Control Engineering Practice 19 (2011) 649–657

Contents lists available at ScienceDirect

Control Engineering Practice

0967-06

doi:10.1

$(DIS

case nu� Corr

E-m

vishal.m

mark.de

journal homepage: www.elsevier.com/locate/conengprac

Derivative free filtering in hydraulic systems for fault identification$

Vishal Mahulkar a,�, Douglas E. Adams a, Mark Derriso b

a 1500 Kepner Drive, Lafayette IN 47905, USAb USAF Air Vehicles Directorate, Air Force Research Laboratory, Dayton OH, USA

a r t i c l e i n f o

Article history:

Received 25 March 2010

Accepted 8 January 2011Available online 6 May 2011

Keywords:

Divided difference filtering

Hydraulic system

Fault identification

Estimation

System identification

Leakage

61/$ - see front matter & 2011 Published by

016/j.conengprac.2011.01.003

TRIBUTION A. Approved for public release; di

mber (88ABW-2010-1259).

esponding author. Tel.: +1 765 586 2044.

ail addresses: vmahulka@purdue.edu,

ahulkar@gmail.com (V. Mahulkar), deadams

rriso@wpafb.af.mil (M. Derriso).

a b s t r a c t

Diagnosis of incipient faults in hydraulic systems is of prime importance due to the performance and

reliability demands. This paper outlines the application of derivative free filtering in hydraulic systems

for the purpose of real-time fault identification. A flexible experimental setup is constructed in order to

simulate different types of faults. The method in this paper deals with internal leakage faults. A detailed

non-linear model of the hydraulic actuator experimental setup is developed and validated. Robust

control strategies typically hide the presence of faults during incipient stages, making identification of

the fault difficult. A partial feedback linearization based robust position control strategy is also

presented and faults are identified in the presence of robust control. Faults in the hydraulic systems are

modeled as parametric faults and second order divided difference filtering (DDF) is used to estimate the

states and the parameters. The efficacy of this estimation algorithm is demonstrated using different

fault levels as well as different fault growth profiles. The accuracy and the reliability of the

methodology are also demonstrated by identifying faults as small as .01 l/s.

& 2011 Published by Elsevier Ltd.

1. Introduction

Hydraulic systems are popular in the industrial fields due to thesize to power ratio of hydraulics and the ability to exhibit fastresponse times in the application of large forces and torques (Jelali &Kroll, 2004). Their industrial application include positioning (Shih &Sheu, 1991; Sohl & Bobrow, 1999; Tafazoli, de Silva, & Lawrence,1998), active suspensions (Alleyne & Hedrick, 1995; Alleyne & Liu,2000; Lin & Kanellakopoulos, 1995), material testing, aircraft, andindustrial hydraulic systems. These applications place high impor-tance on reliability and safety of hydraulic systems, which whencompounded with the arduous work conditions, necessitates devel-opment of reliable and efficient fault identification techniques.

Fault diagnosis of electro-hydraulic systems is a very challen-ging task due to the presence of highly non-linear dynamicbehaviors such as dead-band and hysteresis of control valves,non-linear pressure-flow relations, and variation of fluid volumesdue to the movement of the actuator (Merritt, 1967). It is alsodifficult to model any system accurately resulting in parametricuncertainties and uncertain non-linearities, which are observedby the control algorithm as well as the fault identificationalgorithm. Other disturbances such as external forces, friction,

Elsevier Ltd.

stribution unlimited) WPAFB

@purdue.edu (D.E. Adams),

etc. cannot be modeled. The fault identification algorithm thusneeds to be robust to these uncertainties and disturbances.

A variety of methods have been proposed in the literature forearly detection of faults in dynamical systems. These methods arebroadly divided into two main types: hardware redundancy basedand analytical redundancy based methods (Watton, 1992). Analy-tical redundancy based methods are further divided into signalmodel based methods and process model based methods (Isermann,2006; Patton, Frank, & Clark, 2000). An excellent review of faultdetection and diagnosis methods is available in several surveypapers (Dash & Venkatasubhamanian, 2000; Isermann, 1993;Patton, 1991, 1997). The most important of the analytical redun-dancy based approaches fall under parity space techniques, para-meter estimation techniques, statistical techniques, and observerbased techniques (Hammouri, Kabore, Othman, & Biston, 2002).

