applied sciences Article
Integrated Adaptive Cruise Control with Weight Coefficient Self-Tuning Strategy Junhui Zhang 1,2,3 , Qing Li 1,2,3 and Dapeng Chen 1,2, * 1 2 3
*
Automotive Electronics Center, Institute of Microelectronics of Chinese Academy of Sciences, Beijing 100029, China;
[email protected] (J.Z.);
[email protected] (Q.L.) University of Chinese Academy of Sciences, Beijing 100049, China Kunshan Department, Institute of Microelectronics, Chinese Academy of Sciences, Kunshan 215347, China Correspondence:
[email protected]; Tel.: +86-186-1056-8899
Received: 19 April 2018; Accepted: 1 June 2018; Published: 15 June 2018
Abstract: This paper presents a novel multi-objective coordinated adaptive cruise control (ACC) algorithm based on a model predictive control (MPC) framework which can comprehensively address issues regarding longitudinal car-following performance, lateral stability, as well as vehicle safety. During the car-following, vehicle dynamics, illustrating the forces acting on the tire contact patches, are established. To simplify the tightly coupled dynamics system, a state-feedback based disturbance decoupling method is employed, by which longitudinal and lateral dynamics can be completely decoupled. Furthermore, the traditional MPC control with a constant weight matrix will probably not be able to solve time-varying multi-objective coordinated optimization issues, especially in transient scenarios. A weight coefficient self-tuning strategy is therefore suggested by which the weight coefficient for each sub-objective can be adjusted automatically with the change of traffic scenarios, accordingly improving the overall car-following performance. The simulations show that the control algorithm utilizing the suggested self-tuning strategy reaps significant benefits in terms of longitudinal car-following performance, while at the same time maintaining a small lateral stability error range. Keywords: adaptive cruise control (ACC); model predictive control (MPC); direct yaw-moment control (DYC); longitudinal car-following performance; lateral stability
1. Introduction The adaptive cruise control (ACC) system is the commercial implementation of the advanced driver assistance system (ADAS) that can reduce drivers’ workload (e.g., taking over the longitudinal control task), and improve vehicle safety as well as traffic flow [1,2]. The designs of current commercial ACC systems mainly focus on tracking a desired speed or maintaining a desired safe distance from preceding vehicles in the same lane. Additionally, various studies have recently begun to explore additional objectives such as fuel economy, longitudinal ride comfort, lateral stability, and vehicular physical limitations [2–6]. As these mentioned objectives usually conflict with each other [5], coordinating them simultaneously still presents a lot of challenges for designers, especially under complicated and volatile traffic scenarios. Ioannou et al. [7] pointed out that fuel consumption was much higher under conditions of large acceleration/deceleration, lane changing, as well as cut-in/cut-off. On this basis, Jonsson et al. [8] suggested reducing fuel consumption while decreasing tracking capability. With respect to the longitudinal ride comfort, the vehicle longitudinal acceleration and its derivative (jerk) are typically restrained [9]. To improve lateral stability, researchers typically employ a direct yaw-moment control (DYC) system [10,11]. Importantly, an emergency braking, road surface disturbance or a sudden strong side wind might result in vehicle deflection during the longitudinal car-following [11], Appl. Sci. 2018, 8, 978; doi:10.3390/app8060978
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(DYC) system [10,11]. Importantly, an emergency braking, road surface disturbance or a sudden 2 of 15 strong side wind might result in vehicle deflection during the longitudinal car-following [11], and furthermore ACC and DYC are reversely interactive with each other under some traffic scenarios and furthermore ACC and DYC areareversely interactive withis each other under traffic [12]. Zhang et al. [12] therefore present curving ACC system that coordinated with a some DYC system scenarios [12]. Zhang et al. [12] therefore present a curving ACC system that is coordinated with and gives consideration to both the longitudinal car-following performance and lateral stability on acurved DYC system consideration to both provides the longitudinal car-following andimpact lateral roads. and The gives developed control system significant benefits inperformance weakening the stability roads. Thecar-following developed control system provides benefits in weakening of DYC on curved the longitudinal performance while alsosignificant improving the lateral stability. the impact of DYC on the longitudinal car-following performance while also improving the lateral What is more, in order to achieve the multiple objectives mentioned above, it is natural to introduce stability. What is more, in order to framework, achieve the multiple objectives in mentioned above, ithave is natural to a model predictive control (MPC) whose advantages ACC controllers recently introduce a model been proven [2,5,6].predictive control (MPC) framework, whose advantages in ACC controllers have recently been proven [2,5,6]. model predictive control algorithm with a constant weight matrix will However, the traditional However, traditional model control algorithm withscenarios a constant[13]. weight matrix probably not bethe able to improve the predictive overall performance in transient Zhao et al. will [13] probably not