Michael Rudolf Koblischka
email: m.koblischka@gmail.com

Alex Wiederhold

Anjela Koblischka-Veneva

Crosby S Chang

Kévin Berger

Quentin Nouailhetas

Bruno Douine

Masato Murakami

Affiliations

On the origin of the sharp, low-field pinning force peaks in MgB2 superconductors

Various MgB 2 thin films and single crystals were found in the literature to exhibit a sharp, narrow peak at low fields in the volume pinning force, Fp(H)-diagrams. The origin of this peak is associated with a steep drop of the current density when applying external magnetic fields and is ascribed to sample purity. We show here that bulk MgB 2 prepared by spark-plasma sintering also shows the sharp, narrow peak in Fp. The peak is also seen in the volume pinning force scaling, Fp/Fp,max vs h = H/H irr . Furthermore, polycrystalline bulk MgB 2 samples prepared close to the optimum reaction temperature reveal this peak effect as well, but other samples of the series show a regular scaling behavior. The combination of magnetization data with data from electric transport measurements on the same samples demonstrates the origin of this peak effect. On increasing preparation temperature, the pinning force scaling changes from grain boundary pinning to point pinning and the grain connectivity gets worse. Hence, the sharp, low-field peak in Fp vanishes. Therefore, the occurrence of the peak effect in Fp gives important information on the grain coupling in the MgB 2 samples.

I. INTRODUCTION

The metallic superconductor with the highest transition temperature, MgB 2 , is considered as a candidate material for many applications such as trapped field magnets, tapes, and wires due to its simplicity and the simple preparation route possible, even though its Tc is much lower than most of the high-Tc superconductors. 1,[START_REF] Tomsic | of MgB 2 superconductor[END_REF] Even in sintered, polycrystalline MgB 2 samples, it was found to be possible to establish a current flow through the entire sample perimeter as the grain boundaries (GBs) do not act as weak links, but as flux pinning sites, e.g., in Nb 3 Sn. 3,4 Together with modern cryocooling techniques, the lower Tc does not pose a serious problem. [START_REF] Radebaugh | Proc. IEEE[END_REF] On the other hand, MgB 2 has its own characteristic magnetic behavior including dendrite-like flux penetration, [START_REF] Johansen | [END_REF] flux jump effects, 7 and peculiar flux pinning properties, 8 which may influence possible applications. Therefore, these points must be understood properly.

For the flux pinning behavior, the analysis of the flux pinning forces Fp = jc × B (where jc denotes the critical current density, and B = μ 0 H) using the scaling approach by Dew-Hughes (DH) 9 and Kramer 10 is an important issue. The flux pinning in MgB 2 is mainly provided by GB-pinning and point-pinning. Two peculiar observations of the flux pinning analysis on MgB 2 include (i) the change of the main acting pinning mechanism with increasing temperature found in polycrystalline samples prepared with varying reaction temperatures 11 and (ii) the appearance of a sharp, narrow peak in the Fp -H diagrams at relatively low applied fields. [12][13][14][15][16][17][18][19][20][START_REF] Badica | Correlated Functional Oxides: Nanocomposites and Heterostructures[END_REF] Such a peak was observed first in MgB 2 single crystals [START_REF] Shi | [END_REF]23 and then in thin films. [24][25][26] The occurrence of this peak at relatively low fields was ascribed to the strong decrease in jc(B) due to sample purity.

In this contribution, we analyze magnetization data obtained on a spark-plasma sintered (SPS) MgB 2 sample with a highly dense microstructure 27 and compare the results with a series of sintered, polycrystalline bulk MgB 2 samples prepared with varying reaction temperatures. 28 On all samples, we have performed electric transport measurements, 31,32 the behavior of which enable us now to clarify the origin of the peak found in the Fp-diagrams.

II. EXPERIMENTAL PROCEDURE

The MgB 2 sample was prepared by SPS at 1200 ○ C and a uniaxial pressure of 50 MPa. The sample reached a density of 2.61 g/cm 3 (99.2%). 27 Our data are compared to sintered, polycrystalline MgB 2 samples fabricated in the solid state reaction in a pure Ar atmosphere with reaction temperatures between 750 ○ C and 950 ○ C. 28 For superconducting quantum interference device (SQUID) measurements, small samples (2 mm × 2 mm × 1.5 mm) were cut from the big pellets. For the resistance measurements, small bars (10 mm × 1 mm × 1 mm) were prepared and the contacts were fixed with silver paint, ensuring a 5 mm distance between the voltage pads. The magnetic characterization measurements were performed using SQUID magnetometry (Quantum Design MPMS3), and the magneto-resistance characteristics were recorded using an Oxford Instruments 8 T Teslatron system. The critical current densities were calculated from the magnetization loops using the extended Bean model for rectangular samples. 29 Electron backscatter diffraction (EBSD) and transmission electron microcopy (TEM) were performed on TEM-slices prepared using focused ion-beam milling. Details of these procedures are given in Ref. 30 and the supplementary material.

III. RESULTS AND DISCUSSION

Figures 1(a) and 1(b) present jc as a function of the applied field, μ 0 Ha (a), and the volume pinning force, Fp, as a function of the field (b) of the SPS sample at various temperatures. In (b), the definitions of H peak denoting the position of the sharp Fp peak and Hcross marking the crossover field after the peak are given. The peak and shoulder shape of the Fp(H) curves is observable up to 30 K. The inset to Fig. 1(a) gives the jc(H, T) data in a double-log plot. Here, two regions are clearly distinguishable, as indicated by the dashed lines. The flat region at low fields can be considered to be the singlevortex pinning region, whereas the connected sloped region is where weak collective pinning is dominant. [33][34][35] The values of the exponent β in jc(H) ∝ H -β in the sloped region are around 1.0 at 5-25 K, 1.15 at 30 K, and 1.2 at 35 K, indicating that the decreasing slope of jc(H) gradually becomes steeper at temperatures above 25 K due to thermal fluctuation effects. 36,37 Figure 2(a)-2(d) show the peak region of the Fp vs μ 0 Hadiagrams for samples SPS (a), S775 (b), S850 (c), and S950 (d). The signature of the sharp Fp-peak is clearly visible in (a) and (b), whereas only traces are seen in (c), and in (d), the Fp -H behavior is fully regular. The pinning force scaling analysis is based on the approach of DH. 9 The scaled pinning force data f = Fp/Fp,max vs h = Ha/H irr can be fitted to the functional dependence given by f = A ⋅ (h) p ⋅ (1 -h) q , with A being a numerical parameter, and p and q are describing the actual pinning mechanism. The position of the maximum in the Fp plot, h 0 , is given by p/(p + q). For MgB 2 , the pinning at GBs (p = 0.5 and q = 2) and the pinning at non-superconducting point pins (p = 1, q = 2) are the most important functions. Figure 2(e) gives the DH-scaling of the SPS sample. The pinning force scaling is well developed, and the fit (red line) to all data yields a peak position h 0 = 0.21, which clearly points to a dominant flux pinning provided by extended defects, i.e., GBs. The pinning force scaling of all samples studied is given in Fig. S1 of the supplementary material. Now, we turn to the origin of the peak effect in the Fp(H)curves. From the literature, we can extract the following findings concerning the sharp, narrow peak in Fp(H):

(i) Single-crystals and high-quality thin films exhibit the specific Fp-curve shape. (ii) Sintered, polycrystalline samples, normal thin films, and powders do not show any peculiar Fp-curve shape. (iii) The peak shows a clear temperature dependence; therefore, matching effects are excluded. (iv) The peak is located at relatively high fields, which implies that surface effects are excluded. The peak positions, Hp, and the respective crossover fields, Hcross were extracted from the Fp-data. Here, it should be noted that our values of H peak are outside the range of present-day magneto-optics, 38,39 and so no direct observations of the flux patterns around H peak are possible.

The first finding of the present work is that sintered, polycrystalline samples prepared at reaction temperatures close to the optimum value (i.e., ∼800 ○ C 28 ) do show the sharp Fp-peak, whereas all other samples prepared at higher T do not. The SPS sample with its high density also exhibits the sharp Fp-peak.

