Model Predictive Control Using Dual Prediction Horizons for Lateral Control

2013 IFAC Intelligent Autonomous Vehicles Symposium The International Federation of Automatic Control June 26-28, 2013. Gold Coast, Australia

Model Predictive Control Using Dual Prediction Horizons for Lateral Control Bo-Ah Kim ∗ Young Seop Son ∗ Seung-Hi Lee ∗∗ Chung Choo Chung ∗∗ ∗

Department of Electrical Engineering, Hanyang University, Seoul 133-701, Korea (e-mail: [email protected], [email protected]). ∗∗ Division of Electrical and Biomedical Engineering, Hanyang University, Seoul 133-701, Korea (e-mail: [email protected], [email protected]). Abstract: In this paper, we present model predictive control having dual prediction horizons to reduce the length of prediction horizon and obtain the optimal solution rapidly. If prediction horizon is long, it is easy to get optimal solution while assuring closed-loop system stability. Realtime solution is, however, very difficult to calculate within the sample time because the system has complex formulations involving many constraints. On other hand, if prediction horizon is very short, computation overhead is reduced but the stability and performance of closed-loop system are not guaranteed. In this paper, the proposed method reduces the length of prediction horizon as well as maintains the stability and performance. The comparison of performances between the conventional method and the proposed control method are validated via simulations. Keywords: Predictive control, optimal control, autonomous vehicles, prediction method, constraint problems. SUBSCRIPTS

NOMENCLATURE

• {XY Z} : global coordinate frame • {xyz} : local coordinate frames • x, y : longitudinal position of the origin of {xyz} coordinate to the front fixed point along the longitudinal and lateral axis • y : lateral position of the origin of {xyz} coordinate to the rotation center ‘O’ along the lateral axis • Vx : longitudinal velocity of the vehicle at c.g. • ψ : yaw angle • ψ˙ : yaw rate d : lateral position error w.r.t. reference • eyL = yL − yL at look-ahead distance • eψ = ψ d − ψ: yaw angle error w.r.t. road • Cα : cornering stiffness of tire • Iz : yaw moment of inertia of vehicle • m : total mass of the vehicle • l : distance of the tire respective from c.g. of the vehicle • δ : steer angle • L : look-ahead distance from c.g. to look ahead point • ny : number of outputs • nu : number of inputs • NP 1 : length of prediction horizon for prediction sampling period Tc • NP 2 : length of prediction horizon for prediction sampling period Nn Tc • NC : control horizon • Nn : ratio of prediction time interval • Q, R : MPC weighting • u : input (=δ) 978-3-902823-36-6 © 2013 IFAC

280

• f : front • r : rear • L : value at look-ahead distance SUPERSCRIPTS • d : desired value 1. INTRODUCTION Recently, autonomous vehicle has drawn attention in the automotive industry. Autonomous vehicle makes driving safer and more convenient than ever. Autonomous vehicle control can increase traffic capacity and reduce fuel consumption [Huang (2000), Wu (2008)]. Due to these potential benefits, there has been a lot of works going on for autonomous vehicle control methods. Anti-lock braking system and electronic stability program enhance the vehicle stability by controlling the brake systems. Lane change control (LXC) and lane keeping system (LKS) control the front steer angle in order to improve driving comfortability. Through implementation of the LXC and LKS, the steer angle can be assisted by getting information of road environment and vehicle’s states. Moreover, LXC can be implemented in the autonomous vehicle systems to track reference trajectory. Many control methods has been developed to control autonomous vehicle [Wu (2008), Hatipoglu (2008), Varaiya (1993)]. For autonomous driving, the control objective is following the trajectory under various constraints according to vehicle physical limits and industrial control specifications. Model predictive control (MPC) is an effective method to treat the tracking system by incorporating 10.3182/20130626-3-AU-2035.00054

