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Abstract— This brief describes an observer-based state feedback tracking controller for vehicle dynamics with a four-wheel active steering system as well as an ...
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Observer-Based State Feedback Control for Vehicle Chassis Stability in Critical Situations H. Dahmani, O. Pagès, and A. El Hajjaji

Abstract— This brief describes an observer-based state feedback tracking controller for vehicle dynamics with a four-wheel active steering system as well as an active suspension system. The objective of the proposed controller is to improve the vehicle behavior by forcing the lateral dynamics and the load transfer ratio to achieve the desired vehicle behavior in critical situations. A Takagi–Sugeno (TS) representation of the lateral forces has been used in order to take the nonlinearities into account. Based on the obtained fuzzy model, a TS observer has been designed with estimated membership functions in order to consider the unavailability of the sideslip and the roll angles for measurement. Based on the Lyapunov function and the H∞ approach, the observer and controller design has been formulated in terms of Linear Matrix Inequality constraints. The proposed techniques have been evaluated through a fishhook test conducted in the CarSim professional software package. Index Terms— Estimation, H∞ , state feedback control, Takagi–Sugeno (TS) fuzzy model, tracking control, vehicle dynamics.

I. I NTRODUCTION

D

RIVER assistance systems are classified into two categories: active and passive safety systems. Active safety systems are designed to prevent or mitigate the vehicle instability risks and avoid accidents. Passive safety systems are designed to prevent or mitigate the negative consequences after an accident has occurred. Restraint systems such as seat belts and airbags are examples of passive safety systems which have been developed in early years. In recent years, many active driver assistance systems have been developed: Electronic Stability Program, Adaptive Cruise Control, Forward Collision Warning System, Active Rollover Protection, and many other systems are already available in production vehicles. Active assistance systems are generally based on the knowledge of the vehicle dynamics and the design of controllers and programs capable of improving the behavior of the vehicle in critical situations. Many researchers are interested in this topic [1]–[3]. For example, the influence of the front wheel steering angle as well as the velocity on the vehicle lateral dynamics has been presented in [4] with a focus on stability and bifurcation phenomena. In [5], a robust controller has been designed to improve the yaw stability. Emergency lateral

Manuscript received June 19, 2014; revised November 13, 2014 and March 9, 2015; accepted May 17, 2015. Manuscript received in final form May 19, 2015. This work was supported in part by the Conseil Règional de Picardie and in part by the European Regional Development Fund within the ACADIE Project. Recommended by Associate Editor U. Christen. The authors are with the Motion Imagery Standards Laboratory, University of Picardie Jules Verne, Amiens 80039, France (e-mail: hamid.dahmani@ u-picardie.fr; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2015.2438191

control has been proposed in [6] using the linear matrix inequality (LMI) approach. An integrated electronic stability control system wherein the objectives of yaw stability and rollover prevention are simultaneously addressed is presented in [7]. The authors explore how the use of steer-by-wire technology can address the tradeoff between yaw stability, speed, and rollover prevention performance. To improve the vehicle stability, an adaptive steering-control system for a steer-by-wire system, which consists of a vehicle directionalcontrol unit and a driver-interaction unit, is proposed in [8]. In this brief, a robust control method is proposed to ensure the vehicle stability and to avoid skid and rollover in critical driving situations by operating on the front and rear wheel-axle as well as on an active suspension system. The main objective of the proposed approach is to design a robust controller for the tracking of reference trajectories. The three degrees of freedom model including lateral and roll dynamics is approximated by a Takagi–Sugeno (TS) fuzzy model which is widely applied to solve control and estimation problems for a large class of nonlinear systems [9]–[11]. This brief gives the latest progress in our study already published in [12]. An observer is designed to estimate the sideslip and roll angle of the vehicle by considering the yaw rate and roll rate measurement. This problem has already been addressed in some papers in which the authors do not take into account the influence of the sideslip angle estimation on the membership functions of the TS model [13], [14]. A fuzzy controller is also designed with the main objective of forcing the yaw rate, the sideslip angle, and the load transfer ratio (LTR) to track the desired values. The proposed control system design is represented in LMI terms which can be solved efficiently using the existing LMI solvers. In order to illustrate the effectiveness of the developed techniques, a fishhook test which is often used for vehicle stability testing [15], has been performed using the CarSim professional software package [16]. This brief is organized as follows. The description of the nonlinear lateral and roll dynamics model with its TS approximation is presented in Section II. The fuzzy observer and controller design procedure will be presented in Section III. Section IV gives the numerical results and illustrates the conducted fishhook test in the CarSim simulator. Finally, some conclusions are given in Section V. II. V EHICLE M ODEL A NALYSIS A. Nonlinear Model Analysis The model used in this brief describes the vehicle lateral and roll dynamics, which are obtained by considering the well-known single-track (bicycle) model with added degree of freedom for rolling (Fig. 1). With the assumption of

