Novel Two-stage Nonlinear Interconnected Unknown Input Observer Design: Hardware & Experimental Vehicle Validation


 This paper concerns both vehicle lateral and longitudinal nonlinear dynamics estimation. Consequently, the interlinked vehicle models dependency and the hurdle coupling features will be overcome here thanks to the NONLINEAR INTERCONNECTED UNKNOWN INPUTS OBSERVER (NI-UIO) framework. This interconnection scheme extends the estimation of the lateral motion to the longitudinal one with the unknown accelerator, brake pedal and driver steering torque inputs, as well as tire-ground pneumatic efforts to reduce conservatism and observability problems. The aspects of immeasurable real-time variation in the forward speed and tire slip velocities in front and rear wheels are taken particularly into account. Consequently, TAKAGI-SUGENO (TS) fuzzy form is undertaken to deal with these nonlinearities in the observer synthesis. The INPUT TO STATE STABILITY (ISS) of the estimation errors is exploited using Lyapunov stability arguments to allow more relaxation and additional robustness guarantee regarding the disturbance term of immeasurable nonlinearities. Therein, sufficient conditions of the ISS property are formulated as an optimization problem in terms of linear matrix inequalities (LMIs). Finally, hardware and experimental validation with robustness test are performed with the well-known SHERPA dynamic interactive driving simulator as well as TWINGO vehicle prototype to highlight performances and applicability of the outlined observer.


