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Int. J. Vehicle Design, Vol. 48, Nos. 3/4, 2008
Vehicle parameter estimation and stability enhancement using sliding modes techniques Hassan Shraim∗ and Mustapha Ouladsine Laboratory of Sciences of Informations and of Systems, LSIS UMR 6168 University of Paul Cézanne, Aix-Marseille III Av, Escadrille de Normandie, Niemen 13397, Marseille Cedex 20 E-mail:
[email protected] ∗ Corresponding author
Leonid Fridman Department of Control, Division of Electrical Engineering, National Autonomous University of Mexico (UNAM), Ciudad Universitaria, 04510, DF, Mexico E-mail: lfridman@verona.fi-p.unam.mx
Monica Romero Facultad de Cs. Exacts, Ingenieria y Agrimensura, Departemento de Electronica, Universidad Nacional de Rosario – Argentina E-mail:
[email protected] Abstract: In this paper, tyres longitudinal forces, vehicle side slip angle and velocity are identified and estimated using sliding modes observers. Longitudinal forces are identified using higher order sliding mode observers. In the estimation of the vehicle side slip angle and vehicle velocity, an observer based on the broken super – twisting algorithm is proposed. Validations with the simulator VE-DYNA pointed out the good performance and the robustness of the proposed observers. After validating these observers, controller design for the braking is accomplished using a reduced state space model representing the movement of the vehicle centre of gravity in the (X, Y ) plane. Driver’s reactions are taken into account. The performance of the closed loop system is carried out by means of simulation tests. Keywords: automotive; estimation; sliding mode observer; vehicle parameters. Reference to this paper should be made as follows: Shraim, H., Ouladsine, M., Fridman, L. and Romero, M. (2008) ‘Vehicle parameter estimation and stability enhancement using sliding modes techniques’, Int. J. Vehicle Design, Vol. 48, Nos. 3/4, pp.230–254.
Copyright © 2008 Inderscience Enterprises Ltd.
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Biographical notes: Hassan Shraim received his Diploma in the Mechanical Engineering from the Lebanese university 2003. In 2004 he received the master degree from the University Paul Cézanne, polytechnic Marseille in the control and simulation of complex systems, PhD in the diagnosis and control for non-linear systems and their applications in the vehicle domain. He is now an industrial scientific consultant. Mustapha Ouladsine received his PhD in Nancy 1993 in the estimation and identification of non-linear systems. In 2001, he joined the laboratory of sciences of information and systems in Marseille France. His research interests include non-linear estimation and identification, neural networks, diagnosis and control and their applications in the vehicle and aeronautic domains, he published more than 50 technical papers. Leonid Fridman received his MS in Mathematics from Kuibyshev (Samara) State University, Russia, PhD in Applied Mathematics from Institute of Control Science (Moscow), and DSc in Control Science from Moscow State University of Mathematics and Electronics in 1976, 1988 and 1998 correspondingly. In 2002, he joined the Department of Control and Robotics, Division of Electrical Engineering of Engineering Faculty at National of Autonomous University of Mexico, Mexico. His research interests include variable structure systems, nonlinear observation, singular perturbations, and systems with delay. He is an editor of three books and five special issues on sliding modes. He published over 200 technical papers. Monica Romero is Electronic Engineer Graduated at National University of Rosario (UNR), Argentina, and PhD Graduated at National University of La Plata, Argentina. Now, she is Adjunt Professor at the Control Department of Facultad de Cs. Ex., Ingenieria y Agrimensura, UNR. Her research interest includes motion control, power electronics, and fault detection and isolation systems with applications in mechatronics.
1 Introduction 1.1 Preliminaries and motivations In the recent years, important research has been undertaken to investigate the safe driving conditions in both normal and in critical situations. Safe driving requires the driver to react extremely quickly in dangerous situations, which is generally very difficult unless for experts, that result the instability of the system. Consequently, the improvement of the vehicle dynamics by active chassis control is necessary for such catastrophic situations. Increasingly, commercial vehicles are being fitted with micro processor based systems to enhance the safety and to improve driving comfort, increase traffic circulation, and reduce environmental pollution associated with vehicles. Considerable attention has been given to the development of the control systems over the past few years, authors have investigated and developed different methods and different strategies for enhancing the stability and the handling of the vehicle such as, the design of the active automatic steering (You and Chai, 1999) and the
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wheel ABS control (Petersen, 2003; Unsa and Kachroo, 1999), or the concept of a four-wheel steering system (4WS) which has been introduced to enhance vehicle handling. Some researchers have shown disadvantages on (4WS) vehicles (Nalecz and Bindemann, 1989). In terms of vehicle safety, and in order to develop a control law for the vehicle chassis, accurate and precise tools such as sensors should be implemented on the vehicle, to give a correct image of its comportment. Difficulties in measuring all vehicle states and forces, due to high costs of some sensors, or the non existence of others, make the design and the construction of observers necessary. In the field of automotive engineering, the estimation of vehicle side slip angle and wheel interaction forces with the ground are very important, because of their influence on the stability of the vehicle. Many researchers have studied and estimated vehicle side slip angle, using a bicycle model as in Stephant (2004), or by using an observer with adaptation of a quality function as in Von Vietinghoff et al. (2005) which requires a certain linearised form of the model. Moreover, an extended Kalman filter is used for the estimation of wheel forces (Samadi et al., 2001). In the previous work (Shraim et al., 2005a), we have proposed a non-linear control strategy based on the principles of predictive control for vehicle trajectory tracking, the proposed controller was not robust to outside disturbance in a certain given horizon, and in addition to that drivers’ reactions were not considered. In this paper we propose a robust sliding mode controller together with estimation of the states, forces and parameters, in order to ensure safety in the critical situations and considering driver’s reactions (steering angle and torques applied at the wheels).