be able to improve the overall performance in transient scenarios [13]. Zhao et al. [13] pointed out that the traditional model predictive control algorithm with a constant weight matrix pointed out that traditional model predictive control algorithm with comfort, a constant weight matrix cannot meet the the improvement requirement in fuel economy and ride and accordingly cannot meet the improvement requirement fuel time-varying economy andmulti-objective ride comfort,control and accordingly proposed a real-time weight tuning strategy toinsolve problems, proposed real-time weight strategy torespect solve time-varying multi-objective control where theaweight matrix cantuning be adjusted with to different operating conditions. Inproblems, addition, where the weight weight coefficient matrix canfor be adjusted with respect to different conditions. In addition, when the each sub-objective is constant, it isoperating easy to cause the distribution of when the to weight coefficient for each sub-objective it is easy to causethe theglobal distribution of solutions be more concentrated, and accordingly is it constant, will be difficult to achieve optimum solutions to be more concentrated, and accordingly will be difficult achievethe thelongitudinal global optimum [14]. [14]. For instance, when the current traffic scenarioit is emergent andtosevere, tracking For instance, when the current traffic scenario is emergent andwhich severe, the longitudinal tracking capability should be most important to avoid a rear-end collision, indicates that the importance capability shouldcoefficient) be most important avoid a rear-end collision, indicates that the (namely weight of such to a sub-objective in the MPC which cost function should beimportance paid more (namely coefficient) such a sub-objective in the MPC cost function should be paid more attentionweight to. Similarly, whenof the tire-road friction coefficient is small, or the road curvature is large, attention to. Similarly, when the tire-road friction coefficient is small, or the road curvature is large, priority should be given to the lateral stability for driving safety consideration. priority be given to the lateral stabilityvehicle for driving safety consideration. As should the lateral stability of an automotive in a steering maneuver is critical to the overall thean lateral stability of an cruise automotive a steering is critical to the safetyAs[11], integrated adaptive controlvehicle (IACC) in model which maneuver takes the longitudinal interoverall [11], andynamics integratedinto adaptive cruise control which takes theto longitudinal vehicle safety and lateral consideration as a(IACC) wholemodel is suggested so as achieve a inter-vehicle lateral dynamics into consideration as awhile wholeguaranteeing is suggested as tostability. achieve maximizationand of the longitudinal car-following performance thesolateral aFurthermore, maximization the longitudinal car-following performance while guaranteeing lateral stability. toofsolve the time-varying multi-objective coordinated optimizationthe issues explained Furthermore, to solve the time-varying multi-objective coordinated optimization issues explained above, a weight coefficient self-tuning strategy is thus proposed, where the weight coefficient for above, a weight coefficient self-tuning strategy is thus where the weight coefficient for each each sub-objective can be adjusted automatically withproposed, the change of traffic scenarios. sub-objective canthis be paper adjusted automatically with theSection change2 of traffic scenarios. The rest of is organized as follows. presents the modeling of an integrated The rest ofsystem. this paper is organized as follows. Sectionindexes 2 presents the modeling integrated car-following Section 3 quantifies performance for IACC. Sectionof4 an suggests the car-following system. Section adjustment 3 quantifies strategy performance indexes for IACC.optimization Section 4 suggests thealgorithm dynamic dynamic weight coefficient as well as receding control weight coefficient adjustmentand strategy as well recedingin optimization controlconclusions algorithm based on MPC. based on MPC. Simulations analysis areas discussed Section 5. Lastly, are drawn in Simulations Section 6. and analysis are discussed in Section 5. Lastly, conclusions are drawn in Section 6. Appl. Sci. 2018, 8, 978
2. Modeling of of Integrated Integrated Car-Following Car-Following System 2. Modeling System Section is mainly mainly organized organized as as shown shown in in Figure Figure 1. 1. During During the the car-following, car-following, vehicle vehicle dynamics, dynamics, Section 22 is illustrating the forces To simplify simplify the the tightly illustrating the forces acting acting on on the the tire tire contact contact patches, patches, are are established. established. To tightly coupled dynamics system, a state-feedback based disturbance decoupling method [15] is employed, coupled dynamics system, a state-feedback based disturbance decoupling method [15] is employed, by which longitudinal longitudinal and lateral dynamics dynamics can completely decoupled. After that, that, the the longitudinal longitudinal by which and lateral can be be completely decoupled. After inter-vehicle dynamics model are derived, as well linearized, respectively. Finally, inter-vehicle model modeland andlateral lateral dynamics model are derived, as as well as linearized, respectively. the integrated car-following state space model based on the MPC framework is designed. Finally, the integrated car-following state space model based on the MPC framework is designed.
