FDI by Extended Kalman Filter Parameter Estimation for the Industrial Actuator Benchmark

Copyright @IFACFaultDetection, Supervision and Safely for Technical Processes, Espoo, Finland, 1994

FDI BY EXTENDED KALMAH FILTER PARAMETER ESTIMATION FOR THE INDUSTRIAL ACTUATOR BENCHMARK

Bl"UCe K. Walker' KIS1g-yang HIS1g Dept . of Aerospace Engineering & Engineering Mechanics , Mail Loc. 343, Univeraity of Cincinnati . Cincinnati, OH 45221-0343 , USA

Abstract. The extended Kalman filter (EKF) algorithm is formulated as a parameter estimator and used to estimate sensor bias and actuator bias signals for the industrial actuator benchmark test system. These bias estimates are then compared instantaneously to a threshold for failure detection and identification (FDI). The paper reports the results of applying this method to given actuator benchmark data and shows that its Fol performance is good for detecting position sensor and actuator drive circuit faults in the presence of an unmodeled load change under both small and large ~litude signal conditions when the EKF in.,lementation includes parameter pseudonoise and a slow decay in the parameter dynamics. Examination of results for different in.,lementations of the parameter estimator indicate that an EKF that uses a linear model for the system dynamics does not yield acceptable FoI performance, especially for large signal ~litudes, which justifies the use of a more con.,lex, nonlinear system model in the algorithm. Key words. Failure detection; monitoring; identification; Kalman filters; nonlinear filtering; parameter estimation; servomotors

1. IIITRmUCTI(JI

k(C) • f(x(C),u(C),e,C) + w(C)

During the past two decades, numerous approaches have been suggested for detecting and identifying failures of co...,onents in dynamic systems. Survey papers describing many of these techniques include Willsky (1976), Walker (1983), Gertler (1988), Patton, et al. (1989), and Frank (1990). One particular category of failure detection approaches for dynamic systems is based upon estimating the parameters of a model describing the dynamics of the system and monitoring these estimates for changes that reflect failures. Isermann (1984) surveys these approaches. Relative to other failure detection methods, approaches based on system parameter estimation have the advantage that they produce a system dynamics model as a byproduct. However, they have the disadvantage that they often require a computational in.,lementation of the system model, which can result in considerable burden to the computational resources available, especially when failure monitoring must be accon.,lished in real time.

(1)

where the parameters to be estimated are denoted then the EKF dynamic model is:

e,

k(C)] _[f(X(C),u(c),e,c»] + [ c) F.

e(

e

[w1(C)]

(2)

wa ( C)

where the EKF state vector is partitioned into a "state" part x and a "parameter" part e. The measured outputs of the system are assumed to be given by: y( Ck )

-

[C~

: <1]

[:~ ::~] + v( C

k)

(3)

where the left·hand partition of the output matrix represents the dependence of the outputs on the system state and the right·hand partition represents the direct dependence of the outputs (if any) on the parameters to be estimated. The EKF parameter estimation algorithm then takes the following form. Assume that x(O) is random with mean Xo and covariance Px(O), that w1(t) and w2(t) are white noise processes with zero means and mean square intensities Q1(t) and Q2' and that v(k) is a white noise sequence with zero mean and positive definite covariance matrix R. The intensity matrix Q2 is the pseudonoise covariance that the EKF designer must choose, divided by the time between measurement s~l ,s. Th, augrpented state vector z is defined as z = [x le].

