State estimation for predictive maintenance using Kalman filter

Transcription

1 Reliability Engineering an System Safety 66 (1999) State estimation for preictive maintenance using Kalman filter S.K. Yang, T.S. Liu* Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 31, Taiwan, ROC Receive 7 July 1998; accepte 19 January 1999 Abstract Failure can be prevente in time by preventive maintenance (PM) so as to promote reliability only if failures can be early preicte. This article presents a failure preiction metho for PM by state estimation using the Kalman filter on a DC motor. An exponential attenuator is place at the output en of the motor moel to simulate aging failures by monitoring one of the state variables, i.e. rotating spee of the motor. Failure times are generate by Monte Carlo simulation an preicte by the Kalman filter. One-step-ahea an two-step-ahea preictions are conucte. Resultant preiction errors are sufficiently small in both preictions Elsevier Science Lt. All rights reserve. Keywors: Kalman filter; Failure preiction; Preventive maintenance; DC motor Nomenclature (X) k : The value of (X) at time kt ^ a=b : The estimate of ( ) at time at base on all known information about the process up to time bt a: Parallel path number for armature wining A: A matrix A c : Coefficient matrix of the state equation for a continuous system A : Coefficient matrix of the state equation for a iscrete system A T : Transpose matrix of A A 1 : Inverse matrix of A B: Damping coefficient B c : Coefficient matrix of the state equation for a continuous system B : Coefficient matrix of the state equation for a iscrete system B k : Coefficient matrix for the input term of a iscrete state equation C: A matrix C c : Coefficient matrix of the state equation for a continuous system C : Coefficient matrix of the state equation for a iscrete system * Corresponing author; fax: aress: tsliu@cc.nctu.eu.tw (T.S. Liu) D c : Coefficient matrix of the state equation for a continuous system D : Coefficient matrix of the state equation for a iscrete system E: Applie voltage e b : Motor back emf e f : Fiel voltage e k /k 1: Prior estimation error E r : Estimation error for mean value, or confience interval for estimate mean value E[X]: Expecte value of X f(t): Distribution function of life H k : Matrix giving the ieal (noiseless) connection between the measurement an the state vector h(t): Failure rate i a : Armature wining current i f : Fiel current J: Moment of inertia of rotor an loa k 1 : Motor constant k b : Back emf constant k f : Fiel flux constant K k : Kalman gain k m : Motor gain constant k T : Motor torque constant L a : Armature wining inuctance L f : Fiel inuctance L 1 : The inverse Laplace transform N: Sample size P: Magnetic pole number P k/k 1 : Estimation error covariance matrix /99/$ - see front matter 1999 Elsevier Science Lt. All rights reserve. PII: S (99)15-

