Actuator Fault Diagnosis and Isolation in Vehicle Active Suspension System

S.M.Hur Rizvi1, Muhammad Abid2, Abdul Qayyum Khan3

Department of Electrical Engineering Pakistan Institute of Engineering and Applied Sciences, P.O. Nilore, 45650

Islamabad, Pakistan hur89@hotmail.com1, mabid@pieas.edu.pk2, aqkhan@pieas.edu.pk3

Abstract—In this paper actuator fault diagnosis problem in active vehicle suspension system is investigated. Full car model is used in the study. A fault diagnosis approach based on Unknown input observer (UIO) is used for detection of actuator faults. A bank of observers is used to design a fault isolation scheme for actuator faults.

Index Terms—Active Vehicle suspension system, Un-known Input observer, Actuator fault, Fault Tolerant control

I. INTRODUCTION

Vehicle suspension system connects the vehicle body to the wheels. Suspension system improves the ride quality by reducing the effect of road irregularities on human body. There are three types of suspension systems, namely passive suspension system, semi-active suspension system and active suspension systems. A passive suspension system consists of springs and dampers to improve ride quality, dampers in passive suspension have fixed damping coefficient. In semi-active suspension the dampers have variable damping coefficient. In active suspension system there is a force actuator in addition to springs and dampers. With efficient control techniques force actuator in active vehicle suspension can improve the ride comfort and vehicle stability even further. Due to requirement of superior system performance, reliability and product quality the complexity of the systems is steadily increasing. With this increase in complexity systems are becoming more prone to faults and active vehicle suspension is no exception. Bounds on system performance demand fault free operation, but even with the most dedicated effort one can not guarantee fault free operation. One way to ensure reliable operation of system during occurrence of faults is to rely upon fault tolerant control. In fault tolerant control the first step is fault detection and identification and then the controller takes the required action. Many researchers have treated the control problem in active vehicle suspension system but few have treated the fault diagnosis problem. Sensor faults have been considered but not much attention has been paid towards the actuator faults. Alain Yatendje et al. have considered the loss of actuator fault but the deviation of actuator from its operating value

has not been discussed [2], Hassan Noura and Abbas have discussed sensor fault detection and identification along with fault tolerant control in active vehicle suspension system for full car model. Fischer et al. have used parameter estimation and parity equations for model based fault diagnosis in active vehicle suspension system for quarter car model [1, 8, 9]. P. Mettadalis, G. Verros et al. have applied statistical fault diagnosis techniques to non-linear quarter car model [5] and few other techniques for fault detection in active vehicle suspension for quarter, half and full car models have been studied. In this paper work has been presented related to actuator fault diagnosis and isolation. Perfect Unknown input decoupling has been discussed for active vehicle suspension system and Unknown Input observer has been designed for detecting actuator faults. Fault isolation scheme for actuator faults has also been presented. This paper is organized as follows. In section II system model is presented. In section m PUIDP (Perfect unknown input decoupling problem) has been discussed. In section IV an UIO (Unknown Input Observer) is presented for detection of actuator faults in active vehicle suspension system. In section V isolation scheme for actuator faults has been presented. In section VI simulation results are given and finally future works and conclusion are presented.

II. SYSTEM DESCRIPTION The system model consists of car chassis (Sprung mass)

connected with the help of active vehicle suspension system to the four wheels (Un-sprung masses). Active vehicle suspension system is modelled by a linear spring, linear viscous damper and a force actuator. The dynamics of force actuator have not been considered in this paper.

