Sensor fault-tolerant control for active suspension using sliding mode techniques Abbas Chamseddine, Hassan Noura, and Mustapha Ouladsine Laboratoire des Sciences de l’Information et des Systèmes LSIS - UMR CNRS 6168 Université Paul Cézanne, Aix-Marseille III Domaine Universitaire de St Jérôme, Ave. Escadrille Normandie-Niemen, 13397 Marseille Cedex, France Email: [email protected], Tel: 04 91 05 60 59

Abstract—In this paper, a nonlinear full vehicle active suspension system and its actuation modules are considered for the purpose of sensor fault tolerance. A sliding mode controller is designed. Sensor fault effects on the controlled system are shown and a fault tolerance strategy is proposed. The effectiveness of the control and the fault tolerance strategy is analyzed in simulation.

I. INTRODUCTION

D

UE to its importance in vehicle industry, active suspension system is being the subject of many researches concerning its design, control, fault diagnosis and fault tolerance problems. The active suspension system made it possible to surpass the compromise between ride comfort and vehicle stability, which was not possible with the passive suspension. The idea of the active suspension is to insert a force actuator between the sprung and the unsprung masses in addition to the spring and the damper, replacing the damper or replacing both the spring and the damper. The advantage of the active suspension resides in its capability to control the attitude of the vehicle, to reduce the effects of braking and to reduce the vehicle roll during cornering maneuvers in addition to increase the ride comfort and vehicle road handling [1]. The control design of the active suspension system has witnessed a major development; nowadays, many control laws were applied and designed for linear and nonlinear models of quarter, half and full vehicle systems. One may cite: PID controller [2], state and output feedback scheduled controller [3], CRONE [4], mixed H2/H∞ [5], modular adaptive robust control technique [6], stochastic optimal [7], proportional – integral sliding mode [8], combined filtered feedback and input decoupling transformation [9], and fuzzy logic [10] controllers. Depending on the applied control law, active suspension system uses a set of sensors: gyrometers to measure the vehicle pitch and roll angular velocity, position sensors or accelerometers to measure the vehicle heave position and the vertical position of each wheel, accelerometers to determine when the vehicle is accelerating, braking or cornering. Some systems use

‘road preview’ sensors to provide information about the road irregularity before the front wheels reach them. Like any other physical components, the sensors are subject to faults such as noise, outliers, breakdown with a constant value, offsets and gains. Variables measured by faulty sensors are inaccurate and erroneous measurements, which may cause an undesired behavior of the controlled system and dangerous consequences on the passengers and the system. The presence of a fault detection and isolation (FDI) system is then necessary to detect the occurrence of a fault and isolate it. Normally, the task which comes after the isolation of a fault is its repairing. However, maintenance cannot always be done immediately. Thus, when implementing the control law, one should take into consideration the possibility of sensor faults occurrence and propose a set of strategies to compensate for it: which is the idea of the fault tolerant control system (FTCS). The main objective of the FTCS is to insure, in the presence of a fault, the highest possible performance of the controlled system like stability, rapidity and precision. When the FDI informs the FTCS of the presence of a fault and its location, this latter takes the decision to stop the system or to continue running in degraded mode. This decision takes into consideration the severity of the occurred fault, the degree of redundancy, the remained resources and the possibility to manage them, the required performances…etc. To the best of our knowledge, few are the researches which treat the fault diagnosis problem on the suspension system: the parity method is used to detect the sensor faults and the least square method to detect mechanical failures of quarter-car passive suspension system [11]. Statistical methodologies are applied to perform fault detection in nonlinear two degree of freedom quarter-car model and complete vehicle models [12], [13]. Model based fault detection and identification methods are developed in [14], [15], and analytical redundancy techniques are used for fault detection for active heavy vehicle suspension [16]. Fewer treat the suspension fault tolerance problem: a design of a bank of Kalman filters, one for each possible sensor failure configuration,