1.2 Methodology The problem of observation has been actively developed within Variable Structure Theory using sliding mode approach. Sliding mode observers (see, for example, the corresponding chapter in the textbooks by Edwards and Spurgeon (1998) and Utkin et al. (1999) and the recent tutorials by Slotine et al. (1987), Barbot and Floquet (2004), Edwards et al. (2002) and Poznyak (2003) are widely used due to their attractive features: •
insensitivity (more than robustness) with respect to unknown inputs
•
possibilities to use the values of the equivalent output injection for the unknown inputs identification (Utkin et al., 1999)
•
finite time convergence to the reduced order manifold.
In Levant (1998, 2003) robust exact differentiators were designed ensuring finite time convergence to the real values of derivatives, as an application of super-twisting algorithm (Levant, 1993). A new generation of observers based on the high order sliding mode differentiators are recently developed (see Davila et al., 2005; Floquet and Barbot, 2006; Bejarano et al., 2007; Fridman et al., 2007, 2008). Those observers: •
provide a finite time convergence to the exact values of states variables
•
allow the finite time identification the unknown inputs without filtration
•
guarantee the best possible accuracy of the state estimation w.r.t. to the sampling steps and deterministic noises.
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In this paper the sliding mode controller is used to ensure safety in the critical situations and considering driver’s reactions (steering angle and torques applied at the wheels). In the paper the results are validated by the simulator VE-DYNA developed by the group of companies TESIS, which is an independent expert team for the simulation of virtual vehicles in real time, their products are employed by almost all German automakers and sought after worldwide. Their software comprises high precision models for simulation of vehicle dynamics VE-DYNA, engine dynamics EN-DYNA and brake hydraulics (RT Brake Hydraulics). The simulator VE-DYNA that we use in our study is a software especially designed for the fast simulation of vehicle dynamics both in real-time applications (Hardware-in-the-Loop, Software-in-the-Loop) and concept studies on a standard desktop PC, its computational performance enables its usage for the optimisation e.g., for the parameter identification. VE-DYNA vehicle model is fully parametric and has a modular architecture with the following program modules: Vehicle multi-body system (chassis), axles (axle kinematics, compliance), tyre model TM-Easy, transmission and drive line, engine, aerodynamics and braking system. All vehicle Degrees of Freedom (DoF) are nonlinearly modelled. The number of overall DoF depends on the number of additional bodies and the suspension type. There are at least 15 DoF up to 65 DoF in the base vehicle model; the global vehicle model and each sub model are validated by real experiments with different operation conditions.
1.3 Paper contribution The main contributions of this work reside in two important points: •
The estimations of wheels contact forces with the ground, side slip angle and the velocity of the vehicle which avoid the use of expensive sensors. These estimations preview also some critical situations that may occur while rolling, such as excessive rotation around Z axis and also excessive side slipping, inappropriate lateral acceleration, . . . The proposed observers are characterised by the rapid convergence to real values, robustness and they do not require extensive computation load. Both observers are validated by the simulator VE-DYNA. Several validations were made to cover most of the driving cases, such as a double lane trajectory, straight line motion with strong variation in acceleration and deceleration, strong change in the steering angle.
•
The control of the yaw rate, side slip angle and velocity of the vehicle by controlling wheels braking systems. In this part, a reference value is generated for each of the controlled parameters at each time step, and then a robust sliding mode control is applied. This controller functions only in the critical situations and can generate only braking torques on the four wheels.
The paper is organised as follows: In Section 2, problem statement is proposed and the model used for the vehicle is shown. In Section 3, observer design is made, in this section, two sliding mode observers are proposed: a third order sliding mode is used for the estimation of wheels velocities and the identification of the longitudinal forces, and a sliding mode observer based on broken super-twisting algorithm (Davila et al., 2005)
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for the estimation of the side slip angle and the velocity of the vehicle. In Section 4, the controller design is made and the equations of reference trajectories are shown. In Section 5 simulations are shown and finally a conclusion for the work is presented. All the solutions of the differential equations mentioned in the paper are understood in the sense of Filippov (1988).
2 Problem statement The presented problem is the problem of assistance for the driver in critical situations, i.e., when the vehicle goes out from its safety regions and enters in the dangerous situations. So accurate tools to represent vehicle states and parameters are required. These accurate representations need many precise and expensive sensors. The use of many sensors requires an important diagnosis system to avoid false data. To overcome these problems, robust virtual sensors are proposed which estimate vehicle parameters, forces and states. These virtual sensors are embedded in a closed sliding mode control loop. The model used for estimation and control is a non-linear one obtained by applying the fundamental principles of dynamics at the centre of gravity on Figure 1 (Von Vietinghoff et al., 2005): 1 cos(β) FL − sin(β) FS M 1 FL − ψ˙ cos(β) FS − sin(β) β˙ = M vCOG
v˙ COG =
(1) (2)
with
FL = Fxwind + cos(δf )(F x1 + F x2 ) + cos(δr )(F x3 + F x4 ) − sin(δf )(F y1 + F y2 ) − sin(δr )(F y3 + F y4 )
and
FS = sin(δf )(F x1 + F x2 ) + sin(δr )(F x3 + F x4 )
+ cos(δf )(F y1 + F y2 ) + cos(δr )(F y3 + F y4 ) ¨ IZ ψ = tf {cos(δf )(F x2 − F x1 ) + sin(δf )(F y1 − F y2 )} + L1 {sin(δf )(F x2 + F x1 ) + cos(δf )(F y1 + F y2 )} + L2 {sin(δr )(F x3 + F x4 ) − cos(δr )(F y3 + F y4 )} + tr {cos(δr )(F x4 − F x3 ) + sin(δr )(F y3 − F y4 )}.
