Fault Diagnosis and Fault-Tolerant Control for a Nonlinear Electro-hydraulic System

8/3/2019 Fault Diagnosis and Fault-Tolerant Control for a Nonlinear Electro-hydraulic System

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Fault diagnosis and fault-tolerant control for a nonlinear

electro-hydraulic system

Liang Chen and Steven Liu

Abstract— This paper proposes a fault-tolerant control strat-egy for a nonlinear electro-hydraulic system. The fault-tolerantcontrol scheme consists a fault diagnosis system and a reconfig-urable controller. The fault diagnosis system is designed basedon differential-geometric approach and is robust to systemdisturbance. Adaptive backstepping method is utilized forrealizing the reconfigurable controller. The control performanceof the electro-hydraulic system can be guaranteed even in thepresence of considered faults.

I. INTRODUCTION

Electro-hydraulic systems are widely used in today’s in-

dustry. Compared to electrical drives they can generate largeforces or torques very fast with simple structures. Control of

electro-hydraulic systems is a challenging task due to inher-

ent nonlinearities from complicated flow properties, friction

in actuator, varying external load, etc. Advanced control

strategies are necessary for highly demanding application [1].

Meanwhile, reliability and safety are most important issues

of controlled electro-hydraulic systems. They often work in

critical places, like in automobiles or aircrafts. A small fault,

such as leakages in cylinders or valves, sensor faults (noise,

offset), etc, can bring serious problems. For example, if the

landing gear can not work properly due to a leakage in the

actuation cylinder, the airplane may crush during the landing.To improve safety and reliability of electro-hydraulic systems

much research on fault diagnosis in this area has been done:

in [2] the adaptive robust approaches is applied to construct

the fault detection observer; in [3] Unscented Kalman filter

is used for detecting the faults in a hydraulic actuator; in

[4] linear observer and adaptive threshold are employed

for robust fault diagnosis of a nonlinear electro-hydraulic

system.

Moreover, if a controller is able to tolerate possible faults

automatically, the control is known as fault-tolerant control

(FTC) [6], which can broadly classified into two types:

passive FTC and active FTC. The passive FTC systems

usually can be designed with robust control approaches [7].

The active FTC contains a fault diagnosis system and reacts

to faults actively by reconfiguring control actions based on

the fault information. Thus the pre-specified performance of

the system can be ensured. Various active FTC approaches

are already available, such as pseudo inverse [8], model

predictive control [9] and adaptive control [10].

This paper proposes a novel fault diagnosis integrated

fault-tolerant control for a nonlinear electro-hydraulic sys-

Liang Chen and Steven Liu are with Control Systems Research Group,University of Kaiserslautern, Erwin-Schroedinger-Str. 12/332, 67663 Kaiser-slautern, Germany. Email: [email protected]

tem. The fault diagnosis system based on differential-

geometric approach is robust to the system disturbance and

provides updated fault information for the controller. The

controller is designed with adaptive backstepping technique

and can be easily reconfigured to accommodate the faults.

This scheme make the whole system more intelligent and

reliable. Furthermore, it is not extreme complicated and can

be implemented in practice. The experimental results are

shown to illustrate the advantages.

II . SYSTEM DESCRIPTION

A. System modeling

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Fig. 1. The hydraulic circuit of the testbed

The electro-hydraulic system used in our laboratory, whose

hydraulic diagram is shown in Fig. 1, simulates the extrud-

ing machine and is a pump-controlled hydraulic system. It

consists of two identical cylinders, which are named work-

ing cylinder and load cylinder, respectively. The working

cylinder represents the press stem of an extrusion machine.

Its fluid power is supplied by a variable displacement axial

piston servo-pump. The load cylinder simulates the resistance

force, which is generated when the material is pressed. With

a proportional pressure relief valve an arbitrary load can be

realized. The load cylinder is connected with a gear pump

so that load pressure can be built up quickly. The direction

change of fluid flow is realized through several directional

valves, which are labeled with DV i in Fig. 1. The system

dynamics consists mainly of two parts: pump model and

cylinder model.

