# Second Order Sliding-Mode Observer for Estimation of Vehicle Dynamic Parameters

N. M'Sirdi, Abdelhamid Rabhi, L. Fridman, J. Davila, Y Delanne

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N. M'Sirdi, Abdelhamid Rabhi, L. Fridman, J. Davila, Y Delanne. Second Order Sliding-Mode Observer for Estimation of Vehicle Dynamic Parameters. International Journal of Vehicle Design, 2008. hal-01966663

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Second Order Sliding-Mode Observer for
Estimation of Vehicle Dynamic Parameters

N.K. M'sirdi¹, A. Rabhi¹, L. Fridman², J. Davila² and Y. Delanne³

¹LSIS, CNRS UMR 6168. Dom. Univ. St Jérôme, Av. Escadrille Normandie - Niemen
13397. Marseille Cedex 20. France. e-mail: msirdin@lsis.org

²UNAM Dept of Control, Division of Electrical Engineering, Faculty of Engineering, Ciudad Universitaria,
Universidad Nacional Autonoma de Mexico, 04510, Mexico, D.F., Mexico

³ LCPC Nantes: Division ESAR BP 44341 44 Bouguenais Cedex

***Abstract***—This paper uses second-order sliding mode observers to build up an estimation scheme allowing to identify the tire longitudinal equivalent stiffness and the effective wheel radius using the existing ABS angular sensors. This estimation strategy, based on use of the proposed observer could be used with data acquired experimentally to identify the longitudinal stiffness and effective radius of vehicle tires. The actual results show effectiveness and robustness of the proposed method.

***Index Terms***—Sliding modes, nonlinear observers, robust state and stiffness estimation, wheel slip estimation.

I. INTRODUCTION

Car accidents occur for several reasons which may involve
the driver or components of the vehicle or environment.
Such situations appears when the vehicle is driven beyond
the adherence or stability limits. However new active safety
systems are developed, improved, and installed on vehicles
for real-time monitoring and controlling the dynamic stability
(Electronic Braking Systems (EBS), Anti-lock Braking Sys-
tems (ABS), Electronic Stability Program (ESP)). The active
safety becomes more important in recent research on Intelli-
gent Transportation Systems (ITS) technology. Nevertheless,
the possibility of rectifying an unstable condition can be
compromised by physical limits. Therefore, it is extremely
important to detect (on time) a tendency towards instability.
This has to be done without adding expensive sensors, so it
requires quite robust observers looking forward based on the
physics of interacting systems (the vehicle, the driver and the
road).

The tire forces properties affect the vehicle dynamic perfor-
mance. The control of ground - vehicle interactions becomes
important due to research efforts on intelligent transportation
systems, and specially, on automated highway systems. The
design of traction controller is based on the assumption that ve-
hicle and wheel angular velocities are both available on-line by
direct measurements and/or estimations. Thus the knowledge
of tire parameters and variables (stiffness, forces, velocities,
wheel slip and radius) is essential to advanced vehicle control
systems such as ABS, Traction Control Systems (TCS) and
ESP [1]–[3]. However, tire forces and road friction are dif-
ficult to measure directly and to represent precisely by some
deterministic model equations. In the literature, their values are

often deduced by some experimentally approximated models
[4]–[10]. This work is focused to the on-line estimation of
the tires sleep, adherence, stiffness and effective radius. The
vehicle state is estimated and the tire forces are identified[3].
The main contribution is the robust on-line estimation of the
tire effective radius, wheel sleep and velocities, needed for a
control, by using only simple low cost sensors (ABS sensors).

Recently, many analytical and experimental studies have
been performed on estimation of the frictions and contact
forces between tires and road [11], [12], [6]. Tire forces
can be represented by the nonlinear (stochastic) functions
of wheel slip. The deterministic tire models encountered
are complicated and depend on several factors (as load,
tire pressure, environmental characteristics, etc.) [10],[13]–
[17]. This makes on-line estimation of forces and parameters
difficult for vehicle control applications and detection and
diagnosis for driving monitoring and surveillance [18]. In [19],
[20], [18], application of sliding mode control is proposed.
Observers based on the sliding mode approach have been also
used in [22]. In [11] an estimation based on least squares
method and Kalman filtering is applied for estimation of
contact forces. In [6] presented a tire/road friction estimation
method based on Kalman filter to give a relevant estimates of
the slope of µ versus slip (λ), that is, the relative difference
in wheel velocity. The paper [21] presented an estimator for
longitudinal stiffness and wheel effective radius using vehicle
sensors and Global Positioning System (GPS) for low values
of slip.

