fault diagnosis for a hydraulic drive system using a parameter-estimation method

Pergamon

PII:S0967-0661 (97)00103 -2

Control Eng. Practice, Vol. 5, No.9, pp. 1283-1291, 1997

Copyright © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved

00967-0661/97 $17.00+0.00

FAULT DIAGNOSIS FOR A HYDRAULIC DRIVE SYSTEM USING A PARAMETER-ESTIMATION METHOD

D . Y u

Control Systems Research Group, School of Electrical Engineering, Liverpool John Moores University, Byrom Street. Liverpool L8 8AF, UK ([email protected])

(Received January 1997; in final form June 1997)

Abst rac t . Fault diagnosis using a paxameter-estimation method is investigated in this paper. A digital state variable filter is employed to obtain derivatives of the variables, and an interpolation technique is used to approximate the values of the variables between samples. The method is applied to a hydraulic test rig based on real data, mad the simulated faults - changes in the physical parameters - are diagnosed. The simulation results demonstrate that faults having the same direction in the state space of the system mbdel can be isolated using this technique, which is impossible using any observer-based method or any parity equation method. Fault size can also be approximately identified.

Copyright © 1997 Elsevier Science Ltd

Ke3rwords: Fault diagnosis, parameter estimation, hydraulic systems, continuous system identification.

1. INTRODUCTION

Model-based fault detection has been studied for two decades. A number of observer-based approaches have been proposed for linear systems, for example, the eigenstructure assignment method (Pat ton and Chen, 1991), the unknown input observer (Kudva et al., 1980; Frank and Wunnenberg, 1989; Hou and Muller, 1994; Chen et al., 1996), and recently for bilinear systems the bilinear fault detection observer (BFDO) (Yu and Shields, 1996; Yu et al., 1996a) and a bilinear reduced-order observer (Hac, 1992; Yu et al., 1996b). Anothdr kind of method is the parity space method (Chow and Willsky, 1984; Lou et al., 1986; Gertler and Kunwer, 1995). A common feature of these methods in isolating faults is that they utilize the different directions of the different faults in the state space of the model. As a result, a disadvantage of these methods is that they are not capable of isolating faults that have the same direction in the system state space.

The parameter-est imation method for fault diagnosis (Isermann, 1984, 1993) can detect and iso-

late faults, and may diagnose fault size, even for faults having the same direction in the stat e space of the model. Parameter estimation for continuous systems is usually employed in fault diagnosis since an explicit relationship between the physical parameters and the model parameters of a plant can be easily obtained. A limitation to the parameter estimation method is that the number of physical parameters must be less than or equal to that of model parameters, so as to transform the model parameters uniquely to physical parameters. This condition, however, can sometimes not be satisfied in practice (this is the case for the application example in this paper). To solve this practical problem, a combination of the observer-based method and the parameter- estimation method is considered in this paper, and applied to a hydraulic test rig. First, a bilinear model was established for the system. Then, a bilinear fault-detection observer was applied to this model to detect and isolate faults (Yu et al., 1996a). Having the same direction in the system state space, some physical parameters, such as the efficiency of the hydraulic motor, r/m, and the efficiency of the hydraulic pump, r/p, cannot be iso-

1283

1284 D. Yu

lated from one another. Therefore, the parameter- estimation method was used for some local linear models of the system, in order to isolate these faults. For simplicity, the observer-based fault detection part is not described here; the details can be found in (Yu, 1995) and (Yu et al., 1996a). Only the parameter-estimation part is described in this paper.

In continuous parameter estimation the state variable filter (SVF) is often used to estimate derivatives in the system differential equations (Iser- mann, 1984, 1993). Peter and Isermann (1989) used the interpolation techniques in a digital SVF to estimate model parameters for a continuous system. This method is used in this paper to diagnose faults in a hydraulic system. The method is briefly introduced in Section 2, and the hydraulic system is described in Section 3. The simulated faults - the changes in a number of physical parameters of the system - were diagnosed by monitoring the model parameters. This is described in Section 4.

