Modelling and Nonlinear Robust Control of Longitudinal Vehicle Advanced ACC Systems

The main objective of this book is to present important challenges and paradigms in the field of applied robust control design and implementation. Book contains a broad range of well worked out, recent application studies which include but are not limited to H-infinity, sliding mode, robust PID and fault tolerant based control systems. The contributions enrich the current state of the art, and encourage new applications of robust control techniques in various engineering and non-engineering systems.


Introduction
Safety and energy are two key issues to affect the development of automotive industry. For the safety issue, the vehicle active collision avoidance system is developing gradually from a high-speed adaptive cruise control (ACC) to the current low-speed stop and go (SG), and the future research topic is the ACC system at full-speed, namely, the advanced ACC (AACC) system. The AACC system is an automatic driver assistance system, in which the driver's behavior and the complex traffic environment ranging are taken into account from high-speed to low-speed. By combining the function of the high-speed ACC and low-speed SG, the AACC system can regulate the relative distance and the relative velocity adaptively between two vehicles according to the driving condition and the external traffic environment. Therefore, not only can the driver stress and the energy consumption caused by the frequent manipulation and the traffic congestion both be reduced effectively at the urban traffic environment, but also the traffic flow and the vehicle safety will be improved on the highway. Taking the actual traffic environment into account, the velocity of vehicle changes regularly in a wide range and even frequently under SG conditions. It is also subject to various external resistances, such as the road grade, mass, as well as the corresponding impact from the rolling resistance. Therefore, the behaviors of some main components within the power transmission show strong nonlinearity, for instance, the engine operating characteristics, automatic transmission switching logic and the torque converter capacity factor. In addition, the relative distance and the relative velocity of the inter-vehicles are also interfered by the frequent acceleration/deceleration of the leading vehicle. As a result, the performance of the longitudinal vehicle full-speed cruise system (LFS) represents strong nonlinearity and coupling dynamics under the impact of the external disturbance and the internal uncertainty. For such a complex dynamic system, many effective research works have been presented. J. K. Hedrick et al. proposed an upper+lower layered control algorithm concentrating on the high-speed ACC system, which was verified through a platoon cruise control system composed of multiple vehicles [1][2][3] . K. Yi et al. applied some linear control methods, likes linear quadratic (LQ) and proportional-integral-derivative (PID), to design the upper and lower layer controllers independently for the high-speed ACC system [4] . In ref. [5], Omae designed the model matching control (MMC) vehicle high-speed ACC system based on the H-infinity (H inf ) robust control method. To achieve a tracking control between brake system is a typical one with the assistance of the compressed air. On-board millimetric wave radar is used to detect the information from the inter-vehicles (i.e., the relative distance and the relative velocity), which is installed in the front-end frame bumper of the controlled vehicle.

Fig. 1. Block diagram of LFS
x l , x df , v l , v df are absolute distance (m) and velocity (m/s) between the leading vehicle and the controlled vehicle, respectively. d r =x l -x df is an actual relative distance between the two vehicles. Desired relative distance can be expressed as d h,s =d min +v df t h , where, d min =5m, t h =2s. v r =v l -v df is an actual relative velocity. The purpose of LFS is to achieve the tracking of the inter-vehicles relative distance/relative velocity along a desired value. Therefore, a dynamics model of LFS at low-speed condition has been derived in ref. [15], which consists of two parts. The first part is the longitudinal dynamics model of the controlled vehicle, in which the nonlinearity of some main components, such as the engine, torque converter, etc, is taken into account. However, this model is only available at the following strict assumptions:  the vehicle moves on a flat straight road at a low speed (<7m/s)  assume the mass of vehicle body is constant  the automatic transmission gear box is locked at the first gear position  neglect the slip and the elasticity of the power train The second part is the longitudinal dynamics model of the inter-vehicles, in which the disturbance from frequent accelartion/deccelartion of the leading vehicle is considered. In general, since the mass, road grade and the gear position of the automatic transmission change regularly under the practical driving cycle and the traffic environment, the longitudinal dynamics model of the controlled vehicle in ref. [15] can only be used in some way to deal with an ideal traffic environment (i.e., the low-speed urban condition). In view of the uncertainties above, in this section, a more accurate longitudinal dynamics model of www.intechopen.com the controlled vehicle is derived for the purpose for high-speed and low-speed conditions (that is, the full-speed condition). After that, it will be integrated with a longitudinal dynamics model of the inter-vehicles, and an LFS dynamics model for practical applications can be obtained in consideration of the internal uncertainty and the external disturbance. It is a developed model with enhanced accuracy, rather than a simple extension in contrast with ref. [15].

