# New Nonlinear Takagi–Sugeno Vehicle Model for State and Road Curvature Estimation via a Nonlinear PMI Observer

Zedjiga Yacine, Dalil Ichalal, Naima Ait Oufroukh, Said Mammar and Said Djennoune

# Abstract

The present article deals with an observer design for nonlinear vehicle lateral dynamics. The contributions of the article concern the nonconsideration of any force model and the consideration that the longitudinal velocity is time varying, which is more realistic than the assumption that it is constant. The vehicle model is then represented by an exact Takagi–Sugeno (TS) model via the sector nonlinearity transformation. A proportional multiple integral (PMI) observer based on the TS model is designed to estimate simultaneously the state vector and the unknown input (lateral forces and road curvature). The convergence conditions of the estimation error are expressed under LMI formulation using the Lyapunov theory, which guaranties a bounded error. Simulations are carried out for comparison between the conventional PI observer, the enhanced PI observer, and the PMI observer. Finally, experimental results are provided to illustrate the performances of the proposed PMI observer.

## 1 Introduction

Safety driving systems are integrated in today’s vehicles. Their efficiency and effectiveness are no longer to be proved. For their elaboration and conception, some data linked to the vehicle dynamics are required to be available, such as the sideslip angle, lateral velocity, tire–road friction coefficient, and lateral wind. However, most of these data are unavailable for measurement, either because the sensors are too expensive or they are unappropriate from a technological point of view. The alternative commonly used is to replace the hardware by software through the estimation of the unmeasurable variables via observers.

Systems are subject to different types of inputs; some are known, such as the control inputs, whereas others are unknown, identified as disturbances, uncertainties, faults, or even noise measurement. The vehicle system is affected by numerous inputs that are known, such as the steering angle and the driving torque, or unknown, such as the crosswind, tire–road friction coefficient, road curvature, and modeling uncertainties. These unknown inputs are very important to estimate as they strongly influence, and in a bad way, the system. Afterward, their effects should be minimized through the elaboration of controllers that use the synthesized estimations. The design of observers for systems subject to unknown inputs has been intensively undertaken these last decades [2, 10]. The vehicle is a highly nonlinear system, mainly in dangerous situations like cornering and lane changing where some nonlinear modes are excited, and is subject to various inputs, external unknown and control ones. Thus, designing observers for the vehicle is still challenging. Most of the work undertaken to estimate vehicle parameters and its unknown entries had dealt with linear or simplified vehicle models [5].

Many works have been realized to provide estimation tools of the states in the presence of unknown inputs. Some approaches tend to eliminate or decouple these unknown inputs and reconstruct the system’s states [24]. This practice is not efficient and leads to erroneous estimation, as the states evolve influenced by the neglected inputs. Other approaches have been then explored to simultaneously estimate the states and the unknown entries of the system. For this, the unknown input observer (UIO) is developed [12, 18]. Unlike the category of observers that mask the effect of the unknown inputs, UIO observers try to include them in the modeling phase and the estimation one.

Extensions of the UIO have been considered from different points of view. In particular, the proportional integral (PI) UIO has been developed for SISO systems [25] extended to the MIMO case [13]. The main advantage of the PI observer is the integral action that contributes to a good estimation of the unknown input and reduces the conservatism of the convergence conditions by introducing an additional degree of freedom. The condition of a constant unknown entry is required in theory and is seen as an inconvenience; however, in practice, this can be overcome with augmenting the value of the observer gain [12]. Nevertheless, increasing the observer gain may introduce sensitivity to the noise. To estimate a more general class of signals, multiple integrals may be used, which is best known as the proportional multiple integral (PMI) observer, with the advantage of estimating signals in time polynomial forms.

As stated above, the cases treated are mainly the linear ones; therefore, a lot of methods are provided to estimate unavailable variables for that type of systems [5, 23]. However, considering a simplified representation of a highly nonlinear model does not reflect the real behavior of the initial system, as it is well known that linear models are valid only locally around a specific operating point; furthermore, as estimation methods are model based, consequently, this has an effect on the quality of estimation, especially when accuracy is required, in particular in the objective of control and elaborating safety driving systems.

There is abundant literature on observers developed for linear systems. The Luenberger observer has been the first idea investigated and from where all others have emerged [17, 23]. Unlike the linear case, the issue of designing observers for nonlinear systems is still a field of intensive investigations [1, 22]. This case is difficult to address as there is no general framework for nonlinear systems. Three main approaches have been introduced for nonlinear systems, such as geometric methods [15], high-gain observers [9], and sliding-mode observers [2, 8]. All these approaches have their own advantages and disadvantages.