The electric transport measurements elucidate the grain connectivity K, which plays an important role for the current flow through the sample (see Table I). K is calculated from the resistance data obtained via K = Δρg/Δρ, with Δρg = 6.31 μΩ cm for randomly oriented 3D samples, and Δρ = ρ(300 K) -ρ(40 K). 40 In our series of samples, K was found to increase steeply with increasing reaction temperature from 775 ○ C to 805 ○ C and then decreases again when increasing the temperature up to 950 ○ (see Fig. S2 of the supplementary material and Ref. 32). The SPS sample shows K = 0.96 due to its high density.

Finally, Fig. 4 gives the analysis of the scaled flux pinning curve for one selected sample (S775). To elucidate the acting flux pinning mechanism, one could combine two of the pinning mechanisms of DH and try to fit the data using a linear combination. It is straightforward to assume that the GB pinning is related to the low-field Fp-peak. This is symbolized by the red dashed line (fit 1), obtained using the fitting parameters p = 1.17 and q = 4.98. The resulting fit is perfect until Hcross is reached. Now, we subtract this fit from the experimental data and rescale them with a new Hmax. The result of this procedure is drawn with a blue line. Finally, these data are again fitted using a DH function with p = 3.04 and q = 1.76 (fit 2). All this implies that we can fit the experimental data showing the sharp, low-field peak with a combination of two DH pinning functions. The low-field peak in Fp is caused by the GB pinning, whereas the large shoulder is due to pinning at point defects.

Together with the information from the electric measurements revealing the grain coupling, we can now come to the conclusion for the origin of the Fp-peak.

As described before, one of the advantages of MgB 2 is the current flow through the entire sample perimeter even in polycrystalline samples. This implies that the currents run across various GBs, affected by the crystallographic orientation of the grains. The first point enables pinning at GBs, which is beneficiary, e.g., in Nb 3 Sn. Due to the second point, the currents and flux pinning forces are affected by the given anisotropy of the material. 41 Following the calculations of Eisterer, 42 the position of the GB pinning peak may be obtained in the range between 0.1 and 0.2, which is, indeed, the case for the sharp, low-field peaks observed here. In this range, the dominating flux pinning is provided by the GBs, and the current flow is affected by the crystal anisotropy and percolation. When a field-induced decoupling occurs, the currents will only cycle in small regions with the minimum size of an individual grain. Then, anisotropy and percolation do not play the important role anymore, and we will have a situation corresponding to polycrystalline YBa 2 Cu 3 Ox. As there is no texture of the MgB 2 grains, the differences in the local currents will average out and we obtain a peak in the Fp diagram for point pinning at h ≈ 0.33, following a simple DH pinning function with p = 0.52 and q = 0.98. 11 To observe the sharp, narrow Fp-peak, it is essential that the sample is free of additions providing pinning at small, nonsuperconducting inclusions, as this would strengthen the secondary pinning mechanism. The GB pinning may then only appear as a small shoulder on the low field side, if at all. Furthermore, the sample preparation must ensure that the MgB 2 grains are well coupled (i.e., having a high grain connectivity). Such situation is realized in the SPS sample studied here. Figures 5(a is clean, while others are obscured by MgO and MgB 4 particles, as demonstrated in Ref. 30. The decay of the current flow will then not influence the Fp diagram that much as the peak positions are close to each other. In this sense, the previous explanation that only pure samples will exhibit the sharp, narrow Fp-peak is correct.

Table I demonstrates, further, that the peak field, H peak , scales with the sample size when comparing the thin film data with ours. Therefore, this effect will not influence the currently achieved trapped fields in the bulk MgB 2 samples (typical dimensions of 3-5 cm diameter and thickness of 10-20 mm) as the current trapped field values are still relatively low. 18,[43][44][45] However, when attempting to trap much larger fields, then the change of current flow at H peak will pose a serious problem as one would obtain not one large central trapped field peak, but several smaller ones when the current flow is reduced to smaller islands. This situation would get worse the higher the applied field would be. H peak is a limiting factor for the achievable trapped field. As the grain connectivity cannot be tuned to be much better (the sharp peaks crystals, thin films, and the SPS sample), the only way out is to improve the flux pinning inside the MgB 2 grains. Therefore, an ideal MgB 2 sample for applications needs small, nano-sized grains with embedded, strong flux pinning provided by nano-sized pinning centers. In this case, the sharp, low-field peak will be overwhelmed by the flux pinning provided by point-like defects.

IV. CONCLUSIONS

To conclude, we have performed magnetic and electric measurements on a series of sintered, polycrystalline MgB 2 samples. The samples prepared using a low reaction temperature and the highdensity, spark-plasma sintered samples exhibit the sharp, low-field peak in Fp, while the other samples do not. The electric transport measurements performed on the same samples reveal a change of the grain connectivity on increasing the preparation temperature, whereas the highly dense, spark-plasma sintered sample shows a well-developed grain connectivity. As a result, from this combination of magnetic and electric measurements, we can conclude that the change of length scale of the current flow at H peak is responsible for the Fp-peak. When the grains are well coupled together, the current flow through the entire sample perimeter is enabled. In this case, the flux pinning is mainly provided by GB pinning, and the crystal anisotropy and percolation play a role. In case the coupling of the MgB 2 grains is worse, we only observe the flux pinning provided by small, non-superconducting inclusions, and the current flow is reduced to grain clusters or grains. This finding may have important consequences for the achievable trapped fields in MgB 2 .

SUPPLEMENTARY MATERIAL

See the supplementary material for specific information about the pinning force scaling of all samples, the grain connectivity, and experimental details about the EBSD analysis.

FIG. 1 .

 1 FIG. 1. Data of the MgB 2 SPS sample. (a) jc as a function of the applied field, μ 0 H ext , in the temperature range 5 K ≤ T ≤ 35 K. The inset shows the same data in a double-log plot, together with the linear behavior as indicated by the dashed lines. (b) Pinning force, Fp, as a function of the applied field, μ 0 H ext , calculated from the jc(Ha)-data for T = 10 K, 20 K, and 30 K. At the 10 K curve, the peak position, H peak , and the crossover field, Hcross, are indicated by arrows.

FIG. 2 .

 2 FIG. 2. [(a)-(d)] Pinning forces vs applied field for samples: (a) SPS, (b) S775, (c) S850, and (d) S950. (e) Pinning force scaling, f vs h of the SPS sample.

Figure 3

 3 Figure3presents the temperature dependence of the extracted critical fields. H peak and Hcross are well below H irr . The values of H peak clearly show a sample size dependence as the values here are much larger as compared to the thin-film data of Ref. 26; thus, demagnetization effects play a role. Here, it should be noted that our values of H peak are outside the range of present-day magneto-optics,38,39 and so no direct observations of the flux patterns around H peak are possible.The first finding of the present work is that sintered, polycrystalline samples prepared at reaction temperatures close to the optimum value (i.e., ∼800 ○ C 28 ) do show the sharp Fp-peak, whereas all other samples prepared at higher T do not. The SPS sample with its high density also exhibits the sharp Fp-peak.The electric transport measurements elucidate the grain connectivity K, which plays an important role for the current flow through the sample (see TableI). K is calculated from the resistance data obtained via K = Δρg/Δρ, with Δρg = 6.31 μΩ cm for randomly oriented 3D samples, and Δρ = ρ(300 K) -ρ(40 K).40 In our series of samples, K was found to increase steeply with increasing reaction temperature from 775 ○ C to 805 ○ C and then decreases again when increasing the temperature up to 950 ○ (see Fig.S2of the supplementary material and Ref. 32). The SPS sample shows K = 0.96 due to its high density.Finally, Fig.4gives the analysis of the scaled flux pinning curve for one selected sample (S775). To elucidate the acting flux pinning mechanism, one could combine two of the pinning mechanisms of DH and try to fit the data using a linear combination. It is straightforward to assume that the GB pinning is related to the low-field Fp-peak. This is symbolized by the red dashed line (fit 1), obtained using the fitting parameters p = 1.17 and q = 4.98. The resulting fit is perfect until Hcross is reached. Now, we subtract this fit from the experimental data and rescale them with a new Hmax. The result of this procedure is drawn with a blue line. Finally, these data are again fitted using a DH function with p = 3.04 and q = 1.76 (fit 2). All this implies that we can fit the experimental data showing the sharp, low-field peak with a combination of two DH pinning

FIG. 3 . 3 © Author(s) 2020 FIG. 4 .