Model based techniques have received much attention in the lastfew decades Isermann (2005). A number of observer-based techni-ques have been proposed for linear systems, for example, eigen-structure assignment (Patton & Chen, 1992, 1997), unknown inputobserver (Kudva, Vishwanadham, & Ramakrishna, 1980; Patton,Clark, & Frank, 1989; Patton et al., 2000), etc., and for non-linearsystems, for example, robust observer based methods (Hammouriet al., 2002; Preston, Shields, & Daley, 1996). A drawback of observer-based methods in isolating faults is that they utilize the differentdirections of the different faults in the state space of the model, and,as a result, these methods are not capable of isolating faults that havethe same direction in the system state space (Yu, 1997). Parameterestimation techniques (Isermann, 1984), on the other hand, candetect and isolate faults, and may diagnose fault size, even for faults

V. Mahulkar et al. / Control Engineering Practice 19 (2011) 649–657650

having the same direction in the state space of the model. Applica-tion of parameter estimation techniques to linearized models havebeen discussed in the literature (Yu, 1997). Extended Kalman Filters(EKF) (An & Sepehri, 2003) and adaptive observers (Cho & Rajamani,1997; Garimella & Yao, 2005; Rajamani & Hedrick, 1995) have alsofound application in hydraulic fault detection.

In this paper, faults are modeled as parametric faults andadvances in derivative free filtering (Norgaard, Poulsen, & Ravn,2000) are used to estimate the unknown parameters. The initialfault considered is a leakage fault, which is a general symptom ofworn hydraulic components. In most cases, leakage increasesprogressively over a long period of time causing the systemdynamics to also change. Based on the location, the leakage canbe classified into: (i) internal (crossport) leakage, where the fluidleaks to another part of the circuit within the hydraulic system, forexample, from one chamber of the hydraulic actuator to the otherand (ii) external leakage where the fluid leaks out of the hydrauliccircuit. Whereas external leakage can be found through visualinspection, internal leakage caused by seal damage cannot bedetected easily until the actuator seal is severely damaged result-ing in degraded performance. Thus, it is important to identify thisfault as part of any health monitoring strategy. The paper isorganized as follows. The details of the experimental setup capableof simulating different types of faults including leakage faults arepresented in Section 2. The dynamic model of the hydraulicactuator system used in the fault identification process is pre-sented in Section 3. A brief overview of the parameter estimationmethod is presented in Section 4. Experimental results are dis-cussed in Section 5 and the conclusions are discussed in Section 7.

2. Experimental Setup

The picture of the experimental setup is shown in Fig. 1(a). Thesetup consists of two parts. The actuator loop, which simulates theactual hydraulic actuator, and the load loop, which is used tointroduce user-defined loading on the actuator. Fig. 1(b) shows thecircuit diagram of the experimental setup. The part left of the ‘‘Loadcell’’ is the main actuator loop and the part to the right is the load

Fig. 1. Experimental setup. (a) Act

loop. The complete hydraulic system is powered using a pressurecompensated pump (label 1) operating at 6.89 MPa (1000 psi). Thepressure compensation is required to maintain the pressure con-stant in pressure of different supply flow rates or load on thesystem. The pressure in the actuator (label 4) is controlled using aParker D1FH directional proportional valve (label 5). The positionof the actuator piston is measured using an analog position sensor(label 8). The pressures in the two chambers of the cylinder and thevalve spool position can also be measured. Flowmeters (label f1–f4) are added to the circuit to measure the flow rates in the internaland external leakage loops. The load cylinder is controlled with aParker D1FX directional proportional valve (label 9). The load loopalso has safety features such as pressure reducing valve (label 11)and pressure relief valves (label 10). Pressures in the load loop canalso be measured to allow implementation force control loop.

A dSPACE embedded system is used to acquire data and runcontrollers and the fault identification algorithm in real time. Theactuator loop has the ability to simulate different kinds of faultssuch as internal leakage faults using proportional valve (label 6),external leakage faults using needle valves (label 7), supply flowrate and pressure faults at the pump (label 1), etc. The internalleakage fault is simulated by connecting the two chambers of thecylinder externally using a Parker D1FH directional proportionalvalve. This valve can be controlled electronically and is used tosimulate the actual wear process of the internal seal.

Physics based models of such a setup are well documented inthe literature, so the focus in this paper is utilizing these modelsfor system identification and control. The next section gives anoverview of the dynamic model and the robust controllers usedfor position control.