Introduction
Recently, the self-driving traffic of intelligent vehicles has attracted the attention of researchers and automakers around the world to exploit the promising capabilities and to accomplish the safety challenges. Nonetheless, the vehicle complexity, coupling, nonlinear motions, time-variance and, lack of knowledge on dynamical states and external inputs, lead to highly complicated embedded solutions. To improve intelligent safety technologies, many studies are structured around the use of instrumented vehicles and estimation theory to acquire the vehicle's real-time information from conventional sensors or virtual estimators. Over the past decades, a noteworthy interest in vehicle model-based observer is demonstrated by a large body of literature for potential control and safety applications. The interested reader may refer to [1,2] for the bibliographical review of the existing approaches on this topic. In terms of lateral dynamics observer, relevant works are proposed to design state estimation from inertial measurement unit [3,4]. The simultaneous estimation of the lateral dynamics and the road grades including curvature, slope, or banked roads were treated in [5,6]. Particular attention was paid for the estimation of the tire-ground contact forces in [7] to improve the vehicle safety performance. Different strategies have been investigated to estimate vehicle dynamic states with external disturbances or, unknown inputs. The simultaneous estimation of vehicle lateral dynamics and driver's torque is proposed in [8,9] using unknown inputs observer with linear parameter-varying (LPV) techniques or the Takagi-Sugeno transformation. Adaptive observers have been introduced to study the convergence of state estimation jointly with parameters, refer to [10,11]. The computational and technical complexity in the estimation synthesis forced researchers to invoke restrictive assumptions simplifying the mathematical tasks of stability analysis and estimation design. Non-linear estimation schemes were investigated to face the challenge of the nonlinear vehicle dynamics. The vehicle longitudinal forces are reconstructed using an adaptive neural network nonlinear observer in [12]. Dual unscented Kalman filter algorithm has been used in [13], based on the non-linear least-squares problem and the hybrid Levenberg-Marquardt to identify the Pacejka tire model coefficients, lateral and normal forces, and sideslip angle. [14] designed an adaptive controller to attenuate the vibration of the quarter car suspension system together with a nonlinear observer to estimate an internal state variable namely the damping force based on velocity input, internal state, and voltage input. The work presented in [15] solves the stability problem of a vehicle suspension with stochastic disturbance by designing a fuzzy adaptive nonlinear controller. While [16] outlined nonlinear observer for tracking of vehicle motion trajectories on highways using a radar or laser sensor. Parameters estimation of the nonlinear vehicle dynamics has been deduced from fuzzy Unknown Input Observer (UIO) in [17]. Vehicle online parameter estimation was treated in [18] with a nonlinear adaptive observer, to determine the wheel stiffness and radius to be used for controller synthesis and supervision applications. Also, a nonlinear filter observer was reported in [19] for the estimation of vehicle velocity jointly with the tire-road friction coefficient. A wellknown method to estimate time-varying parameters of nonlinear models is the extended Kalman filter (EKF). [20] adopted a nonlinear extended state observer-based output feedback stabilization controller for a half-car active suspension system, to overcome nonlinearities leading to performance deterioration. Recent researches are conducted to study cascade systems or two-stage structures which are very common configurations in engineering applications. The results reported in [21] for cascade systems have revealed interesting results on parameters identification. An interesting solution is suggested in [22] for robust nonlinear estimation of road grade angles with tire-ground interaction forces using delayed cascade observer structure. This method uses a Two-Stage Extended Kalman Filter, allowing a robust simultaneous estimation of the slow and fast dynamics variables. In [23] authors address a novel two-stage extended Kalman filter algorithm for reaction flywheels fault estimation.
Most of the aforementioned papers consider prevalent assumptions as neglecting fast dynamics and tire-road contact efforts, also, the hypothesis regarding vehicle driving practices as an independent behavior, constant velocity or small speed variations. Assuming such assumptions allows one to derive simplified models only valid within a limited range of operating conditions to employ linear techniques. The study of simplified model systems or independent dynamics significantly simplifies the analysis and can help to elucidate the theoretical feasibility of the estimation design. While this is a very desirable development from a theoretical point of view, this simplification is not an adequate representation of the true physical system when the latter is subjected to large disturbances and external unknown inputs. Also, most of the used assumptions are not physically justifiable in certain driving scenarios, and these assumptions may lead to an inaccurate estimation concerning the real dynamics. A direct consequence of this simplification is that any stability assessment relying on such simplified models may lead to wrong predictions, which could lead to severe degradation of the safety system. Despite the abundant literature on vehicle estimation, the reliable and efficient integration of the vehicle observer presents several theoretical and technical challenges.
In this work, a novel two-stage nonlinear interconnected unknown input observers (NI-UIOs) is presented for the estimation of vehicle motion together with tire-road interaction and unknown external inputs namely: driver traction, braking, and steering torques. The main distinction to existing results is that, instead of the usual simplified model that considers only the forward speed to represent the longitudinal motion. Herein, we provide a practical solution in the case of coupled and dependent lateral and longitudinal nonlinear dynamics, which share common and coupled data of the vehicle behavior. More precisely, the proposed estimation scheme has several merits: -The interconnection configuration presents the novelty and reduces conservatism for a more complete vehicle observer design.
- The variations in the forward speed, tire slip velocities on the front and rear  wheels, and the related variables are particularly considered as immeasurable  nonlinearities in the interconnected scheme and treated via the boundary domain.  -The ISS principle relaxes the usual sign definiteness on the Lyapunov principle. This relaxation allows deriving less necessary and sufficient LMI conditions. It has the advantage of providing further insights and additional robustness guarantees concerning the unknown inputs and the disturbances terms of immeasurable nonlinearities. -The efficiency of the ISS framework and the interconnected configuration of the proposed NI-UIOs algorithm in both hardware "SHERPA driving simulator" and experimental "TWINGO car" platforms.
The remainder of the paper is structured as follows. Sec. II describes the vehicle interlinked model with the tire-ground contact forces. While Sec. III presents this model through the TS interconnected fuzzy model. Then, Sec. IV illustrates the observer design and presents the convergence analysis based on ISS-Lyapunov theory. Sec. V discusses the results compared to a real-world experiment. Finally, some concluding remarks and an outlook on future work are given in the last section.

