IET Control Theory & Applications Research Article
Model predictive control allocation for stability improvement of four-wheel drive electric vehicles in critical driving condition Haiyan Zhao1,2 , Bingzhao Gao1 , Bingtao Ren2 , Hong Chen1,2
ISSN 1751-8644 Received on 4th November 2014 Revised on 25th July 2015 Accepted on 24th August 2015 doi: 10.1049/iet-cta.2015.0437 www.ietdl.org
, Weiwen Deng1
1
State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130062, Jilin, People’s Republic of China Department of Control Science and Engineering, Jilin University (Campus NanLing), Changchun 130062, Jilin, People’s Republic of China E-mail:
[email protected] 2
Abstract: To improve the vehicle stability of an electric vehicle (EV) with four in-wheel motors, the authors investigate the use of a non-linear control allocation scheme based on model predictive control (MPC) for EVs. Such a strategy is useful in yaw stabilisation of the vehicle. The proposed allocation strategy allows a modularisation of the control task, such that an upper level control system specifies a desired yaw moment to work on the EVs, while the control allocation is used to determine control inputs for four driving motors by commanding appropriate wheel slips. To avoid unintended side effects, skidding or discomforting the driver in critical driving condition, the MPC method, which permits us to consider constraints of actuating motors and slip ratio, is proposed to deal with this challenging problem. An analytical approach for the proposed controller is given and applied to evaluate the handing and stability of EVs. The experimental results show that the designed MPC allocation algorithm for motor torque has better performance in real time, and the control performance can be guaranteed in the real-time environment.
Nomenclature β, γ δf , δ Mz V,M Fyf , Fyr h, Re Iω ax , ay Iz Lf , Lr L d Cf , Cr Ti Ka , Kb αf , αr Fxi Fzi
1
sideslip angle/yaw rate front wheel steering angle from controller/driver direct yaw moment input vehicle velocity/mass lateral force of front/rear tires height of centre of gravity/wheel radius moment of inertia of the wheels longitudinal/lateral acceleration yaw moment of inertia distance from the centre of gravity to front/rear axle distance from the front axle to rear axle tread of vehicle cornering stiffness at front/rear tire motor torque on the wheel curve fitting coefficients tire sideslip angles of the front/rear wheel longitudinal force static loads at front and rear wheels
Introduction
In recent years, electric vehicles (EVs) driven by electric motors have recently been widely recognised and adopted in many countries as an effective mean to reduce pollution [1, 2]. Compared with traditional internal combustion engine drive vehicles (ICVs), motor torque generation is fast and accurate [3, 4]. This advantage enables us to realise high performance control of EVs. However, in four-wheel drive (4WD) case, due to four driving motors, control system has four permitted independent control inputs [5], which are all available to control yaw rate and sideslip angle. If we design control strategy for 4WD EVs, one issue faced is how to deal with such actuator redundancy, which means how to allocate torque for four driving motors [6]. The subject of resolving the redundancy of over-actuated systems has been an active field of research, see [7] for a comprehensive overview. Most existing algorithms neglect the actuator dynamics or deal with the actuator
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dynamics separately, thereby, a static relationship between actuator dynamic output and plant inputs is assumed [8]. Especially, some constrained control allocation methods are based on some scaling of the unconstrained optimal control allocation, such that the resulting control allocation is projected onto the boundary of the set of attainable generalised forces. The static control allocation (SCA) method starts with the unconstrained control allocation computed using some pseudo-inverse. Following the seminal work of [9], several algorithms for linear control allocation and adaptive control allocation have been developed [10–13]. In addition, actuator dynamics can be considered by model predictive control (MPC) allocation as shown by the authors [14, 15]. The main objective of this paper is to present a control allocation scheme for EVs, which permits us to consider the constraints of actuating motors, and simultaneously consider the characteristic of wheel slip. Building on the results of [16], this paper provides results for an MPC-based dynamic control allocation algorithm for asymptotically stable tracking of non-constant input commands. Here, we also propose an integrated yaw stability control algorithm for 4WD EV. The whole control strategy consists of two control loops. The upper controller is concise and comparable to the practice structure of the modern automotive control [16], which benefits to bridge the gap between advanced model-based control and engineering implementation. The main contribution is a real-time MPC allocation strategy to provide the additional yaw torque. The MPC problem herein is posed as a sequential quadratic programming problem with the constraints [17, 18]. In order to control tire slip on low-friction coefficient road, the soft constraint of tire longitudinal slip ratio is translated into a penalty function to enforce the slip changing within the stable slip range. To evaluate the improvement in performance achieved when EV drives in critical conditions, we also compare the MPC allocation scheme with the SCA scheme. The simulation results also extend those of [16] in that controller evaluation method is used, thereby increasing the importance of accounting for actuator dynamics, and highlighting the requirement for sophisticated control allocation. At last, to improve the real-time performance of controller based on MPC for torque allocation, a scheme for the controller implementation based on FPGA (field-programmable gate array) is adopted.