2 3 S.K. Yang, T.S. Liu / Reliability Engineering an System Safety 66 (1999) Fig. 1. Block iagram of a iscrete system. Q k : Covariance matrices for isturbance R: Armature wining resistance R k : Covariance matrices for noise t: Time variable T: Motor output torque u i : Stanar uniformly istribute ranom numbers U k : Control input of a iscrete state equation at state k U k : Input vector at state k V k : Noise, measurement error vector. It is assume to be a white sequence with known covariance. W k : Disturbance, system stochastic input vector. It is assume to be a white sequence with known covariance an having zero crosscorrelation with V k sequence x, X: Variable of a istribution function X D : Initial states resulting from eterministic input X k : System state vector at state k X S : Initial states resulting from stochastic input Y k : System output vector at state k Z: Conuctor number of armature wining Z k : Output measurement vector Z a /2 : Z-value of the stanar normal istribution, that resulting the cumulative probability between an Z is a/2 a: Confience level u: Motor angle isplacement l: Constant failure rate s: Stanar eviation of a istribution function t: Failure time constant of the motor t m : Motor time constant f: Air gap flux F k : Matrix relating X k to X k 1 in the absence of a forcing function. It is the state transition matrix if X k is sample from a continuous process. methos have been propose for failure etection in ynamic systems [2]. Fault etection base on moeling an estimation is one of the methos [3]. However, the Kalman filter is useful not only for state estimation but also for state preiction. It has been wiely use in ifferent fiels uring the past ecaes, such as on-line failure etection [4], real time preiction of vehicle motion [5], an preiction for maneuvering target trajectories [6]. The Kalman filter is a linear, iscrete-time, an finite-imensional system [7]. Its appearance is a copy of the system that is estimate. Inputs of the filter inclue the control signal an the ifference value between measure an estimate state variables. By minimizing mean-square estimation errors, the optimal estimate can be erive. As a result, the output of the filter becomes optimal estimates of the next step time-state variables. If a evice is juge to know that it is going to fail by the preicte future state variables, the failure can be prevente in time by PM. However, future state variables shoul be accurately preicte at a reasonably long time ahea of failure occurrence. This stuy proposes the state estimation an preiction for PM using the Kalman filter. In Section 2, a iscrete system moel with eterministic control input an white noise isturbance an noisy output measurement will be constructe first. Equation formulation for state estimation of the Kalman filter then follows. Deterministic inputs are consiere in the formulation. Moreover, equations for N-step-ahea preiction are erive. Section 3 presents the transfer function, continuous state moel, an the iscrete state moel of a DC motor that is employe as an example in this article. Section 4 presents the simulation system with prescribe parameters, Monte Carlo simulation an ARMA moel use to generate necessary ata for failure preiction simulation, an the exponential attenuator use to simulate aging failure moe. Results an iscussions are in Section Introuction Preventive maintenance (PM) is an effective approach to promoting reliability [1]. Time-base an conition-base maintenance are two major approaches for PM. Irrespective of the approach aopte for PM, whether a failure can be etecte early or even preicte is the key point. Many 2. Kalman filtering 2.1. System moel The block iagram of a iscrete system is shown in Fig. 1. The state equations [7] are: X k 1 ˆ F k X k B k U k W k ; 1

3 S.K. Yang, T.S. Liu / Reliability Engineering an System Safety 66 (1999) Fig. 2. Block iagram of Kalman filter. Y k ˆ H k X k ; Z k ˆ Y k V k : 3 Substituting Eq. (2) into Eq. (3) yiels Z k ˆ H k X k V k : 4 Let E[X] be the expecte value of X, thus, covariance matrices for W k an V k are given by: ( E W k Wi T Šˆ Qk; i ˆ k ; i k ; 5 ( E V k Vi T Šˆ Rk; i ˆ k ; i k ; E W k Vi T Šˆ; for all k an i: 7 It follows that both Q k an R k are symmetric an positive efinite [8] State estimation State estimation aims to guess the value of X k by using measure ata, i.e. Z ; Z 1 ; ; Z k 1 : Let a b, an efine the notation ^ a=b as the estimate of ( ) at time at base on all known information about the process up to time bt. Accoringly, ^X k=k 1 is calle the prior estimate of X, an ^X k=k is calle the posterior estimate of X [8]. The prior estimation error is efine as e k=k 1 ˆ X k ^X k=k 1 : 8 As W k an V k are assume to be white sequences, the prior estimation error has zero mean. Consequently, the associate error covariance matrix is written as P k=k 1 ˆ E e k=k 1 e k=k 1 T Š ˆ E X k ^X k=k 1 X k ^X k=k 1 T Š: 9 The estimation problem begins with no prior measurements. Thus, the stochastic portion of the initial estimate is zero if the stochastic process mean is zero; i.e. ^X = 1 is riven by 2 6 eterministic input X D only. It follows from Eq. (8) that e = 1 ˆ X ^X = 1 ˆ X X D ˆ X S : 1 Employing Eqs. (9) an (1) yiels P = 1 ˆ E X S XSŠ; T 11 where X D an X S are initial states resulting from eterministic input an stochastic input, respectively. The Kalman filter is a copy of the original system an is riven by the estimation error an the eterministic input. The block iagram of the filter structure is shown in Fig. 2. The filter is use to improve the prior estimate to be the posterior estimate by the measurement Z k. A linear blening of the noisy measurement an the prior estimate is written as given in Ref. [8] ^X k=k ˆ ^X k=k 1 K k Z k H k ^X k=k 1 ; 12 where K k is a blening factor for this structure. Once the posterior estimate is etermine, the posterior estimation error an the associate error covariance matrix can be erive as e k=k ˆ X k ^X k=k ; 13 P k=k ˆ E e k=k e k=k T Š ˆ E X k ^X k=k X k ^X k=k T Š: 14 The optimal blening factor is written as given in Ref. [8] K k ˆ P k=k 1 Hk T H k P k=k 1 Hk T R k 1 : 15 This specific K k, namely, the one that minimizes the meansquare estimation error, is calle Kalman gain. Substituting Eq. (15) into Eq. (12), the posterior error covariance matrix can be erive as follows: P k=k ˆ P k=k 1 P k=k 1 H T k H k P k=k 1 H T k R k 1 H k P k=k 1 ˆ P k=k 1 K k H k P k=k 1 H T k R k K T k ˆ I K k H k P k=k 1 : 16 As epicte in Fig. 2, the one-step-ahea estimate is

