Oscillation Annealing and Driver/Tire Load Torque Estimation in Electric Power Steering Systems Javier Tordesillas, Valentina Ciarla, Carlos Canudas De Wit

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Oscillation annealing and driver/tire load torque estimation in Electric Power Steering Systems Javier Tordesillas, Valentina Ciarla, Carlos Canudas de Wit

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Javier Tordesillas, Valentina Ciarla, Carlos Canudas de Wit. Oscillation annealing and driver/tire load torque estimation in Electric Power Steering Systems. MSC 2011 - IEEE Multi-Conference on Systems and Control, Sep 2011, Denver, United States. pp.s/n. hal-00642035

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Oscillation annealing and driver/tire load torque estimation in Electric Power Steering Systems

J. Tordesillas Ill´an, V. Ciarla, and C. Canudas de Wit

Abstract— The paper presents several aspects of modeling, wheel. In that way the low frequency feelings of the driver observation and control towards a new generation of Electrical will not be affected. The observer in this first version was Power Steering(EPS) systems. In particular we design an derived assuming that the exogenous inputs (driver and tire- optimal control to reject oscillations of the steering column, then we device a new observer to estimate the internal state friction torques) were known. variables of the steering column, the driver applied torque b) Improved LuGre Tire-friction model: The dynamic (steering wheel torque), and the load torque (tire/ground contact LuGre Friction model proposed in [5], [6] is here modified friction). Finally, we also revisited the LuGre tire dynamic friction model by improving the transient behavior between to trade the transition between the stick friction component the sticking phases and the dynamic ones. Simulation of the (that dominated when vehicle is stand still), and the self- proposed control and observer are shown at the end of the alignment torque (produced when vehicle in motion). To paper using the improved LuGre-tire friction model. have a complete tire-friction model is mandatory to test EPS Index Terms— Electric Power Steering systems (EPSs), LQ systems. control, LuGre friction model, observer. c) Driver and load torque estimation: The observer I.INTRODUCTION previously designed was extended to the case where the Nowadays, the existing Electric Power Steering systems exogenous inputs are unknown. This was done assuming the (EPSs) are developed basing on a general driver profile [1], standard hypothesis for observation under unknown input, [2]. Future generation of EPSs should be able to propose a namely a slow rate of variations of these inputs. In spite torque assistance which may be adapted for different kinds of this hypothesis, our results show that these exogenous of driver population: young drivers, aged people, disabled torques where correctly estimated with a very good precision people, etc. and very little phase lag. A first important step towards the above goals consists in setting a control framework that includes a realistic model 2 φ˙ of a steering column accounting for all other torque loads Tyre/Road involved in a real driving situation (torsion torque due to Friction Model β force sensors flexibility, applied driver forces, and tire-road V contact friction forces). The control framework should also include observers that allow to recover signals which are not τa τv 3 y sensed and a control law that compensates for the column Mechanical flexibilities. This is completed with a booster-torque-law Reference Model u u (power steering torque) specific to the driver population Model ass in question. The general architecture that we propose is shown in Fig. 1. This paper concerns with the block 1 xˆ (observer/control), and the block 2 (tire-friction model) of −K Observer 1 Fig. 1, with the additional contribution that the observer also estimate the driver and load torque. The design of the specific Fig. 1. General architecture of the EPSs: block 1 concerns the observer and reference model (block 3) for non-standard drivers is under the control; block 2 includes the tyre-friction model and block 3 includes current investigation. the reference model. The contributions of this paper are: a) Optimal output control feedback: Based on the steer- The paper is organized following the 3 main items as ing column model proposed by [3], [4], we redesign a linear indicated above. optimal control that seeks to cancel oscillation due to column stiffness. This results in an output optimal feedback with an observer included. In addition to cancel oscillation the control design seeks to preserve the open-loop gain between II. OSCILLATIONS ANNEALING the applied driver torque and the motion of the steering To begin with, it is essential to define the mechanical C. Canudas-de-Wit and V. Ciarla are with the Laboratoire d’Automatique de Grenoble, UMR CNRS, Grenoble 5528, France (e-mails: carlos.canudas- system that we will take into account. Fig.2 shows an [email protected], [email protected]) explanatory schema of the system. TABLE I where T = − Bv 0 − k . Fig. 3 shows the frequency CONSTANT PARAMETERS OF THE EPSS Jv Jv response and puts in evidence that the open loop system has Symbol Value Description a significant peak for a frequency of about 12 Hz that cause 2 Jv steering wheel inertia 0.025 [kg · m ] 2 Jm motor inertia 0.00033 [kg · m ] substantial oscillations on the steering wheel and that should Jc steering motor inertia neglected Jw aggregated wheel and rack inertia neglected be avoided so as to improve the driving comfort. k steering column stiffness 100 [N · m] N1 vehicle steering angle to steering colon ratio 13.67 N2 motor-steering column gear ratio 17 B. Full-state optimal feedback control Bv damping coeff. associated to the steering wheel 0.01 [N · s/m] Bm damping coeff. associated to the motor axis 0.003 [N · s/m] To compensate the oscillations an optimal LQR controller is designed. This controller is computed so that the state- τa feedback law u = −Kx (with K the state-feedback gain) = τ θs θm/N2 minimizes following cost function: v θv ∞ T 2 Jv 1 J = (x Qx + ru ) dt (5) k N 0 N2 Z Jm u where the constant matrices Q > 0 and r > 0 are θm the weighting matrices. For the purpose of annealing the Bv Bm oscillation without impact the low frequency gains of the resulting closed loop, we can select Q such that only the Fig. 2. Mechanical model of the EPSs torsion angle and its time derivative is penalized, i.e. ∞ 2 2 2 J = (q1(x1 − x2) + q2x3 + ru ) dt 0 A. Column model Z The mechanical equations governing the system explained Normalizing this cost with r =1, the problem is simplified above are (see [3]) by selecting Q as q1 −q1 0 Jvθ¨v = τv − k(θv − θs) − Bvθ˙v (1) Q = −q1 q1 0 (6) ¨ 2 ˙ τa   JT θs = −k(θs − θv) − N2 Bmθs + + N2u (2) 0 0 q2 N1 This results in a cost function with only two parameter to 2 Jw with JT = Jc + N2 Jm + 2 . The constants of the N1 be tuned. The closed-loop transfer function is given by the model are defined in Table I, while θv, θs and θm are, following equation: respectively, the steering wheel, the motor-shaft and the ¨ motor angles. Both Jc and Jw are neglected for simplicity, θv −1 Gcl = = T (sI − (A − BK)) G1 (7) but to obtain more precise results it is advisable to introduce τv these parameters in the calculations. Let with A − BK the closed-loop state space matrix. T T T x = x1, x2, x3 = θ˙v, θ˙s,θv − θs (3) Bode Diagram The model can be formulated into the state-space form 80