A MATHEMATICAL MODEL

Mathematical model of the system is derived using Euler-Lagrange equations. Modelling equations are given as follows

. IpO = • Z¿rl) + C u f r T f ( Z s f r - ZÜfr) + CufiTf(z¿fi -

KurrTpi^Zsrr -^nrr) -^url^pi^^srl %url) "I" KufrTf(zsfr - z u f T ) + K u f i T f ( z s f i - zufi) + T f f r r +

978-1-4799-3457-7/13/$31.00 ©2013 IEEE

TABLE I SYSTEM PARAMETERS represented in the state space model as

Parameter Description Value ms Sprung mass 1500 [kg]

murr Rear right Un-sprung mass 59 [kg] muri Rear left Un-sprung mass 59 [kg] mufr Front right Un-sprung mass 59 [kg] mufi Front left Un-sprung mass 59 [kg] mu Un-sprung mass 59 [kg]

Kufl Front left tyre stiffness 35000 [N/m] Kufr Front right tyre stiffness 35000 [N/m] Kurr Rear right tyre stiffness 35000 [N/m] KUrl Rear left tyre stiffness 35000 [N/m] Kfi Front left tyre stiffness 190000 [N/m] K f r Front right tyre stiffness 190000 [N/m] Krr Rear right tyre stiffness 190000 [N/m] Krl Rear left tyre stiffness 190000 [N/m] I p Pitch axis moment of Inertia 460 [Kg-m2] Ir Roll axis moment of Inertia 2160 [Kg-m2]

T f Distance of front side from pitch axis 1 [m] Tp Distance of Rear side from pitch axis 1 [m] a Distance of left side from Roll axis 1.5 [m] b Distance of Right side from Roll axis 1.5 [m]

Tïl-fijj- (^Z-fij-j- ) Cgrr(Zg r r

f r r 7' Zrrr)

x = Ax + Bu + Edd (1)

Tf frl - Tf f f r - Tf f f l

Iv(j) — C'urrb^Zsrr -^urr) ^url^i^^srl %uri)

C u f r b ( Z g f r - ZÙfr) + Cufia(Zgfi - zùfl) - - Kurrb{zsrr

**urr ) + Kurla{zsrl — zurl) ~ K u f r b ( z s f r ~ Z u f r ) +

K u f l b { z s f l - Z u f i ) - bfrr + a f r l ~ b f f r + d f f l

mufi(züfi) = Csfi(zsfi - zùfi) + K u f i ( z s f i - zufi) + f f l - K f l ( z u f l - Z r f l )

TYlufr (-^u/r) — ^ s f r i^^sfr ^ u f r ) -^ufri^^sfr %ufr )

f f r ~ Kfr(%ufr - Z r f r )

y = Cx

Here the order of the system is 14 where, x\= 6- Pitch angle x2= 6- Pitch angular velocity x3= <j>= Roll angle x4= 4>= Roll angular velocity 005= zUrr= Rear right un-sprung height xe= ¿urr= Rear right un-sprung mass vertical velocity X7= zufr= Front right un-sprung mass height Xs= ¿ u f r - Front right un-sprung mass vertical velocity Xg— zufi= Front left un-sprung mass height xio= zufi= Front left un-sprung mass vertical velocity xu= zUri= Rear left un-sprung mass height Xi2= zuri= Rear left un-sprung mass vertical velocity xi3= zs= Chassis (sprung mass) heave position Xi4= Zg— Chassis (sprung mass) heave velocity A, B, and Ed are system matrices of proper dimensions. C has been selected to be Identity assuming availability of all states, u = [ f r r , f f r , f f l , frl]' is the control input vector and d = [zrrr, zrfr, zrfi, Zrri]' ' s the disturbance vector due to road irregularities. System matrices are given as follows.

Zurr) Kurr{,Zgfr ZUrr)

A =

f ai,i ai,2 0,2,1 a 2,2 a3,i a 3,2 04,1

\ ai4,i

ai,i4 \ a2,i4 a3,i4 a4,i4

ai4,i4 /

(2)

Wlurl i%url) — ^srli^^srl ^url) -^urli^^srl %uri) "I"

f r l — Krl (Zurl Zrrl) Values of entries of matrix A are given in appendix

• ms(Zg) = -Cgfl(Zgfl ~ ZUfl) ~ Kufl(Zgfl ~ ZUfl) + f f l - C s f r ( z s f r Z u f r ) K u f r ( z s f r - Z u f r ) + f f r -

C'srr srr %urr ) urr s f r %urr) frr ^srli^^srl

Zurl) Kuri(zsri Zurl) frl

where + Tpe + bcj)

Zsrl = Zs + TpO + CLCj)

Zsfr = Zs - T f O - bcj>

zsfi = Zs - Tf6 - cu¡>

The values of different parameters in system equations are given in TABLE I.

