Model predictive control of vehicle roll-over with experimental verification

Control Engineering Practice 77 (2018) 95–108

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Model predictive control of vehicle roll-over with experimental verification Milad Jalali a, *, Ehsan Hashemi a , Amir Khajepour a , Shih-ken Chen b , Bakhtiar Litkouhi b a b

Department of Mechanical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada Global Research and Development Center, General Motors Company, Warren, MI 48090-9055, United States

ARTICLE

INFO

Keywords: Roll angle estimation Roll-over prevention Vehicle stability Model predictive control (MPC) Soft constraints

ABSTRACT This paper presents a model predictive approach to directly control untripped vehicle roll-over. First, a novel real-time estimation scheme is designed to provide the vehicle roll angle using an observer on a combined model of vehicle kinematics and roll dynamics. This angle is used in an integrated directional and roll dynamics model for the prediction of vehicle states and roll-over index. The roll-over prevention objective is specified as a soft constraint to ensure persistent feasibility. If the controller foresees impending vehicle roll-over, it intervenes and reduces the roll-over index by torque vectoring. Software simulations are used to assess the effectiveness of the proposed roll-over control method in the industry standard fishhook maneuvers. In addition, the accuracy and performance of the suggested estimation and control algorithms are verified in double-lane change and flick maneuvers on dry asphalt with an instrumented test vehicle. The results show accurate estimation of the vehicle roll angle and excellent control on roll-over index with the proposed predictive roll-over controller.

1. Introduction Vehicle stability control systems have been in the market for decades. Research indicates that these systems have reduced single vehicle accidents (Lyckegaard, Hels, & Bernhoft, 2015). In particular, roll-over incidents are violent in nature. Fig. 1 shows the severity of crashes involving roll-over in various vehicle types. In general, stability control programs modify a vehicle’s response to correct severe understeer or oversteer behavior and keep the driver in control of the vehicle. Electronic Stability Control (ESC) and rollover mitigation/avoidance are some of the features which have been deployed in production. One advantage of the stability control systems is limiting vehicle’s yaw rate. This also helps to mitigate vehicle roll-over (Rajamani, 2011). However, it is not equivalent to controlling vehicle’s roll motion directly. In addition, keeping tire slip angles within the linear range is one of the primary objectives of stability control systems and prevents vehicle drift (as well as understeer). Vehicle assumes a larger lateral acceleration in presence of ESC compared to when drifting. Large lateral acceleration is a common contributor in roll-over incidents. Therefore, having a dedicated control system that directly controls vehicle roll-over can be beneficial from a safety perspective. Roll-over control systems have been extensively studied in the literature. Kang, Yoo, and Yi (2011) designed a multi-level driving control algorithm for a four-wheel drive electric vehicle. The desired *

traction force and yaw moment were calculated in a high-level controller and a numerical control allocation scheme was used to distribute the calculated control actions among the existing actuators, considering the actuator constraints. They used computer simulations to show the effectiveness of the proposed controller. Rajamani and Piyabongkarn (2013) studied an integrated yaw rate and roll-over control scheme and used software simulations to analyze various approaches such as speed reduction before cornering, speed reduction during cornering as well as counter-steering. They proposed a transient steer-by-wire algorithm to improve roll-over prevention performance. Alberding, Tjønnås, and Johansen (2014) adopted a new approach for coordinated yaw and roll-over stability control. They used a twolevel control structure where the high-level controller stabilizes the yaw motion and the roll-over prevention objective is introduced as a constraint on the control allocation problem. They validated their method using software simulations. Licea and Cervantes (2017) studied a switched predictive roll-over mitigation strategy using a lateral skid index. They studied robustness of the controller with regards to modeling uncertainties. The performance of the proposed roll-over mitigation scheme was evaluated in hardwarein-the-loop simulations. Dahmani, Pages, El Hajjaji, and Daraoui (2014) designed a vehicle dynamics fuzzy controller to prevent roll-over and improve vehicle

Corresponding author. E-mail address: [email protected] (M. Jalali).

https://doi.org/10.1016/j.conengprac.2018.04.008 Received 1 September 2017; Received in revised form 8 April 2018; Accepted 14 April 2018 0967-0661/© 2018 Elsevier Ltd. All rights reserved.

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Control Engineering Practice 77 (2018) 95–108

Cao, Jing, Guo, and Yu (2013) designed a nonlinear model predictive controller for vehicle yaw and roll-over stability. They used a 3 degreesof-freedom vehicle model to predict yaw rate and roll angle. Using computer simulations, they compared the performance of their controller with the performance of separate yaw and roll-over controllers and concluded that the coordinated control of both objectives has superior performance. Lee, Yakub, Kasahara, and Mori (2013) designed a switching model predictive scheme to prevent vehicle roll-over using differential braking and active rear steering. They used software simulations to show the effectiveness of the proposed switching MPC controller. Although active rear steering system was shown effective in simulations, such an actuation system is typically not available in production vehicles. Since very few production cars are equipped with suspension travel sensors, estimation and prediction of vehicle roll motion is an integral part of predictive roll-over control. Yoon and Yi (2006) proposed a rollover mitigation control scheme using a roll-over index based on phaseplane analysis of the vehicle roll dynamics. They designed a model-based roll estimator using a linearized 3 degrees-of-freedom vehicle model and extended Kalman filter (EKF). Yoon, Cho, Koo, and Yi (2009) proposed a unified chassis control approach to prevent vehicle roll-over without loss of vehicle lateral stability by integrating electronic stability control and continuous damping control (CDC). In their approach, the vehicle roll was estimated by a Kalman filter on a combined vehicle lateral-roll dynamics model. Rajamani, Piyabongkarn, Tsourapas, and Lew (2011) proposed a roll-over index estimation method by including vehicle’s suspension into the roll dynamics and estimation of the height of the vehicle’s C.G. by recursive least square, and combining a linear observer and a low-frequency roll angle estimator from the roll dynamics. A nonlinear vehicle dynamic model was represented by a TS fuzzy model suggested by Dahmani et al. (2014). An 𝐻∞ approach was used to design an observer-based robust controller. The designed observer provides vehicle roll and sideslip angle, considering the road bank angle as an unknown input. The performance of the controller was studied in software simulations. Phanomchoeng and Rajamani (2014) studied real-time estimation of the roll-over index to detect both tripped and untripped roll-overs and proposed a method to estimate unknown disturbance inputs based on nonlinear observer design and the mean value theorem to express the estimation error dynamics as a convex combination of known matrices. Phanomchoeng and Rajamani (2013) introduced a new rollover index that utilizes vertical accelerometers in addition to a lateral accelerometer and predicts roll-over in spite of unknown external inputs. A sliding mode observer is designed by Tafner, Reichhartinger, and Horn (2014) for on-line identification of the roll dynamics. A TS observer with estimated membership functions is developed by Dahmani, Pagès, and El Hajjaji (2016) to estimate vehicle’s sideslip and roll angles. The main contribution of this paper is to develop a model predictive controller that directly controls vehicle’s roll-over index. An integrated model of vehicle’s directional and roll dynamics is used as the prediction model. The roll-over index is considered as a soft constraint on system states (see Section 3.3) to prevent closed-loop infeasibility. Therefore, there is no switching behavior in the controller and it naturally activates when the roll-over index is about to exceed the defined thresholds. Another contribution of this paper is a novel roll angle estimation scheme, which utilizes an observer on the roll rate measurement and uses adaptive stationary roll angle value based on the vehicle kinematics and roll dynamics. The proposed estimation and predictive control schemes are verified with software simulations as well as experimental tests. An instrumented electric all-wheel drive Chevrolet Equinox is used to conduct the experimental part of this research. This paper is organized in 6 sections. In Section 2, the vehicle roll angle estimation scheme is introduced. In Section 3, the prediction model, objective function and constraints of the model predictive rollover control problem are described. Software simulation and experimental verification are studied in Sections 4 and 5, respectively. Section 6 summarizes the work done in this paper and its findings.

