Explicit non-linear model predictive control for electric vehicle traction control

Transcription

1 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 Explicit non-linear moel preictive control for electric vehicle traction control D. Tavernini, M. Metzler, P. Gruber an A. Sorniotti, IEEE Member Abstract This stuy presents a traction control system for electric vehicles with in-wheel motors, base on explicit non-linear moel preictive control. The feeback law, available beforehan, is escribe in etail, together with its variation for ifferent plant conitions. The explicit controller is implemente on a rapi control prototyping unit, which proves the real-time capability of the strategy, with computing times in the orer of microsecons. These are significantly lower than the require sampling time for a traction control application. Hence, the explicit moel preictive controller can run at the same frequency as a simple traction control system base on Proportional Integral (PI) technology. High-fielity moel simulations provie: i) a performance comparison of the propose explicit non-linear moel preictive controller with a benchmark PI-base traction controller with gain scheuling an anti-winup features; an ii) a performance comparison among two explicit an one implicit non-linear moel preictive controllers base on ifferent internal moels, with an without consieration of transient tire behavior an loa transfers. Experimental test results on an electric vehicle emonstrator are shown for one of the explicit non-linear moel preictive controller formulations. Inex Terms traction control, wheel slip, moel preictive control, PI control, electric vehicle, in-wheel motors. T I. INTRODUCTION HE aoption of electric rivetrains, an in particular of inwheel motor layouts, has the potential of significantly enhancing the performance of wheel slip control systems, i.e., anti-lock braking systems (ABS) an traction control (TC) systems [1]. This is cause by the higher control banwith an precision in torque moulation that electric rivetrains can offer, with respect to the more conventional internal combustion engines an hyraulic / electro-hyraulic braking units. [2] an [3] inclue experimentally measure reuctions in stopping istances an acceleration times, achieve through the continuous moulation of the electric rivetrain torques. However, further work can be one in terms of control esign to enhance the slip ratio tracking performance an the seamless blening of the regenerative an issipative braking contributions. In parallel to sliing moe control [4] an maximum transmissible torque estimation [5] algorithms, the recent literature (see [6]-[18]) on the topic of ABS an TC shows growing interest in moel-base control, with focus on moel preictive control (MPC). For example, [6] iscusses a gain scheule linear quaratic regulator (LQR) approach for ABS control, with experimental results on an internal-combustionengine-riven vehicle with electro-mechanical brakes. [7] an [8] inclue ifferent approaches to ABS control, i.e., linear quaratic Gaussian (LQG) regulation an generalize preictive control, which is re-propose in [9] for a TC implementation. A linear MPC strategy is evelope in [10], where the ABS slip regulation is achieve through torque blening between the friction brakes an in-wheel motors. Similarly, [11], [12] an the very recent research [13] combine ABS control an torque blening, by using a linear MPC formulation. [14] presents an MPC-base ABS, with test results on a harware-in-the-loop rig. The internal moel inclues a tire force ynamics formulation; however its effect on the controller performance is not iscusse in the stuy, nor, to the authors knowlege, in any other stuy in the literature. [15] presents a non-linear moel preictive controller (NMPC) for ABS an TC. The formulation consiers all four wheels in the same internal moel. Reference tracking is not use, since the slip ratio is solely controlle through the constraints of the NMPC formulation. Moreover, the tire-roa friction coefficient is consiere to be known a-priori, which introuces further challenges for a real vehicle implementation. For an internalcombustion-engine-riven vehicle, [16] introuces four linear MPC TC strategies that are compare with a hybri explicit MPC. The hybri esign aopts a piecewise linear approximation of the non-linear longituinal tire force characteristic as a function of the slip ratio. Simulation an experimental results show the performance enhancement of the hybri strategy with respect to the linear approaches. In the case of implicit NMPC, a non-linear programming (NLP) problem is solve on-line at each sampling time. The resulting computational loa makes implicit NMPC ifficult to implement in real automotive applications, if the require sampling frequency is high. In this respect [17] provies an example of real-time capable NMPC for an ABS with torque blening, incluing a comparison with a linear MPC approach. The results show that the computational time of the implicit This work was supporte by the European Union's Horizon 2020 Programme uner Grant Agreement No (SilverStream project, Social Innovation an Light electric VEhicle Revolution on STREets an AMbient). D. Tavernini ( .tavernini@surrey.ac.uk), M. Metzler ( m.metzler@surrey.ac.uk), P. Gruber ( p.gruber@surrey.ac.uk) an A. Sorniotti ( a.sorniotti@surrey.ac.uk) are with the University of Surrey, Guilfor, GU2 7XH, Guilfor, UK.

2 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 2 NMPC, i.e., 3-4 ms on a esktop personal computer, is within the selecte sampling interval of 5 ms. In [15] the implicit NMPC strategy is run on a rapi control prototyping unit, with a computational time of 4-5 ms an an implemente sampling interval of 10 ms. The stuy of this paper presents an explicit NMPC (enmpc in the remainer) for TC on electric vehicles with in-wheel rivetrains. The explicit solution is compute off-line by using a multi-parametric (mp) quaratic programming approximation of the mp-nlp problem. The control action is evaluate on-line at each sampling time starting from the current values of system states an parameters, an the off-line explicit solution, store in the memory of the control unit. This rastically reuces the require computational power. The other avantage is that the complete feeback law is available beforehan in its explicit form, which allows its analysis for the range of states an reference parameters. Another important aspect is the performance comparison an critical analysis of ifferent TC implementations. In this respect, [16] claims that the performance of the propose MPC is comparable with that of a well-tune PID (proportional integral erivative) controller. The same authors state that the simulation an test results emonstrate that the l 1-optimal hybri controller use in this problem can lea to about 20% reuction in peak slip amplitues an corresponing spin uration when compare to best case linear MPC counterparts. Similarly, [17] shows the superiority of NMPC over linear MPC in terms of slip control performance. The necessity of objective benchmarking technologies in the fiel of ABS / TC was pointe out in the survey stuy in [19]. In orer to unerstan where the strategies of the ifferent papers stan with respect to each other, a comparison is well neee. [20] partially covers this knowlege gap, but limits the analysis to on-boar electric rivetrains, characterize by significant torsional ynamics. [13] inclues also an MPC-PI experimental comparison, but for an ABS application combine with torque blening. Base on the previous iscussion, the points of novelty of this stuy are: The esign of TC systems base on enmpc, implementable at the