There are many approaches to system parameter estimat i on that can be adapted to the failure detection application. See Isermann (1984) for a summary. Among these approaches is the extended Kalman filter (EKF), which was originally formulated as a nonlinear filter for recursive state estimation when the dynamics and measurement equations of a dynamic system are nonlinear but linear i zable (Jazwinsky, 1969). It is straightforward to extend the EKF formulation to parameter estimation by considering as additional system states the parameters to be estimated (Cox, 1964). The EKF for parameter estimation then uses the nonlinear differential equations describing the dynamics of the system augmented by the dynamic equations describing the behavior of the quantities to be estimated. If the differential equations for the system are:

Let 1- (k) be the estimate of z just before incorporating the measurement at time tk and let 1 ' (k) be the estimate of z just after incorporating the measurement at time tk' and let the respe~tive erro~ covariance matrices be denoted P (k) and P (k). Note that the bottom partition of 1(k) is then the current estimate of the parameter vector e. The algorithm then proceeds as follows: 481

1. Initialize ~·Ck) at [XoT : 6 0 T]T, where 6 0 is

Note that the formulation just given for the EKF requires the assumption of dynamics describing the time behavior of the parameters. The natural assumption in most applications is that the parameters are constant except possibly for occasional sudden changes (as when a fault occurs), in which case Fs and w2 are both assumed to be zero. However, it has been shown (Ljung, 1979) that standard EKF-based parameter estimators with the static parameter assumption generally do not provide convergent parameter estimates. Ljung (1979) suggests a modification to the algorithm to guarantee convergence of the parameter estimates that essentially requires estimation of the filter gains (or some parameterization of them) in addition to the system state and the system parameters. This can result in a very complex implementation, however, which merely exacerbates the computational burden that the standard EKF already carries.

an initial parameter estimate vector that must be chosen by the designer, which can be something other than the true mean of x(O), if desired. Initialize the expected error covariance matrix as p+(O) as a block diagonal matrix with upper left block Px(O) and lower right block P9(0), where both Px(O) and P9(0) must be chosen by the designer (though Px(O) is typically chosen to be its true value, if known). Set k=O. 2. Integrate (numerically) the nonlinear equations (2) from tk to tk+l using R-Ck) as the initial condition for the states and using O·Ck) as the initial parameter values, where the noises Wl(t) and w2(t) are assumed to be zero for this calculation. The results are the propagated estimates R-Ck+l) and O-Ck+l). Simultaneously, form the linearized system dynamics matrices: afCx, u, 6, t) Ix." , afCx, u, 6, t) Ix." (4)

as

,.8

ax

Another modification to the EKF algorithm that improves the convergence properties of its parameter estimates without adding to the computational burden is the assumption of a nonzero covariance Q2 for the "pseudonoise ll process w2 driving the values of the parameters. The relationship between the pseudonoise covariance and the convergence behavior of the parameter estimates is studied quantitatively for simple linear system identification problems in (Hill & Walker, 1991; Hill, 1993; Hill & Walker, 1994). Investigation of the relationship for more complex cases remains a topic of investigation. This modification will be used here.

,.8

(These matrices can be computed once at time tk us i ng the est imates R- (k) and 0- (k) , or for more accurate integration, they can be evaluated at numerous time points between tk and tk+l using the result of the state equation integration for R and O-(k) _) Use these matrices to construct F as: af(x,u,e, e) af(x,u,e,t)l

ae

ax

F •

(5)

o

F,

,.,."

As noted above, the natural assumption for the parameter dynamics matrix FS is zero, corresponding to static parameters. However, this assumption results in marginally stable parameter estimate behavior between measurement samples and this can be undesirable, especially when the system includes unmodelled effects. Therefore, FS is sometimes chosen to be a stable matrix with eigenvalues very near zero but with slightly negative real parts. This has the advantage of aiding the parameter estimate behavior but the disadvantage of producing biased parameter estimates when convergence takes place. In the failure detection context, this bias can often be tolerated provided the threshold selection process accounts for its presence.

and form the exponential matrix • • e,,·t where At is (tk+l-tk) (or smaller if the matrices are evaluated at intermediate time points)_ Use Ql to form the mean square contribution of the noise wl(t) to the state covariance (usually just Ql(tk) At) and likewise the mean square contribution of the pseudonoise w2 to the covariance of 6 (usually just Q2 At), then use these matrices to form the upper left and lower right (respectively) partitions of the matrix Q_ Use. and Q to propagate the error covariance matrix as: P- (k+l) ... po (k) +0 (6) 3_ Set k = k+1 and read the new measurement y(k). Form the measurement residual r(k) as:

.2'

r(k)

= y(k)

- 9(k)

- y(k)

-

e R(k)

The EKF parameter estimator using the pseudonoise assumpt i on (but with FS cO) has been used previously by the authors and their colleagues to identify parameters in dynamic models of complex dynamic systems including high performance aircraft (Garcia-Velo and Walker, 1992) and the Space Shuttle Main Engine (Walker and Baumgartner, 1990). In the latter application, as well as in this paper, the parameter estimates produced by the EKF are used to determine the failure status of the system.