B. SYSTEM MODEL

The mathematical model of the system was obtained using Lagrange's equation. The mathematical model can be

B =

0 0 0 0 Tp/Ip Tp/Ip - T f / I P - T f / I P

0 0 0 0 -b/Ir a/Ir -b/Ir a/Ir

0 0 0 0 l / m „ 0 0 0

0 0 0 0 0 i K 0 0 0 0 0 0 0 0 l / m „ 0 0 0 0 0 0 0 0 l / m „ 0 0 0 0

l/ms l/ms l/ms l / r a s

(3)

Fig. 1. Full Car model with Active Vehicle Suspension System

Gyu = C(sI-A)~1B + D

Gyd = C(sI-A)~1Ed + Fd

Gyu = C(sI-A)~1Ef+Ff

Where, Gyu is transfer function between input u and output y. Gyd is the transfer function between output y and disturbance d. Gyf is the transfer function between output and fault / . For the above system residual generator in general form is given as.

r(s) = R(s)Mu(s)[Gyd(s)d(s) + Gyf(a)f(a)] (11)

Then the Ideal residual generator must fulfil following conditions so that the residual is perfectly decoupled from disturbances.

Ed = l / m „

(0 0 0 0 \ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 )

(4)

G = /i4X14 (5)

D = 0 (6)

Where I represents identity matrix.

C. Actuator Fault model In this paper we have considered only actuator fault, in the

presence of actuator fault system takes the form

x = Ax + Bu + Edd + E f f (7)

y = Cx + Du + F f f

For actuator fault, system matrices Ef and Ff are given as

R(s)Mu(s)Gyd(s) = 0 and R(s)Mu(s)Gyf(s) ± 0

A. PUIDP Existence Condition For system given by Eq. 10, Ideal residual generator which

satisfies PUIDP condition exists if and only if.

Rank[Gyf(s) Gyd(s)] > Rank{Gyd(s))

The geometric interpretation of existence condition is that the subspace spanned by Gyf(s) is different from that spanned by Gyd(s). With respect to system structure, number of unknown inputs that are observable from output y(s) should be smaller compared to number of sensors. For the purpose of ideal residual generation the residual must be decoupled from the effect of unknown inputs.

B. Unknown Input Observer In this section basic theory of UIO (Unknown Input Ob-

server) is presented, consider an LTI system given by the following equation

x = Ax + Bu + Edd + Eff (12)

y = Cx + Du + Fdd + F f f

Then there exists an UIO for the system if PUIDP solvability conditions are satisfied. Unknown input observer(UIO) has received much attention due to its application in robust state estimation and observer based robust control. UIO is a Luenberger type observer that delivers state estimation independent of unknown input d such that

Ef = B (8) Ff = D (9)

III . PERFECT U N K N O W N INPUT DECOUPLING

In this section basic concepts related to Perfect Unknown Input Decoupling are discussed. An ideal residual generator should be decoupled from disturbances or unknown inputs and should be sensitive only to faults. Consider the system given by following equation.

y(s) = Gyu(s)u(s) + Gyd(s)d(s) + Gyf(s)f(s) (10)

limt^m{(x(t) — x(t)) = 0 for all u(t) and d(t)

This is the way that is widely used in design of Unknown input diagnostic observer because the algorithms for designing UIO are well established

Existence Conditions for existence of UIO: Given a system model (12) there exists an UIO exists if and

only if

1) rank(CEd)=Ed 2) (A, Ef,C) has no unstable transmission zero.