Nomenclature

SWAFH 𝑎𝑦 𝑏𝜙 𝐶𝛼𝑖 𝐹𝑥𝑖𝑗 𝐹𝑦𝑖𝑗 𝐹𝑧𝑖𝑗

Slip angle of front (𝑖 = 𝑓 ) or rear (𝑖 = 𝑟) tires Front wheel steering angle Slack variable for roll-over soft constraint Vehicle’s roll angle Steering wheel angle in 𝑗 direction resulting in |𝑎𝑦 | = 0.3𝑔 Average of left and right steering wheel angles resulting in |𝑎𝑦 | = 0.3𝑔 Maximum steering input in fishhook maneuver Vehicle’s lateral acceleration Vehicle suspension’s roll damping constant Cornering stiffness of axle 𝑖 Longitudinal force of tire 𝑖𝑗 Lateral force of tire 𝑖𝑗 Vertical force of tire 𝑖𝑗

𝑔 ℎ𝑅 𝐼𝑧 𝐼𝑥𝑥 𝑘𝜙 𝐿𝑖 𝑙𝑠 𝑚 𝑚𝑠 𝑀𝐹𝑥 𝑁𝑐 𝑁𝑝 𝑟 𝑅𝑒 𝑅𝐼 𝑇𝑠 𝑇𝑖𝑗 𝑇𝑖𝑗𝑑𝑟𝑣 𝑢 𝑣 𝑤𝑖 GPS IMU SISM

Gravitational constant (9.81 m/s2 ) Vehicle’s C.G. height from the roll axis Vehicle’s yaw moment of inertia Vehicle’s roll moment of inertia Vehicle suspension’s roll stiffness constant Distance from C.G. to the center of axle 𝑖 Distance between left and right suspension springs Vehicle’s mass Vehicle’s sprung mass Yaw moment of longitudinal tire forces Number of points in the control window Number of points in the prediction window Vehicle’s yaw rate Effective radius of tires Roll-over index Controller’s sample time Total torque applied to wheel 𝑖𝑗 Torque at wheel 𝑖𝑗 demanded by driver Vehicle’s longitudinal velocity at C.G. Vehicle’s lateral velocity at C.G. Trackwidth of axle 𝑖 Global Positioning System Inertia measurement unit Slowly increased steering maneuver

𝛼𝑖 𝛿𝑓 𝜖 𝜙 SWA𝑗0.3𝑔 SWAave 0.3𝑔

stability. Nonlinearities of the tire forces and road friction uncertainties were considered in the proposed approach. A Takagi–Sugeno (TS) observer was designed to consider the unavailability of the sideslip angle measurement. The observer and controller gains were obtained from linear matrix inequalities. Model predictive control (MPC) approaches continue to gain popularity with constant improvement of powerful hardware that can handle computational burden of MPC controllers (Beal & Gerdes, 2013; Jalali, Hashemi, Khajepour, ken Chen, & Litkouhi, 2017, 2018; Jalali, Khajepour, Chen, & Litkouhi, 2016; Jalali, Khosravani, Khajepour, ken Chen, & Litkouhi, 2017; Shakouri & Ordys, 2014). The main advantage of model predictive controllers is the ability to explicitly consider system and input constraints and suggest control actions that can proactively satisfy these constraints. To predict the likelihood of vehicle roll-over, Larish, Piyabongkarn, Tsourapas, and Rajamani (2013) developed a predictive real-time lateral load transfer ratio (PLTR), which employs steering input and available measurements from the vehicle’s electronic stability control system. They claimed that PLTR index can predict roll-over tendency which is essential for closed-loop roll-over prevention systems. They examined the benefit of using PLTR in software simulations and experimental tests. 96

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Fig. 2. (a) Bicycle vehicle model (top view) (b) Planar roll model (rear view). Fig. 1. Vehicle roll-over occurrence percentage, by vehicle type and crash severity (NHTSA, 2015).

system is shown by 𝐼𝑥𝑥 . Ignoring the effect of the roll rate 𝜙̇ and acceleration 𝜙̈ at the steady-state condition, one can write the second stationary roll value obtained by the roll dynamics as:

2. Roll angle estimation

𝜙𝑠2 =

This section provides an approach to estimate the vehicle body’s roll using a nonlinear observer and acceleration measurements, which are typically available on most production vehicles. The estimator includes a compensation for the difference between filtered estimated values and stationary roll values. The stationary value is a convex combination of two static values and adaptively changes with respect to the excitation level. The measured lateral acceleration 𝑎𝑦 = 𝑣̇ + 𝑟𝑢 + 𝑔 sin(𝜙 + 𝜙𝑟 ), with the vehicle yaw rate 𝑟 and longitudinal and lateral velocities 𝑢, 𝑣 contain the roll angle of the sprung mass (𝜙) and the effect of the road bank (𝜙𝑟 ). Estimation of the road bank angle is not the concern of this paper and can be done separately without road friction information by employing unknown input observers (Hashemi, Zarringhalam et al., 2017), or using disturbance observers and assuming known road friction (Ryu & Gerdes, 2004), or designing nonlinear observer with modulated gains (Grip, Imsland, Johansen, Kalkkuhl, Suissa, et al., 2009), or employing sliding mode observers (Menhour, Lechner, & Charara, 2012). Therefore, the stationary value for the vehicle roll angle can be expressed as follows (Rehm, 2010): 𝜙𝑠1 = sin−1 (

𝑎𝑦 − 𝑣̇ − 𝑟𝑢 𝑔

)

(3)

𝑘𝜙 − 𝑚𝑠 𝑔ℎ𝑅

The stationary roll angle (1) together with the stationary values from the vehicle roll dynamics (3) are employed in an observer to correct the errors produced by integration of the roll rate measurement sensor. The convex combination of the first and second sets of stationary values (1) and (3) are employed for the following adaptive stationary roll that is used in the roll observer: (4)

𝜙𝑠 = 𝛤𝜙1 𝜙𝑠1 + 𝛤𝜙2 𝜙𝑠2 ,

such that the roll coefficients are 𝛤𝜙1 , 𝛤𝜙2 ∈ {R+ |𝛤𝜙1 + 𝛤𝜙2 = 1}. The roll coefficients change adaptively based on the level of excitation to incorporate the effect of roll dynamics (2) as well as the lateral dynamics (1). For high excitation maneuvers, the time derivatives of the vehicle longitudinal and lateral velocities (i.e., 𝑢,̇ 𝑣) ̇ may impose large noises, thus the stationary values from the roll dynamics (3) are more reliable. On the other hand, the identified roll stiffness and damping in (3) may have uncertainties that leads to more tendency in using the first set of stationary values. The adaptive gain 𝛤𝜙2 changes between predefined upper and lower bounds 𝛤𝑢 , 𝛤𝑙 according to the driving conditions as