providing an estimate of the system state when a sensor fault occurs, is carried out for a quarter car test rig [17]. A methodology for the comparison among different alternative fault tolerant architectures, based on risk evaluation has been applied to a full active suspension control system [18]. Generally speaking, researches treat separately the active suspension and the actuation module. However these two systems are coupled, thus taking into consideration that interconnection makes the study more realistic. In [17] a sensor fault tolerant strategy is proposed for a quarter car test rig using an electrohydraulic actuator. In this paper a full nonlinear vehicle active suspension with its hydraulic actuation system are considered for the purpose of fault tolerant control study. A sliding mode control is designed for the whole system. Sensor faults effects on the behavior of the controlled system are analyzed and a fault tolerant control strategy to compensate for the sensor faults is proposed. This paper is composed as follows: the system model is presented in section II. The control strategy is then discussed in section III and simulation is made to apply the proposed control on the system. In section IV the effect of sensors faults on the attitude of the vehicle is showed and a fault tolerant control strategy is proposed in section V. Finally a conclusion and future work are given.

Each active suspension system is modeled as a linear viscous damper, a linear spring and a force actuator. Each wheel is modeled as a linear spring. The car body is free to heave, pitch and roll, and the wheels are free to bounce vertically with respect to the car body. Supposing that the force actuators use the hydraulic technology to generate their forces, the dynamics of the electro-hydraulic servo-valves given by Gaspar et al in [19], will be taken into consideration in this study. This will be useful to understand more the occurrence of sensor faults and its propagation. Both an actuator and its corresponding servo-valve are represented in Fig. 2. Zsij

ms Zvij R S Ksij

fij

Bsij

Zuij

Actuator

muij Kuij Zrij

R

uij Servo-valve

Road

Fig. 2. Representation of an actuator with its servo valve with i = f , r ; j = r, l

Each actuation module consists of a cylinder (actuator) and a four way servo valve. The function principle of the actuation module is to generate a force u ij on the spool

II. SYSTEM MODEL Since the dynamics of the vehicle active suspension system are nonlinear in reality and since faults may lead the linear or the linearized system model out of the region where the linear model and its control are valid, a nonlinear model of the full vehicle active suspension system given by Güçlü [2] was adopted for this study. For the sake of simplification, the dry friction forces and the passenger seat presented in the original model will not be considered here. The full vehicle model consists of the car body (sprung mass) connected by the suspension systems to four wheels (unsprung masses), this system is illustrated in Fig. 1.

which is able to move forward or backward in the direction of the spool valve. The variation of the spool position results in the variation of the amount of fluid entering (or leaving) the cylinder rooms, thus changing the pressure drop across the piston and generating the actuator force f ij . For more details concerning the physical models, the reader can refer to [2] and [19]. The concatenation of the vehicle active suspension and the servo-valves models results in an input affine nonlinear model of twenty-two states, which can be written as a state space representation, as following:

Z z s rl

z s rr K s rr B s rr

m u rr

K u rr

z rrr

θ

K sfr Bs fr

f fr

X

B s rl

f rl

K u rl

z sfl K sfl Bsfl

K u fr

K u fl

Fig. 1. Model of a full vehicle

z rrl

Where x(t )∈ℜ 22 is the state vector of the following components: x1 = zufr : front-right unsprung mass height,

f fl

m u fl

m u fr

(1)

z u rl

m u rl

ms

φ

z sfr

z rfr

K s rl

z Y

z u rr

z u fr

f rr

x& (t ) = f ( x(t )) + Gu (t ) + Fd (t ) y (t ) = Cx(t )

z u fl z rfl

x2 = z&ufr : front-right unsprung mass velocity, x3 = A fr : front-right actuator load pressure, x4 = zvfr : front-right spool valve position,

x5 = zufl : front-left unsprung mass height,

III. SLIDING MODE CONTROL STRATEGY

x6 = z&ufl : front-left unsprung mass velocity,

Sliding mode control (SMC) is used to control the vehicle’s heave position, pitch and roll angles. The main advantage of SMC is its insensitivity to variation in system parameters, external disturbances and modeling errors. SMC was applied to vehicle active suspension system by many researchers [22], [23] and [24]. However, in this paper a novel SMC strategy is proposed and analyzed. When applied to (1), three SMC will be used as illustrated in Fig. 3; the first controller determines the

