Model Predictive Control Approach to Design Practical Adaptive Cruise Control for Traffic Jam

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1 Taku Takahaa et al./international Journal of Autootive Engineering Vol.9, No.3 (218) pp Research Paper Model Predictive Control Approach to Design Practical Adaptive Cruise Control for Traffic Ja Taku Takahaa 1) Daisuke Akasaka 2) 1) Hitachi Autootive Systes, Ltd Onna, Atsugi, Kanagawa, , Japan (E-ail: 2) The MathWorks GK Akasaka, Minato-ku, Tokyo, 17-52, Japan Received on October 12, 217 ABSTRACT: This paper presents a design ethod of a Model Predictive Control (MPC) with low coputational cost for a practical Adaptive Cruise Control (ACC) running on an ebedded icroprocessor. Generally, a proble with previous ACC is slow following response in traffic jas, in which stop-and-go driving is required. To iprove the control perforance, it is iportant to design a controller considering vehicle characteristics which significantly changes depending on driving conditions. In this paper, we attept to solve the proble by using MPC that can explicitly handle constraints iposed on, e.g., actuator or acceleration response. Furtherore, we focus on decreasing the coputational load for the practical use of MPC by using low-order prediction odel. Fro these results, we developed ACC with high responsiveness and less discofort even for traffic ja scene. KEY WORDS: electronics and control, Adaptive cruise control, Model predictive control [E1] 1. Introduction ACC with a stop-and-go function, has been widely coercialized. It enables drivers to be free fro driving stress in traffic jas and has the potential to enhance safety driving. However, during the traffic ja, conventional ACCs have a slower response tie than noral drivers traffic flow. The response delay is caused by using general control ethods designed based on a linear tie invariant odel, and not aking full use of output characteristics of an engine and brake actuators. Using such ACC, other cars cutting in front of own car fro adjacent lane tend to be increased due to the response delay. As a result, traffic flow will be disturbed, and the risk of traffic accident will increase. Up to now, as for the longitudinal control using the MPC (1),(2), researches related to stop-and-go (3), fuel consuption, vehicle platooning, and ulti-objectives (4) are thriving, however any of the are not able to ipleent with an ebedded icroprocessor and are not practical. Since the MPC needs to solve the optiization proble considering future prediction in every sapling period, there is a disadvantage that the calculation load becoes large. Therefore, in this paper, in order to suppress the calculation load of MPC, we propose a designing ethod for prediction tie and prediction odel. Furtherore, we devised MPC's weights setting ethod to relieve discofort during ACC traffic ja driving, and worked to solve these probles. Fro these results, we derived a MPC controller with a lighter coputation load that can be executed in real tie with an ebedded icroprocessor, and developed an ACC that can be used for traffic ja with high responsiveness and less discofort. In the nuerical siulation of the ACC, we show the advantages of MPC with high responsiveness copared with an existing control ethod. We also show that the calculation load proble of MPC can be solved fro the easureent result of the processing tie of an ebedded icroprocessor. Moreover, we show the ACC that has good response even during traffic ja fro experiental results in a real traffic. It is expected that the coputing power of ebedded icroprocessors will continue to increase in the future, however, since functions ipleented to an autonoous driving syste will also increase, it is desirable that it is a saller control syste, and for popularization it is inexpensive icroprocessor is required. This research is also useful in that sense. The paper is organized as follows. Section 2 describes the ACC proble with a plant odel. Section 3 proposes the MPC design ethod including prediction odeling, optiization proble, and controller tuning strategy. In Section 4 and 5, the result of coputer siulation and experiental validation will be shown, respectively. Finally, Section 6 concludes the paper with soe future work Control Proble of ACC 2. Proble Forulation The purpose of ACC is to follow a desired inter-vehicle distance or follow a desired velocity between a preceding vehicle and a host vehicle. In this paper, we consider the control proble C1-C3 as below: (C1) Inter-vehicle distance following error Δd, defined as a following error between an inter-vehicle distance d and a desired distance d r, i.e., Δd = d d r is controlled to converge to zero. (C2) Velocity following error Δv, defined as a following error between the preceding vehicle s velocity v p and the host vehicle s velocity v h, i.e., Δv = v p v h is controlled to converge to zero. 99

