Antilock Braking System (ABS) is an important active safety feature in preventing accidents during
emergency braking. Electrified vehicles which include both hydraulic and regenerative braking
systems provide the opportunity to implement brake torque blending during slip control operation.
This study evaluates the design and implementation of a new torque allocation algorithm using a
Nonlinear Model Predictive Control (NMPC) strategy that can run in real-time, with results showing
that wheel-locking can be prevented while also permitting for energy recuperation

Basrah, M. Sofian

Cao, Dongpu

Longo, Stefano

Siampis, Efstathios

Velenis, Efstathios

English

CERES Research Repository (Cranfield Univ.)

Integration of Torque Blending and Slip Control using Nonlinear Model Predictive Control  M. Sofian Basrah*, Efstathios Siampis, Efstathios Velenis, Dongpu Cao, Stefano Longo Cranfield University *E-mail: s.s.basrah@cranfield.ac.uk   Antilock Braking System (ABS) is an important active safety feature in preventing accidents during emergency braking. Electrified vehicles which include both hydraulic and regenerative braking systems provide the opportunity to implement brake torque blending during slip control operation. This study evaluates the design and implementation of a new torque allocation algorithm using a Nonlinear Model Predictive Control (NMPC) strategy that can run in real-time, with results showing that wheel-locking can be prevented while also permitting for energy recuperation.  Topics / Brake, active safety, full-electric vehicles and hybrid electric vehicles 1. INTRODUCTION   Electrified vehicles are equipped with redundant braking actuators, namely hydraulic brakes and a regenerative braking system. This creates the opportunity for research into brake torque blending for Antilock Braking System (ABS) in a hybrid braking system.  Several articles report torque blending algorithms using optimization methods. In [1] an adaptive slip controller is proposed and the brake torque allocation is designed using Control Allocation (CA). In [2] a linear slip controller is designed and a linear Model Predictive Control (MPC) strategy is employed to allocate the brake torque between the two actuators. The cascaded strategy in [2] is replaced with a combined strategy for slip control and torque blending using linear MPC in [3]. Static brake torque allocation by Daisy Chain (DC) which requires less computational effort as compared to optimization methods is proposed and tested on a hardware-in-the-loop simulator in [4]. The problem of integrating slip control and brake torque allocation includes important nonlinearities in both the system dynamics and constraints. The emergence of real time nonlinear solvers make it possible to treat the optimization as a constrained nonlinear problem instead of using linear approximations. To the authors’ best knowledge, there is currently no work on integrated slip control and brake torque allocation using Nonlinear Model Predictive Control (NMPC) with real time implementation capability. This work therefore presents a slip control and torque blending strategy incorporated in single NMPC formulation.  2. PREDICTIVE CONTROL STRATEGY  For the NMPC formulation a single-wheel model is employed. Assume that the continuous-time model is  ?̇?𝑥 =𝐹𝑥𝑚,                                  (1𝑎) ?̇?𝑥 = −𝐹𝑥𝑉𝑥(𝑅𝑤2𝐽𝑤+𝑠𝑥𝑚+1𝑚) +𝑅𝑤𝐽𝑤𝑉𝑥𝑇𝑡𝑜𝑡 ,       (1𝑏)  with 𝑉𝑥 the wheel’s forward velocity, 𝑠𝑥 the longitudinal slip, 𝐹𝑥 the tyre’s longitudinal force, 𝑇𝑡𝑜𝑡 the total torque applied on the wheel and 𝑅𝑤, 𝐽𝑤 and 𝑚 the wheel’s radius, moment of inertia and mass respectively. In the above model, 𝐹𝑥 is set as a function of 𝑠𝑥 through a simplified version of Pacejka’s Magic Formula (MF) [5]:  𝐹𝑥 = 𝐹𝑧𝐷𝑠𝑖𝑛(𝐶𝑡𝑎𝑛−1(𝐵𝑠𝑥)),                 (2)  where 𝐵, 𝐶 and 𝐷 are the MF’s factors and 𝐹𝑧 the vertical force on the tyre. For our blending strategy, the 𝑇𝑡𝑜𝑡 is the summation of the torque from the electric motor 𝑇𝑒 and the torque from the hydraulic brake 𝑇ℎ. Then if the single wheel model (1) is augmented with the trivial equalities 𝑑𝑇𝑒 = 𝑇𝑒𝑟𝑎𝑡𝑒 and 𝑑𝑇ℎ = 𝑇ℎ𝑟𝑎𝑡𝑒, the following nonlinear system is obtained  ?̇?  =  𝑓(𝑥, 𝑢),                                (3)  with 𝑥 = [𝑉𝑥   𝑠𝑥   𝑇𝑒   𝑇ℎ]𝑇 and 𝑢 = [𝑇𝑒𝑟𝑎𝑡𝑒   𝑇ℎ𝑟𝑎𝑡𝑒]𝑇. Using system (3) the NMPC problem with sampling time 𝑇𝑠 and prediction horizon 𝑁 is  min𝑢    ∑ ‖[𝑉𝑥𝑘 − 𝑉𝑥𝑟𝑒𝑓𝑠𝑥𝑘 − 𝑠𝑥𝑟𝑒𝑓]‖𝑄2+ ‖𝑢𝑘‖𝑅2𝑁−1𝑘=0  s. t.        