60 x˙ = Ax + Bu + Gw 40 Open Loop Peak T with w = τv, τa and 20 at 12 Hz

− Bv 0 − k 0 Jv Jv 0 2 N2 Bm k N2 Magnitude (dB) −20 A = 0 − − ; B = J ;  JT JT   T  1 −1 0 0 −40  1    −60  J 0  v 1

G = 0 = g1 g2 ; −80  N1JT  0 0 −100 −1 0 1 2 3 4 5 10 10 10 10 10 10 10   (4) Frequency (Hz) It is interesting to observe how the model responds to a variation of the torque exerted by the driver on the steering Fig. 3. Open loop (blue) vs closed loop frequency response. Weighting q1 = 3 q2 = 12 r = 1 wheel. To do this, it seems appropriate to compute the parameters are: and , . transfer function of the steering wheel’s angular acceleration The beneﬁts of this design can be observed in Fig. 3. θ¨ to the driver’s exerted torque τ . Such a transfer function v v The resonance peak that might cause undesirable oscillations is easily computed from the state-space matrices as follows: have been eliminated thanks to the proposed optimal linear ¨ θv −1 control. Note also that the low-frequency gain has kept Gol = = T (sI − A) G1 τv unchanged. Profile of driver’s torque input C. Optimal ”output” feedback controller 1.5

1 (Nm)