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Fig. 2.

Tire slip angle regions of the fuzzy model.

B. TS Model Description Fig. 1.

Definition of vehicle lateral variables.

small angles and a constant vehicle speed, the resulting three degrees of freedom model can be described by the following differential equations [17]: ⎧ ⎪ ⎨m s a ys + m u a yu = 2Fy f + 2Fyr (1) Iz ψ¨ = 2Fy f l f − 2Fyr lr ⎪ ⎩ ¨ Ix φv = m s ghφv + m s a ys h − kφ φv − Cφ φ˙v + Mx where a ys and a yu are, respectively, the sprung and unsprung mass lateral accelerations given by ˙ − h φ¨v , a yu = v(β˙ + ψ). ˙ a ys = v(β˙ + ψ)

(2)

In order to improve the model precision, the sprung and unsprung masses of the vehicle are considered separately. This separation is important in the estimation of the roll dynamics which are caused only by the sprung mass. Most of the vehicle dynamic linear models used consider the sprung mass of the vehicle as the total mass and ignore the unsprung mass. However, this assumption can introduce an imprecision in the roll estimation, since the unsprung mass is approximately 10% of the vehicle mass [18]. In (1) β, ψ, and φv are the sideslip, the yaw, and the roll angle of the vehicle, respectively. Mx is the active roll torque, F f and Fr are, respectively, the cornering forces of the two front and rear tires. For further description of the parameters appearing in the vehicle dynamics model, refer to Table I and Fig. 1. Several formulations are proposed to define these tire forces (Pacejka, Kienke, Dugoff, etc.) [19]. They are given as a function of the tire slip angles using nonlinear relationships; however, these models are hardly usable in model-based control, since they lead to very complex models. The lateral forces are assumed to be proportional to the slip angle in some research works [20], but this assumption leads to a linear model which is valid only when the slip angles are very small. However, for large slip angles, a nonlinear model must be considered. To overcome this problem, we propose approximation of the nonlinear behavior of the cornering forces by a fuzzy TS model with four rules which describes all of the operating regions.

Let us consider fuzzy sets M f i and Mri (i = 1, 2) defined for the two tire slip angle regions, i.e., M f 1 and Mr1 are the fuzzy symbols for the low front and rear tire slip angle regions, respectively, and M f 2 and Mr2 for the high front and rear tire slip angle regions, respectively (Fig. 2). The front and rear cornering forces may be described by the following TS fuzzy rules: If |α f | is M f 1 then Fy f = C f 1 α f If |α f | is M f 2 then Fy f = C f 2 α f If |αr | is Mr1 then Fyr = Cr1 αr If |αr | is Mr2 then Fyr = Cr2 αr

(3)

where C f i and Cri are the front and rear tire cornering stiffnesses, respectively, which depend on the road friction coefficient and the vehicle parameters. Variables α f and αr are the slip angles of the front and rear tires (Fig. 1) which are given by the following equations: αf = δf −

l f ψ˙ lr ψ˙ − β, αr = δr + − β. v v

(4)