(3)
The model representing the dynamics of each wheel i is found by applying Newton’s law to the wheel and vehicle dynamics Figure 2: Iri Ω˙ = −r1i F xi + torquei
i = 1 : 4.
(4)
In this paper, the task is to design a virtual sensor (observer) for the vehicle to estimate the states, parameters and forces which need expensive sensors for their measurement.
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Figure 1 2D vehicle representation
Figure 2 Wheel and its contact with the ground
But due to the fact that it is not easy to apply an observer for the global model, equation (4) are taken at first, a high order sliding mode observers are proposed for each equation (4) to observe the angular velocity and to identify the longitudinal force of each wheel. After having the longitudinal forces, we substitute their values in equations (1)–(3). From these equations, it is seen that if we substitute the longitudinal forces, we will still have as complex terms the lateral forces. In order to model the lateral force, we use a brush model for the contact with the ground (Shraim et al., 2005b). The brush model divides the surface of contact into two parts: a sliding part and an adherent part. Then the lateral force generated at the surface of contact will be the sum of forces generated at each part of the surface. These lateral forces are
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represented as in Shraim et al. (2005b): F yi =
tan(αi )2 +
+
tan(αi )µi F zi Xslidingi 2 F zi Ωi K2i K1i + × × Xti −1 V xi pi 1000
Ai 3 × Xadherencei 3
(5)
with Ai =
1 F yslidingi × ϕ 2Ki tan(αi ) − B Xadherencei Xslidingi
(6)
and B =C +D 1 (1 + exp(−β1 Xadherencei )(sin(β1 Xadherencei ) − cos(β1 Xadherencei ))) C= β1 D = X(1 + exp(−β1 Xadherencei ) cos(β1 Xadherencei ))
(7) (8) (9)
with β1 =
4
Ki E × Iri
(10)
and ϕ=1+F +G+H
(11)
F = − exp(−β1 Xslidingi ) cos(β1 Xslidingi )
(12)
G = − exp(−β1 Xadherencei ) cos(β1 Xadherencei )
(13)
H = exp(−β1 Xti ) cos(β1 Xti )
(14)
tan(αi )µi F zi Xslidingi F yslidingi = . 2 Ωi K2 F zi 2 tan(αi ) + K1 + × × Xti −1 V xi pi 1000
(15)
From this representation of the lateral force, we need to define some of the variables, all the equations for these variables are detailed in Shraim and Ouladsine (2006) and Uwe and Nielsen (2005), and they are given by: •
the coefficient of adherence µ: µi =
F xi F zi
(16)
Vehicle parameter estimation and stability enhancement •
the side slip angle of each wheel: vCOG sin(β) + L1 ψ˙ −1 α1 = tan − δf vCOG cos(β) − tf ψ˙ vCOG sin(β) + L1 ψ˙ −1 α2 = tan − δf vCOG cos(β) + tf ψ˙ vCOG sin(β) − L2 ψ˙ α3 = tan−1 − δr vCOG cos(β) − tr ψ˙ vCOG sin(β) − L2 ψ˙ α4 = tan−1 − δr . vCOG cos(β) + tr ψ˙
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(17) (18) (19) (20)
The velocity of each wheel, ˙ cos(δf ) + (vCOG sin(β) + L1 ψ) ˙ sin(δf ) V x1 = (vCOG cos(β) − tf ψ) ˙ cos(δf ) + (vCOG sin(β) + L1 ψ) ˙ sin(δf ) V x2 = (vCOG cos(β) + tf ψ)
(22)
˙ cos(δr ) + (vCOG sin(β) − L2 ψ) ˙ sin(δr ) V x3 = (vCOG cos(β) − tr ψ) ˙ cos(δr ) + (vCOG sin(β) − L2 ψ) ˙ sin(δr ). V x4 = (vCOG cos(β) + tr ψ)
(24)
(21) (23)
For the vertical forces, cheap sensors can be found for their measure (or they can be estimated as shown in Uwe and Nielsen (2005)), the determination of the contact patch and its repartition into a sliding part and adhesion part can be found as shown in Shraim and Ouladsine (2006). The system described by the equations (1)–(3) is observable if we consider that we measure only the yaw rate (and by supposing the longitudinal forces as inputs). A sliding mode observer based on broken super-twisting algorithm is used to estimate vehicle side slip angle and velocity. By these estimations, the longitudinal and lateral velocities of the centre of gravity, the lateral forces of the wheels are then directly deduced. By these estimations, the driver (or the controller) knows if the states and parameters are in the safe region or not. These regions depend on the velocity, coefficient of friction and the steering angle (Uwe and Nielsen, 2005; Gillespie, 1992). In this study it is supposed that we can measure: •
angular positions of the wheels
•
front wheel angle
•
yaw rate
•
torque applied at each wheel
and it is required to estimate: •
angular velocity of the wheels
•
contact forces
•
vehicle velocity
•
vehicle side slip angle.
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3 Observer design 3.1 Estimation of wheels angular velocities and longitudinal forces In this part, sliding mode observers are proposed to observe the angular velocity wi and to identify the longitudinal force of each wheel F xi . Dynamical equations of wheels (4) are written in the following form: x˙ 1 = x2 x˙ 2 = f (x1 , x2 , u)
(25)
where x1 and x2 are respectively θi (which is measured) and wi (to be observed) (appears implicitly in F xi ), and u is torquei . In fact this torque may be measured as shown in Rajamani et al. (2006), and it can also be estimated by estimating the motor and the braking torque, the motor torque may be estimated as in Khiar et al. (2006), while the braking torque is estimated by measuring the hydraulic pressure applied at each wheel (existing on the most of the vehicles).