2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia the constraints into the controller formulation. Therefore, MPC has been widely used in the lateral control of vehicle [Li (2011), Falcone (2007)]. MPC computes a sequence of the control inputs to optimize the future behavior under various constraints [Garcia (1989), Qin (2003)]. The predictive horizon of MPC must be long enough to guarantee stability and performance [Maciejowski (2002)]. Generally, one method to achieve stability with a finite horizon is to add a terminal constraint [Rawlings (1993), Kwon (1997)]. However, real-time solution for MPC of long predictive horizon is very difficult to calculate in the sample time because the formulation of the system is too complex. Thus the MPC having complex formulation can not be applied to low cost implementation. Therefore a simple and fast method to compute a optimal solution is developed [Lee (2012)]. But this method is also difficult to obtain the optimal solution when horizon is either very long or model is complex. On the other hand, when horizon is too short to reduce the complexity of formulation, the optimal solution could not be existed. Therefore, in this paper, we propose the method reducing the length of predictive horizon while keeping the stability and performance. We present dual prediction horizons of MPC for an autonomous vehicle. The dual prediction horizons consist of near prediction horizon having short time period and far prediction horizon having long time period. By using the dual prediction horizons, the MPC can consider near future more than important than far future. Also, number of horizon for dual prediction horizons method is less than conventional single prediction horizon method, computation time can be shortened, and this method can be applied to low cost implementation. < 

  

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T  dynamics in terms of the state vector x = eyL e˙y eψ ψ˙ ,  T the control input u = δ, and the output y = eyL eψ ψ˙ is obtained as follows: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 1 0 −L 0 L   ⎢0 a22 a23 a ⎥ ⎢b21 ⎥ ⎢Vx ⎥ ˙d 24 ⎢ ⎥ x˙ = ⎣ x + ⎣ ⎦u + ⎣ ⎦ψ 0 0 0 −1 ⎦ 1 0  0 b41 0 a42 a43 a44







(1) Bq B2 A   10 0 0 y= 0 0 1 0 x 00 0 1

 C

where 2Cαf + 2Cαr a22 = − , a23 = −a22 Vx , mVx 2Cαf lf − 2Cαr lr  , a24 = (a24 − 1)Vx , a24 = −1 − mVx2 2Cαf lf − 2Cαr lr  a42 a42 = − , a42 = , a43 = −a42 , Iz Vx 2 2 2Cαf lf + 2Cαr lr 2Cαf  , b21 = , b = b21 Vx , a44 = − Iz Vx mVx 21 2Cαf lf . b41 = Iz The discretization of (1) using the zero-order hold method at electronic control unit (ECU) sampling period Tc leads to the discrete-time matrices (Φ, Γ2 , Γq ) from (A, B2 , Bq ) such that x(k + 1) = Φx(k) + Γ2 u(k) + Γq ψ˙ d (k) (2) y(k) = Cx(k) where T T Φ = eATc , Γ2 = 0 c eA(Tc −v) B2 dv, Γq = 0 c eA(Tc −v) Bq dv. Likewise, in order to apply dual prediction horizons, we define matrix of sampling period Nn Tc   the discrete-time ˜ Γ ˜ 2, Γ ˜ q from (A, B2 , Bq ) such that as Φ, ˜ ˜ 2 u(k) + Γ ˜ q ψ˙ d (k) +Γ x(k + Nn ) = Φx(k) y(k) = Cx(k)

 [

/

(3)

where  ˜ = eANnTc = ΦNn , Γ ˜ 2 = Nn Tc eA(Nn Tc −v) B2 dv = Φ 0   Nn −1 i ˜ q = Nn Tc eA(Nn Tc −v) Bq dv = Nn −1 Φi Γq . Φ Γ , Γ 2 i=0 i=0 0

; 

Fig. 1. Lateral position and velocity error at the lookahead distance point 2. LATERAL DYNAMICS MODEL OF VEHICLE The vehicle lateral dynamics can be described by using a simple bicycle model [Rajamani (2006)]. Figure 1 shows the lateral vehicle dynamics on the global coordinate. The reference trajectory shows center of load lane. We treat the lateral control system using a vision processing system. Thus the lateral dynamics in terms of lateral offset at a look-ahead distance is useful [Lee (2012)]. The controller is designed with the output measurements eyL , eψ from the vision system, and ψ˙ from an inertia measurement unit. Then, the state-space model of the vehicle lateral 281