 3320204 FIG. 3.Irreversibility field, H irr , peak field, H peak , and the crossover field, Hcross of sample S775 as a function of temperature. Note that H peak and Hcross are considerably smaller than H irr .

  ) and 5(b) present TEM images of GBs in this and Fig.5(c) gives a phase map obtained using electron backscatter diffraction (EBSD). A majority of the GBs

FIG. 5 .

 5 FIG. 5. GBs in the SPS-sample. TEM images of clean GBs (a) and GBs with secondary phase particles (b). (c) presents an EBSD phase map, where green indicates MgB 2 , red MgO, and yellow MgB 4 . More details on their EBSD analysis are found in the supplementary material and in Ref. 30.

  Table I gives the data of all bulk MgB 2 samples and those of Ref. 26 for comparison. The smaller values of Ref. 26 demonstrate the demagnetization effects. The values of Hp are sufficiently large to be above the full penetration field.

TABLE I .

 I Values for Hp and Hcross for all MgB 2 bulk samples studied here and those of Ref. 26 (MgB 2 thin film, 700 nm thickness), together with data for Δρ and the connectivity, K. The scaling parameters A, p, and q are found in TableS1of the supplementary material. All magnetic data are for T = 10 K. For details of the electric measurements, see Ref.32. The * indicates that these values are for the normal peak in Fp, not the sharp peak as in the case of the other samples. Correspondingly, there is no value for Hcross. n.a. indicates that there are no data available.

	Sample
	T ( ○ C)	Hp (mT)	Hcross (mT)	Δρ (μΩ cm)	K (%)
	S775	643	907	140	4.51
	S800	826	1305	77	8.195
	S805	798	1525	37	17.054
	S850	802 *	. . .	64	9.859
	S950	897 *	. . .	113	5.584
	SPS	883	1447	6.5	97.076
	Reference 26	54	122	n.a.	n.a.

AIP Advances 10, 015035 (2020); doi: 10.1063/1.5133765 10, 015035-1 © Author(s) 2020

AIP Advances 10, 015035 (2020); doi: 10.1063/1.5133765 10, 015035-2 © Author(s) 2020

AIP Advances 10, 015035 (2020); doi: 10.1063/1.5133765 10, 015035-4 © Author(s) 2020

ACKNOWLEDGMENTS

We would like to thank J. Noudem (CRISMAT, Caen, France) for the spark-plasma sintered MgB 2 and K. Inoue and M. Muralidhar (SIT, Tokyo, Japan) for the sintered MgB 2 samples. This work is part of the SUPERFOAM international project funded by ANR and DFG under Grant Nos. ANR-17-CE05-0030 and DFG-ANR Ko2323-10, respectively.


Hongzhi Yang

Marc Brunel

Marc Vallet

Haiyang Zhang

Changming Zhao

Pulse Doublets Generated by a Frequency-Shifting Loop Containing an Electro-Optic Amplitude Modulator

published or not. The documents may come

Introduction

Frequency-Shifted feedback Loops FSL, both active or passive, are promising solutions for pulse generation with high repetition rate. FSL usually contains an acousto-optic frequency shifter AOFS for single-side band modulation [1-3]. This leads to limited tunability and low modulation frequency f m .

Conversely: the use of in-loop electro-optic amplitude modulators EOM permits high modulation, wide bandwidth and integration [4][5].

However, due to dual-side band modulation, EOM lead to new regimes Here, theoretical and experimental study of FSL with EOM

Theoretical model

E in1 cw: ( ) sin(2 ) m m t f t       1 11 12 21 1 22 1 1 ( ) sin( ( )) p N p p out in p q E t t t t t t q E                            phase retardance  m modulation depth   11 11 22 12 21 1 1 22 sin ( ) ( ) 1 sin ( ) out in t t t t t t E t E t t         For integer Talbot condition [1-2]: f m =n/ If N >>1
With modulation at f m :

Partial fundings : CPERSOPHIE-PHOTONIQUE

Mode-locked double-pulse regime Conclusion

•

Forthcoming

• Phase modulation vs intensity modulation • Frequency-to-time mapping (as in [6] with AOFS)

• Repetition-rate tunability from MHz to GHz • Fractional Talbot condition f m =p/q f c : towards high repletion rates

Experimental results

Waveform generation

1/ = f c = 6.737 MHz

  Observation of original double-pulse regime • Theoretical model in agreement with experiments • Repetition-rate tunability from MHz to GHz • frequency comb width up to 40 GHz • waveform generation: rectangle, sawtooth,… [1] H.G. de Chatellus et al, Phys. Rev. A 2013 [2] C. Schnébelin et al, CLEO 2017 [3] H. Yang et al, IEEE Photon. J. 2017 Up: V b =4 V; Down: V b =2 V

  for (t)≈/2 2 pulses per period 1/f m

SYN


Hongzhi Yang

Marc Brunel

Marc Vallet

Haiyang Zhang

Changming Zhao

Ghz-Rate

GHz-rate pulse pairs generated

come

Introduction

Frequency-shifted feedback loops, both active or passive, are promising solutions for pulse generation with high repetition rate. FSFL usually contains an acousto-optic frequency shifter AOFS for single-side band modulation [1-3]. This leads to poor tunability and low modulation frequency f m .

Conversely: the use of in-loop electro-optic amplitude modulators EOM permits high modulation, wide bandwidth and integration [4][5].

However, due to dual-side band modulation, EOM lead to new regimes Here, theoretical and experimental study of FSFL with EOM

Mode-locked Double-pulse regime Conclusion

•

Forthcoming

• Phase modulation vs intensity modulation • Frequency-to-time mapping (as in [6] with AOFS)

• Repetition-rate tunability from MHz to GHz • Fractional Talbot condition f m =p/q f c : towards high repletion rates

  Observation of original double-pulse regime • Theoretical model in agreement with experiments • Repetition-rate tunability from MHz to GHz • frequency comb width up to 40 GHz • waveform generation: rectangle, sawtooth,… [1] H.G. de Chatellus et al, Phys. Rev. A 2013 [2] C. Schnébelin et al, CLEO Techn. Digest 2017 [3] H. Yang et al, IEEE Photon. J. 2017 Up: V b =4 V; Down: V b =2 V

Theoretical model E in1 cw + modulation at f m

	1 out E	( ) t	11 t	12 21 t t	N	t	22 p p	p	sin( ( t q	))	1 in E	phase retardance
								p	1		q	1	m modulation depth
	( ) t		m	sin(2	m f t	)				[ ] t	0.44 0.65 0.43 0.67 i i
													30
	For integer Talbot condition [1-2]: f m =n/
	If N >>1	1 out E	( ) t			11 t		11 22 22 t t 1 t	12 21 sin ( ) sin ( ) t t t t	1 in E	Pulses for (t)≈ /2

N

Partial fundings : CPER "Sophie-photonique"

Experimental results

Waveform generation


H Zhang

H Yang

M Brunel

M Vallet

C Zhao

S Yang

Fiber frequency-shifting loop for RF upconversion and waveform generation Time domain: waveform depends on the RF harmonics phases (nπf AO τ) Model: time delayed interferences Introduction Conclusion

n/2 T 12 (1-Γ) 2 + 4Γ sin 2 (nπ f AO τ ) (1-ΓT 11 ) 2 + 4ΓT 11 sin 2 (nπ f AO τ ) E out1 E out2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = t 11 t 12 t 21 t 22 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ E in1 E in2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ E out 2 (t) = t 21 ( t 22 γ ) p E in1 (t -p τ )e i 2π f AO p(t-τ ) e -iπ f AO p 2 τ p=0 ∞ ∑ E out1 (t) = t 11 E in1 (t) + t 12 E in2 (t) = t 11 E in1 (t) + t 12 γ E out 2 (t -τ )e i 2π f AO (t-τ ) = t 11 E in1 (t) + t 12 γ t 21 (t 22 γ ) ( p-1) E in1 (t -pτ )e -iπ f AO pτ e i 2π f AO pt e -iπ f AO p 2 τ p=1 ∞ ∑ γ = ηG e iϕ Γ = γ 2
Output field:

Inside the loop:

Frequency domain: harmonics vs gain

-RF enhancement x15 (here up to ~4 GHz) -programmable and integrable scheme Perspec-ves -+ SHG => green source for underwater [5-6] -Ultra-low Doppler frequency shios -All-fibered experiment @ 1550 nm -Above threshold (solitons, mode-locking, etc)

Output power: Influence of the loop length on the waveform (steps of 15 cm) Influence of frequency shio on the waveform -ring interferometer + frequency-shioer + amplifier [1][2][3][4] => self-heterodyning, Frequency-Shioed Feddback laser, real-^me Fourier Transform, wavelength scan, etc.