Observers robust to unknown inputs are efficient for esti-
mation of road profile and the contact forces [18], [22]. Ac-
celeration and braking maneuvers modify the wheel slip. This
phenomenon could be controlled by means of its regulation
while using sliding mode approach [2], [22]. This methods
enhances the road safety leading better vehicle adherence
and maneuvers ability but the vehicle controllability in its
environment along the road admissible trajectories still remain
an important open problem.

¿From the other hand, it is necessary to remark that ob-
servers for mechanical systems with unknown inputs based
on standard first order sliding mode approach (as for example
[23], [24], [25]) has the following disadvantages:

* for observation of the velocity a filtering is needed cor-
rupting the results;

*the need of filtering in the observation process destroys the finite time convergence property, and the separation principle must be taken into account to design a control;

*for the uncertainties and parameters identification a second filtering is necessary. This leads to a bigger corruption of results.

A robust exact differentiator [30] based on super twisting algorithm ([27]) ensures a finite time convergence to the values of the corresponding derivatives and provides the best possible accuracy of the derivatives for the given value even considering deterministic noise, sampling step and in the case of discrete measurements.

In this paper, a nominal model of the vehicle is considered and the super-twisting based robust exact observer [26] is applied for estimation of rotational velocities. The stiffness and effective radius are identified by application of a dynamical identification algorithm. The robust exact observer [26] used in this paper allows

*to make the velocity observation without filtering;

*to provide finite time convergence to the exact value of the rotational velocity, ensuring separation principle;

*to identify the uncertainties with only just one filtering;

*to apply a continuous time parameter identification algorithm for system parameters identification.

This work deals with a simple vehicle model coupled with wheel-road contact. It is proposed a vehicle model for the online estimation using robust observers. The main characteristics of the vehicle longitudinal dynamics were taken into account in the developed model. The obtained dynamics equations may be written in a state space allowing to define an observer based on the sliding mode approach (as presented in [18], [2], [22]). The observer has been used to reconstruct the global system state components and then to estimate the tires forces [2], [22]. The use of sliding mode approach has been motivated by its robustness with respect to the parameters and modeling errors and has been shown to cope well with this problem.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Wheel dynamics and the ABS system</figcaption></figure>

It is presented, a method to estimate the wheel angular velocities by considering the wheel angular position measurements (produced by an ABS variable reluctance sensor as

shown in figure 1). As a second step, we estimate the longitudinal stiffness and wheel effective radius using additional sensors for the accelerating torque and the linear velocity of the vehicle. The proposed method of estimation is verified through one-wheel simulation model with a "Magic formula" tire model and then application results (on a Peugeot 406) show an excellent reconstruction of the velocities, tire forces and radius estimation.

The developed estimations can be used to detect critical driving situations and then improve the security. It can be used also in several vehicle control systems such as ABS, TCS, diagnosis systems, etc...

## II. PROBLEM STATEMENT

Consider the simplified motion dynamics of a quarter-vehicle model, capturing only nominal behavior. This model retains the main characteristics of the longitudinal dynamic. For a global application, this method can be easily extended to the complete vehicle and involve the four wheels.

Applying Newton's law to wheel and vehicle dynamics, the equations of nominal motion is given by

$$ \dot{\theta} = \omega \qquad (1) $$

$$ J\dot{\omega} = J\ddot{\theta} = T_f - R_e F_x \qquad (2) $$

$$ m\dot{v}_x = F_x \qquad (3) $$

where $m$ is the vehicle mass and $J, R_e$ are the inertia and effective radius of the tire, respectively. $v_x$ is the linear velocity of the vehicle, $\theta$ is the angular position of the considered wheel, $\omega$ is the angular velocity of the considered wheel, $T_f$ is the accelerating (or braking) torque, and $F_x$ is the tire/road friction force. The tractive (respectively braking) force, produced at the tire/road interface when a driving (braking) torque is applied to a pneumatic tire, has an opposed direction to relative motion between the tire and road surface. This relative motion determines the tire slip properties. The wheel-slip is due to deflection in the contact patch. The longitudinal wheel-slip $\lambda$ is generally called the slip ratio and can be described by a kinematic relation like [13]:

$$ \left\{ \begin{array}{ll} \lambda = \frac{R_{ef}\omega}{v_x} - 1 & \text{if } v_x > R_{ef}\omega \text{ (braking)} \\ \lambda = 1 - \frac{R_{ef}\omega}{v_x} & \text{if } v_x < R_{ef}\omega \text{ (traction)} \end{array} \right. \qquad (4) $$

<figure><img src="image_3.png" /><figcaption>Fig. 2. Wheel slip - forces steady state characteristics</figcaption></figure>

During ordinary driving, however, the tire slip rarely exceeds 5%. By linearizing the model in a small region (around
origin), the force slip relation can be characterized as follows

$$F_x = C_x \left( \frac{v_x - R_e \omega}{v_x} \right) \qquad (5)$$

Where $F_x$ and $C_x$ are, respectively, the force and the longitudinal stiffness of the tire(s). The dynamic equation of the whole system can be written in state space form by defining the following state variables. The angular position $x_1 = \theta$ is measured by the ABS sensor. The angular velocity $x_2 = \dot{\theta} = \omega$ is not measured and can be obtained by observer application. The vehicle velocity $x_3 = v_x$, and the accelerating torque $u = T_f$ are assumed measurable. Note that these expressions assume that velocity is non zero by definition. We can write the system model

$$ \begin{aligned} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \frac{u}{J} - \frac{R_e C_x}{J} + \frac{R_e^2 C_x}{J} \frac{x_2}{x_3} \\ \dot{x}_3 &= \frac{C_x}{m} - \frac{C_x R_e}{m} \frac{x_2}{x_3} \\ y &= \begin{bmatrix} x_1 & 0 \\ 0 & x_3 \end{bmatrix} \end{aligned} \qquad (6) $$

The task is to reconstruct the angular velocity ($x_2$) of the system by using $x_1$ and $u$. The equivalent output injection will be used for parameters identification. An auxiliary system will be introduced for the variable $x_3$ in order to obtain an equivalent output injection for this variable.

### III. STATE OBSERVATION

#### A. States $x_1, x_2$

Consider the subsystem with state variables $x_1 = \theta, x_2 = \dot{\theta} = \omega$, and the control input $u = T_f$ (may be computed in function of the system states or their estimates), this submodel of (6) can be rewritten in the state space form as follows:

$$ \begin{aligned} \dot{x}_1 &= x_2, \\ \dot{x}_2 &= f_1(t, x_1, x_2, u) + \xi_1(t, x_1, x_2, u) \\ y &= x_1, \end{aligned} \qquad (7) $$

where the nominal part of the system dynamics is represented by $f_1(t, x_1, x_2, u) = \frac{u}{J}$ containing the known nominal functions, while uncertainties are concentrated in the term $\xi_1(t, x_1, x_2, u) = -\frac{R_e C_x}{J} + \frac{R_e^2 C_x}{J}$. The system (7), understood in Filippov's sense [31] is assumed such that the functions $f_1(t, x_1, x_2, u)$ and the perturbation $\xi_1(t, x_1, x_2, u)$ are Lebesgue-measurable and uniformly bounded in any compact region of the state space.

Our task is then to design a finite-time convergent observer of the angular velocity $x_2 = \dot{\theta} = \omega$ assuming that the position $x_1 = \theta$, the torque $u$, and the nominal model are available. Only the scalar case $x_1, x_2 \in R$ is considered for simplicity. In general case the observers are constructed in exactly the same way for each wheel position variable $x_{1j}$ in parallel.

