Nonlinear analysis and control of a variable-geometry suspension system

Control Engineering Practice xx (xxxx) xxxx–xxxx

Contents lists available at ScienceDirect

Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Nonlinear analysis and control of a variable-geometry suspension system ⁎

Balázs Németh , Péter Gáspár Institute for Computer Science and Control, Hungarian Academy of Sciences, Kende u. 13–17, H-1111 Budapest, Hungary

A R T I C L E I N F O

A BS T RAC T

Keywords: Variable-geometry suspension system SOS programming method LPV control design Performance and stability analysis Tire characteristics

The paper proposes methods for both the analysis and the synthesis of variable-geometry suspension systems. The nonlinear polynomial Sum-of-Squares (SOS) programming method is applied in the analysis and it gives the optimal utilization of the maximum control forces on the tires. Moreover, the construction of the system can be based on the nonlinear analysis. The variable-geometry suspension system affects the wheel camber angle and generates an additional steering angle, thus the coordination of steering and wheel tilting can be handled. An LPV (Linear Parameter-Varying) based control-oriented modeling and control design for lateral vehicle dynamics are also proposed. The novelty of the method is the combination of the LPV-based control design and the SOS-based invariant set analysis. The simulation example presents the efficiency of the variable-geometry suspension system and it shows that the system is suitable to be used as a driver assistance system. In the SIL (software-in-the-loop) simulation both the dSPACE-AutoBox hardware and the CarSim simulator are used as standard industrial tools.

1. Introduction and motivation The variable-geometry suspension system is a novel mechanism with which road holding can be improved. The control input of the system is the camber angle of the front and rear wheels, with which the driver is supported to perform the various vehicle maneuvers, such as a sharp cornering, overtaking or double lane changing. By changing the camber angles the yaw rate of the vehicle is modified, which can be used to reduce the tracking error relative to the reference yaw rate. The suspension determines such components as the height of the roll center and the half-track change. The roll center can be modified by setting the camber angle of the wheels. Thus, during maneuvers the control system must guarantee various crucial vehicle performances such as trajectory tracking, roll stability and geometry limits. The advantages of the mechanism are the simple structure, low energy consumption and low cost compared to other mechatronic solutions, see Evers, van der Knaap, and Nijmeijer (2008). Several papers for various kinematic models of suspension systems have been published. A review of the first variable geometry systems was presented by Sharp (1998). A nonlinear model of the McPherson suspension system was published by Fallah, Bhat, and Xie (2009) and Németh and Gáspár (2012), while Hong, Jeon, and Sohn (1999) proposed a linearized model of the suspension, which can be used for active suspension design. In Habibi, Shirazi, and Shishesaz (2008) the effect of unnecessary steering due to the chassis roll angle was considered. An optimization method of the McPherson suspension ⁎

system was introduced in Lee, Won, and Kim (2009). By using this model the kinematic parameters, such as camber, caster and king-pin angles, were examined. The kinematic design of a double-wishbone suspension system was examined by Sancibrian, Garcia, Viadero, and Fernandez (2010). The performance requirements often lead to conflicts and require a compromise considering the kinematic and dynamic properties, see Vukobratovic and Potkonjak (1999). The vehicle-handling characteristics based on a variable roll center suspension were proposed by Lee, Lee, Han, Hedrick, and Catala (2008). A rear-suspension active toe control for the enhancement of driving stability was proposed by Goodarzia, Oloomia, and Esmailzadehb (2010). A series active variable-geometry suspension which is able to improve the pitch attitude control of the chassis was found in Arana, Evangelou, and Dini (2015). This solution offers several advantages compared to the semi-active suspension, e.g. fail-safe behavior and negligible unsprung mass increment. Another field of variable-geometry suspension is the steering of narrow vehicles. These vehicles require the design of an innovative active wheel tilt and steer control strategies in order to perform steering similarly to a car on straight roads but in bends they tilt as motorcycles, see Suarez (2012). The active tilt control system, which assists the driver in balancing the vehicle and performs tilting in the bend, is an essential part of a narrow vehicle system, see Piyabongkarn, Keviczky, and Rajamani (2004). The intervention of variable-geometry suspension systems requires the lateral motion of the suspension arm. In a real implementation it is realized using an electro-hydraulic actuator (Iman, Esfahani, &

Corresponding author. E-mail addresses: [email protected] (B. Németh), [email protected] (P. Gáspár).

http://dx.doi.org/10.1016/j.conengprac.2016.09.015 Received 14 May 2015; Received in revised form 19 August 2016; Accepted 26 September 2016 Available online xxxx 0967-0661/ © 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Németh, B., Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.09.015

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

space representation of the LPV model is valid in the entire operating region of interest. The advantage of LPV methods is that the controller meets robust stability and nominal performance demands in the entire operational interval, since the controller is able to adapt to the current operational conditions (Bokor & Balas, 2005; Packard, Seiler, Hjartarson, & Balas, 2014). In the control design both the steering angle and wheel tilting are handled. The novelty of the method is the combination of the LPV-based control design and the results of the analysis based on the SOS-based invariant set. The paper is organized as follows. In Section 2 the nonlinear vehicle model with nonlinear characteristics of tire forces is formed and the mechanism of variable-geometry suspension system is presented. In Section 3 the SOS programming method is applied to analyze the nonlinear polynomial system. An iterative method for Maximum Controlled Invariant Set is proposed and the actuator efficiency of the variable-geometry suspension system is presented as a demonstration example. In Section 4 the LPV-based control design of the variable-geometry suspension system is developed. In Section 5 the efficiency of the variable-geometry suspension system is presented using a simulation scenario. Section 6 contains some concluding remarks.

Fig. 1. Scheme of the suspension construction.