Figure 1. Flow chart of integrated car-following decoupling and linearization; LPV: linear Figure 1. Flow chart of integrated car-following decoupling and linearization; LPV: linear parameter-varying. parameter-varying.
2.1. Vehicle Dynamics As illustrated in Figure 2, the longitudinal (x-direction), lateral (y-direction) and yaw dynamics during the car-following can be described as the equations below.
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J f + Jr 1 vx = Fxf − Fyf sin δ + Fxr − mgf cos θ − mg sin θ − CD Aρ vx2 m(vx − v yωr ) + 2 2 r 2.1. Vehicle Dynamics ( ) cos m v + v = F + F ω δ (1) y x r yf yr As illustrated in Figure 2, the longitudinal (x-direction), lateral (y-direction) and yaw dynamics I zωr = aFyf coscan bFdescribed δ −be yr + M z , des as the equations below. during thecar-following . J f + Jr . 2 v x = Fx f − Fy fcontact sin δ + patch Fxr − mg f coswhile θ − mgFsin θand − 21 CFD Aρv + longitudinal x − vF y ωr )are x where Fxfm(vand tire forces, r2 . xr yf yr are lateral (1) m(vy + v x ωr ) = Fy f cos δ + Fyr . ones. J f Izand J are the moment of inertia of the front and rear tires, whose rolling radius is r. r ω r = aFy f cos δ − bFyr + Mz,des vx is the longitudinal speed, vy is the lateral speed, and ωr is the yaw rate. m is the mass of the where Fx f and Fxr are longitudinal tire contact patch forces, while Fy f and Fyr are lateral ones. J f and f tires, δ longitudinal is the of gravitational acceleration, is the rolling resistance is the road vehicle, Jr are the gmoment inertia of the front and rear whose rolling radiuscoefficient, is r. v x is the θ isωthe wheel of the front and and roadyaw slope. themass dragofcoefficient, thegravitational air density, speed, angle vy is the lateral speed, rate.CmD isisthe the vehicle,ρg isisthe r is the acceleration, the rolling coefficient, δ isbtheare road anglefrom of thethe front and θofisthe themass road a and area resistance of the vehicle. thewheel distances center and A is thef isfrontal slope. CD is the drag coefficient, ρ is the air density, and A is the frontal area of the vehicle. a and b are (CM) of the vehicle to the front and rear axles respectively, I z is the yaw inertia, and M z ,des is the the distances from the center of the mass (CM) of the vehicle to the front and rear axles respectively, desired Iz is the yaw yaw moment. inertia, and Mz,des is the desired yaw moment.
Figure 2. Schematic representing vehicle dynamics. Figure 2. Schematic representing vehicle dynamics.
Now the assumptions below are made to simplify Equation (1): (1):
•• ••
road wheel angle δδis tiny so so that sin δsin ≈ δδ, ≈ δ cos δ ≈δ1.≈ 1 . , cos is tiny that road slope θ is assumed to be neglected. road slope θ is assumed to be neglected.
Then we obtain
.
vx =
1
(F + F + F )
xr d 1M x f ( Fxf + Fxr + Fd ) ( . vx =1 vy = mM ( Fy f + Fyr − mv x ωr ) .
ωr =
1 Iz ( aFy f
− bFyr + Mz,des )
1 v y = (1Fyf + Fyr2 − mvxωr ) Fd = − Fy f δ − mg f m− 2 CD Aρv x + mvy ωr . where M = m + ω r = 1 (aFyf − bFyr + M z , des ) 2.2. Disturbance Decoupling Iz J f + Jr , r2
(2) (2) (3)
(3)
Furthermore, the lateral forces Fy f and Fyr are assumed to linearly vary with the tire slip angles α f and αr respectively J +[16]. J 1 M = m + f 2 r Fd = − Fyf δ −(mgf − C D Aρ vx2 + mv yωr F = yf 2 kf αf r , . where (4) Fyr = kr αr
where k f and kr are the cornering stiffness of the front and rear tires respectively. Besides, the tire slip angles associated with each wheel can be related to the body side-slip angle β, road wheel angle δ and the yaw rate ωr by the relationships below. (
r α f = δ − β − aω vx bωr αr = − β + v x
(5)
α = − β + bωr r vx Integrating Equations (3)–(5), we obtain a 2-DOF vehicle model, which can be used for a lateral stability controller design.