3. DUAL PREDICTION HORIZONS MPC FORMULATION In this section, we introduce the dual prediction horizon to reduce complexity of formulation. For the lane change system, the control objective is following the desired lateral offset trajectory with the fulfillment under various constraints reflecting the vehicle physical limits and industrial control specifications. The MPC is an effective method to treat the tracking system by incorporating the constraints into the controller formulation. The MPC computes a set of optimal inputs that will drive the plant to the desired trajectory without violating constraints [Qin (2003)] as shown in Fig. 2. The input constraints for the LXC such as steer angle and steer angle ratio came from the attributes

2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia \

 

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Then, prediction time is calculated by (NP 1 + Nn NP 2 )Tc . And we define NP = NP 1 + NP 2 . Usually the output prediction horizon NP is longer than the control horizon NC [Camacho (2004)]. To simplify formulation, we assume NP = NC . Then, we can define lifting matrices ˜ (k), Δ˜ ˜ (k), y ˜ (k) for the input u (k), incremental u u (k), x input Δu (k) = u (k) − u (k − 1), state x (k), and output prediction y (k) such as ⎡ ⎤ ⎤ ⎡ y (k + 1) x (k + 1)

 

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⎤ u (k) .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ u (k + NP 1 − 1) ⎢ ⎥ ˜ (k) = ⎢ u ⎥, u (k + NP 1 ) ⎢ ⎥ ⎢ ⎥ .. ⎣ ⎦ . u (k + NP 1 + NP 2 Nn − Nn ) ⎡ ⎤ Δu (k) .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ Δu (k + NP 1 − 1) ⎢ ⎥ Δ˜ u (k) = ⎢ ⎥. Δu (k + NP 1 ) ⎢ ⎥ ⎢ ⎥ .. ⎣ ⎦ . Δu (k + NP 1 + NP 2 Nn − Nn ) Then, the lifted input and output matrices can be rewrit˜ (k) − u ˜ (k − 1), y ˜ = C˜ x ˜ with C˜ = ten as Δ˜ u (k) = u diag (C, . . . , C) where C is output matrix. The variable r (k + i) denotes reference trajectory at time k + i based on the information available at time k. Then, the solution of MPC problem, i.e. Δ˜ u∗ (k), at time k is obtained by minimizing the following dynamic objective ˜ function J: T T J˜ = (˜ y (k) − ˜r (k)) Q (˜ y (k) − ˜r (k)) + Δ˜ u (k) RΔ˜ u (k)

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= Fig. 3. Controller state at the k-th sampling instant of dual prediction horizons of steering electric power steering. The constrained optimization problem can be solved by online quadratic programming (QP) at each sampling time, k, using the current states and previous input value [Camacho (2004), Maciejowski (2002)]. The MPC use a large horizon to guarantee the stability and performance [Maciejowski (2002)]. But, a long horizon entails long computation time. Generally, it is used a large horizon to predict long time because conventional MPC predicts the output of single prediction horizon as shown in Fig. 2. We propose the method that predicts the output of dual prediction horizons as shown in Fig. 3. Then, in the same horizon, prediction time of the proposed method can be extended.

+

N P1  i=1 N P2 

{y (k + i) − r (k + i)}T Q{y (k + i) − r (k + i)}

{y (k + NP 1 + Nn i) − r (k + NP 1 + Nn i)}T Q

i=1

+

+

N P 1 −1 i=0 N P 2 −1

{y (k + NP 1 + Nn i) − r (k + NP 1 + Nn i)} T

Δu (k + i) RΔu (k + i) T

Δu (k + NP 1 + Nn i) RΔu (k + NP 1 + Nn i) ,

i=0

subject to ˜ (k) ≤ u ˜ max ˜ min ≤ u u Δ˜ umin ≤ Δ˜ u (k + i) ≤ Δ˜ umax ˜ (k) ≤ y ˜ max ˜ min ≤ y y (4)

3.1 Objective function using lifting matrices Let NP 1 and NP 2 denote lengths of prediction horizons for prediction sampling period TC and Nn TC , respectively. 282

The objective function involves two contributions. The first and second terms in (4) represent the penalty on trajectory tracking error and the third and fourth terms

2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia in (4) penalize the steering effort. From lateral dynamics model (1), the output prediction is described by ˜ (k) = C˜ x ˜ (k) y = Ψx (k) + ΘΔ˜ u (k) + Γu (k − 1) , where ⎡

⎡

⎤

Φ .. .