2. A CONTINUOUS PARAMETER-ESTIMATION METHOD

Continuous-time parameter-estimation methods have been developed for quite a long time (Young, 1981). The method using the state variable filter can be briefly described as follows. Consider a single-input single-output system described by a linear differential equation

m

y(t) = - + Z bju(J)(t) + i=1 j = 0

+ v(t), n > _ m (1)

where u(t), y(t) and v(t) are the input, output and noise, respectively. The basic idea of the SVF method is to transform the original system model into an estimated model by introducing a set of identical linear filters, operating on each term in the original model. Let g(t) be the transfer function of the filter. The transformed system model is then given by

ot y(~-)g(t - T)dT =

° fot -- E ai y(i)(T)g(t -- T)dT i=1

m f0 t + E b j UU)(T)g( t - - r )dT j = 0

f + v(~-)g(t - ~')d~- (2)

where y(i)(r) and u(J)(r) denote derivatives of order i and j respectively. Introducing the state

variables yFi(t), UFj(t) and vFo(t), equation (2) can be simplified into

n

YFo(t) -- E a~yFi(t) i=1

m

+ E b j u F j ( t ) +VFo(t) . (3) j = 0

From filter theory it is well known that the following identity holds for any linear filter g(t):

f = X(O(T)g(t - r)dT

fo t (t - T)dT + Xoi(t) x(T)g (0

with

i - 1 = )

k=0

_

Here, filter g(t) can be chosen such that xo~(t) converges to zero (for choosing g(t) see (Peter and Isermann, 1989)). The state variables yFi(t) and uFi(t), therefore, can be obtained by operating a variety of filters characterized by weighting func- tions g(O(t) on u(t) and y(t) respectively.

In the case of a digital filter, the state-space representation is

£(t) = Az(t) + bx(t) (4) xFo(t) = c%(t) (5)

and the output of the filter at t = kT , where T is the sample interval, is given by

z (kT) = eATz((k -- 1)T) kT

q- f eA(kT-r)Dx(T)dT (6) J( k-1)T

where e At is the transition matrix of the filter. Since only the samples of signal x(t) , x ( kT) , are available, the state variables can only be calculated by an approximation where x ( ( k , 1)T) is used to approximate X(T) ((k - 1)T < T < kT):

£(kT) = eAT~.((k -- 1)T)

+ eA(kT- ' )bdTx( (k - 1)T)(7) k-1)T

Approximations of the filter output in (7) are precise only when the sample interval is small enough. In order to reduce the approximation error, Peter and Isermann (1989) applied interpolation techniques to x(r) to produce a polynomial function of order r

P(~ =Po + p i t + "'" + p r ~ (8)

Fault Diagnosis for a Hydraulic Drive System

where t" = k T - r , which can be used as an approximation of the unknown z(r). Substituting x(r) in (6) by p(r) yields

where

f ~(kT) = eAr 2((k - 1)T) + ~'~pibi (9)

i=0

jr0 T

bi = eMtidt, i = 1 , . . . , r . (10)

The integral term in equation (10) depends on the filter chosen in (4)-(5) and polynomial function (8) so that it can be solved analytically. Alternatively, bi can also be calculated iteratively, by using the following algorithm,

f = A-leAWb

bc = f - A - l b

bi = T i f - i A - Y b i - 1 , i = l , . - - , r .

1285

Here, A is chosen to be full rank in the filter design. For the detail see (Peter and Isermann, 1989). Model parameters, ai and bj, then can be estimated on-line using the recursive least-squares algorithm with a forgetting factor A.

3. HYDRAULIC TEST RIG

The test rig was purpose-built as a vehicle for test- ing advanced control and monitoring strategies (Daley, 1987). It consists of a stiff shaft which is driven by a hydraulic motor and loaded with a hydraulic pump. The oil flow to the motor is controlled by an electro-hydraulic servo-valve, and the pressure differential across the pump can be changed to increase or decrease the toad on the shaft. The rig is instrumented so that the input to the servo-valve, the shaft speed and the pressure differentials across the motor and the pump can all be monitored. The schematic of the hydraulic test rig is displayed in Fig.1.

v(t)

.[ control

logic

,1 servo

m o t o r

xs(t)

Ps

valve monitor

Pm

Qv = K ( P s - Pro) °5

hydraulic

motor

gear

Tm ~

monitor

Pp

hydraulic

pump

torque

s e n s o r

speed

sensor

Fig. 1. Schematic of the hydraulic test rig

1286 D. Yu

3.1 The Mathematical Model

Considering the second-order dynamics of the valve, the spool valve displacement, X~(t), sat- isfies

T1T2J(8(t)+(TI+T2)28(t)+X,(t) = K,v( t ) (11)

where v(t) is voltage input to the valve, T1 and T2 are electro-magnetic and electro-mechanic time constants of the servo-valve respectively and /{8 is the valve gain. The flow rate, Qy(t), through the valve can be approximated by a square-root relation of the orifice (Merritt, 1967):

Qv(t) = KoXs(t)(P~(t) - Pro(t))½ (12)

where Ps(t) is the supply pressure, Pro(t) is the pressure differential across the motor and Ke is the valve flow coefficient.