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k f + kr bk − ak k β + ( r 2 f − 1)ωr + f δ β = − mvwe mvx vehicle model, mvx which can be used for a lateral Integrating Equations (3)–(5), x obtain a 2-DOF (6) 2 2 stability controller design. ω = bkr − ak f β − a k f + b kr ω + ak f δ + M z ,des r . r bkr − f +kr I zak vf Ik f Iz β = I−z kmv β + ( mv 2 x − 1) ωr + mvz x δ x
x (6) . = bkr − ak f β − a2 k f +b2 kr ω + ak f δ + Mz,des Regarding Equation (2), ω there is aIz time-varying item: F , which changes with the road wheel r r Iz v x Izd Iz
δ , longitudinal angleRegarding vx ,islateral speed vyitem: andFthe yaw rate ωr . The idea of disturbance Equationspeed (2), there a time-varying d , which changes with the road wheel angle decoupling, as speed shownv xin Figurespeed 3, is trying find a rate state-feedback soof asdisturbance to constructdecoupling, a closed-loop δ, longitudinal , lateral vy andto the yaw ωr . The idea as systemin [15]. shown Figure 3, is trying to find a state-feedback so as to construct a closed-loop system [15].
Figure 3. 3. Disturbance Disturbance decoupling decoupling via via aa state-feedback. state-feedback. Figure
Then the decoupled longitudinal dynamics is rewritten as Then the decoupled longitudinal dynamics is rewritten as
.v v xx
1 = 1 ( Fxf + Fxr ) ax = M ( Fx f + Fxr ) , a x M
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2.3. Longitudinal Dynamics Model 2.3. Longitudinal Dynamics Model As illustrated in Figure 4, three state variables are defined: the distance error ∆d, speed error As illustrated in Figure 4, three state Δ variables defined: the distance error Δd , speed error d = d −are d des ∆v and derivative of the longitudinal acceleration (jerk for short) j, which can be described as Δv and derivative of the longitudinal acceleration (jerk for short) j, which can be described as (8) Δ v = v − v p x = d − ddes ∆d j = ax ∆v (8) = v p − vx j = a. where d is the actual inter-vehicle distance, d x is the desired inter-vehicle distance, v , a des
p
p
and vdx ,is the ax actual are the longitudinal speed/acceleration of the precedingdistance, vehicle vand the following where inter-vehicle distance, ddes is the desired inter-vehicle p , a p and v x , a x are the longitudinal speed/acceleration of the preceding vehicle and the following vehicle respectively. vehicle respectively.
Figure 4. Sketch of longitudinal car-following.
With , the constant time headway (CTH) spacing policy [17] is employed, which With respect respecttoto dddes des , the constant time headway (CTH) spacing policy [17] is employed, which changes with the of speed of the following vehicle. changes linearlylinearly with the speed the following vehicle.
= ττhhvvxx + + dd00 des = dddes
(9) (9)
the nominal time headway and is zero-speed the zero-speed clearance. where ττ his is where the nominal time headway and d0 d is0the clearance. h In addition, the decoupled decoupled longitudinal longitudinal dynamics dynamics of of the the vehicle vehicle (see (see Figure Figure 3) 3) is is still still nonlinear, nonlinear, mainly due to the nonlinearity of the engine torque maps, time-varying gear position, as well as due to the nonlinearity of the engine torque maps, time-varying gear position, as well aerodynamic drag force [18]. In order to linearize the longitudinal dynamics of the vehicle, we introduce a first-order inertial system [5], whose transfer function is described as
H (s) =
KL TL ⋅ s + 1
(10)
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as aerodynamic drag force [18]. In order to linearize the longitudinal dynamics of the vehicle, we introduce a first-order inertial system [5], whose transfer function is described as H (s) =
KL TL · s + 1
(10)
where K L is the system gain and TL is the time constant. Integrating Equations (8)–(10), a four state space model for longitudinal car-following dynamics can be formulated as x1 (k + 1|k) = A1 x1 (k) + B1 u1 (k) + G1 w1 (k ) (11) where x1 (k) = [∆d(k), ∆v(k), a x (k), j(k)] T , u1 (k) = a x,des (k ), w1 (k) = a p (k ), k represents the kth sampling point, A1 , B1 and G1 are the state matrices, mathematically expressed as A1 =
Ts 1 0 0
1 0 0 0
−τh Ts − Ts 1 − Ts /TL −1/TL
0 0 0 0
B1 = [0, 0, Ts · K L /TL , K L /TL ] T G1 = [0, Ts , 0, 0] T 2.4. Lateral Dynamics Model As shown in Equation (6), a 2-DOF model is always used for the lateral stability controller design [12,19]. Additionally, the matrix representation of Equation (6) is described as .
