The matrix EM0 corresponds to the measured disturbances which are the Usup and psup, hence it is written as follows:
EM0 =
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0
. (2.71)
The performance output is generated from the measured output by the following simple equation:
zM0 = [ 0 0 0 0 0 1 0 0 ]yM0. (2.72)
follows:
xM0 =
0 0 0 0 0 0 0 0 0 0 0 0 6.82·10−4 105 0 0T
. (2.73)
The disturbance vector has been considered with its nominal values as follows:
dM0 =
24 9.5·105 293 105 293 T
. (2.74)
Using the given initial conditions above, from Eq. (2.22) which describes the current change of the SMVs, the time function of the currents can be derived, but the switching moments are shown only in the time graphs to focus attention on the details of the transient behavior (see Fig. 2.7).
It can be seen that the current goes towards to its steady state in exponential way after the solenoid terminal voltage becomes higher than zero. When the electro magnetic force, gener-ated by the solenoid current (see Eq. (2.37)), exceeds the sum of the return spring force (see Eq. (2.38)) and the force generated by the pressure difference between the input and output ports of the valve (see Eq. (2.39)) the armature of the SMVs starts to move. The armature movement causes induced voltage due to the mutual inductance, which decreases the current of the solenoid according to the L1dLdxdxdt iterm ofEq. (2.22). When the armature reaches its limita-tion and its velocity becomes zero, the current starts to increase towards the steady state in an exponentially way again. This means that the SMV is opened completely. The second exponen-tial current change is slower due to the change of the inductance of the solenoid (seeEq. (2.40) and Eq. (2.35)). The current level in the steady state, from Eq. (2.22), where di/dt = 0, is uterm/Ras expected.
When the solenoid terminal voltage is switched off the current starts to decrease immediately and this current change induces voltage due to the self inductance of the solenoid. This voltage
0.045 0.05 0.055 0.06 0.065 0.07
−50 0 50
uterm[V]
0.245 0.25 0.255 0.26
−50 0 50
uterm[V]
0.0450 0.05 0.055 0.06 0.065 0.07
1 2 3
i[A]
0.2450 0.25 0.255 0.26
1 2 3
i[A]
0.045 0.05 0.055 0.06 0.065 0.07
−0.5 0 0.5
v[m/s]
0.245 0.25 0.255 0.26
−0.5 0 0.5
v[m/s]
0.0450 0.05 0.055 0.06 0.065 0.07
0.5 1 1.5x 10−3
x[m]
time [s]
0.2450 0.25 0.255 0.26
0.5 1 1.5x 10−3
x[m]
time [s]
SMVsl SMV
bl
Figure 2.7: Transient of the valve signals during disengagement
is limited by the power stage of the SMV to protect its switching circuit. The SMV armature is returned by the return spring when the electro magnetic force decreases below a certain level and the SMV becomes closed again. The responses of the load SMVs are similar, since only some parameters are different from each other. The deviations originated from the different solenoid type, which causes different current dynamics and steady state.
The chamber pressure starts increasing when the load SMVs become open (seeFig. 2.8). The load SMVs are activated for constant 200ms intervals.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 2 4 6x 10−3
mch[kg]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 2 4 6x 105
pch[Pa]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.2 0.4
vpst[m/s]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.01 0.02
xpst[m]
time [s]
SMVsl SMVbl SMVsl+bl
Figure 2.8: Transient of the chamber/piston states during disengagement
First the solenoid of the small load SMV is excited only (red lines), then the big load SMV (green lines) and in the end both of them are activated (black lines) in order to see the dynamic responses of the disengagement with different flow cross sections. In all the three cases 100%
PWM duty cycle are applied. As expected the bigger flow cross section causes higher mass flow rate, more dynamic pressure build up and higher piston velocity, which causes shorter disengagement process. After the SMVs have been opened completely the sum of the valve mass flow rates (σ), which equal to the change of the gas mass in the chamber (see Eq. (2.2)), have constant values due to the constant supply pressure, temperature and the overcritical pressure ratio, i.e. sonic flow (seeEq. (2.28) and Tab. 2.3 for details). Moreover it can be seen that the pressure curves are not increased monotonically. On one hand this is the consequence of the piston dynamic depending on its inertia, on the other hand arising from the nonlinearity of the static characteristic of the clutch mechanism (see Fig. 2.6).
2.8.2 Engagement process verification
This process has been simulated also with constant disturbances, starting from the disengaged state of the clutch mechanism (currentless valves but filled up chamber) as opposed to the
previous case. The initial state vector is as:
xM0 =
0 0 0 0 0 0 0 0 0 0 0 0 43.93·10−4 4·105 0 15.43·10−3 T
. (2.75) The disturbance vector is considered to be the same as before.
The responses of the exhaust and load SMVs are similar to each other (seeFig. 2.9).
