Design of a Vehicle Steering Assist Controller Using ... - CiteSeerX

Design of a Vehicle Steering Assist Controller Using Driver Model Uncertainty Liang-Kuang Chen and A. Galip Ulsoy Department of Mechanical Engineering University of Michigan, Ann Arbor, MI 48105-2125, USA

Abstract Single vehicle road departure (SVRD) accidents lead to over 1/3 of highway fatalities in the United States. A steering assist controller to reduce SVRD accidents is proposed. Previous studies using driving simulator data illustrate that the parametric uncertainty can represent the change in driver steering behavior during long driving task. The uncertainty results show that occasionally the driver can perform very poorly. A serial steering control approach is investigated. The controller is used to attempt robust performance under the influence of driver uncertainty. A robust Smith predictor controller is employed to handle the delay in the driver model. The results indicate that the poor stability situations are successfully avoided and robust stability is achieved. However, due to large variations in driver behavior, robust performance is not achieved. An adaptive version of the robust Smith predictor controller is shown to reduce the uncertainty in the driver model. The designed controllers are evaluated using frequency domain analyses and computer simulations.

1. Introduction The steering control of vehicle motion is crucial for vehicle safety. Single vehicle road departure (SVRD) accidents account for approximately 20 percent of all vehicle crashes, and nearly 40 percent of fatalities (NHTSA, 1999). In order to prevent SVRD accidents, it is desirable to design a steering assist controller. In previous work the authors have investigated modeling driver uncertainty from driving simulator data (Chen and Ulsoy, 1999). The data is collected from the Ford Driving Simulator, as shown in Fig. 1 (Pilutti and Ulsoy, 1999). Twelve subject drivers were asked to drive the simulator for two hours, in order to ascertain the effect of fatigue. The experiments were scheduled to begin at around 1:30 pm, just before the “diurnal (circadian) dip” (Kryger et. al, 1987) in the alertness level. The parametric uncertainty is used to represent change in the driver steering behavior with time. The uncertainty results show that although the average driver model exhibits acceptable performance, (i.e., in terms of phase margin, gain margin, and crossover frequency), the driver can perform very poorly during the two-hour test. This can be seen from the Nyquist plots of the driver-vehicle system, as shown in Fig. 2. The different curves in Fig. 2 are computed from the time-varying driver models. The occasional low stability margins are evident. The degradation in stability margins and response speed may be an indication of low alertness level of the driver. Therefore, the uncertainty in the driver can potentially cause road departure accidents. Unfortunately, the low stability margins cases do not necessarily correspond to more fatigued drivers, as expected. The results show that the stability margins do not 1

reduce as simulation time increases. That is, the stability margins are not worse during the later portion of the simulation. However, it is observed that the average crossover frequency reduces as time increases. The average crossover frequency is 0.74 rad/sec for the first 30 minutes and 0.59 rad/sec for the last 30 minutes. This might explain the fact that the standard deviation of lateral position error increases with time, as discussed in (Pilutti and Ulsoy, 1999). One might conjecture that as the driver becomes tired, he will sacrifice tracking performance to maintain reasonable stability margins. The goal of this work is to design a steering assist controller to improve stability margins and crossover frequency under the influence of parametric uncertainty. Fig. 3 shows the serial controller structure that is of interest in the article. It is noted that other structures can also be employed, e.g., using a parallel controller. Fig. 4 shows a more general control structure that contains both a serial and a parallel controller to provide more design flexibility. The parallel controller has autonomous steering capability, however, the price that must be paid is the extra sensors and actuators needed to implement a parallel system. Although the parallel controller is expected to be less sensitive to the delay and uncertainty in the driver model, the interaction between the parallel controller and the driver is still a major issue that needs further investigation. On the other hand, the implementation of serial controller requires fewer sensors and actuators, but it does not have independent steering capability. The serial controller is selected as the starting point for the design of the steering assist controller and is the focus of this paper.