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4. Let: H

= [ e" : e. I

(8)

Compute the Kalman gain matrix K(k) as:

1-1 (9) Finally, use the Kalman gain matrix to update the state and parameter estimates as: K(k) -P"(k) H2'[HP"(k) H2'+R(k)

~. (k)

• ~- (k)

+ K(k)

rCk)

The subsequent section formulates an EKF-based parameter estimator for the industrial actuator benchmark problem described in Nielsen et al (1993). The parameter estimation and failure detection results obtained with the EKF-based parameter estimator are sllll1\8rized in Section 3. Section 4 presents some brief conclusions.

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and to update the expected error covariance as: po (k) .. [I-K(k) Hl P" (k) [I-K(k) HJ T (11) +K(k) R(k) KT(k)

5. Return to Step 2 unless the end of the data has been reached. The speed of the EKF by the rate at which the system equations dynamics required in

algorithm tends to be limited the numerical integration of and the linearization of the Step 2 can be accomplished.

2. APPLlCATUIf TO

IIIXJSTRIAL BElCIIWtI:

The industrial actuator hardware that provides the setting for the benchmark problem is described in

482

The parameters to be estimated are the position sensor bias f. and the actuator current bias fa' Thus: (13) 81' '" [fa fa]

Nielsen et al. (1993). Complex and simplified models for the actuator dynamics are also provided in Nielsen et al. (1993). Essentially, the actuator is an electromechanical position control servo that uses both internal velocity feedback through a proportional-plus-integral (PI) control law and position feedback through an external PI controller to drive the position of its output wheel. This wheel, through a geartrain, drives a rotary load that includes inertia and friction and an additional load torque that is unknown and can change. Measurements, possibly noisy, are available for the motor velocity and the position of the servo wheel every 0.01 seconds (frequency lOO Hz). Saturation limits on input current, gear backlash, and stiction and coulomb friction provide nonlinear elements in the system. The position sensor and the motor input current are both subject to additive faults.

The differential equations for the linear and nonlinear models are given in Nielsen et al. (1993) and are not repeated here. L,t the measurement vector y be defined as y C ["m so), Then, for the three-state linear model, the output relationship yields: 0 1 0 ] C" - [ 0 0 1 ;

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while for the six-state nonlinear model: C" - [

Nielsen et al. (1993), recorded measurement and speed reference input data are provided for 300 samples (3.0 seconds) representing various levels of complexity for the servo dynamics (linear nonlinear but differentiable, and complex nonline~r) and for the signals (small amplitude with no noise, small amplitude with noise, and large amplitude with noise). In each case, the position sensor failure is present from 0.7 sec to 0.9 sec and the current failure is present from 2.7 seconds to the end of the data at 3.0 sec. Also, an additional (unknown) load torque is present from 1.2 sec to 2.3 sec. ~ith

~ ~ ~ ~ ~ ~];

C. -

[~ ~]

(15)

For the benchmark application, the EKF parameter estimator is now set up. Several design choices must still be made to specify the EKF, and these will be choices will be discussed in the results section, which follows. Also, the FOI performance results for simple threshold logic applied to the current and sensed position bias estimates will be presented. 3_ RESUlTS

An EKF parameter estimator was first designed using the linear model of the dynamics given in Nielsen et al. (1993) as its embedded system model. Henceforth, this estimator is referred to as EKF-L. Later, an EKF parameter estimator was constructed that uses the Usoft" nonl inear system model of Nielsen et al. (1993) as its embedded system model. This estimator is hereafter referred to as EKF-NL. The results of applying these two EKF-based parameter estimators to the data provided with Nielsen et al. (1993) will be described in this section. All of the coding to implement the algorithms was done in MATLAB/Simul ink.