All the calculations carried out until now assumed that v 0.5 the whole set of state-space variables was measurable. In τ 0 commercial products only the motor angle and the torsion −0.5 torque are measured. This is due to the fact that installation −1 of these additional sensors implies an important extra cost Driver’s torque −1.5 0 5 10 15 20 25 30 Time (s) that should be, if possible, avoided. (a) The previous control problem can be reformulated by Steering wheel angular velocity in open loop assuming only that the motor speed (approximated from 2 motor position sensor) is available. This means that we 1.5 dispose of the output 1 0.5 ˙ 0 y = Cx = (0, 1, 0)x = θs −0.5

−1

Assuming as before (this hypothesis will be relaxed latter) −1.5 SW angular velocity (m/s) −2 0 5 10 15 20 25 30 that w is measured as well, then the proposed observer has Time (s) the following form (b)

Steering wheel angular velocity in closed loop xˆ˙ = (A − LC)ˆx + Bu + Gw + Ly (8) 4

3 where L is the observer gain to be designed. By the sepa- 2 ration principle, the control K is kept as before, and L can 1 0 be designed either via pole placement method, or via the −1 loop-recovering strategy. In any case, it is suited that the −2 −3

˙ SW angular velocity (m/s) dynamics of the observation error xˆ − x˙ =e ˙ = (A − LC)e −4 0 5 10 15 20 25 30 Time (s) has a fast dynamics and hence a fast convergence of the (c) observer towards the real states. Estimated steering wheel angular velocity 4

D. Simulation results 3 In order to see the performances of the proposed controller, 2 1 following simulations are carried out using as input to the 0 system the driver’s exerted torque profile shown in Fig. 4-a. −1 1) Open loop behavior: The first simulation concerns the −2 −3 SW angular velocity (m/s) −4 system in open loop. As shown at the Fig. 4-b, once the 0 5 10 15 20 25 30 Time (s) steering wheel is released by the driver at the time instant (d) 16s, the steering wheel suffers significant oscillations that Fig. 4. (a) Real driver’s steering torque used for the simulation. (b) Profile would cause an undesirable driving feeling and might even of the steering wheel speed in open loop. (c) Profile of the steering wheel be dangerous when we are in driving situations at fast speeds. speed in closed loop. (d) Estimation of the steering wheel speed. 2) Closed-loop behavior under closed-loop: The optimal linear controller computed before has the aim to eliminate the oscillations found in open loop thanks to the assistance vehicle dynamics are coupled. However to avoid unnecessary motor. As shown at the Fig. 4-c, the oscillations derived complexity (implementing a complete vehicle model it is not from the release of the steering wheel have been satisfactorily essential for the development of our simulations), the model compensated by the optimal controller. The behavior of the chosen is a simplified model of the tires assuming that the observer is shown by Fig. 4-d. From this figure we see that normal force on them is equal to a quarter of the total weight the performance of the observer is completely satisfactory. of the vehicle. III. TIRE/ROADCONTACTFRICTIONTORQUE Fig. 5 illustrates the longitudinal and the lateral velocity The model used until now takes into account all the that might appear when driving under the effect of centripetal elements that go from the steering wheel to the steering forces. This lateral force appearing during the turning ma- shaft. Load torques exerted by tire/road contact need to be noeuvres, is the origin of the slipping, in the direction of the consider explicitly in the model. The contact friction force so-called slipping angle β. is an important element of the driving it is present in all the So as to prove the validity of the observer, it is imple- different driving situations that a driver may find, and it is mented in simulation. In the bottom of fig. 4 it can be the main responsible of the ultimate feeling that the driver stated that the performance of the observer is completely will feel during his driving maneuvers. satisfactory. To sum up, an observer has been calculated In general, models for contact forces need to be completed such that the whole state-space has been rebuilt from both with the mechanical model of the vehicle dynamics as the the control input and the measurable output. Once all the states/variables between the contact forces and those of the different states have been estimated, the full state feedback T carried out in section II-B is implemented obtaining very where vr = [ vrx vry ] is the vector of the relative satisfactory results. speeds:

vrx = ωr − vcos(β) (11) v y vc vry = −vsin(β) (12) with ω that corresponds to the angular velocity of the wheel β of radius r, while v is the velocity of the vehicle and β is vx the slip angle. The scalar function C0i(vr) (with i = x, y) is peculiar of the LuGre model and is given by: Fig. 5. Forces action on the tire when turning. Slip angle β show the direction of the vehicle resulting velocity λ(vr)σ0i C0i(vr)= 2 (13) µki