Longitudinal dynamics model of the controlled vehicle
Based on the vehicle multi-body dynamics theory [19] , modeling principles, and the above assumptions, two nominal models of the longitudinal vehicle dynamics are derived firstly according to the driving/braking condition:  (1) where two state variables are x 1 =ω t (turbine speed (r/min)) and x 2 =ω ed (engine speed (r/min)); a control variable is α th (percentage of the throttle angle (%)); definitions of nonlinear items f av1 (X), f av2 (X), g av1 (X) and g av2 (X) are presented in Appendix (1) where x 3 =a b is a braking deceleration (m/s 2 ); u b is a control variable of the desired input voltage of EBS (V); definitions of nonlinear items f dv1 (X)~f dv3 (X) and g dv1 (X)~g dv3 (X) are presented in Appendix (2). As mentioned earlier, models (1) and (2) Braking condition: In terms of multiple factors of the uncertain matrixes, it is difficult to estimate the upper and lower boundaries of Eqs. (3) and (4) precisely by using the mathematical analytic method. Therefore, a simulation model of the heavy-duty vehicle is created at first by using the MATLAB/Simulink software, which will be used to estimate the upper and lower boundaries of the uncertain matrixes. To determine the upper and lower boundaries, an analysis on extreme driving/breaking conditions of models (3) and (4) is. At first, the analysis of Eq. (3) indicates that with the increase of the mass M, road grade φ s and the gear position, the item of f av1 (X) converges reversely to its minimum value relative to the nominal condition (at a given ω t , ω ed ). Similarly, the extreme operating condition for the maximum value of f av1 (X) can be obtained. The analysis above can be applied equally to other items of Eq. (3), and can be summarized as the following two extreme conditions: (a1) If the vehicle mass is M =10,000kg, the road grade is φ s =-3° and the automatic transmission is locked at the first gear position, then the upper boundary of Δf av1 can be estimated. (a2) If the vehicle mass is M =25,000kg, the road grade is φ s =-3° and the automatic transmission is shifted to the third gear position (supposing that the gear position can not be shifted up to the sixth gear position, since it should be subject to a known gear shift logic under a given actual traffic condition), then the lower boundary of Δf av1 can be estimate. On the analysis of Eq. (4), two extreme conditions corresponding to the upper and lower boundaries can also be obtained: (b1) If the vehicle mass is M=10,000kg, the road grade is φ s =-3°, the braking deceleration is a b =0m/s 2 and the gear position is locked at the first gear position, then the upper boundary of Δf dv1 can be estimated. (b2) If the vehicle mass is M=25,000kg, the road grade is φ s =3°, the braking deceleration is a b =-2m/s 2 (assuming it as the maximum braking deceleration commonly used) and the gear position is locked at the third gear position, then the lower boundary of Δf dv1 can be estimated. By the foregoing analysis, the extreme and nominal operating conditions will be simulated respectively by using the simulation model of the heavy-duty vehicles. In order to activate entire gear positions of the automatic transmission, the vehicle is accelerated from 0m/s to the maximum velocity by inputting a engine throttle percentage of 100%. After that, the throttle angle percentage is closed to 0%, and the velocity is slowed down gradually in the following two patterns: 1. according to the requirement of (b1) condition, the vehicle is slowed down until stop by the use of the engine invert torque and the road resistance. 2. according to the requirement of (b2) condition, the vehicle is slowed down until stop through an adjoining of a deceleration a b =-2m/s 2 generated by the EBS, as well as the sum of the engine invert torque and the road resistance.
According to the above extreme conditions (a1), (a2), (b1), (b2), the variation range of each uncertainty can be obtained by simulation, as shown in Figures 2 and 3. For removing the influence from the gear position, the x-coordinates in Figures 2 and 3 have been transferred into a universal scale of the engine speed. For instance (see solid line in Figure 2), during the increase of the engine speed in condition (a1), the upper boundary of the item Δf av1 increases gradually, while the items Δf av2 , Δg av2 change trivially. As to the increase of the engine speed in condition (a2) (see dashed line in Figure 2), the lower boundary of the item Δf av1 increases rapidly at the beginning, and then drops slowly. The minimum value appears approximately at the slowest speed of the engine (i.e., the idle condition). The items Δf av2 , Δg av2 decrease during the engine speed increases.   To verify the proposed models, some profiles are prepared in Figure 4 according to the aforementioned extreme conditions. They include the throttle angle percentage, EBS desired braking voltage and the road grade containing two values of 3   . Figures 5 and 6 are the  comparison results corresponding to 10,000kg and 25,000kg, respectively. The dashed lines and the solid lines are the results of the control models (3) and (4) and the simulation models, respectively. It can be seen from the comparison results that the control models (3) and (4) are able to approximate the simulation models very closely, even in the case of a wide variation ranges of the velocity (0m/s~28m/s), mass (10,000kg~25,000kg) and the gear positions of the automatic transmission (1~6 gears). Because the models (3) and (4) only present the longitudinal dynamics of the controlled vehicle, the inter-vehicles dynamics has to be considered furthermore such that a completed dynamics model of the LFS at fullspeed can be obtained.