Mosayebi, 2010; Lee, Sung, Kim, & Lee, 2006; Schiehlen & Schirle, 2006) or an electric motor (Arana et al., 2015; Evers et al., 2008). In the paper the electro-hydraulic construction is considered as the real actuator of the system, see Fig. 1. In the illustrated example, the tilting and steering positioning of the wheel are yielded by the lateral motion of the upper arm. It is realized by the position control of the piston in the hydraulic cylinder (Németh, Varga, & Gáspar, 2014). The cylinder and the upper arm are connected through a crankshaft, whose rotation results in the lateral motion of the arm. The role of the crankshaft is to define the profile of the arm motion in lateral and vertical directions. Moreover, it has safety reasons in case of a hydraulic fault. The paper proposes methods for both the analysis and the synthesis of the variable-geometry suspension system. The efficiency of variablegeometry suspension has been presented in preliminary works (Németh, 2013; Németh & Gáspár, 2014). In several automotive applications the linearized model of lateral dynamics is sufficient to design controllers for driver assistance systems. However, the comprehensive analysis of the suspension system requires nonlinear techniques to understand its operation in detail. In the analysis the nonlinear polynomial Sum-of-Squares (SOS) programming method is applied to calculate the shape of the Controlled Invariant Sets of control systems. It is an efficient tool to find feasible solutions. Important theorems in SOS programming, such as the application of Positivstellensatz, were proposed in Parrilo (2003). The analysis provides information on the optimal utilization of the maximum control forces on the tires. The vertical position of the actuator can be calculated using the result of the nonlinear analysis. In an earlier paper a simultaneous design of robust control and the construction of a variable-geometry suspension was proposed in order to enhance vehicle stability, see Németh and Gáspár (2013). It was shown that there was a trade-off between the control design and the construction design, thus an optimization criterion which contained both the performances of the suspension construction and the performances of control design was formulated. In the current paper the construction parameter is fixed using the result of the nonlinear SOS-based analysis. It provides an optimal balance between the camber angle and the control system. The orientation of wheels is modified by the variable-geometry suspension system, which affects both the steering angle and the camber angle. Thus, the integration of steering and wheel tilting can be handled by the variable-geometry suspension system. A further contribution of the paper is the LPV (Linear Parameter-Varying) based control-oriented modeling and control design for lateral vehicle dynamics. Based on the LPV modeling approach the nonlinear effects can be considered in the state space description. Furthermore this state

2. Modeling of lateral vehicle dynamics for purposes of analysis The variable-geometry suspension system has different effects on the wheels: the modification of the steering and the camber angles. The relationship between the two angles is determined by the construction of the suspension system. In this section the nonlinear vehicle model is created and the mechanism of variable-geometry suspension system is presented. 2.1. Nonlinear vehicle model The modeling of tire forces is a crucial point of vehicle dynamics. Several tire models have already been published, see e.g. Pacejka (2004), Kiencke and Nielsen (2000), and de Wit, Olsson, Astrom, and Lischinsky (1995). In this paper the lateral tire force characteristics using polynomial functions are approximated. The advantage of this model is its formulation, which can be efficiently used in the analysis methods, e.g. Lyapunov-based stability and controllability examinations. Furthermore, the polynomial functions fit the lateral tire force characteristics appropriately. Some further papers also motivate the use of polynomial descriptions, such as Hirano, Harada, Ono, and Takanami (1993), Sadri and Wu (2013), and López, Olazagoitia, Moriano, and Ortiz (2014). The lateral dynamics of the vehicle is formulated by the following dynamic model, see Fig. 2:

Jψ¨ = Flat ,1 (α1) l1 − Flat ,2 (α2 ) l2 = F1 (α1) l1 − F2 (α2 ) l2 + G (α1) l1 γ

(1a)

mv (ψ˙ + β˙) = Flat ,1 (α1) + Flat ,2 (α2 ) = F1 (α1) + F2 (α2 ) + G (α1) γ

(1b)

where m is the mass of the vehicle, J is the yaw-inertia, l1 and l2 are geometric parameters. β is the side-slip angle of the chassis, ψ˙ is the yaw-rate, δ is the steering angle and γ is the camber angle of the wheel. Flat,1 (α1) and Flat,2 (α2 ) represent lateral tire forces, which depend on tire side-slip angles α1 and α2. The relationships between the tire side-slip angles for the front and rear axles, the steering angle of the vehicle and the side-slip angle of the chassis are tan(δ − α1) = (l1 ψ˙ + v sin β )/(v cos β ) and tan(α2 ) = (l2 ψ˙ − v sin β )/(v cos β ). At stable driving conditions the tire side-slip angle αi is normally not greater than 10° and the equations can be simplified by substituting sin β ≈ β and cos β ≈ 1. Moreover, the relative error of these simplifications is less than 1%. Thus, the following side-slip angles of the front and rear axles can be approxi2

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

Fig. 2. Scheme of lateral vehicle model.

mated:

α1 = δ − β −

α2 = −β +

ψl ˙1 , v

Fig. 3. Modeling of lateral tire force Flat.

(2a)

ψl ˙2 . v

polynomial state-space representation x˙ = f (x ) + gu , where x is the state vector, u is the control input signal, f and g are matrices. The rearrangement of vehicle model shows that the new states of the model are tire slip angles α1 and α2: x = [α1 α2]T . In this way the nonlinearity of the lateral tire forces F1, F2 and G can be considered. However, (5) incorporates the time-derivative of the front-wheel steering angle. In the actuator efficiency analysis the limit of intervention max(|δ|) has relevance. The detailed vehicle model is used for actuation range determination. For actuation limit purposes the following approximations are applied:

(2b)

From the above expressions the yaw-rate and the side-slip angle of the vehicle can be expressed in the following forms:

ψ˙ = v

α2 − α1 + δ , l1 + l2

(3a)

α1 l2 + α2 l1 − l2 δ l1 + l2

(3b)

β=−

In the case of the variable-geometry suspension system the nonlinearities of the tire characteristics must be considered in a given operation range.

•

•

Lateral tire force F (α ) depends on the lateral tire slip α nonlinearly: n F (α ) = ∑ j =1 cj α j . Although in several control applications the lateral forces are approximated with linear functions, which results in a simple description, it can be used in a narrow tire side-slip range. Vehicle motion is significantly characterized by this nonlinearity. The generated lateral tire force from camber angle G (α ) depends on m α nonlinearly: G (α ) = ∑k =0 gk α k . Thus, the efficiency of actuator intervention is influenced by the tire side-slip range.

n

j =1

k =0

α˙1 l2 + α˙2 l1 = v (α2 − α1) − −

l1 v [F2 (α2 ) l2 − F1 (α1) l1] + (α2 − α1) − Jv l1 + l2 l v f2 = 2 [F1 (α1) l1 − F2 (α2 ) l2] + (α2 − α1) − Jv l1 + l2 ⎛ l2 v v + ξ, h21 = , h12 = −⎜ 1 + h11 = l1 + l2 l1 + l2 ⎝ Jv ⎛l l 1 ⎞ h22 = ⎜ 1 2 − ⎟ G (α1). ⎝ Jv mv ⎠ f1 =

(4)

(5a)

(7)