IET Control Theory Appl., 2015, Vol. 9, Iss. 18, pp. 2688–2696 © The Institution of Engineering and Technology 2015
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System model for controller design
To design control law, a non-linear model based on vehicle dynamics is developed in this section, which mainly consists of vehicle, tire and in-wheel motor. The symbols in modelling procedure and the related physical meanings are listed in Nomenclature section. To design the controller, a widely used simplified single track vehicle model is considered to capture the essential vehicle lateral steering dynamics as follows Fyf + Fyr − γ, MV Lf Fyf − Lr Fyr + Mz γ˙ = . Iz β˙ =
Fig. 1
2
Vehicle dynamics control structure
Integrated yaw stability control algorithm
The concept of the proposed control algorithm is shown in Fig. 1. The upper control loop is used for calculating virtual control effectors Mz , which is required for realising steady and dynamic characteristics of control system. The control allocation task is to distribute this generalised force to the individual tyres, in an optimal manner by mapping Mz to physical control inputs for motor. The controlled output variables are γ and β, and a reference model is used for calculating desired input commands γr and βr in accordance with driver’s commands. A reference model is designed based on the linear two-degreesof-freedom vehicle model, which is formed as follows 2Cf (δ − β − {Lf γ /V }) + 2Cr (−β + {Lr γ /V }) − γ , (1a) MV 2Lf Cf (δ − β − {Lf γ /V }) − 2Lr Cr (−β + {Lr γ /V }) , (1b) γ˙ = Iz β˙ =
where the responses of γ and β to the front wheel steering input δ are second-order systems. From the perspective of frequency response analysis, we derive the transfer function from steering angle δ to γ as mentioned in [19] (see (2)) where ωn is the natural frequency, ξ is the damping coefficient and Gγ (s) is the steady gains transfer function. To achieve an ideal closed-loop performance, the reference model is defined as follows γr (s) =
ωd2 Gkγ (s) s2 + 2ωd ξd · s + ωd2
· δ(s),
(3)
where ωd = kωn ωn , ξd = kxi ξ and Gkγ (s) = kg Gγ (s). To improve the yaw rate phase delay and make response quick, we should increase the bandwidth of system, which means frequency factor kωn > 1. To reduce the volatility of second-order system response, the damping coefficient should be larger than the previous, as damping factor kxi > 1. Due to the conflict of rapidity and stationarity performance exists, kωn and kxi need to coordinate. In addition, γr should be considered by its adhesion saturation value |γr | ≤ μg/V, and the sideslip angle β is desired to be zero to improve handling stability, i.e. βr = 0. It should be noted that the modularisation offered by the control allocation would to a large extent simplify the design of such a controller.