Fig. 1: Ford Driving Simulator screen display

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2. Literature Review Vehicle active safety systems are a subclass of advanced vehicle control systems that are designed to improve driving safety while the driver is still in primary control of the vehicle motion. Steering assist is one of the active safety systems designed to prevent road departure. Naab and Reichart (1994) reported on the driver assistance system developed by BMW in the framework of PROMETHEUS. They implemented a parallel steering wheel control by comparing the calculated steering wheel angle generated from computer vision system with the driver’s steering angle. Birch (1995) described a steering assist system designed by Jaguar/Lucas. Steering aids are introduced via power steering. A warning is issued if the car starts to move off course. These systems are actually extensions of automated steering systems in that the driver input is treated as a disturbance to the controller. An upper limit is set on the controller authority so that the driver can override the controller when necessary. The interaction between the driver and the controller is not addressed in the control system design. Pilutti and Ulsoy (1998) show that a lane departure warning system can be improved by considering variations in driver state. An active controller that uses direct intervention of vehicle motion will lead to driver-controller interaction issues, and generally requires a thorough investigation of driver behavior before active safety controllers can be implemented, (e.g., (LeBlanc et al., 1996(a)), (LeBlanc et al., 1996(b)) and (Pilutti, 1997)). Developments such as steering through differential braking (Pilutti et al., 1998) and yaw rate estimation (Sivashankar and Ulsoy, 1998) also support the development of steering assist systems, as well as systems for warning and/or full steering intervention. The design of active safety systems requires a good representation of driver behavior. Although driver steering control models have been studied extensively, driver model uncertainty has received relatively little attention. Previous research has discussed systematic methods to compute the driver model and its uncertainty using a system identification approach (Pilutti and Ulsoy, 1999; Chen and Ulsoy, 1999). In general, model uncertainty can be divided into structured uncertainty (e.g., parametric uncertainty) and unstructured uncertainty (e.g., additive uncertainty). The unstructured uncertainty can represent the modeling error due to model order and unmodeled dynamics. The parametric uncertainty can be used to represent the driver variation with time. The computation of both types of uncertainty has been presented in (Chen and Ulsoy, 1999). Fig. 5 shows a sample parametric variation of a second order driver model of the form

b1q −1 + b2 q −2

1 + a1q −1 + a2 q − 2

, where a delay of one sampling interval (0.1 sec) is used.

An important characteristic of existing driver steering control models is that they usually include a time-delay element. An effective control technique for systems with time delay is to employ the predictor control scheme due to O.J.M. Smith (1959). However, the Smith predictor controller is designed based on the assumption that an exact model of the system is available. In reality, the model of a physical system is never exact. The Smith predictor can be very sensitive to model uncertainty (Palmor, 1980). Therefore, a robust Smith predictor scheme has been reported to provide robustness to the Smith predictor control (Whalley and Zeng, 1996; Abdelrahman and Moore, 1997). This robust Smith predictor structure will be applied to the steering assist controller design

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presented in this paper. A brief illustration of the robust Smith predictor structure will follow. The real system with the time delay is divided into two parts: a delay-free part and a pure time delay part. For the driver-vehicle system, the open loop transfer function for the driver (Gd) and the vehicle (Gv) can be represented as GdGv = Gdo Gv e −Ts := Go e −Ts

(1)

where Gdo is the delay free part of the driver model, and Go represents the delay free part of the driver-vehicle combination. T is the amount of time-delay that is determined from identification of the driver model. The control structure of the Smith predictor is shown in Fig. 6. The controller C(s) is designed for the system without the delay, i.e., Go. It can be shown that the equivalent transfer function for the Smith predictor controller is Gs (s) =

C (s) 1 + C ( s )Go ( s )(1 − e −Ts )

(2)

The closed-loop transfer function becomes y C ( s )Go ( s )e −Ts = y d 1 + C ( s )Go ( s )

(3)

The poles of equation (3) do not suffer from the excessive phase lag caused by the time delay. However, the “model matching” requirement is needed for this predictor to function properly. Equation (2) indicates that the controller is dependent on the system model Go. Due to the presence of driver model uncertainty, the performance of the overall system cannot be guaranteed. Therefore, the robust Smith predictor is introduced to enhance system robustness. The idea used here is to combine the Smith predictor with a controller that is robust to system uncertainty. The combined scheme has been shown to benefit from both the robustness of the robust controller and the large phase lead of the Smith predictor (Whalley and Zeng, 1996; Abdelrahman and Moore, 1997). Therefore, the first step is to design a robust controller for the no-delay part of the system. After the robust controller is designed, the Smith predictor structure will be inserted to complete the robust Smith predictor structure.