1 [F1 (α1) + F2 (α2 )], mv 1 [F1 (α1) + F2 (α2 )], mv 1 ⎞ ⎟ G (α1), mv ⎠

2.2. Mechanism of variable-geometry suspension construction

l1 + l2 [F1 (α1) + F2 (α2 )] + vδ + l2 δ˙ mv

l1 + l2 G (α1) γ mv

(6b)

where

where γ is the camber angle of the wheel. An example of the nonlinear characteristics in the function of tire side-slip α is illustrated in Fig. 3. Since (1) contains the time derivatives of ψ˙ and β, they must be differentiated by using (3). At constant velocity v their derivatives are ψ¨ = v (α˙2 − α˙1 + δ˙)/(l1 + l2 ) and β˙ = −(α˙1 l2 + α˙2 l1 − l2 δ˙)/(l1 + l2 ). Now the vehicle model (1) is reformulated:

⎡ l + l2 ⎤ l (l + l2 ) α˙2 − α˙1 = ⎢ 1 (F1 (α1) l1 − F2 (α2 ) l2 ) ⎥ − δ˙ + 1 1 G (α1) γ ⎣ Jv ⎦ Jv

δ˙ ≈ ξ·δ,

⎡ α˙ ⎤ ⎡ f (α , α ) ⎤ ⎡ h h ⎤ ⎡ ⎤ x˙ = ⎢ 1 ⎥ = ⎢ 1 1 2 ⎥ + ⎢ 11 12 ⎥ ⎢ δγ ⎥ ⎣ α˙2 ⎦ ⎣ f2 (α1, α2 )⎦ ⎣ h21 h22 ⎦ ⎣ ⎦

m

∑ cj α j + ∑ gk α kγ

(6a)

where parameter ξ represents the relationship between the maximum steering value and the variation speed of δ. Since max δ is a given fixed limit at the actuator analysis, a high ξ value represents a fast-changing steering signal, while a slow-changing steering signal is modeled with low ξ. Note that the proposed modeling formula is only valid for actuation limit computation. The polynomial state-space representation of the system is formulated using (5) and the substitution of (6) is as below:

The nonlinear model of the tire is constructed from the polynomial approximation of the previous two effects, F (α ) and G (α ):

Flat (α ) = F (α ) + G (α ) γ =

⎛ |δ˙| ⎞ max(|δ˙|) = max ⎜ ⎟ ·max(|δ|) = ξ·max(|δ|), ⎝ |δ | ⎠

In the following the relationship between δ and γ is proposed. The scheme of the variable-geometry suspension system is illustrated in Fig. 4. The modification of the lateral position of A (ay) affects the rotation of the front wheel around axis KD. Thus, the camber angle γ and steering angle δ are simultaneously changed. The position of the

(5b)

In the following these expressions are used to transform (1) into a 3

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

Fig. 4. Wheel position related to the steering and the camber angle.

3. Nonlinear approach for actuator efficiency analysis

steering track-rod connection point K has an important role in the distribution of δ and γ. Thus, it is necessary to choose a vertical position Kz, by which the lateral force on the tire is improved the most effectively. The deduction of the relationships δ = fδ (a y , Kz ) and γ = fγ (a y , Kz ) is detailed in Németh (2013). In the model the arms and bodies of the system are elements which are connected to the vehicle chassis by joints. Figs. 5(a) and (b) show angles δ and γ at different Kz heights. The aim of the example is to illustrate the relationship between signals. The variation of Kz has a great influence on angle δ, however, it modifies γ slightly. KD is the axis of wheel rotation during the actuation ay, therefore its orientation influences the relationship between these angles. Since generally δ and γ are in conflict, it is necessary to find an appropriate solution to parameter Kz. In the analyzed construction Kz has a significant influence on δ and with an increased Kz it is possible to achieve a high lateral tire force. Consequently, the steering angle and the camber angle are functions of the actuation, i.e., δ = fδ (a y ) and γ = fγ (a y ). It shows that the construction parameter significantly affects the operation of the variable-geometry suspension system. The example also shows the conflict between the parameters. A high Kz is advantageous since δ can be modified in a wider range by using control input ay, while it is disadvantageous concerning γ.

In this section the fundamental concepts of the Sum-of-Squares (SOS) programming are introduced, which is a suitable method to analyze and control a nonlinear polynomial system. The method is applied to determine actuator limit influences in vehicle dynamics. 3.1. Theoretical background Several papers deal with SOS programming, which has been elaborated in the past decade for control purposes. Important theorems in SOS programming were proposed and summarized in Parrilo (2003). Sufficient conditions for the solutions to nonlinear control problems, which were formulated in terms of state dependent Linear Matrix Inequalities (LMI) was shown in Prajna, Papachristodoulou, and Wu (2004). The application of SOS programming to several control problems, e.g. reachable set computation and control design algorithm, was introduced by Jarvis-Wloszek, Feeley, Tan, Sun, and Packard (2003). The local stability analysis of polynomial systems and an iterative computation method for their region of attraction were presented in Tan and Packard (2008). The SOS method was applied to non-convex problems in Scherer and Hol (2006). Robust performance in polynomial control systems was analyzed in Topcu and Packard (2009). As a new result the Maximum Controlled Invariant

Fig. 5. Influence of Kz on the relationship between δ and γ.

4

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

⎧ ∂V ⎫ ∂V inf ⎨ f (x ) + g · u⎬ < 0 u∈ ⎩ ∂x ⎭ ∂x

Sets of polynomial control systems were calculated in Korda et al. (2013). The following definitions are essential to understand SOS programming (Jarvis-Wloszek et al., 2003). The basic elements of the method are polynomials and SOS as defined below:

(11)

for each x ≠ 0 .According to Definition 3 two cases are distinguished: Definition 1. A Polynomial f in n variables is a finite linear combination of the functions mα (x )≔x α = x1α1 x 2α2 ⋯xnαn for α ∈  +n , n deg mα = ∑i =1 αi :

f ≔ ∑ cα mα = α

∑ cα x α α

• •

(8)

∂V

if ∂x f (x ) < 0 then the system is stable and u ≡ 0 . This stability scenario is contained by the next two stability criteria. ∂V if ∂x f (x ) > 0 then the system is unstable. However, the system can be stabilized 1. the upper peak-bound of control input u stabilizes the system if

∂V g<0 ∂x

with cα ∈  . Define  to be the set of all polynomials in n variables. The degree of f is defined as f ≔maxα deg mα .

and

∂V ∂V f (x ) + g ·umax < 0. ∂x ∂x

(12)

2. The lower peak-bound of control input u stabilizes the system if Definition 2. The set of Sum-of-Squares (SOS) polynomials in n variables is defined as:

⎧ ⎪ Σ n≔⎨ p ∈ 9n p = ⎪ ⎩

⎫ ⎪ ∑ fi2 , fi ∈ 9 n, i = 1, …, t ⎪⎬ ⎭ i =1

∂V g>0 ∂x

∂V ∂V f (x ) − g ·umax < 0. ∂x ∂x

(13)

t

Note that in the inequalities u min = −u max . (9) The Controlled Invariant Set of the system (10) is defined as the levelset of the Control Lyapunov Function at V (x ) = 1. Thus, the fulfilment of the previous stability criterion must be guaranteed at V (x ) ≤ 1. In the following an iterative computation method is proposed to find the maximum Controlled Invariant Set, which, in our experience, can lead to an easier calculation. The practical method contains three steps: Step 1: The region of attraction of the uncontrolled system x˙ = f (x ) is determined as an initial set. In this step the maximum level set of V0 = 1 is found, which is incorporated in the stable region. The SOS based computation of the region of attraction was presented in JarvisWloszek (2003). Step 2: An η parameter is chosen and Vη = V0·η is checked as a Local Control Lyapunov Function. The level-set Vη = 1 represents a Controlled Invariant Set Sη , in which the system can be stabilized using a finite control input u. Depending on parameter η the size of the level-set can be enlarged or reduced. The SOS based computation of the Local Control Lyapunov Function is proposed in Tan and Packard (2008). Step 3: In the final step the acceptability and the enlarging possibility of Sη Controlled Invariant Set must be checked. The peakbounds of the actuation are u min = −u max and umax. ∂V Sinst = {x ∈ 9 n ∂x f (x ) > 0} is the unstable region of the system.

The goal of the nonlinear actuator analysis is the determination of their intervention limits next to a peak-bounded actuation. With an appropriate intervention of the actuators some of the unstable regions can be stabilized. In the following section the largest state-space region where the stability of the system can be guaranteed by a given peakbounded control input is sought. This problem leads to the computation of Controlled Invariant Sets, see also Korda et al. (2013). 3.2. Computation method of controlled invariant sets The state-space representation of the system is given in the following form, see (7):

x˙ = f (x ) + gu

and

(10)

where f(x) is a matrix, which incorporates smooth polynomial functions and f (0) = 0 . In the next analysis one control input is considered. The global asymptotical stability of the system at the origin is guaranteed by the existence of the Control Lyapunov Function of the system defined as follows (Sontag, 1989): Definition 3. A smooth, proper and positive-definite function V: n →  is a Control Lyapunov Function for system if the

Smin = {x ∈ 9 n

∂V f ∂x

Fig. 6. Maximum Controlled Invariant Sets.

5

(x ) −

∂V g ·u max ∂x

> 0} is the region, which cannot be

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár ∂V

∂V

stabilized by umin. Similarly, Smax = {x ∈ 9 n ∂x f (x ) + ∂x g ·u max > 0} is the region, which cannot be stabilized by umax. If Sη is an appropriate Controlled Invariant Set and Vη is an appropriate Control Lyapunov Function, then

Sη ∩ Sinst ∩ Smin ∩ Smax = ∅

(14)

The emptiness of the intersection condition defined below can be checked manually by the plot of Sη , Sinst, Smin and Smax . Additionally, if Sη is appropriate then η value can be reduced in the previous step to maximize the Controlled Invariant Set. 3.3. Demonstration of the control efficiency In the following an example of a Maximum Controlled Invariant Set S analysis is proposed. Since wheel tilting has a great impact on the design of lightweight vehicles, in the example a small vehicle with a double-wishbone suspension is considered. The form of the polynomial tire model is the following, see (4): 6

F1 (α1) =

∑ cj,1 α1j, j =1

Fig. 7. Maximum Controlled Invariant Sets of the systems.

6

F2 (α2 ) =

∑ cj,2 α2j, G (α1) = g0 + g4 α14. j =1

4. Control design based on the LPV method (15) In this section the LPV-based control design of the variablegeometry suspension system is developed. The analysis provides the construction parameter Kz and information about the optimal utilization of the maximum control forces on the tires. Moreover, the effects of the camber angle and the steering angle on the operation regions have also been provided by the Maximum Controlled Invariant Sets. The variable-geometry suspension system affects the wheel camber angle and generates an additional steering angle. Thus the coordination of the steering angle and wheel tilting should be handled by the control input of the variable-geometry suspension system (ay).

In the analysis the influence of Kz on the Maximum Controlled Invariant Sets is examined. The maximum control input of the variable-geometry suspension actuator is |a y, max | = 150 mm , see Fig. 5. Two scenarios are compared in the example: in the first case the variable-geometry suspension with different Kz parameters is considered, the dynamics of the vehicle is influenced during steering and wheel tilting effects. In the second case a front wheel steering system with steering angle limitation |δmax | is assumed, where |δmax | is equal to the maximum steering effect of the variable-geometry suspension, see Fig. 5(a). In Fig. 6 the Maximum Controlled Invariant Sets of each Kz value are compared at fixed velocities. When Kz = 100 mm the wheel tilting intervention is dominant, while steering has a slight counter-influence. Thus, Kz = 100 mm can be analyzed in variable-geometry suspension, because pure steering is inefficient. Kz = 300 mm leads to a balance between the camber and the steering angle, while at Kz = 500 mm steering is preferred. Fig. 6 also shows that construction parameter Kz = 100 mm results in a small S region at both velocities. Thus, camber angle intervention is insufficient by itself, so it is necessary to find a Kz which causes γ and δ angles simultaneously. It can be seen that the simultaneous actuation of steering and wheel tilting leads to the enlargement of S, see Kz = 300 mm and Kz = 500 mm scenarios. However, the enlargement is more significant at Kz = 300 mm . Moreover, S region of Kz = 300 mm is larger than the region of Kz = 500 mm . The reason for these phenomena is the reduction of the maximum γ at high Kz values. The results of the analysis show that Kz influences the Maximum Controlled Invariant Sets significantly. The minimization of γ or δ leads to small sets, thus it is required to find a balance between the tilting and steering effects of the variable-geometry suspension. Since Kz = 300 mm is the best choice in the proposed example, in the following the Maximum Controlled Invariant Sets are computed for different velocities. The results of the variable-geometry suspension with Kz = 300 mm and the results of the pure steering with |δmax | = 18° are illustrated in Fig. 7. Since the unstable regions of the lateral dynamics increase at higher velocity, the controllable region with limited control input is reduced. At v = 65 km/h the steering and the suspension sets reach the limit of the model validity, thus the wheel camber angle has relevance at v > 65 km/h . During the simultaneous actuation of steering and tilting, S regions are enlarged compared to the individual steering. In Fig. 7 the distance of set boundary from zero increases approximately by 10%. The efficiency of the proposed set computation method is conspicuous: the increase of unstable regions and the benefit of variable-geometry suspension are demonstrated.