(4a) (4b)
Assuming that the longitudinal speed V is constant, only the lateral and yaw motion of the vehicle is taken into account. 3.1
Tire model
A widely used semi-empirical tyre model to calculate steady-state tyre force and moment characteristics for use in vehicle dynamics studies is based on the so called Magic formula [20]. Tire lateral forces can be calculated as follows Fyi = −Dy sin(Cy arctan(By αi − Ey (By αi − arctan By αi ))), i = f,r
(5)
where By , Cy , Dy and Ey all depend on the normal vertical forces Fz , so the Fy is a function of α and Fz . To simplify the model and keep tire non-linear characteristic, we expand the magic formula of Fy with the Taylor expansion method, and obtain a cubic polynomial as follows Fyf = −2Cf (1 − Ka αf2 )αf ,
(6a)
−2Cr (1 − Kb αr2 )αr ,
(6b)
Fyr =
where Ka , Kb are coefficients, which can be obtained by the Taylor expansion of the Magic formula model of tires. Tire side slip angles of the front and rear wheels are calculated as αf = β + (Lf /V )γ − δ and αr = β − (Lr /V )γ . The practical longitudinal slip ratio κ is defined as follows κ=−
vx − ωRe vsx =− , vx vx
(7)
where vsx , vx are the slip velocity and the wheel velocity, respectively. To calculate the required motor torque, a torque balance equation is formed for each wheel. The motion dynamics of a single wheel are calculated as follows 1 ˙ M V = Fx , 4 J ω˙ = Te − Tb − Re · Fx ,
(8a) (8b)
where Fx is a tire longitudinal force and Te is the motor drive torque in each wheel. According to (7) and (8), we assume that
γ (s) (2Cf Lf /Iz )s + [{4Cf Cr (Lf + Lr )}/MVIz ] , = 2 2 2 δ(s) s + ([{2(Cf Lf + Cr Lr )}/VIz ] + [{2(Cf + Cr )}/MV ])s + [{4Cf Cr (Lf + Lr )2 − 2MV 2 (Cf Lf − Cr Lr )}/MV 2 Iz ] =
s2
ωn2 Gγ (s) . + 2ωn ξ · s + ωn2
IET Control Theory Appl., 2015, Vol. 9, Iss. 18, pp. 2688–2696 © The Institution of Engineering and Technology 2015
(2)
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wheel longitudinal velocity vx is equal to vehicle longitudinal velocity V , so the wheel slip ratio dynamics is derived to ωR ˙ e V − ωRe V˙ V2 ωRe Fx (Te − Re Fx − Tb )Re − = JV (1/4)MV 2 R2 Re κ +1 = − e − · Fx + (Te − Tb ). JV (1/4)MV JV
κ˙ =
(9)
x(k + 1) = f k (x(k), u(k)) · Ts + x(k), y(k) = C · x(k),
Here the mechanical braking mode is not considered, so Tb = 0. To calculate force easily, the formula can be simplified as, Fx = Ck κ, where Ck is decided by tire vertical load Fz as follows Ck = Fz (pk1 + pk2 dfz ) · e dfz =
Fz − Fz0 , Fz0
pk3 dfz
(10)
Fz0 = 4000 N,
It is important to note that, although all of the variables mentioned above are accessible in a simulation environment, they must be measured or estimated in real life. In-wheel motor model
We assume that there is a motor torque controller for each wheel and the ideal motor closed-loop dynamics is simplified as follows Te =
1 · Tc , τs + 1
(12)
where τ is the closed-loop response time which is a control characteristic of motor torque controller, and Tc is the motor driving torque command. Due to motor torque saturation, the constraints of the control inputs Tc can be written as follows −Te max ≤ Tc ≤ Te max ,
(13)
where Te max is the maximum torque output of the motor. Then, the total drive torque Tt from driver pedal command determines the driving torque of four motors, which is an equality constraint as follows Tt = Tcfl + Tcfr + Tcrl + Tcrr . (14)
4
Design of optimisation function
Combine (9), (12) and (13), we get the state-space model as follows R2 Re xi + 1 Cki xi + ui , x˙i = − e − JV {(1/4)MV } JV
i = fl, fr, rl, rr
d y = (−Ckfl xfl + Ckfr xfr − Ckrl xrl + Ckrr xrr ), 2
(15a) (15b)
where x = [κfl , κfr , κrl , κrr ]T , u = [Tcfl , Tcfr , Tcrl , Tcrr ]T , y = Mz .