An alternative approach to represent the nonlinear model is the Takagi–Sugeno (TS) modeling. The idea behind this approach is to generalize the methods and tools developed in the linear domain to the nonlinear one. It is based on expressing the nonlinear model as an interpolation of several local models. Three distinct methods can be used for this objective. The first one is based on identification, where models are identified from input–output data. The second method considers linearization around operating points, and the last one is based on sector nonlinearity (convex polytopic transformation). The transition between the local models is done through activation functions (weighting functions). Feng et al. [7] have proved that any nonlinear system can be represented with arbitrary precision by a TS model in a compact set of the state space with a good accuracy even exactly using the last approach.

The TS model is still nonlinear but is a set of linear models interpolated to form the global one through weighting functions. If these last are a function of the input and/or output, then the model is said to be with measurable premise variables; however, if they depend on the unmeasured states of the system, then the model is known as an unmeasurable premise variables TS model [22]. The structure of the TS model allows using the design and analysis tools available for the linear theory, which is one of their advantages beyond accurate representation of the nonlinear model.

The design of UIO based on TS models has been investigated, mostly in the case of measurable premise variables [10, 16]. The unmeasurable premise variables case is less addressed, although it is the formalism that often leads to an exact representation of the nonlinear system [22]. We can, as a matter of citing works that had considered the unmeasurable premise variables TS model for UIO, point out the studies by Ichalal et al. [10]. Unlike the measurable decision variable case, observer design for unmeasurable decision variables TS systems is hard to address. The main developed approaches can be classified in the high-gain approaches and in sliding-mode techniques [2]. Recently, some other techniques exploiting the input-to-state stability (ISS) concept were introduced in [26, 27]. For vehicle applications, some works were recently dedicated to that purpose. One can cite Ref. [19] where the authors consider an approximation of the lateral forces by means of TS modeling with constant longitudinal velocity. The purpose of that work is the fault-tolerant control of the lateral dynamics. A similar problem was studied in Ref. [11] where the lateral forces are assumed to be linear with respect to the tire sideslip angles but with time-varying longitudinal velocity. For comparison, our article proposes a new approach that does not need any model of the lateral forces and the longitudinal velocity is considered time varying, which represent a more general case than those studied in the previously cited work. In the theoretical point of view, indeed, the structure of the observer is closely similar to the one proposed by Ichalal et al. [10]; however, the proof of the convergence is more rigorous in this new article with the use of the ISS concept.

The work presented in this article is an extension of the works undertaken in previous reports [26, 27]. We propose here a new representation of the vehicle nonlinear model subject to the road curvature and the lateral forces, such as unknown inputs via the TS formalism. Using a sector nonlinearity approach, this leads to an exact TS model in a large range of longitudinal velocity variations. Furthermore, the obtained model is valid in a compact state space domain defined by the range of the longitudinal velocity. A PMI observer is then designed on the basis of the new model representation. It allows a simultaneous estimation of the states of the system and the road curvature even if this last is time varying. Convergence conditions are expressed as an optimization problem under LMI constraints ensuring bounded error and a lower bound of the ISS gain. Simulations are performed after what experimental validation is provided to illustrate the developed design approach with a comparison to a conventional PI observer, in the case of constant and varying longitudinal velocity. The remainder of the article is organized as follows: the nonlinear vehicle model is described in Section 2. In Section 3, the TS formalism is introduced using the sector nonlinearity transformation. Section 4 describes the PMI observer design and the convergence conditions. Section 5 is dedicated to both simulation and experimental results before ending with a conclusion.

## 2 Description of the Nonlinear Model of the Vehicle

### 2.1 Lateral Dynamics of the Yaw Drift Vehicle Model

For describing the lateral dynamics of the vehicle, the two-state lateral drift and yaw motion model is the one commonly used [6]:

(1) { m v x ( t ) β ˙ ( t ) = F f ( t ) + F r ( t ) m v x ( t ) ψ ˙ ( t ) I z ψ ¨ ( t ) = a f F f ( t ) a r F r ( t ) ,  (1)

where β is the sideslip angle; ψ ˙ is the yaw rate; vx is the longitudinal velocity; m is the mass of the vehicle; Iz is the yaw moment of inertia; af and ar are the distances from the front and rear axle to the center of gravity, respectively; and Ff and Fr are the lateral front and rear forces, respectively. As exploited in the literature, these forces are commonly expressed by well-known and validated models such as Pacejka’s magic formula [20], Dugoff’s model [14], and the polynomial models (Rajamani’s model [21], etc.).