4 32 S.K. Yang, T.S. Liu / Reliability Engineering an System Safety 66 (1999) Fig. 3. One-step estimator an N-step preictor. formulate as ^X k 1=k ˆ F k ^X k=k 1 F k K k Z k H k ^X k=k 1 B k U k ˆ F k ^X k=k 1 K k Z k H k ^X k=k 1 B k U k ˆ F k ^X k=k B k U k 17 Consequently, the one-step-ahea estimation error is erive as e k 1=k ˆ F k X k B k U k W k F k ^X k=k B k U k ˆ F k X k ^X k=k W k ˆ F k e k=k W k : 18 In a manner similar to Eq. (14), the one-step-ahea error covariance matrix is erive as P k 1=k ˆ E F k e k=k W k F k e k=k W k T Š ˆ F k P k=k F T k Q k : 19 Accoring to the aforementione statements, several remarks for the Kalman estimation are conclue as follows: 1. As K k is optimal, the posterior estimate ^X k=k is an optimal estimate. 2. Base on Eqs.(12), (15), (16), (17) an (19), recursive steps for constructing an one-step estimator are summarize in Fig The recursive loop has two ifferent kins of upating. Eqs. (12) an (16) yieling ^X k=k an P k=k from ^X k=k 1 an P k=k 1 are measurement-upate; Eqs. (17) an (19) projecting ^X k=k an P k=k to ^X k 1=k an P k 1=k are timeupate. 4. Initial conitions, i.e. ^X = 1 ; P = 1 ; F ; H ; Q ; an R have to be known to start recursive steps Preiction The estimate resulting from recursive steps in Fig. 3 is a one-step-ahea preiction. Base on the posterior estimate, i.e. (12), the state that is N steps ahea of the measurement Z k can be preicte by using the ARMA moel [8]. From Eqs. (17) an (19), equations for N-step-ahea preiction are erive as! Y ^X k k N=k ˆ F i ^X k=k P k N=k ˆ iˆk N 1 X k N 2 mˆk " # Ym 1 iˆk N 1 B k N 1 U k N 1 ; Y k iˆk N 1 X k N 2 mˆk F i! 2 4 P Ym 1 iˆk N 1 F i!b m U m Y k N 1 F T j jˆk F i! Q m 1 Y k N 1 F T j jˆm 1 13 A5 2 Q k N 1 : 21 The N-step preictor is an appenage of the one-step estimation loop [8]. It is also shown in Fig. 3. As the current