2 Taku Takahaa et al./international Journal of Autootive Engineering Vol.9, No.3 (218) pp (C3) Acceleration of the host vehicle v h is controlled to converge to zero. In addition, there are soe perforance requireents that are necessary to actually use ACC on public roads as follows: (R1) Relative velocity ust be not too large, except for the situation a preceding vehicle is initially detected outside the range. (R2) Acceleration of the host vehicle ust be greater than or equal to -.25 [G]. (R3) Acceleration change ust be slow except for traffic ja scene Vehicle Dynaics A vehicle dynaics odel will be built to design MPC and analyze the control perforance through coputer siulation. The longitudinal dynaics of a host vehicle is given by v h = a f r travel (1) where is the vehicle ass, v h is the vehicle speed, a f is the traction force converted to acceleration and r travel is the travel resistance. The dynaics of a vehicle actuation including engine, transission or brake has nonlinear characteristics. In general, the input/output relationship of the actuation dynaics is described as an ordinary differential equation below: x f = f act (x f, u) a f = h act (x f ) where x f R n f, u R are respectively the state, the input of the actuation syste. u is an acceleration coand, i.e., a control input calculated by adaptive cruise controller. The output of the syste (2) is a f. The travel resistance r travel contains several factors to resist its otion. A odel of the resistive force is expressed as r travel = r air v h 2 + r roll (v h ) + r accel v h + r grad (θ) (3) In the right-hand side of the equation (3), fro left to right, each ter describes the air drag, the rolling resistance, the acceleration resistance and the grading resistance. r air is the air drag coefficient, r accel is the acceleration resistance coefficient and θ denotes the slope angle State-Space Model for ACC Syste To build a plant odel for the ACC syste design, two state variables are defined: inter-vehicle distance following error Δd = d d r and velocity following error Δv = v p v h. The d r is deterined based on the constant tie headway policy given by d r = T hw v h + d (4) where T hw is the constant tie headway and d is the stopping distance for safety argin. Let us define the state variables of the plant as x = [x 1 x 2 x 3 T ] T R 2+n f with x 1 = Δd, x 2 = Δv and x 3 = x f. Then, the state-space odel is forulated as x = f(x, u) + Gv + Hw y = Cx + Jv (2) (5) x 2 T hw x 3 T hw / f(x, u) = [ h act (x f ) ], G = [ 1/ ], H = [ 1] f act (x f, u) 1 C = [ 1 ], J = [ ] 1 1/ where u R and y = [Δd Δv v h ]T R 3 are the input and output of the plant and v = r travel and w = v p represents disturbances into the plant. 3. Model Predictive Control Design 3.1. Prediction Model for Controller Design MPC requires a plant odel to predict the future behavior of the plant along a finite tie horizon. We can use the plant odel (5) reoving uneasurable disturbance ter as a prediction odel. However, a siple prediction odel with lower diension is strongly desired in order to reduce the coputational cost on icroprocessor. Therefore, the actuation syste odel (2) is approxiated to a switched linear tie variant syste as x f = A f (t)x f + B f (t)u a f = C f x f where the state x f R, i.e., n f = 1 and 1, if u(t) a T thr_off A f (t) eng = 1, if u(t) < a T thr_off brk K eng (t), if u(t) a T thr_off B f (t) eng = K brk (t), if u(t) < a T thr_off brk C f = 1. T eng is the tie constant of acceleration using engine, T brk is the tie constant of deceleration using brake, a thr_off is the acceleration generated when the throttle valve is closed, K eng (t) and K brk (t) are the steady-state gains depending on tie. In the syste (6), the dynaics is separated into acceleration and deceleration side and siply odeled as first-order delay syste. However, prediction error due to odeling error soeties degrades control perforance such as vibration phenoenon. To copensate the prediction accuracy, the steady-state gains can be changed. For exaple, K eng (t) is coposed of constant value K c and copensation signal ΔK(t) as below. K eng (t) = K c + ΔK(t) (7) where ΔK(t) is coputed by the filter ΔK(s) = L(s)u(s) with L() =. With the odel (6), the coputation load of the prediction can be reduced while the prediction accuracy is increased by updating the paraeter value at each sapling tie. Considering the above, the prediction odel (5) can be siplified to the linear tie variant syste with three states below: x = A(t)x + B(t)u + Gv + Hw y = Cx + Jv (6) (8) 1