The failure detection objective for this system is to develop and test an algorithm that is designed to quickly detect the current failure and th! position sensor failure without false detectIons induced by the noises, nonl ineari ties, and disturbances (such as the onset and vanishing of the unknown load torque). In the results to be shown below, we accomplish this using an EKF-based parameter estimation scheme, as described above. For the benchmark problem, the failure modes include a sensor fault mode (position) and an actuator fault mode (current into motor). Each of the fault modes involves the addition of an extraneous signal that is low frequency in nature (essentially a bias for the current failure and a ramp for the sensor failure) for a brief time interval. Thus, the appropriate formulation for the EKF parameter estimator is that it estimate the bias present in the measured position signal and in the actuator current signal_ Choosing the state vector for the system as given in Nielsen et al. (1993), we have:

Before the numerical results are presented, it should be noted that the position sensor measurement signals provided with Nielsen et al_ (1993) were scaled by dividing them by 0_978 before they were used in the estimation algorithms. This effectively cancels the position sensor scale factor «. that was present in the simulations in Nielsen et al. (1993) that created the data. Because this scale factor is known exactly, no generality is lost by cancelling it.

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The data provided with Nielsen et al. (1993) consist of four data sets, where each set is characterized as follows in terms of the system model and signal conditions that were used to produce them: Data set 1: Linear model, small signals, no noise. Data set 2: Soft nonlinear model, small signals, no noise. Data set 3: Complex nonlinear model, small signals, noise. Data set 4: Complex nonlinear model, large signals, noise. Each data set uses At = .01 sec.

where nraf , tilt is the filtered motor speed reference (produced by a given first order filter acting on the position controller speed command corrupted by a speed demand noise), i2 is the PI velocity controller internal state variable, "m is the servomotor speed, nl is the load speed, So is the servo position, and sl is the load position (which differs from the servo position due to flexibility and backlash in the geartrain connecting the servo and load). The states designated by a * in (12) are used only for a sixstate EKF based on the "soft" nonl inear model given in Nielsen et al. (1993), as opposed to a three-state EKF based upon the simpler linear model given in Nielsen et al. (1993) that uses only the states not designated with a *.

By observing the typical values of the states, the failure magnitudes, and the noises from data sets 1 through 3, the following values of the state noise intensity Q1 and the measurement noise

483

covariance were used throughout the study for the small amplitude s~gnal ~ases: Q1 = diag[ (.2) (.4) (.005)2 ]/At for EKF-L Q1 = diag[ (.4)2 (.2)2 (.4)2 (.5)2 (.005)2 (.005)2 ]/At for EKF'NL R = diag[ (.4)2 (.005)2] for both where the state vector for EKF·L and EKF·NL is given in (12). For the large signal amplitude da:a (data set 4), the covariances of t~e state nOIse were scaled up by a factor of (3) , reflecting the approximate threefold increase in the signal magnitudes for this case. The initial state estimation error covariance was chosen to be an identity matrix (of appropriate dimension) for all cases, yielding an initial 1 0 uncertainty for each state of 1 unit.

.. ."L-______-.I

.•. _.L----------l

Fig. 1. State and parameter estimates for EKF-L using data set 3

The pseudonoise intensity Q2, as noted above, is an important parameter in determining the convergence behavior of the parameter estimates. Based upon previous studies (Garcia'Velo and Walker, 1992; Walker and Baumgartner, 1990; Hill & Walker, 1991), a rule of thumb is that the assumed parameter driving noise should have a 1 0 value over each time step of approximately K 6nominal where K is a number on the order of .001 to .2, with smaller values desirable for long·term convergence and larger values desirable for fast response to changes in the parameter value. The latter case applies to the failure detection problem, so K was chosen to be .2, and the nominal parameter values were chosen as the approximate maxirrun failure magnitudes that are to be detected. In particular, this translated into pseudonoise inten~ities o~: Q2 = diag[ (.4) (.005) ] for small signals Q2 = diag[ (1.)2 (.015)2] for large signals

