where udr is the disturbance rejection part of the control. Smoothing of control discontinuity is essential to avoid chattering effect thus in Eq. (5.19) the sgn(s) is replaced bysat(s), where sat(.)is a linear saturated function to±1with gradient of one. This achieves a trade-off between tracking precision and robustness to unmodeled dynamics.
Finally, substituting Eq. (B.19),Eq. (B.20)and Eq. (5.16) intoEq. (5.19)the SMC law can be obtained and the controlk andλparameters are tuned for the controlled EPC system.
In order to decrease the load of the computing device the reduction of the equivalent control part to zero (ueq = 0) is examined. In this wayu =−k sat(s) is the control rule, which can be considered as a simplified SMC. This reduction can be done since in static state the equivalent control part is equal to zero. Otherwise the u = −k sat(s) part could ensure the s= 0 sliding condition and through these the prescribed dynamics [86].
pch [P a]
σdem v[kg/s]
psup
pamb 2 3 4 5 6 7 8 9
0 σmaxsl+bl σmaxbl
σmaxsl
σse+bemax σbemax σsemax
0.005 0.01
−0.005
−0.01
−0.015
uslv = 1;ublv = 1;usev = 0;ubev = 0 uslv = 0;ublv = 1;usev = 0;ubev = 0 uslv = 1;ublv = 0;usev = 0;ubev = 0 uslv = 0;ublv = 0;usev = 1;ubev = 1 uslv = 0;ublv = 0;usev = 0;ubev = 1 uslv = 0;ublv = 0;usev = 1;ubev = 0 uslv = 0;ublv = 0;usev = 0;ubev = 0
Figure 5.8: Available mass flow rates plotted against different chamber pressure
the update rate of the valve commands should be high enough in order to the keep a good system controllability. But increasing the update rate of the valve commands yields increasing computation costs and reducing lifetime as well.
With this approach a trade-off must be taken between the number of the valves, the maximum level of the mass flow rate, the realizable accuracy and the update rate of the valve commands.
5.4.2 Dynamic mass flow rate decomposition approach
The other proposed solution is a dynamic mass flow rate decomposition approach, in which dynamic valve states correspond to a given mass flow rate demand. In this approach a low frequency PWM method is used, where the switching frequency is low compared to the frequency required for current control.
From the first mean value theorem for integration the mean mass flow rate (¯σT) of a valve during a period T is as follows:
¯ σT = 1
T ZT
0
σ(t)dt. (5.20)
In case of an ideal switching valve, where the switching transients are instantaneous and the
realizable mass flow rates are zero and maximal (σ(t)∈ {0, σm(t)}), the mean mass flow rate for a givenT period is written as follows:
¯ σT = 1
T
ts
Z
0
σm(t)dt, (5.21)
where the switch durationts, is the product of the smallest switching time step tq >0 and the step counterihence,ts=i·tq. In case of constantσm(t)the mean mass flow rate is proportional to the duty cycle (ts/T) and it is written as follows: σ¯T =ts/T ·σm.
Considering successive valve activation, which means that an additional valve is activated only, when the previous one is opened fully, the realizable number of the mass flow rate quantum levels, isq =n·tq/T + 1in case ofnvalves.
In Fig. 5.9 the mass flow rates of an ideal on/off-, a proportional- and a real on/off valve are shown using tsk−1 and tsk switching times (continuous lines) and τsk−1 and τsk switching times (dashed lines). It can be seen that the mean mass flow rate of the ideal on/off- and the proportional valves forT period are equal each other as opposed to the real on/off valve. Hence, the mean mass flow rate is not proportional to ts/T due to the valve nonlinearity. Thus, the appropriate switching time of a real on/off valve corresponding to a given mass flow rate should be calculated in a different way.
(k−2)T (k−1)T kT
σ[kg/s]
time [s]
ideal on/off proportional real on/off
tk−1s tks
τsk−1 τsk
σm
¯ σtTs
¯ σTτs
Figure 5.9: Mass flow rates of ideal, proportional and real valves
The switching dynamics of the real valve and through this the piecewise continuous mass flow rate is dependent not only on the valve switching time, but on the valve state variables
and disturbance variables as well [88]. Following that the appropriate switching time cannot be calculated accurately or the calculation requires a high computation effort, which is not acceptable in case of an embedded application.
The appropriate switching time for a given mass flow rate can be determined in an empirical way as well. In order to determine the appropriate switching time for a given mass flow rate, the effect of the previous switching and through this the valve states and the disturbances should be considered as well. Since the dynamics of the valves are much higher than the dynamics of the pneumatic actuators it can be assumed that the value of the demanded mass flow rate and thus the appropriatetschanges slowly enough to consider that tsk−1 is equal totsk for a given period.
This assumption makes it possible to collect the valve mass flow rates for different ts values by means of a valve characteristics. The measurements are executed for givents switching times using j·T activation periods, wherej is high enough to reach the satisfying accuracy.
In Fig. 5.10 and Fig. 5.11 the characteristics of the small and the big SMVs are shown re-spectively, where the σ¯T/σm values are shown as a function of the duty cycle ts/T considering differentUsup disturbance input values (left side) and considering different ∆p=pin−pout val-ues (right side), wherepin and pout are the pressures on the inlet- and on the outlet port of the valve. The remaining disturbance input Tamb is assumed to be constant, since it changes slowly compared to the system dynamics.
0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
¯σT/σm
ts/T Usup= 28V
∆p= 2·105Pa
∆p= 4·105Pa
∆p= 6·105Pa
∆p= 8·105Pa
∆p= 10·105Pa
∆p= 12·105Pa
0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
¯σT/σm
ts/T
∆p= 5·105P a
Usup= 18V Usup= 20V Usup= 22V Usup= 24V Usup= 26V Usup= 28V Usup= 30V Usup= 32V
Figure 5.10: Characteristics of small SMVs
The characteristics of the valves can be used to determine the switching time for a given mass
0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
¯σT/σm
ts/T Usup= 28V
∆p= 2·105Pa
∆p= 4·105Pa
∆p= 6·105Pa
∆p= 8·105Pa
∆p= 10·105Pa
∆p= 12·105Pa
0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
¯σT/σm
ts/T
∆p= 5·105P a
Usup= 18V Usup= 20V Usup= 22V Usup= 24V Usup= 26V Usup= 28V Usup= 30V Usup= 32V
Figure 5.11: Characteristics of big SMVs flow rate as follows:
ts=T·f(¯σT/σm,∆p, Usup). (5.22) Since the cross section of the big SMVs is much higher than the cross section of the small ones the big valve is not used alone. Instead, the small SMV switching is applied in the low range of the mass flow rate demand and if the mass flow rate demand is higher than the available maximum mass flow rate of the small SMV, the small SMV is opened fully and the big SMV switching is applied additionally. Thus the big SMV switching without applying the small one is not used.
To allow the calculation of the switching time for arbitrary mass flow rate demand and disturbances, interpolation is used between the measured points. Moreover, the computation cost can be kept at a low level in case of using look-up tables and linear interpolation. Besides, the update rate of the valve commands can be decreased compared to the previous method, since the quantization error is decreased. This provides a method better suited for embedded environment application.