Using the above described fuzzy rules, the overall cornering forces are written as follows:  Fy f = μ f 1 (α f )C f 1 α f + μ f 2 (α f )C f 2 α f (5) Fyr = μr1 (αr )Cr1 αr + μr2 (αr )Cr2 αr where μ f i and μri (i = 1, 2) are bell curve membership functions associated with symbols M f i and Mri given by 1 1 μfi =     2bf i , μri =   2bri .  |α |−c   ri  1 +  fa f i f i  1 +  |αra|−c  ri (6) Parameters of membership functions (ai , bi , and ci ) and the stiffness coefficient values are obtained using an identification method based on the Levenberg–Marquardt algorithm combined with the least square method [21]. Fig. 3 gives the cornering force estimation using the TS model and linear model compared with CarSim, for a fishhook maneuver (see simulation results).

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TABLE I N OMINAL VALUES OF THE V EHICLE PARAMETERS

Fig. 4.

Fig. 3. Cornering force estimation using TS model and linear model compared with CarSim.

Using (3) to approximate the lateral cornering forces, the TS model is obtained in the following form: x(t) ˙ =

4

h i (α)[ Ai x(t) + B f i δ f + Bri δr ] + Bm Mx

(7)

i=1

where x(t) = [ β ψ˙ φ˙v φv ]T is the state vector and membership functions h i (α) are given as follows: ωi (α)

h i (α) = 4

i=1

ωi (α)

where ω1 = μ f 1 μr1 , ω2 = μ f 1 μr2 , ω3 = μ f 2 μr1 , and ω4 = μ f 2 μr2 . Matrices Ai , B f i , Bri (i = 1, . . . , 4) and Bm are given in the Appendix. III. C ONTROL S TRATEGY The main goal of the control strategy is basically to ensure the tracking of a reference model which corresponds to the ideal behavior of the lateral and roll dynamics. The controlled

Observer-based control system.

vehicle is equipped with front and rear wheel active steering as well as an active suspension system (Fig. 4). Even if active suspension was initially designed for better passenger comfort, the suspension controller can have an important impact on load distribution over the four wheels which can be used for rollover control. Compared with the existing methods, the main advantage of the presented approach is the use of the front and rear active steering and the active suspension simultaneously in order to stabilize the whole vehicle chassis in critical situations. A second advantage is the use of integral action in the control law and the use of an H∞ approach which provides robustness and allows the tracking of a desired trajectory even if the reference model is unknown. Considering that the front steering is given by both the driver and controller actions, TS model (7) can be rewritten as follows: x(t) ˙ =

4

h i (α)[ Ai x(t) + B f i (δd + δc ) + Bri δr + Bm Mx ]

i=1

⇔ x(t) ˙ =

4

i=1

h i (α)[ Ai x(t) + Bi u(t) + B f i δd (t)] (8)

where u(t) = [ δc (t) δr (t) Mx (t) ]T and Bi = [ B f i Bri Bm ]δd (t) is the front wheel steering angle due to the driver action, δc (t) and δr (t) are, respectively, the front and rear active steering, and Mx is the active roll torque. The main idea of the control system is actually to force the yaw rate, the sideslip angle, and the LTR to track desired trajectories z ref = [βref , ψ˙ ref , LTRref ] obtained from a reference model which represents the ideal response of the vehicle according to the driver maneuvers (Fig. 4). The LTR gives the load transfer ratio which can be considered as a rollover risk indicator. In order to avoid rollover crashes, the

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LTR must be kept near zero and less than one. This ratio can simply be defined as the difference between the normal forces on the right and left hand sides of the vehicle divided by their sum, and it can also be approximated using roll dynamics with the following equation [22]: 2 Fzl − Fzr (Cφ φ˙v + kφ φv ).  (9) LTR = Fzl + Fzr mgT

with

A. Observer-Based H∞ Controller Design

In order to simplify the stability conditions of the closed-loop i j and  Bwi can be rewritten as follows: system, matrices A