3.2 Broken super-twisting observer structure In the first part of this section, the so-called broken super-twisting algorithm observer proposed in Davila et al. (2005) will be employed (see M’Sirdi et al., 2006). This is both inherently suited to nonlinear plant representations and does not require to have relative degree one with repsect to unknown inputs. x ˆ˙ 1 = x ˆ 2 + z1 x ˆ˙ 2 = f1 (x1 , x ˆ2 , u) + z2
(26)
where x ˆ1 and x ˆ2 are the state estimations of the angular positions and the angular velocities of the four wheels respectively, f1 is a nonlinear function containing only the known terms (which is only the torque in our case), z1 and z2 are the correction factors based on the broken super-twisting algorithm having the following forms: z1 = λ|x1 − x ˆ1 |1/2 sign(x1 − x ˆ1 ) z2 = λ0 sign(x1 − x ˆ1 ).
(27) 1
In the above equations the function | · | 2 and sign(.) should be thought of as componentwise extensions of their traditional scalar counterparts. ˆ1 and x ˜2 = x2 − x ˆ2 we obtain the equations for the error Taking x ˜1 = x1 − x x ˜˙ 1 = x ˜2 − λ|˜ x1 |1/2 sign(˜ x1 ) ˙x ˜2 = F (t, x1 , x2 , x ˆ2 ) − λ0 sign(˜ x1 )
(28)
where F (t, x1 , x2 , x ˆ2 ) = f (x1 , x2 , u) − f1 (x1 , x ˆ2 , u) is the unknown function to be identified. Suppose that the system states can be assumed bounded, then the existence is ensured of a constant f + , such that the inequality: |F (t, x1 , x2 , x ˆ2 )| < f +
(29)
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x2 | ≤ 2 sup |x2 |. As described in Davila et al. holds for any possible t, x1 , x2 and |ˆ (2005), it is sufficient to choose λ0 = 1.1f + and λ = 1.5 f + . A sliding motion occurs ˜2 ≡ 0 in finite time. Furthermore in the error system (27) which makes x ˜1 ≡ 0 and x whilst sliding x ˜2 = 0 and so from equation (27) we get: α sign(x1 − x ˆ1 ) = F (t, x1 , x2 , x ˆ2 )
(30)
where the left hand of the above equation represents the average value of the discontinuous term which must be taken in order to maintain a sliding motion. ˆ1 ) can be obtained by appropriate low-pass filtering of the The quantity sign(x1 − x discontinuous injection signal sign(x1 − x ˆ1 ) and so is available in real time. So in the broken super-twisting observer considered here the estimate of the longitudinal forces must be obtained by filtering the discontinuous injection signal (which may cause a certain delay). In the next, a third order sliding mode observer will be considered to obviate this necessity to filter.
3.3 A third order sliding mode observer The proposed third order sliding mode observer has the form: ˙ ˆ i + λ0 |θi − θˆi |2/3 sign(θi − θ) θˆi = Ω
vo
˙ i = torquei + z1 Ω Iri
(31)
i are the state estimations of the angular positions and the angular where θˆi and Ω velocities of the four wheels respectively, z1 is the correction factor based on the super-twisting algorithm having the following forms: i − vo |1/2 sign(Ω i − vo ) + Z1 z1 = λ1 |Ω
v1
Z˙ 1 = λ2 sign(Z1 − v1 ).
(32)
√ √ with λ2 = 3 3 λ0 , λ1 = 1.5 λ0 et λ0 = 2f + . The sliding occurs in (32) in finite time (see Levant, 2003). In particular, this means that the longitudinal forces can be estimated in a finite time without the need for low pass filter of a discontinuous switched signal. Remark 1: In order to apply a sliding mode observer, only the Euler integration may be used. Remark 2: The broken super-twisting algorithm observer structure (26) and (27) is referred to in the literature as broken super-twisting structure since in the estimation analysis associated with (28) the disturbance term with a amplitude ε appears in the channel where the discontinuous terms acts, whereas typically in the super-twisting 1 controller formulation the disturbance term occurs in the channel associated with |ε| 2 (Levant, 1998).
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3.3.1 Simulations and results Simulations are made and results are compared by those provided by simulator VE-DYNA, the operation condition corresponds to a strong variation in F xi Figure 7 and wi Figure 6 (acceleration, constant velocity, deceleration, constant velocity, acceleration, constant velocity, deceleration, constant velocity) with a zero steering angle. The same observer is applied on the four wheels, but for the similarity, we present only one observer corresponding to the front left wheel (wheel 1). The simulator uses a car with two rear wheel drives. Figure 3 shows the input torque for the two rear wheels and Figure 4 the torque for the two front wheels. In Figures 5 and 6 we see θ1 and w1 (given by the simulator VE-DYNA) and those computed by the proposed observer. In these figures we see the rapid convergence of the observer in spite of ˆ10 = the initial values are: θ10 = 0 radians, θˆ10 = 50 radians, w10 = 0 rad/s and w 100 rad/s. Figure 3 Motor and braking torque (N.m) applied at the two rear wheels (see online version for colours)
Figure 4 Motor and braking torque (N.m) applied at the two front wheels (see online version for colours)
The unknown functions computed from the equivalent output injection is supposed F xi , we suppose that the radius and the moment of inertia of the wheel equal to −r1i Iri are constants, the we can find the longitudinal force. Figure 7 shows a comparison between the longitudinal force (computed from the observer) and that given by VE-DYNA.