### 2.2 Positioning of the Vehicle Relative to the Track

The positioning on road of the vehicle is provided by the position of the front wheels. This is described by the differential equations related to the lateral deviation yL at a target distance ls from the center of gravity and the heading angle ψL of the vehicle (see Ref. [6]), the system is described in the Figure 1.

### Figure 1

Lateral Vehicle Dynamics and Positioning.

(2) { y ˙ L ( t ) = v x ( t ) ( β ( t ) + ψ L ( t ) ) + l s ( ψ ˙ ( t ) v x ( t ) ρ ( t ) ) ψ ˙ L ( t ) = ψ ˙ ( t ) v x ( t ) ρ ( t ) ,  (2)

where ρ is the road curvature. The global nonlinear model describing the vehicle related to the traffic lane and subject to the road curvature as an unknown input is given by the following expression, where the input of the system is the steering angle, implicitly included in the force expressions:

(3) { β ˙ ( t ) = 1 m v x ( F f ( t ) + F r ( t ) ) ψ ˙ ( t ) ψ ¨ ( t ) = 1 I z ( a f F f ( t ) a r F r ( t ) ) ψ ˙ L ( t ) = ψ ˙ ( t ) v x ( t ) ρ ( t ) y ˙ L ( t ) = v x ( t ) ( β ( t ) + ψ L ( t ) ) + l s ( ψ ˙ ( t ) v x ( t ) ρ ( t ) ) .  (3)

## 3 Observer Design

In this article, a new observer strategy is proposed for vehicle lateral dynamics estimation. The advantage of such an approach is that no force model is needed and the longitudinal velocity is considered time varying, which is more realistic than the observers based on a fixed velocity. Concerning the lateral forces, in most published works in that field of research, the authors assume that these forces obey linear or nonlinear models depending on some parameters related to the road and the tire characteristics. The proposed approach aims to estimate the body sideslip angle β, the road curvature ρ, the lateral forces Fyf and Fyr, the tire sideslip angles αf and αr, and the lateral velocity vy, without the knowledge of any force model that renders the observer insensitive to the wheel and road characteristics (friction coefficient, tire stiffness, etc.). The proposed observer is based on the notion of PMI, which takes into account the time varying of the unknown inputs. Indeed, using this type of observers aims to estimate simultaneously the state and the time-varying unknown inputs by estimating, in addition, the successive time derivatives of the unknown inputs, which considerably enhance the quality of estimation compared with the classic PI observer. However, it is also possible to enhance the capabilities of the PI observer by assuming that the first time derivative of the unknown input is bounded instead of the restrictive assumption of zero first time derivative. Thus, it can be seen that the enhanced PI observer may have acceptable performances that tend to be similar to those of the PMI observer in some situations (e.g., with slowly time-varying unknown inputs). The second contribution of the article is the consideration of the longitudinal velocity as time varying. This complexity is handled by transforming the model describing the lateral dynamics into a polytopic model also known as the TS model. This allows taking into consideration a large range of vehicle velocity.

Let us recall the model of the lateral dynamics of the vehicle described by the following equations:

(4) { β ˙ ( t ) = 1 m v x ( t ) ( F f ( t ) + F r ( t ) ) ψ ˙ ψ ¨ ( t ) = 1 I z ( a f F f ( t ) a r F r ( t ) ) ψ ˙ L ( t ) = ψ ˙ ( t ) v x ( t ) ρ ( t ) y ˙ L ( t ) = v x ( t ) ( β ( t ) + ψ L ( t ) ) + l s ( ψ ˙ ( t ) v x ( t ) ρ ( t ) ) .  (4)

The available sensors required for the proposed observer are only the inertial unit providing the yaw rate ψ ˙ and a vision system providing the lateral offset yL and the angle ψL at a look-ahead distance ls. The output equation is then given by

(5) y ( t ) = C x ( t ) ,  (5)

where

(6) C = ( 0 1 0 0 0 0 1 0 0 0 0 1 ) .  (6)

The lateral model (4) is easily transformed in a matrix form as follows:

(7) { x ˙ ( t ) = A ( v x ( t ) ) x ( t ) + E ( v x ( t ) ) d ( t ) y ( t ) = C x ( t ) ,  (7)

where the matrices are longitudinal velocity dependent and described by

A ( v x ( t ) ) = ( 0 1 0 0 0 0 0 0 0 0 0 0 v x ( t ) l s v x ( t ) 0 ) , E ( v x ( t ) ) = ( 1 m v x ( t ) 1 m v x ( t ) 0 a f I z a r I z 0 0 0 v x ( t ) 0 0 v x ( t ) )

and

x ( t ) = ( β ψ ˙ ψ L y L ) , d ( t ) = ( F y f F y r ρ ) .