Φ .. .

(5)

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ N N P1 P1 ⎢ ⎥ ⎥ ⎢ Φ Φ ⎢ ⎥ ⎥ ˜ ⎢ Ψ = C ⎢ NP 1 +Nn ⎥ = ⎢ NP 1 ˜ ⎥, Φ ⎥ ⎥ ⎢ Φ ⎢ Φ ⎥ ⎢ ⎥ ⎢ .. .. ⎦ ⎣ ⎦ ⎣ . . ˜ NP 2 ΦNP 1 +NP 2 Nn ΦNP 1 Φ ⎡ ⎤ Γ2 .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ N P 1 −1 ⎢ ⎥ ⎢ ⎥ i Φ Γ2 ⎢ ⎥ ⎢ ⎥ i=0 ⎢ ⎥ N P 1 −1 ⎢ ⎥ Γ = C˜ ⎢ ⎥, i ˜ ˜ Φ Γ2 + Γ2 Φ ⎢ ⎥ ⎢ ⎥ i=0 ⎢ ⎥ ⎢ ⎥ . .. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ N N P 1 −1 P 2 −1 ⎣ ˜ NP 2 i i ˜ Γ ˜ 2⎦ Φ Φ Φ Γ2 + i=0   Θ 0 11 Θ = C˜ , Θ12 Θ22 with ⎡ ⎤ Γ2 0 0 .. ⎢ ⎥ .. ⎢ . 0⎥ . ⎥, Θ11 = ⎢ P 1 −1 ⎢N ⎥ ⎣ ⎦ i Φ Γ 2 . . . Γ2

⎡

i=0

˜ Φ

Θ12

N P 1 −1

i=0

⎤ ˜2 Φ Γ2 + Γ i

⎤ 0 0 ⎥ ⎢ .. ⎢ . 0⎥ ⎥. ⎢ = ⎢NP 1 −1 ⎥ ⎣  ˜i˜ ˜2⎦ Φ Γ2 . . . Γ ˜2 Γ .. .

i=0

The constraints of the lifted output, steer angle and rate can be put in the form of ymin ≤y (k) ≤ ymax , umin ≤u (k) ≤ umax , (7) Δumin ≤Δu (k) ≤ Δumax . ˜ (k) ≤ The constraint of the lifted output is described by y ymax , −˜ y (k) ≤ −ymin , then the constraint is rewritten as     ˜ max Iny ×NP y ˜≤ y . −Iny ×NP −˜ ymin Likewise, the constraints for the horizon from the control specifications and the physical limits are given by ˜ (k) ≤ g1 , Gu u ˜ (k) ≤ g2 , GΔu Δ˜ Gy y u (k) ≤ g3 , where       Iny ×NP Inu ×Np Inu ×NP Gy = , Gu = , GΔu = , −Iny ×NP −Inu ×NP −Inu ×NP       ˜ max ˜ max y u Δ˜ umax g1 = , g2 = , g3 = . −˜ ymin −˜ umin −Δ˜ umin Then, from the output prediction (5), the constraints for the output are obtained by ˜ (k) = Gy (Ψy y (k) + ΘΔ˜ Gy y u (k) + Γu (k − 1)) ≤ g1 Gy Θ˜ u (k) ≤ g1 − Gy Ψy y (k) − Gy Γu (k − 1) , (8)   T T −1 where Ψy = ΨC CC . The constraints for the steer angle and rate are obtained ⎡ ⎤ u (k − 1)  −1  0 0 ⎢ ⎥ .. Gu INP − Δ˜ u (k) + Gu ⎣ ⎦ . I(NP −1) 0