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d −Ckfl 2
Ckfr
−Ckrl
Ckrr .
According to the main principle of MPC, p is defined as predictive horizon to extend the forecast of the system future output and m is defined as control horizon and set m = p for simplification. Usually horizon p ≥ m, and here p is simply set to be equal to m. The values of p and m determined by trial and error not only ensure control performance, but also reduce optimisation burden. The future control input sequence U(k) is optimisation vector in the following cost function and the predictive control output sequence is Y(k), which can be defined as follows ⎤ u(k|k) ⎢ u(k + 1|k) ⎥ ⎥ U(k) = ⎢ , .. ⎦ ⎣ . u(k + m − 1|k) m×1 ⎡r(k)⎤ ⎡
⎡y(k + 1|k)⎤ ⎢y(k + 2|k)⎥ ⎥ Y(k) = ⎢ , .. ⎦ ⎣ . y(k + p|k) p×1
⎢r(k)⎥ ⎥ R(k) = ⎢ . ⎣ .. ⎦ . r(k) p×1 where each vector of U(k) is an array of control inputs u, and each vector of Y(k) is an array of system outputs y, R(k) is defined as output reference sequence containing m arrays of virtual control input Mz which keeps the same in each predictive horizon. The main control requirement is to ensure the EV good handling and stability on cornering, i.e. to make actual yaw rate γ and sideslip angle β track the required reference output γr and βr quickly. Optimum torque allocation control is used to convert virtual control effector to physical control inputs for available sets of motors. So we define the first cost function as follows J1 = Y(k) − R(k)2Q p [(Mzr (k + i|k) − Mz (k))2 · Q],
(17)
i=1
If we design control strategy for 4WD EVs, one issue faced is how to deal with such actuator redundancy, which leads to a control allocation problem. Due to the strong non-linearities and dynamic constraints, the use of MPC techniques will generally be desired for lateral vehicle control. 4.1
C=
=
Yaw moment control allocation
(16a) (16b)
where output matrix is
· λk ,
where the data of factors pk1 , pk2 and pk3 are derived from [20]. As mentioned above, the slip derivative dynamics can easily be formulated as follows Re κ +1 R2 Ck · κ + Te . (11) κ˙ = − e − (1/4)MV JV JV
3.2
The main task is to determine control input commands for four motors to (i) keep the tire from slipping or locking; (ii) obtain the maximum tire load ratio; (iii) energy consumption as small as possible; and (iv) improve yaw rate stability. The sample time k is chosen as k = int(T /Ts ), where T is running time and Ts is a fixed sampling step. The system model (15) can be described as a discrete non-linear form as follows
where Mzr is the reference control input, Q is the positive weight factors for adjusting tracking performance. As the tyre longitudinal slip κ directly reflects the tyre slip performance, we should restrict κ with a soft constraint of system. However, this output constraint of κ in this paper is a non-linear constraint for the optimisation problem, and it will affect the speed of solving. Hence, we consider a penalty function for anti-slip instead of slip soft constraints to facilitate fast solution, which enforces κ to remain within the stable slip range and is calculated as follows ⎧ κ (k) − κ i max ⎪ , κi (k) ≥ κmax , ⎪ ⎪ ⎪ κmax ⎨ Ei (k) = (18) 0, −κmax ≤ κi (k) ≤ κmax , ⎪ ⎪ ⎪ κi (k) + κmax ⎪ ⎩ , κi (k) ≤ −κmax , κmax where the subscript i denotes fl, fr, rl, rr for each wheel.