### 3.1 Observability Analysis

After an observability analysis, it appears that the state vector and the unknown input vector are observable. Indeed, by taking the three measurements ψ ˙ , ψL, and yL, it is possible to recover the unknown state β and the unknown inputs Fyf and Fyf. Let us go to the initial system (7) before polytopic transformation. From an algebraic point of view, the estimations are given by the following algebraic equations:

(8) { β ^ = 1 v x ( y ˙ L v x ψ L l s ψ ˙ L ) ρ ^ = 1 v x ( ψ ˙ ψ ˙ L ) ,  (8)

and the forces are derived by

(9) { F ^ y f = ( m v x ( t ) a f m v x ( t ) a f + a r ) β ^ ˙ ( t ) + ( m v x ( t ) a f m v x ( t ) a f + a r ) ψ ˙ ( t ) + I z a f + a r ψ ¨ ( t ) F ^ y r = a f m v x ( t ) a f + a r β ^ ˙ ( t ) + a f m v x ( t ) a f + a r ψ ˙ ( t ) I z a f + a r ψ ¨ ( t ) .  (9)

It is then clear that both the unknown state and the unknown inputs are observable from the measured outputs if the longitudinal velocity satisfies vx(t) ≠ 0. From this analysis, one can express the outputs and their time derivatives in the following compact form

(10) Y = Ψ ( X ) ,  (10)

where

Y = ( y ˙ L ψ L ψ ˙ L ψ ˙ ) , X = ( β ρ F y f F y L ) .

As discussed previously, the vector function Ψ ( X ) is invertible for all vx(t) ≠ 0 and

(11) X ^ = Ψ 1 ( Y ) ,  (11)

which confirms the observability of the states and the unknown inputs.

However, as the estimations are obtained by algebraic equation and by time derivatives of measured outputs (sliding-mode techniques, algebraic techniques, or LTV differentiators), there is no correction action on the estimation errors if the model’s parameters are uncertain. For example, if the mass m and the inertial moment Iz of the vehicle change with respect to the number of passengers inside the vehicle, the provided estimation may be severely affected by those uncertainties. Then, the objective of the proposed approach is to estimate the state and the unknown inputs robustly with respect to uncertain parameters and with time-varying longitudinal velocity.

### 3.2 Polytopic Modeling of the System

To transform the system into a polytopic one, the following assumptions are considered:

{ z 1 ( t ) = v x ( t ) z 2 ( t ) = 1 v x ( t ) ,

where

{ 0 < z 1 m i n z 1 ( t ) z 1 m a x z 2 m i n z 2 ( t ) z 2 m a x ,

where z 1 m i n is the lower bound of vx(t), which must be nonzero, and z 1 m a x is the upper bound of vx(t). Similarly, z 2 m i n is the lower bound of 1/vx(t), which corresponds exactly to 1 / z 1 m a x , and z 2 m a x is the upper bound of 1/vx(t), which corresponds to 1 / z 1 m i n . By defining the local weighting functions

(12) { F 1 0 ( v x ( t ) ) = z 1 ( t ) z 1 m i n z 1 m a x z 1 m i n , F 1 1 ( v x ( t ) ) = z 1 m a x z 1 ( t ) z 1 m a x z 1 m i n F 2 0 ( v x ( t ) ) = z 2 ( t ) z 2 m i n z 2 m a x z 2 m i n , F 2 1 ( v x ( t ) ) = z 2 m a x z 2 ( t ) z 2 m a x z 2 m i n  (12)

we define the global weighting functions

h 1 ( v x ( t ) ) = F 1 0 ( v x ( t ) ) F 2 0 ( v x ( t ) ) h 2 ( v x ( t ) ) = F 1 0 ( v x ( t ) ) F 2 1 ( v x ( t ) ) h 3 ( v x ( t ) ) = F 1 1 ( v x ( t ) ) F 2 0 ( v x ( t ) ) h 4 ( v x ( t ) ) = F 1 1 ( v x ( t ) ) F 2 1 ( v x ( t ) )

that satisfy the convex sum property

{ i = 1 4 h i ( v x ( t ) ) = 1 0 h i ( v x ( t ) ) 1, i = 1, , 4, t ,

the system (7) can be described exactly in the compact S = { v x m i n v x ( t ) v x m a x } by