Interlinked Road-Vehicle Lateral and Longitudinal Dynamics
Ground vehicles are highly dynamic, nonlinear and coupled systems, that involves mechanical parts connected by several links such as braking, suspension, steering, powertrain, etc. This section briefly introduces the nonlinear models used herein for observer synthesis. First, we assume the following: 1. Vertical, pitch, and roll dynamics are neglected.
2. Left and right wheels at each axle are grouped to form a single equivalent tire, as shown in Fig. 1. 3. Vehicle have planar motion parallel to the road's surface.
The vehicle dynamics is a single-track model [24] described in the vehicle fixed frame with 12-DoF (twelve degrees of freedom). In which, nonlinear longitudinal, lateral, and yaw motions, the vehicle steering system, accelerator, and brake pedal are respectively taking into account with tire-ground interaction forward and cornering, front and rear forces. Additionally, the positioning of the vehicle on the road is described via a standard vision model [24].
-Longitudinal and lateral motions -Yaw motion -Vehicle positioning on the roaḋ -Electronic power steering system dynamics The main aim of adding a vision system to our structure is to estimate the road curvature which will give us an additional degree of freedom in the reconstruction of the vehicle motion. Moreover, the lateral vehicle model is presented together with a vehicle steering system to estimate the total steering torque (T s ) which is composed of both the electrical assistance torque and the driver torque [25]. Note that the driving torque can be easily determined from the estimation of the total steering torque (T s ) and the known assistance electrical torque. B f , B r are the braking torques applied to the front and rear tires, T B f = B f + T f is the total braking and traction torque, with T f is the engine torque applied on the front wheels. The vehicle parameters and variables, defined in the above equations, are listed in table 1. The vehicles dynamics are subject to the tire-road interaction and pneumatic deformations [26]. These efforts represented by the longitudinal "braking or acceleration" and lateral "cornering" forces are necessary to accelerate, brake, and steer the vehicle.
For small values of the tire side-slip angle α or slip velocities ratio λ, the instantaneous lateral F y and longitudinal F x forces can be approximated by where σ is the tire relaxation which models the transient time. With In the presence of traction or braking forces, the rolling wheel will create a longitudinal skidding throughout the contact surface. In this case, the forward speed of the contact point is different from the tire's tangential velocity. To quantify the proportion of the sliding to the wheel rolling movement, the following switching signal (denoted i , i ∈ {r, f }), called the slip velocity, is considered to compute the tire slip velocity ratios Then Consequently, the slip velocities in (8) are expressed as nonlinear parameters and assumed to be unknown but bounded with a priori known bounds. The variations of these nonlinear switched parameters are treated as premise parameters and transformed in TS representation by the upper and lower bounds. Also, we assume a small variation of the steering angle under normal driving conditions. In the next section, the TS polytopic representation [27] is undertaken, giving rise to an ease transposition of the time-variance dynamics using the well-known sector nonlinearity approach. Yaw rate (rad.s −1 ) and Lateral slip angle (rad) ω f , ωr Angular velocities of the front and rear wheels (rad.s −1 ). δ, α, λ Steering angle, side-slip and longitudinal slip ratio (rad) y L , ψ L Lateral offset and angular displacement F yf , Fyr Cornering front and rear forces (N ).

Fxr, F xf
Longitudinal rear and front forces (N ).

Fa, Frr
Aerodynamic and rolling resistance forces Braking torques applied to the front and rear tires.
Engine torque applied on wheels.

Fw
Lateral wind force due to the effect of the wind gusts(N ) m, Iz Vehicle mass (kg) and inertia about the z-axis (kg.m 2 ) Cα, C λ Cornering and longitudinal stiffness parameters (N.rad −1 ) i f y , iry the wheels moment of inertia R Rolling radius. l f , lr Distances between the C.G. and front and rear axles (m) ls, lw Look ahead distance and distance of wind force action (m) σ relaxation length which models the transient time.