x7 = A fl : front-left actuator load pressure, x8 = zvfl : front-left spool valve position, x9 = zurr : rear-right unsprung mass height, x10 = z&urr : rear-right unsprung mass velocity, x11 = Arr : rear-right actuator load pressure, x12 = zvrr : rear-right spool valve position, x13 = zurl : rear-left unsprung mass height, x14 = z&url : rear-left unsprung mass velocity,

d , Arld ]T desired actuator load pressure A d = [ A dfr , A dfl , Arr

x15 = Arl : rear-left actuator load pressure,

d d T = [ x 3d , x 7d , x11 , x15 ]

x16 = zvrl : rear-left spool valve position,

d d d T P d = [ z d , θ d , ϕ d ]T = [ x17 , x19 , x 21 ] in the presence of

x17 = z : heave position of the sprung mass, x18 = z& : heave velocity of the sprung mass,

to obtain the desired position

external disturbances d (t ) , the second determines the spool

x19 = θ : pitch angle of the sprung mass, x = θ& : pitch angular velocity of the sprung mass,

valve

d d d z vd = [ z vfr , z vfl , z rr , z rld ]T

position

d d T = [ x 4d , x8d , x12 , x16 ] to obtain the desired actuator load

20

pressure Ad and the third controller determines the input u to servo-valve to obtain the desired spool valve

x21 = ϕ : roll angle of the sprung mass, x22 = ϕ& : roll angular velocity of the sprung mass,

position zvd .

u (t ) = [u fr (t ), u fl (t ), urr (t ), url (t )] T ∈ℜ 4 is the vector of the inputs to the four servo-valves, d (t ) = [ z r fr (t ), z r fl (t ), z rrr (t ), z rrl (t )] T ∈ ℜ 4

d(t) Pd

is

the

vector of the external disturbances affecting the system due to the irregularities of the road. As will be shown later, the control of the system requires that all the system states must be measured. Thus fifteen sensors should be used: accelerometers which measure the heave acceleration &z& and the four unsprung masses acceleration &z&uij , velocities and positions are obtained by double integration and filtering of measurements [20]. Gyrometers are used to measure the pitch θ and roll φ angular velocities; angles are obtained by integrating the measurements [20]. The spool valve positions zvij can be measured by linear variable displacement transducers (LVDT) and load cells can be used to measure the actuators force output f ij [21]. The force generated by the hydraulic actuator is given by: f ij = S . Aij with S is the area of the piston

and i = f , r ; j = r , l . Thus, matrix C is the identity matrix of dimension 22. G and F are constant matrices with appropriate dimensions. For brevity, G , F and f ( x) are not detailed in this paper.

y

Load Ad pressure SMC

Spool valve position SMC

zvd

Servo valve input SMC

u

System

y

Fig. 3. Control strategy

The vehicle active suspension is a second order input affine nonlinear system, thus the first sliding mode control law is considered as [25]:

&&d − M sign( s )] u1 = Ad = G1+ [ P 1 1

(2)

&&d = [ &x&d , &x&d , &x&d ] T , s is the sliding surface Where: P 1 1 17 19 21 & defined as: s1 = e1 + λ e1 , e1 is the error given by: d d d T e1 = [ x17 − x17 , x19 − x19 , x21 − x21 ] and M is a diagonal matrix of appropriate dimension. In order to smooth out the discontinuous sign function, a saturation function is used instead of it. The control law is then given by:

&& d − M sat ( s )] u1 = Ad = G1+ [ P 1 1

(3)

Where the saturation function is defined as follows:

sign( s1 ) when s1 > σ  sat ( s1 ) =  s1 when s1 ≤ σ  σ

(4)

And σ is the thickness of a thin boundary layer neighboring the switching surface. The details of the control law are not presented in this paper. However, for more details about the spool valve position and servo valve input sliding mode control, the reader can refer to [21]. The efficiency of the previously proposed controller (3), together with the controllers proposed by Alleyne et al [21] is analyzed in simulation using MATLAB/SIMULINK. The input terrain disturbances vector d (t ) is supposed to have the form:

z r fr (t ) = a fr sin(ω t ) z r fl (t ) = a fl sin(ω t )

