1. Introduction
In the last two decades, the idea of increasing the number of electric devices onboard aircraft has gained more and more popularity. The initial motivation was to replace hydraulic and pneumatic actuators with electric ones, thus increasing efficiency and safety onboard, and possibly reducing weight. This rationale gave birth to the so-called More Electric Aircraft (MEA) paradigm [
1], on which different national and international projects have been funded. For instance, since the Fifth Framework Programme dating back to the beginning of 2000s (e.g., with the project Power Optimised Aircraft, POA [
2]), the European Community has constantly increased the budget for aerospace-oriented projects, with emphasis on employing electric technologies onboard. Since the beginning of 2010s, the creation of the “Clean Sky JU” initiative [
3] has increased the focus on noise and weight reduction. It is worth citing that recently the more ambitious project of an All Electric Aircraft (AEA) has been gaining more interest. For instance, some months ago, Hyundai promised to launch an all-electric flying taxi in 2028 to reduce gas emission and congestion in urban traffic [
4].
From a technical point of view, what really motivates the use of electric actuators in the MEA approach is their efficiency. Pneumatic power, used mainly in wings ice protection systems and in environmental control systems results in poor efficiency and complex maintenance [
5]. On the other hand, hydraulic actuators are robust and able to produce high power, but they are generally heavy and subject to leakage of corrosive liquids [
5]. Moreover, the price paid when transferring mechanical energy from the main energy source (usually, the aircraft gas turbine engine) to the loads (e.g., fuel and oil pumps) is the usage of heavy gearbox systems [
6,
7]. It is apparent that it is possible to improve efficiency by directly using electrically fed devices, e.g., electric pumps for fuel, electrically driven air compressors, and so on. Incidentally, this also increases robustness to faults and allows capabilities of fault detection and diagnosis at the system level.
The increase of electrically operated devices onboard requires advanced automatic control of the electric network since it is no longer possible for the pilot to manually control devices that have to be operated in a coordinated fashion. Moreover, employing automatic control avoids human errors and/or misunderstanding and increases reliability. Actually, two levels of control have to be considered, namely a low-level control addressing the single electric device to be operated and a high-level supervisory control coordinating the network of low-level controllers according to prescribed requirements or control objectives. Possible control objectives are optimal distribution of electric power, reduction of weights onboard, alleviation of stresses on generators (e.g., by smoothening power peaks), optimal response to time-varying loads, and so on. This is possible if the electric actuators are transformed into “smart” devices by a low-level controller acting on the power converters feeding the actuator, which is the interface between the actuator and the electric network. Hence, bidirectional power converters capable of four-quadrant operations are key elements of the MEA paradigm as they act as a bridge between the main aircraft bus and auxiliary power sources, such as batteries or supercapacitors. For instance, the presence of auxiliary batteries is exploited to implement load-sharing policies between the aircraft generator and the battery itself to allow the installation of smaller, thus lighter, onboard generators [
8,
9]. Supercapacitors, instead, can be adopted for their intrinsic capability to absorb or provide fast power peaks, thus allowing stress reduction on the mechanical parts of the aircraft generator [
10,
11].
The power flow between the main aircraft bus and the battery (and/or supercapacitors) is regulated by bidirectional four-quadrant power converters [
12,
13]. A converter for this application has to meet the prevalent aeronautic requirements, such as being a low-cost design and minimizing the component size, weight, and number. Fixed-frequency operation is desired to meet electromagnetic interference (EMI) standards ,and a highly compact design and a low overall weight are required. Efficiency of the dc–dc converter over a wide input and output power range is also a critical issue since the converter is required to work at different operating points. One commonly used converter topology for this application, due to its limited number of components and high efficiency, is the four switches buck–boost converter, which is adopted in this work. As highlighted in [
14], there are a number of alternatives to the four-switches buck–boost converter, such as the constant-frequency zero-current-switching quasi-square-wave (CF-ZVS-QSC) converter [
15], the single ended primary inductor converter (SEPIC) [
16,
17], or the zero-voltage zero-current switching (ZVZCS) converter presented in [
17]. The main drawbacks of these converters are doubled switch-blocking voltage stress and diode recovery losses in case of the SEPIC [
17], a larger number of passive components, and a larger inductance value for the main inductor in the SEPIC and ZVZCS topology [
15]. Furthermore, SEPIC and ZVZCS use a capacitive energy transfer that performs badly in high-power applications. Other drawbacks of resonant converters are variable switching frequency, which complicates EMI filter design, and limitations in operating range for soft switching.