3. Problem formulation and dynamic model

The non-linear equations of motion of a double rod hydraulicactuator are given as (Merritt, 1967)

m €xL ¼ PLA�b _xL�Ffcþ~f ð1Þ

ual setup. (b) Circuit diagram.

1 2 3 4 5 6 7 81.4

1.41

1.42

1.43

1.44

1.45

1.46

1.47

1.48

1.49

x 10−8

Cdw √

(uni

ts)

Run

�

Fig. 2. Identification of flow coefficient.

1 2 3 4 5 6 7 80.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5x 10−12

Vt

4�e

(uni

ts)

Run

Fig. 3. Identification of bulk modulus.

−0.5 0 0.5−1000

−800

−600

−400

−200

0

200

400

600

800

1000

Velocity m/s

Forc

e N

Fig. 4. Identification of cylinder friction.

V. Mahulkar et al. / Control Engineering Practice 19 (2011) 649–657 651

Vt

4be

_PL ¼�A _xL�Ctm

ffiffiffiffiffiffiffiffijPLj

psgn PLþQL ð2Þ

tv _xv ¼�xvþkvi ð3Þ

where xL is the displacement of the piston, PL¼P1�P2 is the loadpressure of the cylinder, A is the ram area of the cylinder, Ffc

represents the modeled Coulomb friction force, ~f represents theexternal disturbances such as unmodeled friction forces, and Ctm

represents the coefficient of the total internal leakage of thecylinder. The spool dynamics are modeled using a simplified firstorder equation. xv is the displacement of the spool of thedirectional proportional valve. QL is the load flow and is relatedto the spool displacement as follows (Merritt, 1967):

QL ¼ Cdwxv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPs�sgnðxvÞPL

r

sð4Þ

A new set of parameters are defined as follows:

PL ¼PL

Sc3

, xv ¼xv

Sc4

, b ¼b

ASc3

K v ¼kv

Sc4

, y1 ¼ASc3

m, C tm ¼

ffiffiffiffiffiffiSc3

pSc4

ffiffiffiffirpCdw

Ctm

y2 ¼ dn, y3 ¼4be

Vt

Sc4ffiffiffiffirp CdwffiffiffiffiffiffiSc3

p , A ¼1

Sc4

ffiffiffiffiffiffiSc3

p ffiffiffiffirpCdw

A ð5Þ

where Sc3and Sc4

are scaling constants used to achieve moreaccurate numerical results. The system dynamic equations canthen be written as

_x1 ¼ x2

_x2 ¼ y1ðx3�bx2�F fcÞþy2þ~d

_x3 ¼ y3ð�Ax2�C tm

ffiffiffiffiffiffiffiffijx3j

px3þg3x4Þ

_x4 ¼�1

tvxvþ

kv

tvu ð6Þ

where ðx1 x2 x3 x4Þ ¼ ðxL _xL PL xvÞ, and g3 is a suitably definedfunction.

3.1. System identification

The parameters of the system were determined by using anopen loop pseudo-random binary signal followed by least-squaresestimation. For example, the parameters Vt=4be and Cdw=r wereidentified using about 30 independent runs and the median wasused in the calculations. Figs. 2 and 3 show the parameterestimation results of eight different runs. The cylinder frictionwas also identified as shown in Fig. 4. The friction characteristicclearly shows the effects of Stribeck, Coulomb, and viscouscomponents typically used to model friction. The values used inthe simulation and control development are

Vt

4be

¼ 1:14� 10�12 m3

Pað7Þ

and

Cdwffiffiffiffirp ¼ 1:47� 10�8 m2

sffiffiffiffiffiffiPap ð8Þ

3.2. Position control

The robust position control strategy is based on feedbacklinearizing control of the pressure dynamics equation (2). Thedesired force trajectory is calculated based on the desired position

15 20 25−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time (s)

Pis

ton

posi

tion

(m)

ReferenceExperimentalSimulated

Fig. 5. Trajectory following for simulated model and experimental setup.

15 20 25−2

−1

0

1

2x 10−3

Time (s)

Err

or (m

)

Simulated

15 20 25−2

−1

0

1

2x 10−3

Time (s)

Err

or (m

)

Experimental

Fig. 6. Trajectory following errors.