The angle β, is known to depend on the vehicle’s speed, with 2 ||Mk vr|| v as well as on the radius of curvature, ρ given by the curve λ(vr)= (14) traced by the vehicle in a given moment, i.e. β(t)= f(v,ρ). g(vr) The radius of curvature can be considered as being inversely In Eq. 14 we recognize the the matrix of the kinetic friction proportional to the steering wheel angle: an infinite radius µkx 0 coefficients Mk = > 0 for the motion along of curvature means the vehicle is going straight and hence 0 µky the steering wheel angle is zero. For the reasons advocated the x and the y directions, respectively; note also that each before, in our simulations a simple relation between β and parameter µki in Eq. (13) corresponds to one element on the steering wheel angle weighted by the vehicle velocity is the main diagonal of matrix Mk. We find also the function considered. g (vr), given by: 2 2 2 γ Mk vr Ms vr Mk vr −( kvrk ) g (vr)= + − e vs A. Dynamic tyre friction model kMkvrk kMsvrk kMkvrk ! (15) We introduce here some modifications to the dynamic µ 0 with M = sx > 0 that is the matrix of static LuGre Friction model proposed in [5], [6]. These modifi- s 0 µ sy cations aim at improving the transition between the sticking friction coefficients. ss ss friction component (that dominates when vehicle is stand To evaluate the constants κi and ν , we do the hypothesis still), and the self-alignment torque (produced when vehicle that the steady-state solution of the lumped model is the is in motion). The most important phenomena (components) same with the steady-state solution of the distributed one. captured by the model are: The complete expression of these parameters is given from: 1 v • Sticking torque Msticking : it is the torque that opposes ss ri κ = − C0i(vr) (16) the movement of the tyres when turning them around the i |ωr| z¯ss i vertical axis. It is especially important for low speeds ss ss 1 1 Gvry |ωr|z¯y 0 (17) of the vehicle. ν = ss + − C y |ωr| zˆy FnL L • Self-alignment torque Mself−alig : it is the torque that ss ss appears as the vehicle speed up, and tends to rotate the with z¯i and zˆy , that are the steady-states of the system; wheels around its vertical axis in order to make them the explicit expression for these parameters is given in the return to the straight position. appendix of [5]. The auto-aligning torque can be written in terms of the mean The dynamic LuGre Friction model proposed in [5], [6] states z¯ (t) and zˆ (t) as follows: derives from a distributed friction model. It is described y y by four differential equations, which capture the average 1 M = F L σ0 z¯ (t) − zˆ (t) behavior of the internal friction states. To introduce this self−alig n y 2 y y model, let z¯i(t) (i = x, y), zˆy(t) denote the internal states h1 + σ1 z¯˙ (t) − zˆ˙ (t) (18) of the system: y 2 y y ss z¯˙i(t)= vri − C0i(vr)¯zi(t) − κi |ωr|z¯i(t) 1 G (9) + σ2y vry − (i = x, y) 2 FnL i

G To calculate the sticking torque we introduce the last zˆ˙y(t)= vry − C0y(vr)ˆzy(t)− FnL equation (10) ˙ ss |ωr| σ0z|φ| − ν |ωr|zˆy(t)+ z¯y(t) z˙z = φ˙ − zz(t) (19) L gz(φ˙) TABLE II of this speed range is due to the fact that an EPSs operate CONSTANT PARAMETERS OF THE LUGRE MODEL in this range. Symbol Value Description r wheel radius 0.38 [m] Sticking torque at different speeds v vehicle speed range from 0 to 30 [km/h] 30 ω angular velocity of the wheel ω = v [rad/s] V = 0.5 km/h r µ kinetic friction coeff. x-axis 0.75 V = 5 km/h kx 20 V = 10 km/h µky kinetic friction coeff. y-axis 0.75 V = 15 km/h µ kinetic friction coeff. z-axis 0.76 V = 20 km/h kz 10 µsx static friction coeff. x-axis 1.35 V = 25 km/h µ static friction coeff. y-axis 1.40 V = 30 km/h sy 0 µsz static friction coeff. z-axis 0.91 vs Stribeck relative velocity 3.96 [m/s] −10 φ˙s Stribeck relative velocity 74 [rad/s] γ steady state constant 1 σ normalized rubber stiffness 6000 [N/m] −20