Longitudinal dynamic model of the inter-vehicles
For the purposes of vehicular ACC or SG cruise control system design, many well-known achievements on the operation policy for the inter-vehicles relative distance and velocity have been intense studied [20,21] . Focusing on the AACC system, the operation policy for the inter-vehicles relative distance and relative velocity should be determined so as to  maintain desirable spacing between the vehicles  ensure string stability of the convoy Inspired by previous research [1], [2], [7] on the design of upper level controller, the operation policy of inter-vehicles relative distance and relative velocity can be defined as where a df is a controlled vehicle acceleration (m/s 2 ); ε d is a tracking error of the longitudinal relative distance (m); ε v is a tracking error of the longitudinal relative velocity (m/s).
As the illustration of the vehicle longitudinal AACC system (see Figure 1), it should be noted that an item a df t h is introduced to define the inter-vehicles relative velocity ε v so as to www.intechopen.com fit the dynamical process from one stable state to another one. In contrast to Eq. (5), conventional operation policy of inter-vehicles relative velocity is often defined as ε v =v l -v df , which only focuses on the static situation of invariable velocity following. However, on account of the dynamic situation of acceleration/deceleration, the previously investigation [15,16] has demonstrated that it is dangerous and uncomfortable for the AACC system to track a vehicle in front still adopted conventional operation policy. Therefore, an item of a df t h is proposed to capture accurately the human driver's longitudinal behavior aiming at this situation. Generally, Eq. (5) can be regarded as the dynamical operation policy. The accuracy of Eq. (5) is validated by the following experimental tests, which is carried out under complicated down-town traffic conditions in terms of five skillful adult drivers (including four males and one female). Two cases including an acceleration tracking and a deceleration approaching are considered. In the case of acceleration tracking, the driver is closing up a leading vehicle without initial error of relative distance and relative velocity. Then, the driver adjusts his/her velocity to the one of the vehicle in front. The headway distance aimed at by the driver during the tracking is essentially depending on the driver's desire of safety. In the case of deceleration approaching, the driver is closing down a leading vehicle with constant velocity. The driver brakes to reestablish the minimal headway distance, and then follow the leading vehicle with the same velocity. The experimental data presented in Figure 7 are the mean square value of five drivers' results. The comparison results confirm that Eq. (5) shows a sufficient agreement with practical driver manipulation, which can be adopted in the design of vehicle longitudinal AACC system.

Inter-vehicles Relative Distance / m Inter-vehicles Relative Velocity / m/s
where l v  is a leading vehicle acceleration (m/s 2 ), which is generally limited within an extreme acceleration/deceleration condition, i.e., ms ms Although the inter-vehicles dynamics is considered in Eq. (6), the dynamics of the controlled vehicle that has great impact on the performance of entire system has been ignored instead. Actually, two aforementioned models are interrelated and coupled mutually in the LFS. To overcome the disadvantages of the existing independent modeling method, a more accurate model will be proposed in the following to describe the dynamics of the LFS reasonably. In this model, the longitudinal vehicle dynamics models (3) and (4) with uncertainty and the longitudinal inter-vehicles dynamic model (6) are both taken into account. As a result, a control system can be designed on this platform, and an optimal tracking performance with better robustness can also be achieved.