Is, Bs
steering system inertia and steering system damping coefficient Rs, ηt steering column-wheels gear ratio and trail length Herein, to derive the observer-based vehicle model, a new representation of the vehicle dynamics, described in section 2, combined with cornering and longitudinal forces dynamics leads to two-stage subsystems assembled in the interconnection scheme with strong three nonlinearities in each subsystem. The mathematical formulation for the time-variance interconnected system (1)- (5) and (6) with its q varying parameters is exactly rewritten as a compact TS representation with r multi-models weighted by membership functions η i ( . ) as follow Therein, the nonlinearities considered here are related to tire slip velocities on the front and rear wheels f , r and forward speeds v x , 1 vx , 1 v 2 x , considered as external immeasurable time varying parameters. Let us consider that the time-varying matri- where matricesΠ i andΠ i are constant for all i ∈ [1, ..., r]. With r = 2 q represents the number of local submodels where the q non-linearities related toθ ∈Θ,θ ∈Θ are captured via membership weighting functions η i ( . ), which satisfy the convex-sum property in the compact set of the state space [27] The bounds of these smooth scheduling variables are defined in an hyper-rectangles ∀θ ∈Θ and ∀θ ∈Θ given bȳ whereθ min  (9) allows decreasing the number of varying nonlinearities which decreases the number of LMI related to the induced sub-models. Consequently, the conservatism problem and computational complexity are reduced when solving the observer. If we consider q i , q j , are respectively the non-linearities of each subsystem on the interconnected models, in this case, we have r = 2 qi = 8 TS lateral sub-models (r = 2 qj = 8 TS longitudinal sub-models) and the induced LMIs for each motion rather than r = 2 q = 32 TS sub-models in case of one global system. Therein, the interconnected approach allow deriving less necessary and sufficient conditions in the LMI problem. This allows to relax the usual optimization problem by exploiting the interconnected scheme. This relaxation is essential to establish the main result of the present paper.

Remark 2
The numerical complexity and LMIs conservativeness can be further reduced by exploiting the relation between the vehicle speed nonlinearities v x , 1 vx and x . This problem has been exploited in [28] using the variable change and firstelement Taylor's series expansion. This idea will be investigated in future work for the design of a shared steering controller with an interconnection configuration.

Observer Design
The objective of this section is to design a two-stage nonlinear interconnected unknown input observer (NI-UIO) with state-dependent matrices and immeasurable nonlinearities in the sequel for the vehicle whole motion estimation. Systems performances under stability and robustness are highly desirable properties from a practical point of view. Therein, our analysis is conducted using the ISS-based-Lyapunov function to guarantee the stability theory of estimated systems whose dynamics depend on unknown perturbations, or other inputs. To begin with, the following assumptions for the existence of the observer are considered: Assumption 1 (i) The state (X 1 ) and (X 2 ) are all bounded. (ii) The pairs (Ȃ η ,C) and (Ā η ,C) are observable or detectable to guarantee solutions to the LMI problem. -The polytopic sub-systems (9) are observable, ie.: -The polytopic LPV systems (9) are detectable, ie.: holds for all complex number s with Re(s) ≥ 0 X(t) ∈ R n , U(t) ∈ R p . (iii) Assume that, for the design of each separate UI observer, the states of the other subsystem are already estimated from the interconnection configuration. (iiii) The matching condition for the model holds: The first assumption holds in open-loop and the vehicle remains in a bounded statespace region to guarantee stability. By assumption (iii), we mean that the estimators request current state information from the neighboring subsystems through the interconnection as it is the case of the vehicle motions because of the physical interactions. The assumptions (ii) and (iiii) can easily be checked numerically. An overall scheme of the system structure linked to the observer is in figure 2. In this structure, observers' outputs are used as inputs or measurements in other observers of the interconnected configuration.

State estimation
Under the fulfilled matching condition, the problem of NI-UIO design can be stated as follows T are the state of the observer, andX(t) = [X(t)X(t)] T are the estimated states and Y(t) = [Ȳ(t)Y(t)] are the output vectors. The parameter varying observer matrices N η , G η , L η , and H η must be well chosen to satisfy a stable convergence of the estimation error dynamics even in the presence of UI. The observer gains are written as (10). In the following, the observer design procedure aims to determine the aforementioned observer's matrices.