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Figure 5 Angular position (rad) by the simulator VE-DYNA, and that estimated by the observer (see online version for colours)
Figure 6 Angular velocity (rad/s) by the simulator VE-DYNA, and that estimated by the observer (see online version for colours)
Figure 7 The unknown input after filtration (N) and the longitudinal force from the simulator VE-DYNA (see online version for colours)
3.4 Estimation of the side slip angle, velocity of the vehicle and reconstruction of the yaw rate In this part, a sliding mode observer based on the super-twisting algorithm is used to estimate the velocity and the side slip angle at the centre of gravity. The model of the vehicle is a non-linear model and it can be written as follows: x˙ = f (x, u) = A(x) + B(x)u
(33)
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where 1 (cos(β)(− sin(δ )(F y + F y ) − sin(δ )(F y + F y )) f 1 2 r 3 4 M − sin(β)(cos(δf )(F y1 + F y2 ) + cos(δr )(F y3 + F y4 )) 1 (cos(β)(cos(δ )(F y + F y ) + cos(δ )(F y + F y )) f 1 2 r 3 4 Mv COG (34) A(x) = − sin(β)(− sin(δf )(F y1 + F y2 ) − sin(δr )(F y3 + F y4 ))) − ψ˙ 1 ({tf sin(δf )(F y1 − F y2 )} + L1 {cos(δf )(F y1 + F y2 )} IZ +L2 {cos(δr )(F y3 + F y4 )} + tr {sin(δr )(F y3 − F y4 )})
and b11 B(x) = b21 b31
b12 b22 b32
b13 b23 b33
b14 b24 b34
(35)
with: cos(β) sin(β) cos(δf ) − sin(δf ), M M cos(β) sin(β) cos(δf ) − sin(δf ) b12 = M M cos(β) sin(β) cos(δr ) − sin(δr ), b13 = M M cos(β) sin(β) cos(δr ) − sin(δr ) b14 = M M cos(β) sin(β) cos(δf ), sin(δf ) − b21 = M vCOG M cos(β) sin(β) cos(δf ) b22 = sin(δf ) − M vCOG M cos(β) sin(β) cos(δr ), sin(δr ) − b23 = M vCOG M cos(β) sin(β) cos(δr ) sin(δr ) − b24 = M vCOG M 1 {L1 sin(δf ) − tf cos(δf )}, b31 = IZ 1 {L1 sin(δf ) + tf cos(δf )} b32 = IZ 1 {L2 sin(δr ) − tr cos(δr )}, b33 = IZ 1 {tr cos(δr ) + L2 sin(δr )} b34 = IZ ˙ x = [vCOG β ψ] b11 =
(36)
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the input: u = [Fx1 Fx2 Fx3 Fx4 ]
(37)
and the measurement vector ˙ y = [ψ].
(38)
Before the design of the sliding mode observer for the model of equation (33), the observability of the model must be investigated and tested. The observability definition is local and uses the Lie derivative (Nijmeijer and Van der Schaft, 1990) (see also Fridman et al., 2007, 2008). It is a function of the state trajectory and the inputs to the model. For the For the system described by equation (33) the observability function is:
c(x)
observability(x, u) = Lf c(x, u) L2f c(x, u) where Lf c(x, u) =
dcj (x) f (x, u). dx
The system is observable if its Jacobian matrix Jobservability has a full rank (which is 3 in our case). Jobservability =
d observability(x, u). dx
By applying these notions to the system described by equation (33), we see that its rank is 3 and hence observable. A complete study for the observability for the system presented by equation (33) (including the cases that when we have an input making the matrix singular) is presented in Fridman et al. (2007, 2008). So, the proposed sliding mode observer based on the hierarchical super twisting algorithm is: ˙ˆ 1/2 ˙ˆ ˆ ψ, ˙ u) + ∆1 |(ψ˙ − ψ)| sign(ψ˙ − ψ) + Z1 x˙ = f (ˆ vCOG , β, yˆ = C x ˆ ˙ ˆ˙ Z1 = ∆ sign(ψ˙ − ψ)
(39)
where ∆ and ∆1 are the gains of the sliding mode observer. The convergence of this observer is proved in Levant (1998).
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3.4.1 Simulations results Once again, the estimated variables, vˆCOG and βˆ are compared to that provided by VE-DYNA. The operation conditions are given by a variation in δf Figure 8 and torquei Figure 9, which constitute a significant driving situation. F xi are estimated from the third order sliding mode observer. In Figures 10–12, we see the observed ˆ vˆCOG , βˆ and ψ˙ and those provided by VE-DYNA. The rapid convergence point out the good performance of the proposed observer. The gains of the observer used are: ∆ = [10, 10, 10]T ; ∆1 = [10, 10, 15]T .