LFS dynamics model
is a vector of the state variables, l wv   is a disturbance variable, and th  is a control variable. The definition of each item in Eq. (7) can be referred to Appendix (2). Similarly, an LFS dynamics model for the braking condition is achieved: is a vector of the state variables, b u is a control variable.
The definition of each item in Eq. (8) can be referred to Appendix (4).
According to the analysis of the extreme driving/braking conditions in 2.1, an approximate ranges of the upper and lower boundaries regarding uncertain items in Eqs. (7) and (8)  where an unit of * f  is m/s 2 , units of 11 , ad gg are m/(s 2 ·%) and m/(s 2 ·V), respectively. The analysis of the dynamics models (7) and (8) indicates that the LFS is an uncertain affine nonlinear system, in which the strong nonlinearities and the coupling properties caused by the disturbance and the uncertainty are represented. These complex behaviors result in more difficulties while implementing the control of the LFS, since the state variables ε d , ε v are influenced significantly by the nonlinearity, uncertainty, as well as the disturbance from the leading vehicle's acceleration/deceleration. However, because the longitudinal dynamics of the controlled vehicle and the inter-vehicles can be described and integrated into a universal frame of the state space equation accurately, this would be helpful for the purpose of achieving a global optimal and a robust control for the LFS. The LFS AACC system intends to implement the accurate tracking control of the intervehicles relative distance/relative velocity under both high-speed and crowded traffic environments. Thus, the system should be provided with strong robustness in view of the complex external disturbance and the internal uncertainty, as well as the capability to eliminate the impact from the system's strong nonlinearity at low-speed. Focusing on the LFS, refs. [22][23][24][25][26][27] presented an NDD method to eliminate the disturbance effectively, which was, however, limited to some certain affine nonlinear systems. Utilizing the invariance of the sliding mode in VSC, the control algorithm proposed in refs. [28,29] can implement the completely decoupling of all state variables from the disturbance and the uncertainty. But, it is not a global decoupling algorithm, and should also be submitted to some strict matching conditions. Refs. [30][31][32][33][34] studied the input-output linearization on an uncertain affine nonlinear system, but did not discuss the disturbance decoupling problem. On a nonlinear system with perturbation, ref. [35] gave the necessary and sufficient condition for the completely disturbance decoupling problem, but did not present the design of the feedback controller. To avoid the disadvantages of those control algorithms mentioned above, a DDRC method combining the theory of NDD and VSC is proposed in regard to the complex dynamics of the LFS.

DDRC method
The basic theory of DDRC method is inspired by the idea of the step-by-step transformation and design. First, on account of a certain affine nonlinear system with disturbance, the NDD theory based on the differential geometry is used to implement the disturbance decoupling and the input-output linearization. Hence, a linearized subsystem with partial state variables is given, in which the invariance matching conditions of the sliding mode can be discussed easily via VSC theory, and then a VSC controller can be deduced. Finally, two methods will be integrated together such that a completely decoupling of the system from the external disturbance, and a weakened invariance matching condition with a simplified control system structure are obtained. www.intechopen.com

NDD theory on certain affine nonlinear system
At first, consider a certain dynamics model of the LFS, where uncertain items of ΔF a (X), ΔG a (X), ΔF d (X) and ΔG d (X) are considered as zero. Hence, a certain affine nonlinear system can be simplified as where XR n and u, w, where μ satisfies In this way, the original closed-loop system (9) can be modified as a following form over the new coordinate  [36] ,   is an orthogonal of"  " [37] . Eq. (10) is a necessary and sufficient condition of the disturbance decoupling problem, which can be expressed in the equivalent form  (19) In this way, a weakened necessary and sufficient condition of the disturbance decoupling problem is achieved as As a result, some existing linear control methods (likes, LQ, pole placement) can be used to implement the pole placement over the linearized decoupling subsystem. In the following, the NDD theory is used to discuss the VSC problem of the affine nonlinear systems under the impact of the uncertainty.