B1wi A i − B i K j S + Ri K P j   , Bwi = Ai j = B2w 0 Ai − L i C

Knowledge of the vehicle states is necessary for the fuzzy state feedback controller design. Since the sideslip and roll angle sensors are very expensive, we propose their estimation by designing an observer using only measurable variables such as the steering angle, the yaw rate, and the roll angle rate. Moreover, from the TS model of the vehicle lateral dynamics (8), we remark that functions h i (α) depend on variables α f and αr , which depend on sideslip angle β. ˆ = h i (αˆ f , αˆ r ) For this reason, let us denote by h i (α) the membership functions calculated from the estimated slip angles. Based on the structure of TS model (8), the following fuzzy state observer is proposed: ⎧ 4 ⎪ ⎨˙

h i (α)[ ˆ Ai xˆ + Bi u + B f i δd + L i (y − yˆ )] xˆ = ⎪ ⎩

i=1

y = C x,

yˆ = C xˆ



0 C= 0

1 0

0 1

0 0

(10)

where y = [ψ˙ φ˙v ]T is the measured output and matrices L i are observer gains to be determined. The proposed fuzzy controller is represented by ⎧

 t 4 ⎪ ⎨u(t) = − h (α) ˆ K x ˆ + K (ˆ z (τ ) − z (τ ))dτ j Pj Ij ref 0 j =1 ⎪ ⎩ zˆ = C z xˆ z = C z x, ⎤ ⎡ 1 0 0 0 ⎢0 1 0 0 ⎥ ⎥ (11) Cz = ⎢ ⎦ ⎣ 2 2 0 0 Cφ kφ mgT mgT z = [β ψ˙ LTR]T is the controlled output. Let us remark that in addition to the state feedback control, an integral action has been added to force the tracking of reference vector z ref . Let us denote the tracking error dynamic by e˙c = zˆ − z ref and denote the estimation error by eo = x − x. ˆ From (8) and (10), the estimation error dynamics are given by e˙o =

4

h i (α)[(A ˆ i − L i C)eo + υ]

(12)

i=1

with υ = 4i=1 (h i (α) − h i (α))[ ˆ Ai x + Bi u + B f i δd ]. From (11), the augmented system with  x (t) = [ x ec eo ]T can be written as follows:  x˙ (t) =

4

4

i=1 j =1

  i j x(t) h i (α)h ˆ j (α) ˆ A ˜ + Bwi W (t)

(13)

where



⎤ Bi K P j Ai − Bi K P j −Bi K I j i j = ⎣ Cz 0 −C z ⎦ A 0 0 Ai − L i C ⎡ ⎤ ⎡ ⎤ Bfi 0 I δd (t)  −I 0 ⎦, W (t) = ⎣ z ref (t) ⎦. Bwi = ⎣ 0 0 0 I υ(t)



Ai 0 Bi Bi 0 , Ri = , Bi = , S= Cz 0 0 −C z 0

Bfi 0 I = , B2w = [ 0 0 I ] and 0 −I 0

Ai = B1wi

K j = [ K P j K I j ]. Let us consider the H∞ performance related to the tracking error ec (t) and the estimation error eo (t) as follows:  ∞   ec (t)T Q 1 ec (t) + eoT (t)Q 2 eo (t) dt 0  ∞ W (t)T W (t)dt. ≤ ρ2 0

This H∞ performance criterion can be rewritten, if the initial state x(0) ˜ is considered, in the following form:  ∞  ∞ T 2 ˜ ˜ x˜ (t) Q x(t)dt ˜ ≤ x(0) ˜ P x(0) ˜ +ρ W (t)T W (t)dt 0

0

(14) with



0 Q˜ = ⎣ 0 0

0 Q1 0

⎤ 0 0 ⎦ Q2

and P˜ is a symmetric positive definite matrix. The fuzzy closed-loop system (13) is globally stable and the H∞ control performance in (14) is guaranteed for a prescribed attenuation level ρ 2 , if there exists a common symmetric positive definite matrix P˜ solution of the following inequalities [23]: T ˜ i j + A iTj P˜ + 1 P˜  P˜ A Bwi  Bwi P + Q˜ < 0, i, j = 1, . . . , 4. ρ2 (15)

The most important task of the observer-based tracking control design is how to find the common solution P˜ = P˜ T > 0 from the inequalities (15). It is not easy to obtain the observer and controller gains directly for (15) since the problem is not convex. The following section presents the matrix inequality conditions in order to obtain the observer and fuzzy controller gains.