-microwave photonics context (lidar-radar for example) [5][6][7][8] => RF up-conversion, waveform genera^on -Here high-harmonic RF beats, sub-threshold opera^on (≠ FSF laser) + all-fibered experiment @ 1064 nm vs simula^on G η

  1064 nm (YDFA with gain G) -Acousto-Op^c Frequency Shioer f AO = 200 MHz ± 5 MHz, η = 0,4 -~20 m loop (τ = 5 ns) Harmonics Linewidth:


Carolina Albea-Sanchez

Germain Garciat

Robust hybrid control law for a boost inverter

Keywords: Hybrid control application, boot inverter, dwell time guarantees, robust control

In this paper, we propose a robust hybrid control for a boost inverter. This control presents the particularity of considering the real nature of the dynamics, which means, the continuous-time dynamics, voltages and currents, and the discrete-time dynamics, the switching of the transistors. Likewise, due to implementation issues a minimum dwell time is introduced in the control loop scheme. Uniform global asymptotic stability is guaranteed, ensuring that the tracking error of the system output enters into a small neighbourhood of zero.

Moreover, robustness guarantee with respect to parameter variations, in terms of an error tracking output regulation, are also provided here. The effectiveness of the proposed hybrid control scheme is illustrated in simulation.

Introduction

In many electronic applications, it is necessary to convert a DC signal, into an AC one. Moreover, it also can be required to boost this signal, as can be found in AC microgrids, for instance. Often, it is made this transformation in two steps. Hence, in a first step, it is used an elevator DC-DC converter, and in a second one, a DC-AC converter [START_REF] Almada | A centralized and heuristic approach for energy management of an AC microgrid[END_REF][START_REF] Manandhar | Efficiency comparison of dc and ac microgrid[END_REF].

The boost inverter proposed by [START_REF] Caceres | A boost dc-ac converter: analysis, design, and experimentation[END_REF] is a particular power converter, that can generate a larger AC signal from a given DC one just in one step, with a simple architecture. Nevertheless, to design control laws for this converter is a challenge, because it is composed by two DC-DC boost converters, being a nonlinear non-minimum phase of fourth order [START_REF] Escobar | An experimental comparison of several nonlinear controllers for power converters[END_REF]. Furthermore, the control goal of this boost inverter is to track a sinusoidal signal. Some control strategies have been proposed by controlling separately each DC-DC boost converter [START_REF] Caceres | A boost dc-ac converter: analysis, design, and experimentation[END_REF][START_REF] Gurpide | A new control strategy for the boost DC-AC inverter[END_REF], in such a way to generate a sinusoidal output voltage and lager than the input one. The used control techniques are PI controllers [START_REF] Sanchis | Boost DC-AC inverter: a new control strategy[END_REF], sliding mode controllers [START_REF] Bajaj | Design and controlling of proposed efficient boost-inverter implemented using boost dc-dc converter[END_REF][START_REF] Vazquez | Some critical aspects in sliding mode control design for the boost inverter[END_REF]. However, stability properties of the system in a whole is not proven.

There have been several attempts to control this inverter, considering the complete system and with the advantage of using only one output voltage reference. Hence, the authors, in [START_REF] Cortes | Dynamical sliding-mode control of the boost inverter[END_REF], design a single sliding mode controller to generate the required sinusoidal voltage on the load. Moreover, they do not guarantee an error tracking output regulation in the sense of parameter variations. In [START_REF] Albea | Adaptive control design for a boost inverter[END_REF] a control law is proposed by using energy-shaping methodology.

Moreover, an adaptive controller is added to guarantee the tracking error with respect to parameter variations. However, this method is focussed on an approximated average model in continuous-time and the control law presents a relative computational cost. In [START_REF] Flores-Bahamonde | Using the sliding-mode control approach for analysis and design of the boost inverter[END_REF] a cascade control focussed on sliding-mode methodology is presented. The internal control loop is based on a switching surface, generated from the difference between the inductor currents, whereas the external control loop consists of a PI compensator in order to reduce the tracking error of the inverter output voltage. Nevertheless, the authors do not consider a minimum dwell time in the control updates, which is required in implementation issues.

In this paper, a novel control law based on the hybrid dynamical system paradigm given in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF], is proposed for the considered inverter. The main advantage of this approach is to take into account the real nature of the dynamics, meaning the continuous-time dynamics, which are voltages and currents, and the discrete-time ones, which are switching signals. The control law is based on a Lyapunov matrix-based min-projection control, as used in [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF][START_REF] Sanchez | Hybrid dynamic modeling and control of switched affine systems: application to DC-DC converters[END_REF][START_REF] Sanchez | Practical stabilisation of switched affine systems with dwell-time guarantees[END_REF]. One of the challenge here is to know the entire reference state corresponding to the tracking signal, which is not a simple task. To deal with this objective, we assume that the variation of the desired reference with respect to time is sufficiently slow, in order to obtain a small enough tracking error. Moreover, we consider a minimum dwell time in the hybrid control scheme, by introducing a time regulation [START_REF] Sanchez | On the practical stability of hybrid control algorithm with minimum dwell time for a DC-AC converter[END_REF]. Stability properties of the tracking error in a small neighbourhood of zero are guaranteed, applying the theory for hybrid system presented in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]. In a second time, we propose a robust hybrid control, whit respect to parameter variations, in such a way, the system output rejects any perturbation. Some validations are performed in simulation. This paper is organized as follows. Section 2 is devoted to the problem formulation. Some assumptions and properties are stated in Section 3. Section 4 presents the main result about the proposed hybrid control and an extension to a robust hybrid control is given in Section 5. Some simulations are performed in Section 6. Finally, the paper is closed with a conclusion section.

Notation: Through out the paper N denotes the set of the natural numbers and R the set of real numbers, R n the n-dimensional euclidean space and R n×m the set of all real n×m matrices. The set of non-negative real numbers is denoted by R ≥0 . M 0 (resp. M ≺ 0) represents that M is a symmetric positive (resp. negative) definite matrix. 0 n×m is a zero matrix of n × m-dimension. λ m (M ) and λ M (M ) represent the minimum and maximum eigenvalues of M .

• represents the norm euclidean.

Conflict of interest -none declared.

Boost inverter

The boost inverter depicted in Fig 1, is a device that can generate, in a single stage, a sinusoidal voltage with a larger amplitude than its input DC voltage,

V in . This converter is composed by a load R 0 differentially connected to two DC-DC converters.

The inductance currents flowing through the inductors L 1 , L 2 are i 1 and i 2 .

L 2 i 2 L 1 i 1 R 1 R 2 C 1 C 2 R 0 V in + - v 1 v 2 S 1 S 3 S 4 S 2 Figure 1: Boost inverter circuit.
Likewise, the capacitor voltages applied to the capacitors C 1 , C 2 are v 1 and v 2 .

The switches are S i ∈ {0, 1}, with i = {1, 2, 3, 4}, and they take the value 1, if they are switched ON, and the value 0, if they are switched OFF, respectively.

Finally, R 1 and R 2 are parasitic resistances.

A Nonlinear model

The boost inverter can be represented by the following nonlinear model:

                     ż =         -R1 L1 -u L1 0 0 u C1 -1 R0C1 0 1 R0C1 0 0 -R2 L2 -1-u L2 0 1 R0C2 1-u C2 -1 R0C2         z +         Vin L1 0 Vin L2 0         w = [0 1 0 -1]z, ( 1
)
where z = [i 1 v 1 i 2 v 2 ] ∈ R 4 is the state vector, w = v 1 -v 2 is
the controlled output and u ∈ {0, 1} is the control input. z, is composed of continuous-time variables, whereas, u is a switching signal, thus, a discrete-time variable.

