Cui Yang

Nicolas Masquellier

Camille Gandiolle

Xavier Sauvage
email: xavier.sauvage@univ-rouen.fr

Multifunctional properties of composition graded Al wires

Keywords: AlMgSi alloy, graded material, mechanical strength, electrical conductivity, wire

come

Architectured and graded materials have attracted a wide interest since they offer new opportunities for the optimization or combination of properties [1,2]. Metallic alloys can be architectured or graded by tuning the local composition or microstructural features. It has been shown for example that surface processing by mechanical treatments may successfully produce gradients of structural defects (dislocations, grain boundaries or twins) and internal stresses that positively affect fatigue resistance [3,4] or evade the strength-ductility trade-off [5][6][7][8][9]. Chemical gradients provide a larger playground to tune locally microstructures and combine properties. They can be created by bottom-up techniques, such as vapor deposition [10], additive manufacturing [11,12] or electrodeposition [13], or by topdown processes like solid state diffusion [14] or oxidation [2].

Materials for electrical conductors that have to sustain mechanical loads are designed so that the trade-off between strength and electrical conductivity is optimized for the envisioned application.

When the weight is a critical factor, aluminum conductors are usually preferred even if they exhibit a maximum conductivity of only 62% of the International Annealed Copper Standard (IACS). For high strength applications, the best combination of properties is usually achieved with alloys from the AlMgSi system, like the 6101 and 6201 industrial alloys [15,16]. However, for applications where the main solicitation is fatigue in torsion or bending (like medium-voltage loop cables of windmills or wires for industrial robots), graded wires with a high Mg and Si content in the outer part (for maximum local strength) and low Mg and Si content in the core (for a maximum local conductivity) should help to optimize the wire performances. Thus, the aim of this work was to investigate the potentiality of composition graded AlMgSi wires for optimized combination of electrical conductivity and torsion strength.

Composition graded AlMgSi wires were processed in two steps (Fig. 1). At first, a rod (diameter 10mm) of 1370 commercial aluminum alloy was inserted in a tube of 6201 commercial aluminum (outer diameter 16 mm). Before assembly, materials were sand blasted and cleaned with ethanol. The nominal composition of these alloys is given Table 1. Then, this assembly was cold drawn at a drawing speed of 0.2 m/s with diameter reductions ranging from 10 to 20% per pass. Intermediate annealing treatments of 1h at 550°C have been performed at a current diameter of 10 and 4mm to recover ductility and allow further plastic deformation to reach a final diameter of 1mm. In a second step, composition gradients were tuned by thermal diffusion at 550°C (duration up to 50h). This temperature corresponding also to the solution treatment temperature of the 6201 alloy, materials were water quenched at the end of this treatment to keep Mg and Si in solid solution. Then, an aging treatment was performed at 190°C during 4h to achieve precipitation.

Figure 1: Schematic representation of the processing route designed to achieve composition graded Al-Mg-Si wires (see text for details).

The gradient of mechanical properties was evaluated thanks to nano-indentation measurements using a TriboIndenter Hysitron TI 950 with a Berkovitch indenter and a penetration depth of 500 nm (corresponding to an indent size of about 4 m). The global mechanical strength and conductivity were then assessed. Tensile properties were evaluated with a gauge length of 8mm at a constant cross-head speed of 0.01mm/s. The torsion behavior was evaluated on a torsion bench with a rotation speed of 60rev/min on a 0.265m long sample under a tensile stress corresponding to 25% of the yield stress.

The torque and length variations were measured as a function of the number of revolutions. Electrical properties were measured at 20°C on wires with a length of 0.5m with a four-point probe method.

Microstructures and especially nanoscaled precipitates were characterized by Transmission Electron

Microscopy (TEM). Samples were sliced in the cross-section of wires and electron transparency was achieved thanks to standard electropolishing procedures followed by ion milling using a Precision Ion Polishing System (PIPS, Gatan) at a voltage of 3 kV. Observations were carried out in a JEOL-ARM 200F microscope operated at 200kV. Si and Mg concentration in the wire core were also evaluated thanks to Atom Probe Tomography (APT). Samples were prepared by standard electropolishing procedures and analyzed using a LEAP 4000HR at a temperature of 30K, with electric pulsing (20% pulse fraction, 200kHz).

To validate the experimental approach and to determine optimal diffusion times required to establish composition gradients, solute diffusion and precipitation were investigated in 4mm diameter wires (Fig. 2). Since dislocations introduced during the drawing process also affects the hardness, a short annealing treatment of one hour at 550°C was applied to recover these defects and achieve a reference state. The resulting nano-hardness gradient is relatively sharp (Fig. 2a) and it significantly extends after 50h at 550°C. Solute elements in aluminum enhance the yield stress due to interactions with mobile dislocations. The yield stress increment  is a function of the solute concentration and can be written as [17,18]:

∆𝜎 = ∑ 𝑘 𝑖 𝐶 𝑖 2/3 𝑖 (1)
Where ki are constants and Ci the concentrations of solutes i. Thus, the hardness being proportional to the yield stress (Tabor's law [20]), hardness profiles measured before the precipitation treatment may provide a qualitative estimation of solute concentration profiles in the composition graded wires. Si and Mg in solid solution do not contribute equally to yield stress increment [17]. However, the Mg/Si ratio should not change much along the gradient since these elements exhibit similar diffusion coefficients at 550°C [19]. Then, an average contribution k * might be considered and assuming proportionality between hardness and yield stress increment (Tabor's law [20]), the hardness increment H might be written from equation (1) as:

∆𝐻 = 𝑘 * 𝐶 𝑆𝑖+𝑀𝑔 2/3 (2)
Where CSi+Mg is the Mg plus Si concentration (wt.%). After one hour at 550°C, the nanohardness difference between the two plateaus corresponding to the 1350 and 6201 alloys is about 300MPa. The solute concentration difference between these two alloys being of about 1.5wt.% (see Table 1), it leads to k*230 MPa wt% -2/3 . Solute concentration profiles displayed on Fig. 2a were estimated thanks to equation (2). After 50h annealing, at 1.5mm from the wire axis, the amount of Mg and Si is still relatively high, between 1 and 1.5 wt.%, while it stands below 0.5 wt.% at a distance of 1mm. Since, the diffusion coefficients of Mg and Si are relatively close at 550°C [19], they can be considered as a unique solute to estimate the composition gradient from the Fick's second law written for a semi-infinite diffusion couple [START_REF] Paul | Thermodynamics, diffusion and the Kirkendall effect in solids[END_REF]:

𝐶 𝑆𝑖+𝑀𝑔 (𝑥) = 𝐶 𝑆𝑖+𝑀𝑔 0 2 [1 + erf ( 𝑥 2√𝐷𝑡 )] (3)
Where C 0 Si+Mg is the initial Si+Mg concentration of the 6201 alloy, D the mean diffusion coefficient, t the time and x the distance from the initial interface. The composition profile estimated after 50h at 550°C on Fig. 2a was fitted with this equation with a diffusion coefficient D = 10 -9 cm 2 .s -1 (black dotted line). This is significantly less than diffusion coefficients reported for Mg or Si in Al (at 550°C, they are relatively similar for these two elements, being respectively 8. 10 -8 and 10 -7 cm 2 s -1 [19]). This difference could be due to interactions between Mg and Si that would reduce their mobility. It may also be the result of a not perfectly bonded interface between the 6201 and the 1350 Al alloys during the codrawing process. Anyway, this mean diffusion coefficient allowed the determination of an optimal annealing time for 1mm diameter wires. To achieve a composition gradient spreading over one third of the radius, two hours at 550°C are necessary.