3 Taku Takahaa et al./international Journal of Autootive Engineering Vol.9, No.3 (218) pp where x R 3 and 1 T hw A(t) = [ 1 ], B(t) = [ ] A f (t) B f (t) The plant dynaics is affected by the travel disturbance r travel and it can cause prediction error. Especially, the acceleration resistance has an ipact on the transient response, i.e., stop and go action. To prevent fro deteriorating control perforance due to such disturbance, it is iportant to explicitly consider the disturbance characteristics in MPC design. The acceleration resistance odel in (3) is a linear function of the acceleration of the host vehicle. Using state transforation, the disturbance characterization can be built into the prediction odel. This siple ethod can reject the disturbance effect without increasing coputation load. The travel disturbance odel (3) can be written as r travel = r accel v h + r travel (9) where the first ter is the acceleration resistance and the second ter is other resistances r travel = r air v h 2 + r roll (v h ) + r grad (θ). Here, let us define a state transforation for x 3 as below. x 3 = 1 1+ r accel x 3 (1) Fro the vehicle dynaics (1), the acceleration of the host vehicle becoes v h = x 3 1 r travel = (1 + r accel ) x 3 1 r travel (11) Substituting (11) to (9) gives r travel = r accel x r accel r travel (12) Thee travel resistance can be described using the new state x 3 instead of v h. Finally, the state transforation (1) and (12) yields the prediction odel with the new state coordination below: x = A(t)x + B(t)u + Gv + Hw y = Cx + Jv where x = [Δd Δv ρ accel x f ] T R 3 v = r travel w = v p and 1 T hw A(t) = [ 1 ], B(t) = [ A f (t) G = ρ T hw accel 1, H = [ ρ accel [ ] 1 C = [ 1 ], J = [ 1 ρ accel = r accel 1] ρ accel 1 ρ accel B f (t) ] ] (13) In the above odel, ρ accel iplies the scaling factor for control input and disturbance input. If ρ accel is increased, i.e., the acceleration resistive force gives a large effect, the agnitude of the control input is decreased. Hereinafter, we use the prediction odel below based on (13) and assue that the disturbances v and w are uneasurable. x = A(t)x + B(t)u y = Cx (14) In the controller, the prediction odel (14) ust be converted to a discrete-tie syste. x t+1 = A d (t)x t + B d (t)u t y t = C d x t (15) One siple ethod of the discretization is forward difference approxiation x (x t+1 x t )/T s in which T s is a sapling tie and A d (t) = (I + T s A(t)), B d (t) = T s B(t) and C d = C are obtained Optiization Proble The control input u, i.e., the acceleration coand, is calculated by solving the constrained optiization proble below during each sapling period: in u s.t. p J = y t+k T Q t y t+k k=1 +Δu t+k T R t Δu Δu t+k + u t+k T R t u u t+k } (16) y t in y t+k y t ax Δu t in Δu t+k Δu t ax u t in u t+k u t ax where t is the current tie, p is the prediction horizon (nuber of interval), Δu is the increent of the control input, and the suffix in and ax in the inequality constraints denotes lower bound and upper bound. Q t, R t Δu and R t u are the positive-sei definite weight atrices for the following error, the change rate and the agnitude of the control input, respectively. The linear MPC optiization proble above results in Quadratic Prograing (QP) proble. Hence, QP solvers are available such as active-set ethods and interior-point ethods to solve the proble. Once an optial solution, i.e., a control input sequence u = [u t u t+1 u t+p 1 ] T along prediction horizon is nuerically obtained, we only use the first eleent u t of the sequence as an actual control input. In our ACC proble, the inter-vehicle distance error y 1, and also the velocity error y 2 and the acceleration y 3 ust be regulated to converge to the origin under the upper and lower liits of the input and output variables, e.g., the acceleration coand input u and output y 3. In (16), let the weight atrices be of the for below: Q t = diag (w y1 (t), w y2 (t), w y3 (t)) R t Δu = w Δu (t) (17) R t u = w u (t) where diag( ) represents a diagonal atrix. Each weight value can be changed at every sapling tie. 11

4 Taku Takahaa et al./international Journal of Autootive Engineering Vol.9, No.3 (218) pp MPC Paraeter Design To achieve satisfactory following perforance without feeling uncofortable, the paraeters in MPC should be appropriately designed and tuned. The MPC paraeter includes prediction horizon, control horizon and weighs in the perforance index. In this subsection, we will show the paraeter design and tuning strategy obtained through a nuber of closed-loop siulation and experient. Let us define a relative horizon distance as r h : = p c in which p is the prediction horizon and c is the control horizon. Let w y (t): = [w y1 (t) w y2 (t) w y3 (t)] be a raw vector of the weights for the output variables. STEP 