The proposed super-twisting observer for the system (7) takes the form [26]

$$ \begin{aligned} \dot{x}_1 &= \hat{x}_2 + z_1 \\ \dot{x}_2 &= f_1(t, x_1, \hat{x}_2, u) + z_2 \end{aligned} \qquad (8) $$

where $\hat{x}_1$ and $\hat{x}_2$ are the state estimations, and the correction variables $z_1$ and $z_2$ are calculated by the super-twisting algorithm

$$ \begin{aligned} z_1 &= \lambda |x_1 - \hat{x}_1|^{1/2} \operatorname{sign}(x_1 - \hat{x}_1) \\ z_2 &= \alpha \operatorname{sign}(x_1 - \hat{x}_1). \end{aligned} \qquad (9) $$

It is taken for ensures observer convergence that at the initial moment $\hat{x}_1 = x_1$ and $\hat{x}_2 = 0$.

Taking $\tilde{x}_1 = x_1 - \hat{x}_1$ and $\tilde{x}_2 = x_2 - \hat{x}_2$ we obtain the error equations

$$ \begin{aligned} \dot{\tilde{x}}_1 &= \tilde{x}_2 - \lambda |\tilde{x}_1|^{1/2} \operatorname{sign}(\tilde{x}_1) \\ \dot{\tilde{x}}_2 &= F(t, x_1, x_2, \tilde{x}_2) - \alpha \operatorname{sign}(\tilde{x}_1) \end{aligned} \qquad (10) $$

where $F(t, x_1, x_2, \hat{x}_2) = f_1(t, x_1, x_2, u) - f_1(t, x_1, \hat{x}_2, u) + \xi_1(t, x_1, x_2, u)$. In our case, the system states are bounded, then the existence of a constant $f^+$ is ensured such that

$$ |F(t, x_1, x_2, \hat{x}_2)| < f^+ \qquad (11) $$

holds for any possible $t, x_1, x_2$ and $|\hat{x}_2| \le 2 \sup |x_2|$. The state boundedness is true, because the system (7) is BIBS (Boundary Input - Boundary State) stable, and the control input $u = T_f$ is bounded. The maximal possible acceleration in the system is a priori known and it coincides with the bound $f^+$. Let $\alpha$ and $\lambda$ satisfy the following inequalities, where $p$ is some chosen constant, $0 < p < 1$.

$$ \begin{aligned} \alpha &> f^+, \\ \lambda &> \sqrt{\frac{2}{\alpha - f^+} \frac{(\alpha + f^+)(1+p)}{(1-p)}}, \end{aligned} \qquad (12) $$

**Theorem 1:** ([26]). Suppose that condition (11) holds for system (7), and the parameters of the observer (8) are selected according to (12). Then, the observer (8) guarantees the convergence of the estimated states $(\hat{x}, \dot{\hat{x}})$ to the real value of the states $(x, \dot{x})$ after a finite time transient, and there exists a time constant $t_0$ such that for all $t \ge t_0$, $(\hat{x}_1, \hat{x}_2) = (x_1, x_2)$. The proof of this theorem is presented in the work [26].

Let $f_1, x, z_1, z_2$ be measured at discrete times with the time interval $\delta$, and let $t_i, t_{i+1}$ be successive measurement times. Consider a discrete modification of the observer (the Euler scheme)

$$ \begin{aligned} \hat{x}_1(t_{i+1}) &= \hat{x}_1(t_i) + (\hat{x}_2(t_i) + \lambda |x_1(t_i) - \hat{x}_1(t_i)|^{1/2} \operatorname{sign}(x_1(t_i) - \hat{x}_1(t_i))) \\ \hat{x}_2(t_{i+1}) &= \hat{x}_2(t_i) + (f_1(t_i, x_1(t_i), \hat{x}_2(t_i), u(t_i)) + \alpha \operatorname{sign}(x_1(t_i) - \hat{x}_1(t_i))) \end{aligned} \qquad (13) $$

where $\hat{x}_1(t_i), \hat{x}_2(t_i)$ are the estimated variables.

**Theorem 2:** ([26]). Suppose that the function $f_1$ is uniformly bounded and condition (11) holds. Then the observation algorithm (13) with parameters (12) ensures the convergence of the estimation errors to the domain $|\tilde{x}_1| \le \gamma_1 \delta^2$, $|\tilde{x}_2| \le \gamma_2 \delta$ where $\gamma_1, \gamma_2$ are some constants, depending on the observer parameters.

This theorem is proved in [26].