therefore the state estimate histories will not be presented hereafter. The threshold for simple, threshold'based failure detection testing is set to C 0 where 0 is the square root of the calculated covariance of the parameter estimation error, which is the appropriate diagonal element of P+(k) produced by the EKF calculations. The multiplier C is determined by approximating the minirrun C for which C 0 exceeds the magnitude of the parameter estimate at all times for all nominal (i.e. no failures) simulations. No effort was made to optimize the thresholds in this study. Applying a threshold of .33 times the calculated 0 of the sensor bias estimate to the EKF-L data shown in Fig. 1, the sensor failure is first detected at t=0.74 sec, or .04 sec after its onset, and continues to be detected ~ti l tc.9O sec, with no false alarms. For the current failure, a threshold of 3 0 yields a detection at t=2.76 sec, or .06 sec after its onset, and continues to indicate the failure is present ~til t=2.85 sec, with no false alarms. Note, however, that the current bias estimate is subject to a relatively large estimation bias when the ~known load is present. Although this bias is smaller than the current failure signal that must be detected, it is a significant fraction thereof and is therefore undesirable.

These values were used throughout the study. Other values were tried, both smaller and larger, but the estimation results were very similar qualitatively until the pseudonoise was either made so small that the corresponding parameter was unable to converge or so large that the random perturbation in the estimate masked the value (if any) to which it was converging. When EKF·L and EKF-NL are applied to data sets 1 and 2, the state estimation and bias estimation results are extremely good with the exception of the current bias estimate, which is biased when the unknown load is present (by approximately 5 A for EKF'L, which is consistent with the values given in Nielsen et al. (1993). This corresponds exactly to the notion expressed in Nielsen et al. (1993) that a detection scheme based upon a linear system model cannot distinguish between current failures and unknown loads. For brevity and because the more complex system model results are of more interest, the results for data sets 1 and 2 are omitted here.

In an effort to reduce the bias error in the current bias estimate, the EKF algorithms were modified to include the estimate decay dynamics represented by F9 in the equations describing the EKF. Trial-and-error testing of both EKF-L and EKF-NL indicated that reasonable results were obtained when F9 was chosen such that the current bias estimate decreases by 2X at each time step and the sensor bias estimate decreases 1~ for each time step in the absence of driving terms.

For data set 3 using EKF'L, Fig. 1 shows the time

Fig. 2 shows the current bias estimate with

histories of the actual and estimated values of the two measured states "m and So (top two plots) and the actual and estimated values of the current bias fa and the sensor bias fs (bottom two plots) using data set 3. The actual values are plotted as solid lines while the estimated values are plotted as dashed lines. For "m and so' the estimates are so close to the actual values (except when failures are present) that it is difficult to distinguish them. This is the case for the state estimates produced by essentially all of the modified versions of EKF-L, and

thresholds at +/- 2 o. The bias estimate now has considerably less bias than the EKF-L result without the parameter estimate dynamics, but this improvement is gained at the cost of more delay in detecting failures. The current failure is now detected at all time steps from tz2.76 sec to t=2.85 sec with no false alarms. Fig. 3 shows the position sensor bias estimate for this case with +/- 1 0 thresholds.

The sensor failure is detected at all times from t=O.77 sec to 0.90 sec

484

be distinguished from the unknown load effect. When nonzero F9 is included, the I.I'Iknown load effect is diminished, but the estimated current bias during the presence of the failure is also diminished. The failure can just barely be distinguished, but a threshold of 9 a is required to avoid false alarms (and a larger I.I'Iknown load could still produce false alarms), and the failure is not detected I.I'Itil t=2.92 sec, or .22 sec after onset .

t . . . . UI .......