Assumption 1. Let us consider

• R=R 1 =R 2 , • C 0 =C 1 =C 2 • L=L 1 =L 2 and
• the state values of the switches given in Table 1. From Assumption 1 and the following variable changes v := 2u -1 and x=

1 1 0 1 0 0 0 1 0 1
[x 1 x 2 x 3 x 4 ] = [i 1 +i 2 v 1 +v 2 i 1 -i 2 v 2 -v 1 ]
, the model (1) can be rewritten as:

                             ẋ = Av         -R L -1 2L 0 v 2L 1 2C0 0 -v 2C0 0 0 v 2L -R L -1 2L -v 2C0 0 1 2C0 -2 R0C0         x + B         2 L 0 0 0         V in y = C [0 0 0 1] x, (2)
where v ∈ {-1, 1} is the control variable.

The objective of this inverter is to guarantee that the output signal y(t), corresponding to v 2 (t) -v 1 (t), follows a sinusoidal reference defined by

V r (t) = V max sin(ωt), (3)
where V max and ω are the amplitude and the angular frequency of the desired sinusoidal signal. Moreover, it is defined w = 2π Te with T e ∈ R ≥0 the time period of the desired signal. In order to impose a such behaviour to system (2), let us to introduce its averaged model, which is

                         ẋa (t) = A λ(t)         -R L -1 2L 0 2λ(t)-1 2L 1 2C0 0 -2λ(t)-1 2C0 0 0 2λ(t)-1 2L -R L -1 2L -2λ(t)-1 2C0 0 1 2C0 -2 R0C         x a (t)+ B         2 L 0 0 0         V in , y a (t) = [0 0 0 1]x a (t), (4)
where x a (t) represents the averaged signal of x and parameter λ(t) ∈ [0, 1] is the duty cycle of v. This averaged model allows to define the desired complete reference of state x associated with V r (t), i.e., the ideal reference associated with infinity switching. One can see, the time-varying reference of ( 2) is given considering the desired output reference (3) for y a (t) of system (4) and its associated variable 0 ≤ λ e (t) ≤ 1, such that,

x ae (t) = -(λ e (t)A 1 + (1 -λ e (t))A -1 ) -1 B
(5)

y ae (t) = V r (t) (6)
is the equilibrium of (4). This desired reference obtained satisfying ( 5)-( 6) is perceived as relaxed solutions in the generalized sense of Filippov. Nevertheless, to obtain an explicit expression of x ae (t) is not a trivial task.

From the introduced boost inverter model (2) and its reference ( 5)-( 6), we are in condition of stating the problem.

Problem 1. The first goal in this work is to find a tracking reference of (2),

x e (t), such that, x e (t) -x ae (t) is bounded and y e (x e (t)) -V r (t) is small enough. A second goal deals with designing a hybrid control law for system (2), such that, for any initial condition x(0) ∈ R 4 , the following holds:

• x(t) -x ae (t) is bounded,

• lim t→∞ y(t) -V r (t) presents an error small enough,

• the output is robust with respect to parameter variations and modelled errors.

Preliminaries

The following property is a standard property found in the literature [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF][START_REF] Feron | Quadratic stabilizability of switched systems via state and output feedback[END_REF][START_REF] Sanchez | Practical stabilisation of switched affine systems with dwell-time guarantees[END_REF], that characterizes the equilibrium in switched affine systems.

Property 1. Given matrices A 1 and A -1 in model (2), a matrix Q 0 ∈ R 4 , there exists λ ∈ Λ and a matrix P 0 ∈ R 4 satisfying

A i P + P A i + 2Q ≺ 0, for i = {-1, 1}. (7)
Note that Property 1 is direct if A -1 and A 1 are Hurwitz.

This property allows introducing the next definition. The averaged model of ( 2) is given by 105

Operation point definition

Let us consider the particular case, corresponding to a constant V r , i.e., ω = 0. Then, the desired state reference generated by ( 5) is

ẋae (t) = 0 4×1 , y ae = V r . (8)
In this particular case, we can formulate the following assumption for the desired operating point of (2).

Assumption 2. Consider the desired output reference (8) and a parameter 0 ≤ λ e ≤ 1, such that,

x e = -(λ e A 1 + (1 -λ e )A -1 ) -1 B
is the equilibrium point of (4).

This equilibrium is also the equilibrium of system (2) related to arbitrarily fast switching. It is worth noting, if Assumption 2 is satisfied and from Property 1, it is easy to see that (1

-λ e )(A -1 x e + B) + λ e (A 1 x e + B) = 0. Proposition 1. Consider that R is negligible (i.e. R R 0 ), Assumption 1
and Assumption 2 are satisfied, then i) the equilibrium point can be expressed as

x e (λ e ) =         (2λe-1) 2 λ 2 e R0(λe-1) 2 -1 λe(λe-1) (2λe-1) λ 2 e R0(λe-1) 2 -(2λe-1) λe(λe-1)         V in , 0 < λ e < 1. (9)
ii) The reference of the constant output voltage, V r , provides (given also in

[3]) V r V in = - 2λ e -1 λ e (λ e -1) . V r λ 2 e + (2V in -V r )λ e -V in = 0,
and their roots are

λ e (V r ) =          1 2 -Vin Vr + 1 4 + V 2 in V 2 r if V r ≥ 0 1 2 if V r = 0 1 2 -Vin Vr -1 4 + V 2 in V 2 r if V r ≤ 0. ( 10
)
Therefore, for a given value of V r , it is possible, from the previous formulation, to determine the corresponding value of λ e (V r ), inducing the equilibrium point x e of (2).

Property of the sinusoidal reference signal

Now, we consider the sinusoidal reference (3) that generates the reference dynamic described in ( 5)-( 6), with ω different to zero.

The next Lemma will be devoted to prove that λ e (V r (t)) given in ( 10) is a periodic function, with the same time period than the reference signal V r (t).

Lemma 1. We have the following properties:

i) The function t → λ e (V r (t)) is a continuous and differentiable function from R to ]0, 1[. ii) If the function V r (t) is a periodic function of period T e , then the function t → λ e (V r (t)
) is a periodic function of period T e , with a mean value equal to 1/2.

Proof. From the function given in [START_REF] Albea | Adaptive control design for a boost inverter[END_REF], we have

lim Vr→∞ λ e (V r ) = 1 and lim Vr→-∞ λ e (V r ) = 0.
The continuity and differentiability is obvious for all V r ∈ R. We have from ( 10)

lim Vr→0 λ e (V r ) = lim Vr→0 1 2 - V in V r ± V in V r .
Then, we can deduce

lim Vr→0 + λ e (V r ) = lim Vr→0 -λ e (V r ) = 1 2
and, therefore λ e (V r ) is continuous at V r = 0. A simple calculation leads also to

dλ e dV r =          Vin V 2 r - V 2 in V 3 r 1 4 + V 2 in V 2 r if V r > 0 Vin V 2 r + V 2 in V 3 r 1 4 + V 2 in V 2 r if V r < 0. Moreover, lim Vr→0 dλ e dV r = lim Vr→0 V in V 2 r ± |V r | V in V 3 r , then, lim Vr→0 + dλ e dV r = lim Vr→0 - dλ e dV r = 0.
Thus, λ e (V r ) is differentiable at V r = 0. Simple calculations show that λ e (V r )

presents an inflection point at V r = 0, noting

dλ e dV r > 0 if V r > 0 and dλ e dV r > 0 if V r < 0.
The proof of i) is complete. Then, the periodicity of λ e (V r (t)) follows by continuity arguments and the fact that

dλ e (V r (t)) dt =                  Vin V 2 r - V 2 in V 3 r 1 4 + V 2 in V 2 r   dVr dt if V r > 0   Vin V 2 r + V 2 in V 3 r 1 4 + V 2 in V 2 r   dVr dt if V r < 0. (11)
Furthermore, note that the mean of V r (t) is 0, hence the mean of λ e (V r (t)) = 1 2 , concluding the proof of item ii). 130 Remark 1. It is worth noting Lemma 1 allows to obtain an approximative expression of the periodic equilibrium of x ae (t), such that, x e (λ e (V r (t))) →

x ae (t), as dxae(t) d t → 0.

An hybrid control Scheme

The inverter model presented in (2) covers two different types of dynamics.

In one hand, the continuous-time variables which are the differences and the sums of the inductor currents and the capacitor voltages. In other hand, the control input, v, is a discrete-time variable. Consequently, the framework given in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] about hybrid dynamical system is well suited.

