Advanced Yaw Control of Four-wheeled Vehicles via Rear Active Differential Braking. Matteo Corno, Mara Tanelli, Ivo Boniolo, Sergio M. Savaresi. AbstractâIn ...
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThBIn6.4
Advanced Yaw Control of Four-wheeled Vehicles via Rear Active Differential Braking Matteo Corno, Mara Tanelli, Ivo Boniolo, Sergio M. Savaresi Abstract— In this paper, the problem of active lateral dynamics control of four-wheeled vehicles during braking is addressed. In certain specific conditions, a vehicle may deviate from straight line trajectory even if the steering wheel is kept still and straight. By applying a differential brake pressure, it is possible to generate a yawing moment to recover the desired trajectory. An analytical model of the vehicle dynamics of interest is derived and validated against experimental data; the model dynamic analysis shows that its behavior is strongly affected by longitudinal velocity and deceleration. Hence, this dependence is taken explicitly into account by devising an LPV controller. The performance of the proposed controller is finally assessed both via simulation and on a test vehicle.
I. I NTRODUCTION AND M OTIVATION Asymmetries in tire pressure or strong cross winds, among other exogenous phenomena, can cause lateral disturbances in the dynamics of four-wheeled vehicles during braking. These disturbances pose a potential safety hazard. Current active stability control systems are designed to intervene only if the lateral stability is compromised, but often lateral exogenous disturbances do not affect stability, but only trajectory. The problem of rejecting these disturbances can be recasted in the lane keeping framework, a well treated subject in the scientific literature. Most of the available works deal with the problem of rejecting disturbances at constant velocity [1], [2] using active steering systems [3] or visual feedback from cameras [4], [5] and other positioning systems. The results are very promising, but the related approaches neglect two important aspects: (1) by focusing on the constant velocity case, load transfer phenomena are not taken into account; (2) the proposed solutions are not cost-effective. The lane keeping problem can also be cast in a more general framework that has been arising in the past few years. In the Global Chassis Control approach [6], [7], the vehicle is analyzed as a Multi-Input-Multi-Output (MIMO) system with several actuators (e.g., semi-active suspensions, active steering, braking and semi-active differentials) and several output variables (e.g., yaw rate, side slip angle, longitudinal velocity). This approach is very promising, but it suffers from the aforementioned problems of high realization costs and actuation complexity. In this paper, a cost-effective approach to the problem of active yaw control of four-wheeled vehicles is presented. This work was supported by MIUR PRIN project Identification and adaptive control of industrial systems. M. Corno, M. Tanelli, I. Boniolo and S. M. Savaresi are with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Milan, Italy [corno,tanelli,
boniolo,savaresi]@elet.polimi.it
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
The idea is to rely on brake actuation. In fact, all modern cars are equipped with ABS systems that can independently modulate the braking pressure at each wheel: by applying a differential braking pressure it is possible to generate a yawing moment. The proposed idea has some similarities with Electronic Stability Control (ESC), as in ESC systems individual wheel braking is employed to control the vehicle. However, while ESC systems are designed to intervene only when stability is compromised, the proposed control system is designed to correct even small variations in the yaw rate which might not cause loss of stability but compromise lane keeping. In this paper experimental tests from a test vehicle for laboratory prototyping are used to derive and validate an analytical model which serves as a basis to design a gainscheduled LPV yaw rate controller. The control system is implemented on the test vehicle and its performance is experimentally assessed. The paper is organized as follows. Section II is devoted to the derivation, identification, validation and parameter sensitivity analysis of the vehicle dynamics analytical model. In Section III the model is employed to design a Linear Parameter Varying (LPV) controller