In general, the simplified model should retain the major dynamic characteristics of the system only and omit all the details that are weakly represented in the outputs where the major dynamic characteristics depend on the intended use of the model.
3.3.1 Simplified nonlinear dynamic hybrid model for simulation purposes (M1)
The following properties, defined by performance (Px) and size (Sx) indices of the simplified EPC model (M1) are considered below to achieve the simplifying goals, which are in this case fast dynamic simulation with reduced computational effort to make it possible to integrate the EPC model into a complete driveline or a vehicle model.
M1P1. The model variables and parameters should preserve the physical meaning (χM1P1 ∈B).
M1P2. The model should be capable of describing the dynamic behavior of the EPC system within 5% deviation in the whole operation domain with respect to the chamber pressure and the piston position (χM1P2 ∈ R, χmaxM1P2 = 0.05). This performance index is calculated by the Euclidean norm as:
χM1P2 =q χ21,M
1P2+χ22,M
1P2, (3.2)
where χ1,M1P2 is the chamber pressure deviation and χ2,M1P2 is the piston position deviation. For computing χi,M1P2 an L2 error has been used to measure the devia-tions of the model response, based on the entries of the model output vectors and the measurement results on the real system as follows:
χi,M1P2 = vu ut1
T Z T
0
yiM eas(t)−yiM1(t)
¯ yiM eas
!2
dt, (3.3)
where the suffix i refers to the ith output of the model, M eas and M1 refer to the corresponding output vector of the measurement and simulation respectively. The over line refers to the integral mean of the particular signal andT is the duration of the test case.
M1S1. The model size and complexity should be significantly reduced (ξM1S1 ∈R). The size index is also calculated by the Euclidean norm as:
ξM1S1 = vu ut
X8
i=1
wiξi,2M1S1, (3.4) whereξi,M1S1 are the number of the balance volumes (ξ1), state variables (ξ2), control inputs (ξ3), disturbance inputs (ξ4), parameters (ξ5), hybrid terms (ξ6), relative degree (ξ7) and finally the relative computational time of the simulated time stretch (ξ8), i.e.
the calculation time divided by the simulated time. Moreover, in order to achieve the weighted sum of the individual items numberswi weights are used. The weights should reflect the different impact of the indices on the change of the model complexity.
To achieve the specified goals above via systematic model simplification the first step is to simplifying the balance volumes then the balance equations and finally the transport mechanisms and constitutive equations. Hence, the following assumptions are done:
M1A1. Remove the power stage / valve subsystem dynamics from the model and apply ideal valves in which the opening and closing processes are instantaneous.
This assumption can be done since the power stage / valve subsystem has much faster dynam-ics than the remained model parts considered for position control of the piston. The dynamdynam-ics of the power stage / valve subsystem is considered for the mass flow rate control (see details later inSection 5.4). This assumption removes the ixx,vxx and xxx terms from the state vector, the Usup term from the disturbance inputs besides, avoid all the parameters related to the power stage / valve subsystem except αxx andAxx.
M1A2. The thermodynamic processes in the chamber can be considered as isothermal instead of polytropic.
This assumption can be done, as well, since the focused dynamical range of the modeled processes should cover the medium dynamic responses only while the air temperature converges to the environment temperature with slow dynamics. In this way the temperature in the chamber can be considered with constant value. Thus, the isothermal model class can cover the required behavior. One has to mention that this assumption eliminates themch state since the chamber balance equation is derived from the gas mass balance only and the gas energy balance equation is not needed anymore. Moreover, eliminates the Tsup disturbance input by usingTamb.
M1A3. The piston cannot reach its limitations during normal operation.
One of the extreme piston positions, corresponding of the completely exhausted chamber, is determined by the stiffness of the disc and helper springs without reaching the stroke limitation (xlim,1pst ). Besides, the piston stroke is large enough to cover the working domain of the clutch mechanism without reaching the other stroke limitation (xlim,2pst ). This assumption results that the piston limiting term (Fpstlim) can be left out of the model, which reduces the number of the hybrid modes and eliminates the computational effort regarding the collisions with the stroke limiters.