𝑥0 = 𝑥𝑐𝑢𝑟𝑟𝑒𝑛𝑡 , 𝑥𝑘+1 =  𝑔(𝑥𝑘 , 𝑢𝑘),    𝑘 = 0, … , 𝑁 − 1, 𝑥 ≤ 𝑥𝑘 ≤ 𝑥, 𝑢 ≤ 𝑢𝑘 ≤ 𝑢,  where we choose to penalize the 𝑉𝑥 and 𝑠𝑥 errors only from a given reference, along with the control effort in the form of a penalty on the torque rates 𝑇𝑒𝑟𝑎𝑡𝑒 and 𝑇ℎ𝑟𝑎𝑡𝑒. In this way we do not explicitly set references for the ???????electric motor torque 𝑇𝑒 and the hydraulic brake torque 𝑇ℎ, but rather leave the NMPC find the appropriate values according to the given 𝑉𝑥 and 𝑠𝑥 references, the torque and torque rate constraints, and the chosen weight matrices 𝑄 ≻ 0 and 𝑅 ≻ 0. To solve the NMPC problem online, the Real Time Iteration (RTI) scheme available as part of the ACADO Toolkit [6] is employed, which allows for small computational times as demonstrated in the simulation study of the next section. Finally, in order to generate the reference wheel speed 𝑉𝑥𝑟𝑒𝑓 we can use steady-state analysis. Given a longitudinal slip reference 𝑠𝑥𝑟𝑒𝑓, a longitudinal force reference can be computed using (2) and from that a constant acceleration target 𝑎𝑥𝑟𝑒𝑓 from (1a). Then the reference wheel speed is simply given by   𝑉𝑥𝑟𝑒𝑓 = 𝑉𝑐𝑢𝑟𝑟𝑒𝑛𝑡 + 𝑇𝑠𝑁𝑎𝑥𝑟𝑒𝑓 .  3. SIMULATION STUDY  In this section we present preliminary results using the NMPC strategy from section 2 on a single-wheel model in Simulink, with the motor and hydraulic brake modelled as simple 1st order delays (time constants of 𝜏𝑒 = 0.03 and 𝜏ℎ = 0.09 respectively). The wheel and tyre parameters used can be found in Table 1.  Table 1 Wheel and Tyre Parameters wheel total mass m 362.5 kg wheel moment of inertia Jw 1.04 kgm2 wheel radius Rw 0.3 m MF’s stiffness factor B 7  MF’s shape factor C 1.6  MF’s peak value D 1   For the NMPC we set 𝑇𝑠 = 10𝑚𝑠 and 𝑁 = 50. The rate limits for the motor and the brake are set to 10000.1 Nm/s and 10000.3 Nm/s respectively and the weights in the cost function are chosen such that we penalize large longitudinal slip errors, while also we give priority to the use of the electric motor for the torque delivery: 𝑄 = 𝑑𝑖𝑎𝑔([10, 100000]),   𝑅 = 𝑑𝑖𝑎𝑔([0.08, 0.072]) . In the presented scenario the wheel is initially free-rolling with initial speed of 30m/s on a dry road (𝜇 = 1) and after 1s a slip target of 𝑠𝑥 = −0.1 is demanded. As we can see from Fig 1, the NMPC strategy successfully regulates the longitudinal slip (Fig 1(b)) by distributing the 𝑇𝑒 and 𝑇ℎ torques (Fig 1(c)), with an obvious preference for the electric motor. It is also worth noting at this point that the commanded torques from the controller are very close to the actual torques as delivered by the actuators, a result of including the torque rate constraints in the NMPC formulation. Finally, as we can see from Fig 1(d), the time to compute the solution using a rather standard laptop (i5-2520M at 2.50Ghz with 8GB of memory) takes around 1ms, which is much lower than the sampling time of 10ms.  4. SUMMARY & ONGOING WORK  A new unified slip controller and brake torque integration strategy has been proposed in this work using NMPC formulation. Preliminary result shows that real time implementation of NMPC for slip control by different braking actuators can be deployed. Actuator dynamics and constraints have been taken into consideration in the optimization formulation. Ongoing work will include brake torque range limit and validate using high fidelity model.  REFERENCES  [1] de Castro, R. et al., “Torque blending and wheel slip control in EVs with in-wheel motors,” Vehicle System Dynamics, vol. 50, no. sup1, pp. 71-94, 2012. [2] Satzger, C. et al., “A Model Predictive Control Allocation Approach to Hybrid Braking of Electric Vehicles”, IEEE Intelligent Vehicles Symposium Proceedings, June 2014. [3] Satzger, C. & de Castro, R., “Combined wheel-slip control and torque blending using MPC”, International Conference on Connected Vehicles and Expo, November 2014. [4] Basrah, M. S., et al., “Four wheel torque blending for slip control in a hybrid electric vehicle with a single electric machine”, International Conference on Sustainable Energy Engineering and Application, October 2015. [5] Bakker E. et al., “Tyre modelling for use in vehicle dynamics studies”, SAE Technical Paper, No. 870421, 1987. [6] Houska B. et al., “An Auto-Generated Real-Time Iteration Algorithm for Nonlinear MPC in the Microsecond Range”, Automatica, Vol. 47, No. 10, 2011, pp. 2279–2285. (a)                                              (b) (c)                                          (d) 0  0.5 1  1.5 2  2.5 3  1214161820222426283032Time (s)Speed (m/s)  VxRw0  0.5 1  1.5 2  2.5 3  -0.12-0.1-0.08-0.06-0.04-0.0200.02Time (s)Longitudinal Slip  sx target0  0.5 1  1.5 2  2.5 3  -800-600-400-2000200400600Time (s)Torque (Nm)  commanded Teactual Tecommanded Thactual Th0  0.5 1  1.5 2  2.5 3  051015Time (s)Computational time (ms)  sampling timeFig. 1 Simulation results ???????

Chapter 192: Integration of torque blending and slip control using nonlinear model predictive control

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Integration of torque blending and slip control using nonlinear model predictive control

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