Fig. 5 presents the analogous comparison of sensor •.•..-_____-----.-.!:,.!..!!'~:..!_~4:..! . •::::_~-=_=.~-------.-----,

..•

...

..•

...

Fig. 2. Estimate of current bias with t thresholds for EKF-L using data set 3

2a I

...

f •• _

..........

, .- ,

....,

..-

~ .t

-..... .....

.L__________........____......______~----:":_----_:!

Fig . 5. EKF-L estimates of sensor bias using data set 4 showing improvement due to parameter decay

..

,

-' ---

bias estimates for EKF-l applied to data set 4. As for the current bias estimates above, the unknown load effect on the sensor bias estimates is severe, though mitigated somewhat by the use of nonzero F9' However, another problem now becomes apparent, namely that the sensor bias estimates are now "fooled" by the actuator failure near the end of the data history, and the use of nonzero F9 has little effect on this_ A threshold of 5 a avoids false alarms due to the actuator failure effect, but does not detect the sensor failure I.I'Itil t=_85 sec, or .15 sec after onset. Because larger actuator failures could produce even larger effects on this estimate, this performance is probably not acceptable. One of the sources of error in using EKF-l is the simpl ification to a linear model of the system dynamics embedded in the EKf-l algorithm. The EKf-Nl algorithm uses a more sophisticated nonlinear system model for its calculations, and therefore has the potential to yield improved performance relative to the EKF-l algorithm at the expense of a more complex implementation.

0

Fig. 3. Estimate of sensor bias with t thresholds for EKF-L using data set 3

1a

with no false alarms. Thus, the sensor failure detection time is delayed slightly relative to the previous case. We next applied EKF-L to the large signal data in data set 4. Fig. 4 compares the time histories of the current bias estimates for EKf-L with F9=O ' . . h , . . , . . . . . ., .

, ...... . . . . . . . . . . . . .u .... .

o /

-t •

. ____ ~.L

~

__________________

~

____

~.~.~

__

~

Fig. 4. EKF-L estimates of current bias using data set 4 showing improvement due to parameter decay (solid line) vs. EKF-L with the nonzero F9 discussed above (dashed line). Clearly, the unknown load effect is more severe when F9=O is assumed, so much so that the failure effecf ' cannot

fig. 6. Comparison of EKF-L and EKF-Nl current bias estimates for small signals 485

Figs. 6 and 7 compare the estimates of the current bias (Fig. 6) and of the sensor bias (Fig. 7) produced by the EKF-L and EKF-NL algorithms using nonzero Fe applied to the small signal amplitude

, __ . . .. . . . . . . . . . . . . 1. . . . . . . . 4

• data in data set 3. The figures show the time histories of the absolute values of each estimate, f . _ _ . ....... _

......... . . . . _ • •

.... .. .....

Fig. 8. Comparison of EKF-L and EKF-NL current bias estimates for large signals From Fig. 8, it is clear that the unknown load has considerably less impact on the EKF-NL current bias estimate than on the EKF-L estimate. Furthermore, the EKF-NL estimate more reliably indicates the presence of the current failure, though the magnitude of the estimated size is rather small. A threshold of 5 q produces a current failure detection for all time samples between t=2.75 sec (.05 sec after onset) and t z3.0 sec. However, false current failure detections are indicated for time samples between tz1.54 sec and t=1.56 sec and between t=2.56 sec and t=2.63 sec. These time intervals correspond to changes in the commanded position, and the false failure indication is as much due to the decrease that occurs in q for the current bias estimate (due to more input excitation) as they are due to the transient behavior of the estimate itself. Raising the threshold to 7.5 q can eliminate these false excursions over the threshold, but the detection of the current failure is then delayed to t=2.76 sec.