(1)

Longitudinal and lateral velocities may be available from GPS or estimated by nonlinear observers (Sun, Huang, Rudolph, & Lolenko, 2015), robust LPV observers (Hashemi, Khosravani et al., 2017; Hashemi, Pirani et al., 2017), Kalman filter (Gadola, Chindamo, Romano, & Padula, 2014; Katriniok & Abel, 2016), or force measurement based Kalman estimator (Madhusudhanan, Corno, & Holweg, 2016) in Vehicle Dynamic Control (VDC) systems. Inevitable uncertainties in the calculated stationary roll angle from (1), due to estimated velocities, can be handled with robust gain allocation. Fig. 2 shows longitudinal/lateral forces of each axle as well as roll motion of the sprung mass 𝑚𝑠 in the 𝑦𝑧 coordinates attached to the vehicle body. Using the measured lateral acceleration 𝑎𝑦 and the roll angle of the sprung mass, the roll dynamics of the sprung mass can be expressed in the coordinates shown in Fig. 2: (𝐼𝑥𝑥 + 𝑚𝑠 ℎ2𝑅 )𝜙̈ + 𝑏𝜙 𝜙̇ + (𝑘𝜙 − 𝑚𝑠 𝑔ℎ𝑅 )𝜙 = 𝑚𝑠 𝑎𝑦 ℎ𝑅

𝑚 𝑠 ℎ 𝑅 𝑎𝑦

𝛤𝜙2 = 𝛤𝑢 −

1 √ 𝑒 𝜏𝑒 2𝜋

−

𝜉𝑒𝜙 𝜎𝑒𝜙 2𝜏𝑒2

where 𝜏𝑒 =

1 √ (𝛤𝑢 2𝜋

− 𝛤𝑙 ) and 𝜎𝑒𝜙 represents

variance of the vehicle’s accelerations 𝑎𝑥𝑘 and 𝑎𝑦𝑘 at time 𝑘 over a moving window with size 𝑁𝑎 , i.e. 𝜎𝑒𝜙 = var{|𝑎𝑦𝑘 | ∶ 𝑞 − 𝑁𝑎 ≤ 𝑘 ≤ 𝑞}, ∀𝑞 ∈ N, 𝑞 ≥ 𝑁𝑎 . The rate of transition between the predefined upper and lower thresholds 𝛤𝑢 , 𝛤𝑙 is denoted by 𝜉𝑒𝜙 for the roll estimator. The filtered value of the stationary roll angle from (4) is denoted by 𝜙̄ 𝑠 and used for the observer in Rehm (2010) to estimate the vehicle body’s roll 𝜙̂ as: ̄̂ 𝜙̂̇ = 𝜙̇ 𝑚 + 𝜂2𝜙 (𝜙̄ 𝑠 − 𝜙)

(5)

in which 𝜙̇ 𝑚 is the measured roll rate signal and 𝜂2𝜙 is the observer gain. The estimated angles from (5) are then filtered to reach 𝜙̄̂ by (6) with ̄̂ 𝑇 : the states 𝐱̂ 𝜙 = [𝜙̂ 𝜙] [ ] [ ] ] [ ̂] [ 𝜙̂̇ 𝜙 0 −𝜂2𝜙 𝜙̇ 𝑚 + 𝜂2𝜙 𝜙̄ 𝑠 = + (6) ̇ 𝑓2𝜙 −(𝑓1𝜙 + 𝜂1𝜙 ) 𝜂1𝜙 𝜙̄ 𝑠 𝜙̄̂ 𝜙̄̂ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(2)

where 𝑘𝜙 and 𝑏𝜙 are roll stiffness and damping, and the distances between the roll axis and the center of gravity is denoted by ℎ𝑅 . The moments of inertia about the roll axis parallel to the frame coordinate

𝐴𝜙

97

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in which 𝑓1𝜙 and 𝑓2𝜙 are the coefficients of the filters. The observer gains 𝜂1𝜙 , 𝜂2𝜙 are used for compensation of the deviations of the filtered ̄̂ estimated angles from the stationary values, i.e., 𝜙̄ 𝑠 − 𝜙. Estimator’s stability analysis: The observer (5) for the roll angle estimation with the filtered states from (6) is stable and its estimation error is asymptotically bounded.

where 𝑚 is the vehicle’s mass. The slip angles 𝛼𝑖 are expressed in terms of other vehicle kinematic variables: 𝑣 + 𝜉𝑖 𝐿𝑖 𝑟 𝛼𝑖 = 𝛿𝑖 − tan−1 (13) 𝑢 + 𝜁𝑗 𝑤𝑖 𝑟∕2 where 𝜉𝑓 = −𝜉𝑟 = 1 and 𝜁𝑟 = −𝜁𝑙 = 1. In practice, 𝑢 is the dominant term in the denominator of the above equation. Therefore, assuming constant wheel steering angle and longitudinal speed (in the prediction window), the time derivative of the slip angles is derived:

Proof. The proof and stability analysis of error dynamics of the system (5) is provided in Appendix A. □

𝛼̇ 𝑖 ≈𝑘𝑖,𝑣 𝑣̇ + 𝑘𝑖,𝑟 𝑟̇ 1 𝑘𝑖,𝑣 = )2 ( 𝑢0 + 𝑣0 + 𝜉𝑖 𝐿𝑖 𝑟0 ∕𝑢0 −𝜉𝑖 𝐿𝑖 𝑘𝑖,𝑟 = ( )2 𝑢0 + 𝑣0 + 𝜉𝑖 𝐿𝑖 𝑟0 ∕𝑢0

Finally, the estimated angle 𝜙̂ from (6) is employed for roll-over control in the next section. The estimated vehicle roll angle can also be used to correct the measured accelerations for the force or velocity estimators as done by Hashemi et al. (2016, 2017). 3. Controller design

𝑅𝐼 ≈

3.1. Integrated directional and roll prediction model

𝛼𝑓

𝛼𝑟

𝜙̇

𝜙

]𝑇

(7)

where 𝑟 is vehicle’s yaw rate, 𝑣 is vehicle’s lateral velocity, 𝛼𝑓 and 𝛼𝑟 are respectively the slip angle of the front and rear tires and 𝜙 is the vehicle’s roll angle. The inputs of the prediction model are: [ 𝐮 = 𝑇𝑓 𝑙

𝑇𝑓 𝑟

𝑇𝑟𝑙

𝑇𝑟𝑟

𝜖

]𝑇

2ℎ𝑅 2ℎ sin 𝜙 + 𝑅 𝑎𝑦 cos 𝜙 𝑙𝑠 𝑙𝑠 𝑔

The objective function of the roll-over control problem is defined below: 𝑁𝑝

) 1 ∑( ‖𝐮𝑘 − 𝐯‖2𝐑 + ‖𝐮𝑘 − 𝐮𝑝 ‖2𝐓 + 2𝐰𝑇 𝐮𝑘 2 𝑘=1 [ ]𝑇 𝑇𝑓drv 𝑇𝑟𝑙drv 𝑇𝑟𝑟drv 0 𝐯 = 𝑇𝑓drv 𝑙 𝑟 [ ]𝑇 𝐰 = 0 0 0 0 𝑟1𝜖 𝐽=