The main idea of dealing with Problem 1 is to use Proposition 1 to generate a time-periodic reference for x(t), from V r (t) given in (3), as noted in Remark 1.

Moreover, if the variation of x e (λ e ) with respect to time is sufficiently slow, we can expect that the output y(t) = x 4 (t) will be closed to the desired reference

V r (t) given in (3).
This basic idea is illustrated by the bloc scheme represented in Fig. 2. It is easy to see that the generated output of this control process, y(t), will present an error in steady state from the desired reference, V r (t). Indeed, for one side, there is an error between the output generated from the averaged model y ae = V r (t) and the one generated from the computed reference y e := [0 0 0 1]x e (λ e ), given in Proposition 1. For another side, due to implementation issues, it must have a minimum dwell time in the control updates, inducing an error in steady state between x(t) and the provided reference x e (λ e ).

CONTROL SYSTEM λ e (V r ) V r x e (λ e (V r )) v x(t)

Hybrid scheme

As proposed in [START_REF] Sanchez | Practical stabilisation of switched affine systems with dwell-time guarantees[END_REF], a new state variable τ is added to ensure a minimum positive dwell time, T . Then, the closed loop system is proposed in the hybrid formulation H given below.

Hybrid model

H :                                                                                      ẋ v τ ẋae λe ẋe               =               A v x + B v 0 1 -dz( τ T ) A λe x ae + B f 1 (V r ) f 2 (λ e )               , ζ ∈ C               x+ v + τ + x + ae λ + e x + e               ∈                x argmin i∈{-1,1} x P (A i x + B v ) 0 x ae λ e x e                , ζ ∈ D, ỹ = C(x + x e -x ae ) (12)
where x := x -x e , ζ := [x v τ x ae λ e x e ] ∈ H, such that,

H := {x ∈ R 4 , v ∈ {-1, 1}, τ ∈ [0, T ], x ae ∈ R 4 , λ e ∈ [ν, 1 -ν], x e ∈ R 4 },
with a parameter ν > 0 small enough. Likewise,

B v := A v x e + B, f 1 (V r ) := dλe(Vr(t)) dt
given in [START_REF] Flores-Bahamonde | Using the sliding-mode control approach for analysis and design of the boost inverter[END_REF] and f 2 (λ e ) is the derivative of ( 9) with respect to time.

160 dz defines a dead-zone function, for all s ≥ 0, dz(s) := max{0, s -1}.

Inspired by [START_REF] Sanchez | Practical stabilisation of switched affine systems with dwell-time guarantees[END_REF], we select the so-called flow and jump sets

C := {ζ : x P (A v x + B) ≤ -x Qx, τ ∈ [0, T ]} (13)
D := {ζ : x P (A v x + B) ≥ -x Qx, τ = T }, (14)
ensuring that all maximal solutions are complete. Note that, τ introduces a minimum dwell time, T , i.e., the solutions are forced to flow at least T ordinary time after each jump. Then, the following condition must hold:

T T e . ( 15
)
Proposition 2. The hybrid dynamical system (12)-( 14) satisfies the basic hybrid conditions [12, Assumption 6.5], then it is well-posed.

Proof. The hybrid dynamical system ( 12)-( 14) satisfies the basic hybrid conditions because

• sets C and D are closed.

• f is a continuous function, thus it is outer semicontinuous and convex.

Moreover, it is locally bounded.

• G is closed, then it also is outer semicontinuous [12, Lemma 5.1] and, it is locally bounded.

Finally, following [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]Theorem 6.30], we conclude that the hybrid dynamical system is well posed.

In order to achieve the objectives stated in Problem 7, we will provide a hybrid control scheme establishing stability properties for x = x e , considering

x e constant and a minimum dwell time in the implementation. Then, we will assume the time varying of x e (t), finding a bound of y e (t) -y ae (t) . Finally, stability properties of y(t) = y ae (t), will be ensured.

Stability properties for V r constant

The objective here is to regulate the state vector, x, to a neighbourhood of a constant point x e (λ e (V r )), providing suited stability properties by using the paradigm presented in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]. Then, for any parameter X > 0 and a given T > 0 the goal is to ensure UGAS of the compact attractor

A := {ζ ∈ H : x -x e ≤ X, τ ∈ [0, T ]}.
To this end, we invoke the next lemma.

Lemma 2. [START_REF] Sanchez | On the practical stability of hybrid control algorithm with minimum dwell time for a DC-AC converter[END_REF] Consider that Property 1 is satisfied, then the eigenvalues of matrix P -1 Q are positive and e Avt ≤

λ 1/2 M λ 1/2 m e -αt , whit α = λ m (P -1 Q).
Moreover, we need to provide a practical minimum dwell-time property for H due to implementation issues. Then, next property is established.

Property 2. There exists a positive scalar T * , such that, for any chosen T ∈ [0, T * ], the solutions to hybrid system H flow for, at least, T ordinary time after 185 the jump, before reaching set D.

Proof. From the proof given in [START_REF] Sanchez | On the practical stability of hybrid control algorithm with minimum dwell time for a DC-AC converter[END_REF] and without loss of generality, we consider that the first jump occurs in t = t 0 , defining t := t -t 0 . Then, the trajectories of the error dynamics of (2), given by x = x -x e flowing in C, as follows,

x( t) = e Av t(x 0 + A -1 v B v ) -A -1 v B v .
From Lemma 2, we have

x( t) -A -1 v B ≤ λ 1/2 M (P ) λ 1/2 m (P ) e α t x0 + A -1 v B v .
Then, for all 0 ≤ t ≤ T , we get

0 ≤ t ≤ 1 α Ln λ 1/2 M (P ) x0 + A -1 v B λ 1/2 m (P ) x( t) -A -1 v B .
It is worth noting that in the instant time t = T the solution to H jumps, i.e., the solution is in D. Thus, it is possible to define a dwell-time upper bound, T * ≥ T ≥ 0, as follows,

T * = max i∈{-1,1} 1 α Ln λ 1/2 M (P ) x0 + A -1 i B i λ 1/2 m (P ) x(T * ) -A -1 i B i .
Note that for a maximum possible chattering given by x0 and x(T * ), we get the upper bound T * .

The next Lemma is a fundamental step to prove stability results. Then, we evoke [14, Lemma 1], which is necessary to provide stability properties of ( 12)-( 14). x P

(A i x + B) ≤ -x Qx, (16)
with x = x -x e .
Now, we are able to introduce the asymptotic stability result for the regulation problem, i.e., for a constant reference x e .

Theorem 1. Consider an operating point x e ∈ R 4×1 that satisfies Assumption 2. Moreover, consider that matrices P ∈ R 4 0 and Q ∈ R 4 0 satisfy Property 1. Then, for any X > 0 and a given scalar T ∈ [0, T * ] the following holds:

1. set A is compact, 2. set A is UGAS for hybrid system (12)-( 14), 3. the system output is bounded by y -V r < X, 4. set A 0 := {ζ : x = 0, u ∈ {0, 1}, τ ∈ [0, T ]} is globally practically asymptotically stable for system (12)-( 14). As long as, T is small enough, set

A can be arbitrarly close to A 0 .

Proof. First, we will prove the compactness of attractor A.

Proof of item 1) Hybrid system ( 12)-( 14) satisfies Property 2, then we can define an upper bound of x in C

x(t) ≤ max i∈{-1,1} λ 1/2 M (P ) λ 1/2 m (P ) e α t x(t 0 ) + A -1 i B i + A -1 i B i := X,
such that, t 0 is any time instant where occurs a jump in the steady state. Then, we ensure that A is compact. Now, from Proposition 2, it is possible apply useful well-posed hybrid results.

Proof of item 2) Let us consider the following Lyapunov function,

V 1 (x) := 1 2 x P x, (17)
with x = x -x e . Then, from Lemma 3, it is easy to see that during flows, the solutions are in C, i.e., in [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF], therefore, the following holds

∇V 1 (x), f (ζ) = x P (A v x + B v ) ≤ -x Qx. V 1 (x + ) -V 1 (x) = 0, ( 18
)
since x+ = [x + -x e ] = x.
Then, UGAS of A is shown from the proof given in [14, 15, Theorem 1].

Proof of item 3) The proof is direct from item 2).

Proof of item 4) From Property 2 is easy to see that as T goes to zero, the minimum dwell time, T goes also to zero. Then, from a sufficiently small T , set A shrinks to A 0 .