Figure 3: Nanohardness maps of the cross section of the 1mm wire after 2h annealing at 550°C to achieve a composition gradient (a) and after subsequent aging at 190°C during 4h to achieve precipitation hardening (b).

Once diffusion gradients are established, a classical precipitation treatment of 4h at 190°C can be applied to achieve mechanical strengthening like in standard 6201 alloys. The large difference in Mg and Si concentrations along the gradient naturally leads to very different nanoscaled precipitates [START_REF] Marioara | [END_REF]23]. At a distance of about 1.5mm from the axis of a 4mm-wire, a relatively high density of needle shape nanoscaled precipitates was indeed clearly revealed using bright field TEM (Fig. 2b). While shifting toward the wire axis, they progressively disappear: At 1mm (Fig. 2c), a much lower number density of nanoscaled particles could be seen and most of them are not needle shape like the classical ' phase [START_REF] Marioara | [END_REF]. The hardening resulting from the nucleation of theses precipitates is exhibited on the nano-hardness maps of the 1mm wires (Fig. 3). The original axisymmetric structure of the composite created with the 1350 alloy rod inserted in the 6201 alloy tube was relatively well preserved during the drawing process. In the precipitated state (Fig. 3b), the gradient of properties is nicely exhibited on the map, with a nano-hardness ranging from a mean value of 1.2 GPa at the wire edge, down to 0.4 GPa at the center. Thus, as expected, a strong hardening occurred due to precipitation [17,24] in the solute rich outer part of the wire. The core (about 0.5mm in diameter) did not significant harden because only a very small fraction of solutes has migrated into this region during the diffusion annealing treatment. APT analyzes carried out in the wire core region revealed indeed that as compared to the original 1350 alloy, the amounts of Si and Mg have increased of only about 0.1 and 0.01wt.% respectively (see Table 1).

The tensile behavior of this composition graded wire was compared to the conventional 6201 and 1350 aluminum alloys (Fig. 4a and Table 1). Before the precipitation treatment, when only a solute concentration gradient was created by annealing, the yield stress is slightly higher than the 1350 alloy (30 MPa against 15MPa), but the strain hardening rate is much larger, giving rise to a significantly larger Ultimate Tensile Stress (UTS) (90 MPa against 40 MPa). This is the result of the interaction between moving dislocations and solute elements that typically gives rise to highest dislocation densities. After the precipitation treatment during 4h at 190°C, the yield stress dramatically increases (up to 115MPa) but does not reach the yield stress of the 6201 alloy treated in similar conditions (250MPa). The uniform elongation of the graded material is however twice larger than that of the 6201 alloy. This might be attributed to the soft and ductile wire core whose properties were relatively unchanged (Fig. 3) and which might help to delay necking and thus damage propagation. As shown on Fig. 4b, the combination of yield stress and electrical conductivity of the composition graded wire stands close to the prediction based on a linear rule of mixture and on properties of the original 1350 and 6201 alloys. Thus, the gain in electrical conductivity obtained thanks to the low Mg and Si concentration in the wire core is fully balanced by the drop of yield stress. The combination of properties is even slightly below the rule of mixture prediction, probably because of the small amount of Mg and Si that have migrated in the core region of the wire. Indeed, the contribution of Si and Mg in solid solution to the electrical resistivity of aluminum is relatively similar (about 5 10 -7 cm at.% -1 [18,25]), and an increase of 0.1at.% of solute as measured by APT, would shift the conductivity of the wire core by about -1%IACS. However, the torsion behavior evaluated thanks to the torque at yield in torsion shows that the graded materials stands above the prediction based on a linear rule of mixture with a positive shift of about 1%IACS (Fig. 4b). This is obviously the result of the hardness gradient displayed on Fig. 3b, since a wire under torsion stress has to sustain the maximum load in the skin while the shear stress linearly decreases down to zero on the axis. Besides, like under tensile stress, a larger uniform plastic deformation (20% more) could be achieved for the graded wire under torsion as compared to the 6201 alloy (Table 1).

In conclusion, this work demonstrates the potentiality of composition graded aluminum wires for applications where an optimal combination of electrical conductivity and mechanical strength in torsion is required. It is envisioned that tuning the Mg and Si contents of the starting outer AlMgSi alloy (especially using higher concentrations than in the 6201 alloy of the present study) could open a broad panel of Al wires with optimized multifunctional properties.

Tables

Figure 2 :

 2 Figure 2: Figure 2: (a) Nanohardness profiles (green) in the cross section of the 4mm wire after 1h (circles) and 50h (squares) annealing at 550°C, and corresponding concentration gradients (blue) estimated from Eq. (2) -see text for details. The black dotted line is a fit done with Eq. (3) with C 0 = 1.5 at.%, t=50h and D= 10 -9 cm 2 .s -1 ; TEM bright field images (SAED patterns insets) showing a relatively high density of needle shape nanoscaled precipitates (circled) in Zone A located at a distance of 1.5mm of the wire axis (b) and significantly less in Zone B located at 1mm (c), after 50h annealing at 550°C and 4h aging at 190°C.

Figure 4 :

 4 Figure 4: (a) Strain stress plots of 1mm wires of 6201AA (homogenized and aged 4h at 190°C), 1350AA (annealed at 550°C), composition graded material (annealing 2h at 550°C followed by quenching), and composition graded material after precipitation treatment at 190°C during 4h. (b) Yield stress and torque at yield versus the electrical conductivity of the wires. Doted lines correspond to extrapolations based on a linear rule of mixture between the two original alloys.

Table 1 :

 1 Tensile, torsion and electrical experimental data of the two reference materials (6201 AA homogenized and aged 4h at 190°C, 1350AA annealed at 550°C) and of the 1mm composition graded wire (annealed 2h at 550°C, quenched and aged at 190°C during 4h). The yield stress is the conventional engineering stress measured at 0.2% strain and UTS stands for Ultimate Tensile Stress. The conductivity is given in % of international annealed copper standard (IACS). Si and Mg concentrations are nominal compositions for 6201 and 1350 aluminum alloys and composition measured by APT at the center (±0.1mm) of the 1mm composition graded wire.