1. First, deterine the sapling tie T s, the prediction horizon p and the control horizon c by tuning MPC with a linear invariant prediction odel approxiated to either of engine or brake dynaics. Set w y (t) = [1 ] and w Δu (t) = w u (t) = to ephasize the inter-vehicle distance error. The plant odel in siulation is the sae as the siplified prediction odel. The value of T s, p and c are chosen in reference to the knowledge obtained as below: The larger r h is, the slower the following response is. The saller r h is, the faster the following response is. Especially when p is saller, the response tends to suffer vibration like windup phenoenon, depending on the control input constraint. The saller the sapling tie is, the faster the following response is. Also, the weights w y (t), w Δu (t) and w u (t) have the effects on the ACC syste behavior as below: If w y2 (t) is larger, MPC focuses on the velocity following to the previous vehicle. If w y3 (t) is larger, the agnitude of the acceleration v h is reduced and the overall following responses are slower. If w Δu (t), w u (t) is larger, the agnitude of Δu, u is reduced and the overall following responses are slower. These intuitive relationship between the weight and the resulting behavior helps us tune the controller for various scenes. STEP 2. Using a phase plane as depicted in Fig.1, divide the operating region and specify a set of the weights w y (t), w Δu (t), w u (t) in each region. The purpose of control is to ake the state keep or converge to the origin in the region #9. The basic ethod of the weighting based on the behavior on the phase plane is as follows: In the region #9, set w y3 (t) as a coparatively larger value to oderate accelerating or braking anipulation and avoid uncofortable feeling in the steady state. The area of the region becoes larger according to the host vehicle velocity. In the region #1-3, set w y1 (t) and w y2 (t) as coparatively saller values to prevent the host vehicle fro accelerating with full throttle when a preceding vehicle is detected faraway In the region #5 and #8, set w y3 (t) as coparatively saller values to quickly respond to the sudden deceleration of a preceding vehicle. In the region #4 and #6, set w y1 (t) as coparatively saller values to avoid uncofortable feeling due to deceleration of the host vehicle when the inter-vehicle distance is close and the relative velocity is increasing. Fig. 1 The operating region division in phase plane for the weight design. 4. Nuerical Siulation Nuerical siulation was carried out to evaluate the perforance of the MPC applied to the ACC syste. The control perforance was copared with linear quadratic regulator (LQR). In the siulation scenario, assuing a stop-and-go traffic ja scene, the both of the host and preceding vehicle are initially stopped and the initial inter-vehicle distance is 6.1. Then, the preceding vehicle rapidly accelerates at 2 /s 2 fro to 1 /s and finally deaccelerates to stop. In the siulation, the controller s sapling tie T s is.5 sec. The constant tie headway T hw is 1.3 sec. The prediction horizon p = 2 and control horizon c = 1 are chosen to be as sall as possible. The control input and its change rate have the lower and upper liit 2.5 u 1.5 and 1.5 Δu 1.5. Table 1 shows the paraeters of the actuation syste odel (6) for prediction. As seen in the table, there is a difference in the tie constant and the steady-state gain between engine acceleration and braking deceleration. In the detailed odel (2), we assued an overshoot occurs in the engine acceleration response. To copensate the prediction error, Δ K(t) is added and coputed by the 2-order filter F(s) like a band-pass filter. Table. 1 The paraeters of the actuation syste odel. T eng.46 K eng (t) Δ K(t) where Δ K(t) is obtained through the filter Δ K(s) = F(s)u(s). 1.5s F(s) = s 2 + 3s + 4 T brk.193 K brk (t).979 MATLAB, Siulink and Model Predictive Control Toolbox (5),(6) were used in the siulation. The optiization proble is solved by a QP solver, based on KWIK algorith (7), in Model Predictive Control Toolbox. We used Adaptive MPC Controller block provided by the toolbox to odel the controller and run the closed-loop siulation with the plant odel in Siulink. C-code can be generated fro the MPC block for the ipleentation to an ebedded icroprocessor. Fig. 2 and 3 illustrates the siulation result of MPC and LQR. Each figure shows, fro above, (a) inter-vehicle distance following error y 1, (b) relative velocity y 2 and (c) preceding vehicle s acceleration v p, host vehicle s acceleration y 3 and control input u. As can be seen in Fig. 2, the MPC achieves that all the plant output 12