The standard strategy in controlling power converters is to consider a linear (typically, a proportional–integral PI or proportional–integral–derivative PID) controller. This is justified by a preliminary linearisation of the mathematical model of the network, device, and converter around a prescribed operating point. However, if the operating point drastically changes (e.g., due to a load change), there is no guarantee that the linear controller still behaves correctly, and even stability may be lost, as is well-known, for instance, in the presence of constant power load (CPL) [
18,
19,
20,
21]. Note that often the action of low-level controllers is to exactly make the load a CPL; hence the risk of destabilising the electric network is a critical issue. For this reason, nonlinear controllers able to directly address the nonlinear nature of the controlled electrical network, thus avoiding linearisation, are becoming more common [
22,
23].
1.1. Contributions
In this paper, the control problem of a standard aeronautic electric network comprising a high-voltage DC (HVDC) generator and a low-voltage DC (LVDC) battery is considered. Usually, the HVDC is found in an aircraft gas turbine engine acting as an AC electric generator and followed by a voltage rectification stage. As mentioned above, the key element to control is the DC/DC converter that acts as an interface between HVDC and LVDC busses. By using a suitable control strategy, it is possible to accurately impose the flow of energy both from the generator to the battery (thus recharging the battery) and from the battery to the HVDC, thus helping the generator if some extra load requires more power than the rated generator power. In this work, a buck–boost converter is considered and stability analysis and control design are thoroughly addressed. Specifically, stability in the Lyapunov sense is proved resorting to the notion of Input-to-State Stability [
24], while the control design is carefully addressed for the buck and the boost mode based on saturated Suboptimal Second Order Sliding Mode Control (SOSMC) [
25]. In fact, it is observed that while the boost mode satisfies the required assumptions for SOSMC implementation, the
nonlinear input gain function in buck mode is not sign definite, thus not enabling direct application of the SOSMC. In fact, the nonlinear input gain being sign definite is a fundamental hypothesis in most sliding mode algorithms [
26]. As a solution to this challenge, a switching control strategy inspired by
uniting control [
27] has been designed and implemented. This technique is particularly indicated for systems characterized by difficulties (or the impossibility) in the design of a unique controller capable of guaranteeing stability, performance, and robustness. For instance, a recent application of uniting control was presented in [
28] for the case of stabilization on
space. Furthermore, the adopted SOSMC also takes into account the duty cycle characteristic to be constrained in the interval [0,1] [
29].
Finally, the main contributions of this work can be summarized in the following points:
The stability of the buck–boost converter is addressed and demonstrated through the ISS framework;
A (monotonic) Saturated Suboptimal Second Order Sliding Mode Control is designed to achieve finite time control of the converter current to the given reference;
The overall designed control architecture is designed as the combination of the (monotonic) Saturated Suboptimal Second Order Sliding Mode Control and a feedback-based monotonic control algorithm. The two control laws are then orchestrated through a switching control strategy based on uniting control. Here, monotonicity is fundamental to avoid repetitive switching among the two control laws.
1.2. Structure of the Paper
After the Introduction, the electrical network that synthetically describes the aeronautic power grid is presented in
Section 2 together with the dynamic equations describing the grid behavior. Stability analysis and converter control design are presented in
Section 3 and
Section 4, respectively. In
Section 5, the outcome of the proposed control strategy is presented in a detailed simulation environment, while conclusions are drawn in
Section 6, with a list of symbols afterwards.