		6201AA	1350AA	Composition graded
		(4h190°C)		(4h 190°C)
	wt.% Mg	0.6 -0.9	---	at wire center
				0.01±0.008
	wt. % Si	0.5 -0.9	< 0.1	at wire center
				0.2±0.001
	Yield stress (MPa)	250 ± 20	15 ± 5	115 ± 10
	UTS (MPa)	260 ± 20	40 ± 5	150 ± 10
	Uniform elongation	3.5 ± 0.5	20 ± 5	7 ± 1
	(%)
	Torque at yield	25 ± 5	7 ± 1	15 ± 5
	(mN.m)
	Maximum torque	40 ± 10	17 ± 2	30 ± 5
	(mN.m)
	Torsion at torque max	100 ± 10	800 ± 80	120 ± 10
	(rev/m)
	Electrical conductivity	52 ± 0.5	62 ± 0.5	56.5 ± 0.5
	%IACS

Acknowledgements

Prof. Jean-Hubert Schmitt is gratefully acknowledged for fruitful discussions.


Antonino Sferlazza
email: antonino.sferlazza@unipa.it

Carolina Albea-Sanchez

Germain Garcia
email: germain.garcia@laas.fr

A Hybrid Control Strategy for Quadratic Boost Converters with Inductor Currents Estimation

Keywords: Quadratic boost converter, hybrid dynamical systems, switching systems, observer, linear matrix inequalities

This paper deals with a control strategy for a DC-DC quadratic boost converter. In particular, a hybrid control scheme is proposed to encompass a control law and an observer for the estimation of the system states, based only on the measurements of the input and output voltages. Differently from classical control methods, where the controller is designed from a small-signal model, here the real model of the system is examined without considering the average values of the discrete variables. Using hybrid dynamical system theory, asymptotic stability of a neighborhood of the equilibrium point is established, ensuring practical stability of the origin, which contains estimation and tracking errors. Experimental results show the effectiveness of the proposed approach.

Introduction

Nowadays, the development of renewable power sources drives the interest towards DC-DC conversions, in particular to control DC-DC converters. Indeed, the most part of these sources are characterized by a low voltage output, i.e. photovoltaic panels, fuel cells, etc, as well as, the most part of storage systems, i.e. batteries supercapacitors, etc. This means that a DC-DC conversion stage is needed in order to allow the distribution of such energy [START_REF] Carrasco | Powerelectronic systems for the grid integration of renewable energy sources: A survey[END_REF]. For this reason, a relevant effort has been done in order to increase both the efficiency of DC-DC converters and their steady-state gain. The improvement of the efficiency of these kind of systems is important, because it allows to reduce losses, such as switching and Joule losses. The steady-state gain is also relevant to reduce the number of conversion stages when a high conversion ratio is required. Together with these characteristics, that are strictly related to circuit topology and selected devices, it is very important to reduce:

• switching frequency,

• transient times, as well as, current and voltage overshoots,

• current ripples, which are influenced by the selected control technique.

A converter particularly suitable for the applications mentioned above is the quadratic boost converter, because it presents a high voltage ratio. Among the existing configurations of these converters, it is possible to distinguish two main topologies. The first one consists in two independent boost stages with two switching devices [START_REF] Chen | Buck-boost PWM converters having two independently controlled switches[END_REF] (in the following referred as CBC), while the second one is composed of only one conversion stage with only one switching device [START_REF] Baek | High boost converter using voltage multiplier[END_REF][START_REF] Zhao | High-efficiency, high step-up DC-DC converters[END_REF] (in the following referred as QBC). Multi-stage configurations can be also found, which achieve a very high input-output ratio [START_REF] Luo | Positive output cascade boost converters[END_REF]. There are some works in the literature, where QBC and CBC are compared. For example, [START_REF] Choudhury | Comparison and analysis of cascaded and quadratic boost converter[END_REF] shows that both of them can deliver a high voltage gain, but the stress in QBCs is higher, increasing the cost of these converters. Nevertheless, the switching losses for the QBC are reduced with respect to the CBC, allowing to obtain a higher efficiency.

In [START_REF] García-Vite | A DC-DC converter with quadratic gain and input current ripple cancelation at a selectable duty cycle[END_REF], the authors propose a QBC that allows to increase the voltage gain and to drain the ripple-free current from the source. These features have a special interest in renewable energy systems, that are composed of low voltage generation sources and are sensitive to current ripples, such as in fuel cells applications. Other works related with the design of QBC with low current and low voltage stress for fuel-cell applications are presented in [START_REF] Al-Saffar | A high voltage ratio and low stress DC-DC converter with reduced input current ripple for fuel cell source[END_REF]; [START_REF] Wang | A novel quadratic boost converter with low current and voltage stress on power switch for fuel-cell system applications[END_REF]. Moreover, in [START_REF] Morales-Saldaña | Modelling and control of a DC-DC quadratic boost converter with r 2 p 2[END_REF], the authors provide a QBC based on the reduced redundant power processing (R 2 P 2 ) principle, as well as, on a controller design methodology using current-programmed control satisfying the specifications about output voltage regulation. [START_REF] Chen | Variable duty cycle control for quadratic boost PFC converter[END_REF] shows a QBC with a higher Power Factor Correction (PFC), obtained from a variable duty-cycle control. [START_REF] Alonge | Dynamic modelling of a quadratic DC-DC single-switch boost converter[END_REF] proposes a transformer-less single switch topology of QBC. In [START_REF] Patidar | High step-up converters based on quadratic boost converter for micro-inverter[END_REF], the authors deal with QBCs for micro-inverter applications that integrate a high efficiency in the step-up capabilities.

