cally, the fault detection residual (solid line) and its thresh-1.0old (dashed line) associated with 13 are shown in the topleft plot of Fig. 1. As can be seen, the fault is detectedalmost immediately at approximately Ta = 5.03 s. Then,the three FIEs are activated to determine the particular fault一: Residual (3); --: Threshold (W)type that has occurred. Selected fault isolation residualsFig.2Case of an actuator fault (fault type 1): esidual componentsand their corresponding thresholds generated by FIE #2 and their thresholds generated by the FIE for actuator fault(designed based on the fault structure of sensor bias in y2)and FIE #3 (designed based on the fault structure of sen-.4厂sor bias in y3), respectively, are shown in Fig. I. It canFault detected.2be seen that the residual components associated with y3Fault detected,0.2-一: Residual (w); --: Threbhold (m)(间) FDE0.0248i0Time/8(a) FDE00.5-a 0.010一: Residual (h); -一: Theshold (加)Time/g(b) FIE for sensor fault in hFault isolated(c) FIE for sensor fault in h一: Residual (w); -: Threshold (b)Residual (uw); --: Threshold (y)中国煤化工>r faultFig.1 Case of an actuator fault (fault type 1): seleted residuals andFig3C ve 2): selected residualstheir thresholds generated by the FDE, FIE for sensor bias in y2, andand th:MYHcNMHGreorataterlal,FiE for sensor bias in ysand FIE for sensor bias in 13Xiaodong Zhang et al: Diagnosis of process faults and sensor faults in a class of nonlinear uncertain systems31actuator fault (i.e. fault type 1) can be concluded. The faultis isolated at approximately 7.1 s.Analogously, Fig. 3 and Fig. 4 show the simulation re-sults when a sensor fault in y2 with 02 = 0.18 occurs atTy= 58. The case of a sensor fault in y3 withθ= 0.15; Residual (); .: Threshold (y)and Ty = 5s is reported in Fig. 5 and Fig.6. In each of.0-these cases, the fault is successfully detected and isolated..5-.0Time/soL; Residual (s); --: Thresbold (u)02.5 r1.0- : Residual (n); --: Thresbold (n)100.5-一: Residual (v); =:Threshold (x)6Fig 6 Case of a sensor blas in y3 (fault type 3): residual com-ponents and their thresbolds generated by the FIE for sensor bias: Residual (h); --: Threshold (n)in3.7. Concluding remarks0.5里。In this paper, a robust fault diagnosis method for a classof nonlinear uncertain system is presented. The FDI ar-chitecture consists of an FDE and a bank of adaptive FIEs一- : Residual (蝴); -: Threshold (5)each corresponding to a particular fault type. The fault de-Fig.4 Case ofa sensor bias tn 1n (fault type 2): residual com-tectability conditions for sensor faults and process FIEs areponents and their thresbolds gernted by the FIE for sensr bias investigated, respectively. A simulation example is givenin unto ilustrative the effectiveness of the presented method.One important topic of future research work is to in-0.2pvestigate the fault isolability conditions for different faults0.1-under consideration (for instance, see [18] for some pre-liminary work). Additionally, the extension of the results8to distributed fault diagnosis of large-scale systems is an-other interesting direction for future research [24]. More-- : Residual (n); : Treshold (n)over, the integration of fault diagnosis with fault-tolerant(回) FDEcontrol to automatically compensate for the effect of faults回1.0Fault isolatedusing on-line diagnostic information deserves further in-vestigation.References: Residual (b); --: Threshold (h)[1] M. Blanke, M. Kinnaert, J Lunze, et al. Diagnosis and fault-(b) FIE for sensor fault in htolerant control. Berlin: Springer, 2006.[2] J. Chen, R. J. Patton. Robust model-based fault diagnosisfor dynamic systems. London: Kluwer Academic Publishers,[3] P. M. Frank. Fault diagnosis in dynamic systems using ana-lytical and knowledge-based redundancy - a survey and somenew results. Automatica, 1990, 26(3): 459- 474.- - : Reeidual (x); = Threshold (W)[4] J.J. Gertler. Fault detection and diaenosis in engineering s5y8-(c) FIE for actuator fault中国煤化工[5]mpot. Fault diagnosisFlg.5 Case of a sensor bias in y3 (fault type 3): selected residualsYHC N M H G of nonlinear systemsand their thresholds generated by the FDE, FIE for actuator fault,wiu JONIUWII put ulcicis. sruer rurad Joumal of Control,and FIE for sensor bias in 722004, 77(4): 415- 426.32Joumal of Systems Egineeing and Eetronis Vol. 22, No.1.Febnuary 2011(6] C. de Persis, A. Isidr. A gcometrie approach to nonlinear(22) P. A. Ioannou, J. Sun. Robust adaptive control. Englewoodfault detection and isolation. IEEE Trans. on Automatic Com-Ciffs: Prentice Hall, 1996trol, 2001, 46(6); 853- 865.