0.045 0.05 0.055 0.06 0.065 0.07
−50 0 50
uterm[V]
0.845 0.85 0.855 0.86
−50 0 50
uterm[V]
0.0450 0.05 0.055 0.06 0.065 0.07
1 2 3
i[A]
0.8450 0.85 0.855 0.86
1 2 3
i[A]
0.045 0.05 0.055 0.06 0.065 0.07
−0.5 0 0.5
v[m/s]
0.845 0.85 0.855 0.86
−0.5 0 0.5
v[m/s]
0.0450 0.05 0.055 0.06 0.065 0.07
0.5 1 1.5x 10−3
x[m]
time [s]
0.8450 0.85 0.855 0.86
0.5 1 1.5x 10−3
x[m]
time [s]
SMVse SMV
be
Figure 2.9: Transient of the valve signals during engagement
As expected, the chamber pressure starts to decrease when the exhaust SMVs become open (see Fig. 2.10). The exhaust SMVs are activated for constant 800ms intervals and for all the cases 100% PWM duty cycle has been applied. First the solenoid of the small exhaust SMV is excited (blue lines), then the big exhaust SMV (magenta lines) and in the end both of them are activated (black lines) in order to see the dynamic responses of the engagement with different flow cross sections similarly as before.
Although the flow cross sections of the load and exhaust valves are equal, the engagement processes show slower dynamic piston movement compared to the disengagement, caused by the available mass flow rate. This is influenced by the pressure ratio differences (see Eq. (2.28)) between the ambient-chamber and chamber-reservoir pressures. The pressure curves do not change in a monotonic way, similarly to the disengagement process, due to compression effects.
2.8.3 Validation
In order to validate the model, the model output vector has been compared to measurements on the real EPC system. The measurement set-up includes a measurement PC with data acquisition unit that is connected to the measurement sensors. The data acquisition unit is able to measure analogue-to-digital converted (ADC) channels and has digital I/O channels that can be used as
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0
2 4 6x 10−3
mch[kg]
0 0.2 0.4 0.6 0.8 1 1.2 1.4
1 2 3 4 5x 105
pch[Pa]
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.4
−0.2 0
vpst[m/s]
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 0.01 0.02
xpst[m]
time [s]
SMVse SMV
be SMV
se+be
Figure 2.10: Transient of the chamber/piston states during engagement
control outputs to switching the power stages of the SMVs properly. The resolution of the ADC channels is 14 bits and the sampling frequency can be set up to 20kHz.
The target is to control the real system by its input vector to execute predefined test sequences and to measure all the characterizing signals which are available. Hence, all the members of the input and the output vector are measured on the test bench, which introduces ten signals to the measurement system that are as follows: uterm,sl,uterm,bl, uterm,se,uterm,be,isl,ibl,ise,ibe, pch and xpst. Note that, instead of the real inputs (md) the terminal voltage of the SMVs are measured, since they represent not only the presence or absence of the supply voltage between the terminals of the SMVs but also show the brake down voltage (UBR) in the avalanche state.
To predict the behavior of the real system the disturbance variables should be known as well.
From the five signals of the disturbance vector three can be considered as quasi-constant signals (Tsup, pamb and Tabm) that changes slowly, so these values are assumed to be constants (293K, 105P a and 293K). The remained (usup and psup) can change dynamically, so they should be acquired as well. The accuracy and the measurement range, according to the specification of the sensor suppliers, are seen in Tab. 2.7.
Table 2.7: Accuracy and range of measured signals Signal u [V] i[A] p [Pa] x [m]
minima -60 0 0 0
maxima 40 4 16×105 0.04
accuracy [%] ±1 ±1 ±1.6 ±2
For validation purpose the simulated test cases above are acquired on a real EPC bench. Then an additional test case has been acquired that includes a real clutching procedure. Through these
the first six test cases investigate short term dynamic behavior with one excitation of different SMV combination. The seventh one checks the long term dynamics with SMV excitation for a real clutching procedure.
The disengagement and engagement with various SMV combinations are presented below.
The measured and the simulated signals are represented with continuous- and dashed lines re-spectively. The transient states of the SMVs in the switching moment are shown in seeFig. 2.11 and Fig. 2.12.
0.045 0.05 0.055 0.06
−50
−40
−30
−20
−10 0 10 20 30
uterm[V]
SMVsl on
0.0450 0.05 0.055 0.06 0.5
1 1.5 2 2.5 3
i[A]
time [s]
0.248 0.25 0.252 0.254 0.256 SMVsl off
0.248 0.25 0.252 0.254 0.256 time [s]
0.395 0.4 0.405 0.41 SMVbl on
0.395 0.4 0.405 0.41 time [s]
0.598 0.6 0.602 0.604 0.606 SMVbl off
0.598 0.6 0.602 0.604 0.606 time [s]
SMVmes
sl SMVsim
sl SMVmes
bl SMVsim
bl SMVmes
se SMVsim
se SMVmes
be SMVsim
be
Figure 2.11: Transient of the terminal voltages and currents during disengagement The deviations in the SMV currents between the measurement and simulation caused by the simplifying assumptions, in which the nonlinearity and the dynamics of the SMVs are neglected and reduced respectively (see AssumptionsA7,A10andA11). In spite of the deviations between the simulated and the measured currents in opening direction the valve opening times are close to each other. This can be seen from the change of the current curve, as it has been described in the verification above, where the armature of the valve reaches its stroke limit i.e. opened fully.