VSC of uncertain affine nonlinear systems based on NDD
Considering Eqs. (7) and (8) with uncertainty, they can be simplified as a more general forms for the analysis, i.e., where F, G, P, h indicate the certain part of the system, and they are defined as Eq. (8) (30) www.intechopen.com It can be noticed from Eq. (29) that for the state variables z i of the first r-1 dimensions, the linearization and the disturbance decoupling have been achieved, except for the remaining z r (Eq. (30)). By virtue of the invariance of the sliding mode in VSC [28] , it will be used in consequence to eliminate the disturbance and the uncertainty on z r . Based on the VSC theory [28] , a switching function is designed easily by taking advantage of the linearized decoupling subsystem (29) over the new coordination (31) where C=[c 1 ,…,c r-1 ,1] is a normal constant coefficient matrix to be determined. Once the system is controlled towards the sliding mode, it satisfies (32) yielding the following reduced-order equation Clearly, a desired dynamic performance of each state variable in Eq. (33) can be achieved by configuring the coefficient C.
As the desired dynamic performance of the sliding mode has already been achieved, an appropriate VSC law is to be defined so as to ensure the desired sliding mode occurring within a finite time. It is convenient to differentiate the switching function (31) (38) may lead to the following inequality: Namely, the convergence condition of the sliding mode is achieved. From the above verification, the desired sliding mode is achievable under the VSC law (35), as long as the matching condition (c2) and the constraints (38) are satisfied. Since Eqs. (31) and (35) (19)). To summarize, for an uncertain affine nonlinear system, if the disturbance decoupling condition (17) or (20) and the matching conditions of (c1) and (c2) hold respectively for the certain part and the uncertain part, the DDRC method with the combination of NDD and VSC theories can be figured out as the following design procedure: Step 1. According to the NDD theory of affine nonlinear systems, the feedback control law (Eq. (18) or (19)) and the coordinate transformation (Eqs. (12) and (13)) are derived to transfer the original system into the linearized decoupling normal form (Eq. (15)) over the new coordinate.
Step 2. Give the VSC matching conditions (c1) and (c2) for the uncertain part of the affine nonlinear systems. Step 3. Utilize the linearized decoupling normal form (Eq. (15)) over the new coordinate to design the switching function (Eq. (31)), and determine its coefficients accordingly. Step 4. Design the VSC law (Eq. (35)) based on the perturbation boundary (37) of the uncertainty part, and the convergence condition of the sliding mode (39).
Step 5. Define the coordinate transformation (12) to transfer the switching function (Eq. (31)) and the VSC law (Eq. (35)) from the new coordinate Z back to the original coordinate X. Step 6. Substitute the VSC law (Eq. (18) or (19)) over the original coordinate into the feedback control law, and yield the DDRC method. A block diagram of the closed-loop system for the aforementioned DDRC method is shown in Figure 8, which includes two feedback loops. The nonlinear loop (i.e., the NDD loop) is used to achieve the disturbance decoupling and the partial linearization, regarding the system output y from the disturbance w . On the other hand, the linear loop (i.e., the VSC loop) is used to restrain the system's uncertainty and regulate the closed-loop dynamic performance. www.intechopen.com

LFS AACC system
In this section, the proposed DDRC method will be used to design the LFS AACC system with respect to the driving and the braking conditions.

LFS AACC system for driving condition
Recall the procedure in 2.2, the disturbance decoupling problem on the LFS dynamics model without the impact of the uncertainty is considered (i.e., for the uncertain items of Eq. (7) let ΔF a (X)=0, ΔG a (X)=0). On the purpose of LFS AACC system, the following affine nonlinear system with the output variable is defined: By adopting the NDD theory of certain affine nonlinear system, the relative degree of system (40)  The purpose is to ensure the diffeomorphism relationship of the coordinate transformation between the original and the new one (in other words, it is a one-to-one continuous coordinate transformation between the original and the new one, the same is for the inverse transformation). Obviously, one solution of the partial differential Eq. (47) is  By substituting the decoupling state feedback u=α th (Eq. (45)) into model (7), and making use of the coordinate transformations (46) and (49), a linearized subsystem below can be achieved, in which the certain part is completely decoupled from the disturbance.
Part of uncertain and disturbance Certain part 11 Besides, a nonlinear dynamic internal subsystem without separating from the disturbance and the uncertainty is yielded Then, the parameter b as =250 can be determined, and a as =10 is achieved separately by the condition of a as >0.
By the procedure (Step5) in 3.2, the coordinate transformations Z a =ψ a (X) and μ a =ϕ a (X) will be used to transfer the new coordinates (Z a , μ a ) back to the original coordinate X. In this way, the switching function over the original coordinate becomes The control laws designed above only satisfy the convergence stability and the robustness of the linearized decoupling subsystem. In order to ensure the stability of the total system, the stability of the remaining nonlinear internal dynamic subsystem has to be verified, so that the problem of tracking control can be solved completely. Based on ref. [38], the study on the stability of nonlinear internal dynamic subsystem can be turned into the study on its zero dynamics correspondingly. Therefore, let ΔQ a =ΔK a =0, i.e., ignore the tiny impact of the uncertain part. Then the zero dynamics of the nonlinear internal dynamic subsystem (53) owing to z a1 , z a2 , w=0 is obtained as follows   Assembling Eqs. (84) and (85), the following inequality is hold Thus, the zero dynamics of the nonlinear internal dynamic subsystem (78) is asymptotically stable as long as z d1 , z d2 , w=0. www.intechopen.com