Fig. 7. Comparison for EKF-L and EKF-NL sensor bias estimates for small signals with the solid line produced by EKF-NL and the dotted line produced by EKF-L. (The state estimate histories for the three states that are common to the two models, i 2 , Om, and so' are similar and are not shown. Also, the results for Fe=O are similar to Figs. 6 and 7 except that the unknown load effect is larger for both algorithms. These results are also not shown.) From Figs. 6 and 7, the results for EKF-L and EKFNL are similar except that EKF-NL produces more accurate estimates of the current bias when the current failure is present. In fact, Fig. 6 shows that the current bias estimate produced by EKF-L decays to zero well before the duration of the current failure is complete, while the EKF-NL estimate remains significantly nonzero until the end of the data. A threshold of 5 q applied to the EKF-NL current bias estimate yields a current failure detection at all times from t=2.76 sec (.06 sec after onset) to t=3.0 sec with no false alarms. For the sensor bias estimate, no failure data suggests a threshold of 5 q, and when this threshold is applied to the EKF-NL sensor bias estimates, the sensor bias is detected for all time samples from t=.77 sec (.07 sec after onset) to t=.91 sec. However, the effect of the actuator failure on the sensor bias estimate is large enough to produce false sensor failure detections at t=2.82 and 2.83 sec. No effort was made to modify the EKF-NL algorithm to correct this behavior, since it probably represents a realistic performance result.

•.•• r---~-....!::'..~-~.!! '-:"':-=''':'';'';:'''::''~'':'':''-=-:::.;-::...!.'--..----.,

....

....

....

..• --'",- -'

The advantages of using a nonlinear embedded system model in the EKF algorithm should be more pronounced for large signal amplitude cases, and this hypothesis is tested here by comparing the performance of EKF-NL to that of EKF-L when they are applied to data set 4 using nonzero Fe. (The results for Fe=O are similar to the nonzero Fe results shown here except that the effect of the .- unknown load is more severe, particularly for the EKF-L results, as noted in the discussion above.) Figs. 8 and 9 present this comparison for the magnitudes of the current bias estimates (Fig. 8) and the magnitudes of the sensor bias estimates (Fig. 9). The EKF-NL results are depicted as solid lines while the EKF-L results are depicted as dashed lines.

Fig. 9. Comparison of EKF-L and EKF-NL sensor bias estimates for large signals The sensor bias estimate comparison in Fig. 9 shows that both sensor bias estimates are affected considerably by the unknown load, which produces bias in the estimates even with nonzero Fe. The bias is so large relative to the calculated q for the estimate (by a factor of approximately 40) that it is meaningless to apply a threshold based upon q to these estimates to perform failure detection. Note that the EKF-NL result is not perceptibly affected by the onset of the actuator failure, as opposed to the EKF·L estimate, which is affected enough that a false sensor failure detection could be induced by the actuator failure effect. A constant threshold of .07 red yields 486

frank, P.M. (1990). Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy - A survey and some new resul ts. Automatica. 26. 459-474. Garcia-Velo, J. and ~alker, B.K. (1992). Aircraft parameter identification by nonlinear filtering techniques," report to NASA Langley Research Ctr. on Grant NCC1·165, U. Cincinnati. Gertler, J. (1988). Survey of model based failure detection and isolation in complex plants. IEEE Control Systems Magazine. 8. 6, 3·

detection of the sensor failure by the EKF-NL estimate for all time samples between t=.87 sec (.07 sec after onset) and t=.90 sec. The results in Fig. 9 suggest that an alternative approach to detecting sensor failures should be considered.

The results for data set 4 were verified by using the MATLAB/Simulink codes obtained from the authors of [1] to produce large signal amplitude data sets similar to data set 4 but with different random noise sequences. The results of using EKF· NL on these data were essentially identical to the results reported here.