0y Sticking torque (Nm) σ1y normalized rubber damping 0.3568 [N/m/s] −30 σ2y normalized viscous relative damping 0.0001 [N/m/s] 0 5 10 15 20 25 30 L patch length 0.15 [m] Time (s) ζL left patch length 0.0030 [m] (a) ζR right patch length 0.1155 [m] Fmax max value of normal load distribution 1900 [N] Fn normal value of normal load distribution 249.37 [N] Self−Alignment torque at different speeds 15 α1 coeff. for Self-Align. torque 63000 N/m V = 0.5 km/h α2 coeff. for Self-Align. torque −55000 N/m V = 5 km/h β2 coeff. for Self-Align. torque 8260 N 10 V = 10 km/h σ0z coeff. for Stick. torque 20 V = 15 km/h V = 20 km/h σ coeff. for Stick. torque 0.0023 5 1z V = 25 km/h σ2z coeff. for Stick. torque 0.0001 2 V = 30 km/h G load distribution function 16.83 Nm 0 κss function used to approx. steady behavior 11.9 ssi ν function used to approx. steady behavior −0.8 −5 SA torque (Nm)

−10

−15 0 5 10 15 20 25 30 where the angular velocity of the wheel rim is φ˙ = θ˙s/N1, Time (s) and and the function (b)

φ˙ 2 Fig. 6. (a) Sticking torque for different driving speeds (b) Self-Alignment − ˙ gz φ˙ = µkz + (µsz − µkz) e φs (20) torque for different driving speeds where µkz and µsz are, respectively, the kinetic and static friction coefﬁcients across the z-axis, while φ˙s is the Stribeck The results obtained from those simulations are shown velocity. by Fig. 6. Fig. 6(a) shows the different curves obtained The sticking torque can be evaluated as follows: for the sticking torque for different driving speeds, while Fig. 6(b) shows the different curves obtained for the self- M = −LF (σ0 z (t)+ σ1 z˙ (t)+ σ2 φ˙) (21) sticking n z z z z z alignment torque. Results are as expected: the sticking torque Hence, the total torque generated by the contact of the tyres decreases exponentially as the velocity increases, so as the and the road is: contribution of the self-alignment is more important as the speed increases. τa = Mself−alig + Msticking (22) The parameters used in this model are reported in Table II. IV. EXOGENOUS TORQUES ESTIMATION

B. Friction model improvements The purpose of this section is to develop an estimator that includes estimation of the exogenous torques that appear One possible improvement of the previous model con- in the system. Both τ and τ are not sensed a priori. cerns the the dependence of the sticking torque to the v a By constructing an observer for the ”extended” systems vehicle velocity, v. In fact, it can be proved experimentally the implementation of the previous controller can be done that, as the speed of the vehicle grows, the self-alignment under the relaxing hypothesis on the known and sensed torque dominates over the sticking torque, and inversely. information. Another interesting issues, which is a side The previous model in its actual form, does not respect this effect, is the observer not only provide the control states observation. It is then necessary to weight the sticking torque estimates but also provide an estimate of the driver delivered as a function the velocity of the vehicle. Thus, a possible torque, and the contact tire/road friction forces. modiﬁcation along these observations is

−|v|/vk Msticking = −LFn(σ0zzz + σ1zz˙z + σ2zφ˙)e (23) A. Extended state-space representation where vk is a positive constant. Let consider that both exogenous torques are slowly time- varying: τ˙a =τ ˙v ≈ 0. The extended state-space representa- C. Dependency on the vehicle’s speed tion with z now deﬁned as, It is also important to check if the results follow the T z˜ = θ˙ , θ˙ ,θ − θ , τ , τ (24) expected logic when we vary the vehicle’s speed. In order v s v s v a to do this, several simulations were carried out for velocities in which both exogenous torques are added as state variables, going from 0 to 30 km/h and over a range time of several is give by seconds, in order to see the effects at steady-state. The choice z˜˙ = A˜ez˜ + B˜eu (25) 1.5 The state matrices for this system are shown below: Real driver’s torque Estimated driver’s torque 1 B ˜ A G ˜ 0.5 Ae = O O ; Be = 0 ; 2×3 2×2   (26) 0 0