#### B. State $x_3$

Consider the subsystem with state variable $x_3 = v_f$, in this case an observer will be introduced in order to obtain an
equivalent output injection, in the same form that the states $x_1$ and $x_2$, the dynamic equation of $x_3$ could be written as

$$ 
\begin{aligned}
\dot{x}_3 &= f_2(t, x_3, u) + \xi_2(t, x_2, x_3, u) \\
y_2 &= x_3,
\end{aligned}
\quad (14) $$

in this case, the dynamic of the system is considered as unknown $\xi_2(t, x_2, x_3, u) = \frac{C_x}{m} - \frac{C_x R_e}{m} \frac{x_2}{x_3}$ in consequence $f_2(t, x_3, u) = 0$. The system (14), understood in Filippov's sense [31] is assumed such that the perturbation $\xi(t, x_1, x_2, u)$ is Lebesgue-measurable and uniformly bounded in any compact region of the state space.

Our task is then to design a finite-time convergent observer of the linear velocity $x_3$. The proposed sliding mode observer is given by

$$ \dot{x}_3 = z_3 \quad (15) $$

where $z_3 = \beta \operatorname{sign}(x_3 - \hat{x}_3)$. Defining $\tilde{x}_3 = x_3 - \hat{x}_3$, the dynamic of the error for $x_3$ becomes

$$ \dot{\tilde{x}}_3 = \xi_2(t, x_2, x_3, u) - \beta \operatorname{sign}(\tilde{x}_3) \quad (16) $$

where $\beta$ is chosen such that $\beta > \max(\frac{C_x}{m} - \frac{C_x R_e}{m} \frac{x_2}{x_3}) = \eta$.

**Theorem 3:** Suppose that $|\dot{x}_3| \le \eta$, and the parameter $\beta$ is chosen such that $\beta > \eta$. The observer (15) guarantees the convergence of the estimated state $(\hat{x}_3)$ to the real value of the states $(x_3)$ after a finite time transient, and there exists a time constant $t_1$ such that for all $t \ge t_1$, $\hat{x}_3 = x_3$.

*Proof:* Consider the Lyapunov function

$$ V(\tilde{x}_3) = \frac{1}{2} \tilde{x}_3^2 $$

its time derivative

$$ \dot{V}(\tilde{x}_3) = \tilde{x}_3 \dot{\tilde{x}}_3 = \tilde{x}_3(\xi_2(t, x_2, x_3, u) - \beta \operatorname{sign}(\tilde{x}_3)) \quad (17) $$

If $\beta$ is chosen as was given in the theorem (3), then $\dot{V}(\tilde{x}_3) < 0$. This shows that $\tilde{x}_3$ goes to zero in a finite time, then, there exist a constant $t_1$ such that for all $t \ge t_1$ holds $\tilde{x}_3 = 0$.

## IV. EQUIVALENT OUTPUT INJECTION ANALYSIS

For the time $t_2$ where $t_2 = \max(t_0, t_1)$ and for all $t \ge t_2$ the error dynamics (10) and (16) holds

$$ \dot{\tilde{x}}_2 = 0 = F(t, x_1, x_2, \hat{x}_2) - \alpha \operatorname{sign}(\tilde{x}_1) \quad (18) $$

$$ \dot{\tilde{x}}_3 = 0 = \xi_2(t, x_2, x_3, u) - \beta \operatorname{sign}(\tilde{x}_3) \quad (19) $$

Notice in (18) that at this time $\hat{x}_2 = x_2$ and $f_1(t, x_1, x_2, u) \equiv f_1(t, x_1, \hat{x}_2, u)$ in consequence $F(t, x_1, x_2, \hat{x}_2) = \xi_1(t, x_1, x_2, u)$.

It was assumed that the terms $z_2, z_3$ change at a high (infinite) frequency. However, in reality, various imperfections make the state oscillate in some vicinity of the intersection and components of $z_2, z_3$ are switched at finite frequency, this oscillations have high and slow frequency components.

The high frequency terms $z_2, z_3$ are filtered out and the motion in the sliding mode is determined by the slow components [32]. It is reasonable to assume that the equivalent control is close to the slow component of the real control which may be derived by filtering out the high-frequency component using low pass filter.