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Fig. 6: Smith predictor control structure

3. Robust Controller Design There are several powerful tools for robust controller design available in the literature, e.g., H∞, QFT, and µ synthesis. Although in many cases the advantage of these tools are more noticeable for MIMO systems, they can still be applied to SISO systems and provide useful insights into the design problem. In this study two robust control design techniques are investigated, namely, Quantitative Feedback Theory (QFT) and H∞ control. To begin the robust controller design, the designer must provide model of the uncertainty and desired performance specifications. 3.1 Design Specifications The driver model and its uncertainty obtained from simulator data (Chen and Ulsoy, 1999) is used in the controller design process. Although the system identification method can compute both structured and unstructured uncertainty, only structured uncertainty, (i.e., parametric uncertainty), is used because it represents the change in

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driver behavior being addressed in this paper. An example plot of the parametric variation has been shown in Fig. 5. For QFT design, a model template at selected frequencies to represent the model uncertainty can be obtained directly from the structured uncertainty. Fig. 7 shows an example of such a template for the driver-vehicle system where the selected frequencies are 0.1, 0.4, 1, 4, and 10 rad/sec. The same uncertainty can also be transformed into frequency domain uncertainty bounds for H∞ control design. Fig. 8 shows the multiplicative uncertainty (|∆M|). A weighting function W3 that bounds the uncertainty is also included in Fig. 8. W3 is also the weighting function that penalizes the complimentary sensitivity function of the system. Therefore, W3 is chosen to be improper deliberately so that the resulting H∞ controller can be proper. The model uncertainty shown in Figures 7 and 8 are computed from the same set of parametric variations. It is seen that the driver exhibits large variation in low frequency gain. This is reasonable for a real driver, but it causes the robust controller to be too conservative. Adaptive control with on-line estimation is investigated to resolve this. Templates of GdGv 30

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Figure 7: Sample model template of driver-vehicle system The parametric uncertainty is found to be significant and a controller that achieves robust performance is difficult to find. Specifically, for the subject driver investigated here, the averaged performance can be summarized as about 32 deg phase margin, 12 dB gain margin, and a crossover frequency of 0.68 rad/sec. However, the same driver may

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show only 6.5 deg phase margin, 4.8 dB gain margin, and a crossover frequency of 0.26 rad/sec during the test. The variation is significant and the robust performance specifications cannot be too demanding. Therefore, the desired robust performance is chosen to be similar to that of an average driver’s performance. The following specifications for the overall system are eventually selected: (1) (2) (3) (4)

Phase margin > 20° Gain margin > 6 dB Crossover frequency above 0.6 rad/sec Open loop gain (|GdGsGv|) is large (i.e., >20 dB) at low frequency (i.e., at 0.1 rad/sec)

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Figure 8: Multiplicative uncertainty and weighting W3 These specifications need to be modified when designing the robust controllers because the delay element will appear after the Smith predictor is inserted, as seen in equation (3). This delay will introduce additional phase lag to the system. Considering the time delay (T) to be 0.1 sec and known, the new specifications are: (1) (2) (3) (4)

Phase margin > 24° Gain margin > 6 dB Crossover frequency above 0.6 rad/sec Open loop gain (|GdGsGv|) is large (i.e., >20 dB) at low frequency (i.e., at 0.1 rad/sec)

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The above performance specifications can be further transformed into a weighting function (1/W1) on the sensitivity function (S) so that it can be used in the QFT and H∞ design algorithms. The controller is designed so that the resulting sensitivity function (S) can be below 1/W1 for all frequency. The resulting weighting function on the sensitivity function is shown in Fig. 9. Although this weighting function is not completely equivalent to the specifications on phase margin and gain margin, it provides similar performance results. |1/W1| 5