2. The Electrical Network
The schematic of the electrical network that describes the aeronautic power grid is shown in
Figure 1. The DC aircraft electric generator is modeled as an ideal voltage source,
, with internal resistance
, while, the DC aircraft battery is represented here as an ideal voltage source,
, with internal resistance
. Aeronautical loads connected on the network side can be modeled as a time-varying resistor denoted by
. As we are interested in active power, only resistive loads will be considered in this work. Connecting the two power sources, there is a four-switch buck–boost bidirectional converter, where the switches
and
operate in anti-phase as well as the two switches
and
. Such switches are controlled by Pulse Signal Modulation (PWM) signal generators driven by the duty cycle signals
and
, the former regulating switches
and
, the latter regulating switches
and
. The converter comprises two capacitors, namely
and
and an inductor,
L, and we indicate the generator current with
and the battery current with
.
Considering the four possible configurations of the switches, namely:
on, off, on, and off,
off, on, on, and off,
on, off, off, and on,
off, on, off, and on,
different operating principles can be observed. The converter will operate as a traditional synchronous boost if switch
keeps the
on state and
keeps the
off state, while
and
have a switching behavior. Similarly, it operates as synchronous buck when keeping
on and
off, while
and
are switching. Specifically, when
and
are turned on and
and
are turned off, the inductor gets charged during buck mode, while it discharges the stored energy during boost mode. When
and
are turned on and
and
are turned off, the inductor is charged during boost mode. Instead, when
and
are turned on and
and
are turned off, the inductor discharges energy during buck mode. Finally, the case when
and
are turned on and
and
are turned off is utilized in neither buck nor boost mode, but only when the converter needs to be disconnected from the grid for safety reasons [
13].
Considering the four possible configurations of the switches, it is not difficult to derive the set of four topologies (and the corresponding sets of linear differential equations), each one describing a specific configuration. The power grid dynamic equations can be written in a compact way as follows:
Let us define the state vector , where is the voltage across the capacitor , is the current flowing through the inductor L(characterized by inner resistance ), and is the voltage across the capacitor .
Therefore, system (
1) can be reorganized as
The main objective of the bidirectional converter is to regulate the power flow between the main aeronautical bus and the battery side. Several objectives can be achieved through regulation of the power flow, for instance: regulation of the bus side voltage in case of generator failure, control of converter input current in order to limit the generator current below a given critical value, or recharge the battery in constant current or constant power mode. In this paper, the control of converter input current
, aimed at generator current limitation, is tackled. Such a challenge can be described as the problem of regulating current
, indicated in
Figure 1 and calculated as
to a given piecewise constant reference current
.
This control objective can be achieved by properly regulating the duty cycle signals
and
. Several modulation strategies are available for the control of the class of bidirectional converters as in
Figure 1, e.g., hard-switching and soft-switching [
30]. In the case of hard-switching, the two legs commute alternatively, that is
is kept to 1,
operates in boost mode and operates,
is kept to 1, and
operates in buck mode. The soft-switching modulation is instead characterized by the simultaneous switching of the two legs. As detailed in [
31], where a similar application is investigated with a focus on the network parameters optimization, both modulation strategies have their pros and cons; however, hard-switching is preferred over soft-switching modulation because of its lower losses. For this reason, also in this work hard-switching modulation is considered.
3. System Stability
In this section, it will be formally proven that system (
2) cannot undergo any unstable behavior despite the selected control law (note that here the instability notion is intended in the sense of Lyapunov [
32]). This result is achieved by resorting to the concept of
Input-to-State Stability (ISS). Preliminarily, let us recall the definition of ISS and its Lyapunov characterization, where comparison function definitions from [
32] are adopted.
Definition 1 (Input-to-State Stability [24]).The systemwith , piecewise continuous in t and locally Lipschitz in x and u, is said to be ISS if there exist a class function β and a class function γ, such that, for any initial state and any bounded input , the solution exists for all and satisfies Definition 1 provides an estimation of the bound of the system state norm provided that the input norm is bounded. The bound depends on the sum of two terms, the first depending on the initial condition of the system and decreasing with time, while the second depends on the input infinite norm. A sufficient condition for the ISS property is given by the following Lyapunov-like theorem [
24].