The servo-pump itself is a closed-loop control system. The

output flow of the pump Q p is linearly adjusted by a non-



rotating swashplate. The model of the pump is governed by

the dynamics of swashplate, which can be approximated by

the following second order system

α = vα (1)

vα = − p1α − p2vα + p1k puuin, (2)

Q p = K Qα (3)

where α and vα are the angle and angular velocity of the

swashplate respectively, uin is the input voltage, p1, p2 and

k pu are constant. K Q is a constant determined by the pump

geometry.

The cylinder piston motion can be described as:

xc = vc (4)

vc =1

mc

(P AA1 − P BA2 − F f − F L), (5)

where xc is the piston position, vc is the piston velocity,

A1 and A2 are cross-section area of piston and ring-side

respectively, P A and P B are the pressures in chamber A and

B respectively, mc denotes the moving mass, F f representsthe friction force and F L is the varying external load. By

ignoring the static friction the mathematical description of

F f is

F f = Dvvc + f c,

where Dv is the coefficient of viscous friction and f c is

the Coulomb friction. Dv and f c are constant and can be

identified through experiments.

The pressure dynamics in cylinder chambers can be ex-

pressed as follows [1]

P B =E B

V 20 − A2xc(A2vc

−QB) (6)

P A =E A

V 10 + A1xc(−A1vc + QA). (7)

Here E i is the effective bulk modulus and dependent on the

pressure, Qi denotes the volume flow, V 10 and V 20 are the

initial volume of chamber A and B respectively. The leakage

flows of cylinder are omitted in fault free case.

The volume flow QA is the same as the output flow of

the servo-pump due to the pump-controlled structure of the

hydraulic system. The volume flow QB is dependent on the

characteristic of the directional valves and can be described

by

QB = √Γ if Γ > 0,

0 if Γ ≤ 0,

where Γ = kqb1(P B − P o) + kqb2 with tank pressure P o,

constant parameters kqb1 and kqb2.

The effective bulk modulus has significant effect to the

pressure dynamics and can not be simplified as a constant.

It can be calculated by an empirical formula

E i =P i/P at + kair

kairE o/P i + P i/P at, (8)

where E o is the bulk modulus of oil, P at is the atmospheric

pressure and kair is the air/oil volume ratio.

Combining the equations (1)-(6) the model of the system

in fault free case can be derived

xc = vc

vc =1

m(P AA1 − P BA2 − F f − F L)

P B =E B

V 20−

A2xc(A2vc − QB)

P A =E A

V 10 + A1xc(−A1vc + K Qα)

α = vα

vα = − p1α − p2vα + p1k puuin.

(9)

All system variables can be directly measured except the vcand vα, which are calculated through numerical differentia-

tion of xc and α followed by a low-pass filter respectively.

The external load F L varies during the motion of cylinder

piston and is viewed as an unknown disturbance of the

system model.

B. Considered faults

The usual faults of hydraulic systems include pulsation and

pressure drop of pumps, leakages in valves and cylinders,

sensor noise or offsets, oil pollution, etc. The considered

faults of the testbed are internal leakage of the cylinder QLin

and the offsets of pressure sensors, which are denoted by

∆P A and ∆P B .

The artificial internal leakage is realized through the

bypass pipeline between the chamber A and B of working

cylinder, as shown in Fig. 2. The leakage flow can be

controlled by the proportional direction valve. The dynamics

Fig. 2. Artificial internal leakage of cylinder

of pressures are rewritten as

P B =E B

V 20

−A2xc

(A2vc − QB + QLin) (10)

P A = E AV 10 + A1xc

(−A1vc + QA − QLin). (11)

The rest of system equations remains the same.

The sensor offsets are simply accomplished by addition of

a constant after A/D conversion.

III. MODEL-BASED FAULT D IAGNOSIS

A. Disturbance decoupling

The system (9) is obviously highly nonlinear and subject

to the unknown disturbance F L. Therefore, it is reasonable

to decouple the disturbance first for keeping the robustness

of the residual generator. Among various robust residual



generation methods the differential-geometric approach is

appropriate for the system because it can also be applied to

nonlinear systems. The principle of the differential-geometric

approach is to find a distribution, with which a system

transformation can be carried out. The transformed system is

only affected by the considered faults. Then fault detection

observers can be designed for the transformed system [5].