same sampling interval as a typical PI controller for TC, but with better tracking performance. The observation of the explicit feeback control law, an its epenency on the vector of parameters from the plant. The simulation-base analysis of the performance avantages of the propose enmpc compare to a welltune benchmark PI TC system with gain scheuling an anti-winup features. The sensitivity analysis of the performance of TC algorithms with respect to their sampling interval. The iscussion of the benefit of consiering transient tire response an vertical loa transfers in the internal moel for the NMPC formulation. The presentation of experimental test results base on explicit non-linear moel preictive control applie to a fully electric vehicle prototype with in-wheel rivetrains. II. EXPLICIT NON-LINEAR MODEL PREDICTIVE CONTROL A. Problem formulation Similarly to the NMPC, the enmpc requires the formulation of an optimization problem, potentially incluing constraints on the control inputs an system states. A generic non-linear optimal control problem for a finite horizon in the time interval [t k, t f ] can be efine as the minimization of the following cost function: V(x[t k, t f ], u[t k, t f ], p(t k ), ν[t k, t f ]) t k t f L(x(t), u(t), p(t k ), ν(t)) t + M(x(t f ), p(t k ), t f ) where x, u, p an ν are the state, input, parameter an slack variable vectors, respectively. L is the stage cost, an M is the terminal cost. The problem is subject to inequality constraints of the form: x min x(t) x max (2) u min u(t) u max (3) g(x(t), u(t), p(t k ), ν(t), t) 0. (4) The equality constraints are represente by the orinary ifferential equations (ODEs) escribing the system ynamics: t x(t) = f(x(t), u(t), p s (t k ), t) (5) where p s is the vector of the system parameters. The initial conition x(t k ) is assigne to the state vector. The infinite-imensional optimal control problem in (1)-(5) is iscretize, thus becoming an NLP problem, which is solve through numerical methos. This approach is known as Direct Metho [21]. In this operation, the equality constraints (5) are represente by finite approximations. The infinite-imensional unknown solution, u[t k, t f ], an the slack variables, ν[t k, t f ], are replace by a finite number of ecision variables. The preiction horizon t p = t f t k is efine as t p = N p t s, where N p is the number of preiction steps an t s is the characteristic iscretization interval of the internal moel. The input signal, u[t k, t f ], is assume to be piecewise constant along the horizon. It is calculate through the function μ an is parameterize through the vector of control parameters, U, such that u(t) = μ(t, U). Similarly, the piecewise constant slack variable trajectory is parameterize through the vector of slack variables, N. The technique known as Direct Single Shooting ([21], [22]) is use for the management of the equality constraints. It consists of eliminating the ODE equality constraints by substituting their iscretize numerical solution into the cost function an constraint formulations. Starting from the continuous constraint equations (5), the numerical solution is erive by iscretization an integration of the ODE: x(t k+j ) = φ(x(t k ), U, p s (t k ), t k+j ), j = 1,, N p. (6) To obtain the function φ, an explicit integration scheme is selecte: x(t k+j+1 ) = F(x(t k+j ), μ(t k+j, U), p s (t k ), t k+j ) (7) with given initial conition x(t k ). If the whole horizon is consiere, the state trajectories are all mappe into a single (1)

3 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 3 function, an the system ynamics o not appear any more as equality constraints: x(t k+j ) = F(x(t k+j 1 ), μ(t k+j 1, U), p s (t k ), t k+j 1 ) x(t k+2 ) = F(F( F(F(x(t k ),, t k ),, t k+1 ),, t k+j 2 ), ) x(t k+1 ) x(t k+j 1 ) The optimal control problem is now in its generic multiparametric mp-nlp form: V (x(t k ), p(t k )) = min U,N V(x(t k ), U, p(t k ), N) (9) subject to: G(x(t k ), U, p(t k ), N) 0 (10) where p inclues the system an controller parameters, which are consiere constant for the uration of the preiction horizon. Two aitional vectors are efine: i) the vector of parameters x p (t k ) R n p, where n p = n +, i.e., n p is the sum of the number of states an the number of parameters: x p (t k ) = [ x(t k ) p(t k ) ] (11) an ii) the vector of ecision variables, z R s : z = [ U N ]. (12) Base on (11) an (12) it is possible to reformulate the optimization problem as: V (x p (t k )) = min z V (z, x p (t k )) (13) subject to: G (z, x p (t k )) 0. (14) The minimization is performe with respect to z an is parameterize with x p (t k ). B. Off-line solution The mp-nlp problem is not solve irectly, but through its approximation (see [23]). In this stuy a multi-parametric quaratic programming (mp-qp) formulation is aopte, as suggeste in [21] an implemente in [24]. The mp-nlp in (13) an (14) is linearize aroun a preefine point (z 0, x p,0 ) by means of Taylor series expansion, such that the cost function is approximate with a quaratic function (15)-(16) an the constraints assume a linear formulation (17): V 0 (z, x p ) 1 2 (z z 0 )T H 0 (z z 0 ) + (D 0 + (x p x p,0 ) T F 0 ) (z z 0 ) + Y 0 (x p ) Y 0 (x p ) 1 2 (x p x p,0 ) T 2 xp x p V(z 0, x p,0 )(x p x p,0 ) T + ( xp V(z 0, x p,0 )) (x p x p,0 ) + V(z 0, x p,0 ) (8) (15) (16) G 0 (z z 0 ) E 0 (x p x p,0 ) + T 0. (17) The ifferent terms are compute as follows an evaluate at the linearization point (z 0, x p,0 ): H 0 zz 2 V(z 0, x p,0 ) D 0 ( z V(z 0, x p,0 )) T G 0 ( z G(z 0, x p,0 )) T T E 0 ( xp G(z 0, x p,0 )) T 0 G(z 0, x p,0 ) (18) F 0 1 (( 2 zx 2 T p V(z 0, x p,0 )) + 2 xp zv(z 0, x p,0 )). The mp-qp formulation is employe to compute local approximations of the original mp-nlp problem in the exploration space. The final space is represente by a number of hyper-rectangles, on which single mp-qp problems are solve. Each hyper-rectangle is further partitione into polyhera, i.e., the critical regions for the mp-qp problem. Finally, the mp-qp solution is represente by a piecewise affine function that is continuous across the bounaries among ifferent polyhera, but iscontinuous across the hyperrectangles. In this stuy the mp-qp problems are compute by means of the Multi-Parametric Toolbox 3.0 [25]. The solution is evaluate in points of interest within each hyper-rectangle an compare with the solution of the NLP problem at the same points, where the initial state conitions are the coorinates of the points themselves. The NLPs are compute by means of IPOPT, a software package for non-linear optimization [26]. Base on the maximum error between the evaluate mp-qp an compute NLP solutions for all the points, a ecision is mae whether to sub-partition the hyper-rectangle into smaller ones, or to stop the process an accept the mp-qp approximating solution. The algorithm in [21] that implements this concept is summarize. For all the unexplore hyper-rectangles the following steps are implemente: Compute the hyper-rectangle volume (a minimum volume is efine to ecie whether the hyper-rectangle can be further split). Compute the NLP solution (or recover it from previous steps) at the points of interest. Compute the mp-qp solution on the whole hyper-rectangle, using the NLP solution at the Chebyshev center plus its coorinates, as the linearization point for the terms in (15)- (18). Evaluate the mp-qp solution for all the aforementione points. Calculate the maximum error between the NLP-compute solutions an the mp-qp-evaluate solutions. Base on this information each hyper-rectangle is either store or marke to be split with a heuristic splitting rule similar to the one in [21]. When all the tolerances are fulfille or the minimum allowe volume has been reache, the algorithm terminates an the solution is available for any point insie each hyper-rectangle. With respect to the stability of the resulting controller, common schemes in the literature for implicit MPC inclue stabilizing terminal constraints or terminal costs, which nee to satisfy Lyapunov function-type conitions (see [27] an [28]). Alternatively, [29] an [30] present a stability an performance analysis technique for unconstraine non-linear implicit MPC schemes. However, all these approaches are for implicit MPC. To the best of the authors knowlege, there is no comparable practical non-linear MPC theory in the literature aressing the stability an sub-optimality for explicit non-linear MPC. Therefore, in this stuy the enmpc stability will be verifie