(13) { x ˙ ( t ) = i = 1 4 h i ( v x ( t ) ) ( A i x ( t ) + E i d ( t ) ) y ( t ) = C x ( t ) .  (13)

## 4 Observer Construction

For the PMI observer design, the system (13) is transformed by choosing an augmented state containing the state vector, the unknown input vector d(t), and the successive (k + 1)th time derivatives of d(t).

To estimate simultaneously the state and the road curvature, the road curvature is assumed to have a polynomial form:

(14) d ( t ) = A 0 + A 1 t + A 2 t 2 + + A k t k + O ( k + 1 ) ,  (14)

where A k are constant unknown vectors and O ( k + 1 ) is the remaining term containing the high-order terms. Two cases can be considered, the first one assumes that the (k + 1)th derivative of d(t) is zero (i.e., O ( k + 1 ) = 0 ), and the second is when the (k + 1)th derivative is not zero but bounded (i.e., O ( k + 1 ) 0 , but O ( k + 1 ) < ξ , where ξ > 0 is a positive scalar).

Under the consideration that d(t) takes the polynomial form, the system (13) becomes

(15) x ¯ ˙ ( t ) = i = 1 4 h i ( v x ( t ) ) ( A ¯ i x ¯ + D ¯ ω ( t ) ) y ( t ) = C ¯ x ¯ ,  (15)

where

A ¯ i = ( A i E i 0 0 0 0 I 3 0 0 0 0 I 3 0 0 0 0 ) , D ¯ = ( 0 0 0 I 3 ) , x ¯ = ( x d d 1 d k ) C ¯ = ( C 0 0 0 )

and d1(t), …, dk(t) are the successive time derivatives of d(t) and ω ( t ) = O ( k + 1 ) .

After verification, all the pairs ( A ¯ i , C ¯ ) are observable for all 0 < v x m i n v x ( t ) v x m a x . Furthermore, the observability analysis of the nonlinear system is also observable for all vx(t) > 0, which ensure the existence of an observer.

Let us consider the following observer

(16) { x ¯ ^ ˙ ( t ) = i = 1 4 h i ( v x ( t ) ) ( A ¯ i x ¯ ^ ( t ) + L ¯ i ( y ( t ) y ^ ( t ) ) ) y ^ ( t ) = C ¯ x ¯ ^ ( t ) ,  (16)

where the gains L ¯ i are described by the following matrix

L ¯ i = ( L P i L I i 1 L I i k 1 L I i k ) .

The state estimation error given by e ( t ) = x ¯ ( t ) x ¯ ^ ( t ) obeys the following differential equation:

(17) e ˙ ( t ) = Φ h ( t ) e ( t ) + D ¯ ω ( t ) ,  (17)

where

(18) { Φ h ( t ) = i = 1 4 h i ( v x ( t ) ) Φ i Φ i = A ¯ i L ¯ i C ¯ .  (18)

Now, the objective is to compute the matrices L ¯ i in such a way to stabilize the system (17) and to minimize the effect of the higher time derivatives of the unknown input vector described by ω(t). Note that such a strategy aims to reduce the number of time derivatives of d(t) considered for the observation problem of time-varying unknown inputs and have a minimal transfer of the higher-order bounded time derivatives of the unknown inputs.

### 4.1 Convergence Analysis

Let us consider the Lyapunov candidate function

(19) V ( e ( t ) ) = e T ( t ) P e ( t ), P = P T > 0  (19)

satisfying

(20) α 1 e ( t ) 2 V ( e ( t ) ) α 2 e ( t ) 2 ,  (20)

where 0 < α1 < α2. Its time derivative along the trajectory of the error (17) is given by

(21) V ˙ = e T ( t ) ( Φ h T P + P Φ h ) e ( t ) + 2 e T ( t ) P D ω ( t ) .  (21)

By adding and subtracting the term –αeT(t)Pe(t) + γωT(t)ω(t), eq. (21) is equivalent to