As the introduced faults only affect the cylinder dynamicsresidual generator is constructed based only on the cylinder

model governed by equations (4)-(7). Following the observ-

ability codistribution algorithm proposed in [5] the state

variables of the transformed system are

x = [xc P B P A]T = [x1 x2 x3]T .

Its dynamics is

x1 = vc (12)

x2 =E B

V 20−

A2x1(A2vc − QB) (13)

x3 =E A

V 10 + A1x1(−A1vc + K Qα) (14)

with input signals α and vc. It can be observed that the

system with equations (12)-(14) is only affected by the

considered faults QLin, ∆P A and ∆P B . Furthermore, as xcis just an integration of a input signal in transformed system,

it can reflect none of the introduced faults. Thus, the residual

generation is only based on the system equations (13)-(14).

B. Residual generation

Now the fault detection observer can be designed for the

system (13)-(14). As all state variables are measurable and

the nonlinearities can be compensated a linear observer is

employed for fault detection. The structure of the observer

is

˙x = Ax + f (y) + g(y)u + Lo(y − y)

y = C x,(15)

where x is the state vector, u is the input and y is the output.

A, f (·) and g(·) are the system matrix or vectors. C is the

output matrix. Lo is the observer gain and should ensure (A−LoC ) stable. x is the estimation of x. Define the observer

error eo = x− x. The residual is chosen as r = y− y. Thenthe residual generator can be obtained as

eo = (A − LoC )eo

r = Ceo.(16)

In fault free case the residual r satisfies limt→∞ r = 0. If

the fault occurs, r will diverge from zero. Then the fault can

be detected.

It should be noted that for the presented electro-hydraulic

system the step-like sensor offsets can be detected with the

introduced method. A slowly changing sensor fault, like a

drifting, can not be detected due to the system structure.

C. Fault isolation

Let r = [rP B rP A ]T corresponding to [P A P B]T . rP Band rP A are both affected by the fault QLin. However the

fault ∆P A only affects the residual rP A and ∆P B affects

the rP B merely. Then the three faults can be isolated from

each other theoretically. In practice the leakage possibly can

not make |rP A| and |rP B | larger than respective thresholdssynchronously. A proper latency time tw should be set

for isolating the faults. For example, if |rP A | exceeds the

threshold it can not concluded immediately that the fault

∆P A occurs. The diagnosis system will wait for tw. Then,

if |rP B | also exceeds its threshold during the period tw, the

fault QLin is occurred. Otherwise the fault ∆P A is present.

The same rule is also applied to isolate QLin from ∆P B .

IV. FAULT-TOLERANT CONTROLLER DESIGN

The design of the reconfigurable controller is based on

backstepping approach, which can deal with nonlinear sys-

tems and achieve regulation and tracking properties without

cancellations of stabilizing nonlinearities [11]. The backstep-

ping controller can be easily reconfigured through parameter

adaption or model modification. For the electro-hydraulic

testbed adaptive backstepping is applied to compensate for

the effect of time-variant external load in fault free case.

Accommodation of the fault QLin can be accomplished

by another parameter adaption. The sensor faults affect

the system through the controller. The accommodation of

such faults is usually realized through replacing the faulty

measurement by estimation. But our system is not observable

without the measurement of P A or P B . In this paper thesensor fault tolerance is achieved by model modification.

The FTC structure based on backstepping approach is

shown in Fig. 3. The control input un is designed based

on the fault free model. If the fault occurs and is identified,

reconfigured control input uf based on the model of faulty

system is active. Thereby the control performance can be

recovered.

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Fig. 3. FTC structure of the system

A. Adaptive backstepping

The nonlinear system (9) should be reformulated in a

strict feedback form for the construction of backstepping

controller. Assume a new state variable F HY = P AA1

−



P BA2. Then the original system (9) can be rewritten as

xc = vc

vc =1

m(F HY − F f − F L)

F HY =E AA1

V 10 + A1xc(−A1vc + K Qα) (17)

−E BA2

V 20 − A2xc (A2vc − QB)

α = vα

vα = − p1α − p2vα + p1k puuin.