Parameter M r1i Iri lo Cij
Value
Parameter
1296 kg 0.28 m 0.9 kg.m2 −0.03 m 50000 N/rad
L1 IZ tf l1 AL
Value
Parameter
Value
L2 H tr ρ
1.53 m 0.52 m 0.75 m 1.25 kg/m3
0.97 m 1750 0.7 m 0.12 m 2.25 m2
Figure 8 Front steering angle (rad) (see online version for colours)
Figure 9 Motor and braking torque (N.m) applied at the two rear wheels (see online version for colours)
Vehicle parameter estimation and stability enhancement Figure 10 Estimated vehicle velocity (m/s) and that of the simulator VE-DYNA (see online version for colours)
Figure 11 Estimated side slip angle (rad) using sliding modes and that of the simulator VE-DYNA (see online version for colours)
Figure 12 Reconstructed yaw rate (rad/s) and that of the simulator VE-DYNA (see online version for colours)
Figure 13 Estimated Vy (m/s) and that of the simulator VE-DYNA (see online version for colours)
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By estimating the slip angle and the velocity of the centre of gravity, the velocities of the centre of gravity in (X, Y ) can be found by Uwe and Nielsen (2005): the lateral velocity (see Figure 13): V y = vCOG sin(β)
(40)
and the longitudinal velocity which coincides with that of the simulator (see Figure 14): V x = vCOG cos(β)
(41)
and the lateral force (rear left) (see Figure 15): Figure 14 Estimated Vx (m/s) and that of the simulator VE-DYNA (see online version for colours)
Figure 15 Estimated lateral force (N) for the front left wheel and that of the simulator VE-DYNA (see online version for colours)
4 Controller design In this section a sliding mode control strategy is presented. For the design process, the model presented by equation (33) is used. As we have described, the state vector ˙ T and the input u = [F x1 , F x2 , F x3 , F x4 ]T (the steering angle is is x = [vCOG , β, ψ] supposed given by the driver). As the only available actuators are the brakes, then only braking torques can be generated by the controller (Uwe and Nielsen, 2005;
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Alvarez et al., 2005). These braking torques are directly related to the hydraulic pressure applied at each wheel (Uwe and Nielsen, 2005): torquecontroller = −r1i kbi pBRi
i = 1 : 4.
(42)
To design a sliding mode controller, a sliding surface is proposed as: s = x − xref
(43)
with xref = [vCOG ref , βref , ψ˙ ref ]T represents the reference states. The control objective is to derive the state vector x to the reference state vector xref . In order to ensure the stability, let us suppose that Lyapunov candidate is given by Utkin (1992): V =
1 T s s > 0. 2
Its derivative can be written as: V˙ = sT s˙ where s˙ = x˙ − x˙ ref .
(44)
Substituting equation (33) into equation (44): s˙ = A(x) + B(x)u − x˙ ref .
(45)
Then we have: V˙ = sT (A(x) + B(x)u − x˙ ref )
(46)
x˙ ref = A(xref ) + B(xref )uref
(47)
with
let us suppose the following control law: ∆u = u − uref = −K sign(B(x)T s)
(48)
substituting ∆u in the above equation, we get: V˙ = sT (A(x − xref ) − B(x − xref )K sign(B(x)T s)). In order to satisfy the conditions of stability in the Lyapunov sense, V˙ should be negative, that means: sT (A(x − xref ) − B(x − xref )K sign(B(x)T s)) < 0
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since K is scalar, we can write: sT A(x − xref ) − K sT B(x − xref ) sign(B(x)T s) < 0 K sT B(x − xref ) sign(B(x)T s) > sT A(x − xref ) the dimensions of dim(sT B(x − xref ) sign(B(x)T s)) = dim(sT A(x − xref )) = 1 then we can write: k>
sT A(x − xref ) sT B(x − xref ) sign(B(x)T s)
and then k>
sT A(x − xref ) |sT B(x − xref )|
a necessary condition for the existence of the control law is that sT B(x − xref ) = 0. where sgn(s) is a sign function which equals to 1 when s > 0 and −1 if s < 0. The chattering of the function sgn(s) may be reduced by sat Φs , where Φ is a design parameter denoting the boundary layer thickness (Stephant, 2004).
4.1 Reference values The estimated vˆCOG , βˆ and ψ˙ should follow reference values. But due the fact that the controller is designed to assist the driver only in the critical situations, the reference values are chosen to be equal to the estimated values when the vehicle is in the safety region and equal to certain defined values other wise. They can be described as (Uwe and Nielsen, 2005): For the β: βmax = 10◦ –7◦ × βref = β
2 vCOG
(49)
2
(40 m/s )
if |β| ≤ |βmax |
βref = ±βmax otherwise.
(50)
˙ For the ψ: ψ˙ max =
1 (aY vCOG cos(β)
max
− v˙ COG sin(βref ))
(51)
with aY
2
max
= µ.8 m/s .
(52)
The derivative of the velocity is found by using a robust exact differentiator proposed by Levant (1998).
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˙ two cases are considered, in the case of over For the reference value of the ψ, steering: ˙ ≤ |ψ˙ max | if |ψ| ψ˙ ref = ψ˙ ψ˙ ref = ±ψ˙ max otherwise.
(53)
In the case of under steering, the rear tyre side slip angle shall be used as a reference to determine when the front tyre side slip angle reach a critical value (Uwe and Nielsen, 2005). αi i = 1 or 2 ˙ ˙ ≥ 1.5 ψref = ±ψmax if αi i = 3 or 4 ψ˙ ref = ψ˙ otherwise.