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Figure 9: Sensitivity function weighting 1/W1

3.2 The Robust Controller To design the controller using the H∞ technique, a “mixed sensitivity synthesis problem” is formulated based on the weighting functions W1 and W3. It is observed that robust performance cannot be met by using the H∞ controller. Alternatively, a controller that can minimize the specified “mixed sensitivity cost function” is generated. The resulting H∞ controller can be presented in Bode plots, as shown in Fig. 10. The H∞ controller can provide significant phase lead and robustly stabilize the lateral dynamics. The H∞ controller is found to have high gain in high frequency range. However, the gain increase above 2 rad/sec is not necessary because the crossover frequency seldom exceeds 2 rad/sec. Since the problem addressed here is a SISO system, it is easy to further modify the controller using traditional loop shaping or other robust control techniques. 9

The H∞ controller has another disadvantage: the controller order. Based on the way the problem is formulated, the resulting H∞ controller shown in Fig. 9 is a 9th order one. The high order of the H∞ controller makes it difficult to implement on physical systems. The problem becomes worse when the Smith predictor is added to the system. The high complexity of the H∞ robust Smith predictor reduces its plausibility in practical applications. It is decided to design a 4th order controller that approximates the H∞ controller in low frequency range, using the QFT tool. The QFT technique provides a graphical tool to show the tradeoff between the performance and uncertainty. An example of the QFT graphical design template is shown in Fig. 11. The resulting QFT controller is also shown in Fig. 10. It is seen that the QFT controller can satisfy what is expected from the robust controller; therefore, it will be used in the robust Smith predictor structure. One advantage of the QFT control design is the nominal compensated system response can be seen directly on the Nichols chart. This feature helps reducing the design effort significantly. The H∞ control design has the advantage that the frequency domain weighting functions can be obtained directly from the model uncertainty and performance specifications. Therefore, the controller design involves less trial-and-error routines. On the other hand, the QFT control algorithm involves significant trial-and-error work and the design procedure is time-consuming. This is the price that must be paid for the extra flexibility of the QFT control algorithm. The H∞ control algorithm provides a convenient approach to obtain a controller template for comparison with QFT control designs. H and QFT controllers ∞

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4. Robust Smith Predictor Controller Evaluation The controller C shown in Fig. 5 is replaced by the designed robust controller to complete the robust Smith predictor structure. Equation (2) can be used to compute the frequency response of the robust Smith predictor controllers based on the nominal system model. The Bode plots of the resulting controller (Gs) are shown in Fig. 12. The controller without the Smith predictor is also included in order to compare the effect of the Smith predictor. It is observed that the Smith predictor only provides small change to the robust controller. This suggests that the extra effort with the Smith predictor structure may not be worthwhile for the lateral control system considered here. The robust Smith predictor controller is evaluated in the frequency domain. First the nominal driver model is used and the resulting open-loop frequency response is shown in Fig. 13. Fig. 13 shows that the robust Smith predictor controller can satisfy the desired performance specifications for the nominal system model. This implies that if the driver performs consistently like the nominal model, the desired performance is met. It is also observed that the controller only provides slight improvement in stability margins for the nominal system. However, the purpose of the controller is to avoid the poor stability margin cases that usually have much slower crossover frequency (i.e., around 0.2 rad/sec). Fig. 12 shows that the controller is very effective in providing phase lead in those frequency ranges.

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Given that the nominal performance is met, the robust performance is investigated. The source of the driver model uncertainty is the parametric variations of driver models. The parametric variation represents the time varying behavior of driver steering control. The open loop frequency response is plotted for all the driver models that contribute to the parametric variation. The resulting open loop frequency responses for the system with robust Smith predictor controller are shown in Fig. 14. Fig. 14 shows that the resulting crossover frequencies are between 0.3 rad/sec and 1.5 rad/sec, which is slightly higher than the case without the controller. In order to compare the robustness of the robust Smith predictor controller, a Nyquist plot is used to show the phase and gain margins.