For simplicity the dynamics of F HY is given by

F HY = f (xc, vc) + g(xc)α (g(xc) = 0), (18)

where E A and E B can be calculated by equation (8).

The controller for fault free case can be derived step by

step as follows.

Step 1: Define

e1

= xc − xr, e2

= vc − β2

, e3

= F HY − β3

e4 = α − β4, e5 = vα − β5,

where βi (i = 2, 3, 4, 5) are the stabilizing functions, xr is

the reference position trajectory and is 5th order differen-

tiable. Then the Lyapunov function in this step is

V 1 =1

2ρ1e21 (ρ1 > 0). (19)

Then the derivative of V 1 is

V 1 = ρ1e1e1 = ρ1e1 (β2 + e2 − xr) . (20)

Choose β2 = −k1e1 + xr, with k1 > 0. This follows that

V 1 = −ρ1k1e21 + ρ1e1e2. (21)

Step 2: The design of stabilizing function in this step

requires the information of F L. The parameter adaption

should be applied. Define θ1 = F L and the estimation error

eθ1 = θ1 − θ1. The Lyapunov function of this step is

V 2 = V 1 +1

2ρ2e22 +

1

2ρθ1e2θ1 (ρ2, ρθ1 > 0). (22)

The derivative of V 2 is

V 2 = −ρ1k1e21

+ ρ1e1e2 + ρθ1eθ1 eθ1

+ ρ2e2−F f

m +β3 + e3

m −eθ1 + θ1

m − β2

. (23)

Then β3 is chosen as

β3 = F f + θ1 − ρ1ρ2

me1 − mk2e2 + mβ2 (24)

with the adaption law of θ1

˙θ1 = − ρ2

mρθ1e2. (25)

This leads to

V 2 = −ρ1k1e21 − ρ2k2e22 +ρ2m

e2e3. (26)

Step 3: Define the Lyapunov function of this step

V 3 = V 2a +1

2ρ3e23 (ρ3 > 0). (27)

The derivative is

V 3 = −ρ1k1e21− ρ2k2e2

2+

ρ2m

e2e3

+ ρ3e3 [f (xc, vc) + g(xc)(β4 + e4)] . (28)

Because g(xc) = 0, β4 can be designed as

β4 =1

g(xc)

−f (xc, vc) − ρ2

ρ3me2

− k3e3 + β3

(k3 > 0). (29)

This leads to

V 3 = −ρ1k1e21− ρ2k2e2

2− ρ3k3e2

3+ ρ3g(xc)e3e4. (30)


V 4 = V 3 + 12

ρ4e24 (ρ4 > 0). (31)

Choose β5 as

β5 = −ρ3ρ4

g(xc)e3 − k4e4 + β4 (k4 > 0). (32)


V 4 = −ρ1k1e21 − ρ2k2e22 − ρ3k3e23 − ρ4k4e24 + ρ4e4e5. (33)


V 5 = V 4 +1

2ρ5e2

5(ρ5 > 0). (34)

Chose the control input as

uin =1

p1k pu( p1α + p2vα

− ρ4ρ5

e5 − k5e5 + β5) (k5 > 0). (35)


V 5 = −5

i=1

ρikiei2 (ki, ρi > 0). (36)

For the construction of backstepping controller the derivative

of the stabilizing function is necessary. Here the numeri-

cal differentiations followed by low-pass filters replace the

analytic derivatives for simplicity. Following the LaSalle-

Yoshizawa theorem [11] the error term ei will asymptotically

converge to zero with the controller (35). Meanwhile the

estimation error eθ1 is bounded. Thus, the desired position

tracking can be achieved with good transient performance.

Theoretically the asymptotic stability can be ensured by

positive ki and ρi. But every system sate has its physical

limitation. Thus, the parameters ki and ρi should be so cho-

sen that the stabilizing functions are limited in the physical

constraints.