−0.5 Ce = C 0 0  

−1 If only the motor velocity is used as a available output, then Driver’s torque (Nm) −1.5 0 5 10 15 20 25 30 the observability matrix for this new extended system has Time (s) rank 4 in spite of dealing with a system of dimension 5. (a)

50 Real load torque B. Using the torsion force as a additional output 40 Estimated load torque

30 Although it has been so far considered that the only mea- 20 10 surable variable was the angular velocity of the assistance 0 motor, it is as well possible to measure a second signal: the −10 −20

−30 torsion force, Load torque (Nm)

−40

−50 2 3 0 5 10 15 20 25 30 y = k(θv − θs)= kx . Time (s) (b) In this case, the output matrix Ce of the extended system becomes, Fig. 7. (a) Real driver’s torque (solid) vs. estimated driver’s torque (dotted). 01000 (b) Real load torque (solid) vs. estimated load torque (dotted). C˜ = . e 0 0 k 0 0 Computing the new resulting observability matrix, it can be VI. ACKNOWLEDGEMENTS easily check that it has now full rank. The system is then observable and it is concluded that both exogenous torques This work was partially funded by the project VOLHAND can be correctly estimated. The method used to design the ANR-09-VTT-14. observer’s gain is analog to the one used in section II-B, REFERENCES i.e. the poles of the observer’s dynamic (matrix ˜ ˜ ) Ae − LeCe [1] Parmar, M.; Hung, J.Y.; , ”A sensorless optimal control system should be fast enough compared to those resulting from the for an automotive electric power assist steering system,” Industrial linear controller. Electronics, IEEE Transactions on , vol.51, no.2, pp. 290- 298, April The performances of the observer have been tested in simu- 2004 [2] Morita, Y.; Torii, K.; Tsuchida, N.; Iwasaki, M.; Ukai, H.; Matsui, N.; lations. The results obtained for the estimation of τv and τa Hayashi, T.; Ido, N.; Ishikawa, H.; , ”Improvement of steering feel are shown in Fig. 7(a) and Fig. 7(b). The maximum value of Electric Power Steering system with Variable Gear Transmission reached by the observation error can be reduced at the price System using decoupling control,” Advanced Motion Control, 2008. AMC ’08. 10th IEEE International Workshop on , vol., no., pp.417- of increasing the observer gain. Measurement noise will limit 422, 26-28 March 2008 at certain point the maximum possible value of the observer [3] C. Canudas de Wit, S. Guégan, and A. Richard ”Control design gain. for an electro power steering system: part I The reference model”, ECC’2001, European Control Conference, Porto(Portugal), septembre 2001 (invited) V. CONCLUSIONS [4] C. Canudas de Wit, S. Guégan, and A. Richard ”Control design for an electro power steering system: part II The control design”, ECC’2001, European Control Conference, Porto(Portugal), septembre This paper presented simulation results of a detailed model 2001 (invited) of an EPS system including a dynamic model that repro- [5] E. Velenis, P. Tsiotras, C. Canudas de Wit and M. Sorine ”Dynamic duces the physical phenomena involved in driving. From Tire Friction Models for Combined Longitudinal and Lateral Vehicle Motion”, Vehicle System Dynamics, Volume 43, Issue 1 January 2005 an adequate mechanical model of the steering column, the [6] C. Canudas de Wit, P. Tsiotras, E. Velenis, M. Basset, and G. natural oscillations that exist were eliminated by using a Gissinger, ”Dynamic friction models for road/tire longitudinal inter- suited optimal linear control design. Furthermore,a highly action”, Vehicle System Dynamics, vol. 39, no. 3, pp. 189 to 226, 2003 reliable modified dynamic friction model has been devised from previous author works. This model will allow to carry out trustworthy simulations that will be used to evaluate the different typologies of the potential drives for the new generation of EPSs. Finally, a satisfactory observer for the internal state variables needed for control, but also for the exogenous torque estimation due to the driver and tire/road friction contact was proposed. This observer was evaluated via simulation on the context of closed-loop showing good results and good estimation characteristics.