Stability properties for

V r (t) = V max sin(ωt)
For λ e (V r ) fixed, the proposed hybrid control allows to globally stabilize the sampled equilibrium ( 9), as seen above. Now, we assume a time-varying nonsampled reference, V r (t), in particular take λ e = λ e (V max sin(ωt)). From the property of this function given in Proposition 1 and by continuity of ( 9) with respect to λ e , we can deduce that x e (λ e ) is also a periodic continuous function of time. Intuitively, if the variation of x e with respect to time is sufficiently slow, we can expect that the output y e (x e ) be close to y ae = V max sin(ωt).

Stability properties of the tracking problem of system 2 are established in next lemma.

Lemma 4. Consider the averaged model (4) and x e = x e (V max sin(ωt)), being

x e (λ e ) given by [START_REF] Cortes | Dynamical sliding-mode control of the boost inverter[END_REF], such that, we have

A λe(Vmax sin(ωt)) x e (t) + B = 0. ( 19
)
Assume that there exist parameters ρ > 0 and > 0, such that, Q = C C + I and a positive definite symmetric matrix P R 4 satisfying min

P,ρ ρ s.t. ρ > 0 P > 0   A v P + P A v + 2Q -P -P -ρI   ≤ 0, v = -1, 1 (20)
then, for each solution to

   dxae dt = A λe(t) x ae + B y ae = Cx ae , (21)
with x ae (0) = x e (0) and λ e (t) = λ e (V max sin (wt)). When t → ∞, it holds, for

γ 2 = ρ, that t+Te t |y e ( t) -V max sin(ω t)| 2 d t ≤ γ 2 t+Te t dx ae ( t) d t 2 d t,
where y e (t) := Cx e (λ e ).

Proof. The proof is inspired by the proof of [ Consider now the Lyapunov function V 2 (x e , x ae ) = 1 2 (x e -x ae ) P (x e -x ae ), then the derivative of V 2 along the trajectories of ( 4) is given as:

dV 2 dt = (x e -x
To complete the proof, it suffices now to remark that y ae (t) = V max sin(ωt).

From this lemma, we can provide our main result about controlling the boost inverter working as DC-AC converter.

Theorem 2. For a given output reference V r = V max sin(ωt), consider x e (λ e (V r (t))), defined in Proposition 1 and some given parameters T e and T , such that condition (15) is satisfied. Moreover, consider there exist matrices P ∈ R 4 0,

Q ∈ R 4
0 and parameters γ 2 = ρ > 0 and > 0, such that, all Lemma 4 assumptions are satisfied. Then, for any parameters X t > 0, X > 0 and for

ε 0 (t) := γ 2 2 t+Te t dx ae ( t) d t 2 d t.
and any scalar T ∈ [0, T * ] the following holds:

1. set

A t := {ζ ∈ H : x -x e ≤ X + X t , τ ∈ [0, T ]}.
is compact, 2. set A t is UGAS for hybrid system (12)-( 14), 3. the system output is bounded by

t+Te t |y(t) -V r ( t)| 2 d t < T e X 2 + ε 0 , (12)- (14)
, 4. set A t,0 := {ζ : x -x e ≤ X t , u ∈ {0, 1}, τ ∈ [0, T ]} is globally practically asymptotically stable for system (12)-( 14). As long as, T is small enough, set A can be arbitrarily close to A 0 .

Proof. The proof mainly follows the proof of Theorem 1.

Proof of item 1) From the proof of Theorem 1, here, the proof is completed from Lemma 4, which provides a bound of y e -V r (t) , which implies the bound

x e -x ae ≤ ψ(ε 0 (t)) ≤ X t . Then, x -x e ≤ x -x e + x e -x ae ≤ X + X t .
Proof of item 2) In this item, we will consider the following Lyapunov candidate

V (x, x e , x ae ) := V 1 (x, x e ) + V 2 (x e , x ae ) (24)
defined in [START_REF] Feron | Quadratic stabilizability of switched systems via state and output feedback[END_REF] and Lemma 4 proof. It is easy to see the solution to the optimisation problem given in Lemma 4, satisfy the assumptions of Theorem 1. Then, the solution to hybrid system ( 12)-( 14) goes to the interior of A t , guaranteeing that this set is UGAS.

Proof of item 3) The output error is |y -

y ae | 2 = |y -V max sin(ωt)|
|y e ( t) -V r ( t)| 2 d t < T e X 2 + ε 0 .
Proof of item 4) Direct from the proof of Theorem 1.

Remark 2. If x ae (t) is constant, the right term in the inequality ( 23) is equal to zero and y e (t) → V r (t). As expected, the error depends on the variations of

x ae (t) and γ 2 can be interpreted as the rejection gain when dxae(t) dt is considered as a perturbation signal.

Remark 3. The approach proposed above is based on the assumption R R 0 .

If this assumption is not satisfied, it is not possible to obtain an analytical expression for λ e (V r ). In such case, the value of λ e (V r ) is characterized by:

V r = -λ e R 0 V in (2λ e -1)(λ e -1) λ 2 e R 0 (λ e -1) 2 + R(2λ 2 e -2λ e + 1)
.

Recall that only the admissible values of λ e are real, meaning 0 < λ e < 1. The previous equation can also be written as

1 + V r R 0 V in λ 2
e R 0 (λ e -1) 2 + R(2λ 2 e -2λ e + 1) λ e (2λ e -1)(λ e -1) = 0.

Invoking the simple root locus building rules, we can easily see that depending of the sign of V r , only one root is of interest and varies with V r around 1/2.

Then it is possible to compute the value of λ e for a given value of V r . All the values of λ e for a given function V r (t) can be determined off-line. The approach developed above can then be applied.

It is worth noting that Theorem 2 provides a solution for almost all items stated in Problem 1. However, the last item related with robustness is not still 265 considered.

Robustness

In many occasions, these electronic systems suffer variations in the load, R, in the voltage input, V in , or in other components, needing to guaranty output robustness with respect to parameter variations, or even, to modelled errors. To this deal, we introduce an extra dynamic to regulate the voltage output, y(t).

This dynamic is given by

˙ = -K 1 -K 1 K 2 x4 , (25)
where K 1 and K 2 are predefined positive parameters, such that, Property 1 holds. The variable gathers the current reference variations to compensate any perturbation occurred in the voltage due to any parameter change. Therefore, it is needed to add to the current references. In particular, to the reference of x e,1 = i 1 + i 2 , and x e,3 = i 1 -i 2 . Thus, we can define

where

Āv :=            -R L -1 2L 0 v 2L -K 1 K 2 -K 1 + R L 1 2C0 0 -v 2C0 0 v-1 2C0 0 v 2L -R L -1 2L -K 1 K 2 -K 1 + R L -v 2C0 0 1 2C0 -2 R0C0 v-1 2C0 0 0 0 -K 1 K 2 -K 1            Bv :=            2Vin L 0 0 0 0            +            A v x e (λ e ) 0            C := [0 0 0 1 0].
From this dynamic, we can formulate the robust hybrid system as follows:

H :

                                                                                           χ v τ χae λe χe               =                  Āv χ + Bv 0 1 -dz( τ T ) Āλe χ ae + B f 1 (V r ) f 2 (λ e ) 0                  , ζ ∈ C               χ+ v + τ + χ + ae λ + e χ + e               ∈                χ argmin i∈{-1,1} χ P ( Āi χ + Bv ) 0 χ ae λ e χ e                , ζ ∈ D, ỹ = Cχ (27) where ζ = [ χ v τ χ ae λ e χ e ] ∈ H, such that, H := { χ ∈ R 5 , v ∈ {-1, 1}, τ ∈ R + , χ ae ∈ R 5 , λ e ∈ [ν, 1 -ν]
, χ e ∈ R 5 }, with a parameter ν > 0 small enough.

Moreover, Āλe = Āv , where v is replaced by λ e and B := [ B 0 ]. Likewise, the flow and jump sets are

C := { ζ : χ P ( Āv χ + Bv ) ≤ -χ Q χ, τ ∈ [0, T ]} (28) D := { ζ : χ P ( Āv χ + Bv ) ≥ -χ Q χ, τ = T }. (29)
with matrices P , Q 0 ∈ R 5 .