−1

0

1

2

3

4

5

Inpu

ts

SimulatedExperimental

V. Mahulkar et al. / Control Engineering Practice 19 (2011) 649–657652

trajectory. The spool dynamics are almost 10 times faster than thehydraulic actuator dynamics and, hence, the spool dynamics canbe neglected and xv ¼ K vu for position control purposes. Byneglecting the internal leakage dynamics, the force dynamicsequation can be derived as follows:

F ¼ PLA ð9Þ

) _F ¼ _PLA ð10Þ

¼4be

VtAð�A _xLþQLÞ ð11Þ

¼4be

VtA �A _xLþCdwxv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPs�sgnðxvÞPL

r

s !ð12Þ

Define

B¼ 4be

VtACdw

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPs�sgnðxvÞPL

r

sð13Þ

and let Fd be the desired force trajectory to be achieved. Thefollowing control input is then considered:

xv ¼1

B4beA2 _xL

Vtþ _F d�kf ðF�FdÞ

� �ð14Þ

A Lyapunov type function is also defined as

V ¼ 12klðF�FdÞ

2ð15Þ

and then differentiated,

_V ¼ klð_F� _F dÞðF�FdÞ ð16Þ

_V ¼ kl4beA2 _xL

VtþBxv�dotFd

� �ðF�FdÞ ð17Þ

_V ¼�klkf ðF�FdÞ2

ð18Þ

producing a negative semi-definite function. Using Barbalatt’sLemma, it can show that the error F�Fd goes to zero exponen-tially. The desired force trajectory is chosen as follows:

Fd ¼m €xd�kvð _xL� _xdÞ�kpðxL�xdÞþ f ð19Þ

where xd is the desired position trajectory and f is the estimate offrictional forces on the cylinder. Using this trajectory, it can beshown that the position error is upper bounded by er 1

2

� �jdj at

steady state, where d¼ f�~f .

3.3. Model validation and verification

To validate the developed model, the designed control strategywas implemented on the model and the experimental setup. Thecontrol parameters that were used are given in Table 1. The kf

parameter used in the experiments is smaller because thatparameter defines the time constant of the error signal. Largerkf results in a smaller time constant requiring larger systembandwidth for chatter free operation. This choice of kf leads tochatter free operation in the experimental response characteris-tics. Figs. 5–7 show the comparison of the experimental andsimulated results for a randomly generated input signal. From the

Table 1Control parameters.

Simulation Experiments

kv 1�102 1�102

kp 1�106 1�106

kf 80 30

15 20 25−4

−3

−2

Time (s)

Fig. 7. Input voltage comparison.

V. Mahulkar et al. / Control Engineering Practice 19 (2011) 649–657 653

error plots (Fig. 6) and the input comparisons (Fig. 7), it isconcluded that the model that has been developed is sufficientlyaccurate and the model and control strategy are assumed to beverified.

4. Fault identification

In this section, a brief introduction of divided differencefiltering is given and then this method is modified to identifythe actuator states and parameters.

4.1. Divided difference filtering

Although the Kalman Filter (KF) has been used to solve theproblem of optimal state filtering in linear models with Gaussiannoise, there is no such solution available for non-linear systems.Instead, a number of estimators may be found in the literaturethat are basically extensions of the KF based on approximations ofthe non-linear system equations. The most prominent and widelyused method among these is the Extended Kalman Filter (EKF)(Boutayeb & Aubry, 1999; Gunther, Yaz, & Unbehauen, 1999;Ljung, 1979), which is obtained by first order linearization of thesystem equations so that the traditional KF can be applied at eachstep. Extensive use of the EKF over the past few decades hasshown that, in several important practical problems, the firstorder approximation provides insufficient accuracy, resulting insignificant bias or even convergence problems (Simon, 2006).Moreover, the derivation of the Jacobian matrix, which is neces-sary for the first order approximation, is non-trivial in manyapplications.

Recently, efficient derivative free filtering techniques, whichdo not require evaluation of the Jacobian matrix, have beenproposed. These filters have a computational complexity similarto the EKF but are much simpler to implement. Schei (1997) usesa first order approximation of a non-linear transformation thatimplements divided differences for the propagation of the meanand covariance of the state distribution. Quine (2006) uses aminimal ensemble set of state vectors to propagate the first twomoments of the state distribution to obtain a filter that is shownto be equivalent to the EKF in a limiting case. More accuratefilters, which are called Sigma Point Kalman Filters (SPKFs) (vander Merwe, Julier, Bogdanov, Harvey, & Hunt, 2004), have alsobeen proposed that use second or higher order approximations ofthe non-linear transformation. This class of filters include theUnscented Kalman Filter (UKF), which is based on the unscentedtransform, a technique for propagating the mean and covarianceof a distribution through a non-linear transformation (Julier &Uhlmann, 2004), and the divided difference filter (DDF), which isbased on a second order approximation using Stirling’s interpola-tion formula (Norgaard et al., 2000).