The filter time constant should be sufficiently small to preserve the slow components undistorted but large enough to eliminate the high frequency component.

Thus the conditions $\tau \to 0$ where $\tau$ is the filter time constant, and $\delta/\tau \to 0$, where $\delta$ is the sample interval, fulfilled to extract the slow component equal to the equivalent control and to filter out the high frequency component.

The above reasons allows us to write the equivalent output injection as

$$ \bar{z}_2 = \xi_1(t, x_2, x_3, u) \quad (20) $$

$$ \bar{z}_3 = \xi_2(t, x_2, x_3, u) \quad (21) $$

where $\bar{z}_2$ and $\bar{z}_3$ are the filtered versions of $z_2$ and $z_3$ respectively.

## V. SYSTEM IDENTIFICATION

Assuming that $J$ and $m$ are known, and defining $a_1 = \frac{1}{J}$, it is possible to write the system (6) as follows

$$ 
\begin{aligned}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= a_1 u + \vartheta_1 \varphi_1 \\
\dot{x}_3 &= \vartheta_2 \varphi_2 \\
y &= \begin{bmatrix} x_1 & 0 \\ 0 & x_3 \end{bmatrix}
\end{aligned}
\quad (22) $$

with

$$ 
\begin{aligned}
\varphi_1(x) &= \begin{bmatrix} \frac{-1}{J} \\ \frac{x_2}{Jx_3} \end{bmatrix} \quad \text{and} \quad \varphi_2(x) = \begin{bmatrix} \frac{1}{m} \\ -\frac{x_2}{mx_3} \end{bmatrix} \\
\vartheta_1 = \begin{bmatrix} R_e C_x \\ R_e^2 C_x \end{bmatrix} \quad \text{and} \quad \vartheta_2 = \begin{bmatrix} C_x \\ R_e C_x \end{bmatrix}
\end{aligned}
\quad (23) $$

Notice (22) is in a regression form with regressor vectors $\varphi_1(x), \varphi_2(x)$ in (23), and parameters vectors $\vartheta_1, \vartheta_2$ in (23).

Using the regression notation of (22) the observer (8) could be written as

$$ 
\begin{aligned}
\dot{\tilde{x}}_1 &= \tilde{x}_2 + z_1 \\
\dot{\tilde{x}}_2 &= a_1 u + \bar{\vartheta}_1 \varphi(t, x_1, x_2, u) + z_2
\end{aligned}
\quad (24) $$

where $\bar{\vartheta}_1$ is a parameters vector with nominal values of $\vartheta_1$. For all $t \ge t_2$ equations (20), (21) become

$$ \bar{z}_2 = \Delta\vartheta_1\varphi_1(t, x_1, x_2, u) \quad (25) $$

$$ \bar{z}_3 = \vartheta_2\varphi_2(t, x_2, x_3, u) \quad (26) $$

where $\Delta\vartheta_1 = \vartheta_1 - \bar{\vartheta}_1$.

To proceed we will consider, for clarity of presentation only, the estimation procedures in two steps, one for $x_2$ and one for $x_3$ in order to estimate respectively $\vartheta_1$ and then $\vartheta_2$.

### A. Identification of $\vartheta_1$

It is possible to apply a dynamic form of the Least Square identification algorithm to estimate the parameter vector with the knowledge of $\bar{z}_2$ the regression vector deduced from the measurements and observations of $\varphi_1$.

The model structure for the linear regression [35],[36],[26] can be written as in equation (25) where $\bar{z}_2$ is a measurable quantity, $\varphi_1(t)$ is a regression vector made of known quantities and $\Delta\vartheta$ is the unknown parameters vector (difference to the
nominal parameters). The application of linear regression algorithms like the Least Squares parameter estimation algorithm can be written as

$$ \bar{z}_2 = \hat{\Delta} \vartheta \varphi_1(t, x_1, x_2, u) \quad (27) $$

$$ \varepsilon(t) = h(t) - \hat{h}(t) = \hat{\Delta} \vartheta \varphi_1(t, x_1, x_2, u) \quad (28) $$

$$ \hat{\Delta} \vartheta = \frac{\sigma}{\gamma_t} \Gamma_t \varphi_1(t, x_1, x_2, u) \varepsilon(t) = \Gamma_t \varphi_1 \varphi_1^T \hat{\Delta} \vartheta \quad (29) $$