Theorem 3. For a given output reference V r = V max sin(ωt), consider χ e (x e (λ e (V r ))),

being x e (λ e ) defined in Proposition 1 and some given parameters T e and T , such that condition (15) is satisfied. Moreover, consider there exist matrices P ∈ R 5 0, Q ∈ R 5 0 and parameters γ 2 = ρ > 0 and > 0, such that, all Lemma 4 assumptions are satisfied. Then, for any parameters X t > 0, X > 0 and for

ε 0 (t) := γ 2 2 t+Te t dχ ae ( t) d t 2 d t.
and any scalar T ∈ [0, T * ] the following holds:

1. set

A t := {ζ ∈ H : χ -χ e ≤ X + X t , τ ∈ [0, T ]} is compact, 2. set A t is UGAS for hybrid system (27)-(29), 3
. the system output is bounded by

t+Te t |y(t) -V r ( t)| 2 d t < T e X 2 + ε 0 , (12)- (14)
, 4. set A t,0 := {ζ : χ -χ e ≤ X t , u ∈ {0, 1}, τ ∈ [0, T ]} is globally practically asymptotically stable for system (12)-( 14). As long as, T is small enough, set A can be arbitrarily close to A 0 .

Proof. The proof is direct from Theorem 2 proof. Now, all items of Problem 1 are satisfied.

Simulations

In this section, we validate our hybrid control approach to the boost inverter (2) in simulations. These simulations are performed in MATLAB/Simulink by exploiting the HyEQ Toolbox [START_REF] Sanfelice | A toolbox for simulation of hybrid systems in Matlab/Simulink: Hybrid equations[END_REF].

The boost inverter parameters are given in Table 2, satisfying Assumptions 1.

Moreover, we take K 1 = 1 and K 2 = 1000. Then, the related optimization problem of Theorem 3 provides = 9 • 10 -8 , ρ = 1.14 • 10 -5 , and

P =            3
          
.

The upper bound of T for a chattering of 10A in the currents, 4V in the voltages and 2 • 10 5 in ρ is T * = 7.92 • 10 -4 . Hence, the selected minimum dwell-time was T = 10 -6 s. This scenario is characterized by a perturbation of the load resistance, R 0 , of 100% with respect to its nominal one, i.e., it changes from 100Ω to 200Ω at 0.02s. Figure 4 shows the evolution of the voltages, currents and variable of [START_REF] Manandhar | Efficiency comparison of dc and ac microgrid[END_REF] when the controlled hybrid system (27)-( 29) is applied. Note, after the change of load R 0 at 0.02s, the currents i 1 and i 2 converge to a different equilibrium.

Moreover, Fig. 4 performs the system output. Note, the extra state evolves, such that, the system output gets to reject the perturbations and the modelled errors.

Scenario II

In this second scenario, the input voltage V in changes a 30% with respect to its nominal value, i.e., from 48V to 100V at 0.02s. Figures 5 and6 show the state and output system evolutions. Note, the system output rejects the perturbations and modelled errors with the introduced extra state ρ, converging conclude the error tracking system output is regulated, such that, is robust with respect to the considered perturbation.

Conclusion

In this paper, a hybrid control law is proposed for a DC-AC converter also known as a boost inverter. This control takes into account the real nature of the inverter signals, meaning the continuous-time dynamics represented by the voltages and the currents and the discrete-time dynamic, which is the discrete input signal. Moreover, a minimum dwell time is also considered. UGAS is guaranteed to an attractor, ensuring the convergence of y(t) to a neighbourhood of its reference, small enough. An extension of this result is given to provide robustness properties in terms of error tracking output regulation. Here, it is shown the ability of the system output to reject any parameter change.

The future work is to obtain experimental results in a prototype.

Figure 2 :

 2 Figure 2: Block diagram of the control mechanism.

Lemma 3 .

 3 Consider matrices P, Q ∈ S 4 satisfying Property 1, for each point x e ∈ R 4 satisfying Assumption 2, then, min i∈{-1,1}

x

  e, := x e (λ e ) -Υ , where x e (λ e ) is given in Property 1 and Υ := [1 0 1 0] . Defining the augmented vectors χ := [x ] , χ ae := [x ea 0] and χ e := [x e, 0] , as well as, χ := χ -χ e , we get the following error system dynamics    χ = Āv χ + Bv , ỹ = C( χ + χ e -χ ae ),

Remark 4 .

 4 Remark 3 makes reference to modelled errors, when considering R 0 R. This can be compensated from the hybrid scheme (27)-(29) and applying Theorem 3.

Figure 3 :

 3 Figure 3: Evolution of the states with a 100% variation of R 0 w.r.t. its nominal one.

y

  (t) to V r (t) = V max sin(ωt) after the change of V in . Nevertheless, we can

Figure 4 :

 4 Figure 4: Output with a 100% variation of R 0 w.r.t. its nominal one.

Figure 6 :

 6 Figure6: Output system with a 30% variation of V in from its nominal one.

Table 1 :

 1 Boost inverter switching logic.

  -x ae ) (A λe(t) ) P + P A λe(t) )(x e -x ae ) -2(x e -x ae ) P dx ae dt . I and on the right by its transpose, we obtain (x e -x ae ) (A v P + P A v + 2 I + 2C C)(x e -x ae )

	ae ) P -(x e -x ae ) P and due to (19), we can write dV 2 dt = (x e Multiplying (20) on the left by (x e -x ae ) dxae dx ae dt + dx ae dt dx ae dt -dx ae dt P (x e -x ae ), P (x e -x ae ) dt , using the selected Q = C C + -2(x e -x ae ) P dx ae dt -γ 2 dx ae dt dx ae dt ≤ 0, t+Te t dx ae ( t) d t 2 d t ≤ 0. From equation (22), we have: V 2 (x t+Te t |y e ( t) -y ae ( t)| 2 d t -γ 2 2 t+Te t dx ae ( t) d t 2 d t ≤ 0, and by the periodicity of x ss (t), dx Then, t+Te t |y e ( t) -y ae ( t)| 2 d t ≤ γ 2 2 t+Te t 2 dx ae ( t) d t d t.

a dt = A λe(t) (x e -x ae ).

with v = {-1, 1}. Note that A 1 = A(1) and A -1 = A(0). Now, multiplying the first inequality by λ(t) and the second one by 1 -λ(t), and summing the two resulting inequalities, we obtain: (x e -x ae ) (A λe(t) P + P A λe(t) + 2 I + 2C C)(x e -x ae ) -2(x e -x ae ) P dx ae dt -γ 2 dx ae dt dx ae dt ≤ 0. Then, by the positivity of (x e -x ae ) 2 I(x e -x ae ) we have dV 2 dt + 2|y e (t) -y ae (t)| 2 -γ 2 dx ae (t) dt 2 dt ≤ 0, which in turns by integration on [t, t + T e ] gives V 2 (x e (t + T e ), x ae (t + T e )) -V 2 (x e (t), x ae (t))+ t+Te t 2|y e ( t) -y ae ( t)| 2 d t -γ 2 e (t + T e ), x ss (t + T e )) -V 2 (x e (t), x ss (t))+

  2 , integrating from t to t+T e and applying Theorem 1 and Lemma 4 to bound |y -y e | 2 and |y e -y ea | 2 respectively, we have

	t+Te t	|y( t)-V r ( t)| 2 d t <	t+Te t	|y( t)-y e ( t)| 2 d t+
	t+Te
	t

Table 2 :

 2 Simulation parameters

	Parameter	Notation Value Units
	DC input voltage	V in	70		V
	Reference peak voltage	V max	220	√	2	V
	Nominal angular	ω	120π	rad/s
	frequency
	Nominal load resistance	R 0	100		Ω
	Estimated series	R	0.02		Ω
	resistance
	Inductor	L	44		mH
	Output capacitor	C	200		λF
	Note that Property 7 and condition (15) are satisfied. Some simulations are
	performed considering two scenarios:
	6.1. Scenario I

  Figure5: Evolution of the states with a 30% variation of V in from its nominal one.

	-5									i 1
										i 2
	-10
			0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
										v 1
										v 2
			0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
	1.5	10	5
	0.5
	-0.5
	-1
			0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04
	v_1-v_2 (V)
	-100
	-200								Reference
									v 1 -v 2
	-300		0.005	0.01	0.015	0.02	0.025	0.03	0.035	0.04

Acknowledgements

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