For continuity of flow, the flow rate, Qv(t), can be expressed as

Q,(t) = CrS~(t) + P,~(t) + KiPm(t) (13)

where Ss(t) is the shaft angular velocity, Cr is the motor displacement, Vt is the total trapped volume, 13 is the oil bulk modulus and Kl is a leakage coefficient. The motor torque is

Tin(t) = C~lmPm(t) (14)

where 7/m is the efficiency of the motor. Ignoring static and coulomb friction,

Tin(t) = I & ( t ) + DS,( t ) + Tp(t) (15)

where I is the total inertia of the pump, motor and the shaft, D is the viscous friction coefficient and

Tp(t) = C~ ~---Pp(t) (16) Vp

where Pp(t) is the pressure differential across the pump and r/p is the efficiency of the pump.

3. 2 Real data collection

Real data from the test rig were collected when the system was in the following conditions. The test rig was subjected to a voltage input to the valve, v(t), and a pump pressure, Pp(t), while t.he supply pressure, Ps, was kept almost constant. The motor pressure, P,~(t), the shaft speed, Ss(t), and the inputs were measured using analogue transducers. These input and output signals were then sampled with a sample interval, T~ = 10 ms. Three thou- sand samples were collected, and are displayed in Fig. 2.

150

145

s u p p l y p r e s s u r e P s ( b a r )

140

135

130 0

I I

1000 2000 3000

s a m p l e t i m e

Fig. 2a. The supply pressure Ps(t).

0 vo l t age i n p u t to valve: v (volt) 1 i

- 0 . 5

-I

- 1 . 5

- - 2 '

0 I L

I000 2000 3000

s a m p l e t i m e

Fig. 2b. The voltage input to the valve: v(t).

40 PUmP p r e s s u r e Pp(bar).

35

30 0

s a m p l e t i m e

Fig. 2c. The pump pressure Pp(t).

I I

1000 2000 3000

Fault Diagnosis for a Hydraulic Drive System 1287

I

60

50

40 0

m o t o r p r e s s u r e P m bar ) 70

1000

i i

I

H

2000

s a m p l e t i m e

Fig. 2d. The motor pressure Pm(t).

3000

of the system. The second is that the parameter- estimation method cannot be used to diagnose all the possible faults on its own, since there are more physical parameters than model parameters. However, the method can be used to detect and isolate faults in cooperation with an observer- based method. In fact, most of the possible faults in this system have been diagnosed using the bilinear fault detection observer (BFDO) method (see Yu et al., 1996a). Neverthless, some changes in the physical parameters of the system, such as the efficiency of the hydraulic motor (~m) and the efficiency of the hydraulic pump (~p), the total inertia of the pump (I) and the viscous friction coefficient (D), give rise to the same fault directions in the system state space and therefore cannot be isolated by any observer-based method or parity space method. In this section, the parameter- estimation method is employed to do this work.

1000 s h a f t speed Ss ( rpm)

!l i l i I

0 0 1000 2000 3000

s a m p l e t i m e

Fig. 2e. The shaft speed Ss(t).

Fig. 2. Measurements of the hydraulic test rig

Note that realistic conditions were simulated in the real data collection, where a typical environment was designed which included: (i) step changes in the voltage input and the load pressure; this was used to simulate a dynamic environment for FDI; (ii) system operation over a typical operating region such that the system's non-linear behaviour was exhibited; Off) data collection with disturbances and noise present. All these features were reflected in the collected real data.