Theorem 1 (ISS Lyapunov Characterization [24]).Consider system (4), and let be a continuously differentiable function, such thatholds for all , where , , α are class functions, and χ is a class function. Then, system (4) is ISS. It is now possible to characterize the stability of system (
2) with respect to its uncontrolled input
. This stability result is summarized in the following theorem.
Theorem 2 (ISS Property of System (2)). The switched system (2) is ISS with respect to the uncontrolled input vector Γ for any arbitrary switching signal. Proof. In order to prove the theorem statement, a reasonable choice for the Lyapunov function is the system energy function, that is
Its time derivative along the trajectory of system (
2) is
Consider the property for the square of a binomial
, and apply it to the last two terms of (
9). More precisely, it holds that
Therefore, the time derivative of the Lyapunov function is bounded by
which implies that system (
2) is ISS with respect to the input
with
□
The ISS proof for system (
2) has dual importance. Firstly, it can be seen that the time derivative of the selected Lyapunov function does not depend on the input terms
and
. This, in turn, implies that stability of the system (intended as boundedness) is not affected by the duty cycle trajectories
and
. Secondly, having proved the ISS property implies that the norm of the system state is bounded with a bound dependent on the norm of
. This means that there exist positive scalars
and a negative scalar
, such that
,
, and
. Note that positivity of
and
is not directly implied by the ISS property, but it is merely based on physical and application considerations (in fact,
and
represent capacitor voltages and can be assumed to be always positive). Hereafter, we will indicate with
the set containing the state trajectories, that is
.
Remark 1. As described in [24], the ISS property of system (2) could have been equivalently demonstrated by proving that: (1) system (2) is zero globally asymptotically stable (0-GAS) and (2.i) system (2) has the asymptotic gain property (AG) or (2.ii) system (2) has the limit property (LIM). However, while verification of the 0-GAS property is trivial (note that the unforced system is linear), proving AG or LIM for system (2) is more complex than the proof provided in Theorem 2. As a consequence of the above discussion, the control law can be designed with the sole aim of tracking the reference rather than concerning about the stability in the Lyapunov sense of the overall system.
5. Simulation Results
The proposed control algorithm for current control of a buck–boost converter and the associated supervisory strategy were tested in a detailed MATLAB/Simulink/ SimPowerSystem simulator, shown in
Figure 4, and it is composed of six blocks:
Supervisor: implements the supervisory finite-state machine designed according to
Figure 3.
Controller: implements the control algorithms presented in Theorems 3 and 4.
Modulator: realizes the PWM switching modulation for switches , , , and .
Reference Generator: this block selects the proper reference converter input current to be tracked by the active controller.
Loads: this block contains a set of resistors which can be activated or deactivated during the operating time.
Battery: this is the implementation of an accurate battery model [
35].
The converter and network parameters are shown in
Table 1.
Specifically, V indicates the nominal value of the aeronautic generator voltage connected to the main bus, while V indicates the nominal value of the battery voltage. The switching frequency of the PWM generator is indicated here with f. In nominal conditions, i.e., when both generator and battery operate with nominal voltage, the converter acts as a buck converter, thus control laws (37) and (38) are adopted. However, there may be some cases when the bus voltage becomes smaller than the battery voltage. For instance, this may occur in case of a generator operating with reduced capabilities, or in the case of over-voltage of the battery. In such scenarios, the converter will perform as a boost converter. In both buck and boost mode, the control goal is to control the input converter current to a given reference value, despite sudden load variations and parameter uncertainty. The reference value of the current is provided by the reference generator block. Typically, in the framework of the MEA, the reference generator block is implemented as an external higher-level supervisor responsible for selecting the adequate operating set points for each electrical subsystem. A possible logic behind the selection of the current reference is the following:
If the generator current is below a threshold value (indicated with
in
Table 1), the objective is to control the converter input current to a prescribed positive value in order to charge the battery.