M1A4. Subsonic flow in both of the intake and exhaust direction cannot be reached simultane-ously.
The necessary conditions for achieving subsonic flow in both of the two directions are fulfilled, when Πl>Πcrit andΠe>Πcrit, is pamb/psup >Π2crit. But this condition has not been satisfied since it is out of the (disturbance) input constraints of the EPC (see the input constraints in Section 2.3).
M1A5. The friction force term (Ff r) can be approximated using sigmoid function as follows:
Ff r =µpst(pch−pamb)Apst
2
1−e−uf rvpst −1
, (3.5)
where uf r determines the friction force steepness at small velocities. This helps to solve the numerical problems for eliminating the hybrid modes regarding the friction force.
M1A6. The cross sections and the contraction coefficients can be lumped together.
The new, i.e. effective, cross section terms are obtained as follows: Aeffxx=αxxAxx.
M1A7. For the sake of simplicity, the spring term shsp(xpst,0−xpst) can be integrated into the Fl(xpst) term.
Applying these assumptions above the state space description of the simplified model for simulation purposes is obtained. From the simplified DAEs of the model the state vector is composed as follows:
xM1 = [ pch vpst xpst ]T. (3.6)
The control input vector (uM1), that includes the duty cycle of the PWM control signals of the SMVs, is not changed (see Eq. (2.48)), while the disturbance input vector includes only the compressed air pressure, ambient- pressure and temperature respectively:
dM1 =
psup pamb Tamb T
. (3.7)
The output vector includes the chamber pressure, the piston position and the supply pressure:
yM1 =
pch xpst psup T
. (3.8)
The simplified state space model has the form of Eq. (2.50), as well, where the nonlinear state functions are written as follows (the entries that depend on the hybrid modes are boxed, too):
f1,(k)M1 = R
md,sl Aeffsl +md,bl Aeffbl
ξl Tamb Vchd +xpstApst −
− R
md,se Aeffse +md,be Aeffbe
ξe Tamb Vchd +xpstApst −
− pchvpstApst Vchd +xpstApst,
(3.9)
f2,(k)M
1 = (pch−pamb)Apst−vpstkpst−Fl(xpst) mpst
−
−
µpst(pch−pamb)Apst
2
1−e−ufrvpst −1 mpst
,
(3.10)
f3,(k) =vpst, (3.11)
where
ξl = s
2κ κ−1
p2sup RTamb
Πl
2 κ − Πl
κ+1 κ
and (3.12)
ξe = s
2κ κ−1
p2ch RTamb
Πe
2 κ − Πe
κ+1 κ
. (3.13)
The output equation has the form of Eq. (2.69), as well, where
CM1 =
1 0 0 0 0 1 0 0 0
(3.14)
and the EM1 matrix corresponds to the measured disturbance which is the psup, hence it is written as follows:
EM1 =
0 0 0 0 0 0 1 0 0
. (3.15)
The performance output is generated from the measured output by the following simple equation:
zM1 = [ 0 1 0 ]yM1. (3.16) The simulation result of the simplified model (M1) is shown inFig. 3.4.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 105
pch[Pa]
Measurement M0 M1 M2
0 0.2 0.4 0.6 0.8 1 1.2
0 2 4 6 8 10 12 14
xpst[mm]
time [s]
Figure 3.4: EPC measurement and simulation results in case of modelM0,M1 and M2
Besides, the measurement on the real system and the response of the detailed model (M0) for the same inputs are shown, as well. The input signals are derived from a real clutching.
The disengagement has been executed with maximal dynamics to reach the disengagement state as quickly as possible. Then, the engagement has been started with fast dynamics as well to reach the touch point, where the pressure plate and the friction disc have been connected. In the clutching domain the stroke has been decreased slowly to increase the transmitted torque smoothly. Finally, in the end of the engagement the engagement state has been set up with maximal dynamics as well.
The most important results of this simplifying procedure are as follows. All the retained system variables have preserved their physical meaning meanwhile, some model parameters (αxx
and Axx) have changed slightly their meaning (Aeffxx = αxxAxx) due to the lumping (χM1P1 = true). The pressure deviation of M1 comparing with M0 has been increased from 1.23% to 2.38%, the position deviation has been increased from 2.63% to 3.43% and the total deviation has been increased from 2.9% to 4.18% hence, the required accuracy has been fulfilled (χM1P2 <
χmaxM1P2).