B. Fault accommodation

Scenario 1: System fault QLin

If the internal leakage occurs, the model is different from

fault free case as shown in equations (10) and (11). It can

be described as

F HY = f (xc, vc) + ψ(xc)θ2 + g(xc)α, (37)

where θ2 denotes the leakage QLin. An extra parameter

adaption can be utilized to compensate the effect of the

leakage, which is same as handling the problem F L. Define

the estimation error eθ2 = θ2 − θ2. The new Lyapunov

function is

V 3a = V 3 +1

2ρθ2e2θ2 (ρθ2 > 0). (38)

The stabilizing function is chosen as

β4 =1

g(xc)

−f (xc, vc) − ψ(xc)θ2

−ρ2

ρ3m e2 − k3e3 + β3

(k3 > 0), (39)

with the adaption law of θ2

˙θ2 =

ρ3ψ(xc)

ρθ2g(xc)e3. (40)

This leads to V 3a = V 3.

The rest of the design is the same as in the no fault case.

The asymptotic convergence of the position tracking error

can be guaranteed even with the internal leakage in system.

Scenario 2: Sensor fault ∆P AThe Sensor fault ∆P A can bring more serious consequence

than QLin, especially if the system is running under highpressure. Fortunately, the reconfiguration can be easily ac-

complished thanks to the structure of adaptive backstepping.

∆P A affects two parameters of the backstepping controller:

error term e3 and the bulk modulus E A.

Firstly, it can be proved that the sensor offset is estimated

by the adaption law of F L (θ1). The faulty measurement can

be modeled as F HY = F HY + ∆F HY , where ∆F HY =∆P AA1. Assume there exist a stabilizing function β3a,

which satisfies F HY − β3a = F HY − β3 = e3. Then

F HY = β3a + e3 − ∆F HY . (41)

Substitute (41) into equation (23). Then

V 2 = −ρ1k1e21 + ρ1e1e2 + ρθ1eθ1 eθ1 + ρ2e2

−F f

m

+β3a + e3

m− θ1 + ∆F HY

m− β2

. (42)

∆F HY is a step-like signal and its derivative can be viewed

as zero. So the term (θ1 + ∆F HY ) can be viewed as one

unknown constant and estimated by the same adaption law

of θ1. Also β3a can be obtained as

β3a = F f + θ1 + ∆F HY − ρ1ρ2

me1 − mk2e2 + mβ2. (43)

It leads to F HY − β3a ≈ F HY − β3 = e3. The design of

β3a is same as β3. The error term e3 is almost unaffected

by ∆P A.

The parameter E A, which is also affected by the ∆P A,

is replaced by a constant. The calculation of E A can not

be accurate when the value of P A is wrong. Furthermore, if

the pressure is high enough, then the bulk modulus can be

viewed as a constant. Thus, the reconfiguration of controllerwith respect to ∆P A is accomplished by setting the bulk

modulus E A as constant.

Scenario 3: Sensor fault ∆P BIn principle dealing with ∆P B is similar to ∆P A. However

compared with the external load and P A the pressure P B is

much smaller. P B can be viewed as a part of external load

and compensated by parameter adaption of F L. Therefore

the accommodation of ∆P B is simply realized by assuming

P B is zero.

V. EXPERIMENTAL RESULTS

The proposed fault-tolerant control is implemented on

the electro-hydraulic testbed and tested with the considered

faults. For proof of the robustness of the residual generator

the external load varies slowly from 1.5 Bar to 120 Bar.

The sampling frequency of the experiments is set to 200

Hz. The initial errors have significant effect to the fault

detection observer. Therefore, the fault diagnosis is activated

as t = 2 second. The QLin is designed as a slop-like signal

and saturated with 5 L/min. The sensor offset is 20% of the

measurement. The latency time tw is 0.5 second. The choice

of tw and the vaules of thresholds is a compromis between

the robustness and sensitivity of the diagnosis system. Here

they are determined conservativly based on the test in faultfree case to avoide false alarms.

0 20 40 600

0.2

0.4

0.6

0.8

Time (sec.)