(54)
5 Simulation results In this section, computer simulations are carried out to verify the effectiveness of the proposed observers and controller. Simulation is made, in which driver’s inputs are given by the simulator VE-DYNA. The vehicle model used is validated by VE-DYNA (Shraim and Ouladsine, 2006). The driver wants to move on a ‘chicane’ (double lane) trajectory described by driver’s steering angle Figure 8 and wheel torques Figure 9. In fact, in this simulation we see that driver’s inputs make the yaw rate exceeds its limit value. In Figure 16 we see three curves, the reference yaw rate, the yaw rate for the system without the controller and that with the sliding mode controller, it is seen how the controller pushes the controlled yaw rate to its reference value. In Figure 17, three curves also are shown for the side slip angle which are: the reference side slip angle, the side slip angle without the controller and that with the sliding mode control. In Figure 18, the response of the controller is shown, it is seen that in the normal cases where the side slip angle and the yaw rate are in their safety regions, the controller gives zero, other wise, when they exceed their limits, the controller tries to regulate this problem giving different torques on the different wheels. Figure 16 Yaw rate with and without the controller and the reference yaw rate (see online version for colours)
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Figure 17 Side slip angle with and without the controller and the reference side slip angle (see online version for colours)
Figure 18 The four outputs of the controller (see online version for colours)
6 Conclusions Sliding mode observers are proposed in this work to estimate vehicle parameters and states which are not easily measured. These observers have shown a short time of convergence and robustness in the automotive applications that we have proposed. The validation of the proposed observers is realised by comparing the observers output with the outputs of the simulator VE-DYNA, reasonable and acceptable results have been shown. In the second part of this work, sliding mode controller is designed. This controller shows its strong and fast reactions on the braking systems in the critical situations where we need the controller to work.
References Alvarez, L., Yi, J., Horowitz, R. and Olmos, L. (2005) ‘Dynamic friction model-bases tyre road friction estimation and emergency braking control’, Journal of Dynamic Systems, Measurement and Control, Vol. 127, pp.22–32.
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Bejarano, F.J., Fridman, L. and Poznyak, A. (2007) ‘Exact state estimation for linear systemswith unknown inputs based on a hierarchical super twisting algorithm’, International Journal of Robust and Nonlinear Control, Vol. 17, No. 18, pp.1734–1753. Davila, J., Fridman, L. and Levant, A. (2005) ‘Second-order sliding mode observer for mechanical systems’, IEEE Transactions on Automatic Control, Vol. 50, No. 2, pp.1785–1789. Edwards, C., Spurgeon, S.K. and Hebden, R.G. (2002) ‘On development and applications of sliding mode observers’, in Yu, X. and Xu, J-X. (Eds.): Variable Structure Systems: Towards XXIst Century, ser. Lecture Notes in Control and Information Science, Springer Verlag, Berlin, Germany, Vol. 274, pp.253–282. Edwards, C. and Spurgeon, S.K. (1998) Sliding Mode Control, Taylor and Francis, London. Filippov, A.F. (1988) Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic Publishers, Dordrecht, The Netherlands. Floquet, T. and Barbot, J. (2006) ‘A canonical form for the design of unknown input sliding mode observers. In advances in variable structure and sliding mode control’, in Edwards, C., Fossas, E. and Fridman, L. (Eds.): Lecture Notes in Control and Information Science, Springer Verlag, Berlin, Vol. 334, pp.271–292. Fridman, L., Levant, A. and Davila, J. (2007) ‘Observation of linear systems with unknown inputs via high order sliding modes’, International Journal of System Science, Vol. 38, No. 8, pp.773–791. Fridman, L., Shtessel, Y., Edwards, C. and Yan, X-G. (2008) ‘Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems’, International Journal of Robust and Nonlinear Control, Vol. 18, pp.399–413. Gillespie, T.D. (1992) Fundamentals of Vehicle Dynamics, Published by Society of Automotive Engineers, Inc, USA. Khiar, D., Lauber, J., Floquet, T., Guerra, T., Coline, G. and Chamillard, Y. (2006) Estimation of the Instantaneous Torque of a Gasoline Engine, CIFA, Bordeaux, France (In French). Levant, A. (1998) ‘Robust exact differentiation via sliding mode technique’, Automatica, Vol. 34, No. 3, pp.379–384. Levant, A. (1993) ‘Sliding order and sliding accuracy in sliding mode control’, International Journal of Control, Vol. 58, pp.1247–1263. Levant, A. (2003) ‘Higher order sliding modes, differentiation and output control’, International Journal of Control, Vol. 76, pp.924–941. M’Sirdi, K.N., Rabhi, A., Fridman, L., Davilia, J. and Delanne, Y. (2006) ‘Second order sliding-mode observer for estimation of vehicle parameters’, American Control Conference, Minneapolis, Minnesota, USA, pp.3316–3321. Nalecz, A.G. and Bindemann, A.C. (1989) Handling Properties of Four Wheel Steering Vehicles, SAE P. 890080, pp.63–81. Nijmeijer, H. and Van der Schaft, A.J. (1990) Nonlinear Dynamical Control Systems, Springer-Verlag, Berlin. Orlov, Y. (2000) ‘Sliding mode observer-based synthesis of state derivative free model reference adaptive control of distributed parameter systems’, ASME J. Dyn. Sys. Meas. and Cont., Vol. 122, pp.725–731. Petersen, I. (2003) Wheel Slip Control in ABS Brakes Using Gain Scheduled Optimal Control with Constraints, Thesis submitted for the Degree of Doctor Engineer Department of Engineering Cybernetics, Norwegian University of Science and Technology Trondheim, Norway. Poznyak, A.S. (2003) ‘Stochastic output noise effects in sliding mode estimations’, International Journal of Control, Vol. 76, pp.986–999.