(9)

where 𝐼𝑧 is the vehicle’s yaw moment of inertia, 𝛿𝑓 is the front wheels’ steering angle, 𝐶𝛼𝑖 is the cornering stiffness of the front (𝑖 = 𝑓 ) or rear (𝑖 = 𝑟) axle, 𝐿𝑖 is the distance from C.G. to the center of axle 𝑖 and 𝑀𝐹𝑥 is the yaw moment of the longitudinal tire forces and is defined below: ( ) 𝑤𝑓 𝐿𝑓 𝑤 𝑀𝐹𝑥 = − cos 𝛿𝑓 + sin 𝛿𝑓 𝑇𝑓 𝑙 − 𝑟 𝑇𝑟𝑙 + … 2𝑅𝑒 𝑅𝑒 2𝑅𝑒 ( ) 𝑤𝑓 𝐿𝑓 𝑤 + cos 𝛿𝑓 + sin 𝛿𝑓 𝑇𝑓 𝑟 + 𝑟 𝑇𝑟𝑟 (10) 2𝑅𝑒 𝑅𝑒 2𝑅𝑒

(11)

where 𝑎𝑦 is vehicle’s lateral acceleration corrected for the roll angle and 𝑢 is assumed constant in the prediction horizon. The lateral acceleration can be expressed in terms of the lateral forces of the front and rear axles: 𝑎𝑦 =

) 1 ( 𝐶 𝛼 cos 𝛿𝑓 + 𝐶𝛼𝑟 𝛼𝑟 𝑚 𝛼𝑓 𝑓

(18a) (18b) (18c)

𝐑 = diag(𝑟𝑇 , 𝑟𝑇 , 𝑟𝑇 , 𝑟𝑇 , 𝑟2𝜖 )

(18d)

𝐓 = diag(𝑡𝑇 , 𝑡𝑇 , 𝑡𝑇 , 𝑡𝑇 , 𝑡𝜖 )

(18e)

where 𝑁𝑝 denotes the number of points in the prediction window, 𝑇𝑖𝑗drv is the driver’s torque request for wheel 𝑖𝑗,1 𝑟𝑇 > 0 and 𝑟2𝜖 > 0 are respectively the weights of the quadratic terms of control actions and slack variable, 𝑟1𝜖 ≥ 0 is the linear weight of the slack variable, 𝑡𝑇 ≥ 0 and 𝑡𝜖 ≥ 0 can be used to enforce proximity to the previous solution of the QP problem (𝐮𝑝 ) and prevent oscillations in the control actions. In order to reduce the computational burden of the MPC controller, it is common practice to adopt a shorter control window of size 𝑁𝑐 , after which the control action remains constant:

where 𝑅𝑒 is the effective radius of tires, 𝑤𝑖 is the trackwidth of axle 𝑖 and the longitudinal force of tire 𝑖𝑗 is assumed to be linearly dependent on 𝑇𝑖𝑗 by 𝐹𝑥𝑖𝑗 = 𝑇𝑖𝑗 ∕𝑅𝑒 . Vehicle’s lateral velocity and lateral acceleration are related by the following dynamics: 𝑣̇ = 𝑎𝑦 − 𝑟𝑢

(16)

3.2. Objective function

(8)

where 𝑇𝑖𝑗 is the total torque exerted on wheel 𝑖𝑗 and 𝜖 is the slack variable for the roll-over index soft constraint, further discussed in Section 3.3. The vehicle’s yaw acceleration can be expressed as: 𝐼𝑧 𝑟̇ = 𝐶𝛼𝑓 𝛼𝑓 𝐿𝑓 cos 𝛿𝑓 − 𝐶𝛼𝑟 𝛼𝑟 𝐿𝑟 + 𝑀𝐹𝑥

(14c)

The above approximate of RI is the output of the prediction model and can be expressed in terms of the states with the assumption of cos 𝜙 ≈ 1 and sin 𝜙 ≈ 𝜙. The prediction model defined in this section can be expressed in the standard state-space format as shown in Appendix B. After discretization, the discrete-time representation of the prediction model is obtained: { 𝐱𝑘+1 = 𝐀𝐱𝑘 + 𝐁𝐮𝑘 (17) 𝐲𝑘 = 𝐂𝐱𝑘

The prediction model is based on a bicycle vehicle model (Fig. 2) and planar roll model and consists of the following states: 𝑣

(14b)

where 𝑢0 , 𝑣0 and 𝑟0 are the measured/estimated longitudinal, lateral and yaw velocities of the vehicle (i.e. initial conditions of the prediction model). Prediction of the roll angle and roll rate are done using the roll dynamics of (2). Roll-over index (RI) is a measure of lateral load transfer and is widely used to assess vehicle’s closeness to roll-over. RI is defined as: 𝐹𝑧 − 𝐹𝑧𝑙 (15) 𝑅𝐼 = 𝑟 𝐹𝑧𝑟 + 𝐹𝑧𝑙 ∑ ∑ where 𝐹𝑧𝑟 = 𝐹𝑧𝑖𝑟 and 𝐹𝑧𝑙 = 𝐹𝑧𝑖𝑙 . Since the vertical tire forces cannot be easily measured in production vehicles, approximations of the rollover index are used (see Chapter 15 Rajamani, 2011):

In this section, a model predictive controller is designed to maintain the vehicle roll-over index within the permissible range by torque vectoring. In this paper, it is assumed that the vehicle is equipped with four independent electric motors allowing full torque vectoring; however, the designed controller can be easily adapted to other driveline configurations such as front-wheel drive or rear-wheel drive, or actuation mechanisms such as differential braking or a hybrid of torque vectoring and differential braking (see Jalaliyazdi, 2016). First, an integrated roll and directional prediction model is developed. Next, the cost function and the constraints are specified. The control problem is then cast in terms of a quadratic programming (QP) problem to be solved in real-time by a numerical QP solver.

[ 𝐱= 𝑟

(14a)

𝐮𝑘 = 𝐮𝑁𝑐

for 𝑘 > 𝑁𝑐

(19)

1 Without loss of generality, it can be assumed that the requested torque is uniformly distributed among all wheels.

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3.3. Constraints Two types of constraints are considered: input constraints and output constraints. Input constraints result from a finite amount of torque deliverable by the electric motors and the output constraints are imposed to model a soft constraint on vehicle roll-over index: |𝐲𝑘 | < 𝑅𝐼max + 𝜖𝑘

(20)

where 𝑅𝐼max > 0 is the selected upper bound on the roll-over index and 𝜖𝑘 > 0 is the slack variable associated with this soft constraint. The input constraints can be written as: [ min ] 𝐓 + 𝐓brake 𝐥𝐛 = 𝑚 (21a) 0 [ max ] 𝐓 + 𝐓brake 𝐮𝐛 = 𝑚 (21b) ∞

Fig. 3. Block diagram of controller and estimation module in software simulations. The controller provides the torque adjustments that are added to the driver’s torque request and then applied to the vehicle.