$$ \dot{\Gamma}_t = -\frac{\sigma}{\gamma_t} \Gamma_t \varphi_1^T(t) \varphi_1(t) \Gamma_t \quad (30) $$

where $\hat{\Delta} \vartheta$ is the estimation of $\Delta \vartheta$ the parameters vector and $\hat{h}(t)$ the prediction of the signal $h(t)$. In general $\gamma_t$ is a normalization term $\gamma_t = 1 + \varphi_1(t) \Gamma_t \varphi_1^T(t)$ and $\sigma \in [0.9, 1]$ a forgetting factor. The initial conditions of the RLS algorithm are $\Gamma_0 = \rho^{-1}I$ initial gain matrix and $\hat{\Delta} \vartheta = \hat{\Delta} \vartheta_0$ initial parameters values.

**Theorem 1:** for the system (27) using the RLS algorithm (27) ensures the following properties:

(i) $\hat{\Delta} \vartheta^T \Gamma_t^{-1} \hat{\Delta} \vartheta$ is a non increasing function and we have $\|\hat{\Delta} \vartheta_t\|^2 \le \frac{\lambda \min(\Gamma_o)}{\lambda \max(\Gamma_t)} \|\hat{\Delta} \vartheta_0\|^2$

(ii) $\tilde{\varepsilon}(t) = (\frac{\sigma}{\gamma_t})^{1/2}(h(t) - \hat{h}(t)) \in L_2$

**Remark 2:** The use of equations (27) ensures the asymptotic convergence of $\hat{\Delta} \vartheta$ to $\Delta \vartheta$ under the persistent excitation condition [36],[35].

**Remark 3:** In application, we have considered the delta operator for approximation of the derivation [35].

## B. Identification of $\vartheta_2$

The low frequency components of the signal $z_3$ satisfies (26), using the notation in (22) takes the form

$$ \bar{z}_3 = \beta \overline{sign}(\hat{x}_3) = \vartheta_2 \varphi_2 = \frac{\vartheta_{21}}{m} - \vartheta_{22} \frac{1}{m} \frac{x_2}{x_3} $$

**Remark 4:** In the same way $\vartheta_{21}$, assuming $\vartheta_{22}$ known or already estimated, can be identified using the Least Squares algorithm.

**Remark 5:** Note also that both parameters in $\vartheta_2$ can be estimated by the Least Square Algorithm at this step. This correspond to estimating twice $\vartheta_{22}$, assuming at this step as previous estimation the value produced by the previous step.

**Remark 6:** Note also that depending on the expression formulated for the forces and wheel slip in (5), (4) several ther variables can be estimated like adherence or longitudinal forces.

# VI. EXPERIMENTAL RESULTS

In this section, we present some experimental results to validate our approach. Several trials have been done with a vehicle (P406 of LCPC) equipped with sensors for wheels angular position measurement.

Measures have been acquired with the vehicle rolling at several speeds. The experimental data used here are those of the rear wheel drive. The installed sensors at each wheel are the variable reluctance ones of the Antilock Braking System (ABS see figure ). Their resolution is 29 dot per revolution

<figure><img src="image_4.png" /><figcaption>Fig. 3. Vehicle used for experiments</figcaption></figure>

(ie $\theta(i) = \frac{2\pi n(i)}{29}rad$). An additional encoder (with 1000dot by turn ie $\theta(i) = \frac{2\pi n(i)}{1000}rad$) have been installed for angular position measurements control and validation.

The Fig. 5 shows the installed laser sensor used for measurement of the wheel radius. Data are sampled at 1kHz frequency and several trials have been considered a different running speeds (40 Km/h, 60 Km/h, 80 Km/h, 100 Km/h and varying velocity) with and without using the ABS system.

<figure><img src="image_5.png" /><figcaption>Fig. 4. 4 Sensors used by ABS 29dot/2π</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Sensor for wheel radius measurement</figcaption></figure>

The Fig. 6 shows the measured displacement using the (high and low resolution) sensors installed on the vehicle and the observed one. We can remark that the curve are well superposed despite resolution.