5 Taku Takahaa et al./international Journal of Autootive Engineering Vol.9, No.3 (218) pp y i (i = 1, 2, 3) converges to zero while those input constraints are satisfied. To follow the preceding vehicle fast, the MPC tries to ake full use of the actuator capability. On the other hand, as shown in Fig. 3, the LQR prefors slower response copared with the MPC. The weight of the LQR is tuned on a trial and error basis to eet the input constraints for this specific scenario. The response is conservative due to the sall feedback gain to reduce overall agnitude of the control input by using such linear controller. The advantage of MPC for ACC is that we can realize high control perforance since such constraints can be explicitly dealt with and the tuned weights can be intuitively and flexibly designed as the function of tie or if-then rule, considering various driving situations. (a) inter-vehicle distance following error y 1 (b) relative velocity y 2 (c) acceleration v p, y 3, and u Fig. 2 The siulation result of MPC. (c) acceleration v p, y 3, and u Fig.3 The siulation result of LQR. 5. Experient The syste configuration of the experiental vehicle which we prepared is shown in Fig. 4. A stereo caera is used for relative distance sensor, and relative velocity is obtained fro a digital derivation of the relative distance. The other signals such as vehicle velocity are obtained via CAN counication. The desired engine torque is obtained by converting acceleration coand of the MPC with an engine characteristic ap and an estiated travel resistance. The desired brake pressure is obtained by converting acceleration coand of the MPC with an engine brake characteristic ap and an estiated travel resistance. The MPC controller was ipleented in an ebedded icroprocessor (Renesas SH-4A, 32-bit processor), we confired the processing tie of the MPC. The easureent result is shown in Fig. 5, the average tie of the ACC function was 1.1s. The C- code is autoatically generated fro a Siulink odel using Ebedded Coder. Evaluation of the ACC was conducted on the etropolitan expressway because there are any traffic conditions such as traffic ja and high speed scene. Results of the experientation are shown in Fig. 6 (MPC) and Fig. 7 (noral driver). Fro the coparing the both results at tie=21 26[s] which is the acceleration scene during traffic ja, we were able to obtain the equivalent response delay (approx. 1.5s) as a noral driver. Also, it can be seen that sudden acceleration/deceleration is suppressed in the high speed driving scene in Fig. 8. This is because the weight for the change in the control input is functioned effectively. Fro the above, it is understood that the response is good even in the traffic ja scene, and the controller is functioning with less discofort in the high speed driving scene. d, v Caera Ctrl unit T EngCMD, P BrkCMD Engine Brake Fig. 4 Syste configuration of the experiental vehicle (a) inter-vehicle distance following error y 1 Fig. 5 The processing tie of the proposed MPC (b) relative velocity y 2 13

6 Taku Takahaa et al./international Journal of Autootive Engineering Vol.9, No.3 (218) pp Fig. 8 Experiental result during high speed driving As future tasks, reduction of discofort in tight curves by cooperation with lateral control syste, iproveent of operability by cooperation with a driver, and so on are considered. This paper is written based on a proceeding presented at JSAE FAST-zero'17 Meeting References 6. Conclusion In this paper, we proposed a ethod of designing a practical MPC suitable for ebedded processors corresponding to various traffic situation. Coputer siulation showed that the MPC is superior to conventional controllers because it treats actuator constraints and has higher response as ACC. Experiental verification results showed that the proposed MPC controller can be ipleented in ebedded icroprocessors and can achieve high responsiveness and less discofort. (1) J. Rawlings and D. Mayne:Model Predictive Control: Theory and Design, Nob Hill Publishing (21). (2) J. Maciejowski, Predictive Control with Constraints. Harlow, UK: Prentice Hall (22). (3) G. Naus, R. Bleek, J. Ploeg, B. Scheepers, R. Molegraf, and M. Steinbuch:Explicit MPC Design and Perforance Evaluation of an ACC Stop and GO, in Proc. Aer. Contr. Conf., pp (28). (4) S. Li, K. Li, R. Rajaani, and J. Wang:Model Predictive Multi-Objective Vehicular Adaptive Cruise Control, IEEE Trans. Contr. Systes Technology, Vol. 19, No. 3, pp (211) (5) A. Beporad, M. Morafi, and N. Ricker:Model Predictive Control Toolbox User s Guide, The MathWorks, Inc., (Accessed ) (6) A. Beporad:Model Predictive Control Design: New Trends and Tools, in Proc. 45th IEEE Conf. Dec. Control, pp (26). (7) C. Schid and L.T. Biegler:Quadratic prograing ethods for reduced Hessian SQP, Coputers & Cheical Engineering, Vol. 18, No. 9, pp (1994). 14