4 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 4 empirically, through the simulate scenarios an experimental test results of the Sections V an VI. C. Implementation of the explicit solution for real-time applications Once the solution is compute off-line, the next step is to efine the most efficient way to access it on-line. This is performe through point location an piecewise control function evaluation. In particular, the former problem becomes challenging if the total number of regions composing the final solution is large (> ). Two families of methos are available: i) sequential search methos, checking in the worst case every region to ientify the one containing the consiere point; an ii) binary-search-tree methos [31], proviing a fast solution for the location of the point with a limite number of mathematical operations, which is logarithmic in the number of regions, for a balance tree. As rawback, binary-search-tree methos require significant off-line processing, which makes them unsuitable for a large number of regions [31]. The specific application (i.e., the 4-imensional case of Section III.B) has a total number of 85 hyper-rectangles, obtaine with approximation tolerances of 0.50, 0.10 an 0.10 for the values of the cost function, the normalize solution an the maximum normalize constraint violation, respectively. A two-layer solution is propose. The top layer inclues a binarysearch-tree to etermine the inex of the hyper-rectangle the measure point lies in. This information is then passe to the bottom layer, consisting of functions, one for each hyperrectangle, which ientify the correct critical region within the hyper-rectangle, an evaluate the piecewise control function. In the bottom layer either binary-search-tree or sequential search methos can be use, as the number of polyheral critical regions is usually limite (i.e., < 100 for this TC application), which makes both methos viable in terms of processing buren an searching time. TABLE I. Internal moel parameters. Symbol Description Value Unit m Quarter car mass kg r Wheel rolling raius m J w Wheel mass moment of inertia 1.5 kgm² B MF coefficient: stiffness factor 40 - C MF coefficient: shape factor D MF coefficient: peak value f z Tire vertical loa 1104 N ref σ x Slip ratio reference value σ Longituinal relaxation length (*) m (*) 5-imensional problem only III. TRACTION CONTROL DESIGN This section iscusses the structure an formulation of the propose moel preictive TC strategies, firstly by eriving the internal moel, an then by formulating the optimal control problem. In particular, three internal moels with increasing complexity are propose an use with the same cost function an constraints. The values of the main vehicle ata use for internal moel parameterization are reporte in Table I. They refer to the electric vehicle simulate in Section V. A. Traction control structure Fig. 1 shows the traction control structure. The torquevectoring controller of the electric vehicle calculates the total reference wheel torque an reference yaw moment. The control allocation (CA) algorithm outputs the iniviual wheel torques for the in-wheel motors, inicate as T CA, to achieve the references. A state preictor (SP) compensates for the system elays on the states, e.g., cause by the CAN bus. The correcte parameter vector with the upate states, x p, is provie to the core block of the TC, i.e., the on-line implementation of the enmpc, which outputs the torque correction T, to be subtracte from T CA. Fig. 1. Simplifie architecture of the implemente TC strategy. B. 4-imensional problem: internal moel The controlle variable is the wheel slip velocity s: s = ωr V (19) where ω is the angular wheel spee, r is the rolling raius of the wheel an V is the linear spee of the vehicle, so that the slip ratio is: ωr V σ x = = s ωr ωr. (20) The time erivative of (19) is given by: t s(t) = r t ω(t) V(t). (21) t The first term on the right-han sie results from the wheel moment balance: t ω(t) = 1 (T J CA T(t) F x r) (22) w where J w is the wheel mass moment of inertia. T CA is kept constant over the preiction horizon, an thus is a system parameter. F x is the longituinal tire force, estimate through a simplifie version of the Pacejka magic formula (MF) [33]: F x = μ x F z (23) μ x = D sin(c arctan(b σ x )) (24) where F z is the vertical tire loa, consiere as a constant, an μ x is the longituinal tire force coefficient, with B, C an D being the MF parameters [33]. The longituinal vehicle ynamics are moele by consiering a mass, m, equal to a quarter of the total vehicle mass: t V(t) = 1 m F x. (25) By substituting (22) an (25) into (21) the wheel slip ynamic equation, i.e., the first equation of the internal moel, is obtaine: r2 s(t) = ( 1 ) Dsin (Carctan (Bs(t) t J w m ω(t)r )) F z + (T CA T(t))r J w. (26)

5 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 5 An integral action is incorporate to tackle the steay-state error. This consiers the integral of the error, e int, between the actual slip velocity, s, an the reference slip velocity compute from the target value, σ ref x, of the slip ratio. The respective ifferential equation, i.e., the secon equation of the internal moel, is: t e int (t) = s(t) σ ref x ω(t)r. (27) By substituting (23) an (24) into (22), the thir equation of the internal moel is obtaine: t ω(t) = 1 (T J CA T(t) w Dsin (Carctan ( Bs(t) (28) ω(t)r )) F zr). The moel state vector, input vector an parameter vector are respectively x = [s, e int, ω], u = [ T] an p = [T CA ]. Unless otherwise specifie, in the following analyses the explicit solution is reporte for N p = 4 an t s = 2 ms. The parametric problem inclues 4 parameters (4-imensional problem), i.e., x p = [s(t k ), e int (t k ), ω(t k ), T CA (t k )], an 5 ecision variables, i.e., z = [ T(t k ), T(t k+1 ), T(t k+2 ), T(t k+3 ), ν(t k )]. The receing horizon control input that is applie to the system is u(t k ) = T(t k ), which will be inicate as u in the remainer. The other elements of z(t k ) are neee only for the off-line calculations of the 4-imensional enmpc, which will be referre to as enmpc 4. During the control system esign, the iniviual components of x p an z are normalize through ivision by their maximum expecte value. C. 5-imensional problem (a): internal moel The moel of Section III.B consiers instantaneous generation of the longituinal tire force. In this section the moel is enhance to account for the tire force ynamics, by incluing the concept of tire relaxation length, σ. A first orer ifferential equation calculates the slip ratio for the MF in (26) an (28), starting from the wheel spee an vehicle spee. If a linear epenency between longituinal tire force an vertical loa is assume, this is equivalent to first orer longituinal tire force ynamics. The resulting internal moel is escribe by the ifferential equations (29)-(32): r2 s(t) = ( 1 t J w m ) Dsin (Carctan (Bσ x rel (t))) F z (29) + (T CA T(t))r J w t e int (t) = s(t) σ x ω(t)r (30) t ω(t) = 1 (T J CA T(t) w Dsin (Carctan (Bσ rel x (t))) F z r) (31) t σ x rel (t) = (ω(t)r s(t)) ( s(t) σ ω(t)r σ x rel (t)). (32) In this case the state vector, input vector an parameter vector are respectively x = [s, e int, ω, σ x rel ], u = [ T] an p = [T CA ]. The problem inclues 5 parameters (5-imensional problem), i.e., x p = [s(t k ), e int (t k ), ω(t k ), σ x rel (t k ), T CA (t k )], an 5 ecision variables, i.e., z = [ T(t k ), T(t k+1 ), T(t k+2 ), T(t k+3 ), ν(t k )]. The respective explicit controller will be calle enmpc 5a in the remainer. D. 5-imensional problem (b): internal moel The moel of Section III.B consiers a constant value of the vertical tire loa. In this section a more accurate case is consiere, where the vertical tire loa is compute as a function of the vehicle longituinal an lateral accelerations. The estimate vertical loa value becomes a slowly varying parameter for the control problem, thus increasing its imension. In this case the equations of the system are exactly the same as in Section III.B, but the state vector, input vector an parameter vector are respectively x = [s, e int, ω], u = [ T] an p = [T CA, F z ]. The problem now inclues 5 parameters (5- imensional problem), i.e., x p = [s(t k ), e int (t k ), ω(t k ), T CA (t k ), F z (t k )], an 5 ecision variables, i.e., z = [ T(t k ), T(t k+1 ), T(t k+2 ), T(t k+3 ), ν(t k )]. The respective implicit controller will be calle NMPC 5b in the remainer. E. Control problem formulation The three internal moels of Sections III.B-III.D share the same optimal control problem formulation. The continuous form of the cost function is: V = t f q x1 t k 2 w (s(t) σ x x1 ref ω(t)r) 2 + q x2 2 e int (t)2 w x2 + r u w u 2 T(t)2 + r ν w ν 2 ν(t k )2 t (33) + p x1 2 w (s(t f) σ ref x ω(t f )r) 2 x1 + p x2 2 w e int(t f ) 2 x2 where q x1, q x2, r u, r ν, p x1, p x2 are the weights of the ifferent terms, an the notations w i inicate scaling factors. As a consequence, a tracking problem is set for the first state, s, an a regulating problem is set for the secon state, e int. The choice of aopting the slip velocity, s, as state an tracking variable, rather than the more commonly use slip ratio σ x, fins its motivation in the algorithm for the computation of the explicit solution. In fact, the aoption of σ x woul lea to a feeback law that is scale with the angular wheel spee. The higher variability of the feeback control law woul imply a finer partition of the space, to reach a goo approximation of the non-linear problem. Hence, the choice of ifferent internal moels, although equivalent from the viewpoint of the represente physics, influences the efficiency of the generation of the explicit solution. Careful consieration of this aspect in the esign phase leas to a reuction of the off-line computational buren an the on-line memory requirement. The minimization of (33) is subject to state an input boun constraints: s min ν s s max + ν (34) 0 T T CA. (35) IV. ENMPC-BASED TC IMPLEMENTATION An avantage of enmpc with respect to implicit NMPC is the availability of the feeback control law beforehan. This

6 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 6 allows the analysis of the control action for any value of the vector of parameters. Fig. 2. Normalize control action, u, for the 4-imensional problem as a function of x p (1) (normalize wheel slip velocity) an x p (4) (normalize torque eman from the river). The solution of the enmpc 4, i.e., the 4-imensional enmpc (see Section III.B), is presente in Fig. 2. To plot the 3- imensional surface in Fig. 2, two parameters have been fixe, i.e., the normalize integral of the wheel slip error, x p (2), which is set to zero, an the normalize wheel angular velocity, x p (3), which is set to The re line reference inicates the wheel slip velocity corresponing to the reference slip ratio for the specific x p (3). x p (4) is the normalize torque eman from the CA. The solution essentially consists of three planes: i) a plateau of zero control action for low values of slip velocity, inicate as input lower constraint in Fig. 2. Accoring to (35), the TC torque correction must be positive; ii) an incline plane, parallel to the x p (1)-axis, inicate as input upper constraint in Fig. 2, which expresses that, accoring to (35), the regulating torque cannot be larger than the torque eman; an iii) another incline plane, i.e., the non-saturate feeback law, which is saturate by the previous two. The analysis of the control action shows that no regulation is applie until the reference slip is reache, if the normalize torque eman is small. On the other han, for high values of x p (4), a regulation is prescribe even before reaching the reference, base on the preiction available to the controller. Beyon the reference a regulation that is below the maximum possible value is applie for the whole range of torque emans, as long as the slip velocity is lower than a specific non-constant value (see the surface non-saturate feeback law ). Above this value the regulating control action is equal to the torque eman, i.e., u= x p (4). The effect of the normalize integral of the slip velocity error, x p (2), is presente in Figs. 3a-b, corresponing to a negative value an a positive value of x p (2), respectively. The whole surface of the feeback law shifts along the x p (1)-axis, while the reference oes not move. This acts as a compensation for the initial positive or negative value of x p (2). Figs. 3c- show the variation of the feeback law with the normalize wheel spee, x p (3). Although the shape of the surface oes not change, it translates with the reference slip velocity along the x p (1)-axis. Fig. 4 shows that the piecewise affine feeback law is actually evaluate from a number of ifferent regions of the parametric problem, i.e., hyper-rectangles an polyheral critical regions, espite the control action mainly consists of only three planes. The analysis of Figs. 2-4 suggests that the whole feeback law coul be realize as a rule-base strategy that efines the ifferent planes intersections an translations, given the input measurements from the plant. Alternatively, a rigorous metho for the reuction of the memory requirements of explicit moel preictive controllers is presente in [34]. During the implementation phase of the enmpc, as shown in Fig. 1, a specific strategy was applie for the compensation of δ m an δ CAN, i.e., the pure time elays associate with the electric motor rive an the CAN bus, respectively. The aopte technique is base on the concept use in [16] for a hybri explicit MPC implementation of a TC. An SP, employing the same moel formulation escribe in Section III.B, an a buffer, containing part of the past control history, are use to preict the trajectory of the input parameters to the enmpc, for a horizon length of δ m +δ CAN. Thus, the inputs to the controller are projecte into the future, an the control action is compute base on this preiction. (a) (c) Fig. 3. Effect of x p (2) an x p (3) on u for the 4-imensional problem: (a) negative value of x p (2); (b) positive value of x p (2); (c) low value of x p (3); an () high value of x p (3). The solution of the enmpc4 was teste on a SPACE MicroAutobox II (900 MHz, 16 MByte) rapi control prototyping unit. An exploration of the parameter space was performe to assess the computational time for a fine an comprehensive gri of possible inputs. The computational time for the combination of the two function evaluation layers was in the range of ~5-25 μs. These values are very low compare to the implemente sampling time of 2 ms, which is not achievable with more conventional implicit NMPC technology on the same harware. Hence, the enmpc can run in real-time at any frequency within the range typical of TC applications. (b) ()