NI-UIO Stability & Convergence Analysis
Let us consider the following suitable state estimation error According to the observer (18) and the system (9) equations, the dynamics of the estimation errors iṡ To satisfy the stability of the error dynamics (20), the following conditions must be guaranteedΦ T ηBη = 0,T ηBη = 0 (23) Consequently, the estimation error dynamics e(t) becomeṡ This is a fundamental prerequisite for the main ISS analysis to verify the impact of inputs and perturbation on the asymptotic bound of the solutions. The following steps in the design approach are followed to satisfy the stability of the error dynamics (25): (1) Condition (22) allows computing the Hurwitz gains (2) To make the state estimation error independent to the UI, the equality constraint (23) can be equivalently written as the decoupling condition in (17). This leads to find matrices H η , ie. (H η ,H η ) with (·) † = ((·) T (·)) −1 (·) T denotes the left pseudo-inverse of the matrix (·). (3) After computing H η , we obtain: T η = I n + H η C η .
The following theorem 1 states the main result in terms of LMIs ensuring ISS convergence of the state vector.
Theorem 1 In view of the two-stage longitudinal and lateral subsystem subject to unknown inputs, if the polytopic interlinked models (9) satisfies the stated Assumptions 1, a NI-UIO observer is designed by (18) and the ISS convergence of the estimation errors between the system and the interconnected observers is ensured, then, the origin of the system will be practically finite-time stable, i.e., the system states will converge to the neighborhood of the origin in finite time.
Step 2: For a given real positive scalar α and matrices G = {Ḡ,G}, if there exist two symmetric positive definite matricesP andP , and gains matricesΩ i and Ω i , i = 1, ..., r, positive scalars η = diag{η 1 , η 2 } solutions of the following LMI optimization problem Step 3: The NI-UIO guarantees the state estimation with an ISS convergence Step 4: The observer gains are given bȳ

LMI-Based Optimization for NI-UIO Observer Design
The Lyapunov method is shown to be very useful for nonlinear model stability analysis. Proof 1 To demonstrate the theorem 1, the observer stability is studied by using the following quadratic storage Lyapunov function to prove the ISS stability with respect to the unknown inputs.
V = e T P 0 0P P e, P = P T > 0 Its time derivative is expressed as follows: Lemma 1 For every positive definite matrix G > 0, the following property [29] holds: Applying Lemma (33), replacing the suitable terms, and by adding and subtracting the term αe T P e, with α is a positive scalar, the inequality (32) yieldṡ and Now, if Ψ < 0, then the time derivative of the Lyapunov function (34) can be bounded as followsV and the following definition holds.

Definition 1 (The Schur's complement Lemma)
Given the matrices Ψ and G with appropriate dimensions, where G = G T , then the following equivalence holds [29] This inequality represent the time-dependent LMI.
By integrating (37) over the interval [0, t], we get Knowing that V (t) is a Lyapunov function, it can be bounded by λ min e(t) 2 2 and λ max e(t) 2 2 , where λ min and λ max are the min and max eigenvalues of the matrix P . Under this condition, the state estimation error is reduced to Definition 2 [30] The state estimation error dynamics verifies the ISS if there exists a KL function f 1 : R n × R −→ R, a K function f 2 : R −→ R such that for each input ξ(t) satisfying ∆ ξ (t) ∞ < ∞ and each initial conditions e(0), the trajectory of the error associated to e(0) and ∆(t) satisfies: Hence, when t → ∞ the exponential converge to zero, implies the straightforward inequality (29) from the ISS property.
From the boundedness of ∆ ξ (t) and thanks to definition (2), it is shown that the errors dynamics are stable and verify the ISS property from the perturbation term ∆ ξ(t) to the estimation error e(t). Assuming λ min (P ) ≥ 1 (P > I) and since G can be imposed, minimizing the ISS gain is equivalent to minimizing a positive scalars η = diag(η 1 , η 2 ) such that By applying the Schur's complement lemma, inequality (42) can be written as the LMI constraint (28c). The positive quantities η 1 and η 2 are minimized in the following objective function min P ,P ,η1,η2 The Lyapunov formulation of the ISS property and the convex sum property of the weighting functions in (38), allow reaching the time-independent condition of the optimization problem obtained in (28a)-(28b)-(28c). Finally, the gains of the NI-UIO observer are computed from (30) given in theorem 1 and the proof is complete. This achieves the observer synthesis. We present hereafter a method to estimate the unknown inputs.