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Given symmetric positive definite matrices Q 1 and Q 2 . If there exist common symmetric positive definite matrices X, P2 , matrices V j , Wi , K i and scalars ρ > 0, solution of the following inequalities: ⎡ ⎤ B1wi i j S + Ri K P j ⎣ ∗ i j P2 B2w⎦ < 0, i, j = 1, . . . , 4 ∗ ∗ −ρ 2 I (16) T

T

i j = Ai X − B i V j + X Ai − V jT B i + XC1T Q 1 C1 X i j = P2 Ai − Wi C + AiT P2 − C T WiT + Q 2 V j = K j X, Wi = P2 L i



0 0 Q1 0 C1 = , and Q 1 = 0 I 0 Q1 where ∗ indicates symmetric blocks. Then the fuzzy closedloop system (13) is globally asymptotically stable and the H∞ control performance in (14) is guaranteed for a prescribed attenuation level ρ 2 [see the Appendix for further details of how to obtain inequality conditions (16)]. The above-described matrix inequalities can be solved by the following two-step procedure. In the first step, note that (16) implies that ⎤ ⎡ T T Ai X + B i V j + X Ai + V jT B i ⎢ 1 XC1T ⎥ ⎥ ⎢ T ⎥ < 0. (17) ⎢ + 2 B1wi B1wi ⎦ ⎣ ρ −1 C1 X −Q 1

Fig. 5.

Steering angles of the driver and the controller in the fishhook test.

Fig. 6.

Active roll torque delivered by the controller.

Parameters X and V j (thus K j = V j X −1 ) can be obtained by solving (17). In the second step, by substituting K j = [K P j K I j ] and X, (16) becomes a standard LMI. Parameters P2 and Wi (thus L i = Wi P2 ) can then be obtained by solving (16). Remark: The above results have been developed so that they can be easily extended to the case where there are uncertainties in the model. IV. S IMULATION R ESULTS The observer-based state feedback controller proposed in this brief has been implemented in the CarSim professional software package. This simulator provides a complex and realistic behavior of different vehicle classes and animates simulated tests and outputs. In our case, a fishhook test, where a driver first steers to the left and then to the right into a circle, has been conducted with a constant vehicle speed of 50 km/h (Fig. 5) and the vehicle parameters given in Table I. A highcenter-of-gravity vehicle has been equipped with the developed controller and then compared in this test with the same vehicle without the controller. In order to generate the reference model, a low-center-of-gravity vehicle, which is more stable in critical situations, has been used. The input steering angle used in the fishhook test is shown in Fig. 5. This figure also shows the front and rear active steering angle delivered by the controller. Parameters of the vehicle model are obtained for the CarSim vehicle tested in a critical situation described by the fishhook test with the road adhesion coefficient μ = 0.7. Observer and

Fig. 7. Comparison of the controlled, uncontrolled, and reference vehicle trajectories in the fishhook test.

controller gains have been obtained by solving (17) and (16) using MATLAB LMI Control Toolbox, with Q 1 = Q 2 = I and ρmin = 0.24. Desired values of the sideslip angle, the yaw rate, and the LTR which correspond to the ideal behavior of the vehicle in this specific situation have been obtained from the lowcenter-of-gravity vehicle in CarSim. To force the behavior of the vehicle to follow these three references, three actions calculated by the controller are necessary. The active roll torque is shown in Fig. 6. Fig. 7 shows a comparison of the controlled, uncontrolled, and reference vehicle trajectories in the fishhook test. Estimation results of the TS observer compared with the actual vehicle state given from CarSim and a linear observer are shown in Fig. 8. These results clearly show the effectiveness of the TS observer which provides a good estimation of the lateral and roll vehicle dynamics,

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Fig. 10. Comparison of the LTR for the controlled and uncontrolled vehicles.