Unknown Input Observer design: In this section basic theory of UIO design has been pre-

sented. Consider an LTI system given by Eq. 12 such that it satisfies the existence conditions for UIO that is

rank(CEd) =Ed = kd (13)

Then there exists a matrix Mce satisfying

Mce(CEd) = Ikdxkd (14)

and matrix T is given as

T = I - EdMceC

Then the Unknown input observer is given by following equations

z = (TA — LC)z + TBu + ((TA- LC)EdMce + L)y (15)

and the residual equation of UIO is given as

r = v((I - CEdMce)y - Cz), v^O

L is selected in such a way that A — LC — EdMceCA is stable.

IV. APPLICATION TO ACTIVE VEHICLE SUSPENSION SYSTEM

In this section an unknown input observer (UIO) is designed to detect actuator faults in active vehicle suspension system. First of all PUIDP solvability is checked for active vehicle suspension system and then Unknown input observer based approach is used to design a residual generator.

A. PUIDP Solvability Perfect unknown input decoupling problem solvability is

checked for the vehicle active suspension system for full car model. System is represented in state space form by Eq. 12. We check the existence conditions for PUIDP solvability

Rank(Gyd) = 4

and

Rank([Gyd Gyf] = 7

It is seen that the system satisfies the PUIDP solvability conditions, i.e.

Rank[Gyf(s) Gyd(s)] > Rank(Gyd(s))

Since the system satisfies the PUIDP conditions, hence an unknown input residual generator can be designed in order to detect actuator faults.

B. Design of UIO for Active Vehicle Suspension System In this section UIO is designed for detecting actuator faults

in active vehicle suspension system. First of all existence conditions are checked i.e.

Rank(CEd) = Rank(Ed) = 4

The existence condition is satisfied, there exists an UIO for the system given by Eq. 15 Mce is found according to Eq. 14,

it's value is found to be Mce=

/ 0 . . . 59 0 0 \ 0 . . . 0 0 59 0 0 0 . . . 0 0 0 0 59 0 0

i 0 . . . 0 0 0 0 0 0 59 0 . . . / (16)

Matrix T is found according to equation

T = I - EdMceC

L is selected ensuring stability of A — LC — EdMceCA. Since the system satisfies the PUIDP conditions, the designed UIO is sensitive only to actuator faults.

V. FAULT ISOLATION SCHEME There are four actuators in active vehicle suspension system

for full car model. Fault isolation is important in order to know in which actuator fault has occurred.

Theorem: Perfect fault isolation is possible if and only if

Rank(Gyf ) = ra/=Number of faults

In order to make fault isolation possible a bank of 4 UIO is designed such that each UIO is sensitive to only one actuator fault. Each UIO treats one actuator fault and treats faults in rest of the actuators as un-known input. There are four actuators in the system and hence there can be four actuator faults, consider that the matrices Ef and Ed can be represented as.

• Ef=lefi e /2 e/3 e/4], Ed=[edl ed2 ed 3 ed4] Residual signal r\(t) indicates fault in actuator 1, ideally it should be sensitive only to fault f 1. Since the system satisfy the PF1 (Perfect fault isolation) condition so in order to isolate / 1 (fault in actuator 1) from other actuator faults we reconstruct Ed and Ef as

• Êf=[efi], Êd=[Ed ef2 ef3 e / 4 ] ^ and then the UIO is designed using Ef and Ed which would be sensitive only to actuator fault (fl) and would treat faults (f2, f3 and f4) as unknown inputs thus the residual 7*1 (t) would only be sensitive to actuator fault (fl). In the same way residual generators are designed to isolate other actuator faults.

VI. SIMULATION RESULTS In order to check the effectiveness of fault diagnosis scheme

simulation has been done in MATLAB/SIMULINK. Fig 2. shows the comparison of a passive suspension system and active suspension system in the presence of disturbance. The Fig 3. shows the output of residual generator for fault in rear-right force actuator.

VII. CONCLUSION AND FUTURE WORKS A. Conclusion

Simulations show that active suspension system further improves the ride comfort as compared with passive sus-pension system. Actuator fault detection is necessary since

f 'i P

J 0.Í j ! i i j 1 1 1 ' i

I I IL

m l í i

S -15

5

Í ! ! í I i 1 ' l i î j

y

í ! ¡ j !