and to validate the design phase. Section IV concludes the paper with some remarks and an outlook to future work. II. D OUBLE -T RACK M ODEL In this Section an analytical nonlinear model of the vehicle dynamics of interest is derived. Further, some unmeasurable parameters are identified from data and the model is validated and discussed. In the literature (see for example [8]) many vehicle dynamics models have been proposed; in this work, the attention is focused on describing the dynamic relationship linking a differential braking pressure input applied at the rear wheels and the resulting lateral vehicle motion. In order to model all the relevant dynamics, the classical double-track model has been extended by introducing a nonlinear tire-road friction description and load transfer phenomena. As the attention is focused on a specific maneuver, namely braking performed with small steering angle at nearly constant longitudinal decelerations, some assumptions can be made. Namely, (A.1) the longitudinal velocity dynamics varies slowly with respect to heave dynamics. We can hence model the load transfer as dependent on the longitudinal deceleration only, neglecting suspensions dynamics. (A.2) lateral load transfer is neglected as only small steering angle values are considered. (A.3) rolling and aerodynamic resistance as well as transmission and engine friction are
5176
ThBIn6.4 neglected, as the forces they generate are smaller than those generated by the brakes in the braking maneuvers of interest. (A.4) the braking torque Tb at each wheel is considered proportional to the local brake pressure pb , i.e., Tb = Kb pb . Fig. 1 schematically represents the main parameters and conventions adopted in the analytical derivation of the considered vehicle model. αi j , i = { f , r}, j = {l, r} are the wheel
Wheels
ω˙ i j = 1/Jw (−Tb i j − rw Fx i j ), i = { f , r}, j = {l, r}.
(4)
The tire longitudinal forces are modeled according to Burkhardt model, [9]; thanks to the small steering angles hypothesis (A.2), a linear model has been employed for the lateral forces. Fx i j = −Fz i j c1 (1 − e−c2 λ i j ) − c3 λ i j Fy i j = Fz i jCα (λ i j )α i j , i = { f , r}, j = {l, r},
(5)
where c1 , c2 and c3 are the tire-road friction parameters, λi j are the longitudinal wheel slips (defined as positive during braking). The tire-road friction coefficient, i.e., Fx i j /Fz i j , as a function of the wheel slip is shown in the left plot of Fig. 2. For small vehicle side slip angles, the tire side slip angles can 1.4
45 asphalt,dry
Double-track vehicle model nomenclature.
40 35
1
pressure [bar]
Fig. 1.
friction coefficient
1.2
concrete, dry
0.8 0.6
asphalt,wet
0.4
v˙ = 1/m Fxrl + Fxrr + (Fx f l + Fx f r ) cos(δ ) +(Fy f l + Fy f r ) sin(δ )
(1)
u˙ = 1/m Fyrl + Fyrr + (Fy f l + Fy f r ) cos(δ ) −(Fx f l + Fx f r ) sin(δ ) + rvm
(2)
Lateral
Yaw r˙ = 1/J((Fyrl + Fyrl )b + (−Fxrl + Fxrl )w + (Fy f l + Fy f r ) cos(δ )a − (Fx f l + Fx f r ) sin(δ )a + (−Fx f l + Fx f r ) cos(δ )w + (−Fy f l + Fy f r ) sin(δ )w)
(3)
0
25 20
identified response
15
0.2
side-slip angles, β is the vehicle side-slip angle, Fx i j , Fy i j , Fz i j the longitudinal, lateral and vertical forces, respectively, δ the steering angle at the wheels, ψ and r the yaw angle and yaw rate, v and u the longitudinal and lateral velocity in body-fixed coordinates. The vehicle geometric parameters are a, b, w and h, which stand for – respectively – the distance from the front axle to the center of mass, the distance from the rear axle to the center of mass, half the track width and the height of the center of mass. Other symbols are J, m, Jw which represent the yaw moment of inertia of the vehicle, the total vehicle mass and wheel moment of inertia, respectively; ωi j are the longitudinal wheel angular velocities, rw is the wheel radius (assumed equal for all wheels) and Tb i j the braking torque at each wheel. The vehicle model is obtained via force and momentum balances relative to the model 7 degrees of freedom, i.e., 2 linear (longitudinal and lateral) and 5 rotational (yaw and the four wheels) degrees of freedom. This yields Longitudinal
18 bar
30
10 0
0.2
0.4
λ
0.6
0.8
1
5
25 ms 0
0.05
0.1
0.15 0.2 time [s]
0.25
0.3
Fig. 2. Longitudinal friction coefficient for different surfaces (left plot). Measured (dot dashed) and simulated pressure (solid) at the brake caliper in response to a step variation of the set-point (solid line) (right plot).