Focusing on the control techniques adopted for the QBC, it is useful to mention some non-standard robust control solutions [START_REF] Olalla | Robust optimal control of bilinear DC-DC converters[END_REF]. For example, a double loop sliding-mode controller is proposed in [START_REF] Lopez-Santos | Robust sliding-mode control design for a voltage regulated quadratic boost converter[END_REF]. In partic-ular, the proposed controller presents an inner loop based on sliding-mode control, whose sliding surface is defined for the input inductor current. The current reference value of the sliding surface is modified by a Proportional-Integral (PI) compensator in an outer loop that operates over the output voltage error. [START_REF] López-Santos | Efficiency analysis of a sliding-mode controlled quadratic boost converter[END_REF] presents an efficiency analysis for such converters, while an interesting description of the steadystate behavior is shown in [START_REF] Lopez-Santos | Steady-state analysis of inductor conduction modes in the quadratic boost converter[END_REF]. Note that the idea of a double control loop, with an inner current loop, is often applied by using linear averaged models to design a controller under peak current-mode control [START_REF] Leyva-Ramos | Switching regulator using a quadratic boost converter for wide DC conversion ratios[END_REF][START_REF] Morales-Saldana | Multiloop controller design for a quadratic boost converter[END_REF], or whose goal is to obtain a fixed-frequency PWM-based sliding mode controller [START_REF] Chincholkar | Design of fixed-frequency pulsewidthmodulation-based sliding-mode controllers for the quadratic boost converter[END_REF]. Nevertheless, in this work the authors use a reduced number of state variables, specifically they only use one current for its implementation instead of two. Finally, related with robust control solutions, there is [START_REF] Alonge | Identification and robust control of a quadratic DC-DC boost converter by hammerstein model[END_REF], where the authors consider the framework of Hammerstein's model for proposing a robust controller. Indeed, a set of fourth-order transfer functions are identified with the Hammerstein's approach, and the set of identified transfer functions are used to design a suited robust control technique that allows to deal with converter parameter uncertainties and load variations.

All these mentioned works, except few cases [START_REF] Chen | Variable duty cycle control for quadratic boost PFC converter[END_REF][START_REF] Lopez-Santos | Robust sliding-mode control design for a voltage regulated quadratic boost converter[END_REF], are based on a constant switching frequency, changing the duty cycle for tracking the voltage reference. Moreover, the design of the controller is based on a small-signal model obtained from the averaged statespace system. This means that the design specifications hold only in a neighborhood of the equilibrium point. For these reasons, theories about hybrid dynamical systems present a suited framework for modeling switching power systems, since it is possible to capture both continuous-and discrete-time dynamics and combine them in a unique model, without approximations. Moreover, the constraint about a fixed switching frequency is not more strictly necessary in this kind of context. Some preliminary results about the application of these hybrid theory to buck-boost converters are presented in [START_REF] Iannelli | Subtleties in the averaging of a class of hybrid systems with applications to power converters[END_REF]; [START_REF] Mariethoz | Comparison of hybrid control techniques for buck and boost DC-DC converters[END_REF]. Likewise, [START_REF] Albea-Sanchez | Hybrid dynamic modeling and control of switched affine systems: application to DC-DC converters[END_REF]; [START_REF] Sferlazza | Min-type control strategy of a DC-DC synchronous boost converter[END_REF] propose control solutions for DC-DC converters in which stability guarantees are established using recent tools from hybrid dynamical systems theory. Moreover, [START_REF] Sanchez | Practical stabilization of switched affine systems with dwell-time guarantees[END_REF] provides practical stabilization of operating points for switched affine systems, ensuring a minimum dwell-time associated with an LQR performance level during the transient response and an admissible chattering around the operating point. Using the same framework, in Albea [START_REF] Sanchez | Practical stabilization of switched affine systems with dwell-time guarantees[END_REF] a similar controller is proposed, but for a half-bridge inverter.

This work is inspired from Albea- [START_REF] Albea-Sanchez | Hybrid dynamic modeling and control of switched affine systems: application to DC-DC converters[END_REF], for controlling a QBC, whose topology, from the electronic architecture point of view, is more complex than the standard boost converters selected in Albea- [START_REF] Albea-Sanchez | Hybrid dynamic modeling and control of switched affine systems: application to DC-DC converters[END_REF]. In particular, the advantage of this QBC, with respect to a classical boost converter, is the higher steady-state gain (two times higher, approximately). Indeed, to obtain the same steady-state gain with a boost converter, one has to implement two converters con-nected in cascade, which requires two switching, increasing the commutation losses. In the QBC, one can obtain the same gain using only one conversion stage increasing the efficiency.

The novelty of this paper is to integrate a state observer in a hybrid control scheme of the converter to avoid using current sensors. To the best of Authors' knowledge, an observer has not been designed before for a switched affine system controlled with a Lyapunov matrix-based min-projection control, in a hybrid dynamical model. This fact is particularly important because it allows to remove current sensors. From a practical point of view, the lack of current sensors can improve the production cost of converters. Indeed, if currents are not measured, it is not needed to employ neither current sensors, nor Analog to Digital Converters (ADC) any more; and the dimensions of the board could be better optimized. Moreover, the fact of using a QBC, which is four dimensions, justifies the use of an observer to reduce the number of sensors. The stability of the complete system is established in terms of Uniform Global Asymptotic Stability (UGAS) of a bounded and closed set that includes the operating point, providing UGAS of both tracking error of the equilibrium point and estimation error. It is useful to point out that the property of UGAS of such set implies the robustness of the complete control loop with respect to small perturbations around the operating point.

Finally, experimental results show the effectiveness of the proposed method. Indeed, the proposed hybrid control technique is compared with a PI controller, which is designed from an averaged model in continuous time, showing that the proposed strategy presents two main advantages in transient times. The first one is a direct control of the voltage and current trajectories, allowing to obtain no-inrush current at start-up, as well as, a better tracking of the voltage reference. The second advantage is the possibility of choosing the trajectory by tuning a pre-defined parameter, which allows to act directly in the number of switching, and consequently to act on the ripple of the inductor current.

The rest of the paper is organized as follows. The QBC model and the problem statement are described in Section 2. An observer for the unmeasured variable, as well as a hybrid control strategy are designed in Section 3 and 4 respectively. Experimental results are presented in Section 5. The paper closes with a section of conclusions.

Notation: Through out the paper R denotes the set of real numbers, R n the n-dimensional euclidean space and R n×m the set of all real n × m matrices. M 0 (resp. M ≺ 0) represents that M is a symmetric positive (resp. negative) definite matrix. The symbol "ˆ" connotes the estimated variables. The Euclidan norm of vector x ∈ R is denoted by x . λ m (M) and λ M (M) represent the minimum and maximum eigenvalues of M.

Mathematical model and problem formulation

The electric scheme of the QBC under analysis is depicted in Fig. 1. It exhibits the presence of two inductors (L 1 , L 2 ), two capacitors (C 1 , C 2 ), four diodes (D 1 , D 2 , D 3 , D 4 ) and one electronic switch (S 1 ). The parasitic resistances of the inductors (r L 1 , r L 2 ) and of the capacitors (r C 1 , r C 2 ) are also considered.