IET Control Theory Appl., 2015, Vol. 9, Iss. 18, pp. 2688–2696 © The Institution of Engineering and Technology 2015
In addition, we also consider energy consuming problem of the motors. As the larger motor torque means more energy provided from the battery, we should make the sum of torque command squares Tc2 as small as possible to save energy while ensuring stable vehicle motion. Therefore, we define an additional cost function in a form as follows
4.2
J2 = U(k)2R =
m−1
where P is a positive penalty matrix. It should be noted that terminal penalty matrix P can be determined by offline, which can reduce the computing time of controller. Equation (21) guarantees closed-loop stability by extending the finite prediction horizon to infinite horizon.
To this end, the total control objective is written as follows 2
(Tcfl (k + j|j) + Tcfr (k + j|j)
2
Jmpc (x(k), U(k)) = J1 + J2 + J3 + J4 ,
j=0
+ Tcrl (k + j|j)2 + Tcrr (k + j|j)2 ) · R,
J3 = U(k)2S =
= Y(k) − R(k)2Q + U(k)2R
(19)
where R = diag(R0 , R0 , R0 , R0 ) is positive weight factor to build a suitable form of the normalised function for Tci , and affect the gaps among four motor torque commands. To reduce the rate of change of the control action and ensure driving of the vehicle, a cost function is considered to control the rate of change of the control action as follows
m−1
Total control objective
+ U(k)2S + x(k + p|k)2P ,
(22)
subject to (16), (18) and − Te max ≤ Tci (k + j|k) ≤ Te max , i = fl, fr, rl, rr,
j = 0, 1, 2, . . . , m − 1.
(23)
Tt = Tcfl (k + j|k) + Tcfr (k + j|k) + Tcrl (k + j|k) + Tcrr (k + j|k), 2
( Tcfl (k + j|j) + Tcfr (k + j|j)
2
j = 0, 1, 2, . . . , m − 1.
(24)
j=0
+ Tcrl (k + j|j)2 + Tcrr (k + j|j)2 ) · S,
(20)
where the control input changing sequence U(k) is calculated by u(k) = u(k) − u(k − 1) and S = diag(S0 , S0 , S0 , S0 ) is positive weight factor to coordinate the rate of change of control inputs Tci . To extend the finite prediction horizon to infinite horizon, a terminal cost is needed to penalise the states at the end of the finite horizon. This quadratic terminal cost in (21) controls the non-linear EV system by local linear state feedback to guarantee closed-loop stability [18] J4 = x(k + p|k)2P = x(k + p|k)T · P · x(k + p|k),
Fig. 2
(21)
The objective function in (22) includes four conflicting contributions. The relative importance of the objective function contributions is controlled by setting the time dependent weight matrices Q, R, S, P, these are chosen to be positive definite.
5
Simulation results
For verifying the derived model and testing the designed controller, a more realistic electric vehicle system simulation model is established in AMESim environment as shown in Fig. 2. The AMESim model include body model, electric wheel model, in-wheel model, steering model, suspension model, battery model and so on.
Electric vehicle model based on AMESim
IET Control Theory Appl., 2015, Vol. 9, Iss. 18, pp. 2688–2696 © The Institution of Engineering and Technology 2015
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5.1
Simulation and parameter tuning
Table 1 Parameters of virtual vehicle model
In this study, the simulation time T = 15 s, the fixed sample step Ts = 0.01 s. According to the data of vehicle, we set the factors of expected response γr , βr of ideal vehicle, weight factors, prediction horizon and control horizon as follows kωn = 1.9,
kxi = 1.3,
Q = 8550,
R0 = 7500,
kg = 0.83, ωd = 8.47, S0 = 1050,
ξd = 0.96.
p = m = 3.