x2 = Ψx2 + Πu2 + Γw2
(12)
where x2 = [ β, ωr ] T , u2 = Mz,des , w2 = δ, Ψ, Π and Γ are the state matrices, mathematically expressed as k +k bkr − ak f r f − mv − 1 mv2x Ψ = bk −akx a2 k f + b2 k r r f − Iz Iz v x h iT 1 Π = 0 , Iz i h k ak T Γ = mvf x , Izf Unlike the longitudinal dynamics model, the lateral dynamics model, whose state matrices change with the longitudinal speed v x , is time-varying. This means that the discretization of the state matrices should be carried out in every control cycle, with p (the length of the predictive horizon) times increased on this basis if the receding horizon control (e.g., MPC) method is employed. Additionally, the main problem this causes is that the system’s real-time performance will be greatly worsened. To reduce the considerable computational burden, a linear parameter-varying (LPV) method [20] is applied, changing the nonlinear model into a linear parameter-varying one. Firstly, the state matrix Ψ can be described as " Ψ =
0
−1
bkr − ak f Iz
0
#
+
= Ψ0 + Ψ1 p1 + Ψ2 p2 2
= Ψ0 + ∑ Ψ i P( i ) i =1
where p1 =
1 vx ,
p2 =
1 , v2x
P = [ p1 , p2 ].
−
k f +kr m
0
0
−
a2 k f + b2 k r Iz
" 1 vx
+
0 0
bkr − ak f m
0
# 1 v2x
(13)
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Secondly, two boundary vectors can be obtained because p1 and p2 achieve maximum / minimum values at the same time. P1 = [ p1,max , p2,max ], P2 = [ p1,min , p2,min ] (14) Then, the local linearization of Equation (13) in the boundaries can be expressed as 2
Ψb,j = Ψ0 + ∑ Ψi P j (i ), j = 1, 2.
(15)
i =1
Thirdly, employing the zero-order hold (ZOH) method [5] with a sample time Ts , the discretization of the state matrices in the boundaries can be expressed as ∞ Ψi Tsi b,j i! i =0 ∞ Ψi−1 Tsi Bb,j = ∑ b,ji! Π, i =1 ∞ Ψi−1 Tsi Gb,j = ∑ b,ji! Γ i =1
Ab,j = ∑
j = 1, 2.
(16)
Finally, a two state space model for the lateral dynamics can be formulated as x2 (k + 1|k) = A2 x2 (k) + B2 u2 (k) + G2 w2 (k )
(17) 2
2
j =1
j =1
where x2 (k ) = [ β(k ), ωr (k)] T , u2 (k) = Mz,des (k), w2 (k) = δ(k), A2 = ∑ wb,j Ab,j , B2 = ∑ wb,j Bb,j , 2
2
j =1
j =1
G2 = ∑ wb,j Gb,j , wb,j is weight coefficient, and ∑ wb,j = 1 , wb,j > 0. During the MPC control process, the discretization of the state matrices, shown as Equation (16), is just carried out once. Additionally, in each control cycle, only the weight coefficient wb,j needs to be adjusted. Therefore, the LPV method has its merits in solving the problem earlier mentioned. 2.5. Integrated Car-Following Model Via the integration of the longitudinal dynamics model (refer to Equation (11)) and lateral dynamics model (refer to Equation (17)), the IACC control plant model is built as (
x(k + 1|k) = Ax(k) + Bu(k) + Gw(k) y(k ) = Cx(k)
(18)
where x(k) = [∆d(k), ∆v(k), ax (k), j(k), β(k), ωr (k)]T , u(k) = [ax,des (k), Mz,des (k)]T , w(k) = [a p (k), δ(k)]T , A, B, G and C are the state matrices, mathematically expressed as " A=
#
A1 A2
" ,B=
#
B1 B2
" , G=
#
G1 G2
, C = I6×6 .
3. Quantification of Performance Indexes 3.1. Cost Function for Longitudinal Car-Following Performance It is accepted that the longitudinal car-following performance is usually evaluated via tracking errors: ∆d and ∆v. However, the strong convergence of tracking errors will result in a frequent acceleration or deceleration, which might weaken the lateral stability and also reduce the longitudinal ride comfort or fuel economy, while the poor convergence will also cause a great amount of problems
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such as frequent cut-in, unsatisfied driver response, or even rear-end collision. Thus, the longitudinal performance indexes are designed to achieve:
• •
• • •
During the steady car-following scenario, the tracking errors should converge to tiny values; If the preceding vehicle accelerates, the tracking errors should be in a driver-permissible tracking error range to reduce frequent cut-in or driver intervention, and while it decelerates, rear-end collision must be avoided; As the fuel consumption increases with the absolute increase of the vehicle acceleration [7], the fuel economy can be improved indirectly via penalizing acceleration; To promote longitudinal ride comfort, the longitudinal jerk should be decreased as much as possible [5]; The level of longitudinal acceleration should be restrained to some extent to guarantee sufficient lateral stability [12].