The velocities can be deduced by several ways from the displacement measurements. Here we compare three of them, two standard computation of derivatives and the proposed observer:
<figure><img src="image_7.png" /><figcaption>Fig. 6. Measured angular displacements</figcaption></figure>

$$ \dot{\theta}(i) = \frac{\theta(i) - \theta(i-1)}{T} \qquad (31) $$

$$ \dot{\theta}(i) = \frac{\theta(i+1) - \theta(i-1)}{2T} \qquad (32) $$

$$ \hat{x}_2 = observed(\dot{\theta}) \qquad (33) $$

In the upper left and right Fig. (7) we can see that estimation of velocity signal derivation, using (31) and (32) respectively, needs a filter to reduce the noise effect. In Fig. (8) corresponding to low resolution encoders the problem is worse and amplitude of noise has a higher level. Filtering this data will affect the measurement precision.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Estimated velocities using the high sensors (1000dot/t)</figcaption></figure>

We remark that when using the proposed observer (bottom left curve in the two figures) that the estimation remain precise despite the bad resolution of the sensors used by ABS. The observed and reconstructed velocities are compared to the measure provided by a high resolution encoder.

These curves show the robustness of our observer based on second order sliding modes and super twisting algorithm versus measurement noise and additional perturbations. Recall that the term $\xi(t, x_1, x_2, u) = \varphi_1(z)\vartheta_1$ is not known and correspond to a perturbation to be rejected in a first step;

<figure><img src="image_9.png" /><figcaption>Fig. 8. Estimated velocities using the ABS sensors (29dot/t)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Estimated velocity using the ABS sensors</figcaption></figure>

thank to the finite time convergence. In a second step (after the convergence time) this perturbation is retrieved by use of a low pass filtering and then the parameters $\vartheta_1$ can be estimated. The second step estimations are the wheel radius and its longitudinal equivalent stiffness. The estimations are shown in Fig.(10), Fig.(9)

<figure><img src="image_11.png" /><figcaption>Fig. 10. Estimated wheel Stiffness and Radius using the ABS sensors and observations</figcaption></figure>

The estimated parameters are quite good and the algorithm is very easy to apply and is not difficult to tune its parameters.
## VII. CONCLUSION

The super-twisting second-order sliding-mode algorithm is modified in order to design a velocity observer for vehicle using only the ABS sensors already placed in standard vehicles nowadays. The finite time convergence of the observer is proved and consequently the separation principle can be considered as avoided. The gains of the proposed observer are chosen very easily ignoring the system parameters. This observer is compared, using experimental data, to classical derivation methods and is proven robust despite the bad resolution of the encoders. Its robustness combined with a sliding mode estimation of the vehicle velocity allow us to reconstruct the wheel sleep. In this way, the observability problems are avoided by means of cascaded finite time converging observer instead of additional sensors. It can be shown that contact forces can also be estimated by this way.

The finite time convergence of state observations in the same time as robustness and perturbation rejection allows to solve the problem of parameter identification using the equivalent control method (by retrieval of the rejected signal). The use of the equivalent control, which provides a linear regression model, allows to apply the classical parameter identification methods (RLS) to estimate the systems dynamic parameters like the tire longitudinal equivalent stiffness and the effective wheel radius.

The estimation scheme build up using a Second Order Sliding Mode observers and a Sliding Mode velocity estimator has been tested on experimental data (acquired with a P406 vehicle) and shown to be very efficient using only standard sensors. The actual results prove effectiveness and robustness of the proposed method. In our further investigations we consider also the case of complete vehicle in a road with changing adherence. The estimations produced on-line will be used to define a predictive control to enhance the safety.

## ACKNOWLEDGMENTS

J. Davila and L. Fridman gratefully acknowledge the financial support of the Mexican CONACYT (Consejo Nacional de Ciencia y Tecnología), grant no. 43807-Y, the Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica (PAPIIT) UNAM, grant no. 117103, as well as, DGSCA - DTD. This work has been done in a collaboration managed by members of the LSIS inside the GTAA (Groupe Thématique Automatique et Automobile). The GTAA is a research group supported by the CNRS. Thanks are addressed to the LCPC of Nantes for experimental data and the trials with their vehicle Peugeot 406.

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