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Rajamani, R., Piyabongkarn, D., Lew, J.Y. and Grogg, J.A. (2006) Algorithms for Real Time Estimation of Individual Wheel Tire Road Friction Coefficients, American Control Conference, Minneapolis, Minnesota, USA, pp.4682–4687. Rabhi, A., M’Sirdi, K.N. and Ouladsine, M. (2006) Observateur Différentiel pour l’Estimation des Vitesses Angulaires des Roues et l’Estimation de l’adhérence, CIFA, Bordeaux, France. Rabhi, A. (2005) Estimation de la Dynamique d’un Véhicule en Interaction avec son Environment, Thesis presented to have the doctor degree from the University of Paul Cezanne AIX Marseille III. Samadi, B., Kazemi, R., Nikravesh, K.Y. and Kabganian, M. (2001) ‘Real-time estimation of vehicle state and tyre friction forces’, Proceedings of the American Control Conference (ACC), Minneapolis, Minnesota, USA, pp.3318–3323. Shraim, H., Ouladsine, M. and El Adel, M. (2005a) ‘A new nonlinear control strategy for a vehicle trajectory tracking in the presence of faults’, 44th IEEE Conference on Decision and Control CDC and European Control Conference ECC, Seville, Spain, pp.1994–1999. Shraim, H., Ouladsine, M., El. Adel, M. and Noura, H. (2005b) ‘Modeling and simulation of vehicles dynamics in presence of faults’, 16th IFAC World Congress, Prague, Czech Republic, pp.687–692. Shraim, H. and Ouladsine, M. (2006) A Non-linear Control Strategy Based on a Validated Model for Vehicle Trajectory Tracking in the Presence of Faults, SAE International, No: 2006-01-3527. Slotine, J.J., Hedrik, J.K. and Misawa, E.A. (1987) ‘On sliding observers for nonlinear systems’, Asliding ModeE J. Dynam. Syst. Meas, Vol. 109, pp.245–252. Slotine, J.J. and Li, W. (1991) Applied Nonlinear Control, Prentice-Hall, Inc., Englewood Cliffs, NJ, USA. Stephant, J. (2004) Contribution à l’étude et à la Validation Expérimentale d’observateurs Appliqués à la Dynamique du Véhicule, Thesis presented to have the Doctor Degree from the UTC, University of Technology Compiègne. Unsa, C. and Kachroo, P. (1999) ‘Sliding mode measurement feedback control for antilock braking systems’, IEEE Transactions on Control System Technology, Vol. 7, No. 2, pp.271–281. Utkin, V., Guldner, J. and Shi, J. (1999) Sliding Modes in Electromechanical Systems, Taylor and Francis, London. Utkin, V. (1992) ‘Sliding mode control design principles and application to electrical drives’, IEEE Trans. Ind. Electron., Vol. 40, pp.23–36. Uwe, K. and Nielsen, L. (2005) Automotive Control System, Springer-Verlag, Berlin, Germany. Von Vietinghoff, A., Hiemer, M. and Uwe, K. (2005) Non-linear Observer Design for Lateral Vehicle Dynamics, 16th IFAC World Congress, Prague, Czech Republic. You, S-S. and Chai, Y-H. (1999) ‘Multi-objective control synthesis: an application to 4WS passenger vehicles’, Mechatronics, pp.363–390.
Bibliography Fridman, L. and Levant, A. (2002) ‘Higher order sliding modes’, in Perruquetti, W. and Barbot, J.P. (Eds.): Sliding Mode Control in Engineering, Marcel Dekker, New York, pp.53–103. Hiemer, M. et al. (2004) ‘Cornering stiffness adaptation for improved side slip angle observation’, Proceedings of the First IFAC Symposium on Advances in Automotive Control AAC04, Italy, pp.685–690.
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Barbot, J. and Floquet, T. (2004) ‘A sliding mode approach of unknown input observers for linear systems’, 43th IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, pp.1724–1729. Young, K.D., Utkin, V.I. and Ozguner, U. (1999) ‘A control engineer’s guide to sliding mode control’, IEEE Trans on Control Systems Technology, Vol. 7, No. 3, pp.328–342. Yi, J., Alvarez, L. and Horowitz, R. (2002) ‘Adaptive emergency braking control with underestimation of friction coefficient’, IEEE Trans. on Control Systems Technology, Vol. 10, No. 3, pp.381–392.
Website Simulator VE-DYNA, http://www.tesis.de/index.php
Nomenclature Symbol
Physical signification
Ωi M ri COG r1i Fzi Fxi Fyi Cf i Cmi torquei IZ ψ ψ˙ δf δr δi Vx Vy Iri vCOG L1 L2 L hCOG tf
Angular velocity of the wheel Total mass of the vehicle Radius of the wheel i Centre of gravity of the vehicle Dynamical radius of the wheel i Vertical force at wheel i Longitudinal force applied at the wheel i Lateral force applied at the wheel i Braking torque applied at wheel i Motor torque applied at wheel i Cmi − Cf i Moment of inertia around the Z axis Yaw angle Yaw velocity Front steering angle Rear steering angle Deflection in the tyre i Longitudinal velocity of the centre of gravity Lateral velocity of the centre of gravity Moment of inertia of the wheel i Total velocity of the centre of gravity Distance between COG and the front axis Distance between COG and the rear axis L 1 + L2 Height of COG Front half gauge
254 tr l Fxwind Fywind AL ρ Caer αi β µi Xt Xadherencei Xslidingi Vxi pBRi kbi pi K1i K2i
H. Shraim et al. Rear half gauge tf + tr Air resistance in the longitudinal direction Air resistance in the lateral direction Front vehicle area Air density Coefficient of aerodynamic drag Slip angle at the wheel i Side slip angle at the COG Friction coefficient at the wheel i Length of the contact patch for the wheel i Length of the adhesion patch for the wheel i Length of the sliding patch for the wheel i Longitudinal velocity of the wheel i Braking pressure at the wheel i Brake coefficient of the wheel i Inflation pressure of the tyre i Constant depending on the deformation of the tyre Constant depending on the deformation of the tyre
