error by a relatively low number of zeros and poles, so an easy implementation is possible in the analog or digital domain. The closed-form formulas are valid for sν when 0 < ν < 1. Then, since ν > 1 in the non-integer order integrator of the FOPI controller, given that sν = s sν−1 , only sν−1 is approximated. To synthesize, if λ = ν − 1, the approximation is as follows:

where N ≥ 1 is the number of zero-pole interlaced pairs and the coefficients αN,j (λ) = βN,N −j(λ), for j = 0, . . . , N , depend on λ and are determined by the following closedform formula:

in which the so-called Pochhammer functions are (λ + j + 1)(N −j) = (λ + j + 1)(λ + j + 2) . . . (λ + N ) and (λ − N )(j) = (λ − N )(λ − N + 1) . . . (λ − N + j − 1), with

(λ + N + 1)(0) = (λ − N )(0) = 1. Simple algebraic manipulations allow us to express the previous formula differently [37, 38]:

where (N − λ)(j) := (N − λ)(N − λ − 1) . . . (N − λ − j + 1) and (N − λ)(N −j) := (N − λ)(N − λ − 1) . . . (j − λ + 1) are falling factorials. It holds (N − λ)(0) = 1.

5.Simulation Results

This section shows the effectiveness, performance and robustness of the described control strategy. To this aim, a detailed non-linear simulation model is created by the commercial AMESim developing package [22]. AMESim is a simulation tool based on a virtual prototyping environment. It enables modeling and integration of different physical components, supports and combines different levels of abstraction. The polymorphism concept of AMESim allows the analysis of a multi-disciplinary system as a whole and the representation of many different scenarios. All its peculiarities and properties make AMESim capable of replacing real experiments for validation, so that the non-linear model of this study accurately represents the complex fluid-dynamic phenomena characterizing the injection system at different working points.

The AMESim model makes some assumptions. First, the distribution of pressure in the control chamber, the common rail and the injectors is uniform; the elastic deformations of solid parts by pressure changes are negligible; the pipes, which are considered incompressible ducts with friction, are subject to non-uniform pressure distribution. Vice versa, the model considers temperature variations affecting the pressure dynamics in each component and only takes into account heat exchanges through pipes, by properly computing a thermal exchange coefficient. The tank pressure plays the role of a maintenance input and it is modeled by a constant pneumatic pressure source. Finally, to simplify the AMESim model construction, some supercomponents have been created to group multiple elements.

The operating ranges of the main variables in the system are the following:

Complimentary Contributor Copy

•pressure in the tank: ptk [30, 200] bar;

•pressure in the control chamber: x1 [4, 25] bar;

•pressure in the common rail: x2 [4, 25] bar;

•injector opening time: tinj [025] ms;

•duty-cycle of the command signal for the valve: [0, 100]%;

•engine speed: ωrpm [1000, 4000] rpm.

As a consequence, the model parameters vary in the following ranges: K [134, 1129] bar, T [1.059, 18.546] seconds.

In the most representative simulation tests, the tank pressure, the engine speed and the injector opening time are kept constant at ptk = 50 bar, ωrpm = 2500 rpm and tinj = 5ms, respectively. However, Table 3 shows the system parameters in several conditions.

Table 3. Engine parameters in working points

To test their efficiency and robustness, the gain-scheduled FOPI controllers, with ν = 1.3, 1.4, 1.5, are compared to a standard integer order PI that is typically used for injection control and usually tuned by the open-loop Ziegler-Nichols rules. The PI is gain-scheduled for fairness of comparison. Values of ν < 1.3 or ν > 1.6 are discarded because they lead to too high or low (below 36◦) phase margins. With reference to the typical working conditions in Table 3, two different cases are simulated.

In the first one, the reference pressure undergoes a small variation from 5 to 6 bar and the system variables are perturbed slightly from their working point. Hence, the local behavior can be analyzed by the linearized model. The parameters of the injection system are K = 200 bar, T = 1.96 seconds (see Table 3). A single FOPI or PI controller is used and designed according to these parameters associated with the final reference pressure that must be reached.

Figure 3 shows the closed-loop step response.

With respect to a standard PI, the FOPI controllers improve the performance indices: even if the rise time is slightly longer, the overshoot is highly reduced, so that a better injection accuracy is achieved; the settling times are comparable for ν ≤ 1.5. With the PI controller, the performance is influenced primarily by disturbances and nonlinearities

Complimentary Contributor Copy

Figure 3. Response to a 5-to-6 bar step in the reference pressure.

related to operations of injectors, by PWM modulation of the solenoid valve command. Large variations of the control signal cause high frequency pressure oscillations around the equilibrium point. Moreover, the saturation in the control signal, which is associated with a complete closure of the valve, determines large rail pressure overshoots and undershoots. Conversely, performances of fractional controllers are less sensitive to system disturbances and nonlinearities.

The second case simulates a large variation in the reference pressure from 6 to 10 bar, with a step of 2 bar, and then back from 10 to 6 bar, with the same step. Then, the scheduling switches between several controllers. In particular, since the variation is divided into two steps, two FOPI/PI controllers are used. This choice is justified by simulation. Several tests, indeed, verified that stability is achieved if variations of the rail reference pressure and of injector opening times are within operating ranges.

This second experiment aims at testing control robustness. Namely, the emptying and filling of the common rail follow different dynamics. This is due to several reasons. First, the gradient between the tank pressure and the main chamber is higher than between the common rail and the discharging manifold. Secondly, the rail fills more rapidly because the feeding tank has a higher pressure than the rail. In contrast, the rail empties more slowly because of the much lower rail pressure and the injection outflow. However, injection timings do not vary because they are set by the ECU based on the amount of fuel to inject. Therefore, the emptying process can be accelerated only by closing the solenoid valve.

Figure 4 shows that, for rising transients, FOPIs remarkably reduce the overshoots and responses are similar to the first case. FOPIs are very prompt in resetting the rail pressure

Complimentary Contributor Copy

Figure 4. Response to large variations in the reference pressure.

to the set-point.

Conversely, the PI controller reacts poorly to larger variations, and the previously described non-linear phenomena considerably affect overall performance.

For descending transients starting from t = 13 s, owing to the request of a quick pressure reduction, a duty cycle equal to zero closes the valve completely. Thanks to injections, the rail pressure decreases to the final reference values (8 and 6 bar, respectively), with a time constant depending on the system geometry. The error cannot be reduced with a higher rate than that achieved, due to the saturation of the actuation variable. This is reflected by the same slope of pressure plots by FOPIs or PI. Figure 4 clearly shows a better tracking of the PI during the descending transient with respect to the rising transient. At the same time, a large undershoot of the rail pressure still remains before reaching the steady state.

Therefore, FOPIs achieve an overall better quality of the response, so they may also improve injection pressure regulation for large perturbations of working conditions.

All the achieved results confirm the higher robustness and performance of the FOPI controllers in all simulated conditions. The tests show that FOPIs significantly improve the usual performance indices for injection pressure control, by reducing undershoots/overshoots, steady-state errors and settling times [14].

This remarkable result can be explained by the proposed control approach. Namely, on one side, it is already well known that fractional order control may help to improve robustness and closed-loop performance. But the added value here is the robust stability properties enforced by the D-decomposition. If one considers the variation of the working points of the injection system, then the approach guarantees that, for each new working

Complimentary Contributor Copy