Accordingly, employing a 2-norm for tracking errors, actual acceleration, jerk and desired acceleration, the longitudinal cost function is built as Jlongi = w∆d ∆d2 + w∆v ∆v2 + wax a2x + w j j2 + wax,des a2x,des
(19)
where w∆d , w∆v , wax , w j and wax,des are the corresponding weight coefficients. 3.2. Cost Function for Lateral Stability Generally, the direct yaw control (DYC) is introduced to improve the vehicle lateral stability which is critical to vehicle safety [11,12]. The purpose of DYC is to make the yaw rate error and side-slip angle error convergent to zero by means of adding an external yaw moment. However, the yaw moment is always generated from a differential braking, which will weaken the longitudinal tracking capability. Thus, the yaw rate error, side-slip angle error, as well as desired yaw moment should be penalized simultaneously. Similarly, the lateral cost function is built as 2 Jlat = w∆ωr ∆ωr 2 + w∆β ∆β2 + w Mz,des Mz,des
(20)
where w∆ωr , w∆β and w Mz,des are the corresponding weight coefficients. ∆ωr = ωr − ωr,nom , ∆β = β − β nom , ωr,nom , β nom is the nominal vehicle lateral dynamics response [12]. 3.3. Rear-End Safety To avoid rear-end collisions, an inequality for rear-end safety is specified as d ≥ dsa f e = max{t TTC ∆v , ds }
(21)
where d is the actual inter-vehicle distance, dsa f e is the safe inter-vehicle distance, ds is the critical minimum safe distance, and t TTC is the time-to-collision (TTC) value [5]. The constraint, shown as Equation (21), should be considered as a hard constraint, owing to the fact that a rear-end collision cannot be tolerated at all. 3.4. Adhesion Workload As the longitudinal tracking capability, along with the lateral stability, is closely related to road adhesion, the adhesion workload is defined as q a2x + a2y η= λmax
(27)
is the adjustment factor of the weight coefficient of the corresponding
sub-objective, λmin together with λmax are the design parameters, and wk+1 (·) is the tuned weight coefficient for the cost function optimization in the next receding horizon. 4.3. Final Expression of Predictive Control Algorithm In order to prevent the slack variables from increasing toward infinity, an additional quadratic term of slack variables is defined to penalize the violation of hard constraints so that it will be possible to make a balance between computing feasibility and the relaxation of constraints. The cost function in the finite predictive horizon, shown as Equation (24), is rewritten as J (y, u, ε) = J + ε T ρε
(28)
where ε = [ε 1 , ε 2 , ε 3 , ε 4 , ε 5 ] T is the slack variable vector, and ρ = diag(ρ1 , ρ2 , ρ3 , ρ4 , ρ5 ) is the penalized matrix. Finally, the predictive optimization problem is to minimize the cost function Equation (28) subject to I/O constraints Equations (21), (22) and (25), which is eventually equivalent to solving Equation (29), as follows. ( min J (y, u, ε) u, ε, Q, R (29) s.t. constraints (21) , (22) , (25) 5. Simulations and Analysis To validate and evaluate the improved performance of the proposed IACC controller with the weight coefficient self-tuning strategy, several numerical comparative simulations are conducted utilizing Matlab/Simulink and Carsim. The main parameters of the IACC controller are shown in Table 1. The test scenario is supposed to be that the preceding vehicle, with an initial speed of 30 m/s, slows into the curve road with an emergent deceleration of −4.0 m/s2 , then maintains a speed of 10 m/s for 15 s, and afterwards speeds up with an acceleration of 1.5 m/s2 to drive away, provided that the radius of the curve road is 350 m and that the tire-road friction coefficient is 0.8. What is more, the MPC based integrated control algorithm with tunable weight coefficients is simply called MPC-TW for short, while MPC-CW is short for the comparative one with constant weight coefficients. During the steering maneuver, priority should be given to lateral stability for driving safety consideration. Lateral stability (maintaining a small lateral stability error range) should be guaranteed for both MPC-TW and MPC-CW. As shown in Figure 7, both control methods can achieve a better performance in lateral stability.
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Table 1. Parameters of the IACC Controller. IACC: integrated adaptive cruise control. Para.
Value
Para.