Algebraic Reconstruction of Unknown Inputs
In this section, we address the problem of unknown input reconstruction of the vehicle coupled dynamics. We focus our interest on the front and rear braking and traction torques, the steering torque, and the road curvature since they play a key role to guarantee vehicle stability in driving maneuvers. In order to avoid using the derivative information of the output directly, we first consider a high-order differentiator that can provide the exact estimates of the output derivatives.

Assumption 2
The state vector (X 1 ) and (X 2 ), the unknown inputs, the measurement and their derivatives are bounded.
Theorem 2 Under Assumptions 2, the following super-twisting algorithms (high order differentiators), based on the work of Levant [31] is used to estimate the derivatives of Y(t) = [Ȳ(t)Y(t)] T . For a given signal y i (t), let us denote the ξ i 2 of y i as the estimates of the first derivative applied for each output variable and ξ i 3 is the second derivative, sign refer to the sign function and γ i j are positive scalars and are chosen according to the limits values of the derivatives of the output variables as suggested in [31].ξ From vehicle dynamics equation (9), it is straightforward to find an approximate estimation of the unknown inputs using a direct method or a dynamics inversion approach. From the output equationŶ = CX(t), the time derivative ofŶ(t) iṡ After estimating the states of the system and from the design of the derivatives estimates in Theorem 2, the unknown inputs can be reconstructed by an algebraic inversion of the previous equation. However, the feasibility of this inversion is conditioned by the fulfill rank condition rank(CB η ) = rank(B η ), the UI estimation is obtained by the following dynamics inversion U : In the other hand, the convergence ofÛ toward U can be analyzed by defining the unknown part estimation error From equation (25) and replacing ∆ ξ = G † η (ė − N η e), the unknown input error is equivalent to Knowing that e satisfy the ISS performance, then e U also guarantee the ISS property. The unknown inputs converge toward a small region to achieve a more accurate estimation.

Experimental Results and Discussions
The NI-UIO approach is tested first against hardware experiments collected using the SHERPA-LAMIH dynamic driving simulator and also validated using a real-world driving experiment using a RENAULT TWINGO prototype.

Hardware Experiments
The practical performance of the proposed NI-UIO is validated via a series of driving maneuvers conducted with a human driver in the SHERPA driving simulator. This interactive car simulator reproduces the vehicle dynamics taking into account a wide variety of parameters such as weather condition, grip, and the road surface [32]. It includes a full car mock-up PEUGEOT 206 vehicle installed on a six-DoF Stewart platform, the setup is placed in front of visual feedback displays of 240 deg wide projection screen offering a panoramic view, presented in Fig. 3a (for more details refer https://www.uphf.fr/LAMIH/en/SHERPA). Therewith, the driving simulator is used in co-simulation with the SCANeR module interfaced with Matlab/Simulink. The optimization problem in Theorem 1 under LMI conditions is solved using the Yalmip toolbox in Matlab software.
(a) SHERPA-LAMIH driving simulator.  Herein, the test maneuver was performed on a Satory test track for normal driving behavior on urban scenic dry asphalt road of a high friction coefficient equal to µ = 1. The road track presented in Fig. 3b, is composed of straight lines followed by several narrow and big bends profiles. This configuration allows the observer evaluation including a wide spectrum of the vehicle dynamics under and over its linearization interval. The NI-UIO estimates the coupled lateral and longitudinal dynamics using the measured vehicle states provided by the driving simulator. These data are depicted in Fig. 4 with their counterpart estimated yaw rate, steering angle and wheels angular velocities as well as the vehicle positioning on the road defined by the lateral deviation and the heading errors. Since these signals are measured and used in the observer design, the state estimation demonstrates a finite-time estimation convergence. In the observer synthesis, the forward speed is explicitly considered as an unmeasured time-varying parameter to improve the estimation performance under different driving conditions. The vehicle in the software simulation undergoes the driving maneuvers on the Satory test track at a varying speed from 0 km/h to 55 km/h managed by the human driver to generate the cornering and forward dynamics involving both traction and braking phases on flat road surfaces. Hence, Fig. 5 depicts the estimation results of unmeasured state variables namely the lateral and forward speeds v y , v x , the front/rear lateral tire forces F yf , F yr and the front/rear longitudinal tire forces F xf , F xr . Comparing the estimated states with those provided by the software simulator, we can see that the observer has a fast dynamic transition and good estimation convergence. Also, we note that in t = 40(s) the lateral speed and the front lateral force estimation suffer slightly. It should be noted that this fluctuation corresponds to the largest curve in the test track as it is shown in the road curvature profile, this is caused by lateral slip motion. It can be appreciated that ISS performance is guaranteed for simultaneous longitudinal and lateral states dynamics estimation. For more faithful validation, the unmeasured states (F xi , F yi ) are used to reconstruct the lateral and longitudinal accelerations a y , a x at the center of gravity CoG given by the following set of equations mâ y = F yi and mâ x = F xi with i = {f, r}. The results reported in Fig. 5d show a finite-time asymptotic estimation even for an aggressive and strongly coupled driving maneuver which excite the vehicle away from the straightline dynamics linearization. On the other hand, unknown input reconstruction is done assuming that all the states are available (either measurable or estimated). Henceforth, the unknown input vector namely the two braking and accelerating torques on both front and rear wheels applied to manage the forward speed and the total driver's steering torque applied on the lateral model are well estimated from the model inversion together with road curvature depicted in Fig. 6, compared to nominal values obtained from the simulator. Note that considering the longitudinal and lateral torques as unknown inputs is more consistent with real constraints, because they depend on several parameters (tire pressure, tire adhesion, mass transfer between the two tires, etc.). According to these results and despite modeling assumptions, it can be appreciated that the proposed observer provides a good estimation accuracy under highly dynamic maneuvering, and proves the reliability of the approach to estimate simultaneously the dynamic states and the unknown inputs of the vehicle longitudinal and lateral motions with ISS performances of the estimation errors.