! V 1

1 2 3 4 5 6 7 8 time 9 11

Fig. 2. Comparison of Chassis heave position between passive (...) and active suspension system (-)

u 1

0.3

I 06

EC

D.4

Ü?

1 t Î 5 6 r tima

[6] D. Fischer, E. Kaus, RJsermann, "Fault Detection for an Active Vehicle Suspension", American Control Conference, Proceedings of the 2003, Vol.5, June 4-6, 2003, pp 4377-4382.

[7] B. P. Jeppesen and D. Cebon, "Analytical Redundancy Techniques for Fault Detection in an Active Heavy Vehicle Suspension", 6th International Symposium on Advanced Vehicle Control, AVEC2002, Hiroshima, Japan, September, 2002.

[8] A. Chamseddine, H. Noura, M. Ouladsine, "Sensor Fault Detection, Iden-tification and Fault Tolerant Control: Application to Active Suspension", American Control Conference ACC'06, June 2006.

[9] A. Chamseddine, H. Noura, M. Ouladsine, "Sensor location for actuator fault diagnosis in vehicle active suspension", Control Applications, 2008. CCA 2008. IEEE International Conference, Sept. 2008.

[10] T.S Ghiasi and Zarabadipour, "Interval fuzzy modelling applied to model based fault detection of an active suspension system", Fuzzy Systems (FUZZ), 2011 IEEE International Conference, June 2011.

[11] Marcus Brner, Harald Straky, Thomas Weispfenning, Rolf Isermann, "Model based fault detection of vehicle suspension and hydraulic brake systems", Mechatronics, Volume 12, Issue 8, October 2002, Pages 999-1010, ISSN 0957-4158

Fig. 3. Residual output (...) of UIFDF for rear right actuator fault (-)

malfunction of actuator can result in system getting unstable. Fault diagnosis and isolation scheme based on Unknown input observer (UIO) makes detection of any actuator fault possible.

B. Future works In this paper active vehicle suspension system for linear full

car model has been considered, the dynamics of force actuator have not been taken into account in actuator fault diagnosis. Future works will focus on actuator fault diagnosis taking into consideration dynamics of force actuator.

REFERENCES

[1] A. Chamseddine and H. Noura, "Control and Sensor Fault Tolerance of Vehicle Active Suspension", IEEE Transactions on Control Systems Technology, vol.16. No.3, pp. 416 - 433, May 2008.

[2] Yetendje A., Seron M., De Dona J., "Diagnosis and actuator fault tolerant control in vehicle active suspension", Information and Automation for Sustainability, 2007. ICIAFS 2007. Third International Conference on Date 4-6 Dec. 2007

[3] Steven X. Ding, Model-based fault Diagnosis techniques- design schemes, algorithms and tools. Springer, 2008.

[4] N. Zbiri, A. Rabhi and N.K. M'Sirdi, "Diagnosis on the Vehicle Suspen-sion", Conference Proceedings, ACD Workshop, Karlsruhe, Proc. IEEE, November 17-18, 2004, pp 148-153.

[5] P. Metallidis, G. Verras, S. Natsiavas, C. Papadimitriou, "Paramet-ric Identification and Fault Detection in Vehicle Models with Non-linear Suspension", Proc. of the 18th Biennial ASME Conference on Mechanical Vibration and Noise, . Pittsburgh, Pennsylvania, paper No. DETC2001ATB21411, September 9-13, 2001.

The « 1 , 2

« 2 , 1

« 2 , 2

«2,3 «2,4 «2,5

« 2 , 1 0

« 2 , 1 2

«2,13 « 2,14 Û4,l Û4,2 «4,3 «4,4 «4,5 «4,8 04,10 «4,13 «4,14 « 6 , 1

«6,3 = ß6,5 «6,7 • • «6,13 « 8 , 1 =

«8,3 = as,7 ß8,9 • • «8,5 = « 1 0 , 1

«10,3 «10,9 Gl0,5 • « 1 0 , 1 1

« 1 2 , 1

«12,3 = « 1 2 , 1 1

APPENDIX entries in system Matrix A are given below

= «3,4 = «5,6 = «7,8 = «9,10 = Gil,12 = «13,14 = 1 (—KuriT — KurrT — KufrTţ — KufiTf)/Ip