This section gives a brief summary of the DDF. The reader isreferred to Norgaard et al. (2000) for a detailed description andanalysis of the DDF. The class of systems for which this procedureis applicable is given below:

xðkþ1Þ ¼ fmðxðkÞ,uðkÞÞþvðkÞ ð20Þ

yðkÞ ¼HxðkÞþwðkÞ ð21Þ

where xðkÞARn is the state vector, yðkÞARp is the output vector,uðkÞARm is the input vector and fmðxðkÞ,uðkÞÞ is the non-linearplant model. It is assumed that fm is either a globally Lipschitzcontinuous function or that it is locally Lipschitz continuous withxðkÞ restricted to a compact domain DARn. vðkÞ and wðkÞ areassumed to be independent, identically distributed Gaussian

random variables with vðkÞ �N ð0,Q ðkÞÞ and wðkÞ �N ð0,RðkÞÞcalled the process and measurement noise, respectively.

The DDF filter provides an estimate of the evolution of aGaussian distribution through a non-linear transformation bymaking use of Stirling’s formula to obtain a second order approx-imation of the non-linear system around the current stateestimate. Stirling’s formula may be derived from the multi-dimensional Taylor’s series expansion by replacing the derivativeswith divided differences. The equations for computing the secondorder approximation along with those for propagating a multi-dimensional Gaussian random variable x with mean x throughthe approximated function, fsðx,xÞ, using special vectors, whichare columns of the matrix square root of the covariance of x, aregiven in the literature (Norgaard et al., 2000). Based on thisformulation, the DDF filter equations for the system in (20) can besummarized as in (26)–(32).

The filter equations have been suitably simplified based on themodel structure with a linear output equation and the assump-tion of additive Gaussian noise. In the following equations, all thequantities with a ‘‘bar’’ denote a priori estimates (before theactual output is observed) while the quantities with a ‘‘hat’’denote a posteriori estimates (after the output is observed).Further, let SxðkÞ, SxðkÞ, SvðkÞ, and SwðkÞ denote the matrix squareroots of PðkÞ (the a priori state covariance estimate), PðkÞ (the a

posteriori state covariance estimate), Q ðkÞ (the process noisecovariance) and RðkÞ (the output noise covariance), respectively.These matrices are calculated using the following equations:

PðkÞ ¼ SxðkÞSxðkÞT

ð22Þ

PðkÞ ¼ SxðkÞSxðkÞT

ð23Þ

Q ðkÞ ¼ SvðkÞSvðkÞT

ð24Þ

and

RðkÞ ¼ SwðkÞSwðkÞT

ð25Þ

Let xðkÞ denote the a priori state estimate and let sx,p denote thepth column of Sx. Then, the a priori state estimate calculated bythe DDF is given by

xðkþ1Þ ¼h2�n

h2fmðxðkÞ,uðkÞÞ

þ1

2h2

Xn

p ¼ 1

ffmðxðkÞþhsx,p,uðkÞÞþ fmðxðkÞ�hsx,p,uðkÞÞg ð26Þ

Two temporary matrices are defined, Sð1ÞxxðkÞ and Sð2Þ

xxðkÞ, whose pth

columns are computed as

Sð1ÞxxðkÞðpÞ ¼

1

2hffmðxðkÞþhsx,p,uðkÞÞ�fmðxðkÞ�hsx,p,uðkÞÞg ð27Þ

Sð2ÞxxðkÞðpÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffih2�1p

2h2ffmðxðkÞþhsx,p,uðkÞÞ

þ fmðxðkÞ�hsx,p,uðkÞÞ�2fmðxðkÞ,uðkÞÞg ð28Þ

Then the updated square root of the a priori state covariancematrix, Sxðkþ1Þ, is given by

Sxðkþ1Þ ¼HTðSð1ÞxxðkÞ Sð2Þ

xxðkÞ SwðkÞÞ ð29Þ

where HTðSÞ denotes a Householder transformation (Golub &Loan, 1996) to convert the matrix S into a square triangular formsuch that HTðSÞHTðSÞT ¼ SST . The gain matrix, Kðkþ1Þ, is calcu-lated using (30), and the square root of the a posteriori state