The number of the balance volumes for which the balance equations are applied has been reduced from 10 to 2. The hierarchy structure of the simplified model is shown in Fig. 3.5 where, the term which has hybrid behavior is depicted by dashed line. The number of the
balance volume level
balance equation level
transport mechanism and constitutive level
Simplified nonlinear dynamic model
for simulation purposes
Piston balance volume
Chamber balance volume
Momentum balance vpst,xpst
Mass balance
pch
Pressure term Apst,pamb
Damping term
kpst
Load term Fload(xpst)
Friction term µpst,Apst,
pamb,uf r
Convective term κ,R, Aeffxx,Vchd, Apst,psup, Tamb,pamb
Figure 3.5: Hierarchical structure of the simplified nonlinear dynamic hybrid model for simulation purposes
state variables has been reduced from 16 to 3, the control inputs have been unchanged under this simplification process and the disturbance variables has been cut to 3 from the original 5.
The variable structure graph of the simplified model is shown in Fig. 3.6. The number of the parameters has been reduced from 83 to 12 where theFl(xpst)term, i.e. the center characteristic line of the clutch mechanism, has been considered as a third order polynomial [39]. The equations have been simplified considerably by eliminating the power stage / valve subsystem and the gas
md,sl
md,bl
md,se
md,be
pch vpst xpst
psup
pamb Tamb
Figure 3.6: Structure graph of the differential variables of the simplified nonlinear dynamic model for simulation purposes
energy equations hence; the total number of the hybrid terms has been reduced from 12 to 1 and trough this the total number of the hybrid modes has been reduced from 746496 to 4. Finally the relative computational time, which corresponds to 1.4s time stretch, has been decreased from 35.8 to 12.6. For computing the model size and complexity index (ξ), based on Eq. (3.4), the following weights are usedw1= 4,w2 = 2,w3= 2,w4 = 1,w5 = 1,w6= 2,w7 = 1and w8= 1.
These weights emphasize the different effects of the reduction of the various size indices (ξi).
E.g. the presence of a balance volume is a greater complexity factor, than that of a disturbance input.
Through these the model size and complexity index has been decreased from 382.9 to 77.7.
Thus, the simplifying goals have been achieved and the simplifying process has been finished. In Fig. 3.7the performance and size indices are depicted in graphical form.
3.3.2 Simplified nonlinear dynamic model for control design purposes (M2) A model for control design purposes should retain all major dynamic characteristics of the real plant (such as its stability and main time constants) but omit all details that are weakly rep-resented in the state variables and not related to the control aims. Hence, the simplified EPC model for control design purposes (M2) should have the following properties:
M2P1. The model variables and parameters should preserve the physical meaning (χM2P1 ∈B).
M2P2. The model should be capable of describing the dynamic behavior of the EPC system within 15% deviation in the whole operation domain (χM2P2 ∈R, χmaxM2P2 = 0.15).
M2P3. The model should contain continuous parts only i.e. the discrete elements should be omitted or a nominal hybrid mode should be selected to be able to apply the continuous control methods (χM2P3 ∈B).
M2S1. The model size should be further reduced using the size norm as before (ξM2S1 ∈R).
To achieve these goals above the following assumptions are considered:
M2A1. Remove the flow property term and use the total mass flow rate of the valves as input.
According to M2A1 the total mass flow rate of the valves is considered as follows:
σv =
md,sl Aeffsl +md,bl Aeffbl ξl−
−
md,se Aeffse +md,be Abeeff ξe .