4.1. Control-oriented modeling of lateral dynamics In the following the polynomial description of the tire and the lateral model are transformed into a control-oriented form. The linearizing of Flat (α ) around a given α0 leads to the following expression:

F (α ) α0 = F0 (α0 ) + c (α0 ) α + G0 (α0 ) γ

(16)

where c (α0 ) is the cornering stiffness. In (16) the parameters F0 (α0 ) and c (α0 ) depend on the slip value α0 in the tire model. These parameters are derived from the fitted polynomial model (4). F0 (α0 ) = F (α0 ) and G0 (α0 ) = G (α0 ) are the values of the lateral tire force at α0, while

c (α 0 ) =

dF (α ) dα

α0

(17)

represents the linear slope at α0. In this way the polynomial model (4) is transformed into a linear parameter-dependent (α0) tire form. The linearization of the tire force is illustrated in Fig. 8. The lateral dynamics of the vehicle is formulated by the following dynamical model:

Jψ¨ = F1 (α1) α0,1 l1 − F2 (α2 ) α0,2 l2 + G0 (α0,1) l1 γ

(18a)

mv (ψ˙ + β˙) = F1 (α1) α0,1+F2 (α2 ) α0,2 + G 0 (α0,1) γ

(18b)

where m is the mass of the vehicle, J is the yaw inertia, l1 and l2 are geometric parameters. β is the side-slip angle of the chassis, ψ˙ is the yaw rate. F1 (α1) α0,1 and F2 (α2 ) α0,2 represent the lateral tire forces, which are linearized around α0,1 and α0,2 . Using the tire characteristics (16) and the side-slip angles the lateral vehicle dynamics (18) is formed as follows: 6

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

⎛ ψl ˙ ⎞ mv (ψ˙ + β˙) = (F0,1 (α0,1) + F0,2 (α0,2 )) + G0 (α0,1) γ + c1 (α0,1) ⎜δ − β − 1 ⎟ ⎝ v ⎠ ⎛ ψl ˙ 2⎞ − c2 (α0,2 ) ⎜ −β + ⎟. ⎝ v ⎠ (19b) In the control design the real physical input of the variablegeometry suspension (ay) is considered instead of separated δ and γ. The linear relations between ay and δ and γ are δ = fδ (a y ) and γ = fγ (a y ). Further information is detailed in Németh (2013). Using these relationships the state-space representation of the system is given as

x˙ = A (α0,1, α0,2 ) x + B1 (α0,1, α0,2 ) w + B2 (α0,1, α0,2 ) u + W (α0,1, α0,2 ), (20) where the state vector contains x = [ψ˙ β ]T , the control input is the camber angle u = a y and w is the disturbance. Moreover, W is a static disturbance on the system, which incorporates the constant terms, such as:

Fig. 8. Modeling of lateral tire force.

⎡ F0,1 (α0,1) l1 − F0,2 (α0,2) l2 ⎤ ⎢ ⎥ W = ⎢ F (α ) +J F (α ) ⎥ 0,1 0,1 0,2 0,2 ⎢⎣ ⎥⎦ mv

(21)

The representation (20) is a control-oriented LPV model of the lateral vehicle dynamics, which depends on the signals α0,1, α0,2 . The measurement of these signals is necessary to compute the current values of the matrices A, B1, B2 and W. 4.2. LPV-based control design The main performance specification of the control system is trajectory tracking. In trajectory tracking control the vehicle must follow the reference yaw rate, i.e., the purpose is to minimize the difference between the actual yaw rate of the vehicle and the reference yaw rate:

Fig. 9. Closed-loop interconnection structure.

z = |ψ˙ref − ψ˙ | → min

(22)

The performance of the system is the tracking of a reference yaw rate ψ˙ref . The reference signal is generated by the driver himself in a driver assistance system or it can be computed from the curvature of the road. For example, the desired yaw rate can be calculated by a firstorder transfer function, see Rajamani (2005):

ψ˙ref =

where δd is the steering angle of the driver, d = l1 + l2 + ηv 2 / g depends on the velocity and geometry parameters, η is an understeer gradient, g is the gravitational constant and τ is the time constant. The control design is based on the parameter-dependent model (20). The scheduling variables of the system are chosen as follows:

Fig. 10. Architecture of the system.

Table 1 Data of vehicle and tire model. m J l1 l2 g0 g4

1833 kg

2765 kg m2 1.402 m 1.646 m 8848.317 −0.442

t v ·(1 − e− τ )·δd d

c1,1 c2,1

3110.52 −12.46

c1,2 c2,2

2662.605 −10.778

c3,1 c4,1 c5,1 c6,1

−30.86 0.5 0.15 −0.005

c3,2 c4,2 c5,2 c6,2

−17.528 0.284 0.085 −0.003

ρ1 = α0,1

(23a)

ρ2 = α0,2

(23b)

ρ3 = v

(23c)

In the following the equation α0, i = αi is assumed, which is an accurate linear approximation of the nonlinear tire model. Moreover, it is assumed that signals ρi are smooth, since both the lateral side-slip angles αi and the velocity are smooth signals. The control-oriented LPV model is the following:

x˙ = A (ρ1 , ρ2 , ρ3) x + B (ρ1 , ρ2 , ρ3) u + W (ρ1 , ρ2 , ρ3)

⎛ ψl ˙ ⎞ Jψ¨ = (F0,1 (α0,1) l1 − F0,2 (α0,2 ) l2 ) + G0 (α0,1) l1 γ + c1 (α0,1) l1 ⎜δ − β − 1 ⎟ ⎝ v ⎠ ⎛ ψl ˙ 2⎞ − c2 (α0,2 ) l2 ⎜ −β + ⎟ ⎝ v ⎠ (19a)

(24)

The LPV model contains disturbance W, which must be handled by an appropriate control input. For this reason, two signals concerning the suspension arm actuation are defined as:

a y = a y,0 + a y,1

(25)

where the purpose of the control motion a y,0 is to handle disturbance 7

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

Fig. 11. Paths of the LPV controlled and the uncontrolled vehicles. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

The closed-loop LPV system is exponentially and asymptotically stable and with its 32 -gain less than γ, if there exists an X (ρ) > 0 satisfying the following linear matrix inequality for all ρ:

W, while the control motion a y,1 guarantees the performance (22). a y,0 can be considered as a direct feedforward control input, which is defined by the first components of both W and B in the following way:

a y,0 = −

F0,1 (ρ1) l1 − F0,2 (ρ2 ) l2 W1,1 . =− B1,1 c1 (ρ1) l1

⎡ A T X + XA + d / dt (X ) XB γ −1CclT ⎤ cl cl ⎢ cl ⎥ T ⎢ Bcl X − I γ −1DclT ⎥ < 0 ⎢ ⎥ γ −1Ccl γ −1Dcl − I ⎦ ⎣

(26)

Using this equation, the static disturbance of the system W can be reduced to the following term:

⎡ ⎤ 0 l (ρ2 , ρ3) = ⎢ F0,2 (ρ2 )(l1 + l2) ⎥ , W ⎢⎣ ⎥⎦ ml1ρ3

If a solution exists, is a parameter-dependent Lyapunov function for the closed-loop system for all ρ. The proof is found in Yu and Sideris (1997). The solution of an LPV problem is based on the set of LMIs being satisfied for all ρ ∈ F7 . In practice, this problem is set up by gridding the parameter space and solving the set of LMIs that hold on the subset of F7 . The number of grid points depends nonlinearly on the operation range of the system. Weighting functions are defined in all of the grid points, see Packard et al. (2014), Rough and Shamma (2000), and Wu, Yang, Packard, and Becker (1996). With respect to the robustness requirement, the same frequency weighting functions are applied in the whole parameter space and the effect of the scheduling variable is ignored. It is a reasonable engineering assumption, since unmodelled dynamics does not depend on the gridding parameters. When the LPV controller has been synthesized, the relation between the state, or output, and the parameter ρ = σ (x ) is used in the LPV controller, such that a nonlinear controller is obtained. The architecture of the entire control system is illustrated in Fig. 10. The feedback LPV controller requires the measurements of the scheduling variables. The side-slip angles and forward velocity are estimated using measured signals of the vehicle, see e.g. Baffet, Charara, and Stéphant (2006), Venhovens and Naab (1999), and Song, Uchanski, and Hedrick (2002). The feedforward component of the controller also uses the ρ1, ρ2 signals. The control input of the vehicle is a y = a y,0 + a y,1. During the operation of the controlled system the controller requires the measurement of the reference yaw rate, the current yaw rate and all the three scheduling variables, i.e., the forward velocity and the current values of α0,1, α0,2 .

(27)

which depends on ρ2 and ρ3. The modified control-oriented LPV model is:

l (ρ2 , ρ3) x˙ = A (ρ1 , ρ2 ) x + B (ρ1 , ρ2 ) a y,1 + W

(28)

The control input of the achieved system is a y,1. The purposes of the control design are to guarantee the performance (22) and to eliminate l . The measured signal of the system is the the reduced disturbance W yaw rate ψ˙ . The closed-loop architecture of the system is illustrated in Fig. 9. In the LPV control design several weighting functions are used, which scale the gain and the frequency range of the signals. The role of the weighting function Wref is to scale ψ˙ref . It is chosen in the form A

Wref = T s +1 1 , where A1, T1 are constant parameters. In this way the low1 pass filter based Wref smoothes the sharp changes in the reference l . Since it signal. The weighting function Ww scales the disturbance W has low dynamics, Ww = A2 constant form is used in the design. A We = T s 3+ 1 scales the sensor noise on the measurement e, in which 2 the fast change of the sensor noise signal is incorporated. The l e]T . The tracking performance of disturbance is defined as w = [W the system is guaranteed by the weighting function Wp. It is chosen in A s+A the form Wp = 24 5 , where A4 , A5 , T3, T4 are design parameters. T4 s + T3 s + 1

The control design is based on the LPV method that uses parameter-dependent Lyapunov functions, see Bokor and Balas (2005) and Packard et al. (2014). The quadratic LPV performance problem is to choose the parameter-varying controller K (ρ1 , ρ2 , ρ3) in such a way that the resulting closed-loop system is quadratically stable and the induced 32 norm from the disturbance w and the performances z is less than the value γ. The induced 32 -norm of the LPV system G F7 , with zero initial conditions is defined as

∥ G F7 ∥∞ = sup

sup

ρ ∈ F7∥ w ∥2 ≠0, w ∈ 32

∥ z ∥2 . ∥ w ∥2

(30)

x T X (ρ ) x

5. Simulation results The efficiency of the variable-geometry suspension control design method is presented using simulation scenarios. The vehicle must perform a double-lane change maneuver with a constant velocity v = 130 km/h . The vehicle must remain within the lanes. The purpose of the controller is the design of the appropriate front-wheel actuation. The selected vehicle is a full-size E-Class passenger car. The vehicle has a 250 kW engine and an automatic transmission with 7 gear positions. The suspension type of the vehicle is independent of the front and the rear wheels. The vehicle data can be found in Table 1. The driver model in the simulations is based on MacAdam (1981). The weighting functions of the LPV design are chosen as follows:

(29)

The 32 -norm level γ = G F7 ∞ for an LPV system represents the largest ratio of disturbance norm to performance norm over the set of all causal linear operators described by the LPV system, hence it is the final performance index of the design. For an appropriate selection of the performance and disturbance weights a successful design means γ < 1. It is noted that the weighting functions must be selected according to the control goals.

Wp = 8

0.35s + 5 s 2 + 12s + 20

(31a)

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

Fig. 12. Motion of the vehicles.

9

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

Fig. 13. Simulation results of the maneuver.

1 0.01s + 1

(31b)

0.005 0.001s + 1

(31c)

Wref =

We =

Ww = 0.01

operation of the vehicle, which controls the integration of steering and wheel tilting is compared to the vehicle, which controls only the wheel camber angle. Fig. 12 shows the results of the double-lane change maneuver of these vehicles. Figs. 12(a), (b) show that the individual steering actuation is insufficient to perform the test, since the vehicle crosses the boundary line at the entry, see Fig. 12(c). The path of the individual wheel tilting control is shown in Fig. 12(b). Although it performs the cornering maneuvers, the lateral motion of the vehicle has an increased overshoot at the straight line and the duration of the oscillation is long, see Fig. 12(b). Fig. 12(e) illustrates the lateral distance of the vehicle from the closest boundary line. When the distance from the boundary line is less than zero, the vehicle crosses the boundary line and fails the test. It can be stated that LPV control is able to guarantee the appropriate distance from the boundaries along the entire trajectory even at the critical sections, while both the uncontrolled vehicle and the individually controlled vehicle fail at the test. The efficient interaction is performed by the joint interaction of the steering and tilting of the front wheels. Fig. 13(a) illustrates the steering of the suspension, while the camber angle is shown in Fig. 13(b). The significant actuation between 150 m and 250 m results in the improvement of vehicle dynamics. The two effects are strongly related, which is determined by the real control input, the motion ay,