Observer Sensitivity against road uncertainties
Remind that the observer was designed for a nominal case with road friction coefficient µ = 1 (dry asphalt). To assess the observer's sensitivity to the road uncertainties, the observer is tested with respect to the friction coefficient variation. To this end, two cases (moderately wet road µ = 0.6 and very wet road µ = 0.4) for the same digital database of the Satory test track, are considered and compared with the nominal case. The effect of road friction changes is evaluated by means of the root-mean-square errors (RM SE % ) and normalized mean-square errors (N M SE % ) considering the difference between the estimated and measured states and UI presented in table 2.
where y is the measurement of length N ,ŷ is the estimate, and indicates the 2-norm of a vector. The errors must not contain any NaN or Inf values. We omit the large picks in the computed metrics. An overview of NI-UIO estimation results are depicted in Fig. 7 and 8. The result illustrates the efficiency and the robustness of the proposed NI-UIO to reconstruct the unmeasured states and UIs unknown parts even in the presence of road uncertainties. It can be noted that the design of the observer still ha good performances and the effect of the road grip variation on the state's estimation is limited, however, it remains barely visible in certain states. From Table 2, the observer gives the better estimation for the nominal case where µ = 1 where the maximal values of (RM SE % ) are the lowest and N M SE % the largest. As expected, the estimation errors increase when the road friction decreases with a maximum degradation of (3%). Moreover, the amplitudes of the deviations errors are more notable for the torques estimations. Also, a x , a y are less sensitive. Otherwise, it can be seen that RM SE % for the yaw rate and curvature remain approximately constant, so, the observer is more robust for parameters uncertainty. However, the quantification of the lateral speed for µ = 0.4 is much more different from the nominal case. Indeed, even with road uncertainties consideration, the deviations amplitude of the errors is quantified with lowers than RM SE % < 9.81% and N M SE % > 84.61% for the very wet road case. Quantification result confirms that the observer robustness is preserved to handle the road parameters uncertainties.