V. Mahulkar et al. / Control Engineering Practice 19 (2011) 649–657654

covariance matrix, Sxðkþ1Þ, is given by (31):

Kðkþ1Þ ¼ Sxðkþ1ÞSxðkþ1ÞT HTðHSxðkþ1ÞSxðkþ1ÞT HT

þRðkþ1ÞÞ�1

ð30Þ

Sxðkþ1Þ ¼HTð½ðI�Kðkþ1ÞHÞSðkþ1Þ Svðkþ1Þ�Þ ð31Þ

The innovation (output prediction error) is defined as gðkþ1Þ ¼yðkþ1Þ�Hxðkþ1Þ and xðkþ1Þ is used to denote the a posteriori

estimate after the kth observation. The state estimate is thenupdated using (32) to get the final a posteriori estimate

xðkþ1Þ ¼ xðkþ1Þ�Kðkþ1Þgðkþ1Þ ð32Þ

0 20 40 60 80 100 120 140 160 1800.075

0.08

0.085

0.09

0.095

0.1

0.105

Time (s)

Pos

ition

(m)

ReferenceResponse

Fig. 8. Square wave reference tracking.

4.2. Fault modeling

The fault in the hydraulic system under consideration ismodeled as a parametric fault (2), This unknown parameter isthen appended to the state space representation of the system.The DDF non-linear filter is then used to estimate this state and,hence, the fault parameter. The internal leakage is given as

Qleak ¼ Ctm

ffiffiffiffiffiPL

psgn PL ð33Þ

where

Ctmpxvleakð34Þ

or

Ctm ¼ kqxvleakð35Þ

This fault parameter is appended to the system (6) and convertedto discrete form using simple Euler difference as follows:

x1ðkþ1Þ ¼ x1ðkÞþTsx2ðkÞ ð36Þ

x2ðkþ1Þ ¼ x2ðkÞþTsy1ðx3ðkÞ�bx2ðkÞ�F fcÞþy2þx6ðkÞ ð37Þ

x3ðkþ1Þ ¼ x3ðkÞþTsy3ð�Ax2ðkÞ�x5ðkÞffiffiffiffiffiffiffiffiffiffiffix3ðkÞ

psgnðx3ðkÞÞ

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPs�sgnðx4ðkÞÞx3ðkÞ

qx4ðkÞÞ ð38Þ

x4ðkþ1Þ ¼ x4ðkÞþTs�1

tvx4ðkÞ

� �þ

Kv

tvu ð39Þ

x5ðkþ1Þ ¼ x5ðkÞ ð40Þ

x6ðkþ1Þ ¼ x6ðkÞ ð41Þ

where x1,y,x4 are states (position, velocity, load pressure andspool position), x5 is the parameter appended as a state forunknown internal leakage fault and x6 is the appended state forestimating friction. It is important to estimate friction onlinebecause of the large uncertainty associated with offline frictionestimation. The addition of the friction parameter improves theestimation accuracy significantly.

To use the divided difference filter, the system model, fm, canbe easily inferred from (36)–(41). yðkÞ is the measured output ofthe system. In this case, the first four states are being measured.Hence,

H¼

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

26664

37775 ð42Þ

vðkÞ �N ð0,Q ðkÞÞ is the Gaussian process noise, with Q ðkÞ beingthe process covariance. The process covariance is estimated basedon the confidence of the system model (36)–(41). For example, for

this application the covariance is assumed to be a constant:

Q ¼ diagð½1� 10�2,1� 10�2,1,1� 10�2,8� 10�6,1�Þ ð43Þ

Eqs. (40) and (41) represent the dynamics of the internal leakagefault parameter and friction parameter, respectively. Since thefault parameter is a slow varying quantity, its process covarianceis very low (6� 10�6), whereas for the friction parameter, whichcan vary suddenly, the process noise covariance is assumed to be 1.

wðkÞ �N ð0,RðkÞÞ represents the Gaussian measurement with Rrepresenting the noise covariance. The measurement noise covar-iance is estimated from experimental data. In the current applica-tion, it is assumed to be constant:

R¼ diagð½1� 10�6 1� 10�2 1� 107 1� 10�2�Þ ð44Þ

The initial variance estimate for the state estimates is assumed to be

P¼ diagð1� 102� ½1 1 1 1 1 1�Þ ð45Þ

5. Experimental result

To ensure the presence of persistent excitation condition duringthe execution of estimation algorithm, all of the estimations wereperformed while the actuator was tracking a square wave refer-ence signal. The reference and response is shown in Fig. 8. Fig. 9shows the results of the fault identification method with stepchanges in the leakage valve commanded input voltage. A stepchange helps understand the response of the filtering algorithmwhen there is a sudden failure in the system. Fig. 9(a) shows theestimated leakage coefficient, C tm, as described in Eq. (5) as well asthe input voltage to the valve controlling the internal leakage. They-axis on the left is for the flow coefficient and the y-axis on theright is for the input voltage. The two axes are scaled differently toallow comparison of the response. As can be seen the estimatedflow coefficient tracks the changes in the input voltages very well.Actual leakage flow rate can then be calculated from the estimatedflow coefficient. This is depicted in Fig. 9(b) along with themeasured flow rate from the flow meters. The agreement betweenthe estimated and the measured flow is very good. The measuredflow rate shows periodic spikes because the flow meters onlymeasure the magnitude of the flow rate. Whenever the pistonchanges direction, the flow rate also changes direction so momen-tarily the flow rate is zero, resulting in spikes. These physicaldynamics are also estimated very well by the filtering algorithm.Furthermore, when the piston stops moving, the pressures in the

0 50 100 150 200 250

0

1

2

Time (s)

Est

imat

ed le

akag

e co

effc

ient

Ctm¯

0

5

Leak

age

valv

e co

mm

ande

d (v

olts

)

0 50 100 150 200 2500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Time (s)

Leak

age

rate

(ltrs

/s)

EstimatedMeasured

Fig. 9. Estimation of leakage coefficient and leakage rate for step changes in leakage control valve input. (a) Estimation of leakage coefficient C tm. (b) Comparison of

estimated and measured leakage.

0 50 100 150 200 250

0

1

2

Time (s)

Est

imat

ed le

akag

e co

effic

ient

0

5Le

akag

e va

lve

com

man

ded

(vol

ts)

0 50 100 150 200 2500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Time (s)

Leak

age

rate

EstimatedMeasured

Fig. 10. Estimation of leakage coefficient and leakage rate for a slow ramp leakage control valve input. (a) Estimation of leakage coefficient C tm . (b) Comparison of

estimated and measured leakage.

V. Mahulkar et al. / Control Engineering Practice 19 (2011) 649–657 655

two chambers of the cylinder start to equalize resulting indecreasing flow rate. This phenomenon is also estimated well bythe algorithm as seen in Fig. 9(b).

The second example considered is for a slow ramp increase in theinput voltage to the valve controlling the internal leakage. Thisexample helps understand the behavior of the fault estimationalgorithm when there is slow fault growth due to physical phenom-ena such as wear. Fig. 10 shows the results for a slow 200 s rampinput to the leakage control valve. Again, Fig. 10(a) shows theleakage coefficient and the input voltage to the leakage valve. Thetwo axes are scaled differently to allow comparison of the trackingability of the fault identification algorithm. Except for the initialerror, the algorithm is able to track the input very well.Fig. 10(b) show the comparison of the measured and the estimatedleakage flow rate. This figure also affirms the two physical phenom-ena of spikes and reduction in flow rate when piston stops movingas described earlier. Both these physical phenomenon are estimatedaccurately by the fault identification algorithm.

These results indicate that the estimation algorithm is accu-rate and consistent over different runs. Very low leakage rates aslow as .025 l/s were identified successfully. These results comparefavorably to a recent publication by Goharrizi, Sepehri, and Wu(2010). Very small leakage rates of up to .01 l/s are difficult toidentify as seen in Figs. 9(b) and 10(a). However, as leakagebecomes greater than .01 l/s, the estimates become very accurate.

Remark 1. The advantage of this methodology is the ability toidentify multiple fault as long as the faults can be modeledparametrically. As long as the faults enter the system equationthrough some physics, and models for those are included in thesystem equation, they will not affect the each other. If the modelsfor the faults are not included in the system equations, they willaffect the estimation. However, the effect of the unmodeleddynamics can be minimized by properly choosing the processcovariance matrix. A larger covariance matrix will mean moreuncertainty in the model or less confidence in the model. Thiswould result in a larger error covariance but the mean estimatewill still be reasonably accurate. If conservative estimates for theprocess covariance are available, the actual error covariance canbe guaranteed to be always less than calculated error covariance.