4. FAULT DIAGNOSIS

In this application , there are two points worth pointing out. The first is that the SVF method cannot be used for the entire system of the test rig, because it is non-linear. However, it can be used for local models (14)-(16) of the linear part

From the dynamics of the test rig (14)-(16), the following linear differential equation is derived:

m ~ S cr

I s + Pp Ivp (17)

which can be expressed as

E ss ] Ss = [an az2 a13] Pm Pp

(18)

where

D Cram Cr (19) all = - - T ' a 1 2 - ---7--, a13 - I~p"

It can be seen that the number of physical parameters in equation (17) is greater than that of the model parameters. It results, therefore, that the changes in the physical parameters in (17) cannot be identified by detecting the changes in the model parameters. However, with the assistance of the BFDO method (Yu et al., 1996a) which gives the fault alarm, the faults can be diagnosed by esti- mating the groups of the physical parameters in (19) from the system continuous model (18). It is evident in (19) that if all changes but a12 and a13 remain unchanged, this then implies a change in D. (Throughout the paper it is assumed that there are never two or more faults occurring simu- taneously in the system.) Similarly, if only a12 or a13 changes, this then implies a change in ~m or Up, respectively. If both a12 and a13 change to the same degree, a change, ACt, can be concluded. If all three model parameters, azl, a12 and a13, change to the same extent, it must be a change, AI, that is occurring. Therefore, faults can be diagnosed by observing changes in aij in cooperation with the BFDO method. Furthermore, the size of a fault can be diagnosed if the estimation is precise.

1288 D. Yu

The parameter-es t imat ion method for the continuous system described in Section 2 was used to est imate a l l , a12 and a13, using three sets of simulated faulty data. In designing the digital SVF, a stable second-order low-pass digital filter was used to est imate the derivative of the shaft speed in (18), and a third-order polynomial function was used in the SVF to do the interpolation. The recursive least-squares algorithm was used to estimate a l l , a12 and a13 on-line. The faulty da ta sets were generated from a non-linear model of the system, which was set up and validated using different sets of real data, and was believed to be a representation of the system with sufficient accuracy (see (Yu, 1995)).

The first set of faulty da ta simulates a change in the efficiency of the hydraulic motor (parameter ~/m) and a change in the efficiency of the hydraulic pump (parameter %),

It can be observed tha t A I causes a decrease in the absolute values of all three model parameters a12, a12 and a13, and A D causes a change in only a n . Therefore, changes in I and D can be concluded respectively. The model parameter changes in the steady-state before and after the faults are

a12 = 828.9 750.0 a13 -1144.9 -1036

AD [ --8.8 --+ 752

-1038

The relative changes in size eters caused by A I are calculated respectively as A a l l / a n = -0.1023, Aal2/al2 = -0 .0952 and Aa13/a13 = -0.0951 and tha t in a n caused by A D is Aal l /a l l = 0.1139. These can be used to roughly diagnose the fault sizes.

] in the three param-

A~m/~?m = 0.1, k > 200; A~p/~ v = 0.1, k > 1600.

By applying the digital SVF method in Section 2, the est imates of a l l , a12 and a13 were obtained, and are shown in Fig. 3. In order to show the three parameters in one graph, the estimates are weighted by multiplying a weighting matrix, W = diag{0.1 0.001 0.001} in all the figures in the rest of this paper.

I t can be seen tha t A?~m causes a significant change in only a12, while A~p causes a change in only a13. So, changes in rlm and ~p can be concluded respectively. The steady-state values of the es t imated model parameters before and after the faults are [alll [ 88 ] [_8.8]

a12 = 828.9 A~'~ 915.1 a13 --1144.9 --1149

A'~v 915.0 . --1044

The percentage increase on a12 caused by A~?m is calculated as Aal2/al2 ---- 0.1040 and the percentage decrease on a13 caused by AT?p is Aal3/al3 = --0.0914. I t is observed tha t the relative change in size of the est imated model parameter is approximate ly equivalent to the relative change in size of the physical parameter for both faults. Therefore, the fault size is diagnosed.

The second set of faulty da ta simulates changes in I and D,

A I / I = 0.1, k > 200; A D / D = 0.1, k > 1600.

The estimates of a l l , a12 and a13 are shown in Fig. 4.

The third set of faulty da ta simulates changes in C~, K0, Kl, Vt and B according to

A C r / C r = 0.1, k > 200; AKe/Ko = 0.1, k > 1400

AK~/KI = 0.1, k > 1800; AVt/Vt = 0.1, k > 2200

ABlE ---- 0.1, k > 2600

in which the changes, AK0, AKI, AVt and A~ are not involved in equation (17). Considering these will demonstrate if the model parameters are robust with respect to these changes. The estimates of a l l , a12 and a13 are shown in Fig. 5.