(3.17)
0 10
χ1,MxP3[%]
M0 M1 M2 Pressure deviation
(1.23) (2.38) (3.58)
0 10
χ2,MxP3[%]
M0 M1 M2 Position deviation
(2.63) (3.43) (14.46)
0 10
χMxP3[%]
M0 M1 M2 Total deviation
(2.90) (4.18) (14.90)
0 5 10
ξ1,MxS1[-]
M0 M1 M2 Num. of balance volumes
(10)
(2) (2)
0 10 20
ξ2,MxS1[-]
M0 M1 M2 Num. of state variables
(16)
(3) (3)
0 5
ξ3,MxS1[-]
M0 M1 M2 Num. of control inputs
(4) (4)
(1)
0 5
ξ4,MxS1[-]
M0 M1 M2 Num. of disturbance inputs
(5) (3)
(2)
0 50 100
ξ5,MxS1[-]
M0 M1 M2 Num. of parameters
(83)
(12) (8)
0 10
ξ6,MxS1[-]
M0 M1 M2 Num. of hybrid terms
(12)
(1) (0)
0 5
ξ7,MxS1[-]
M0 M1 M2 Relative degree
(6)
(3) (3)
0 20 40
ξ8,MxS1[-]
M0 M1 M2 Computation effort
(35.8)
(12.6) (8.6)
0 200 400
ξMxS1[-]
M0 M1 M2 Model size and complexity
(382.9)
(77.7) (52.8)
Figure 3.7: Model performance and size indices in case of model M0,M1 andM2 This assumption can be done since the flow rate change has similar dynamics to the armature of the valves. Moreover, there are no hybrid mode change in the pressure range (2−4.5·105P a) of the position control (see the linear domain of the available mass flow rates in Fig. 5.8). This eliminates the hybrid behavior of the model and reduces the number of the inputs from 4 to 1.
M2A2. The friction term of the piston can be left out.
This assumption can be done since the friction force is much smaller than the force generated by the pressure.
Applying these assumptions above the state space description of the simplified model for nonlinear control design purposes is obtained. The state vector (xM2) is not changed, the resulted control input vector contains the total mass flow rate of the valves only (uM2 =σv) and the disturbance input vector is written as: dM2 = [pamb Tamb ]T. The output vector is obtained as follows: yM2 = [pch xpst ]T.
The state space description of the model for control design purposes can be written into standard input-affine model form as follows:
dxM2
dt = fM2(xM2, dM2) +gM2(xM2,dM2)u. (3.18)
The coordinate functions are written as follows:
f1,M2 =− pchvpstApst Vchd +xpstApst
, (3.19)
f2,M2 = (pch−pamb)Apst−vpstkpst−Fl(xpst) mpst
, (3.20)
f3,M2 =vpst, (3.21)
g1,M2 = RTamb
Vchd +xpstApst, (3.22)
g2,M2 = 0, (3.23)
g3,M2 = 0. (3.24)
The measured output is written as the following linear equation:
yM2 =CM2xM2, (3.25)
where
CM2 =
1 0 0 0 0 1
. (3.26)
The performance output is generated from the measured output by the following simple equation:
zM2 = [ 0 1 ]yM2. (3.27)
The simulation results of this simplified model (M2) are shown inFig. 3.4, as well. It can be seen that the outputs ofM2 have a deviation, which is mainly caused by the neglected friction.
The most important results of this simplifying procedure are as follows. All the retained model variables and parameters have preserved their physical meaning (χM2P1 =true). Since all the discrete switching terms have been eliminated the model became continuous (χM2P3 =true).
The pressure deviation of M2 comparing with M1 has been increased from 2.38% to 3.58%, the position deviation has been increased from 3.43% to 14.46% and the total deviation has been increased from 4.18% to 14.9% (see Fig. 3.7) nevertheless, the required accuracy has been fulfilled (χM2P2 < χmaxM2P2).
The number of the balance volumes has been invariant under this simplification process (see Fig. 3.7). The hierarchy structure ofM2 is shown inFig. 3.8. The number of the state variables has been invariant as well while the control inputs have been reduced from 4 to 1, the disturbance variables have been reduced from 3 to 2 and the number of the parameters has been reduced from 12 to 8. The variable structure graph of M2 is shown in Fig. 3.9. Finally the relative computational time, which corresponds to 1.4s time stretch has also been decreased further from 12.6 to 8.6. Through these theM2 size and complexity index, using the same weights as before, has been decreased from 77.7 to 52.8. Thus, the simplifying goals have been achieved and the simplifying process has been finished.