TS _ rlp — (—CsriTp — Cs

- f jTr Co • CsfTrj - <Jsfi± f)/±p , . , . T])/lP

(—KuriTpb — KurrTpa — KufrTfb — KufiTfà)/Ip

(~CsrlTpb — CsrrTpa — CsfrTfb — CsfiTfà)/Ip

KurrrTp/Ip, Oj2,Q — CsrrTp jIp, a2,7 = - K u f r T f / I p

— KurlFpj Ip = —CsfrTf/Ip, a2,9 = - K u f i T f / I p

= —Ks}iTf/Ip, a2i ii — CsriTpj Ip

— ( Kur{Tp KurrTp KufrTf K u f r T f ) / I p

— ( CsriTp CsrrTp CsfrTf CsfrTf)/Ip

= {—Kur{Tpa — KurrTpb — KufrTfb — KufiTfà)/Ir = (—Csr{Tpa — CsrrTpb — CsfrTfb — CsfiTfa)/Ir = (-Kurla2 - Kurrb2 - Kufrb2 - Ku}la2)/Ir = (-Ksrla? - Ksrrb2 - Ksfrb2 - Ks}la?)/Ir = Kurrb/Ir, 04,6 = Csrrb/Ir, a,4:7 = —Kufrb/Ir = —Csfrb/Ir, a^g = —Kufia/Ir = —Ksfia/Ir, «4,11 = —Kuria/Ir, «4,12 = —Csria/Ir = {—Kuria — Kurrb — Kufrb — Kufra)/Ir = (—Csria — Csrrb — Csfrb — Csfra)/Ir

— KurrTp/TTlu, OjQ, 2 — CSj.j.Tp j Tïlu

—Kurrb/mu, «6,4 = —Csrrb/mu

Kurr j- J JJh., OjQ,Q Csrr/77lu

•«6,12 = 0 — KUTT jTTlu, «6,14 — Csrr/T7lu

= —KufrTf/mu, «8,2 = —CsfrTf/mu

= —Kufrb/mu, «8,4 = —Csfrb/mu

Kufr .«8,12 = 0

- Kfr/mu, «8,8 = -Csfr/mu,

«8,6 = 0, «8,13 = Kufr/mu, «8,14 = Csfr/mu

= -KufiTf/mu, «10,2 = ~CsfiTf/mu

= -Kufib/mu «10,4 = —Csfib/mu

= —Kufr — Kfr/mu, «10,10 = —Csfr/mu

..«10,8 = 0 = «10,12 = 0, «10,13 = K u f i / m u , «10,14 = C s f i / m u

= —KuriTf/mu, «i2,2 = —CsriTf/mu

= —Kurlb/mu, «i2,4 = —Csrlb/mu

= —Kurl — Kri/mu, «i2,i2 = —Csri/mu

ßl2,5 • • • ßl2,9 = 0 ßl2,13 = Kurl/mu, ßl2,14 = Csrl/mu «i4,i = (—KuriTp — KurrTp — KufrTf — KufrTf)/ms ßi4,2 = {-CsrlTp — CgrrTp — CgfrTf — CgfrTf)/mg ßl4,3 = ( — Kurld- — Kurrb — Kufrb — Kufra)/mg

ßi4,4 = {—CgriCL — Cgrrb — Cgfrb — Cgfra)/mg

«14,5 = Kurr/fUs, duß = Csrr/ms, 014,7 = Kufi/ms

«14,8 = Csfi/ms, fll4,g = Kurl/ms, (114,10 = Csrl/ms

ßl4,n = Kufr/ms, ßl4,12 = Cgfr/mg «14,13 = (—Kufr — Kufl — Kurr ~ Kurl)/ms

ßl4,14 = (—Cu fr — Csfi — Csrr ~ Csrl)/ms