where 𝐓max and 𝐓min are respectively the vector of maximum and 𝑚 𝑚 minimum (negative) torques that the electric motors on vehicle corners can deliver and 𝐓brake is the vector of current hydraulic brake torque applied to the wheels. The zero lower bound used in (21a) enforces 𝜖𝑘 > 0 requirement of (20). The constraints of (21) are applied to the prospective control actions 𝐮𝑘 . 3.4. QP optimization problem The objective function of (18) can be rewritten in the following compact form: 1 𝑇 𝐮̄ 𝐇̄𝐮 + 𝐮̄ 𝑇 𝐠 + const. (22) 2 where 𝐮̄ is the vector of prospective control actions 𝐮1 through 𝐮𝑁𝑝 , 𝐇 is a symmetric positive definite matrix in terms of the weighting matrices 𝐑 and 𝐓. Vector 𝐠 additionally depends on the driver’s torque request 𝐯 and the tuning parameter 𝑟1𝜖 . The constant term at the end of (22) is independent of the optimization variable 𝐮̄ and only serves to equate the objective functions in (18a) and (22). The input and output constraints introduced in Section 3.3 can be expressed as following constraints on the optimization vector 𝐮̄ : 𝐽 (̄𝐮) =

𝐋𝐁 < 𝐮̄ <𝐔𝐁 𝐀𝑢 𝐮̄ <𝐔𝐁𝑢

Fig. 4. Steering wheel input and lateral acceleration in simulation of SISM maneuver to the left at 50 mph (80.5 km/h).

delays are assumed negligible, but these lags can be easily incorporated in the design of the MPC controller as previously shown by Jalali, Khajepour, Chen, and Litkouhi (2017). The properties of this vehicle are listed in Table 1. This vehicle by design has the tendency to drift instead of rolling over. Therefore, for the purpose of demonstration, the height of the vehicle’s C.G. is raised by 10 cm in software simulations to increase the roll-over propensity. The model predictive controller designed in Section 3 is implemented in MATLAB/Simulink (Inc, 2011). Fig. 3 shows the block diagram of the control loop used in simulations. The controller receives feedback signals from CarSim and estimation module and calculates the optimal torque adjustments. These torque adjustments are added to the drive torque request by the driver and sent to CarSim to calculate vehicle’s response. The fishhook maneuver consists of steering to one direction, dwelling ̇ falls below 1.5 deg/sec, and then counteruntil vehicle’s roll rate (|𝜙|) steering and dwelling. The initial steering and counter-steering are both performed at the rate of 720 deg/sec. This maneuver is performed at increasingly higher entry speeds from 35 mph (56.3 km/h) to 50 mph (80.5 km/h). If no wheel lift off of more than 2 inches (5.08 cm) occurs during the maneuver, it is deemed successfully completed. The maximum steering input in the fishhook maneuver is determined by performing a slowly increasing steering maneuver (SISM) at 50 mph (80.5 km/h). In this maneuver, the steering input is slowly increased from 0 to 270 degrees at the rate of 13.5 deg/sec. The steering wheel angle and lateral acceleration are plotted and a linear curve is fitted to the data with lateral accelerations in the range of 0.1g to 0.375g. The steering input that corresponds to a lateral acceleration of 0.3g is

(23a) (23b)

The detailed derivations of the objective function in (22) and the constraints in (23) are provided in Appendix C. The objective function along with the constraints constitute a QP problem that can be solved numerically. In this paper, qpOASES toolbox (Ferreau, Kirches, Potschka, Bock, & Diehl, 2014) is used to obtain the optimal sequence of control actions 𝐮̄ ∗ . The optimal torque adjustments are calculated accordingly: [ 𝑇 ] 𝑇 𝑇 𝑇 𝐮̄ ∗2 … 𝐮̄ ∗𝑁 𝐮̄ ∗ = 𝐮̄ ∗1 𝑝 [ ∗] 𝛿𝐓 = 𝐮̄ ∗1 − 𝐯 (24) 𝜖1∗ where 𝛿𝐓∗ is the optimal torque adjustment and is added to the driver’s torque request and then applied to the vehicle (see Fig. 3). 4. Software simulations In this section, software simulations are used to assess the ability of the controller in roll-over prevention. The test scenario includes performing a fishhook maneuver that is the industry standard test scenario for assessing vehicles’ roll-over propensity. Software simulations are performed with a high-fidelity CarSim (CarSim User Manual, 2002) model of the electric 2010 Chevrolet Equinox that is also the test vehicle for the experiments. This vehicle is equipped with four independent electric motors that allow full torque vectoring. The sensor and actuator 99

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(a) Controller off.

(b) Controller on.

Fig. 5. Steering wheel angle and lateral acceleration in fishhook maneuver.

fishhook maneuvers with entry speeds of 35 mph (56.3 km/h), 40 mph (64.4 km/h) and 45 mph (72.4 km/h) are completed successfully without wheel lift off. However, the uncontrolled maneuvers at 47.5 mph (76.4 km/h) and 50 mph (80.5 km/h) result in vehicle roll-over. The controlled and uncontrolled fishhook maneuvers at 50 mph are studied in this section. Fig. 5 shows the steering input and the vehicle’s lateral acceleration. When the controller is off, the vehicle rolls over shortly after countersteering and the simulation is stopped. This is shown in Fig. 6 where the vehicle just before and after roll-over are shown. Fig. 7 compares the estimated vehicle roll angle with the values computed by CarSim. It can be observed that the estimated values show excellent agreement with the roll angle calculated by CarSim in both controlled and uncontrolled maneuvers. In addition, it is observed that in the uncontrolled maneuver, the roll angle increases to 80◦ before the simulation is stopped, thus showing a complete vehicle roll-over. However, when the predictive roll-over controller is active, the roll angle stays at about 6◦ and remains small. Roll-over indexes are compared in Fig. 8. In each graph, the estimated roll-over index from (16) is compared with the actual RI, calculated based on the normal tire forces as reported by CarSim. It is observed that the predictive controller can effectively control the roll-over index and reduce it after the counter-steering action, while in the uncontrolled maneuver, RI remains high after the counter-steer and ultimately results in the vehicle roll-over. The torque adjustments of the controller are shown in Fig. 9. It can be seen that the controller torque vectors to oppose excessive yaw rate and reduce the lateral acceleration and thus the roll-over index.

Fig. 6. Two snapshots of the test vehicle in simulation of uncontrolled fishhook maneuver (after roll-over and 0.4 s before that).

recorded and shown by SWA𝑗0.3𝑔 where 𝑗 = 𝑙 for steering to the left and 𝑗 = 𝑟 for steering to the right. The SISM maneuver is performed in both left and right directions and the average steering wheel angle is denoted by SWAave 0.3𝑔 . The maximum steering input of the fishhook maneuver is selected as 6.5 times the steering wheel angle obtained in 2 the SISM (SWAFH = 6.5 × SWAave 0.3𝑔 ). Fig. 4 shows the steering wheel angle and the lateral acceleration of the vehicle in the SISM maneuver to the left. The red dotted line is fitted to the data points with 0.1𝑔 ≤ 𝑎𝑦 ≤ 0.375𝑔. This gives SWA𝑙0.3𝑔 = 26.76◦ . Similarly, the SISM to the right results in SWA𝑟0.3𝑔 = 26.51◦ . Therefore, the maximum steering input in the fishhook maneuver is obtained as SWAFH = 173.1◦ . Next, the fishhook maneuver is performed with and without the predictive roll-over controller. The parameters of this controller and the roll angle estimation module are listed in Table 2. These parameters are tuned based on various simulations and experimental tests so that satisfactory closed-loop performance is achieved. In particular, the size of the prediction and control horizons (𝑁𝑝 and 𝑁𝑐 respectively) are limited by the processing power of the micro-controller. The uncontrolled