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Fig. 14: Open loop frequency response of compensated system (for different driver models) The effect of the robust Smith predictor controller can be seen in Fig. 15. Fig. 15 shows that the stability margins are within acceptable ranges, indicating that robust stability is achieved. It is seen that the robust Smith predictor can improve phase margin and crossover frequency. The significant improvement in worst-case phase margin indicates the poor stability situations are successfully avoided. Moreover, the phase margin can actually meet the design specification. However, the gain margin is slightly reduced due to the increase in the crossover frequency. It is also seen in Fig. 14 that the improvement in crossover frequency is limited. The low frequency gain is reduced by the controller. This is expected because Fig. 12 shows that the controller has low gain in low frequency ranges. The improvement of the stability margins as well as the comparison of crossover frequency and low frequency gain is summarized in Table 1. Table 1 confirms 13

that the desired performance specifications cannot be met robustly, as has been noticed in the controller design process. The robustness results suggest that the uncertainty associated with driver variation during driving is too significant for a robust controller to achieve the desired performance specification. However, robustness to full uncertainty may not be necessary since the designed controller is simply an assist system. The driver is still the primary controller for the system. with controller GM requirement 0

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frequency and the time-delay are both very small. Therefore, the term (1-e-Ts) in equation (2) is usually very small. Consequently, the resulting Smith predictor controller (Gs(s)) is very close to C(s) in this case. The advantage of the Smith predictor will be more significant when larger delay and higher crossover frequency case is encountered. The time delay for the driver models used here is 0.1 sec and fixed. This is a result from the system identification approach used to obtain driver models from driving simulator data. However, in reality the driver delay is not constant and very difficult to identify accurately. McReur et. al. (1977) have suggested a range of time delay for their driver model. While the range of delay may not be applicable to the driver models used here, the amount of variation in delay can serve as an indication. From their suggestion, it is observed that an increase of delay from 0.1 sec to 0.2 sec is not uncommon. The change in delay is not allowed in the driver models used in this paper, and consequently, not considered in the robust controller design process. It is helpful to examine whether this change in delay can cause catastrophic problems in the compensated system. Therefore, an increase of delay (T) to 0.2 sec is deliberately applied to the system and the robust Smith predictor controller is evaluated in frequency domain. The results of the nominal system frequency response are shown in Fig. 16. It is seen that the increase of delay introduces extra phase lag (e.g., 5.73 deg. at 1 rad/sec). This increased phase lag is the same for any serial controller. For the nominal system shown in Fig. 13, this additional phase lag is not a serious problem. However, when considering model variations, problems may appear when the crossover frequency is higher. The resulting Nyquist plots of varying driver models are shown in Fig. 17. Compare the results with Table 1, it is seen that the mismatched delay reduces the PM and GM slightly (i.e., PM becomes 14.6 deg., GM becomes 2.2 dB), and does not cause significant problem.

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5. Robust Adaptive Control Scheme The significant uncertainty of the driver model causes the robust control approach to achieve robust stability, but not robust performance. Since the driver does exhibit such large variation, an adaptive-robust control approach may be applicable to this problem. It is considered that the large uncertainty in low frequency ranges is responsible for the difficulty in achieving robust performance. Therefore, it is decided to use a simple adaptive controller together with the robust Smith predictor controller. When designing the adaptive algorithm, the robust Smith predictor controller designed in Section 4 is included in the system to be adapted to. The block diagram shown in Fig. 18 illustrates the design problem, where Gs is the Smith predictor controller designed in previous section and Kc is the adaptive controller to be designed. yr

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Gv

To simplify the algorithm, only the proportional gain of the system is estimated and adapted. The adaptive control used here is an integration algorithm that has been proposed in [Koren and Masory, 1981]. The nominal system GdGsGv is represented by koGp, where ko is the DC gain of the nominal driver model, and Gp represents the rest of the transfer function. By this convention the nominal driver model Gd can be written as Gd = koGˆ d . The assumption made for the adaptive control is that the true value of ko is actually unknown and changing, and this changing value is denoted as kp. It is desired to adjust Kc so that Kckp can approximate the nominal system proportional gain, that is, ko. Define the estimation error as e=δ-δe, where δ is the steering command from the driver. This implies that when implementing the adaptive controller, the driver’s steering command needs to be measured. The estimated steering angle (δe) is defined as δe= keGˆ d er . The tracking error (er) is defined as er=yr-y, where yr is the reference signal. ke is the estimated proportional gain to be computed in the estimation algorithm. The estimated gain (ke) is obtained using the following equations: ke = (γ 1e + γ 2e)δdt