0 20 40 60

0

0.1

0.2

0.3

0.4

Time (sec.)

|rPA|

threshold

|rPB|

threshold

Fig. 4. Residuals in fault free case

0 20 40 600

100

200

300

P o s i t i o n ( m m )

xr

xc

0 10 20 30 40 50 60−2

−1

0

1

2

Time (sec.)

e x

( m m )

0 20 40 600

5

10

15

V e l o c i t y ( m m / s e c . )

vr

vc

0 20 40 60−2

−1

0

1

2

Time (sec.)

e v

( m m / s e c . )

Fig. 5. Position tracking in fault free case



0 20 40 600

0.4

0.8

1.2

1.6

Time (sec.)

0 20 40 60

0

0.2

0.4

0.6

0.8

1

Time (sec.)

|rPA|

threshold

|rPB|

threshold

Fig. 6. Residuals with the fault QLin

0 20 40 600

100

200

300


xr

xc

0 20 40 60−3

−2

−1

0

1

Time (sec.)

e x

( m m )

0 20 40 600

5

10

15

V e l o c i t y ( m m / s e c . )

vr

vc

0 20 40 60−2

−1

0

1

2

Time (sec.)

e v

( m m / s e c . )

Fig. 7. Position tracking with the fault QLin

0 20 40 600

0.3

0.6

0.9

1.2

1.5

Time (sec.)

0 20 40 60

0

0.1

0.2

0.3

0.4

Time (sec.)

|rPA |

threshold

|rPB |

threshold

Fig. 8. Residuals with the fault ∆P A

0 20 40 600

100

200

300


xr

xc

0 20 40 60−2

−1

0

1

2

Time (sec.)

e x

( m m )

0 20 40 600

5

10

15

V e l o c i t y ( m m / s e c . )

vr

vc

0 20 40 60−2

−1

0

1

2

Time (sec.)

e v

( m m / s e c . )

Fig. 9. Position tracking with the fault ∆P A

0 20 40 600

0,1

0,2

0,3

0,4

Time (sec.)

| r P

A |

0 20 40 600

0.2

0.4

0.6

Time (sec.)

| r P

B |

Fig. 10. Residuals with the fault ∆P B

0 20 40 600

100

200

300


xr

xc

0 20 40 60−4

−2

0

2

Time (sec.)

e x

( m m )

0 20 40 600

5

10

15

V e l o c i t y ( m m / s e c . )

vr

vc

0 20 40 60−4

−2

0

2

Time (sec.)

e v

( m m /

s e c . )

Fig. 11. Position tracking with the fault ∆P B

Due to the limited space only the residuals and control

errors are shown here. The velocity trajectories are also

displayed for illustrating the transient performance. xr and

vr denote the reference trajectories of cylinder position

and velocity respectively. In fault free case the residuals

are robust to the system disturbance and varies under the

threshold, as shown in Fig. 4. The position tracking is

successfully achieved. It can be observed that the residuals

deviate from zero even through there is no faults. The main

reason is that the parameter E A and E B can not be accurately

approximated, especially in low pressure area. If the faults

are present, they can be detected and isolated correctly by

the diagnosis system, which can be observed in Fig. 6, Fig.

8 and Fig. 10. As soon as the fault is detected and isolated

the corresponding reconfiguration algorithm is activated. The

position error converges to zero quickly as shown in Fig. 7,

Fig. 9 and Fig. 11.

It is a little difficult to evaluate the test results in the

plots. The average errors of position ex and velocity ev areintroduced for comparison. They are summarized in Table I.

The best control performance is archived in fault free case.

The performance has only a very small degradation in the

presence of faults.

TABLE I

THE EXPERIMENTAL RESULTS

No fault QLin ∆P A ∆P Bex (mm) 0.38 0.41 0.49 0.43

ev (mm/sec.) 0.25 0.27 0.27 0.28



V I. CONCLUSIONS

This paper introduces a fault diagnosis integrated fault-

tolerant control for a nonlinear electro-hydraulic system.

High control performance with improved reliability and

safety can be realized with the proposed control strategy.

Two kinds of faults, cylinder leakage and sensor offsets,

can be successfully detected, isolated and accommodated.

The control performance has only a slight depredation in thepresence of faults. The control scheme can be implemented

for real-time application.

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Fault Diagnosis and Fault-Tolerant Control for a Nonlinear Electro-hydraulic System

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