5. Experiments In this section, experimental tests are conducted to assess the accuracy of the roll angle estimator designed in Section 2 and the performance of the predictive roll-over control scheme developed in Section 3 in real-time on an electric SUV. For the purpose of demonstration, an electric 2010 Chevrolet Equinox is used. The main specifications of this vehicle are listed in Table 1. The test vehicle (shown in Fig. 10) is equipped with four independent electric motors on each wheel, that are used for driving the vehicle as well as torque vectoring. The experimental setup is shown in Fig. 11. The vehicle IMU sensor provides vehicle’s longitudinal and lateral accelerations, yaw rate, roll rate and pitch rate. A 6-axis GPS unit (RT2500 inertial and GPS navigation systems by OxTS company) is installed in the vehicle to provide vehicle’s longitudinal and lateral velocities. Furthermore, load wheel sensors are

2 If SISM is to be used for experiments, SISM should be performed three times to the left and three times to the right and the average of these six SWA𝑗0.3𝑔 measurements should be used to determine the maximum steering in the fishhook maneuver.

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(a) Controller off.

(b) Controller on.

Fig. 7. Vehicle roll angle and roll rate in the fishhook maneuver.

(a) Controller off.

(b) Controller on. Fig. 8. Vehicle roll-over index in the fishhook maneuver.

Table 1 Properties of the test vehicle used in simulations and experiments. Parameter

Unit

Value

Description

𝑚 𝑚𝑠 𝐼𝑧 𝑘𝜙 𝑏𝜙 𝐿𝑓 𝐿𝑟 𝑤𝑓 𝑤𝑟 𝐻𝐶𝐺 𝐻𝐶𝐺 𝑅𝑒 max 𝑇m,𝑖𝑗 min 𝑇m,𝑖𝑗

[kg] [kg] [kg.m2 ] [N.m] [N.m/s] [m] [m] [m] [m] [m] [m] [m] [N.m] [N.m]

2270 2060 4690 1.44 × 105 4.51 × 103 1.600 1.577 1.586 1.584 0.747 0.647 0.339 1800 −1800

Vehicle’s mass Vehicle’s sprung mass Yaw moment of inertia Roll stiffness Roll damping Distance from C.G. to front axle Distance from C.G. to rear axle Front trackwidth Rear trackwidth Height of C.G. (in simulations) Height of C.G. (in experiments) Wheel effective radius Maximum torque of electric motors Minimum torque of electric motors

Table 2 Tuning parameters of the roll angle estimator and roll-over controller in simulations and experiments. Parameter

Value (simulation)

Value (experiment)

𝑇𝑠 𝑁𝑝 𝑁𝑐

20 ms 13 3

RImax

0.80

𝑟𝑇 𝑟1𝜖 𝑟2𝜖 𝑡𝑇 𝑡𝜖 𝑓1𝜙 , 𝑓2𝜙 𝜂1𝜙 , 𝜂2𝜙

2 × 10−8 0.4 3.0 4 × 10−8 0.0 183, 152 9.4, 38.5

20 ms 13 3 0.65 in DLC 0.60 in flick 2 × 10−8 0.2 5.0 7 × 10−8 0.0 218, 165 10.6, 52

As mentioned in Section 4, the C.G. of the test vehicle is low enough that it tends to drift instead of rolling, consequently, untripped roll-over and high values of roll-over index (i.e. RI > 0.9) cannot be experimentally achieved. Therefore, a lower threshold is selected as RImax . Furthermore, in absence of automated steering systems, the

installed on four wheels to measure tire’s normal forces and thus rollover index. Suspension travel sensors are also used on all four corners to directly measure the vehicle’s roll angle and verify the accuracy of the roll angle estimation module. 101

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Fig. 10. Electric Chevrolet Equinox vehicle used in experiments (load wheel sensors are visible in this photo).

Fig. 9. Control actions in simulation of the fishhook maneuver.

fishhook maneuver as explained in simulations could not be conducted. For this reason, a human driver is used and two critical driving scenarios on dry pavement are studied. The first maneuver is a double-lane change maneuver on dry pavement at entry speed of 55 km/h. The second maneuver is a flick maneuver at the speed of 45 km/h. Both maneuvers are performed with and without the controller to assess the effectiveness of the roll-over prevention system. 5.1. Double-lane change (DLC) maneuver

Fig. 11. The experimental setup; the communication network, vehicle sensors, and actuators.

In this section, the results of a harsh double-lane change maneuver with entry speed of 55 km/h are studied. Fig. 12 shows the steering wheel input and the lateral acceleration in the controlled and uncontrolled maneuvers. It can be seen that except for minor driver interaction in the controlled maneuver at around 𝑡 ≈ 4 sec, the controlled and uncontrolled maneuvers have similar steering inputs. It is also observed that the lateral acceleration of the controlled maneuver peaks at slightly lower values. Fig. 13 compares the vehicle’s yaw rates and sideslip angles. It is observed that in the uncontrolled maneuver, the vehicle slides more significantly and assumes a sideslip angle of about 13◦ . However, with the controller, the vehicle yaw rate peaks at slightly lower values (especially just before the second lane change) and maintains a smaller sideslip angle. The vehicle roll angle and roll rates are compared in Fig. 14. Excellent agreement between the estimated roll angle and measurements made by suspension travel sensors is observed. In Fig. 15, the roll-over index of the controlled and uncontrolled maneuvers are compared. The roll-over threshold for the controller is selected as RImax = 0.65. Comparison shows that the controller is very effective in limiting the lateral load transfer and controlling the roll-over index. Furthermore, excellent agreement is observed between the actual rollover index (15) calculated using measured normal tire forces and the approximate roll-over index in (16). Fig. 16 shows controller’s torque adjustments on all four corners of the vehicle. The torque adjustments oppose the vehicle’s yaw rate and reduce the peak lateral accelerations when the roll-over index passes the RImax threshold. The required computation time on the dSpace micro-Autobox is plotted in Fig. 17. It can be seen that when the roll-over index is in the vicinity of RImax and some of the constraints in (23) become active, the QP solver used requires more iterations and thus more time is required to calculate the optimal torque adjustments.

5.2. Scandinavian Flick maneuver The Scandinavian flick maneuver consists of coasting with the initial speed of 45 km/h, steering to the right and then counter-steering and holding the steering input. Some drive torque is applied by the driver to maintain the speed throughout the maneuver. This maneuver is also conducted with and without the controller to verify the effectiveness of the predictive roll-over control scheme. Fig. 18 shows the steering inputs and the vehicle’s lateral acceleration in both maneuvers. It can be seen that the steering inputs are very similar, thus making this an ideal case for comparison. Comparing the lateral accelerations, it is observed that when the controller is on, the vehicle’s lateral acceleration peaks at slightly lower values compared to the uncontrolled case. The vehicle’s yaw rate and sideslip angle are shown in Fig. 19. Similarly, it is noted that the vehicle yaw rate is slightly reduced when the controller intervenes, most notably at 𝑡 ≈ 3.5 sec. The vehicle’s roll angle and roll rate are shown in Fig. 20. It can be seen that the roll angle of the controlled maneuver is slightly less than the roll angle in the uncontrolled maneuver. In both cases, the estimated roll angles agree with the measured values; the minor difference is because of using nominal value of the sprung mass, which may vary by the number of passengers. Fig. 21 compares the resulting roll-over index in both maneuvers. It is observed that when the controller is inactive, the rollover index exceeds the RImax threshold. However, when the controller is used, the roll-over index is significantly lower and remains very close to the specified thresholds. The controller’s torque adjustments are shown in Fig. 22. It can be seen that the control actions are in the direction of opposing the vehicle cornering, which results in lower lateral acceleration that is noticed in Fig. 18, and consequently lower roll-over index. 102

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(a) Controller off.