7 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 7 Fig. 4. Normalize control action with the corresponing region inication. x p (2) an x p (3) have been fixe. tracking performance inicator: RMSE = 1 t f (σ t f t x (t) σ ref x ) 2 t (36) i t i where σ x (t) is the actual value of the slip ratio uring the relevant part of the test, efine by the initial an final times t i an t f. The final value of vehicle velocity, V f, i.e., an acceleration performance inicator. The normalize integral of the absolute value of the control action, which gives an inication of the require control effort: IACA = 1 t f T(t) t. t f t (37) i t i V. SIMULATION RESULTS A. Test scenario an evaluation metrics The simulation analysis was carrie out with a high fielity vehicle simulation moel implemente with the software IPG CarMaker. The vehicle ata (see Table II) are those of an electric quaricycle prototype with a front-wheel-rive topology, base on two in-wheel motors (irect rive) with a peak torque of 500 Nm each. Given the limite weight of the vehicle, the available torque is sufficient to provoke front wheel spinning even in high tire-roa friction conitions. The tire moel is the MF (ver. 5.2), an inclues the variation of the longituinal an lateral relaxation lengths as functions of the vertical loa. The electric motor ynamics are moele through a first orer transfer function an a pure time elay. A pure time elay is also consiere on the controller output to moel the CAN bus [32]. Unless otherwise specifie, in the remainer the implementation step size of the controllers, t S,I, is of 2 ms. The consiere acceleration test scenario is base on a straight roa with varying tire-roa friction coefficient, μ. The values of μ are moifie in steps, accoring to the sequence This provies a real challenge to the TC, which has to regulate the slip ratio to a constant reference value of 0.10, while the vehicle is accelerating from an initial spee of 5 km/h, at which a fast torque eman ramp up to the rivetrain peak torque is impose. TABLE II. Main parameters of the simulation moel. Description Value Unit Vehicle mass 450 kg Wheel + motor mass moment of inertia 1.5 kgm² Wheelbase m Wheel raius m Maximum single motor torque 500 Nm Motor time constant (τ m ) 0.5 ms Motor time elay (δ m ) 1 ms CAN bus time elay (δ CAN ) 3 ms To objectively assess the TC performance, a set of performance inicators is ientifie base on [20]: The root-mean square value of the slip ratio error, i.e., a Fig. 5. NMPC 4 an enmpc 4 comparison: actual an reference slip ratios of the front left wheel. B. enmpc 4 benchmarking To prove the effectiveness of the local quaratic approximations of the multi-parametric non-linear problem, the simulation results for the escribe scenario are reporte in Fig. 5, with an overlap between the enmpc 4 solution an the corresponing implicit one. The implicit strategy for the 4- imensional case (NMPC 4) is implemente by solving online the same non-linear optimal control problem with the same solver, IPOPT, employe for the generation of the explicit solution. The implicit strategy, which is not real-time capable, represents the optimal solution, because of the absence of the local quaratic approximations. Fig. 5 shows that the solutions of the NMPC 4 an enmpc 4 are inistinguishable. As this is confirme by all the simulations that were performe uring the stuy, the level of optimality of the enmpc 4 implementation is consiere satisfactory. Fig. 6 reports the inex of the hyper-rectangles that are use by the enmpc 4 in the consiere scenario, an the inex of the polyheral critical regions that are employe within each hyperrectangle. The figure reveals that only a few regions are use in the simulate complex scenario. Moreover, the crossings of ifferent hyper-rectangle bounaries, which imply iscontinuities in the solution, o not bring any significant egraation of the explicit feeback control action.

8 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 8 present unershoots followe by a few oscillations with similar uration. The oscillations have higher amplitues for the PI. In the final μ-transitions the overshoots an unershoots are relatively small an of similar magnitue for the two controllers, although slightly higher for the PI, which also exhibits a slower response. Fig. 6. enmpc 4: hyper-rectangle inex (top) an polyheral critical region inex (bottom) with the vertical lines inicating the hyper-rectangle switching times. Each hyper-rectangle has an inepenent numbering of its polyhera. C. enmpc 4 an proportional integral (PI) controller The results of the enmpc 4 are compare with those obtaine through a simple yet effective PI-base TC system, with gain scheuling on vehicle spee an incluing anti-winup features on its integral contribution. A frequency response-base initial esign of the PI gains was performe with a linearize plant moel for ifferent vehicle spees. This was followe by an empirical fine tuning through simulations in the time omain with the CarMaker moel. The gains obtaine with this process were finally re-assesse by employing the linearize plant to verify gain an phase margins, as well as the sensitivity an complementary sensitivity functions. The comparison of the controller results in terms of slip ratios is reporte in Fig. 7. For both the PI an the enmpc 4 the TC is activate in proximity of the reference slip value, σ x ref. The response of the two controllers to the initial wheel torque eman application presents visible ifferences. The PI overshoots σ x ref, an then reaches the esire value with a ampe oscillatory response. The enmpc 4 presents an initial unershoot cause by the controller activation an the iscrepancy between the tire-roa friction coefficients of the plant an the internal moel. This is promptly recovere by the integral action. Afterwars, the enmpc 4 approaches the reference more gently, with a lower overshoot an less oscillations. The reason for this behavior is that the esign of the enmpc 4 TC is base on tire characteristics for μ = Hence, when the controller operates in higher tire-roa friction conitions (e.g., at μ = 0.9), it tens to be conservative. Nevertheless, σ x ref is reache at approximately the same time as in the PI case. The transition between μ = 0.9 an μ = 0.15 is very emaning for the controllers. The PI respons with an overshoot that is maintaine until the slip ratio reaches σ x ref. For the enmpc 4 the overshoot presents a smaller peak an a faster response leaing to the reference. This is followe by a promptly recovere unershoot. The next ifficult transition is the one that leas back to μ = 0.9. In this case both controllers Fig. 7. PI an enmpc 4 comparison: actual an reference slip ratios of the front left wheel. Fig. 8. PI an enmpc 4 comparison: torques before an after the front left TC block. Fig. 8 plots the wheel torques before an after the TC block. Similarly to the slip ratios, the time histories highlight the marginally faster response of the enmpc 4, together with the more quickly ampe oscillations of its control action. Fig. 9 shows the angular spee of the front left wheel, multiplie by the wheel raius, an the vehicle spee. The time histories of the longituinal vehicle acceleration are reporte in Fig. 10. The wie range of values, i.e., from ~0 m/s² to ~4.5 m/s² uring the relevant part of the test, together with their abrupt variations, confirms the high level of criticality of the selecte scenario. The longituinal acceleration oes not significantly iffer among the two controllers.