be written as a function of the state space variable according to the following expressions
α f l = (v sin(β ) + ra)/(v cos(β ) − rw) − δ α f r = (v sin(β ) + ra)/(v cos(β ) + rw) − δ αrl = (v sin(β ) − rb)/(v cos(β ) − rw) αrr = (v sin(β ) − rb)/(v cos(β ) + rw)
(6)
where the vehicle side slip angle β can be computed as u β = atan . (7) v Note that in the lateral forces expression (5) the cornering stiffness Cα (λ ) is written as a function of the longitudinal wheel slip λ ; thus modeling the loss of lateral force due to braking. (8) Cα (λ ) = Cα (0) − kλ , where k is chosen so as to cause an 80% reduction of the lateral friction coefficient at λ = 0.4. Further, the forces exerted by the tire are a function of the vertical load (see Equation 5). This dependence is crucial to correctly study the effects of the braking maneuver on the lateral dynamics. In view of Assumption (A.1), the vertical forces can be written as mg b hm Fzi = ± v˙ 2 a+b 2(a + b) where the first term represents the static load distribution and the second models the dynamic load transfer.
5177
ThBIn6.4 A. Parameters Identification and Model Validation The proposed dynamic vehicle model depends on several parameters, which can be grouped into two categories: easily measurable parameters (such as lengths, widths and masses) and unmeasurable ones. The second set of parameters must be identified from data. The identified parameters are the cornering stiffness Cα (0) and the yaw inertia J. An accurate model of the tire longitudinal characteristic is available for the test vehicle, and thus its parameters do not need to be identified. The identification of the two remaining parameters, i.e., J and Cα (0) has been performed on runs executed at constant longitudinal velocity by applying a square wave input pressure excitation at the rear brakes. Note that, even though the driver was requested to keep the steering wheel straight, i.e., with δ = 0◦ , it could not be kept always constant. As such, the resulting identification problem is not genuinely a SISO one. In fact, two inputs act on the system: a commanded input (the pressure gradient at the rear wheels) and a non controllable input (the steering angle δ ). Hence, in the identification procedure the measured steering wheel angle δ is treated as a measurable disturbance and the vehicle model using such a δ as an input. By numerically solving the yaw rate simulation error minimization on three test runs carried out at different longitudinal velocities (80, 100 and 120 km/h), a yaw rate moment of inertia J = 1125 kg m2 and a front and rear cornering stiffness values Cα f (0) = 12 rad−1 and Cαr (0) = 15.9 rad−1 , respectively, have been found. The effectiveness of the identification procedure can be appreciated by inspecting Fig. 3 which shows the results obtained in two validation tests at constant velocity. It should simulated yaw rate ∆P
yaw rate [normalized]
yaw rate [normalized]
steer
simulated yaw rate
measured yaw rate
measured yaw rate
∆P
0
0.5
1 time [s]
1.5
steer
2
0
0.5
1
1.5
2