Remark 2. The main drawback of this methodology is that if thereare two faults which can be represented as simple additive para-meters in the same state equation, it would be difficult to distin-guish between them. These two faults can be represented as a singleparameter and its variation will be a combination of both faults.

6. Sensitivity to process noise covariance

The process noise covariance matrix given in Eq. (43) waschosen heuristically based on confidence in the mathematical

V. Mahulkar et al. / Control Engineering Practice 19 (2011) 649–657656

model used for the system as given by Eq. (6). For example for theleakage state estimate, a noise covariance value of 8� 10�6 wasused assuming that the leakage fault is caused by wear which is avery slow process. This value determines the speed at which theleakage coefficient estimate can respond to changes in the faultand the variability of the estimate. Fig. 11 shows the variation ofresponse for different process covariance values. The averagesteady-state value for a 5 V input is around 2.66. Fig. 12 showsthe variation of time required to reach 90% of steady-state valuewith process covariance value used for the leakage coefficientstate estimate. The basic principle of DDF follows the Kalmanfilter with a prediction update and a measurement update. Thebehavior can be explained by looking at the simple updateequations of a discrete Kalman filter. The first set of Eqs. (46)and (47) represent the prediction update and the second set(48)–(50) represent the measurement update:

xðkþ1Þ ¼ AxðkÞþBuðkÞ ð46Þ

Pðkþ1Þ ¼ APðkÞATþQ ð47Þ

0 10 20 30 40 500

1

2

3

4

5

6

Time (s)

Leak

age

coef

ficie

nt e

stim

ate

Ctm

8×10−4

1×10−4

4×10−5

8×10−6

8×10−7Input

Fig. 11. Behavior of leakage parameter Ctm estimation for different process

covariance values.

−1 0 1 2 3 4 5 6 7 8x 10−4

0

5

10

15

20

25

30

35

40

45

Process covariance value

Tim

e re

quire

d to

reac

h 90

% o

f ste

ady

stat

e

Fig. 12. Sensitivity of leakage parameter estimate response to corresponding

process covariance values.

Kðkþ1Þ ¼ Pðkþ1ÞHT ðHPðkþ1ÞHTþRÞ�1ð48Þ

xðkþ1Þ ¼ xðkþ1ÞþKðkþ1Þðyðkþ1Þ�Hxðkþ1ÞÞ ð49Þ

Pðkþ1Þ ¼ ðI�Kðkþ1ÞHÞPðkþ1Þ ð50Þ

As the a priori estimate error covariance Pðkþ1Þ approacheszero, or after the error covariances have converged and if Q issmall, the gain K weighs the residual less heavily. Specifically,

limP ðkþ1Þ-0

Kðkþ1Þ ¼ 0 ð51Þ

When this happens, the actual measurement is trusted less andless, while the prediction is trusted more and more. This results ina large inertia resulting in slow filter response to changes inmeasurements. Hence, as the process noise covariance isdecreased, the time taken to reach the 90% value increases asseen clearly in Fig. 12. This also results in smoother response.

7. Conclusions

An approach based on derivative free non-linear filtering wasdeveloped for detecting internal leakage faults in hydraulic actua-tors. The efficacy of this method was demonstrated by implement-ing it on an experimental testbed. The method was implemented inthe absence of external loading, but the authors speculate that themethod will be robust to any kind of loading that the positioncontrol strategy can accommodate because the estimation strategyworks very well despite the presence of unknown and highlyvariable frictional forces. The strategy identified the leakagecoefficients accurately and can be used in estimation of internalleakage as low as .01 l/s. This capability is very important in theidentification of incipient failures. In addition, when developingprognostics for such actuators, it is critical to identify the fault asearly as possible to ensure that predictive models can be identifiedin a timely manner. The sensitivity of the identification algorithmto the process noise covariance was also evaluated.

Appendix A. Glossary of symbols

xL

hydraulic piston position m mass of the piston A ram area PL load pressure b damping Ffc modeled Coulomb friction force ~f external disturbances

Vt

total hydraulic volume

be

bulk modulus of oil

Ctm

internal leakage coefficient QL load flow Cd valve orifice coefficient of discharge w spool valve area gradient xv spool displacement tv spool time constant kv spool gain Ps supply pressure r oil density tv

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