Experiment validation procedure and trials
This experimental log-data aims principally to point out the performance of the proposed NI-UIO in real-world driving events and to show that the observer estimation fulfills the unknown part reconstruction, which is one of the contributions of this paper. To evaluate the effectiveness of the proposed observer, experiments have been performed using the LAMIH Renault TWINGO experimental vehicle which is instrumented specifically for the study of vehicle dynamics Fig. 9a   On the other hand, this experimental platform is fully equipped with a CORRE-VIT sensor that measures sideslip angle and lateral speed, installed on the right back door at a height of 40cm. The onboard acquisition system includes also an Inertial Navigation System placed near the center of gravity to provide the three axes acceleration, the three Euler angles, and their associated angular velocities in the three directions. The car integrates an odometer to measure the vehicle speed at a sampling rate of 10Hz. The camera and GPS can respectively record the scenario and the path of the test. The front-wheel steering angle has been obtained from an optical encoder, whereas the angular speeds of the wheels are directly obtained from the ABS sensors. The validity of the NI-UIO estimator was investigated on an urban scenic dry road, depicted in Fig. 9b and performed with normal driving behavior and good environmental conditions. This run is composed of big turns and straight lines with a varying vehicle speed to excite the whole vehicle dynamics. The data used in the estimation algorithm are well estimated and depicted in Fig 10. Indeed, braking, traction, and driver steering torques, as well as the curvature are reconstructed from the inversion method and plotted in Fig 12. In the experimental maneuver, it should be noted that the true torques inputs and the curvature are unknown and immeasurable. The state estimation results are predicted in the Fig.11. The lateral and the forward speeds provided respectively by high precision Correvit sensor, serve only for the experimental validation of the observer. The performances of the lateral and forward forces estimator are compared with the ones measured by the sensors through the accelerations. In Fig. 13, we give a comparison between the forward and cornering acceleration obtained from the IMU data and the estimated one from the longitudinal and lateral forces in the contact point between the wheels and the ground from the observer. Experimental results illustrate that the observer quickly and accurately estimates the states with minimal error. Note that the lateral speed has a small value comparing to the other lateral variables. It reveals also the difference between the obtained results from the IMU data and the estimated states is not very perceptible and this confirms that the lateral velocity is less influencing the vehicle reconstruction. Moreover, a high deceleration of the vehicle has occurred which is followed with a strong deformation of the suspension. In this case, there is a strong coupling between the forward and cornering dynamics. Hence, the ISS performances are guaranteed and the estimations are acceptable under high deceleration and soft acceleration. Finally, the interest of using a nonlinear NI-UIO estimation with variable and immeasurable nonlinearities is validated with two test bench, against road uncertainties and different velocities to evaluate the observer sensitivities. Therein, the observer is robust with respect to a large range of variation of longitudinal velocities and to road surface uncertainties.

Conclusion
This paper presented an ISS-LMI based two-stage nonlinear interconnected UIO observers to estimate the full vehicle dynamics. For this purpose, we are interested in estimating simultaneously the lateral, longitudinal, and steering dynamics, as well as the tire/ground forces. The proof of convergence is guaranteed when taking into account real constraints as the variations of the forward speed and the tire slip velocities which were considered immeasurable during the observer's design. The proposed observer gives a very promising solution because it allows the estimation of the unknown braking and traction torques as well as the steering torque and curvature external inputs whose price of its sensors is very expensive. Another technical solution proposed in this paper is the estimation of the tire's forces which are very hard to measure. Moreover, the interconnection structure relaxes the mutual dependence and coupling between the longitudinal and lateral motion, it reduces conservatism and computational complexity. And, ISS property allows guaranteeing the stability and robustness of the estimator against unknown inputs and perturbation term. Both hardware simulation and experimental results allowed the assessment of the performance and the applicability of the proposed estimator under ideal conditions, then with road friction uncertainties based on the acquired data. Finally, the insights that can be gained from our proposed structure can offer valuable conclusions under less restrictive and more realistic assumptions for the interconnected estimation design, robustness and conservatism as well as for the practical applicability of the estimation concepts. In future work, the NI-UIO technique will be used together with a fault detection scheme to detect the abnormal driving behavior based on a fault-tolerant controller.

Conflict of interest
The authors declare that they have no conflict of interest.