order time-invariant linear system with uncertain gain.The closed loop pressure dynamics natural frequency is 29 Hz with a damping coefficient of 0.5 and a pure delay of 25 ms. B. Sensitivity Analysis Once the vehicle model has been derived and validated, it is interesting to study how the operating conditions may affect the vehicle lateral dynamics. This can be done by linearizing the nonlinear model around some specific trim conditions. The operating point is defined by longitudinal velocity and longitudinal acceleration. Thus the trim conditions are found fixing the value of these two variables and considering zero lateral velocity and zero yaw rate. Note that, when the longitudinal acceleration value is non-zero the resulting trim condition is not an equilibrium, so that the linearization is not carried out around a proper steady state condition (see also [10]). Recalling the longitudinal dynamics in (1), it is easy to see that for any given nonzero deceleration value there are infinite trim conditions in terms of wheel velocities, among which that with equal longitudinal slips on the four wheels has been selected as it represents the most balanced braking condition. Fig. 4 shows the Bode diagrams of the transfer function from rear axle pressure gradient to yaw rate for different values of longitudinal velocity and acceleration. Inspecting Fig. 4, the following remarks can be made: (1) the upper bound of the system bandwidth (defined as the frequency at which the frequency response magnitude loses 3dB with respect to the DC gain) is around 3 Hz; (2) the low frequency gain increases as velocity increases; (3) the damping of the complex poles decreases as velocity increases. From the control system design perspective, this causes a loss of phase margin; (4) the low frequency gain increases as deceleration increases; (5) as deceleration increases, the poles move toward the imaginary axis. These results are experimentally confirmed. For example, Fig. 5 shows the open-loop yaw rate response to a step variation of the left rear braking pressure at different velocities. The effect of varying the longitudinal velocity is consistent with the sensitivity analysis both in terms of magnitude and damping properties. The overall analysis highlights that both
2.5
time [s]
120 km/h
∆P set point
be noted that in the identification procedure the actual measured pressure at the brake caliper has been employed. This measurement is available only for identification purposes, as in the final production vehicle pressure sensors are not available. This, as can be seen in the right plot Fig. 2 which shows the measured pressure at the rear brake caliper in response to a step variation of the set-point, introduces an uncertainty in the actuator model. Static errors up to 50% are to be expected. It should be also noted that the actuated pressure is always lower than the set-point. The experimental data also show that the actuator can be modeled by a second
yaw rate [normalized]
Fig. 3. Model Validation. Measured and simulated yaw rate, steering wheel position and pressure gradient request at constant velocity: 100 km/h (left plot) and 120 km/h (right plot).
v = 80 km/h
5
5.5
6
6.5 time [s]
7
7.5
8
Fig. 5. Open loop responses to a step variation of the left rear braking pressure at a deceleration of 0.4 g.
longitudinal velocity and acceleration play an important role in determining the dynamic behavior of the system. Hence, this influence has to be taken into account when designing the controller.
5178
−60
−70
magnitude [dB]
magnitude [dB]
ThBIn6.4
−70
−80
−80
increasing velocity
increasing deceleration
−90
−90
−100 −100 −110 0
10
1
10
2
10
−120 −2 10 0
−1
10
0
increasing velocity
1
10
10
2
10
increasing deceleration
phase [°]
−1
10
phase [°]
−110 −2 10 0
−50
−50
−100 −100 −150 −150 −2 10
−1
10
0
10 frequency [Hz]
1
10
2
−2
10
10
−1
10
0
10 frequency [Hz]
1
10
2
10
Fig. 4. Bode diagrams of the transfer function from rear axle pressure difference to yaw rate at constant velocity from 50 to 120 km/h (left), and at constant acceleration from 0 to -0.8g at 80 km/h.