(31d)

In a double-lane change maneuver usually the final section is critical, when the vehicle reaches one of the boundaries. Since the purpose of the control is to keep within the lanes, the end of the path is a good reference for checking controller efficiency. Fig. 11 illustrates the motions of the two vehicles at the critical final section from a top view. The critical points are at stations 150 m and 180 m, marked with red ellipsoids. The first vehicle is controlled by the proposed LPV control (blue). The second vehicle is uncontrolled, i.e., there is only a driver and no active control (gold). The uncontrolled vehicle fails at the test, because the vehicle crosses the boundary line with the front left wheel at the entry of the final section (first ellipsoid). Moreover, the uncontrolled vehicle also very close to the boundary line with the rear right wheel during the straight section (second ellipsoid). The controlled vehicle, however, performs the maneuver and provides appropriate vehicle dynamics, see Fig. 11. Fig. 12 shows the simulation results of two further scenarios. The 10

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

Fig. 14. Vehicles in the icy curve.

see Fig. 13(c). The scheduling variables of the control are depicted in Fig. 13(d). The values of ρ show that the vehicle moves in the nonlinear tire region, compared with Fig. 3. In this region the proposed LPV

control operates efficiently. In Fig. 14 a 130 km/h cornering maneuver of the vehicle on an icy road bend is illustrated. The paths of the vehicles on the entire road

11

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

Fig. 15. Software-in-the-loop simulation.

HMI (Human Machine Interface), a high-accuracy validated simulation software operated on a workstation and a visual system with real-time graphics. The specific signals, i.e., the position of the accelerator and the brake pedal along with the steering angle, are read through the CAN network by using a standard communication interface. The Driving Simulator of CarSim shows the vehicle maneuvers in real-time graphics projected in front of the vehicle and it provides the signals during the journey. The stationary vehicle can be driven almost exactly the same way as in real life: there is engine sound and screech while skidding; the dash panel displays the current speed and revolution and one can shift gears just like in real life. Various journey scenarios can be generated by the simulation system. Since the car is used for education and research purposes it is run only on test roads.

section with the different control actuations are found in Fig. 14(a). It can be seen that the icy patch (silver ellipsoid in Fig. 14(a)) significantly modifies the motion of the vehicles, and it results in a lane departure, see the uncontrolled scenario in Fig. 14(b). However, the joint actuation of wheel tilting and steering is effective in the avoidance of the lane departure. The lateral slip of the vehicle increases on the front and rear axles simultaneously, and the vehicle reaches the nonlinear tire force region. The control motion a y of the variable-geometry suspension is shown in Fig. 14(d). It yields the control steering δc (Fig. 14(e)) and wheel tilting γ (Fig. 14(f)). The coordinated actuation is able to prevent the lane departure of the vehicle.

5.1. SIL system for controller implementation 6. Conclusion

The proposed simulation scenarios in a software-in-the-loop (SIL) environment have been implemented, see Fig. 15. The SIL consists of a workstation with the vehicle simulator, the dSPACE, in which the controller is implemented. In the workstation the CarSim works together with Matlab/Simulink. The CarSim with the Matlab/ Simulink software, which are standard industrial tools, represent the vehicle dynamics with high accuracy. The suspension controller is implemented in the dSPACE-AutoBox system, which is also standard enclosure for in-vehicle control experiments. The communication between the workstation and the dSPACE is realized through the CAN bus. Before the SIL simulation the designed control system is set on the real-time equipment. The control signal is computed in dSPACE by the discrete-time solver of the differential equations with 0.01 s sampling time. In the future the current dSPACE-based control system will be joined with the real-time simulator environment. For simulation purposes, the control of the vehicle's communication network has been taken over by the simulator unit. The simulation environment contains

In the paper the steering and wheel tilting abilities of the variablegeometry suspension have been analyzed based on the nonlinear polynomial SOS programming method. The construction of the system, i.e., the vertical position of the actuator, is based on the nonlinear analysis. It provides an optimal balance between the camber angle and the control system. The analysis also provides information about the optimal utilization of the maximum control forces on the nonlinear characteristics of the tires. The effects of the camber angle and the steering angle on the operation regions have been analyzed based on Maximum Controlled Invariant Sets. Moreover, the LPV-based control design for lateral vehicle dynamics has been proposed. The control guarantees the coordination of steering and wheel tilting. The main contribution of the paper is the combination of the LPV-based control design and the SOS-based invariant set analysis. The simulation scenarios present the efficiency of the variablegeometry suspension system. The SIL system includes the CarSim 12