u (k − 1) Fd

⎢ ⎢ i=0 ⎢ ⎢ .. =⎢ . ⎢ N N ⎢ P 1 −1 P 2 −1 ⎣ ˜ NP 2 ˜ iΓ ˜2 Φi Γ 2 + Φ Φ ⎡

3.2 Constraints

i=0

u (k) + Fo u (k − 1) ≤ g2 = Fd Δ˜ 

⇔ Fd Δ˜ u (k) ≤ g2 = g2 − Fo u (k − 1) , ⎥ ⎥ (9) ⎥ ⎥ where .. .. , ⎥    . −1 . ⎥ 0 0 T ⎥ Fd := Gu INP − , Fo := Gu [1 . . . 1] , ⎦ 0 I N P 2 (N −1) ˜ ˜2 P ... Φ Γ2 + Γ and u (k) ≤ g3 . (10) GΔu Δ˜ ...

˜ +Γ ˜2 Φ

Then combined constraints from (8), (9), (10) are rewritten as   Gy Θ u (k) GΔ˜ u (k) = Fd Δ˜ i=0 GΔu     Then the objective function J˜ in (4) can be rewritten as  −Gy Ψy −Gy Γ  g1 y (k) 0 −Fo . ≤ b (k) = g2 + u (k − 1) 0 0 g3 J˜ = (˜ y (k) − ˜r (k))T Q (˜ y (k) − ˜r (k)) + Δ˜ u (k)T RΔ˜ u (k) (11) T T = (ΘΔ˜ u (k) − ˜ c (k)) Q (ΘΔ˜ u (k) − ˜ c (k)) + Δ˜ u (k) RΔ˜ u (k) 1 T T 3.3 Standard QP problem u (k) HΔ˜ ≡ Δ˜ u (k) + Δ˜ u (k) f + f0 , 2 (6) The optimization problem (4) can be treated as a standard where QP problem such as   T 1 c˜ (k) = ˜r (k) − Ψx (k) −Γu (k − 1) , H = 2 Θ QΘ + R , u (k)T HΔ˜ J˜ (Δ˜ u (k)) = Δ˜ u (k) + Δ˜ u (k)T f (12) 2 T T f = −2Θ Q˜ c (k) , f0 = ˜ c (k) Q˜ c (k) . subject to GΔ˜ u (k) ≤ b. Θ22

283

2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia Table 1. Simulation scenarios

The vector of optimal solution for QP formulation (12) [Δu∗ (k) , · · · , Δu∗ (k + NP − 1)] (13) is predicted at each sample time k. The superscript * denotes the optimized value. The resulting state feedback control law at k is given by u (k) = u (k − 1) + Δu∗ (k) , then the vector (13) is recalculated at the next computation cycle based on the current measured state values. T

Case

NP 1

NP 2

NP

Nn

method

Case A

8

4

12

3

dual prediction horizon

Case B

N/A

N/A

12

N/A

single prediction horizon

Case C

N/A

N/A

16

N/A

single prediction horizon

4. SIMULATION RESULTS 3.4 Terminal constraints ensure stability Assume that Nn = NP 1 to simplify formulation. Then, ¯ for time interval ˜ (k) we can define the lifting matrix y ¯ t = kNn Tc such as     ¯ ¯ η(k) Δuη (k) ¯ ¯ ˜ (k) = (14) y u(k) = ¯ , ¯ , Δ˜ ¯ (k) Δ¯ u(k) y where ⎡

⎡ ⎤ ⎤ Δu(k) y(k + 1) ⎥ ⎥ .. .. ¯ =⎢ ¯ =⎢ η(k) ⎣ ⎦ , Δuη (k) ⎦, ⎣ . . Δu(k + NP 1 − 1) y(k + NP 1 ) ⎡ ⎤ ⎡ ⎤ ¯ y(k¯ + 1) Δu(k) ⎥ ⎥ .. .. ¯ =⎢ ¯ =⎢ ¯ (k) y u(k) ⎣ ⎦ , Δ¯ ⎣ ⎦. . . ¯ ¯ y(k + NP 2 ) Δu(k + NP 2 − 1)