Value
KL TL Ts
1.0 0.40 0.1 −3 1.5 5 5 5 1444 9.81 0.018 1.10 1.57 0.30 0 2.22 0.37 1750
∆dmax ∆dmin ∆vmax ∆vmin a x,max a x,min jmax jmin ∆d υmax ∆d υmin ∆v υmax ∆v υmin ax υmax ax υmin j υmax j υmin Q R ρ
5 −5 0.9 −1.0 1.0 −4.0 2 −2 3 −3 0.9 −1.0 0.1 −0.1 0.05 −0.05 diag(10,10,1,1,10,10) diag(1,1) diag(3,3,3,3,3)
t TTC τh d0 ds p m g f a b r θ A CD Iz
Figures 8–10 show the longitudinal tracking response of speed, distance, acceleration and jerk, respectively. From Figures 8 and 9, one can see that the MPC-CW cannot provide a substantial braking force in time when the preceding vehicle decelerates emergently at 10 s, and accordingly the inter-vehicle distance becomes shorter than that of MPC-TW. As shown in Figure 10, the MPC-TW can take an immediate brake as a result of tunable weights and also avoid overshooting to some extent, accordingly improving the longitudinal ride comfort. Furthermore, during the steering maneuver, the tracking capability of MPC-CW (from 30 s or so in Figures 8 and 9) is somewhat sacrificed due to the extra brake pressure generated from the direct yaw moment. However, MPC-TW is able to weaken the impact of DYC applied to the longitudinal car-following performance by automatically tuning the weights of ∆d, ∆v and a x,des (see Figure 11), while maintaining a small lateral stability error range (see Figure 7). Moreover, from Figure 9, one can see that the introduction of a slack variable vector into the hard constraints (refer to Equation (23)) is able to satisfy the upper/lower boundary of inequality by increasing the corresponding slack variable when the boundary exceeds, shown as Equation (25), accordingly avoiding computing infeasibility. Figure 11 shows the adjustment factor of the weight coefficient based on the self-tuning strategy suggested as Equation (27). Under the emergent scenario (from 10 s to 20 s or so), the weight of the desired acceleration, as well as the speed error, decreases because the corresponding adjustment factor shifts to less than 1, which indicates that the penalty for a x,des in the cost function becomes weakened, accordingly achieving a substantial braking force (see Figure 10) and also maintaining a better tracking capability (see Figures 8 and 9). Additionally, a small adhesion workload means that the tires are far from the adhesion limit and that the vehicle still has great capacity for acceleration/deceleration and lateral stability [12]. As shown in Figure 12, the peak values for MPC-TW and MPC-CW are about 59% and 52%, respectively, accordingly guaranteeing another hard constraint, shown as Equation (22). Dangerous situations, such as spin or drift, always happen when the adhesion workload tends to its peak [12]. Although the proposed model improves the longitudinal tracking capability and lateral stability, it therefore clearly reduces vehicle safety. That is to say, under the transient scenarios, the pursuit for the longitudinal tracking capability as well as lateral stability should be sought under the premise of sufficient road adhesion capacity. Additionally, this means that the designing of an upper/lower boundary of the adjustment factor, shown as Equation (27), is in relation to the adhesion workload.
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(a) Steering angle input
15 1520
δ sδ (deg) (deg) s δ s (deg)
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(a) Steering angle input (a) Steering angle input
20 20
10 1015 5 510 0 05
-5 -5 00 0 -5
10 10 0
10
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Δω (rad/s) Δω r r (rad/s) Δωr (rad/s)
0.04 0.06 0.04
20 20
30 30
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Time(s) Time(s) 20(b) Yaw rate 30 error 40 (b) YawTime(s) rate error (b) Yaw rate error
0.02 0.04 0.02
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60 60
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MPC-TW MPC-TW MPC-CW MPC-CW MPC-TW MPC-CW
0 0.02 0
-0.02 -0.02 0 -0.04 -0.02 -0.04 0 0 -0.04
10 10 0
10
1 1
30 30
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Time(s) Time(s) 20Sideslip 30 40 (c) angle error (c) Sideslip angle error Time(s) (c) Sideslip angle error
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20 20
0.5 0 0
50 50
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MPC-TW MPC-TW MPC-CW MPC-CW MPC-TW MPC-CW
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Time(s) Time(s) 30
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Time(s) Figure 7. Yawrate rate error and and side-slip angle error. Figure side-slipangle angle error. Figure7.7.Yaw Yaw rate error error and side-slip error. Figure 7. Yaw rate error and side-slip angle error.
Figure 8. Speed. Figure 8. Speed. Figure 8. Speed.
Distance Distance(m) (m) Distance (m)
Figure 8. Speed.
Figure 9. Actual inter-vehicle distance and distance error. Figure 9. Actual inter-vehicle distance and distance error. Figure 9. Actual inter-vehicle distance and distance error.
Figure 9. Actual inter-vehicle distance and distance error.
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j (m/s3 ) j (m/s3 )
2 a x,des (m/s ) a x,des (m/s 2 )
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λ λk k
λ λk k
λ λk k
Figure 10. Acceleration and jerk. Figure Figure10. 10.Acceleration Accelerationand andjerk. jerk.