The mathematical model of this QBC can be expressed as a switched system in a state-space formulation, using the formalism given in [START_REF] Liberzon | Switching in systems and control[END_REF]. In particular, under the assumption that the converter works in Continuous Conduction Mode (CCM), two operation modes can be considered, called, respectively, mode ON and mode OFF. The converter works in mode ON when the electronic switch, S 1 , is turned on, otherwise it works in mode OFF. For this reason, by applying the Kirchhofs's law to the circuit in Fig. 1, it is possible to write the following equations

       ẋ = A u x + B y = C u x, (1)
where

x = i L 1 i L 2 v C 1 v C 2 is the state vector, u ∈ {0, 1}
is the switched control input, such that, u = 1 means that the switch S 1 is turned on, while u = 0 means that the switch S 1 is turned off, and y = V OUT is the output voltage. The matrices of the model ( 1) are

A 0 =                     - r L 1 R t +r C 2 R t 1 L 1 R t r C 1 r C 2 L 1 R t r C 2 L 1 R t - R t 1 L 1 R t r C 1 r C 2 L 2 R t - r L 2 R t +r C 1 R t 2 L 2 R t - R t 2 L 2 R t r C 1 L 2 R t - r C 2 C 1 R t R t 2 C 1 R t -1 C 1 R t -1 C 1 R t R t 1 C 2 R t - r C 1 C 2 R t -1 C 2 R t -1 C 2 R t                     , A 1 =                   - r L 1 L 1 0 0 0 0 - r L 2 R t +r C 2 R t 1 L 2 R t - r C 2 L 2 R t R t 1 L 2 R t 0 r C 2 C 1 R t -1 C 1 R t -1 C 1 R t 0 - R t 1 C 2 R t -1 C 2 R t -1 C 2 R t                   , B =                V IN L 1 0 0 0                , C 0 =                   R 0 r C 1 R t R 0 r C 1 R t R 0 R t R 0 R t                   , C 1 =                  0 R 0 r C 2 R t R 0 R t R 0 R t                  . where V IN is the input voltage, R t = R 0 +r C 1 +r C 2 , R t 1 = R 0 +r C 1 and R t 2 = R 0 +r C 2 .
As mentioned in the Introduction, the most of the works in the literature do not use model (1) for control design purposes, but a State-Space Averaged (SSA) model. The SSA method presents some advantages, but also some drawbacks. The main advantage is the simplicity of the control design. Indeed, the control law can be designed considering a standard linear time-invariant model. However, the drawbacks of this approach are mainly two. The first one, the design is based on a small-signal model, obtained by linearization around the working point. This means that if a wide range of operation is considered, the behaviors of the controlled system could not be appropriated. The second drawback is the constrained structure of the controller, which forces the signal to be updated periodically with a fixed switching frequency. This could facilitate the design of the converter (i.e. the choice of inductances, capacitors, and so on), but it removes a degree of freedom that could be used to minimize the number of commutations, and consequently the energy losses. Conversely, in this work, a different paradigm for approaching the control of the proposed QBC is given, by explicitly considering the switching behavior of the system and by relaxing the constraint of constant switching frequency.

Problem Formulation

The aim of this work is to design a hybrid control scheme for ensuring suitable convergence properties of the plant state, x, to a certain operating point, x e . The specific objectives are:

• to obtain the operating point x * e , which represents the reference state, related with the desired voltage output, V * OUT .

• To design an observer that provides an estimation, x, of the state, x, of the system, based on the input-output voltage measurements, on the input u and on the model of the system (1). The observer has to be designed in order to guarantee the convergence to zero of the estimation error.

• To design a controller, that guarantees the asymptotic stability of the system ensuring the convergence of x to x * e .

One of the main challenge of this work is to provide a controller without having access to the complete state and, considering a more realistic model (differently to the SSA model) that explicitly considers the switching behavior of the system, allowing to manage the switching frequency.

The block diagram of the proposed control scheme is shown in Fig. 2.

Working point selection

As first step, the selection of the operating point x * e is defined, which represents the reference state related with the desired output voltage, V * OUT . Note that a generic equilibrium point x e is not necessarily an equilibrium point for the dynamics (1), but it can be an equilibrium for the switching system in a Filippov's generalization. In the case under study, the set of admissible equilibrium points, Ω e ⊂ R n , is defined as

Ω e := x e ∈ R n | there is λ ∈ Λ s.t. 1 i=0 λ i (A i x e + B) = 0, and 1 i=0 λ i A i is Hurwitz , (2) where Λ = λ ∈ [0, 1] : λ 0 + λ 1 = 1 . (3)
Hence, an equilibrium point is admissible if it is an equilibrium point for the averaged dynamics

ẋ = 1 i=0 λ i (A i x + B) , for [λ 0 , λ 1 ] ∈ Λ, (4)
and stability properties are guaranteed for the corresponding convex dynamics.

Obviously for the case under study the pairs (λ 0 , λ 1 ) and the associated equilibrium points x e ∈ Ω e are infinite. However, the goal is to regulate the output voltage to a fixed value V OUT , which is related to a specific equilibrium point. Then, the following further constraint has to be considered,

y = 1 i=0 λ i C i x e = V OUT .
For these reasons the operating point x * e is defined, with associated values (λ * 0 , λ * 1 ) such that λ * 0 + λ * 1 = 1, as follows:

x

* e = -         1 i=0 λ * i A i         -1 B, for i ∈ {0, 1}, (5)
such that:

V * OUT = 1 i=0 λ * i C i x * e , for i ∈ {0, 1}. (6)
In general, the solution can be founded analytically and/or numerically.

Observer Design

Since the complete system state is not measurable, it is necessary to provide an observer that estimates these quantities, and, particularly, the inductor currents. Note that the system is not observable for all functioning modes. Indeed, when the switch is ON mode, the observability matrix O

1 := C 1 (C 1 A 1 ) (C 1 A 2 1 ) (C 1 A 3 1
) presents a rank equal to 3 for some values of the parameters. Therefore, the system is not observable in this mode. Nevertheless, the QBC is observable when it is working on OFF mode because of the observability matrix

O 0 := C 0 (C 0 A 0 ) (C 0 A 2 0 ) (C 0 A 3 0 )
presents full rank for any values of the parameters. Note that this problem is the same one observed for the standard boost case [START_REF] Liu | Adaptive-gain second-order sliding mode observer design for switching power converters[END_REF], and it presents a physical interpretation. Indeed, it is easy to see in Fig. 1 that when the converter is working on ON mode, there is not information about the current on the inductor, L 1 , which is being inferred from the output. Now, since the system periodically jumps between two modes, i.e. it never remains in ON or OFF mode for an undetermined amount of time, the idea is to design an observer based on (1), driven by the residual

e y = y -ŷ = V OUT -C u x = C u xo , with xo = x -x, (7)
only during the OFF mode.

The proposed switching observer is

ẋ = A u x + B + (u -1)K f C u xo . (8)
Note that when u = 1, the corrective term is equal to zero and (8) becomes the original plant, while for u = 0 the dynamics is driven by the output error (7).

For observer dynamics (8), the following result holds.