(22) V ˙ = e T ( t ) ( Φ h T P + P Φ h ) e ( t ) + 2 e T ( t ) P D ¯ ω ( t ) + α e T ( t ) P e ( t ) γ ω T ( t ) ω ( t ) α e T ( t ) P e ( t ) + γ ω T ( t ) ω ( t ) ,  (22)

where α and γ are positive scalars. In matrix form, we have

(23) V ˙ = ζ T ( t ) Ξ h ζ ( t ) α e T ( t ) P e ( t ) + γ ω T ( t ) ω ( t ) ,  (23)

where

(24) Ξ h = ( Φ h T P + P Φ h + α P P D ¯ D ¯ T P γ I ) , ζ ( t ) = ( e ( t ) ω ( t ) ) .  (24)

If the gains L ¯ i and the parameters α and γ are chosen in such a way to have Ξh < 0, the time derivative of the Lyapunov function may be bounded as follows:

(25) V ˙ < α V + γ ω T ( t ) ω ( t ) .  (25)

The solution of this differential inequality is

(26) V < V ( 0 ) e α t + γ 0 t e α ( t τ ) ω ( τ ) 2 2 d τ .  (26)

By using the definition of the Lyapunov function, one obtains

(27) e ( t ) 2 2 < α 2 α 1 e ( 0 ) 2 2 e α t + γ α 1 α ω ( t ) 2 ,  (27)

and finally, by using the square root function, it follows

(28) e ( t ) 2 < α 2 α 1 e ( 0 ) 2 e α 2 t + γ α 1 α ω ( t ) ,  (28)

which confirms that the state and unknown input estimation error is ISS with respect to the bounded perturbation ω(t). In addition, when t goes to ∞, the estimation error is ultimately bounded by the quantity γ α 1 α ω ( t ) . The ISS gain is given by the term γ α 1 α . In our vehicle application and in all observation problems, the boundedness of the estimation error is often not sufficient to have an accurate estimation. For that reason, the second objective is to minimize the ISS gain. Before doing this, LMI conditions that ensure ISS will be developed.

The ISS property is ensured if Ξh < 0, which is given by

(29) ( Φ h T P + P Φ h + α P P D ¯ D ¯ T P γ I ) < 0.  (29)

By using the convex sum property, sufficient conditions that ensure the negativity of Ξh are given by

(30) ( Φ i T P + P Φ i + α P P D ¯ D ¯ T P γ I ) < 0, i = 1, , 4.  (30)

Finally, by using the change of variables M i = P L ¯ i , LMI conditions are obtained with respect to the variables P and Mi

(31) ( A ¯ i T P + P A ¯ i M i C ¯ C ¯ T M i T + α P P D ¯ D ¯ T P γ I ) < 0, i = 1, , 4.  (31)

Now, to compute the gains Mi that also minimize the ISS gain, let us assume that

(32) γ α 1 α η ,  (32)

where η is a positive scalar. Minimizing the ISS gain results from minimizing η. After calculation, one obtains

(33) γ η α 1 α 0.  (33)

Finally, the problem of observer design ensuring the stability of the estimation error and the minimization of the ISS gain is expressed as an optimization problem under LMI constraints as follows: given positive scalars α, α1, if there exist a symmetric and positive definite matrix P and gain matrices Mi and positive scalars η and γ solution to the following optimization problem

m i n P , M i , γ , η η P α 1 γ η α 1 α 0 ( A ¯ i T P + P A ¯ i M i C ¯ C ¯ T M i T + α P P D ¯ D ¯ T P γ I ) < 0 i = 1, , 4 ,

then the estimation error is ISS with minimal ISS gain. The gains of the observer are computed from L ¯ i = P 1 M i .

## 5 Model Reduction with an Alternative Model

Previously, the model used for estimating the lateral dynamics is transformed into a polytopic one with four submodels. It is known that, the use of quadratic Lyapunov functions for the stability analysis may introduce conservatism in the LMI conditions. In this section, a more general model will be used that reduces the number of submodels. Indeed, the first used model expresses the dynamics of the body sideslip angle β, which is a simplification of the relation β = tan–1(vy/vx)≈vy/vx for small angles and for small lateral velocity. Then, by omitting this assumption, the model (4) is expressed with respect to the lateral velocity instead of β, and the model becomes

(34) { v ˙ y = 1 m ( F f + F r ) v x ψ ˙ ψ ¨ = 1 I z ( a f F f a r F r ) ψ ˙ L = ψ ˙ v x ρ y ˙ L = v y + v x ψ L + l s ( ψ ˙ v x ρ ) .  (34)