(b) Controller on.

Fig. 12. Steering wheel angle and lateral acceleration in DLC maneuver (experiment).

(a) Controller off.

(b) Controller on.

Fig. 13. Vehicle yaw rate and sideslip angle in DLC maneuver (experiment).

(a) Controller off.

(b) Controller on.

Fig. 14. Vehicle roll angle (estimation and measurement) and roll rate in the DLC maneuver (experiment). 103

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(a) Controller off.

(b) Controller on.

Fig. 15. Vehicle roll-over index (estimate and measurement) in DLC maneuver (experiment).

6. Conclusion In this study, a new roll angle estimation scheme was designed by using conventional sensor measurements and a nonlinear observer that can provide an accurate estimate of the vehicle roll angle in various driving scenarios. The accuracy of the roll angle estimator is verified in simulation and with road experiments. Using this roll angle, a model predictive controller was designed based on an integrated model of the vehicle roll and directional dynamics. This controller directly controls the roll-over index which is a measure of how close the vehicle is to a roll-over. The maximum permissible roll-over index is considered as a soft constraint on vehicle states, which alleviates the problem of infeasibility in the MPC approach. The performance of the proposed estimation and control schemes was validated in simulation of a standard fishhook maneuver. Furthermore, experimental tests were conducted with and without the controller using an electric Chevrolet Equinox to verify its effectiveness in roll-over prevention. Direct measurements of the vehicle roll-angle and roll-over index show the effectiveness of the proposed model predictive controller and the accuracy of the roll angle estimator. Acknowledgments The authors would like to acknowledge the financial support of the Automotive Partnership Canada (grant no. APCPJ 395996-09), Ontario Research Fund (grant no. ORF-RE-04-039), and the financial and technical support of General Motors Co.

Fig. 16. Controller’s torque adjustments in DLC maneuver (experiment).

Appendix A. Stability analysis of roll angle estimator The estimation error of the observer (5) yields 𝐞 = 𝐱𝜙 − 𝐱̂𝜙 where ̄̂ 𝑇 from (6) and 𝐱 = [𝜙 𝜙̄ ]𝑇 with the dynamics: 𝐱̂ 𝜙 = [𝜙̂ 𝜙] 𝜙 𝑠 [ ] [ ] ̇ 0 0 𝜙 + 𝛺𝑟 𝐱̇ 𝜙 = 𝐱 + 𝑚 (A.1) 𝑓2𝜙 𝜙𝑠 0 −𝑓1𝜙 𝜙 in which the uncertainty in the roll rate measurement is denoted by 𝛺𝑟 , the adaptive stationary roll signal 𝜙𝑠 is obtained from (4), and its filtered value is 𝜙̄ 𝑠 from 𝜙̄̇ 𝑠 = −𝑓1𝜙 𝜙̄ 𝑠 + 𝑓2𝜙 𝜙𝑠 with 𝑓1𝜙 , 𝑓2𝜙 > 0. Thus, the error dynamics 𝐞̇ = 𝐱̇ − 𝐱̂̇ can be expressed as: [ ] ̄̂ + 𝛺 −𝜂2𝜙 (𝜙̄ 𝑠 − 𝜙) 𝑟 𝐞̇ = ̄̂ + 𝑓 (𝜙 − 𝜙) ̂ −(𝑓1𝜙 + 𝜂1𝜙 )(𝜙̄ 𝑠 − 𝜙) 2𝜙 𝑠 [ ] [ ] 0 −𝜂2𝜙 𝛺𝑟 = 𝐞+ 𝑓2𝜙 (𝜙𝑠 − 𝜙) 𝑓2𝜙 −(𝑓1𝜙 + 𝜂1𝜙 ) = 𝐴𝑒 𝐞 + 𝜴

Fig. 17. Controller’s required computation time in DLC maneuver (experiment). 104

(A.2)

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(a) Controller off.

(b) Controller on.

Fig. 18. Steering wheel angle and lateral acceleration in flick maneuver.

(a) Controller off.

(b) Controller on.

Fig. 19. Vehicle yaw rate and sideslip angle in flick maneuver.

(a) Controller off.

(b) Controller on. Fig. 20. Vehicle roll angle and roll rate in flick maneuver. 105

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(a) Controller off.

(b) Controller on. Fig. 21. Vehicle roll-over index in flick maneuver.

obtained: 0 1 −𝑘𝑓 ,𝑣 −𝑘𝑟,𝑣 0 0

⎡ 𝐼𝑧 ⎢ 0 ⎢ −𝑘 𝐄𝑐 = ⎢ 𝑓 ,𝑟 ⎢ −𝑘𝑟,𝑟 ⎢ 0 ⎢ ⎣ 0

Fig. 22. Controller torque adjustments in flick maneuver.

⎡0 ⎢−𝑢 ⎢ 0 𝐀𝑐 = ⎢ ⎢0 ⎢0 ⎢ ⎣0

0 0 0 0 0 0

⎡𝑏11 ⎢ ⎢0 ⎢ 𝐁𝑐 = ⎢ 0 ⎢0 ⎢0 ⎢ ⎣0

𝑏12

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 𝐼𝑥𝑥 + 𝑚𝑠 ℎ2𝑅 0

𝐶𝛼𝑓 𝐿𝑓 c𝛿𝑓 (𝐶𝛼𝑓 c𝛿𝑓 )∕𝑚 0 0 (𝐶𝛼𝑓 c𝛿𝑓 )𝑚𝑠 ℎ𝑅 ∕𝑚 0

0 0 0 0 0

−𝑤𝑟 2𝑅𝑒 0 0 0 0 0

𝑤𝑟 2𝑅𝑒 0 0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0⎥ 𝑏𝜙 ⎥⎥ 1⎦

−𝐶𝛼𝑟 𝐿𝑟 𝐶𝛼𝑟 ∕𝑚 0 0 𝐶𝛼𝑟 𝑚𝑠 ℎ𝑅 ∕𝑚 0

(B.2)

0 0 0 0 0 1

0 ⎤ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ 𝑎56 ⎥⎥ 0 ⎦

(B.3)

0⎤ ⎥ 0⎥ 0⎥⎥ 0⎥ 0⎥ ⎥ 0⎦

(B.4)

0 ⎡ ⎤ ⎢ ⎥ 0 ⎢ ⎥ 2ℎ𝑅 𝐶𝛼𝑓 c𝛿𝑓 ∕(𝑚𝑙𝑠 𝑔)⎥ ⎢ 𝐂𝑐 = ⎢ 2ℎ𝑅 𝐶𝛼𝑟 ∕(𝑚𝑙𝑠 𝑔) ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎣ ⎦ 2ℎ𝑅 ∕𝑙𝑠

where 𝐴𝑒 is Hurwitz, thus (A.2) is exponentially stable provided that ‖𝜴‖ 𝜂1𝜙 , 𝜂2𝜙 > 0. The estimation error is bounded to sup𝑡≥0 ‖𝐴 ‖2 , which can be 𝑒 decreased to improve the performance by changing the observer gains 𝜂1𝜙 , 𝜂2𝜙 and the filter coefficients 𝑓1𝜙 , 𝑓2𝜙 . Therefore, the error dynamics is stable and non-expansive since its 𝐋2 gain is less than one for some observer and filter gains.