[23) J. Frel M. M. Polycaupou.Adaptive approximation based[7] x. Tang. G. Tao, s. M. Johi. Adapive actuator failure com-conol. HonNi loi We.x0 tion 。o . dasospensation for nonlinear mimo systems with an aircraft control[24] x. D. Zhang, Decentralized fault dapplication. Automatica, 2007, 43(13); 1869-1883.largescale nonlincar input output systems. Proc. of the Amer-[8] H Wang, z. J. Huang, s. Daley. On the use of adapive updat-ican Control Conference, 2010: 5650 -5655.ing rules for actuator and sensor fault diagnosis. Automatica,1997, 32(2): 217-225.Biographies[9] x. Yan, C. Edwards. Nolinear robust fault reconstnucion andestimaion using a sliding mode observer. Automaica, 2007,Xiaodong Zhang rceived the B.S.. degree fromHuazhong University of Science and Technol-([10] X.D. Zhang. M.M.Pogy, Wuhan, China, the M.S. degree fromtion of a class of nonlincar input- output systems. IntemationalShanghai Jiaotong University. Shanghai, China,Joumnal of Control, 2001. 74(13): 1295 -1310.and the Ph.D. degree from University of Cincin-[1] W. Chen, M. Saif. A sliding mode observer-based strategyati, Cincinnati, OH, USA, all in ecrical en-for fault detection isolation, and estimation in a class of lip-gineering, in 1994, 1997 and 2002, respecively.schitz nonliner syscms. Intemational Joumal of System Sci-He has been with the Department of Electricalence, 2007, 38(12): 943- 955.Engincering.Wnight State University, Dayton, OH, as an assistant12] S. S. Li, G. Tao. Feedback based adaptive compensation ofpolssor sine lll 207 Belfore jininig Wrgh SuteBefore joining Wright State University,control sysem sensor uncertainties. Automatica, 2009, 45(2):he had over five years of indutrial experiences.was aSr. reSearcher of General Motors research and development center, War-(13R. Rajamani, A. Ganguli. Sensor fault dagostics for a classren, MI, USA, and a program manager for control and diagnosticsof nonlincar systems usng linear matric inegualties.of Itelligeat Automaion Inc., Rockville, MD, USA, respectively.tional Joumal of Control, 2004, 77(10); 920 930.。His research interests include ieleligent control systems, fault diag-[14} A. T. Vemuri. Sensor bias fault diagnosis in a class of nonlinnosis and prognosis, faul-tolerant control, and cooperative and dis-ear systems. IEEE Trans. on Automatic Control, 200. 46(6):tribued control. He is a member of the IFAC Safeprocess technical949-954.committee and has served as an associate editor of the Conference15] x. D. Zhang, T. Prsini, M. M. Polyarpou. Sensor bias faultEditorial Board of the IEEE Control System Society (CSS) sinceisolation in a class of nonlinear systems. IEEE Trans. on Au-tomatic Control, 2005. 50(3): 370 -376.E.mail: xiaodong .zhang@wrigh.edu[16] x. D. Zhang. Sensor bias fault diagnosis in a class of nonlinsystemns with lipschit nonineaities. Proc. ofear ucrain syotems wih iehihoo 10-276171Qi Zhang was borm in 1983. He received thethe American Control Coference,，(17] x. D. Zhang, M. M. Polyearpou, T. Parisini. Faultiagnosis3.S. and the M.S. degrees in automation fromof a class of nonlinear uncertain system with lipschitz nonlin-East China University of Science andTechnologyearties using adapive esimaion. Automatica, 2010, 46(2):in 2006 and 2009 respctively. He is curently a290- 299.Ph.D. candidate in Wright State University. His[18) X.D. Zhang, M. M. Polycarpou, T. Parisini Design and analy-_research interests include fault diagnosis and insis of a fault isolation scheme for a class of unetan nonlinearelligent control systems.systems. IFAC Anrual Reviews in Conrol, 2008, 32(1); 107-E-mail: zhang .106@ wrighl.cdu121.[19] x. D.Zhang. M. M. Polycarpou, T. Prisi. Arobust dectionand isolation schemeand incipient faults in nonlin-Niharika Sonti received the B.S. degree in elec-an mnrpeta200474ear systems. IEE Trans. on Automatic Control, 2002, 47(4):trical engineering from Jawaharlal Nehru Tech-576-593.nological University, Hyderabad, India in 2008[201 L Tang, x. D. Zhang, 1. A. DeCastro, a al. A unified non-and the M.S. degree in clectrical engineeringlinear adaptive approach for detection and isolation of enginefrom Wright State University, Dayton, OH infaults. ASME Turbo Expo, 2010.2010. Her research interests include itelligeat21] A. Emami-Nacini, M M. Akhter, s. M. Rock. Efct of modelcontrols and robotics.uncerainty on failure detction: the threshold selector. IEEETrans. on Automatic Control, 1988, 33(12); 106-1115.中国煤化工MYHCNMH G