III. LPV YAW C ONTROLLER D ESIGN In the previous Section it has been shown that velocity and acceleration have an important role on the vehicle dynamics. As such, is it clear that a fixed-structure yaw controller cannot yield good results in all working conditions. This Section is devoted to the design of a gain scheduled yaw controller. In particular, a Linear Parametrically Varying (LPV) controller is proposed. LPV control system design techniques, in fact, allow to design gain-scheduled controllers with a-priori stability and performance guarantees. The theory of LPV systems has been extensively documented in e.g., [11]–[13]. For a review of the specific technique applied in this work, the Reader may refer to [14]. The main objective of the proposed control system is to control the yaw rate during straight running braking. As already mentioned, the proposed controller is intended as an advanced controller for lane keeping purposes, which intervenes only if the ABS system is not activated. If the ABS system is triggered, the yaw controller is switched off; this allows to take into consideration only situations where the vehicle dynamics can be regarded as stable. Other more robust but less precise control logics (for example ESC systems) will be activated if stability is compromised. To describe the controller design rationale, refer to Fig. 6, which depicts the interconnections diagram that defines the design objectives. In devising the interconnections one has to take into account the fact that the system is strongly affected by uncertainties. There are mainly two sources of uncertainty, both graphically depicted in Fig. 2. The former is due to the sensor-less pressure control loop. Although the actuator dynamic response to a step variation of requested pressure is fairly repeatable, the static behavior is not. Static uncertainties up to 50% are to be accounted for in the controller design. The second cause of uncertainty is due to the impossibility to accurately estimate the tire-road friction coefficient online. Hence, in order to guarantee safety, the tire-road friction coefficient is treated as an uncertain parameter. Based on these considerations, the weighting functions have been selected.
Fig. 6.
Yaw control system interconnections.
1) The control problem is formulated as a model matching problem, and Wmod (s) represents the second order reference model. A natural frequency of 0.9 Hz and damping coefficient of 0.9 have been chosen so as to guarantee a well damped response with a settling time of about 1 s. The settling time has been chosen coherently with the chassis and actuators dynamics. 2) The model matching error is weighted by an integral-like weighting function to ensures little or none DC error. 3) An output disturbance model is included in the interconnection to increase the robustness of the closed loop system. In the synthesis, the weighting function Wn is kept constant over all frequencies to model white measurement noise. 4) Wact allows to ensure a bounded control action. Specifically, the following actuator weight is used Wact (s) = µ
1/(15 · 2 · π )s + 1 , 1/(300 · 2 · π )s + 1
which penalizes the actuation above 20 Hz. The gain of the weighting function has been tuned so that the maximum available control authority results in a peak of approximately 40 bar in the rear braking pressure difference in the face of a 1◦ /s step variation in the yaw rate. 5) The ABS pressure control loop is modeled as a second order low pass filter with a natural frequency of 29 Hz and a 2nd order Pad´e approximation of a 25 ms pure delay. In the scheme of Fig. 6 it is designated A(s). 6) G(s) models the single input - single output yaw dynamics
5179
ThBIn6.4
X(v, λ ) = X0 + vX1 + aX2 + avX3 + a2 X4
uncontrolled system
yaw rate
fixed controller
0 scheduled controller
14
15
16
17 time [s]
18
fixed controller
19
20
scheduled controller rear left
rear left
front left pressure
front left pressure
adjusted with a scaling and a normalizing factor. 7) Wunc (s) is used to express modeling uncertainty. The input disturbance weighting function is set to a static gain so as to model uncertainties of up to 50% which take into account the effects of both pressure control loop uncertainty and tire-road friction coefficient variations. Note that the gain of G(s) has been reduced so that the pressure loop control uncertainty can be treated as symmetric. The LPV controller synthesis is the carried out on a grid of 24 elements: 6 longitudinal velocity values (v=[40, 60, 80, 100, 110, 120] km/h) and 4 longitudinal acceleration values (a=[-0.2, -0.4, -0.6, -0.8] g). The following basis functions are employed to approximate the infinite dimensional LPV problem
rear right
rear right front right
front right 0
0 14
15
16
17
18
19
14
20
15
16
17
18
19
20
Fig. 7. Closed loop simulation performed at 120 km/h; yaw rate (upper plot), fixed structure controller brake pressures (lower right) and gainscheduled controller brake pressures (lower left).