Symbol Iz d Lf Cr L Re κmax
Unit
Value
Symbol
Unit
Value
kgm2 m m N/rad mH m –
1992.54 1.418 1.0628 23101 0.05 0.29 0.08
m h Lr Cf Rom ρf Te max
kg m m N/rad
Wb Nm
1359.8 0.512 1.4852 23,540 0.0005 0.1 500
(25) Based on the weight factors and data of vehicle as mentioned above, the terminal penalty matrix P is calculated by solving the Lyapunov equation (21) as follows ⎡ ⎤ 0.2597 0 0 0 0.2597 0 0 ⎥ ⎢ 0 . P = 1.0 × 105 · ⎣ 0 0 2.2042 0 ⎦ 0 0 0 2.2042 Setting these parameters requires multiple sets of tests and once determined, do not need to be updated online to adapt to different scenarios. The weight factor Q plays regulatory roles in the following the expectations to maintain vehicle stable. R0 affects the gaps among four motor torque commands. With the increasing of R0 , the differences between four torques become smaller. S0 is positive weight factors to coordinate the rate of change of control inputs δf and Tc . After several commissioning tests, enlarging S0 can make change of control inputs smoothly. In the following sections, the responses are compared with that in the case without any control and the case with SCA. Parameters
of the actual vehicle model used in the simulation are shown in Table 1. 5.2
Simulation results and analysis
The standard double lane-change (DLC) manoeuvre, ISO 38881:1999(E), is established in AMESim. The following scenario is simulated: the first one (0–70 m) is the beginning phase with speed of 80 km/h in straight road topology, the second phase (70–200 m) presents the double lane road, in the third phase (after 200 m), the EV moving up the sloped road of 17% (high friction surface) and 8% (low friction surface). From the above simulation scenarios, all the dynamic characteristics of the vehicle can be adequately described, and the yaw stability of EV and energy performance of the motor can be fully verified. 5.2.1 High friction surface μ = 0.8: The trajectory responses and detailed dynamic response are shown in Fig. 3. EV
Fig. 3 Dynamic response of EV: μ = 0.8 a Sideslip angle and yaw rate b Displacement X –Y –Z of EV c Torque distribution: MPC allocation d Torque distribution: static control
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IET Control Theory Appl., 2015, Vol. 9, Iss. 18, pp. 2688–2696 © The Institution of Engineering and Technology 2015
with SCA and proposed controls can pass the test track with similar responses. Compared to the SCA control, the proposed control shows slightly smaller yaw rate tracking error and sideslip angle deviation. Without control, the vehicle can also driving normally just shows the most oscillatory responses. As shown in Fig. 3b, the vehicle begin climbing at 10 s, the vehicle velocity will be reduced and the motor torque begin to increase for overcoming the influence of slope. Thus, the vibration of the curve at 10 s is large. From Figs. 3c and d, we can see that the motor torque keep close to saturation because of the effort of the control input constraints. Simultaneously, the optimal performance can be obtained and the proposed control algorithm shows smaller energy consumption and good climbing ability. 5.2.2 Low friction surface μ = 0.3: To further evaluate the anti-slip performance of the proposed method, the EV model is driven on low-friction coefficient roads through DLC manoeuvre, which is closed-loop test manoeuvres used to evaluate the lateral dynamic of a vehicle. The simulation is carried out at an initial velocity of about 60 km/h. The road surface is assumed to be wet road with a friction coefficient of 0.3. The trajectory responses and detailed dynamic response are shown in Fig. 4. From Fig. 4a, we can seen that the vehicle without control shows more oscillatory responses and loses the yaw motion stability. The proposed control can track the desired yaw rate while maintaining the sideslip angle within a certain range. Although the SCA control can stabilise the yaw motion, its yaw rate tracking error and sideslip angle deviation are larger than those of the proposed control. It can be seen from Fig. 4b, vehicle trajectory slightly deviates from the desired trajectory. The control input as active steering angle δf is shown in Figs. 4c and d. It can be seen, MPC allocation strategy can better
satisfy the driver’s operation intention and has the better handling performance than SCA strategy. In addition, the result Fig. 5a illustrates that tire longitudinal slip ratio spread quickly and the κ on no-control condition is beyond 0.3, which means that the ground cannot provide enough static friction force for tire. Compared with this, the tire longitudinal slip ratio κ is controlled in the range of 0.02 by MPC allocation and control performance is better than SCA strategy. Distributed torque command Tc as functions of time is shown in Fig. 5b. Compared to the SCA strategy, the proposed control shows smaller energy consumption and the proposed controller has corrected the steering angle δ from driver and the motor torque Tc is under motor constraint Te max . 5.3
Controller evaluation
This section deals with three major performances, the controller’s performance, the driver’s mental workload and the motor actuator’s workload as determining factors for vehicle handling and stability qualities. The task performance J1e is calculated by the following equation as the integral of the square of an error between the desired path and the actual state of the vehicle t J1e = (eβ2 + eγ2 ) dt (26) 0
where eβ = β − βr , eγ = γ − γr . Apparently, the smaller the value of J1e is, the higher is the task performance. The whole control performance is improved by coordination of the upper and lower controllers, i.e. the value of J1e is associated with control objectives J1 , J2 , J3 , J4 .