5 S.K. Yang, T.S. Liu / Reliability Engineering an System Safety 66 (1999) preicte value is assume to be the initial value for the next preiction, the more steps the preictor preicts, the larger error it results in. 3. Armature-controlle DC motor An armature-controlle DC motor is employe in this stuy as the physical moel to perform error preiction. The motor circuit representation is shown in Fig Transfer function Accoring to properties of a DC motor, the following equations can be formulate [9]: f ˆ k f i f ; T ˆ ZP 2pa fi a ˆ k 1 k f i f i a ˆ k T i a ; e b ˆ k b u t ; L a t i a Ri a e b ˆ E; J u B _u ˆ T; Fig. 4. Circuit representation of DC motor where k 1 ˆ ZP=2pa is calle the motor constant, an k T ˆ k 1 k f i f is the motor torque constant. Taking the Laplace transform for Eqs. (24) (26) results in E b s ˆk b su s ; 27 L a s R I a s ˆE s E b s ; 28 Js 2 Bs u s ˆT s ˆk T I a s : 29 Combining Eqs. (27) (29), the transfer function of a DC motor is erive as u s E s ˆ k T s sl a R sj B k T k b Š : 3 Accoringly, the block iagram of a DC motor can be shown in Fig. 5. If L a, (3) can be rewritten as u s E s ˆ k m s st m 1 ; 31 where k m ˆ k T = RB k T k b an t m ˆ RJ = RB k T k b are calle the motor gain constant an motor time constant, respectively Continuous state space moel Define u; _u ; an i a as state variables, so that the state vector is X ˆ u _u i a Š T : As t u ˆ _u ; 32 substituting Eqs. (23) an (32) into Eq. (26) yiels t _ u ˆ 1 J k Ti a B _u ˆ kt J i a B u J _ : 33 Moreover, substituting Eq. (24) into Eq. (25) yiels t i a ˆ 1 E Ri L a e b ˆ E k b _u R i a L a L a L a : 34 a In measurement, the rotating spee _u is the motor output. Accoringly, continuous state equations of the DC motor are u 1 u 6 _u 7 t 4 5 ˆ 6 B=J k T =J _u i a k b =L a R=L a i a E; 35 1=L a 2 3 u Y ˆ 1Š6 4 _u 7 5 : 36 i a Fig. 5. Block iagram of DC motor.

6 34 S.K. Yang, T.S. Liu / Reliability Engineering an System Safety 66 (1999) Fig. 6. Block iagram of simulation system Discrete state space moel The general form of state equations for a continuous system reas [1]: _V t ˆA c V t B c U t ; Y t ˆC c V t D c U t : 37 Let F c t ˆL 1 si A c 1 Š be the state transition matrix for Eq. (37), where L 1 enotes the inverse Laplace transform. The iscrete state equations sample from Eq. (37) by a sample-an-hol with time interval T secons are as follows [11]: X k 1 ˆ AX k BU k ; Y k ˆ CX k DU k ; where A ˆ F c T ; B ˆ ZT C ˆ C c ; D ˆ D c : F c t t B c ; Simulation system 4.1. Parameters Parameters for the DC motor in this stuy are prescribe as follows [12]: E ˆ 1 V; B ˆ :1 N m s; J ˆ :1 kg m 2 ; K T ˆ 1 N ma; K b ˆ :2 V s; R ˆ 1 V; L a ˆ :1 H: Substituting them into Eqs. (35) an (36), the continuous state equations of the motor become u 1 u 6 _u 7 t 4 5 ˆ 6 4 : _u ; i a 2 1 i a u Y ˆ 1Š6 4 _u 7 5 : 43 i a Besies, the following parameters are use to conuct failure preiction: 1. The failure threshol of the motor is efine as 5% less than the normal value, which is set to be the initial estimate in the Kalman preiction proceure. That is, the