Y (v, λ ) = Y0 + vY1 + aY2 + avY3 + a2Y4 . between the controlled system and the uncontrolled system are apparent. The uncontrolled system yaw rate response to a 3 bar difference in the front braking pressure reaches a peak of 0.9◦ /s, whereas the controlled system worst case is 0.3◦ /s, i.e., a peak reduction of 66%. (2) The LPV controller induces a lower yaw rate overshoot; in fact, at 120 km/h the gain of the scheduled controller is higher than that of the fixed structure controller, thus allowing a faster closed-loop yaw rate response. (3) The situation in the yaw rate undershoot is opposite. The scheduled controller causes slightly higher negative yaw rate values. This is a behavior which was not predicted by the linear analysis, and it is a consequence of the time-varying velocity. As far as trajectory following is concerned, however, this yaw rate behavior is to be preferred to that achieved by the fixed-structure controller as it is more symmetric. Further, Fig. 8 shows the results obtained in the same type of maneuver, but performed at a longitudinal speed of 80 km/h and maximum deceleration of -0.5 g. This is the situation where the LPV controller is expected to yield significantly better results as compared to the fixed structure one. uncontrolled system
yaw rate
fixed controller
0
scheduled controller
15
15.5
16
16.5
17 time [s]
17.5
fixed controller
18
18.5
rear left
rear left front left pressure
rear right
front left
rear right
front right 0
front right
0 15
16
17 time [s]
19
scheduled controller
pressure
Finally, the following rate bounds on velocity and acceleration have been employed: γ = [±9.8m/s2 , ±9m/s3 ]. These rate bounds have been derived from analysis of experimental data considering a large number of different braking maneuvers. Linear analysis of the loop transfer functions at the grid vertexes confirms that performance and stability are guaranteed pointwise with a bandwidth of 0.5Hz and a phase margin of 85 ◦ meaning that the design goals are successfully met in the linear domain. In order to assess the advantages of the LPV controller , a H∞ fixed structure controller has been also devised using the same weighting functions. In order to guarantee stability in all working conditions without loosing too much performance, the H∞ controller has been designed for the nominal condition v = 120 km/h and a = -0.4 g. This is not the worst case condition, but it has been selected to represent a trade off between robustness and performance. Furthermore, it represents a realistic condition as even in the case of a hard braking, a few seconds are needed before reaching high decelerations and during that span of time the velocity drops restoring a more favorable vehicle dynamic behavior. To validate the proposed control laws a series of closedloop simulations have been carried out and analyzed. The simulations have been run on the full nonlinear system derived and validated in Section II augmented with the full actuator dynamics. In order to simulate realistic driving conditions, the initial velocity and brake pressures recorded from track tests are used as inputs for the nonlinear simulator. The steering action is kept at 0◦ , as this allows to decouple the effect of the driver action on the steering wheel and the yaw control. In the simulations, the reference yaw rate is kept at 0◦ and a disturbance is applied to the system in the form of a pressure offset on the front left wheel braking pressure. Fig. 7 shows a plot of the yaw rate and brake pressures for a maneuver performed at the initial speed of 120 km/h and with a maximum deceleration of -0.8 g. Inspecting Fig. 7, some remarks are due. (1) The difference
18
19
15
16
17 time [s]
18
19
Fig. 8. Closed loop simulation performed at 80 km/h; yaw rate (upper plot), fixed structure controller brake pressures (lower right) and gain-scheduled controller brake pressures (lower left).
5180
ThBIn6.4 The results of Fig. 8 confirm that in this case the maximum yaw rate obtained with the uncontrolled system is 0.55◦ /s while the fixed controller and the gain-scheduled controller guarantee a maximum yaw rate deviation of 0.3◦ /s and 0.14◦ /s, respectively. As expected, the gain-scheduled controller outperforms the fixed structure controller in this condition which is far from the nominal one used in the fixed structure controller design. Finally, Fig. 9 plots the comparison between the controlled and uncontrolled system on experimental data obtained in a hard braking maneuver at 120 km/h with a highly unbalanced vehicle when the driver action is considered. High speed tests are more meaningful because in this condition the lateral drift due to the unbalanced load is more evident. Analyzing Fig. 9, it can be appreciated that the control system successfully controls the lateral dynamics of the vehicle, thus counteracting the effects of unbalanced load. The effectiveness of the control system is also quantified by the tracking error variance shown in Fig. 9: as can be seen, a reduction of approximately 90% has been achieved with respect to the open loop system behavior.
yaw rate [normalized]
1 controlled system 0.5
0
−0.5 uncontrolled system
−1
normalized tracking error variance
0
0.5
1
1.5
2
2.5 time [s]
3
3.5
4
4.5
5
1 0.8 0.6 0.4 0.2 0
uncontrolled
controlled
Fig. 9. Experimental results of a hard braking at v=140 km/h; yaw rate (upper plot) and tracking error variance (bottom plot).