9 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 9 Fig. 9. PI an enmpc4 comparison: front left wheel spee multiplie by the wheel raius, V FL, an vehicle velocity, V vhl. Fig. 10. PI an enmpc 4 comparison: longituinal acceleration of the vehicle. TABLE III. Summary of performance inicators an respective variations. Case t Controller S,I V No. RMSE (-) f IACA (s) (km/h) (Nm) i (a) PI iv (b) enmpc % w.r.t. (a) % % % v enmpc 4 * % w.r.t. (b) % % % vi enmpc 4 *** % w.r.t. (b) % % % v enmpc 4 * % w.r.t. (b) % % % vi enmpc 4 *** % w.r.t. (b) % % % ii PI ** % w.r.t. (a) % % % iii PI *** % w.r.t. (a) % % % ii PI ** % w.r.t. (a) % % % iii PI *** % w.r.t. (a) % % % * no re-tuning ** anti-win-up gain retuning (otherwise unstable) *** full retuning Table III reports the objective performance inicators efine in Section V.A for Cases i-vi: Case i: the PI TC running at t S,I = 2 ms. During the implementation phase of the controller it was verifie that a further reuction of t S,I within reasonable limits woul not have brought substantial benefits. Case ii: the PI TC running at 4 ms an 8 ms, with the same gains as for Case i, apart from the anti-winup gain. The variation of the anti-win-up gain was necessary to provie control system stability in the selecte test, especially immeiately after the first μ-transition. Case iii: the PI TC running at 4 ms an 8 ms with optimize gains for those time steps. The PI gain optimization was base on CarMaker simulations of the selecte maneuver, an was aime at the minimization of the slip ratio RMSE. Case iv: the enmpc 4 TC running at 2 ms. Case v: the enmpc 4 TC running at 4 ms an 8 ms, with the same weights of the cost function, the same iscretization interval t s of the internal moel an the same preiction horizon t p as for Case iv. In the 4 ms sub-case, in the enmpc 4 off-line process it is impose T(t k ) = T(t k+1 ) an T(t k+2 ) = T(t k+3 ), while in the 8 ms sub-case it is impose T(t k ) = T(t k+1 ) = T(t k+2 ) = T(t k+3 ). Case vi: the enmpc 4 TC running at 4 ms an 8 ms, with a fine-tuning of the weights of its cost function. Similarly to Case iii, the enmpc 4 tuning process consiste of CarMaker moel simulations an iterative computations of the slip ratio RMSE. The comparison between Case i an Case iv shows a 9.2% reuction of the RMSE for the enmpc 4 TC compare to the PI TC, together with a negligible increment on the final velocity an IACA. Both the PI TC an enmpc 4 TC are subject to a significant ecay of the respective tracking performance, when they are implemente at 4 ms an 8 ms without moifying their esign with respect to the cases running at 2 ms. In particular, the RMSE increase is of 15.4% an 185.1% for the PI controller, while it is of 10.1% an 132.4% for the enmpc 4. If the PI TC an enmpc 4 TC are re-tune for the time steps of 4 ms an 8 ms, the performance ecay is still significant, i.e., it amounts to 8.0% an 61.7% for the PI, an 8.6% an 78.6% for the enmpc 4. It is possible to observe that: a) for the specific application significant re-tuning of the controller is neee when changing the time step, which is an important outcome, not reporte in the existing TC literature to the knowlege of the authors; an b) the performance ecay inuce by the increase of t S,I is relatively similar for the two control structures. These results can be justifie through the analysis of the linearize moel of the plant without TC, incluing consieration of tire relaxation. The linearization was carrie out in proximity of the reference slip ratio. At a vehicle spee of 2.5 m/s the slip ratio response to a motor torque step input has a rise time, T r, of sole ~5 ms, which become ~11 ms an ~26 ms respectively at 5 m/s an 10 m/s. The very fast response is relate to the in-wheel layout of the specific electric rivetrains. Base on the inications in [35], the implementation step size shoul range from 6% to 40% of T r. For the average spee of the simulate scenario, i.e., ~5 m/s,

10 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 10 this implies a recommene range of t S,I from 0.7 ms to 4.4 ms. At the initial spee of the simulate tests the recommene step size woul be even significantly lower. Therefore, the system rise time values are consistent with the TC performance egraation for t S,I = 4 ms an t S,I = 8 ms, where the latter is nearly twice the maximum recommene step size at 5 m/s. In summary, a low value of the implementation step size at which the TC is run guarantees a significant enhancement of the results, inepenently of the selecte controller. It shoul be note that in many practical TC applications the step size is of ~10 ms. In particular, the enmpc 4 TCs at 4 ms an 8 ms respectively provie similar an worse results than the PI TC at 2 ms, which means that the appropriate selection of t S,I shoul have higher priority in the TC esign process with respect to the control structure selection, at least for electric vehicles with very responsive in-wheel motors such as that of this stuy. Nonlinear moel preictive control technology can be use to enhance the TC performance, however this is actually beneficial only if the NMPC is run at 2 ms. In such a conition the NMPC provies better results than the PI controller, which can be easily implemente with a very low time step. Moreover, with the available computing harware for automotive applications, an NMPC oes not currently run at 2 ms, an possibly not even safely at 4 ms, accoring to the literature mentione in Section I. This makes the implementation of the enmpc 4, rather than a more conventional implicit NMPC 4, necessary an beneficial to achieve the potential vehicle performance benefits. E. Effect of time-varying vertical loa moeling This section stuies the effect of incluing the variable vertical tire loa in the internal moel of the non-linear moel preictive controller (see Section III.D). Since it has been proven that the generate explicit solution for the enmpc 4 shows no visible ifference from its corresponing implicit solution, i.e., the NMPC 4, the comparison for this particular internal moeling feature will be carrie out through the sole implicit strategy. Fig. 12 shows the results of this comparison along the simulate scenario. The performance of the two controllers is very similar. In the first part of the scenario, when the vehicle is still on ry asphalt, the NMPC 4 shows a slightly better response. In the rest of the test the NMPC 5b provies better tracking. Overall, the ifference is very limite, an it amounts to less than 0.5% in terms of RMSE. It can be conclue that, in this application, to increase the imension of the problem by introucing a time-varying vertical loa oes not provie any major benefit with respect to the 4-imensional problem with constant loa. Future research will focus on the evaluation of alternative selections of the fifth parameter of the controller. For example, aitional parameters coul inclue a time-varying σ x ref, to improve the lateral tire force capability, as shown in [36], or to provie better performance when starting from stanstill. Fig. 12. NMPC 4 an NMPC 5b comparison: actual an reference slip ratios of the front left wheel. Fig. 11. enmpc 4 an enmpc 5a comparison: reference an actual slip ratios. D. Effect of tire force ynamics moeling This section evaluates the effect of consiering the longituinal tire force ynamics in the internal moel for NMPC esign. The simulation results for the enmpc 5a TC, erive from the internal moel of Section III.C, are reporte in Fig. 11 for the consiere μ-varying scenario. The aition of the relaxation length oes not bring any benefit in terms of tracking performance. The reason is relate to the relative fast ynamics of the longituinal tire force generation, especially for higher vehicle velocities an a flat roa surface. The enmpc 5a implementation shows that a 5-imensional problem can also be manage with this control methoology. F. Robustness assessment The robustness against the variation of the tire-roa friction coefficient, μ, has alreay been assesse. In this section further simulations are performe with the enmpc 4 an the PI, with t S,I = 2 ms. Three vehicle parameters have been ientifie to have a potentially relevant effect on control system performance, namely: i) the total vehicle mass, M; ii) the wheel mass moment of inertia, J; an iii) the longituinal slip stiffness of the tires, K x. The results in terms of RMSE an corresponing percentage variation with respect to the baseline conition of the controllers are reporte in Table IV.