3000

1000

z rrl (t ) = arl sin(ω (t + τ d )) Where a fr , a fl , arr and arl are the terrain disturbance amplitudes at the front-right, front-left, rear-right and rear-left wheels respectively, ω is the terrain disturbance frequency, τ d is the time delay given by τ d = L / v , L is the distance between the front and rear axle and v is the speed of the vehicle. In this study the terrain disturbance amplitudes were chosen to be: a fl = arl = 0.05 m and a fr = arr = 0.1 m,

2

Roll acceleration φ ′ ′ (rad/s )

2

40

Pith acceleration θ ′ ′ (rad/s )

2

Heave acceleration z′ ′ (m/s )

whereas the terrain disturbance frequency is ω = 9 rad/s. For the rest of the constants, the reader can refer to [9] for those concerning the active suspension system and to [19] for those concerning the electro-hydraulic servo-valve system. Fig. 4 shows the comparison between the passive and the active suspension system for the heave z , pitch θ and roll φ directions respectively.

20 0 -20

0

1

2 Time (s)

3

4

10 5 0 -5 -10

0

1

2 Time (s)

3

10 5 Passive suspension Active suspension

0 -5 -10

0

1

2 Time (s)

3

Desired force Generated force

2000

Front right force ffl (N)

(5)

z rrr (t ) = arr sin(ω (t + τ d ))

-40

The dotted line represents the time evolution of the acceleration for the passive suspension system, whereas the solid line represents it for the active suspension system. It can be seen that active suspension improves heave, pitch and roll accelerations. The desired force, determined by the first SMC, to improve the heave position, pitch and roll angles, represents the input of the two other sliding mode controllers. These two latter controllers determine firstly the spool valve position and secondly the input to servovalve to track the desired force. Fig. 5 shows the front right actuator force tracking, it is clear that the desired force is perfectly tracked by the force generated by the actuator. The force tracking is the same for the three other actuators but are not shown here for brevity.

4

Fig. 4. Passive and active suspension comparison

4

0

-1000

-2000

-3000

-4000

0

0.5

1

1.5

2

Time (s)

2.5

3

3.5

4

Fig. 5. Actuator force tracking

Although that one of the advantages of the SMC is its insensitivity to variation in system parameters, external disturbances and modeling errors, when applied to the real system, the tracking may not be as perfect as obtained in simulation [21]. IV. INFLUENCE OF SENSOR FAULT Physical system elements: sensors, actuators and components are often subject to failures which affect the overall performance of the system. For example, sensors signals contain the most important information for the feedback of the system controller. The behavior of the closed loop system depends on the information provided by the sensors about the states of the system and the exterior environment, e.g., the heave, pitch and roll of the sprung mass…etc. Then an undesirable or even dangerous behavior can occur due to a faulty sensor. In this paper, the problem of completely broken down sensors is considered; in this case the system controller totally looses the information provided by the broken sensor and the measurement given by the sensor becomes zero. Simulation shows the effect of sensor breakdown on the behavior of the system: Fig. 6 shows the break down effect of the sensor measuring the heave position z on the heave acceleration. When the fault occurs at instant t = 0.5 s, the nominal fault free output of the system (solid line) is degraded (dotted line). The dashed line represents the heave acceleration for the passive suspension.

Fig. 7 shows the effect of front right force f fr sensor breakdown on the roll acceleration: the nominal free fault performance (solid line) is hardly damaged when the fault occurs at the instant t = 0.5 s (dotted line), it is obvious that the effects of such fault is even worse than the passive suspension (dashed line).

In the case of sensor faults, the fault can be compensated by replacing the measured variable by its estimation if this estimation is possible. Observers should be designed to reconstruct the lost variable. In this paper an observer is proposed to estimate the heave position z and the heave velocity z& when the sensor providing these states breaks down.