Otherwise, if the bus current overcomes such a threshold value, the objective is to drive the bus current to the threshold value. In this case, the battery current is either lowered or reversed so that the battery will help the aeronautic generator feeding the connected loads.
Nevertheless, the detailed analysis of such high-level logic is not within the scope of this work, and further details can be found in [
36].
In what follows, two simulation scenarios are presented in order to show the effectiveness of the proposed control architecture and its reference-tracking capabilities both in boost and in buck mode.
5.1. Simulation Results in Boost Mode
Boost mode is activated when
. In this case, the generator and the battery voltage values were set to 300 V and 450 V, respectively, and the controller (
20) was selected. The numerical values of the control parameters for these algorithms are presented in
Table 2.
In the designed simulation test, the power grid undergoes sudden load variations indicated in
Table 2b. During the first 5 s of simulation, the reference current is set to
A, thus the battery is being charged, and as shown in
Figure 5, such reference is tracked with good performance in terms of transient time. During this time interval, the generator feeds both the battery and the connected loads, and its current is equal to
A. At 5 s the resistance of the total connected loads suddenly changes to
, as reported in
Table 2b, the generator current increases above its current threshold
, and a new value of
is provided. In this case, the battery current is negative, meaning that the battery is supporting the generator to feed the loads. The rest of the simulation test proceeds with the same rationale: a current reference
is provided according to the connected load and the control algorithm in (
20) successfully tracks the reference with good performance. The active duty cycle
, the generator side, and the battery side voltages are shown in
Figure 6, while
Figure 7 shows the evolution of the sliding function in (
12).
5.2. Simulation Results in Buck Mode
Buck mode is activated when
. In this case, the generator and the battery voltage values were set to
V and
V, respectively, and the controller (37) and (38) is selected. Specifically, as detailed in Theorem (4), the control law (37) is chosen when
, while control (38) is chosen when
. The numerical values of the control parameters for these algorithms are presented in
Table 3a.
Similar to the case of Boost mode, in the proposed scenario for Buck mode, sudden variations for the total connected loads and the converter reference current occur. Initially, the resistance of the total connected loads equals
, and the current reference is set to
A. As evident from the upper plot in
Figure 8, the control algorithm manages to drive the converter current to the given reference. The plot in the center of
Figure 8 shows that the generator current is below the threshold value of
A, while the battery is charging with a current approximately equal to
A. After 5 s, the total resistance of the connected loads changes to
, while the converter reference current remains equal to
A. In this case, the control algorithm proves its robustness against sudden load variations. Indeed, the two-stage controller manages to almost immediately steer the converter current to
A. As can be appreciated in the middle plot of
Figure 8, the generator current increases to a value approximately equal to
A, still below the generator current.
At time
s, the total resistance of the connected loads is
. In this case, the generator current spikes to a value above the generator threshold. As shown by
Table 3b, the converter reference current is switched to
A. The selected reference value guarantees that the generator current is driven to the threshold value within one second. In this case, as evident from the bottom plot of
Figure 8, the battery current is negative, thus implying that the battery is now supporting the main generator in feeding the loads.
At time s, the total resistance of the connected loads is , while the converter reference current is set to A. Similar to the previous case, the converter input current is driven to the desired reference in a monotonic way, and the generator current is set to its threshold value. The battery current is still negative but with a smaller magnitude.
The total resistance of the connected loads becomes equal to
at
s; this allows for selection of a positive reference value for the converter input current equal to
A. As in the interval
s, the generator current reaches an approximate value of
A, and the converter input current reaches the given reference. It is interesting to notice how, as stated in the theoretical analysis, the current
monotonically converges towards the reference both when
and
(see also
Figure 9).
Finally, at
s, the load resistance is set equal to
, while no variation in the converter input current reference occurs, thus resorting the initial configuration of the network. The active duty cycle
, the generator side, and the battery side voltages are shown in
Figure 10, while
Figure 11 shows the evolution of the sliding function in (
12).