Lemma 1. If there exist matrices P o ∈ R 4×4 0 and Q o ∈ R 4×4 0 and a matrix L ∈ R 4×1 , such that, the following Linear Matrices Inequalities (LMIs) are satisfied,

P o A 1 + A 1 P o ≺ -Q o (9) P o A 0 + LC 0 + A 0 P o + C o L ≺ -Q o , (10)
then by choosing the observer gain K f = P -1 o L, x converges to x asymptotically along the dynamics (1) and (8).

Proof. In order to proof the theorem statement, the switching systems theory given in Liberzon ( 2012) is used. In particular, the following common Lyapunov function

V o ( xo ) = 1 2 x o P o xo (11) is considered.
The derivative of (11) along the dynamics (1) and (8) gives

Vo ( xo ) = x o P o ẋo = x o P o A u + (u -1)K f C u xo =        x o P o A 1 xo if u = 1 x o P o A 0 + K f C 0 xo if u = 0 ≺ -x o Q o xo . ( 12
)
It is worth noting that the two assumptions ( 9)-( 10), as well as, the fact that K f = P -1 o L are considered in (12). Moreover, since x+ o = xo , during jumps, i.e., there is not a jump map in the observer dynamics, then

V o ( x+ o ) -V( xo ) = 0. (13)
Consequently, the statement comes directly from Liberzon (2012, Theorem 2.1).

The control Law

Before introducing the proposed control strategy, it is necessary to satisfy the following assumption.

Assumption 1. Consider a given V * OUT and its associated pair (λ * 0 , λ * 1 ) resulted from Section 2.2 and assume that there exist matrices P ∈ R 4×4 0 and Q ∈ R 4×4 0 that satisfy

A u P + PA u + 2Q ≺ 0. u ∈ {0, 1}. ( 14
)
It is easy to see that if each A u is Hurwitz, then conditions ( 14) sre feasible.

Following the formalism given in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF], wherein the continuous-time behavior resembles the evolution in ( 1) and ( 8), and the discrete-time behavior captures the jump of the control signal u from one mode to another one. The overall dynamics are represented as follows

H :=                                                              ẋ u ẋ τ                  = f (x, u, x, τ), (x, u, x, τ) ∈ C                   x + u + x+ τ +                   ∈ G(x, u, x, τ), (x, u, x, τ) ∈ D, (15)
where

f (x, u, x, τ) =               A u x + B 0 A u x+B+(u-1)K f C u xo 1 -ς( τ T )              
, such that, the scalar T > 0 and the dead-zone function defined as ς(s) := max{0, s -1}, ∀s ≥ 0, denotes the flow map. Moreover,

G(x, u, x, τ) =                 x arg min u∈{0,1} ( x-x * e ) P f 3 (x, u, x) x 0                
, where f 3 (x, u, x) = A u x+B+(u-1)K f C u xo denotes a set-valued map capturing the switching logic, and C and D represent the flow and jump sets, respectively, being defined as

C := x, u, x : x P ẋ ≤ -η x Q x × [0, 2T ] ∪ x, u, x, τ : τ ∈ [0, T ] , (16)
D := x, u, x : x P ẋ ≥ -η x Q x × [T, 2T ], (17)
where x = xs xo , with xs = x-x * e , is the vector error containing both the state and the estimation error, such that,

ẋ = Āu x + B, (18)
with Āu :

= A u (u-1)K f C u 0 A u + (u-1)K f C u
and B := B+A u x e 0 . Finally, matrices P and Q are defined as follows

P := P 0 0 P o , Q := Q Q o 0 Q o ,
and η ∈ (0, 1) is a design parameter chosen to achieve a tradeoff between switching frequency and a LQR-optimization level, as given in Albea- Sanchez et al. (2015, Theorem 2). Note that the dead-zone function ς( τ T ) allows to ensure that all maximal solutions are complete. Then, using the hybrid dynamical theory given in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF], stability property of x is guaranteed. Indeed, UGAS of the compact attractor:

A := {(x, u, x) : x ≤ X, u ∈ {0, 1}, τ ∈ [0, 2T ]} (19)
is established.

The following lemmas are considered, before giving the main result.

Lemma 2. Consider a V * OUT and its associated working point x * e and pair (λ * 0 , λ * 1 ) selected from Eq.s (5)-( 6). Moreover, consider P 0 and Q 0 of suited dimension satisfying Assumption 1, and P o 0, Q o 0 and K f such that, conditions (9)-(10) are satisfied. Then, for each x, min u∈{0,1}

x P

A u x+B+(u-1)K f C u xo A u +(u-1)K f C u xo ≤ -x Q x. ( 20
)
Proof. Considering that (5) is linear in λ u , and λ u belongs to the compact set (3), then the following minimum can be computed at the extreme points, min u∈{0,1}

x P

A u x + B + (u -1)K f C u xo A u + (u -1)K f C u xo = min λ u ∈Λ x P           1 u=0 λ * u A u x * e + B 0           ≤ min λ u ∈Λ x P           1 u=0 λ * u A u x * e + B 0           = 0. ( 21
)
Then, from ( 14), ( 9) and (10), min u∈{0,1}

x P

A u x + B + (u -1)K f C u xo A u + (u -1)K f C u xo = min λ u ∈Λ x P A u ( xs + x * e ) + B + (u -1)K f C u xo A u + (u -1)K f C u xo = min λ u ∈Λ x P A u xs + (u -1)K f C u xo A u + (u -1)K f C u xo ≤ -x Q x, (22)
where the second equality comes from A u x * e + B = 0 for u ∈ {0, 1} and the last inequality is derived from ( 14) and Lemma 1.

Lemma 3. ( Albea Sanchez et al., 2019, Lemma 2) All the eigenvalues of the matrix P-1 Q are positive and e Āu t ≤ λ 1/2 M ( P) λ 1/2 m ( P) e -αt , where α = λ m ( P-1 Q). Now, a practical minimum dwell-time property for H is established.

Property 1. There exists a scalar T * > 0, such that, for any chosen T ≤ T * the solutions to hybrid system H flow for at least T ordinary time after the jump, before reaching set D.

Proof. It is considered that in t = t 0 occurs the first jump, letting define t := tt 0 . Then, the trajectories of the dynamics (18) flowing in C, are defined as follows,

x = e Āu t( x0 + Ā-1 u B) -Ā-1 u B.
From the norm properties and applying Lemma 3, it results

x( t) -Ā-1 u B ≤ λ 1/2 M ( P) λ 1/2 m ( P) e αt x0 + Ā-1 u B .
Then, for all 0 ≤ t ≤ T , it follows

0 ≤ t ≤ 1 α Ln        λ 1/2 M ( P) x0 + Ā-1 u B λ 1/2 m ( P) x( t) -Ā-1 u B        .
Note that the solutions are in D, at time t = T . Therefore, dwell-time upper bound, T * ≥ T ≥ 0, can be defined as

T * = 1 α Ln        λ 1/2 M ( P) x0 + Ā-1 u B λ 1/2 m ( P) x(T * ) -Ā-1 u B        .
Hence, the maximum chattering induced by T * is x0 and x(T * ). Now, the main result can be given.