Control Engineering Practice xx (xxxx) xxxx–xxxx

B. Németh, P. Gáspár

of McPherson suspension systems. In World congress on engineering (pp. 1532– 1537). MacAdam, C. (1981). Application of an optimal preview control for simulation of closedloop automobile driving. IEEE Transactions on Systems, Man, and Cybernetics, 11(6), 393–399. Németh, B., & Gáspár, P. (2012). Mechanical analysis and control design of McPherson suspension. International Journal of Vehicle Systems Modelling and Testing, 7, 173–193. Németh, B., & Gáspár, P. (2013). Control design of variable-geometry suspension considering the construction system. IEEE Transactions on Vehicular Technology, 62(8), 4104–4109. Németh, B., & Gáspár, P. (2014). Set-based analysis of the variable-geometry suspension system. In 19th IFAC world congress (pp. 11201–11206), Cape Town. Németh, B., Varga, B., & Gáspar, P. (2014). Design of a variable-geometry suspension system to enhance road stability. In 2014 22nd Mediterranean conference of control and automation (MED) (pp. 55–60). Németh, B., & Gáspár, P. (2013). Variable-geometry suspension design in driver assistance systems. In 12nd European control conference (pp. 1481–1486), Zürich. Pacejka, H. B. (2004). Tyre and vehicle dynamics Oxford: Elsevier ButterworthHeinemann. Packard, A. K., Seiler, P., Hjartarson, A., & Balas, G. J. (2014). Linear, parametervarying control: Tools and applications. In American control conference, Workshop, Portland. Parrilo, P. (2003). Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming B, 96(2), 293–320. Piyabongkarn, D., Keviczky, T., & Rajamani, R. (2004). Active direct tilt control for stability enhancement of a narrow commuter vehicle. International Journal of Automotive Technology, 5(2), 77–88. Prajna, S., Papachristodoulou, A., & Wu, F. (2004). Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach. In 5th IEEE Asian control conference (Vol. 1, pp. 157–165). Rajamani, R. (2005). Vehicle dynamics and control New York: Springer. Rough, W., & Shamma, J. (2000). Research on gain scheduling. Automatica, 36, 1401–1425. Sadri, S., & Wu, C. (2013). Stability analysis of a nonlinear vehicle model in plane motion using the concept of Lyapunov exponents. Vehicle System Dynamics, 51(6), 906–924. Sancibrian, R., Garcia, P., Viadero, F., & Fernandez, A. (2010). Kinematic design of double-wishbone suspension systems using a multiobjective optimisation approach. Vehicle System Dynamics, 48(7), 793–813. Scherer, C. W., & Hol, C. W. J. (2006). Matrix sum-of-squares relaxations for robust semi-definite programs. Mathematical Programming, 107, 189–211. Schiehlen, W., & Schirle, T. (2006). Modeling and simulation of hydraulic components for passenger cars. Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 44(sup1), 581–589. Sharp, R. (1998). Variable geometry active suspension for cars. IEEE Computing and Control Engineering Journal, 9(5), 217–222. Song, C., Uchanski, M., & Hedrick, J. (2002). Vehicle speed estimation using accelerometer and wheel speed measurements. In Proceedings of the SAE automotive transportation technology, Paris 2002-01-2229. Sontag, E. (1989). A universal construction of Artstein's theorem on nonlinear stabilization. Systems & Control Letters, 13, 117–123. Suarez, L. (2012). Active tilt and steer control for a narrow tilting vehicle: Control design and implementation. Lambert Academic Publishing. Tan, W., & Packard, A. (2008). Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming. IEEE Transactions on Automatic Control, 53(2), 565–571. Topcu U., & Packard, A. (2009). Local robust performance analysis for nonlinear dynamical systems. In Proceedings of the American control conference (pp. 784– 789). Venhovens, P., & Naab, K. (1999). Vehicle dynamics estimation using Kalman filters. Vehicle System Dynamics, 32, 171–184. Vukobratovic, M., & Potkonjak, V. (1999). Modeling and control of active systems with variable geometry. I: General approach and its application. Mechanism and Machine Theory, 35, 179–195. Wu, F., Yang, X. H., Packard, A., & Becker, G. (1996). Induced l 2 -norm control for LPV systems with bounded parameter variation rates. International Journal of Nonlinear and Robust Control, 6, 983–998. Yu, J., & Sideris, A. (1997). / ∞ control with parametric Lyapunov functions. Systems & Control Letters, 30, 57–69.

simulator and the Matlab/Simulink software to represent the controlled vehicle and the dSPACE-AutoBox hardware as a real-time controller realization platform. This is suitable for the standard industrial tools. Acknowledgment The research was supported by the National Research, Development and Innovation Fund through the project “SEPPAC: Safety and Economic Platform for Partially Automated Commercial vehicles” (VKSZ 14-1-2015-0125). The research is also supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. References Arana, C., Evangelou, S. A., & Dini, D. (2015). Series active variable geometry suspension for road vehicles. IEEE/ASME Transactions on Mechatronics, 20(1), 361–372. Baffet, G., Charara, A., & Stéphant, J. (2006). Sideslip angle, lateral tire force and road friction estimation in simulations and experiments. In IEEE international conference on control applications (pp. 903–908), Munich, Germany. Bokor, J., & Balas, G. (2005). Linear parameter varying systems: A geometric theory and applications. In 16th IFAC world congress (pp. 1–6), Prague. de Wit, C. C., Olsson, H., Astrom, K. J., & Lischinsky, P. (1995). A new model for control of systems with friction. IEEE Transactions on Automatic Control, 40(3), 419–425. Evers, W., van der Knaap, A., Besselink, I. & Nijmeijer, H. (2008). Analysis of a variable geometry active suspension. In International symposium on advanced vehicle control (pp. 350–355), Kobe, Japan. Fallah, M. S., Bhat, R., & Xie, W. F. (2009). New model and simulation of MacPherson suspension system for ride control applications. Vehicle System Dynamics, 47(2), 195–220. Goodarzia, A., Oloomia, E., & Esmailzadehb, E. (2010). Design and analysis of an intelligent controller for active geometry suspension systems. Vehicle System Dynamics, 49(1), 333–359. Habibi, H., Shirazi, K. H., & Shishesaz, M. (2008). Roll steer minimization of McPhersonstrut suspension system using genetic algorithm method. Mechanism and Machine Theory, 43, 57–67. Hirano, Y., Harada, H., Ono, E., & Takanami, K. (1993). Development of an integrated system of 4WS and 4WD by H∞ control. SAE Journal, 79–86. Hong, K. S., Jeon, D. S., & Sohn, H. C. (1999). A new modeling of MacPherson suspension system and its optimal pole-placement control. In 7th IEEE Mediterranean control conference on control and automation (pp. 559–579). Iman, M., Esfahani, M., & Mosayebi, M. (2010). Optimization of double wishbone suspension system with variable camber angle by hydraulic mechanism. Engineering and Technology, 299–306. Jarvis-Wloszek, Z., Feeley, R., Tan, W., Sun, K., & Packard, A. (2003). Some controls applications of sum of squares programming. In IEEE conference on decision and control (Vol. 5, pp. 4676–4681), Maui. Jarvis-Wloszek, Z. (2003). Lyapunov based analysis and controller synthesis for polynomial systems using SOS optimization. (Ph.D. thesis). Berkeley: University of California. Kiencke, U., & Nielsen, L. (2000). Automotive control systems for engine, driveline and vehicle Berlin: Springer. Korda M., Henrion D., & Jones C. N. (2013). Convex computation of the maximum controlled invariant set for polynomial control systems. In IEEE conference on decision and control (pp. 7107–7112), Firenze. López, A., Olazagoitia, J. L., Moriano, C., & Ortiz, A. (2014). Nonlinear optimization of a new polynomial tyre model. Nonlinear Dynamics, 78, 2941–2958. Lee, S., Sung, H., Kim, J., & Lee, U. (2006). Enhancement of vehicle stability by active geometry control suspension system. International Journal of Automotive Technology, 7(3), 303–307. Lee, U. K., Lee, S. H., Han, C. S., Hedrick, K., & Catala, A. (2008). Active geometry control suspension system for the enhancement of vehicle stability. Proceedings of the IMechE, Part D: Journal of Automobile Engineering, 222(6), 979–988. Lee, H. G., Won, C. J., & Kim, J. W. (2009). Design sensitivity analysis and optimization

13