Using this model is more interesting because the number of submodels is reduced to only two instead of four owing to the disappearance of the term 1/vx. The polytopic model is then described by the equations

(35) { x ˙ ( t ) = i = 1 2 h i ( v x ( t ) ) ( A i x ( t ) + E i d ( t ) ) y ( t ) = C x ( t ) ,  (35)

where

x ( t ) = ( v y ψ ˙ ψ L y L ) , A 1 = ( 0 z 1 m a x 0 0 0 0 0 0 0 1 0 0 1 l s z 1 m a x 0 ) , E 1 = ( 1 m 1 m 0 a f I z a r I z 0 0 0 z 1 m a x 0 0 z 1 m a x ) ,

A 2 = ( 0 z 1 m i n 0 0 0 0 0 0 0 1 0 0 1 l s z 1 m i n 0 ) , E 2 = ( 1 m 1 m 0 a f I z a r I z 0 0 0 z 1 m i n 0 0 z 1 m i n ) ,

and z1(t) = vx(t), z 1 m a x = v x m a x , and z 1 m i n = v x m i n .

The design procedure follows the same one as in the model presented in the previous section. However, as this technique is LMI based, the conditions may be conservative owing to the scale difference between the magnitudes of the states and the after augmenting the system state with the unknown inputs. Indeed, the forces may take large values from zero to around 5 × 103, while the lateral velocity is very small compared with these forces. To rescale the system, it is possible to introduce constant parameters a and b in the matrix E and rescale the unknown inputs as follows

(36) x ˙ ( t ) = ( 0 v x ( t ) 0 0 0 0 0 0 0 1 0 0 1 l s v x ( t ) 0 ) x + ( a m b m 0 a a f I z b a r I z 0 0 0 v x ( t ) 0 0 v x ( t ) ) ( F y f a F y r b ρ ) .  (36)

Note that the two models are equivalent, and the parameters a and b are scaling factors fixed with respect to the magnitude of the forces (e.g., one can chose a = b = 5000).

Remark 1. Note that the important advantages of the PMI observers compared with the conventional PI observers is, first, the possibility to estimate time-varying unknown inputs and, second, the possibility to estimate unbounded unknown inputs. This second important point can be highlighted as follows. Let us consider the unknown input expressed by d(t) = a0 + a1t + a2t2: a PI observer provides unsatisfactory results and the enhanced PI observer is not possible as the first time derivative of d(t) is not bounded; however, if we consider the P2I observer, it is possible to obtain a correct estimation of d(t) because the second derivative of d(t) is bounded by 2a2, which is considered a perturbation to be minimized. Finally, if a P3I observer is considered, a perfect estimation of d(t) (exponential convergence) is obtained because the third time derivative of d(t) is zero and the optimization problem given in the previous section will be reduced to only LMI conditions:

A ¯ i T P + P A ¯ i M i C ¯ C ¯ T M i T + α P < 0, i = 1, 2 ,

which leads to the exponential convergence of the state and unknown inputs errors to zero

(37) e ( t ) 2 < α 2 α 1 e ( 0 ) 2 e α 2 t .  (37)

## 6 Simulation Results, Experimental Validation, and Discussions

This section is divided into two parts: the first one addresses the simulation results and some discussions related with the theoretical developments. The objective is to compare the classic PI observer with both the enhanced one by minimizing the effect of the first time derivatives of the unknown inputs and the PMI observer by considering two integrals in the observer (P2I). The second part of this section illustrates the PMI observer with two integrals (P2I) with real data obtained from sensors installed on the vehicle.

### 6.1 Simulation Results and Analysis

First, let us consider the nonlinear vehicle model simulated with lateral forces modeled by Pacejka’s model. The steering angle is taken as a sine function δf(t) = 0.15 sin(t), the longitudinal velocity is given by vx(t) = 10 + sin(0.3t), and the road curvature is ρ(t) = 0.01sin(t). The system is simulated with the initial conditions x(0) = [0.5 0.5 0.5 0.5]T. A comparison is made between the conventional PI observer where the first time derivative is considered zero for theoretical developments and stability analysis, and the enhanced PI observer where the first time derivative is assumed to be bounded and minimized with respect to the state and unknown input estimation errors (ISS gain minimization). After solving the LMI conditions for the PI observer and the optimization problem under LMI constraints (with α1 = 0.0001 and α = 1) for the enhanced PI observer and by choosing the initial conditions of the observers as x ¯ ^ ( 0 ) = zeros ( 1, 7 ) , the estimation errors are depicted in Figure 2. One can see clearly that the enhanced PI observer provides more accurate estimations of the state vy and the unknown inputs Fyf, Fyf, and ρ.