7 S.K. Yang, T.S. Liu / Reliability Engineering an System Safety 66 (1999) Fig. 7. Failure time generate by Monte Carlo simulation an preicte by Kalman filter when lea-time ˆ 6 min. motor is juge to fail if the rotating spee rops to 95% of the normal value. 2. Mean time between failure (MTBF) for the motor is 1 h [13]. 3. Sampling interval T is 1 h that is the increment time for every step in Kalman preiction. 4. Disturbance W k has mean an variance.1 V [14]. 5. Measurement error V k for _u has zero mean an stanar eviation of ra s 1, which is 1% full scale accuracy [15] of the measurement. 6. PM lea-time is set at n 6 min, where n is the ahea-step number for preiction. Accoringly, the alarm signal goes on for remining PM to be execute whenever the Kalman filter preicts that the motor spee will be lower than the prescribe threshol n 6 min later.

8 36 S.K. Yang, T.S. Liu / Reliability Engineering an System Safety 66 (1999) Fig. 8. Failure time ifference between Monte Carlo simulation result an Kalman filter preiction when lea-time ˆ 6 min Monte Carlo simulation an ARMA moel Assuming failures of the motor occur ranomly. Monte Carlo simulation (MCS) is aopte to generate failure times of the motor. The relation between failure rate h(t) an istribution function of life f(t) is [1] Zt f t ˆh t exp h t tš: 44 Failures occur ranomly uring the useful life perio of a bathtub curve [1]. The failure rate is constant uring this perio. Let the failure rate in (44) be a constant l, an (44) becomes Zt f t ˆl exp l tš ˆl e lt ; 45 which is an exponential istribution function. Let u i, i ˆ 1,2,3,,m, represent a set of stanar uniformly istribute ranom numbers, the corresponing numbers t i of the ranom variable t in Eq. (45), i.e. simulate failure times, are written as [1] t i ˆ 1 l ln u i; 46 with exponential istribution. The measure ata necessary for the recursive estimation loop of the Kalman filter, as epicte in Fig. 3, are generate by ARMA moel, i.e. Eqs. (1) (3). Simulations in this stuy are performe by using MATLAB [16]. All neee ranom numbers an white sequences with prescribe variances are obtaine using the ranom number generator in MATLAB Exponential attenuator To account for the aging failure moes an the exponentially istribute failure times t i, an exponential attenuator, represente as e t/t, is place at output en of both motor system an the Kalman filter. The block iagram of simulation system is shown in Fig. 6. The symbol t of the attenuator in Fig. 6 enotes the failure time constant of the motor, which varies with failure times that are generate by MCS. 5. Results an iscussions 5.1. Results Two categories of simulation are conucte in this stuy, namely one-step-ahea preiction an two-step-ahea preiction. Accoring to the central limit theorem, estimators follow the normal istribution if the sample size is sufficiently large. The sample size of 3 is a reasonable number to use [17]. The larger the sample size is, the smaller estimate error becomes, which tens to zero when the sample size approaches infinity. Hence, each simulation is execute 1 times. Simulation results for 6 min lea-time, i.e. one-step-ahea preiction, is shown in Fig. 7. Fig. 7(a) shows the results of 1 simulations of failure times generate by MCS, failure times preicte by Kalman filter, an the associate alarm times. Fig. 7(b) shows the results of one of the 1 simulations with properly scale coorinates. The failure time ifferences between MCS an Kalman preiction are shown in Fig. 8. The mean value an the stanar eviation of the ifferences for the 1 simulations are an 65.9 min, respectively. The negative sign of the mean value inicates that the failure time preicte by Kalman filter is prior to the time generate by MCS. Accoring to the Z formula [17], the error for estimating the mean value of the sample population can be calculate