Theorem 1. Consider a V * OUT and its associated working point x * e and pair (λ * 0 , λ * 1 ) selected from Eq.s (5)-( 6). Moreover, consider x * e ∈ R 4×1 , matrices P ∈ R 4×4 0 and Q ∈ R 4 0 satisfying Assumption 1, and matrices P o ∈ R 4×4 0, Q o ∈ R 4×4 0 and vector K f ∈ R 4×1 selected such that Lemma 1 satisfies conditions (9)-(10). Then, for any scalar 0 ≤ T ≤ T * the following holds:

1. set A is compact, 2. set A is UGAS for hybrid system (15),

3. set A 0 := {(x, u, x) : x = {0}, u ∈ {0, 1}, τ ∈ [0, 2T ]} is globally
practically asymptotically stable for system (15). As long as T is small enough, set A can be arbitrarily close to A 0 .

Proof. From the properties of f , G, C and D, system H satisfies the hybrid basic conditions of Goebel et al. (2012, Ass. 6.5), and thus, useful well-posed hybrid results can be applied.

Proof of item 1) Hybrid system (15) satisfies Property 1, then a bound of x in C can be found as follows

x(t) ≤ λ 1/2 M ( P) λ 1/2 m ( P) e αt x(t 0 ) + Ā-1 u B + Ā-1 u B := X,
considering t 0 , a time instant where occurs any jump in the steady-state. Consequently, A is compact.

Proof of item 2) Consider the following Lyapunov function,

V( x) = 1 2 x P x. ( 23
)
During flows

∇V( x), f (x, u, x) = x P A u x+B+(u-1)K f C u xo A u +(u-1)K f C u xo ≤ -x Q x. ( 24
)
This fact comes directly from the flow set C definition, ( 16), and Lemma 1. When a jump occurs,

V( x+ ) -V( x) = 0, ( 25
) since x+ = x+ -x * e x + -x+ = x -x * e x -x = x.
UGAS of A is shown using Prieur et al. (2014, Theorem 1). In particular, since the distance of x to the attractor ( 19

and proving semi-global practical persistence flow for H δ,∆ , for each value of (δ, ∆). In particular, in order to show the practical persistent flow, it needs to be proven that there exist γ ∈ K ∞ and the scalar N ≥ 0, such that all solutions to H δ,∆ satisfy,

t ≥ γ( j) -N, ∀t ∈ j∈dom j ξ I j × { j}, (27)
see [START_REF] Prieur | Relaxed persistent flow/jump conditions for uniform global asymptotic stability[END_REF] for details. To establish ( 27), it results that after each jump,

x P x+ = x P A u + x+B+(u-1 + )K f C u + xo A u + +(u-1 + )K f C u + xo = x P min u∈{0,1} A u + x+B+(u-1 + )K f C u + xo A u + +(u-1 + )K f C u + xo ≤ -x Q x < -η x Q x, ( 28
)
from inequalities ( 9)-( 10), and Lemma 2. Moreover, the strictly inequality is due to the fact that η < 1. Therefore, if any solution to H δ,∆ performs a jump from S δ,∆ it remains in S δ,∆ because x+ = x and x + = x, thus, x does not change. Then, from Proof of item 3) From Property 1, as T tends to zero, the minimum dwell time tends also to zero. Then, set A shrinks to A 0 from a sufficiently small T .

V IN 30V L 1 330µH AGP4233-334ME L 2 470µH AGP4233-474ME r L 1 , r L 1 11.5mΩ C 1 , C 2 20µF MKP1848C62090JP4 r C 1 , r C 1 5mΩ R 0 390Ω D 1,2,3,4 C3D06060A S 1 C3M0065090D Driver 1EDI20N12AF (

Experimental validations

Experimental setup

A test setup was built to validate the proposed hybrid control technique. The complete system is implemented as shown in the block diagram of Fig. 2, where the observer is described by Equation ( 8), and the hybrid controller selects the control variable u according to the dynamics shown in (15). The employed test setup consists of:

• a QBC, whose topology is shown in Fig. 1 and the electrical parameters are given in Table 1.

• An electronic card with current sensors 1 (LEM LTS 15-NP) for the measurements of the inductor currents, and voltage sensors 1 for the measurement of the capacitor voltages. Voltage sensors have been built by means of resistor dividers connected with operational amplifiers in buffer configuration.

• A dSPACE card (DS1103) with a PowerPC 604e at 400 MHz and a fixed-point DSP TMS320F240.

The system was implemented in a dSPACE R card by means of Matlab-Simulink R . Both the QBC and the measurement card were embedded in the electronic card shown in Fig. 3.

1 Current and voltage sensors are used to show the effectiveness of the observer, namely to show only that the estimated variable converge to the measured ones, only the output voltage measurement is used to preform the feedback.

Experimental results

The following results were performed by selecting a desired output voltage V * OUT = 330V. For this value of V * OUT the corresponding operating point is x * e = [10.16 3.56 230.58 99.5] and their associated λ * 0 = 0.292 and λ * 1 = 0.708. Moreover, η = 0.5 is chosen from Albea-Sanchez et al. (2015, Theorem 2) and from the selected x( t) = [1 1 2 2 0.2 0.2 1 1] . T * = 0.001589 was computed from Property 1, considering a current ripple equal to 2.5 A for both i L 1 and i L 2 and a voltage ripple equal to 10 V for both v C 1 and v C 2 . Moreover, T was selected equal to T = 5µs. Then, satisfying Assumption 1, they are obtained: P = 16.356 -0.085 -0.003 0.004 -0.085 23.396 0.011 -0.006 -0.003 0.011 1.002 0.001 0.004 -0.006 0.001 1.000 , Q = 16.194 5.273 -12.211 -8.957 5.273 57.393 -33.235 -16.319 -12.211 -33.235 31.816 18.652 -8.957 -16.319 18.652 13.328 .

Finally, the values of P o , Q o satisfying Lemma 1, that allow to compute the observer gain K f , are Figure 4 shows the evolution of the output voltage in the start-up, applying the proposed control strategy given in (15). Note that the output converges to the desired V * OUT . Figures 5 and6 show the inductor current evolutions, i L 1 and i L 2 , also during the start-up test. Since the feedback is computed from the estimated variables, both measured and estimated variables are shown, as well as, their estimation errors. From these figures, no inrush currents occur, and after a short transient-time the estimated currents track the corresponding measured ones. Finally, both capacitor voltages v C 1 and v C 2 are given in Fig. s 7 and8. In these plots it is also shown that the observer variables converge to v C 1 and v C 2 with a small steady-state error.