IV. C ONCLUSIONS AND F UTURE W ORK In this paper, the problem of cost effective advanced yaw dynamics control for four-wheeled vehicles via active rear
differential braking has been addressed. An ad-hoc analytical and experimentally validated double-track model has been used to design an LPV robust controller. The gain scheduled controller is shown (both in simulation and experimentally) to be superior with respect to a fixed-structure controller. If the disturbances are too intense, the controller may saturate the actuators. As such, a possible extension of the work presented herein lies in considering also front wheel braking pressure modulation. The resulting MISO problem is currently under investigation. R EFERENCES [1] J. Ackermann and T. Buente, “Automatic Car Steering Control Bridges over the Driver Reaction Time,” Kybernetica, vol. 33, pp. 61–74, 1997. [2] B. Guvenc and L. Guvenc, “Robust Two Degree-of-Freedom Add-On Controller Design for Automatic Steering,” Control Systems Technology, IEEE Transactions on, vol. 10, no. 1, pp. 137–148, 2002. [3] T. B¨unte, D. Odenthal, B. Aksun-G¨uvenc¸, and L. G¨uvenc¸, “Robust Vehicle Steering Control Design Based on the Disturbance Observer,” Annual Reviews in Control, vol. 26, no. 1, pp. 139–149, 2002. [4] R. Risack, N. Mohler, and W. Enkelmann, “A Video-Based Lane Keeping Assistant,” in Intelligent Vehicles Symposium, Proceedings of the IEEE, 2000, pp. 356–361. [5] S. Ishida, J. Gayko, H. Ltd, and J. Tochigi, “Development, Evaluation and Introduction of a Lane Keeping Assistance System,” in Intelligent Vehicles Symposium, 2004 IEEE, 2004, pp. 943–944. [6] Y. Shibahata, “Progress and Future Direction of Chassis Control Technology,” Annual Reviews in Control, vol. 29, no. 1, pp. 151–158, 2005. [7] P. Falcone, M. Tufo, F. Borrelli, J. Asgari, and H. Tseng, “A Linear Time Varying Model Predictive Control Approach to the Integrated Vehicle Dynamics Control Problem in Autonomous Systems,” in Decision and Control, 46th IEEE Conference on, 2007, pp. 2980– 2985. [8] T. Gillespie, Fundamental of Vehicle Dynamics. Society of Automotive Engineers, Inc., 1992. [9] U. Kiencke and L. Nielsen, Automotive Control Systems. SpringerVerlag, Berlin, 2000. [10] S. Savaresi, M. Tanelli, and C. Cantoni, “Mixed Slip-Deceleration Control in Automotive Braking Systems,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 129, no. 1, pp. 20–31, 2006. [11] G. Becker, “Quadratic Stability and Performanc of Linear ParameterDependend Systems,” Ph.D. dissertation, Mechanical Engineering, University of California, Berkeley, 1993. [12] F. Wu, “Control of linear parameter varying systems,” Ph.D. dissertation, Mechanical Engineering, University of California, Berkeley, 1995. [13] G. J. Balas, “Linear, Parameter-Varying Control and its Application to a Turbofan Engine,” International Journal of Robust and Nonlinear Control, vol. 12, pp. 763–796, 2002. [14] M. Corno, S. Savaresi, and G. Balas, “On Linear Parameter Varying (LPV) Slip-Controller Design for Two-Wheeled Vehicles,” International Journal of Robust and Nonlinear Control, pre-prints.
5181