Fig. 4 Dynamic response of EV: μ = 0.3 a Sideslip angle and yaw rate b Displacement X –Y –Z of EV c Steering angle d Local expansion of steering angle
IET Control Theory Appl., 2015, Vol. 9, Iss. 18, pp. 2688–2696 © The Institution of Engineering and Technology 2015
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Fig. 5 Response of slip ratio and torques: μ = 0.3 a Tire longitudinal slip ratio b Motor torques
Accordingly, the magnitude of the driver’s physical workload can be determined by the integral of the square of the steering wheel angle as follows [21] J2e
=
t
2
(δ − δf ) dt.
(27)
t 0
( Tfl2 + Tfr2 + Trl2 + Trr2 ) dt,
(28)
where Tij , ij = fl, fr, rl, rr. is the rate of torque change. This means that the smaller the value of J3e is, the lighter the frequent operation or the motor actuator wear is. From the perspective of energy saving, the value of J3e is associated with the selection of weight coefficient of J3 . It is worth emphasising that these evaluation criteria can be used for designing the model predictive control (MPC) allocation controller. According to different control requirements, the weight coefficient of the control objectives J1 , J2 , J3 , J4 need to be correspondingly adjusted. We randomly select 50 sets of control parameters within a certain range for experiments under double lane manoeuvre. Table 2 shows the values of the J1e , J2e , J3e . From Table 2, we can know that the two kinds of controllers can help drivers successfully complete the test in double lane condition, but the MPC allocation (MPCA) shows the best performances. On the one hand, improve the control performance by 47%; on the other hand, improve driver’s physical workload by 41%. In particular,
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Parameter
MPCA
J1e , rad · s−1
0.0318
0.0601
47
J2e , rad2 · s−1
0.0711
0.11970
41
3.108 × 105
2.233 × 106
86
J3e , Nm
0
This means that the smaller the value of J2e is, the lighter is the driver’s physical workload. Moreover, the value of J2e reflect the control performance of the upper controller. It is known that in general the frequent adjustment for motors will cause the motor to overheat and result in thermal protection. Simultaneously, motor actuator wear is inevitable. In addition, the level of smooth operation for motor actuator can be determined by the integral of the square of the rate of torque change as follows J3e =
Table 2 Performance analysis of the considered controller SCA
Improvement ratio, %
the MPC allocation strategy can relieve the motor actuator wear and improve the smooth operation by 86%. Thus, MPC allocation plays an important role for the vehicle’s handling and stability performance. 5.4
Experimental results
To solve the real-time problem of MPC in miniaturised device and embedded system, the MPC controller is built on an FPGA chip. The FPGA implementation allows the particle swarm optimisation (PSO) algorithm to exploit its parallel search capabilities. Therefore, the idea of using PSO to solve non-linear optimisation problem in MPC will be a good alternation. The PSO is employed to achieve realtime operation due to its naturally parallel capabilities. The proposed FPGA-based MPC-PSO controller consists of random number generator, fixed-point arithmetic, PSO solver and universal asynchronous receiver/transmitter (UART) communication interface. The PSO algorithm has three loops which account for most of the computation at each solution; (i) update the particles from generation to generation; (ii) update every particle in one generation; and (iii) update one-dimension of a particle. The FPGA chip is selected as Altera Stratix III EP3SL150F1152 and the clock frequency is set to 50 MHz. The FPGA implementation architecture of the designed MPC-PSO controller for torque allocation control is shown in Fig. 6a. For validating the FPGA-based MPC controller, the testing platform is based on an FPGA board and an xPC-Target real-time system is built up and shown in Fig. 6b. The FPGA board is