Fig. 8.

Comparison of the vehicle sideslip and roll angle estimation.

Fig. 11.

Driver steering angles in the different maneuvers at 70 km/h.

Fig. 12. Comparison of the controlled, uncontrolled, and reference vehicle trajectories for maneuver 1.

Fig. 9.

Comparison of the vehicle lateral dynamics.

except at the time period [2 s, 2.2 s]. This is due to a sudden maneuver in the driver steering which introduces a big load transfer. However, the observer shows good performance with fast convergence to the actual values of the vehicle states. The simulation results in Fig. 9 show the sideslip angle and the yaw rate of the controlled and uncontrolled vehicles

compared with the lateral dynamics of the reference vehicle. The controlled vehicle’s state successfully follows the reference, while the uncontrolled vehicle is completely unstable at t = 2.4 s. In the same comparison for the roll dynamics, the instability of the uncontrolled vehicle can be clearly seen in Fig. 10 which shows the LTR calculated for the vehicle with and without controller compared with the reference model. It shows that the LTR, which represents a rollover risk indicator, is larger in the case of the vehicle without controller, while the controlled vehicle LTR is very close to the desired reference. It should be remembered that LTR = 1 means the beginning of rollover. Without the three control actions, the vehicle is unstable and cannot finish the

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matrix inequality constraints can be solved in two steps. The resulting controller can globally stabilize the closed-loop TS fuzzy system with the minimization of an H∞ performance criterion in order to satisfy the tracking performances. The proposed observer-based controller has been tested using the CarSim software package using a fishhook test. The simulation results show good performance of the developed approach in estimating vehicle dynamics and improving vehicle stability in critical situations. In our future work, we plan to study the feasibility of an experimental test using an instrumented vehicle. Fig. 13. Comparison of the controlled, uncontrolled, and reference vehicle trajectories for maneuver 2.

Fig. 14. Comparison of the controlled, uncontrolled, and reference vehicle trajectories for maneuver 3.

fishhook maneuver in the desired conditions [there is rollover at t = 2.5 s (see Fig. 10)], while the controlled vehicle remains stable during the whole maneuver. In order to show the efficiency of the controller in other maneuvers, three tests have been conducted with a vehicle speed of 70 km/h using different driver steering angles as shown in Fig. 11. A comparison of the controlled, uncontrolled, and reference vehicle trajectories in these maneuvers is shown in Figs. 12–14. These results clearly show that when the driver steering is large, the behavior of the uncontrolled vehicle becomes unstable, but not so for the controlled vehicle. V. C ONCLUSION In this brief we have developed a robust control method operating with active four wheel steering and active suspension in order to ensure vehicle stability in critical situations. The control system improves the vehicle behavior and avoids vehicle skid and rollover when it is subjected to different cornering maneuvers in critical situations. The main objective is not only to stabilize the vehicle but also to force the sideslip angle, the yaw rate, and the LTR to track desired trajectories which represent the perfect vehicle behavior for a specific situation. To achieve this goal, we have proposed an approach in which a fuzzy state feedback controller is designed with an integral action. Since the sideslip and roll angle sensors are too expensive, we have proposed an observer approach by considering the TS fuzzy vehicle representation. The proposed