### Figure 2

Comparison between Conventional PI Observer and Enhanced PI Observer Errors.

Now, consider the same scenario with the enhanced PI observer and a P2I observer, this latter is a PMI observer with two integrations, where the second time derivatives of the unknown inputs are assumed to be bounded. For comparison, the ISS gains of the two observers are fixed to η = 0.1 and the parameters α1 = 0.0001 and α = 1, the errors illustrated in Figure 3, show that the P2I observer provides more accurate estimations than the enhanced PI observer. In Figure 4, the actual state and unknown inputs are depicted with estimated versions provided by the P2I observer.

### Figure 3

Comparison between State Estimation Error and Unknown Input Errors with Enhanced PI Observer and P2I Observer with Fixed ISS Gain to η = 0.1.

### Figure 4

Actual State and Unknown Inputs vs. Estimated Ones.

Note that the oscillatory phenomenon at the transient phase is due to the high imaginary parts of the eigenvalues of the matrices ( A ¯ i L ¯ i C ¯ ) due to the ISS gain minimization. This can be reduced by a simple pole placement in an LMI region where the imaginary parts of the poles are limited (for more details, the reader can refer to an article [4] and a book [3] on LMIs in control). Note also that for α1 = 0.0001 and α = 1, the ISS gain minimization for the enhanced PI observer leads to the bound η = 0.0011 and that of the P2I observer is η = 0.0145. However, the obtained gains are very high and may cause noise amplification in a practical situation; this is why in the above simulations, the minimal ISS gain is limited to η≥0.1.

### 6.2 Simulations with Experimental Data and Validation

In the previous subsection, the P2I observer is designed and the parameters are tuned to have suitable performances. This observer is then tested with real data obtained from a vehicle equipped with required sensors. As explained previously, the minimization of the ISS bound may lead to high gains that amplify the measurement noises. Furthermore, the high imaginary parts of the poles of the observer matrices cause a high oscillatory phenomenon. To overcome these problems in practical situations, an LMI region is defined by a disk centered at the (0, 0) with radius R = 70. The LMIs related to this region are also solved simultaneously with the observer design strategy. Then, solving the optimization problem under both LMI constraints in the defined LMI region provides the results illustrated in Figures 5 and 6. In Figure 5, the state and unknown input error estimations are compared with the ones obtained with the enhanced PI observer designed also in the same LMI region. In Figure 6, it can be seen that the estimation of the unknown inputs are correct. However, the estimation of the lateral velocity is perturbed. This is due to its estimation from the vision system that provides perturbed measures of the lateral displacement yL and the relative sideslip angle ψL. Note that if no model of the forces is considered, then the lateral velocity is not observable without the vision system.

### Figure 5

Comparison PI and P2I errors.

### Figure 6

State Estimation with P2I (Real Data).

## 7 Conclusion

In this article, we proposed an observer for vehicle lateral dynamics estimation. The developed observer is based on a TS model of the vehicle dynamics with the model of the positioning with respect to the road. This last aims to recover the observability of the lateral velocity without assuming any model of the lateral forces. Second, the longitudinal velocity is assumed to be time varying, which is more realistic than the approaches using the assumption of constant velocity. Assuming that the lateral forces and the road curvature are unknown inputs, a PMI observer is proposed to estimate accurately the time-varying unknown inputs. The convergence of the observer is studied according to the Lyapunov theory and the ISS concept. It is then established whether the state and unknown input estimation errors are bounded in the sense of ISS. For more accurate estimation, the ISS gain is minimized. The design technique of the proposed observer is the transformation of the conditions into an optimization problem subject to LMI constraints that ensure ISS stability with minimal ISS gain. Simulations are provided to illustrate the performances of the proposed PMI observer with comparison to the conventional PI and the enhanced PI observer. Some experimental results are also given for illustrating the performances of the observer. For future works, the PMI observer designed will be integrated into a control scheme to reconstruct the unmeasurable vehicle data to perform control strategies, such as state feedback control for yaw stability, in the aim to synthesize driving assistance. The sensitivity of the PMI observer is presently under study.

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