It can be seen that ACT causes a change in both a12 and a13 while the other four faults do not result in any significant change in the model parameters. I t is therefore concluded that a change has occurred in Cr. The estimates of the model parameters before and after the faults are

a12 = 828.9 913.9 a13 -1144.9 -1262.4

z~_~ . . . 9 2 0 . -1271.3

The relative changes in size in a12 and a13 caused by AC~ are calculated respectively as Aa12/a12 = 0.1025 and Aala/a13 = 0.1026. This can be used to diagnose the fault sizes.

In the above three simulations, the changes in the physical parameters are clearly detected and isolated by monitoring the model parameters. Changes in physical parameters which are not involved in equation (17) do not influence the estimated model parameters. This implies tha t the diagnosis is reliable.

Fault Diagnosis for a Hydraulic Drive System 1289

I . ~ ~ - - ................... . . . . . . . . . . . . . . . . . . '- .... ~ i ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 I i

L,

-1 h -r - - - - ~ - r .......... ~ ....... i- ~- .... - ..... ~ . . . . . F ~,- ~i ~t i i

0 500 1000 1500 2000 2500 3000

estimates of a l l , a12 and a13

a l l : , 9; ~112: ~ _ _ , ; a 1 3 : ' . . . . . '

Fig. 3. Diagnosis of faults Ar/,,~ and Ar/p

A I A D i i i ! i

_ ~ . . . . . . . . . . . . . _ . . . . ~ . . . . . . . . . . . . . . . . . : _ _ - . . . . . . . . . . . . . . . . . . . . . . . _ _ . ~ _ _ _ . . . . . . . . _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . i i i i I

0 500 1000 1500 2000 2500 3000


a l l : , 9; a 1 2 : ' - - - ' ", {~13: " " '~

Fig. 4. Diagnosis of faults A I and A D

AC~ AKe AKt AV~ A~

0 ~

i i i I I

0 500 1000 1500 2000 2500 3000


a l l : - - ~ a 1 2 : - ~ a 1 3 : " ' "

Fig.5. Diagnosis of faults ACt and AKe

1290 D. Yu

The fault size (10%) is also roughly diagnosed. Note that the model parameter change is delayed from the physical parameter change for all the faults, due to the convergence of the estimates of the physical parameters. The maximum delay is about 250 sample intervals, or 2.5 seconds, which is allowable in practice. This lag-time is greatly influenced by the size of the forgetting factor, A, in the recursive least-squares algorithm. In this simulation, the factor is chosen as A = 0.95. A small value of A will decrease the fault-detection time but will result in estimates of the model parameters being more sensitive to disturbances and noise.

5. CONCLUSIONS

The parameter-estimation method is applied to a linear part of a hydraulic test rig to diagnose physical parameter changes by monitoring the model parameters. The method operates in combination with an observer-based method which gives the fault alarms. Faults having the same direction in the system state-space can be isolated from one another by using this method, which is impossible using any observer-based method or parity equation method. Furthermore, the fault size may be diagnosed to some degree of accuracy, depending on disturbances and noise. The simulation study suggests that the combination of different methods will be more efficient for fault diagnosis in real industrial systems.

ACKNOWLEDGEMENT

Thanks are given to D.N.Shields for his help in the original work of this research, and to S.Daley for providing the real data.

REFERENCES

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Frank, P.M. and J. Wunnenberg (1989). Robust fault diagnosis using unknown input schemes.

Chapter 3, in Fault Diagnosis in Dynamic Sys- tems, Patton et a!., Prentice Hall.

Gertler, J.J. and M.M. Kunwer (1995). Optimal residual decoupling for robust fault diagnosis. Int. J. Control, Vol. 61, pp. 395-421.

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Fault Diagnosis for a Hydraulic Ddv¢ System 1291

Yu, D.L., D.N. Shields and S. Daley (1996a). A bilinear fault detection observer and its application to a hydraulic drive system. Int. J. Control, Vol. 64, No. 6, pp. 1023-1047.

Yu, ~D.L:, D.N. Shields and K. Disdell (1996b). A simulation study on fault diagnosis of a high- temperature furnace using a bilinear observer method. Control Eng. Practice, Vol. 4, No. 12, pp. 1681-1691.

fault diagnosis for a hydraulic drive system using a parameter-estimation method

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