point associated with a reference rail pressure and injection timing, the representative point P of the designed controller (for each value of ν) belongs to the stability domain. In particular it lies on the relative stability curve associated with the designed phase margin (see Figure 2 for two different points).

This occurs because, for each value of ν, the position and amplitude of both the CRB boundary curves and the relative stability curves change, but their shape and mutual position do not. In addition, consider parametric variations, uncertainties and model inaccuracies that bring the design point P closer to a modified actual CRB curve than to an initially used nominal CRB curve. In this case, the distance between the nominal CRB curve and the relative stability curve on which P lies is sufficient to guarantee stability. If needed, this distance can also be enlarged by shifting the point P along the relative stability curve with the same phase margin specification (see possible different locations in Figure 2) with increasing ω. In this way, P is relocated away from the CRB curve provided that ωc, then ωB , can be increased.

Finally, the employed scheduling strategy guarantees stability when switching the fractional order controllers. Namely, as far as variations in the reference pressure are bounded by 2 bar and those in injection duration by 6 s (which is a good indication for real practice), simulation results always verify closed-loop stable responses.

6.Conclusion

Injection pressure regulation is currently an important control problem in the automotive industry because of the ever-growing focus on reducing pollutant emissions and consumption and increasing engine efficiency. In this chapter, the case of CNG engines was approached by an innovative fractional order control strategy. The typical advantages offered by FOCs are amplified by combining the D-decomposition methodology and appropriate controller scheduling for the specific injection system working points. To synthesize, injection pressure is regulated by tuning different FOPI controllers based on the reference working points to achieve. Variations between working points and disturbances are compensated by scheduling the FOPI controller gains. Moreover, several simulation studies verify that switching between controllers does not lead to stability problems. To sum up, the main benefits of the proposed approach for controlling rail pressure in injection systems of CNG engines are the following.

Closed-form and relatively simple design formulas are applied to obtain the controller gains in terms of frequency domain specifications; such formulas could be used for an automatic synthesis of the controller. As far as relative stability is concerned, a specification on the phase margin can be analytically achieved and is strictly related to the required noninteger order ν and, vice versa, the selected order ν determines the phase margin that can be obtained (see formulas in (29)).

The integral (or derivative) operator of the fractional order controller is realized by an efficient approximation method that prevents numerical problems and leads to a rational transfer function characterized by a low number of zeros and poles. Both reduced approximation errors and easy implementation are guaranteed by the proposed realization. The approximating rational transfer function shows remarkable properties: the poles of the transfer function are stable, the zeros are minimum-phase, and all singularities are interlaced

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58 Paolo Lino and Guido Maione

on the negative real half-axis of the s-plane [35]; interlacing, stability and minimum-phase characteristics are also achieved by the discrete transfer function used for digital realization [34, 36, 37].

Robust stability of the designed FOPI controllers is guaranteed by the D- decomposition, which allows a greater robustness with respect to the usual solutions employed in the automotive industry, even if large variations of working points are considered. Finally, if switching between sufficiently close working points is done such that variations of injection timings are below 6 seconds and changes of rail pressure are below 2 bar, then nonlinearity effects, oscillations and instability problems in the rail pressure are prevented.

Acknowledgment

This work was supported by the Italian Ministry of University and Research under project “EURO6 - Advanced electronic control unit, injection system, combustion strategies, sensors and production process technologies for low polluting diesel engines”.

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