11 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 11 TABLE IV. Robustness assessment: vehicle parameters variation effect on tracking performance. Case Vehicle parameter Controller No. change RMSE (-) i (a) PI iv (b) enmpc vii enmpc 4 * M + 15% % w.r.t. (b) % viii enmpc 4 * M 15% % w.r.t. (b) % xi enmpc 4 * J + 30% % w.r.t. (b) % xii enmpc 4 * J 30% % w.r.t. (b) % xv enmpc 4 * K x + 20% % w.r.t. (b) % xvi enmpc 4 * K x - 20% % w.r.t. (b) % ix PI* M + 15% % w.r.t. (a) % x PI* M - 15% % w.r.t. (a) % xiii PI* J + 30% % w.r.t. (a) % xiv PI* J - 30% % w.r.t. (a) % xvii PI* K x + 20% % w.r.t. (a) % xviii PI* K x - 20% % w.r.t. (a) % * no re-tuning TABLE V. Robustness assessment: noise injection effect on tracking performance. Case Maximum Controller Test conition RMSE (-) No. σ x i (a) PI iv (b) enmpc xix noise on ω enmpc 4 * i with i = FL, FR, RL, RR % w.r.t. (b) % % xx PI* noise on ω i with i = FL, FR, RL, RR % w.r.t. (a) % % * no re-tuning For a +/-15% variation of M, the results show that the RMSE increase/ecrease for the enmpc 4 (cases vii an viii) is confine to +5.1% an -5.4%. The same applies to cases ix an x, i.e., to the PI TC, with +6.1% an -5.6%. Hence, the aition of a passenger or payloa oes not significantly affect the TC tracking performance. When a +/-30% variation of J is impose, the enmpc 4 (cases xi an xii) an the PI (cases xiii an xiv) present the same very marginal performance egraation (i.e., by 0.4% an 2.5%). This means that the controllers will be effective for a wie range of wheel characteristics. Finally, also when K x is varie by +/- 20% to consier ifferent tire properties, the RMSE variation is limite, an it amounts to + 4.7% an 6.2% for cases xv an xvi (enmpc 4), an to + 5.2% an 5.6% for cases xvii an xviii (PI). In conclusion, both controllers are robust for the consiere reasonable range of plant parameter variations, with a limite avantage of the enmpc 4 over the PI. Another aspect of control system robustness is the noise rejection performance. The sensor noise resulting from a real vehicle prototype test, presente later on in the paper, was analyze. Gaussian white noise with ifferent initial sees is ae to the simulate wheel spees of each corner. These are the main input signals of the controller, which are use to compute the slip ratio. The results are reporte in Table V, in terms of RMSE variation an maximum slip ratio throughout the scenario. The comparison is mae with respect to the same controllers without the noise injection. In case xix the enmpc 4 is still able to follow the reference throughout the ifferent μ variations. The RMSE increase is of 142%, an is mainly cause by oscillations aroun the reference. The peak values of slip ratio remain similar to the case without noise injection, with a maximum increase of 28.3%. The PI presents a very ifferent situation. The controller is not able to follow the reference closely in all friction conitions anymore. This is evient from the 290.5% RMSE increase, an the 92.3% increase of the maximum value of σ x. Although the PI controller is still able to eventually recover the tracking of the reference slip ratio, the enmpc 4 presents much better noise rejection characteristics. It must be note that these results were obtaine without any re-tuning of the controllers. This operation is recommene for obtaining esirable performance in case of noisy signals. VI. EXPERIMENTAL RESULTS An experimental testing session was conucte with the enmpc 4 TC on the electric quaricycle prototype of the European H2020 SilverStream project. The vehicle has a mass of 640 kg (river exclue), an is equippe with four in-wheel motors with a peak power of 4.2 kw an a peak torque of 115 Nm each. The prototype is shown in Fig. 13. Fig. 13. The fully electric prototype vehicle uring the traction control an passive vehicle experimental test session on the low-μ metal plates. The bottom picture shows that the vehicle skis laterally when the TC is eactivate. The tests were conucte in front-wheel-rive moe, on a series of smooth steel plates, which were lubricate to further ecrease the friction coefficient. This is estimate to be ~ , which is inee very low an critical. Similarly to the simulation scenarios, the river conucte the vehicle on the

12 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 12 metal plates at 5-7 km/h an then suenly presse the accelerator peal to eman the maximum available torque from the front in-wheel motors. The peal position was maintaine until the en of the metal surface was reache. The enmpc 4 with t S,I = 4 ms was upate in terms of internal moel parameters an input constraints, to take into account the higher vehicle mass an lower motor torque capability, with respect to the simulate scenarios. The slip ratio time histories for the vehicle with the enmpc 4 an the passive vehicle, i.e., the vehicle with eactivate TC, are presente in Fig. 14. In the passive vehicle σ x reaches values of almost 0.9. This affects the uration of the maneuver, since the lateral force capability of the front tires is rastically reuce, because of the coupling effect between longituinal an lateral tire forces. Hence, the river is not able to maintain the vehicle on a straight line. For the enmpc 4, after a first peak of 0.25, σ x goes back to the reference value of 0.10 in the following 0.2 s. The goo tracking performance continues for the uration of the test with limite oscillations aroun the reference. Faster response an closer tracking were obtaine with a ifferent enmpc 4 tuning, at the expense of increase motor torque oscillations. Fig. 15 confirms the criticality of the friction conitions, with the front left tire of the passive vehicle that spins up compare to the rear wheels, which provie the estimate vehicle spee. The vehicle velocity profiles with an without TC present similar trens. In fact, regarless of the consiere roa surface, when the slip ratio moves beyon the peak of longituinal tire force, the longituinal force coefficient, i.e., the longituinal force ivie by the vertical loa, ecreases only by a limite amount, an the longituinal vehicle acceleration is not substantially affecte. In these conitions, the most important effect is the loss of lateral tire force capability, cause by the tire force coupling effect [37], which makes the passive vehicle ski laterally, an go outsie the metal stripes (see the bottom picture of Fig. 13). friction properties along the metal stripes, are reasonable for the specific implementation an the extreme testing conitions. Lower peak-to-peak oscillatory responses were obtaine for higher tire-roa friction levels uring the experimental testing session. Fig. 15. Experimental tests: comparison of vehicle spee (V vhl ) an front left angular