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Hill, B.K., and B.K. ~alker (1991). Approximate effect of parameter pseudonoise intensity on rate of convergence for EKf parameter estimators. Proc. of 30th Conf. on Decision & Control. IEEE, New York, 1690-1697. Hill, B.K., and B.K. ~alker (1994). On approximating the parameter estimate convergence rate of extended Kalman filter parameter estimators. Submitted to IEEE Trans. on Auto. Control. Hill, B.K., Huang, K. and ~alker, B.K. (1991). Effect of parameter pseudonoise intensity on parameter estimators for SSME turbopump parameters. Proc. 3rd Annual Health Monitoring Conf. for Space Propulsion Systems (Cincinnati), pp. 265-295, U. Cincinnati. Hill, B.K. (1993). Approximate effect of parameter pseudonoise intensity on rate of convergence of EKf parameter estimators. Ph.D. thesis, Dept. of Elect. & Computer Eng., U. Cincinnati, July 1993. Isermam, R. (1984). Process fault detection based on modelling and estimation methods. Automatica. 20, 387-404. Jazwinsky, A.H. (1969). Stochastic processes and filtering theory. Academic Press, New York. Ljung, L. (1978). Convergence analysis of parameter identification algorithms. IEEE Trans. on Auto. Control. AC-Z3. 5, 770-783. Ljung, L. (1979). Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems. IEEE Trans. on Auto. Control. AC-24. 1, 36-50. Nielsen, S.B., Patton, R., Blanke, M., and Jorgensen, R.B. (1993). Industrial actuator benchmark test. York Univ. (York, UK) and Aalborg Univ. (Aalborg, DENMARK). Patton, R.J., P.M. Frank, and R.N. Clark (ed.) (1989), Fault Diagnosis in Dynamic Systems: Theory and Application. Prentice Hall International. ~alker, B.K. (1983) . Recent developments in fault diagnosis and accommodation. AIM paper 83-2258, AIM Guidance & Control Conf •• AIM, ~ashington, DC. ~alker. B.K. and Baumgartner, E.T. (1990). Comparison of non I inear smoothers and nonl inear estimators for rocket engine health monitoring. AIM paper 90-1891, 26th Joint Propulsion Conf. (Orlando), AIM, ~ashington, DC. ~illsky, A.S. (1976). A survey of design methods for failure detection in dynamic systems. Automat i ca. 12. 601-611.

As noted above, the improved performance of the EKF-NL algorithm over the EKF-L algorithm is obtained at the cost of more complexity in the implementation. This is reflected by the relative times required to process the 300 measurement data samples through each algorithm after they were implemented in MATLAB/Simulink. The EKF-L algorithm typically required about 2 minutes to run on a time-shared IBM RS6000 workstation. The EKF-NL algorithm with fourth order Runge-Kutta integration of the state equations (via OOE45 of MATLAB), and with nearly twice as many states as the EKF-L algorithm, typically required several hours to run on the same workstation. It should be noted, however, that no effort was made to make the codes efficient either in terms of computational algorithms or memory and disk storage use. 4. aJIICllJS HJlS

The results reported here indicate that the EKF parameter estimator is a viable approach to detecting the motor drive and sensor failures of interest in the industrial actuator benchmark problem. The EKf algorithm produces estimates of the failure magnitudes that converge quickly when sufficient pseudonoise is assumed to be driving the parameters. The assumption of stable parameter dynamics in the EKF system model helps to mitigate the effects of unknown load disturbances. An EKf parameter estimator based upon a I inear model for the system dynamics, whi le efficient computationally, has the drawback that its parameter estimate accuracy is marginal and that its estimate of the sensor failure size is affected considerably by the presence of an actuator drive failure. Use of a nonlinear system model in the EKF mitigates this effect, but at the price of much additional complexity. Simple threshold logic applied to the bias estimates, where the threshold is chosen to be a multiple of the computed parameter error standard deviation large enough that no false alarms are expected based upon data from unfailed cases, yielded fast detection of the actuator and sensor failures, usually with a detection delay of no more than 6 time samples, with no, or very few, false detections. Open questions remain, however, regarding the optimal tuning of the EKF parameter estimator by selection of its assumed noise covariances, the optimal selection of the thresholds, and the possibility that more sophisticated decision logic would result in improved detection performance. 5. REFERENceS Cox, H. (1964). On the estimation of state variables and parameters for noisy dynamic systems. IEEE Trans. on Auto. Control. AC-9, 1, 5·12. 487