Acceleration in the heave direction z (m/s2)

25

Passive suspension Active suspension Active suspension with sensor fault

20

A. Sliding mode observer design

15 10 5 0 -5

-10 -15 -20 -25

0

0.5

1

1.5

2

Time (s)

2.5

3

3.5

4

Fig. 6. Heave position sensor breakdown effect on heave acceleration &z& Acceleration in the roll directionφ (rad/s2)

20

Passive suspension Active suspension Active suspension with fault

15

10

5

0

-5

The study of the observability of the system after the lost of the sensor measuring the heave acceleration shows that the system remains observable and thus an estimation of the lost variable can be used in the place of the lost measurement. To estimate the heave position and the heave velocity, a sliding mode observer (SMO) is designed by using the techniques employed in [26]. To simplify the structure of the observer and to reduce the time of calculations, a reduced order observer will be used. In the observer structure, the lost variables are replaced by their estimations and the other measured variables are considered as inputs to the observer. The structure of the sliding observer is given in the Appendix. h17 , h18 and h20 are the sliding gains, and ~ x = x − xˆ is the error between the measured variable x and its estimation xˆ .

-10

The error dynamics are given by:

-15

-20

0

0.5

1

1.5

2

Time (s)

2.5

3

3.5

4

Fig. 7. Front right force sensor breakdown effect on roll acceleration ϕ&&

In the presence of an FTCS, the system should be able to compensate for the sensor faults when it is possible. However some faults may be compensated for and others may not be: in the case of sensors faults, a fault can be compensated if the lost state can be replaced by its estimation. When the lost state cannot be estimated, the fault is considered as critical. V. SENSOR FAULT TOLERANCE STRATEGY An FDI module monitors the system at each instant and informs the FTCS of the fault location when it occurs and provides, if possible, its amplitude. The main objective of an FTCS is to compensate for failures to maintain the desired performance of the system. If the desired performance cannot be maintained, in the presence of a fault, the FTCS should be able to keep the system running in a safe degraded mode, until it is possible to shut off the system and repair the failed component. Assumption: In this paper, the FDI module is not treated. However a sufficient period of time is left, where the FDI module should have detected and isolated the failure before switching to the compensation for the effect of this failure.

~ x&17 = x&17 − x&ˆ17 = ~ x18 − h17 sign( ~ x20 ) ~ x&18 = x&18 − x&ˆ18 = [−(k sfr + k sfl + k srr + k srl ) ~ x17 ~ − (csfr + csfl + csrr + csrl ) x18 ] / M − h18 sign( ~ x20 )

(6)

(7)

~ x&20 = x&20 − x&ˆ 20 = cos( x19 ){ − [a (k sfr + k sfl ) − b(k srr + k srl )]~ x17 − [a(csfr + csfl ) − b(csrr + csrl )]~ x18 } / I yy (8) − h sign( ~ x ) 20

20

The constants are replaced by their numerical values and the angles are assumed too small. If the sliding surface is defined as s = ~ x20 = x20 − xˆ 20 then the average error dynamics during sliding where s = 0 and s& = 0 is: ~ x 20 = 0 ~ x&20 = 0 ⇒ 14.44 ~ x17 + 0.4352 ~ x18 − h20 sign( ~ x20 ) = 0 ~ ~ &x = ~ x − h sign( x )

(10)

~ x&18 = −97.3~ x17 − 2.8~ x18 − h18 sign( ~ x20 )

(12)

17

18

17

20

(9)

(11)

Substituting (10) in (11) and (12) the two following equations are obtained: ~ x&17 = −14.44 p ~ x17 + (1 - 0.4352 p) ~ x18 ~ ~ x&18 = −(97.3 + 14.44q) x17 − (2.8 + 0.4352q) ~ x18

(13) (14)

The convergence of ~ x17 and ~ x18 depends on the values h h of p and q , where p = 17 and q = 18 . h20 h20 The constants p and q are determined by pole placement technique so as to guaranty the convergence of the estimation error of x17 and x18 . B. Simulation results The observer is tested by simulation. With non zero initial conditions, the estimations of the heave position z and the heave velocity z& tend to track the true evolution of the lost variables as illustrated in Fig. 8. It is clear that the errors of estimation ~ x17 and ~ x18 rapidly converge to zero. Heave velocity z′ (m/s)