A PPENDIX Matrices Ai , B f i , Bri (i = 1, . . . , 4) and Bm of (7) are given as follows: ⎡ ⎤ m s hCφ m s h(m s gh − kφ ) σi Ix1 ρi Ix1 ⎢− m Ix v m Ix v 2 − 1 − m Ix v ⎥ m Ix2 v ⎢ ⎥ 2 2 2 τi ⎢ ρi ⎥ ⎢ ⎥ 0 0 − ⎥ I I v Ai = ⎢ z z ⎢ ⎥ Cφ m s hρi (m s gh − kφ ) ⎥ ⎢ m s hσi ⎢− ⎥ − ⎣ m Ix2 ⎦ m Ix2 v Ix2 Ix2 0 0 1 0 ⎡ 2C I ⎤ ⎤ ⎡ f 2 x1 2C f 1 Ix1 ⎡m h ⎤ s ⎢ m I ⎢ m Ix2 v ⎥ x2 v ⎥ ⎢ ⎥ ⎢m Ix2 v ⎥ ⎥ ⎢ ⎢ 2C 2C l f 2l f ⎥ ⎢ ⎥ ⎢ f1 f ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎥ ⎢ Iz ⎥ Bm = ⎢ 1 ⎥, B f 1,2 = ⎢ ⎥, B f 3,4 = ⎢ Iz ⎢ ⎥ ⎢ ⎥ ⎢2m s hC f 1⎥ ⎢ ⎥ 2m hC ⎥ ⎢ ⎣ ⎦ s f 2⎥ ⎢ Ix2 ⎦ ⎣ mI ⎣ x2 m Ix2 ⎦ 0 0 0 ⎤ ⎡ 2C I ⎤ ⎡ 2Cr2 Ix1 r1 x 1 ⎢ m Ix2 v ⎥ ⎢ m Ix2 v ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ 2Cr1 l f ⎥ ⎢ 2Cr2l f ⎥ ⎥ ⎥ ⎢− ⎢− ⎢ Iz ⎥ Br1,3 = ⎢ Iz ⎥ ⎥ ⎥, Br2,4 = ⎢ ⎢ ⎢ 2m s hCr1 ⎥ ⎢ 2m s hCr2 ⎥ ⎥ ⎥ ⎢ ⎢ ⎦ ⎣ m Ix ⎦ ⎣ mI x2 2 0 0 where Ix1 and Ix2 denote the equivalent roll moments of inertia of the vehicle about the roll axis, which are given by m 2s . m Auxiliary variables σi , ρi and τi are introduced in order to simplify the model description and they are defined as follows: Ix1 = Ix + m s h 2 , Ix2 = Ix + m e h 2 and m e = m s −

σ1 = 2(Cr1 + C f 1 ), σ2 = 2(Cr2 + C f 1 ) σ3 = 2(Cr1 + C f 2 ), σ4 = 2(Cr2 + C f 2 ) ρ1 = 2(lr Cr1 − l f C f 1 ), ρ2 = 2(lr Cr2 − l f C f 1 ) ρ3 = 2(lr Cr1 − l f C f 2 ), ρ4 = 2(lr Cr2 − l f C f 2 )    τ1 = 2 l 2f C f 1 + lr2 Cr1 , τ2 = 2 l 2f C f 1 + lr2 Cr2 )     τ3 = 2 l 2f C f 2 + lr2 Cr1 , τ4 = 2 l 2f C f 2 + lr2 Cr2 . This section gives more details of how to obtain inequality conditions (16). Let us consider the particular matrices P˜ defined as: P 0  and X = P1−1 . P˜ = 1 0 P2

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Taking into account the expressions of matrices i j and  A Bwi , inequality (15) can be expressed in the following form:

i j ϒi j < 0, i, j = 1, . . . , 4 (18) ∗ i j i j = P1 (Ai − B i K j ) + (Ai − B i K j )T P1 1 T + 2 P1 B1wi B1wi P1 + C1T Q 1 C1 ρ i j = P2 (Ai − L i C) + (Ai − L i C)T P2 1 T + 2 P2 B2w B2w P2 + Q 2 ρ 1 T ϒi j = P1 (S + Ri K P j ) + 2 P1 B1w B2w P2 ρ where ∗ denotes the transposed element in the symmetric position.  X 0 Pre- and post-multiplying by and using matrices 0 I V j = K j X and Wi = P2 L i

i j ϒi j < 0, i, j = 1, . . . , 4 (19) ∗ ij i j = Ai X − B i V j T

T

+ X Ai − V jT B i +

1 T B1wi B1wi ρ2

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