In order to highlight some steady-state behaviors of the converter, Fig.s 9-13 show the output voltage evolution, inductor currents and capacitor voltages, at this time period. From Fig. 10 and 11, it is noted an average value of i L 1 and i L 2 around their desired operating point given by x * e , and a current ripple ∆i L 1 = 0.5 and ∆i L 2 = 1.5, which are in the interior of the compact set A, computed from x( t). In these figures, it is noted the effect of the "non-fixed" switching frequency originated from the proposed hybrid control strategy. Moreover, it is possible to reduce the ripple by decreasing the dwell time T w (at steady state). Experimentally, one can note that the switching frequency, in the steady-state, is about 1/(2αT w ), where α ∈ [1 1.5], and this frequency is maintained almost constant despite the equilibrium point changes. For example, T w = 3µs results in a switching frequency of about 150kHz regardless of the equilibrium point. Hence, the above given empirical rule can be used for sizing the passive components such as EMI filters, reactive components, etc.

Therefore, the proposed approach is in a clear-cut contrast with conventional hysteresis-based controllers, where the switching frequency, at steady-state, depends on the load resistance and input voltage. Thus, the proposed strategy combines both a fast transient response of a hysteresis controller with variable switching frequency, and a predictable PWM behavior, at steady-state, with constant switching frequency.

From Fig.s 9-13, it is evident the capability of the observer to estimate the state variables with a small steady-state error.

The robustness of the proposed hybrid control strategy with respect to parameter variations can be pointed out by adding an outer loop for regulating the output voltage, as in [START_REF] Lopez-Santos | Robust sliding-mode control design for a voltage regulated quadratic boost converter[END_REF]. Fig. 14 depicts the converter response to a step variation in the load resistance from 390Ω to 250Ω. The fast response of the output voltage can be observed from Fig. 14, which shows the suited behavior provided by the proposed control strategy. Moreover, show the good tracking of the state variable provided by the observer, which is able to estimate currents i L 1 and i L 2 , and voltages v C 1 and v C 2 with a small error, showing the effectiveness of the proposed estimation strategy. Finally, Fig. 19 shows the converter response to a step variation in the load resistance from 250Ω to 390Ω, respectively. Also in this case same considerations given above can be inferred. Moreover, the proposed control technique is compared with a PI controller (see , which is designed from an averaged model in continuous time. Hence, for comparison purposes, the PI controller is designed for delivering the same settling time, than the one proposed by the hybrid control strategy. However, even if the output voltage reaches the steady-state at 0.04s in both cases (see Fig.s 4 and 20), the proposed hybrid control provides a better dynamic precision because a direct control of the voltage and current trajectories in the phase-plane voltage-current is achieved (differently from PI control where only the output voltage error is considered), allowing to obtain no-inrush current at start-up (see Fig. Moreover, the second advantage of the proposed control algorithm is the possibility of choosing the trajectory in transient time by selecting the tuning parameter η, which allows to act directly in the switching frequency, diminishing the ripple of the inductor current. All these advantages allow to obtain a better performance using the proposed hybrid control, especially during transient times. Finally, the proposed control technique is compared with a PI controller during a step variation of the load from 390Ω to 250Ω (see Fig. 23 and14). In general, the steady state is similar. Nevertheless, the overshoot in the PI case is about 6.97%, while it is about 5.76% with the proposed strategy. This light improvement in the transient time confirms the robustness of the proposed strategy with respect to parameter variations.

Conclusion

This paper proposes a hybrid control for a QBC, based on a Lyapunov matrix-based min-projection control together with the design of an observer, which estimates the state of the system, from the measurement of the input and output voltages.

Differently from classical control methods, where the controller is designed based on small-signal models, here a model based on the hybrid nature of the converter, i.e. on the continuous-time and discrete-time dynamics, is considered. Stability properties of the complete system are established using a hybrid dynamical system theory, in terms of UGAS of a bounded and closed set where both, the estimation and the tracking error converge to a neighborhood of zero, ensuring globally practically stability of these errors equal to zero. The performed experimental results show the efficiency of the proposed approach, and the capability of the observer to track the state variables with small enough errors. In particular, the experimental tests show the better performance achieved using the hybrid controller with respect to conventional control methods such as a PI controller, especially in terms of dynamic precision. Finally, this work provides theoretical and practical results to design a state observer for removing the current sensors in power converters.

Figure 1 :

 1 Figure 1: Electrical scheme of the QBC.

Figure 2 :

 2 Figure 2: Block diagram of the proposed controlled system.

  ) is defined by |xx * e | = | x|, it results that Prieur et al. (2014, Equation (6)) is satisfied from the structure of the Lyapunov function (23) and from (24) and (25). Moreover, Prieur et al. (2014, Theorem 1) requires the construction of the restricted hybrid system H δ,∆ by intersecting C and D with the set, S δ,∆ = {( x, u) : | x| ≥ δ and | x| ≤ ∆},

  17), it jumps to the interior of the flow set C ∩ S δ,∆ . Likewise, from strict inequality in (28), all non-terminating solutions must flow for some time and since C ∩ S δ,∆ is bounded, there is a uniform dwell-time ρ(δ, ∆) between each pair of consecutive jumps. This dwell-time (δ, ∆) implies Prieur et al. (2014, Equation (4)) with the class-K ∞ function γ( j) = ρ(δ, ∆) j and N = 1. Then, all assumptions of Prieur et al. (2014, Theorem 1) are satisfied and UGAS of A is concluded.

Figure 3 :

 3 Figure 3: QBC prototype.

  is K f = -0.354 -0.409 32.436 17.085 .

  Fig.s 15-16 show estimated and measured currents i L 1 and i L 2 , as well as their estimation errors, while Fig.s 17-18 show estimated and measured voltages v C 1 and v C 2 , as well as their estimation errors, during the above mentioned load variation.

Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :Figure 14 :Figure 15 :Figure 16 :Figure 17 :Figure 18 :Figure 19 :Figure 20 :

 456781011121314151617181920 Figure 4: Output voltage during a start-up test.

  s 5-6 and 21), as well as, a good tracking of the voltage reference (see Fig.s 7-8 and 22).

Figure 21 :Figure 22 :Figure 23 :

 212223 Figure 21: Measured currents i L 1 and i L 2 , during a start-up test with a PI control strategy.

Table 1 :

 1 Circuit parameters values ComponentValue Model

Acknowledgement

This work has been partially funded under grant "HISPALIS" ANR-18-CE40-0022-01.