rotational motion into the linear motion that turns the front wheels.
The torque feedback is measured by a sensor mounted between the steering motor and the pinion gear. It corresponds to the signal y3 =τa3 in Fig. 5.2.
Propelling on the NeCS-Car is performed by a DC motor connected to the rear wheels.
The mechanical assembly of the NeCS-Car is shown in Fig. 5.3.
Figure 5.3 – Architecture of the NeCS-Car
5.3 Implementation of the state-space model on the
Chapitre 5. Experimental validation NeCS-Car benchmark Hypothesis 5.3.1
let J and B to be defined as the difference of the values adopted in simulation (respectively, Jv and Bv) and values of the setup system (respectively, J0 and B0)
Jv = J+J0 (5.3.1)
Bv = B+B0 . (5.3.2)
Let us substitute Jv andBv, as resulting from Eqs. (5.3.1) and (5.3.2), in Eq. (2.3.1), as follows :
(J+J0)¨θv + (B+B0) ˙θv =τv−k(θv−θs) (5.3.3) and let us apply the Laplace transform to obtain the value of θv, as function of the driver’s torqueτv and the motor torque Cm, coming from the PC-unit
(J+J0)s2θv+ (B+B0)sθv =τv−k(θv−θs) (5.3.4) Now, let us divide the real inertia and viscosity from residual ones
(J0s2+B0s)θv =τv−k(θv−θs)−(J s2+Bs)θv (5.3.5) It follows that
Cm =−k(θv−θs)−(J s2+Bs)θv (5.3.6) is the simulated torque given to the motor and
θv = 1
(J0s2+B0s)(τv+Cm) (5.3.7) is the real steering wheel angle, obtained from the steering wheel and the column.
To obtain the relation between the steering wheel positionθv and the column-shaft angle θs, let us apply the Laplace transform on Eq. (2.3.2), as follows
JTs2θs+N22Bmsθs =k(θv −θs) + τa
N1 +N2u (5.3.8)
By collecting all the terms in θs, it is possible to obtain
θs= 1
JTs2+N22Bms+k
kθv + τa
N1 +N2u
(5.3.9) Eqs. (5.3.7) and (5.3.9) allow to adapt the state-space representation of the EPS model to the HIL experimental setup. For the same input, these equations produce the same output of the state-space model in Eqs. (2.3.1) and (2.3.2).
5.3.1 Identification of residual viscosity and inertia on the HIL setup
Residual parameters of the inertia and viscosity of the control station B0 and J0 needs to be identified to be able to test the general architecture on the experimental setup.
The key factor of obtaining a good result of identification is selecting the right input signals in order to excite the system over all the frequency range of interest. At this aim, two identification methods are used. They differ each other for the input signal that is used for the identification.
In the first method, a step signal is used as input of the system. Parameters of the system can be easily obtained, starting from the analysis of the transfer function of the system, which behaves as a second order system.
The second method uses one of the signals, which is traditionally used for non-parametric model identification : i.e. the Pseudo-Random Binary Sequence (PRBS) signal. The PRBS signal is a deterministic signal with the property that its auto-correlation function corresponds to the auto-correlation function of the white noise signal ([91], [80], [81]).
5.3.1.1 Step response of the system
The steering model can be described in open loop as a first-order transfer function in Laplace domain
GOL(s) = θ˙v
τv = 1
J0s+B0 (5.3.10)
The corresponding closed loop system with a proportional gaink1 is GCL(s) = θv
τv = k1
J0s2+B0s+k1 (5.3.11) The experimental procedure to obtain values for J0 andB0 with the step input sequence is the following :
– the step input sequence of τmot1, output of the pc-unit, is applied on the electrical motor of the control station in closed loop.
– As consequence of this input, the steering wheel starts to oscillate as a closed-loop system. This signalθv is saved to use it into the identification algorithm. This output is shown in Fig. 5.4. From the analysis of this output, it is possible to identifyJ0 and B0.
At this aim, let us consider the standard form for a transfer function of a second order system
G(s) = 1
1
ω2ns2+2ξω0
ns+ 1 (5.3.12)
where ωn is the natural frequency of the system and ξ0 is the damping coefficient.
Chapitre 5. Experimental validation NeCS-Car benchmark
16 17 18 19 20 21 22
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
time (s) θ v (rad)
Op
Figure 5.4 –Output of the system to the step input.
By inspection of the output of the system in Fig. 5.4, it is possible to obtain the overshoot value of the system Op = 0.854. The equation relating Op with ξ0 is the following
Op = exp
−√πξ0
1−ξ2
0 , (5.3.13)
so the corresponding value forξ0 = 0.05.
The equation relating ωn and ξ0 is
ωn =ω0
q
1−ξ20 (5.3.14)
By comparing Eq. (5.3.11) with Eq. (5.3.12), it is possible to obtain the corresponding values for J0 and B0
J0 k1
= 1
ω2n ⇒J0 = k1
ω2n (5.3.15)
B0
k1
= 2ξ0
ωn ⇒B0 = 2ξ0k1
p1−ξ02 ωn
(5.3.16) By substituting numerical values (k1 = 5 and ωn = 12.56 rad s−1, it is possible to obtain resulting values for the inertia and the viscosity are J0 = 0.0303 kgm2 and B0 = 0.039 N m rad−1s, respectively.
5.3.1.2 Identification via PRBS input sequence
The experimental procedure to obtain values forJ0 andB0with the PRBS input sequence is the following :
– the PRBS input sequence of τmot1, output of the pc-unit, is applied on the electrical motor of the control station in open loop. This sequence is shown in Fig. 5.5.
– As consequence of this input, the steering wheel starts to oscillate with a variable angular speed. This signal ˙θv is saved to use it into the identification algorithm. This output is shown in Fig. 5.6.
0 20 40 60 80 100 120
−1.5
−1
−0.5 0 0.5 1 1.5
time (s)
Input sequence PRBS (Nm)
Figure 5.5 –Sequence PRBS that is given in input for the parametric identification.
0 20 40 60 80 100 120
−8
−6
−4
−2 0 2 4 6 8
time (s)
Angular speed of the motor (rad/s)
Figure 5.6 –Output of the system to the proposed input. This signal is then used for the least squares identification.
Chapitre 5. Experimental validation NeCS-Car benchmark
We make the assumption that the model behaves as a first order transfer function in closed loop with a proportional gain k1
G0(s) = θ˙v
τmot1 = k2
1 +Tps = 1
B0+J0s (5.3.17)
Under this assumption, the parameters to identify correspond to : J0 = Tp
k2 B0 = k1
2 (5.3.18)
The solution of this problem is obtained by applying a non-linear least squares algorithm ([56], [11]), implemented with the software Matlab 2011R. The resulting values for the inertia and the viscosity are J0 = 0.0325 kgm2 andB0 = 0.072 N m rad−1s, respectively.
As concluding remark, it is possible to say that both identification methods produce very similar values. It is possible to conclude that identified values are appropriate. For the following, values obtained with the PRBS method are used.