True heave position Estimated heave position

0.02 0 -0.02 0

0.5 Time (s)

1

0.02 0 -0.02 -0.04 -0.06

0

0.5 Time (s)

1

0.2 0 -0.2 -0.4 -0.6

Estimation error of the heave velocity (m/s)

0.04

-0.04 Estimation error of the heave position (m)

0.4

0

0.5 Time (s)

1

0.6

REFERENCES [1]

0 -0.2 -0.4

0

0.5 Time (s)

1

15

Fault free active suspension Active suspension with fault without FTCS Active suspension with fault with FTCS

5

0

-5

-10

-15

0

0.5

1

1.5

In this paper, a nonlinear full active suspension system and its actuation modules are considered. A sliding mode control is proposed for the whole system. The effects of the sensor breakdown on the attitude of the system are shown and a sliding mode observer is proposed to estimate the lost heave variables and to compensate for the heave sensor breakdown. Future work consists of designing observers for the other sensors and taking into consideration the possibility of the occurrence of multiple sensor faults. The compensation for actuator faults will also be studied.

0.2

To eliminate the chattering phenomenon and to make the estimated variables smoother, the discontinuous function ‘sign’ is replaced by the saturation function as defined in (4). After the identification and the isolation of the sensor fault by the FDI which is supposed to be existed, the FTCS estimates the lost variable and injects it in the controller in the place of the measurement. This sensor fault compensation is shown in Fig. 9.

10

VI. CONCLUSION

0.4

Fig. 8. Estimation of the heave position z and velocity z&

Heave acceleration (m/s2)

Heave position z (m)

0.06

When the sensor fault occurs at instant t = 0.5 s, the performance of the free fault active suspension in the heave direction (dashed line) is badly damaged (dotted line). In the presence of the FTCS the estimated variables are injected in the place of the lost measurements, the system returns then to its pre-fault performance (solid line). However, as mentioned before, not all the sensor faults can be compensated because of the loss of the system observability after the fault occurance and the impossibility to estimate the lost variable. In this case the fault is considered as critical and the replacement of the faulty sensor should be done as soon as possible. In the case of compensable sensor faults, a bank of observers should be designed, one observer for each possible fault.

2

2.5

3

Time (s)

Fig. 9. Sensor fault compensation

3.5

4

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Appendix The observer structure is given by: xˆ&17 = xˆ18 + h17 sign( ~ x20 ) x&ˆ18 = { k sfr x1 + csfr x2 + k sfl x5 + csfl x6 + k srr x9 + csrr x10 + k srl x13 + csrl x14 − (k sfr + k sfl + k srr + k srl ) xˆ17 − (csfr + csfl + csrr + csrl ) xˆ18 − [a (k sfr + k sfl ) − b(k srr + k srl )] sin( x19 ) − [a (csfr + csfl ) − b(csrr + csrl )] cos( x19 ) x20 − [d (k sfl + k srl ) − c(k sfr + k srr )] sin( x21 ) − [d (csfl + csrl ) − c (csfr + csrr )] cos( x21 ) x22 + S ( x3 + x7 + x11 + x15 ) } / M + h18 sign( ~ x20 )

x&ˆ20 = cos( x19 ){ ak sfr x1 + acsfr x2 + ak sfl x5 + acsfl x6 − bk srr x9 − bcsrr x10 − bk srl x13 − bcsrl x14 − [a(k sfr + k sfl ) − b(k srr + k srl )]xˆ17 − [a (csfr + csfl ) − b(csrr + csrl )]xˆ18 − [a 2 (k sfr + k sfl ) + b 2 (k srr + k srl )] sin( x19 ) − [a 2 (csfr + csfl ) + b 2 (csrr + csrl )] cos( x19 ) x20

− [d (ak sfl − bk srl ) − c(ak sfr − bk srr )] sin( x21 ) − [d (acsfl − bcsrl ) − c(acsfr − bcsrr )] cos( x21 ) x22 + S [a ( x3 + x7 ) − b( x11 + x15 )] }/ I yy + h sign( ~ x ) 20

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