Performance of the proposed control method for lane change system was validated via simulations using MATLAB/Simulink, and CarSim. Table. 1 shows the simulation condition of each case. Case A is the proposed method where NP 1 = 8, NP 2 = 4, NP = NP 1 + NP 2 = 12 and Nn = 3, case B is conventional method where NP = 12 and Case C is conventional method where NP = 16. Figure 4 shows reference trajectories lateral offset and heading angle for changing the lane. From 10 [sec] to 15 [sec], the trajectory changes one lane, from 20 [sec] to 25 [sec], it changes two lane. eyL and eψ are belong to the states of model (1), thus we use the edyL and edψ . 4 edy

d

−4 0

5

10

5

10

15

20

25

30

15

20

25

30

5

d

¯ T Rη Δuη (k) ¯ + Δ¯ ¯ T Ru Δ¯ ¯ u(k) u(k). + Δuη (k) (15)   We assume that the terminal constraint y k¯ + NP 2 + 1 = ¯ be the optimal value of V (k) ¯ with the opti0 and V ∗ (k) ¯ Clearly, V ∗ (k) ¯ ≥ 0 and V ∗ (k) ¯ = 0 only if mizer Δ˜ u∗ (k). ¯ ¯ Let us the optimal solution is to set y(k) = 0 for all k.  ¯  ¯ define V (k) and V (k + 1) such as ¯ ¯ = y(k) ¯ T Qη y(k) ¯ +y ¯ T Qy y ¯ (k) ¯ (k) V  (k) (16) T ¯ + Δ¯ ¯ T Ru Δ¯ ¯ ¯ Rη Δu(k) u(k) u(k), + Δu(k)

edyL

0 −2

eψ [deg]

Suppose that predictive control is obtained for the plant (3) by minimizing the objective function ¯ ¯ +y ¯ T Qy y ¯ = η(k) ¯ T Qη η(k) ¯ (k) ¯ (k) V (k)

ey [m]

2

0 −5 −10 0

time [sec]

Fig. 4. Trajectory of lane change system ¯ (k¯ + 1) We assume the vehicle longitudinal velocity V = 30m/s, ¯ (k¯ + 1)T Qy y V  (k¯ + 1) = y(k¯ + 1)T Qη y(k¯ + 1) + y x u(k¯ + 1)T Ru Δ¯ u(k¯ + 1) look-ahead distance L = 10m, sampling time Tc = + Δu(k¯ + 1)T Rη Δu(k¯ + 1) + Δ¯ Tcam = 10ms, and MPC weightings for y and Δu are ¯ + Δ¯ ¯ T Ru Δ¯ ¯ ¯ T Qy y ¯ (k) ¯ (k) u(k) u(k) =y Q = diag (50, 13, 1), R = 200, respectively. And the ¯ − y(k) ¯ T Qη y(k) ¯ − Δu(k) ¯ T Rη Δu(k) ¯ = V  (k) constraints for y, u, Δu are given by −4m ≤ y1 ≤ 4m, ¯ −0.3rad ≤ y2 ≤ 0.3rad, −0.3rad/s ≤ y3 ≤ 0.3rad/s, ≤ V  (k), (17) −0.007rad ≤ u ≤ 0.007rad, −0.0007rad/ΔT ≤ Δu ≤   0.0007rad/ΔT . ¯ because of y k¯ + NP 2 + 1 = 0. And we can find V  (k+1) Figure 5 and 6 show comparison of the general method such as V (k¯ + 1) < V  (k¯ + 1). Then, and proposed method. Fig. 5 shows the simulation results ∗ ¯ V (k + 1) = min V (k¯ + 1) of each state (lateral offset at look-ahead distance, time Δu derivative of lateral offset at c.g., yaw angle error, yaw  ¯ ≤ min V (k + 1) rate) and Fig. 6 shows the steer angle that is control input Δu ¯ ≤ min V  (k) (18) u and the input variation Δu. Because input constraint Δu is −0.007rad(0.4011deg) ≤ u ≤ 0.007rad(0.4011deg), the ¯ ≤ min V (k) steering angle is saturated 20.5-22.5 [sec] and 23-26 [sec]. Δu Dimension of H for QP problem (12) is the same as case A ∗ ¯ = V (k). and case B. Therefore, prediction horizons of case A and ∗ ¯ Thus V (k) is a Lyapunov function and (y, Δu) is stable. B are the same. But case B has poor performance. On 284