9 S.K. Yang, T.S. Liu / Reliability Engineering an System Safety 66 (1999) Fig. 9. Failure time generate by Monte Carlo simulation an preicte by Kalman filter when lea-time ˆ 12 min. by Er 2 ˆ Z2 a=2s 2 : n The Z value for a 99% confience level is [17]. Solving for E r gives E r ˆ 2:575 65:8954 p ˆ 16:97 min : 1 Accoring to the aforementione ata, there is 99% confience to say that the interval for the mean value of the time ifference between MCS an Kalman preiction is ^ min, i.e. from to min. Taking the time ifference into account, the alarm signal will appear at least min prior to failure occurrence. Results for the secon category simulation, i.e. two-stepahea preiction an lea-time for PM is 12 min, are shown in Figs. 9 an 1. The mean value of the failure time ifferences between MCS an Kalman preiction is min, an the 99% confience interval for this mean is 2.6 min. The maximum preiction error for this case is 76.4 min, which is 1.48 times greater than the error of the one-step-ahea preiction Discussions 1. In orer to avoi false alarm, the failure threshol cannot be set too close to the normal value. Otherwise, a

10 38 S.K. Yang, T.S. Liu / Reliability Engineering an System Safety 66 (1999) Fig. 1. Failure time ifference between Monte Carlo simulation result an Kalman filter preiction when lea-time ˆ 12 min. ecision-making algorithm is neee to ientify that a failure inee occurs. 2. The isturbance amplitue shoul be compose of all possible uncertainties of the motor an the environment. 3. The propose metho can not eal with abrupt changes uring a sampling interval. Thus, the sampling interval shoul not be too long. 4. As the preiction is for PM purpose, the preiction time shoul be reasonably long enough for the PM action. 5. In contrast to the eterministic portion, the variance that is riven by the isturbance of the system is small. The ifference of state variables between preiction steps faes very fast. Thus, using the N-step preictor, i.e. (2), only preiction result of the first several steps is of significance. 6. The propose metho in this stuy is exemplifie by a motor system, which is treate as a component. The proceure can be execute on a multi-component system if state equations for the components as a whole can be constructe. Performing the proceure on either the multi-component system or each of the components are both feasible. For a complicate or large system, the propose metho can be performe on those elements in minimum cut sets that are constructe by fault tree analysis or Petri net moel for failure [18]. 7. Regaring multiple failure moes, they can be moele to become moules, such as an attenuator for simulating aging failure moe for an electrical motor exemplifie in this article, an place at the system moel output en to exten the propose metho. As epicte previously, the system moel may be single-component or multicomponent. Whether the failure moules are place in serial parallel or other forms can be etermine by system failure analysis [18]. As for a multi-component system with multiple failure moes the system can be taken apart to several components an place the relate failure moule at the output en of each component to perform state estimation by Kalman filter for each component. 6. Conclusions Failure preiction simulation for PM by state estimation through Kalman filtering has been performe in this paper. Resultant preiction errors are acceptable not only for onestep-ahea preiction but also for two-step-ahea preiction. Consierations for etermining the require PM lea-time an the sampling time contraict to each other. How to compromise them an en up with an optimal value is important. To simulate the aging failure moe, a state variable, i.e. rotating spee, is monitore in this stuy. The more variables are measure, the more complicate failure moes can be simulate. Incorporating with fault tree analysis or Petri net moel for failure, the propose metho can be performe on those elements in minimum cut sets of a complicate or large system instea of on all elements of the whole system. Failure can be prevente in time so as to promote reliability only if failures can be early preicte. References [1] Rao SS. Reliability-base esign. New York: McGraw-Hill, [2] Willsky AS. A survey of esign methos for failure etection in ynamic systems. Automatica 1976;12: [3] Isermann R. Process fault etection base on moeling an estimation methos A survey. Automatica 1984;2(4): [4] Tylee JL. On-line failure etection in nuclear power plant instrumentation. IEEE Transactions on Automatic Control 1983;AC- 28(3): [5] Siar MM, Doolin BF. On th feasibility of real-time preiction of aircraft carrier motion at sea. IEEE Transactions on Automatic Control 1983;AC-28(3):

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