Simulation and analysis
Base on above analysis of the control system under the driving/braking conditions, the LFS AACC system applying the DDRC method can be designed as the block diagram in Figure  9. The system consists of three parts: the controlled object of a convoy with two vehicles, DDRC system, and the input/output signals. In order to verify the control performance of the LFS AACC system, a typical driving cycle of the leading vehicle's aceeleration/deceleration, velocity, as well as the road grade are given in Figure 10. The road grade changes from 0 o~+ 3 o to 0 o~-3 o in a period of 80s~90s and 110s~120s, respectively. Furthermore, the conditions from the high-speed to low-speed SG, and two cases of mass equaling 10,000kg and 25,000kg are included. The initial errors at 0s for the inter-vehicles relative distance and relative velocity are set to 0m and 0m/s, respectively. Table 1 and the solid lines in Figures 11 and 12 are the coefficients and the simulation results, respectively for the proposed control system. In contrast, the coefficients and some simulation results of an upper LQ+lower PID hierarchical control system proposed in ref. [1] are also presented respectively in Table 2 and by the dotted lines in Figures 11 and 12. The comparison results of the throttle angle, desired input voltage of EBS, engine speed, automatic transmission gear position, relations of relative distance/relative velocity tracking error verses time scale, as well as the phase chart of the relative distance/relative velocity tracking error are shown in Figures 11 and 12 in sequences of (a)~(f).
Driving condition c a1 =1 a as =10 b as =250 Braking condition c d1 =1 a ds =10 b ds =185  Table 2. Control parameters of hierarchical control system As illustrated by Figures 11 (a)~(d), for the proposed control system, the throttle angle and the EBS desired input voltage exhibit smooth response characteristic, rapid convergence and small oscillation, even at the moment of gear switching. However, for the hierarchical control system, it shows intense and long time oscillations especially at low-speed condition (shown as dashed border subfigures inside the Figures 11 (a) and (b)), which have impacts on the vehicle's comfortability severely. This is because the small parameters are adopted by the proposed control system as the consequence of applying DDRC method (shown as Tables 1), and thus the unmodeled high frequency oscillation can be effectively eliminated, in contrast with the hierarchical control system adopting large parameters (shown as Tables 2). Moreover, during the time period of 0s ~ 73s and 130s ~ 200s in Figures 11(e) and 12(e), the simulation results of the proposed control system indicate that the errors of the relative distance and the relative velocity are www.intechopen.com constrained within the range of ± 0.02m and -0.05m/s~0.02m/s, respectively. The tracking accuracy of the proposed control system is enhanced and almost frees from the disturbance of the leading vehicle's acceleration/deceleration. However, for the hierarchical control system, it is affected obviously by the change of the leading vehicle's acceleration/ deceleration, and touches the maximum value of ± 0.1m. Finally, the comparison between (e) and (f) in Figures 11 and 12 demonstrates a superior robustness for the proposed control system in spite of the uncertainties caused by the road grade, gear position and the vehicle mass. Particularly, while the road grade changes between ± 3 o in the time period of 80s~120s, the tracking error of the relative distance and the relative velocity for the proposed control system are less than ± 0.05m and -0.04m/s ~ 0.02m/s, in contract to larger than ± 0.15m and ± 0.05m/s of the hierarchical control system.