IET Control Theory Appl., 2015, Vol. 9, Iss. 18, pp. 2688–2696 © The Institution of Engineering and Technology 2015
Fig. 6 Hardware implementation a Implementation architecture b Test platform
the hardware implementation of the MPC-PSO controller. The EV model is established in host PC and then is downloaded to the target PC after compiling. PC1 runs the FPGA design tools (Quartus II and Catapult C) and is used to design the MPC-PSO controller. The result data exchanged between the FPGA-based controller and the EV plant on target PC are transferred through UART RS232 serial link. The synthesis results for the FPGA implementation of the PSO-v1 architecture are presented in Table 3. It is worthy noting that the resource consumption and performance before optimisation and after optimisation are different and the speed of solving the problem is improved greatly at the cost of
Table 3 Synthesis results of the PSO-v1 architecture Architecture
Cycles
Time, ms
Slack
Area (LUTs)
PSO-v1
344,741
6.894820
0.61
62,939
more resource consumption of the FPGA. In this paper, we carry out before optimisation. Experimental parameters are the same as the condition when μ = 0.3 without sloped road. The real-time simulation results are
Fig. 7 Experimental results: μ = 0.3 a Sideslip angle b Yaw rate c Slip ratio response d Torque response
IET Control Theory Appl., 2015, Vol. 9, Iss. 18, pp. 2688–2696 © The Institution of Engineering and Technology 2015
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shown in Fig. 7. It can be seen that the designed shifting controller meets the demand of real time and control. However, compared with the simulation results shown in Section 5.2, there is a certain tracking deviation. First, the optimisation solver is NAG-E04WD when offline simulation, the optimisation tools PSO is used to design the controller when FPGA hardware implementation. Thus, different solvers lead to a certain bias. Second, and noise always exists when the FPGA communicates with the xPC-Target. In addition, the quantisation errors result in low measuring precision of the system output and the controller output. Last, the time needed by once solution of controller calculations is close to the sampling time, thus the speed of the controller still needs to be improved.
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Conclusions and future work
This paper proposed the hierarchical control system of the front steering angle compensation and the traction force distribution for 4WD EVs based on optimal control theory. Simulation study clarified that the vehicle yaw motion and the vehicle attitude, sideslip motion, can be much improved by the proposed hierarchical control system. The upper control loop uses a tripe-step non-linear control method to specify control dynamic which is required for keeping yaw stability. The lower control loop is used to determine control inputs for driving motors by MPC allocation method. Finally, the designed controller is evaluated through simulation model established in the AMESim environment and compared with static allocation method under the same condition. The results show that the proposed control strategy can improve the stability of 4WD EV, especially EV drives in critical conditions.
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This work was supported by the 973 Program (no. 2012CB821202), the National Nature Science Foundation of China (no. 61503149, no. 61520106008) and Jilin Provincial Science Foundation of China (no. 3D513U315420).
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IET Control Theory Appl., 2015, Vol. 9, Iss. 18, pp. 2688–2696 © The Institution of Engineering and Technology 2015