In closing direction the currents have no major deviations, thus the closing times are close to each other as well. The remained assumptions related to the model of the SMVs (Assumptions A6,A12) have no significant influence on the SMV model accuracy. Since the resulted opening and closing time differences between the measurement and simulation have small impact on the remained part of the system, the simplification assumptions for the SMVs can be accepted.
In the next two graphs the pressure and the position signals are presented (see Fig. 2.13and Fig. 2.14).
The deviations between the measured and simulated pressure and piston position signals are caused mainly by the simplifying assumptions (Assumption A18) in which, the clutch nonlinear force characteristics is approximated with its empirical centre curve. The remained assumptions for the clutch mechanism (Assumption A15-A17 and A19) have no significant effect on the
1.095 1.1 1.105 1.11
−50
−40
−30
−20
−10 0 10 20 30
uterm[V]
SMVse on
1.0950 1.1 1.105 1.11 0.5
1 1.5 2 2.5 3
i[A]
time [s]
1.898 1.9 1.902 1.904 SMVse off
1.898 1.9 1.902 1.904 time [s]
2.1 2.105 2.11 SMVbe on
2.1 2.105 2.11 time [s]
2.898 2.9 2.902 2.904 2.906 SMVbe off
2.898 2.9 2.902 2.904 2.906 time [s]
SMVmes
sl SMVsim
sl SMVmes
bl SMVsim
bl SMVmes
se SMVsim
se SMVmes
be SMVsim
be
Figure 2.12: Transient of the terminal voltages and currents during engagement
0 0.05 0.1 0.15 0.2 0.25 0.3
0 1 2 3 4 5
x 105
pch[Pa]
0 0.05 0.1 0.15 0.2 0.25 0.3
0 0.005 0.01 0.015 0.02
xpst[m]
time [s]
SMVmes
sl SMVsim
sl SMVmes
bl SMVsim
bl SMVmes
sl+bl SMVsim
sl+bl
Figure 2.13: Transient of the pressure and position during disengagement
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0
1 2 3 4 5
x 105
pch[Pa]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 0.005 0.01 0.015 0.02
xpst[m]
time [s]
SMVmes
se SMVsim
se SMVmes
be SMVsim
be SMVmes
se+be SMVsim
se+be
Figure 2.14: Transient of the pressure and position during engagement actuator dynamics.
The last test case is a complete clutching procedure applied for gear shifting (seeFig. 2.15).
The deviations in this test case are accumulated during the operation, which cause higher dif-ferences in the piston position.
For comparison purposes individual errors are calculated for each test case based on the entries of the output vectors and measurements results as follows:
ǫyk,j = vu ut1
T Z T
0
yk,jmes(t)−yk,jsim(t)
¯ ymesk,j
!2
dt, (2.76)
where the suffix j refers to the jth output of the model, k refers to the kth test case, mesand sim refer to the corresponding output vector of the measurement and simulation respectively.
The over line refers to the integral mean of the particular signal and T is the duration of the test case. Through this each individual error is an Euclidean signal norm of the error in the particular output compared to the measurement. Moreover a partial error is calculated based on these individual error terms for each test case as:
ǫP,k= vu ut
Xm
j=1
ǫ2yk,j, (2.77)
wherem is the number of the output signals. This error shows the complete error of the corre-sponding test case.
Finally the total error is calculated on the individual errors of thenpiece of test cases using
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
−50 0 50
uterm[V]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 1 2 3
i[A]
SMVmes
sl SMVsim
sl SMVmes
bl SMVsim
bl SMVmes
se SMVsim
se SMVmes
be SMVsim
be
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 2 4
x 105
pch[Pa] pmesch psimch
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 0.005 0.01
xpst[m]
time [s]
xmespst xsimpst
Figure 2.15: Real clutching procedure in case of gear shifting the squared mean as follows:
ǫT = vu ut1
n Xn
k=1
ǫ2P,k. (2.78)
Analyzing the individual errors (see Tab. 2.8), in the test case 1-6, it can be seen that the small SMVs have smaller current deviations than the big ones, moreover the big load SMV has larger current deviations than the big exhaust SMV i.e.
ǫy1,1 ≈ǫy3,1 ≈ǫy4,3 ≈ǫy6,3 < ǫy5,4 ≈ǫy6,4 < ǫy2,2 ≈ǫy3,2 (2.79) It is the consequence of the unmodeled nonlinear magnetic behavior of the SMV material, which nonlinearity depends on the magnitude of the applied current [31]. The magnitude distribution of the current deviations, in the seventh test case, is different. In this test case the deviations distribution has a relation with the switching number of the SMVs, since the steady state current errors are small.
Moreover the deviations between the measured and simulated pressure and position signals, on one hand, are caused by the SMV model deviations, on the other hand, are caused by the reduced dynamics of the gas energy and the reduced dynamics of the piston momentum.
Nevertheless, the total error in the validation is below the specified tolerance limit, thus the model accuracy can be accepted,χMP5a-g are satisfied.