wheel spee multiplie by the wheel raius (V FL ) for the vehicle with the enmpc 4 an the passive vehicle (TC off). Fig. 16. Experimental tests: comparison of motor torque eman before (T CA ) an after the front left TC block for the vehicle with the enmpc 4. Fig. 14. Experimental tests: comparison of actual an reference slip ratios for the vehicle with the enmpc 4 an the passive vehicle (TC off). Fig. 16 shows the electric motor torque regulation, with respect to the torque eman from the river. The reuce torque settles on a value of ~50 Nm, compare to the river eman of 115 Nm, resulting in a 56% torque reuction. The torque oscillations, also cause by the non-perfectly constant tire VII. CONCLUSION The paper presente traction controllers for electric vehicles with in-wheel motors, base on explicit non-linear moel preictive control of the wheel slip velocity. These were compare with more conventional TC strategies base on PI control. The novel conclusions are: The implementation time step of the TC has a more significant impact on the control system performance than the selection of the control system technology. Employing non-linear MPC is not enough to provie better performance than that of a PI running at an appropriate time step. To achieve a performance enhancement, for the case stuy TC application, time steps of ~2 ms are recommene, rather than of 4 ms or 8 ms. Both for the PI TC an non-linear moel preictive control TC, the control system parameters

13 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 13 have to be fine-tune through tests in the time omain for the selecte time step. The presente explicit non-linear moel preictive control implementations are characterize by on-line computational times in the range of 5-25 μs on the aopte SPACE MicroAutoBox rapi control prototyping unit. This means that the strategies coul be potentially implemente at any reasonable frequency for automotive TC applications. On the contrary, base on the literature it woul not be possible to run an equivalent implicit non-linear moel preictive controller at the require time step of 2 ms. The non-linear moel preictive controller allows a 9.2% tracking performance improvement with respect to a PI controller uring the variable tire-roa friction scenario, simulate with a high fielity vehicle moel. The local multi-parametric quaratic approximation of the non-linear problem, typical of the selecte explicit nonlinear moel preictive control metho, oes not bring any perceivable performance ifference with respect to the corresponing implicit non-linear moel preictive controller. The consieration of tire force ynamics an vertical loa transfers in the internal moel for moel preictive control system esign has negligible effects on the TC performance uring the simulate scenario. The interpretation of the non-linear moel preictive control law provies useful information on the effect of the ifferent input parameters on the control action. The piecewise affine control law can be approximate with only three planes. An explicit non-linear moel preictive control strategy for TC has been successfully implemente on a fully electric prototype vehicle for the first time in the literature, to the best of the authors knowlege. Future evelopments of the research will evaluate: i) the increase of the number of parameters of the explicit non-linear moel preictive control problem, an the implications in terms of memory requirements an performance benefits; an ii) the possibility of simpler strategies able to replicate a similar control pattern with reuce memory requirements for the online implementation of the controller. REFERENCES [1] H. Fujimoto, J. Amaa, an K. Maea, Review of traction an braking control for electric vehicle, in 2012 IEEE Vehicle Power an Propulsion Conference, 2012, pp [2] S. Murata, Innovation by in-wheel-motor rive unit, Veh. Syst. Dyn., vol. 50, no. 6, pp , [3] V. Ivanov, D. Savitski, K. Augsburg, an P. Barber, Electric Vehicles with Iniviually Controlle On-boar Motors: Revisiting the ABS Design, IEEE International Conference on Mechatronics (ICM), [4] M. Amoeo, A. Ferrara, R. Terzaghi, an C. Vecchio, Wheel Slip Control via Secon-Orer Sliing-Moe Generation, IEEE Trans. on Intell. Transp. 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14 TCST IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 14 Assessment, IEEE Trans. Veh. Technol., vol. 64, no. 5, pp , [33] H.B. Pacejka, Tire an Vehicle Dynamics, Butterworth Heinemann, [34] T. Geyer, F.D. Torrisi, an M. Morari, Optimal complexity reuction of polyheral piecewise affine systems, Automatica, vol. 44, pp , [35] M.S. Santina an A.R. Stubberu, Basics of sampling an quantization. Hanbook of networke an embee control systems, Birkhäuser Boston, [36] J.H. Park an C.Y. Kim, Wheel Slip Control in Traction Control System for Vehicle Stability, Veh. Syst. Dyn., vol. 31, no. 4, pp , [37] R.W. Allen, T.J. Rosenthal, an J.P. Chrstos, A vehicle ynamics tire moel for both pavement an off-roa conitions, SAE Technical Paper, no , Davie Tavernini receive the M.Sc. egree in mechanical engineering an Ph.D. in ynamics an esign of mechanical systems from the University of Paova (Italy) in 2010 an 2014, respectively. During his Ph.D. he was part of the motorcycle ynamics research group. He is a Lecturer in avance vehicle engineering with the University of Surrey, Guilfor, U.K. His research interests inclue vehicle ynamics moelling an control, mostly applie to electric an hybri vehicles. Mathias Metzler (GSM 17) receive the Dipl.- Ing. egree in mechanical engineering from the Vienna University of Technology, Vienna, Austria, in 2015 (summa cum laue). He is currently pursuing the Ph.D. egree in avance vehicle engineering at the University of Surrey, Guilfor, U.K. His research interests inclue vehicle ynamics control, moel preictive control, optimization, an nonlinear systems. Patrick Gruber receive the M.Sc. egree in motorsport engineering an management from Cranfiel University, Cranfiel, U.K., in 2005, an the Ph.D. egree in mechanical engineering from the University of Surrey, Guilfor, U.K., in He is a Senior Lecturer of Avance Vehicle Systems Engineering with the University of Surrey. His research interests inclue vehicle ynamics an tire ynamics with special focus on friction behavior. Alo Sorniotti (M 12) receive the M.Sc. egree in mechanical engineering an Ph.D. egree in applie mechanics from the Politecnico i Torino, Turin, Italy, in 2001 an 2005, respectively. He is a Professor in avance vehicle engineering with the University of Surrey, Guilfor, U.K., where he coorinates the Centre for Automotive Engineering. His research interests inclue vehicle ynamics control an transmission systems for electric an hybri vehicles.