Figure 11. 11. Adjustment Adjustment factor factor for for weight weight coefficient. coefficient. Figure Figure 11. Adjustment factor for weight coefficient.
Additionally, a small adhesion workload means that the tires are far from the adhesion limit and Additionally, a small adhesion workload means that the tires are far from the adhesion limit and that the vehicle still has great capacity for acceleration/deceleration and lateral stability [12]. As that the vehicle still has great capacity for acceleration/deceleration and lateral stability [12]. As shown in Figure 12, the peak values for MPC-TW and MPC-CW are about 59% and 52%, respectively, shown in Figure 12, the peak values for MPC-TW and MPC-CW are about 59% and 52%, respectively, accordingly guaranteeing another hard constraint, shown as Equation (22). Dangerous situations, accordingly guaranteeing another hard constraint, shown as Equation (22). Dangerous situations,
such as spin or drift, always happen when the adhesion workload tends to its peak [12]. Although the proposed model improves the longitudinal tracking capability and lateral stability, it therefore clearly reduces vehicle safety. That is to say, under the transient scenarios, the pursuit for the longitudinal tracking capability as well as lateral stability should be sought under the premise of sufficient road adhesion capacity. Additionally, this means that the designing of an upper/lower Appl. Sci. 2018, 8, 978 14 of 15 boundary of the adjustment factor, shown as Equation (27), is in relation to the adhesion workload. 0.8 MPC-TW MPC-CW
0.6 0.4 0.2 0
0
10
20
30
40
50
60
Time(s)
Figure12. 12.Adhesion Adhesionworkload. workload. Figure
6. Conclusions 6. Conclusions An IACC control plant model with a weight coefficient self-tuning strategy is suggested so as to An IACC control plant model with a weight coefficient self-tuning strategy is suggested so as to achieve the maximization of the longitudinal car-following performance while guaranteeing lateral achieve the maximization of the longitudinal car-following performance while guaranteeing lateral stability. The main conclusions drawn are as follows. stability. The main conclusions drawn are as follows. (i) During the car-following, lateral stability should be given priority and can be well-guaranteed (i) During the car-following, lateral stability should range be given priority and can be well-guaranteed while satisfying a driver-permissible tracking because input constraints, namely the while satisfying a driver-permissible constraints, longitudinal acceleration level, in the tracking suggestedrange IACCbecause controlinput algorithm, can be namely limited the to a longitudinal acceleration level, in the suggested IACC control algorithm, can be limited to specific range. a specific range. (ii) Compared with the traditional MPC control with a constant weight matrix, the weight coefficient (ii) Compared the traditional MPCsolve control with a constant weight matrix, the weight coefficient self-tuningwith strategy can hopefully time-varying multi-objective coordinated optimization self-tuning strategy can hopefully solve time-varying multi-objective coordinated optimization issues, where the weight coefficient for each sub-objective can be adjusted automatically with issues, where weight coefficient each sub-objective be adjusted performance. automatically with the the change ofthe traffic scenarios, thusfor improving the overallcan car-following change of traffic scenarios, thus improving the overall car-following performance. (iii) In addition, during the steering maneuver, MPC-TW is able to weaken the impact of DYC (iii) In addition, during the steering maneuver, MPC-TW is able to weaken the impact of DYC applied applied to the longitudinal tracking capability by automatically tuning the weight coefficients to longitudinal tracking capability by automatically tuningerror the weight of the sub-objectives, while maintaining a small lateral stability range. coefficients of the sub-objectives, while maintaining a small lateral stability error range. Author Contributions: D.C. and Q.L. conceived and designed the experiments; J.Z. performed the experiments; J.Z. analyzed the data; S.L. from Tsinghua University contributed analysis tools; J.Z. wrote the paper. Author Contributions: D.C. and Q.L. conceived and designed the experiments; J.Z. performed the experiments; J.Z. analyzed the data;The S.L.authors from Tsinghua University analysis tools; J.Z. wroteconstructive the paper. comments Acknowledgments: would like to thankcontributed all anonymous reviewers for their and helpful suggestions on this research work. Meanwhile, the authors would like to thank financialcomments support of Acknowledgments: The authors would like to thank all anonymous reviewers for their constructive STS Plan of Chinese Academy Science (No.: KFJ-STS-ZDTP-045). and helpful suggestions on this research work. Meanwhile, the authors would like to thank financial support of STS Plan of Chinese Academy Science (No.: KFJ-STS-ZDTP-045). Conflicts of Interest: The authors declare no conflicts of interest. Conflicts of Interest: The authors declare no conflicts of interest.
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