(4)

The controller gain Kc is adjusted using a similar integration algorithm. Define another error as ec=ko-keKc. The gain Kc is computed based on the following integral equation: K c = (γ 3ec + γ 4 ec )dt

(5)

The γi, i=1,…,4 in the integrands are adaptation gains that need to be tuned to achieve acceptable performance. The two integrations ensure that at steady state, e and ec will approach zero. This implies that ke equal kp and keKc equals ko. The adaptive control scheme is evaluated using computer simulation. A disturbance rejection scenario is simulated by adding random force disturbance to the vehicle lateral dynamics. Three different levels of proportional gain of the driver have been simulated to test the adaptive controller. The values used for kp are 3.2ko, 1ko, and 0.4ko. They represent high, nominal, and low gain situations respectively. The parametric uncertainty of the selected driver shows that a reduction in proportional gain down to 0.4ko is possible. However, the uncertainty also shows that the maximum increase in gain during the experiment was about 2.2ko. The value 3.2ko is chosen in order to exaggerate the problem that might happen when the driver’s gain becomes too large. This becomes helpful because the robust Smith predictor controller makes the system more robust to gain increase. The simulation results of these three levels of driver gains without the adaptive controller are shown in Fig. 19. The simulation results for the same setting with the adaptive control are shown in Fig. 20. It can be concluded that without the adaptive controller the system becomes unstable when kp reaches above 3.2ko. If kp is very low, e.g., 0.4ko, the disturbance rejection performance becomes worse. The simulation of the adaptive control shows that ke approaches kp and Kckp approaches ko very quickly. An example of the estimated gain ke is shown in Fig. 21. Once Kc converges the system behaves like the nominal system. The unstable response and poor disturbance rejection situations are successfully avoided.

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6. Summary and Conclusion A vehicle steering assist controller is investigated in this paper. The degradation of driver steering performance during long driving is the major concern. It is proposed to use a serial robust controller to help maintain the system performance even when the driver becomes tired and performs poorly. The controllers investigated include a robust Smith predictor controller and an adaptive robust Smith predictor controller. The parametric uncertainty of the driver model is used to represent the change in driver steering behavior over time. The H∞ and QFT techniques have been employed to design the robust controller. The Smith predictor is introduced to handle the nominal delay in the driver models. The adaptive control scheme is used to deal with the large parametric uncertainty in the low frequency ranges. The simple adaptive control scheme only estimates and adapts the proportional gain of the driver. The designed controllers are evaluated in the frequency domain and computer simulation. It is concluded that the serial controller structure can improve the system stability effectively. Under parametric uncertainty of the driver model, robust stability of the system is achieved. However, due to the large uncertainty involved, robust performance cannot be guaranteed. The frequency domain analyses show that the designed robust Smith predictor controller can improve phase margin and crossover frequency, but as a result the gain margin and low frequency gain cannot meet the performance specifications. This is the well-accepted trade-off between performance and robustness. The conclusion that can be drawn is that the serial controller can be used to avoid the poor stability margins situations effectively. However, the serial controller is not capable 19

of achieving robust performance requirements. If robust performance is desired, either the design specifications need to be relaxed, (i.e., reduce the uncertainty or performance specifications,) or other control structures (e.g., consider adding a parallel controller) need to be considered. The simulation of the adaptive controller suggests that the adaptive control approach may be a good candidate to reduce model uncertainty. However, the simulation was performed based on the idealized assumption that only one parameter of the driver model is allowed to change. Future research is needed to test the designed controllers using a driving simulator.

7. Acknowledgement The authors would like to acknowledge the financial support from the Intelligent Transportation System (ITS), Research Center of Excellence (RCE). Also thanks to Tom Pilutti at Ford Research Laboratories for providing the experimental driving simulator data.

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