2013 IFAC IAV June 26-28, 2013. Gold Coast, Australia 5

eyL [m]

0.5

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Case A Case B Case C

−10 5

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δ [deg]

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Case A Case B Case C

0

30 −0.5 5

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0.05 0

Δ δ [deg/Tc]

dey/dt [m/s]

2

−2 −4 5

10

15

20

25

0

30 −0.05 5

time [sec]

eψ [deg]

5 0

Fig. 6. Comparison of the control input for various horizon

S.-J. Wu, H.-H. Chiang, J.-W. Perng, C.-J. Chen, B.F. Wu, and T.-T. Lee, The heterogeneous systems integration design and implementation for lane keeping −10 5 10 15 20 25 30 on a vehicle. IEEE Trans. Intell. Transp. Syst., 9(2): 246-263, 2008. 4 ¨ Ozguner, ¨ C. Hatipoglu, U. and Redmill, K. A., Automated 2 lane change controller design. IEEE Trans. Intelligent Transportation System, 4(1):13-22, 2003. 0 P. Varaiya, Smart cars on smart roads: problems of control. −2 IEEE Trans. Automatic Control, 38(2):195-207, 1993. S. Li, K. Li, R. Rajamani, and J. Wang, Model predictive −4 5 10 15 20 25 30 multi-objective vehicular adaptive cruise control. IEEE time [sec] Trans. Control Syst. Technol., 19(3):556-566, 2011. P. Falcone, F. Borrelli, J. Asgari, H.E. Tseng, and D. Fig. 5. Comparison of the state for various horizon Hrovat, Predictive active steering control for authe contrary, case C has good performance, but prediction tonomous vehicle systems. IEEE Trans. Control Syst. horizon is longer than case A. Through this results, we Technol., 15(3):566-580, 2007. demonstrate that dual prediction horizons of MPC im- C. E. Garcia, D. M. Prett, and M. Morari, Model proves driving performance without increase of length of predictive control : theory and practice - a survey. horizon. Automatica, 25(3):335-348, 1989. S. J. Qin and T. A. Badgwell, A survey of industrial model predictive control technology. Control Engineer5. CONCLUSIONS ing Practice, 11(7):733-764, 2003. In this paper we proposed MPC for dual prediction hori- Maciejowski. J. M. , Predictive control with constraints. Pearson education, 2002. zons. In order to reduce the length of horizon while guaranS.-H. Lee, Y.O. Lee, B.-A. Kim, and C. C. Chung, teeing the stability and the performance, the MPC uses the Proximate Model Predictive Control Strategy for Autwo prediction time intervals. Under a certain modularity tonomous Vehicle Lateral Control. Proc. of American of prediction horizon, we proved the stability of the closedControl Conference, 3605-3610, 2012. loop system using terminal constraints. Through simulaJ. B. Rawlings and K. R. Muske, The stability of tions, the proposed method is shown to be very prospective constrained receding horizon control. IEEE Trans. with significantly fast computation as well as guaranteed Automat. Contr., 38(10):1512-1516, 1993. performance. W. H. Kwon and A. E. Pearson, A modified quadratic cost Acknowledgement This research was supported by Baproblem and feedback stabilization of a linear system. sic Science Research Program through the National ReIEEE Trans. Automat. Contr., 22(5):838-842, 1977. search Foundation of Korea(NRF) funded by the Ministry R. Rajamani, Vehicle dynamics and control. Springer, of Education, Science and Technology(2012R1A6A1029029). 2006. S.-H. Lee, Y.O. Lee, Y. Son, and C. C. Chung, Multirate Active Steering Control for Autonomous Vehicle Lateral REFERENCES Maneuvering. Proc. of Interlligent vehicles symp., 772S. Huang, W. Ren, and S. C. Chan, Design and per777, 2012. formance evaluation of mixed manual and automated E. Camacho and C. Bordons, Model predictive control. control traffic. IEEE Trans. Syst., Man, Cybern. A, Springer, 2004. Syst., Humans, 30(6):661-673, 2000. dψ/dt [deg/s]

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