(B.5)

where 𝑎56 = 𝑚𝑠 𝑔ℎ𝑅 − 𝑘𝜙 , c𝛿𝑓 = cos 𝛿𝑓 , s𝛿𝑓 = sin 𝛿𝑓 , 𝑏11 = (2𝐿𝑓 s𝛿𝑓 − 𝑤𝑓 c𝛿𝑓 )∕(2𝑅𝑒 ) and 𝑏12 = (2𝐿𝑓 s𝛿𝑓 + 𝑤𝑓 c𝛿𝑓 )∕(2𝑅𝑒 ) is used for brevity. Standard state-space representation of the prediction model where 𝐄𝑐 =

Appendix B. State-space representation of the prediction model

𝐈 can be obtained by left-multiplying the first line of (B.1) by 𝐄−1 𝑐 .

The prediction model developed in Section 3.1 can be represented in state-space format as follows: { 𝐄𝑐 𝐱̇ = 𝐀𝑐 𝐱 + 𝐁𝑐 𝐮 (B.1) 𝐲 = 𝐂𝑐 𝐱

prediction model. Therefore, the prediction model in Eq. (17) can be

In this paper, Step-Invariance method is used to discretize the obtained as follows: −1 𝐀

𝐀 = 𝑒(𝐄𝑐

𝑇𝑠

𝐁=

where the subscript 𝑐 is used to differentiate continuous state-space matrices from their discrete counterparts. Using the dynamics of the prediction model described in Section 3.1, the state-space matrices are

∫0

𝑐 )𝑇𝑠

(𝐄−1 𝑐 𝐀𝑐 )𝜏

𝑒

(B.6a) (𝐄−1 𝑐 𝐁𝑐 )𝑑𝜏

where the controller sample time is shown by 𝑇𝑠 and 𝐶 = 𝐶𝑐 . 106

(B.6b)

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Appendix C. Detailed derivation of QP problem

where [ ] ̄ 𝑇 = 𝑅𝐼 𝐑𝐈 𝑅𝐼max … 𝑅𝐼max max ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

Here, the detailed steps of deriving (22) is presented. 𝐓̄ = blockdiag (𝐓, 𝐓, … , 𝐓) ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟

(C.16)

𝑁𝑝

(C.1)

𝑁𝑝

̄ = blockdiag (𝐑, 𝐑, … , 𝐑) 𝐑 ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟

References

(C.2)

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𝑁𝑝

[

] 𝑇

𝐰̄ 𝑇 = 𝐰𝑇 𝐰𝑇 … 𝐰 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(C.3)

𝑁𝑝

[ ] 𝐯 = 𝐯𝑇 𝐯𝑇 … 𝐯𝑇 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ̄𝑇

(C.4)

𝑁𝑝

[ ] 𝐮̄ 𝑇𝑝 = 𝐮𝑇𝑝 𝐮𝑇𝑝 … 𝐮𝑇𝑝 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(C.5)

𝑁𝑝

The cost function of (18a) can now be expressed as: ( )𝑇 ( ) ̄ (̄𝐮 − 𝐯) ̄ 𝑇𝐑 ̄ + 𝐮̄ − 𝐮̄ 𝑝 𝐓̄ 𝐮̄ − 𝐮̄ 𝑝 2𝐽 =(̄𝐮 − 𝐯) + 2𝐰̄ 𝑇 𝐮̄

(C.6)

The above equation can be written in the following compact form: ) ( ) 1 𝑇 (̄ ̄ − 𝐓̄ + Const. (C.7) 𝐮̄ 𝐑 + 𝐓̄ 𝐮̄ + 𝐮̄ 𝑇 𝐰̄ − 𝐑 2 Next, the constraints of (23) are derived. The lower-bounds and upperbounds of (23a) are obtained by simply augmenting those of (21): [ ] 𝐋𝐁𝑇 = 𝐥𝐛𝑇 𝐥𝐛𝑇 … 𝐥𝐛𝑇 (C.8a) [ ] 𝑇 𝑇 𝑇 𝑇 𝐔𝐁 = 𝐮𝐛 (C.8b) 𝐮𝐛 … 𝐮𝐛

𝐽=

The constraint of (23b) requires using prediction model to relate the roll-over index to the control inputs. The outputs of the prediction model over the prediction window can be expressed as: (C.9)

𝐲̄ = 𝐒𝑥 𝐱0 + 𝐒𝑢 𝐮̄ where [ 𝑇 𝐲̄ 𝑇 = 𝐲1

𝐲2𝑇

𝑇 𝐲𝑁

…

[ 𝐒𝑇𝑥 = (𝐂𝐀)𝑇

(𝐂𝐀2 )𝑇

⎡ 𝐂𝐁 ⎢ 𝐂𝐀𝐁 ⎢ 𝐒𝑢 = ⎢ ⋮ ⎢ ⋮ ⎢ 𝑁𝑝 −1 𝐁 ⎣𝐂𝐀

]

… … 𝟎 ⋱ ⋱ …

𝟎 𝐂𝐁 ⋱ ⋱ …

(C.10)

𝑝

(𝐂𝐀𝑁𝑝 )𝑇 … … ⋱ ⋱ …

𝟎 ⎤ 𝟎 ⎥ ⎥ ⋮ ⎥ ⋮ ⎥ ⎥ 𝐂𝐁⎦

]

(C.11)

(C.12)

The constraints of Eq. (20) can be split into two constraints of 𝐲𝑘 < 𝑅𝐼max + 𝜖𝑘 and −𝐲𝑘 < 𝑅𝐼max + 𝜖𝑘 . Therefore, 𝐀𝑢 and 𝐔𝐁𝑢 of Eq. (23b) can be obtained: [ ] −̄𝐛 + 𝐒𝑢 𝐀𝑢 = ̄ (C.13) −𝐛 − 𝐒𝑢 where [ 𝐛= 0 ⎡𝐛 ⎢ ̄𝐛𝑇 =⎢𝟎 ⎢⋮ ⎢𝟎 ⎣ [ 𝐔𝐁𝑢 =

0 𝟎 𝐛 ⋱ 𝟎

0

0

… … ⋱ …

̄ −𝐒𝑥 𝐱0 + 𝐑𝐈 ̄ +𝐒𝑥 𝐱0 + 𝐑𝐈

1

]

𝟎⎤ ⎥ 𝟎⎥ 𝟎⎥ 𝐛⎥⎦ 𝑁𝑝 ×5𝑁𝑝

(C.14a)

(C.14b)

] (C.15) 107

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