Hai Yen Ta

Lucie Perquis

Stéphane Balayssac

Christophe Déjugnat

Alexandre Wodrinski

Fabrice Collin

Véronique Gilard

François Couderc
email: francois.couderc@univ-tlse3.fr

Prof François Couderc

Separation of unsaturated C18 fatty acids using perfluorinated-micellar electrokinetic chromatography: I) Optimization and separation process

Keywords: micellar electrokinetic chromatography, ammonium perfluorooctanoate, fatty acids, surfactant, self-assembly alpha linolenic acid ALA, ammonium perfluorooctanoate APFOA, elaidic acid EA, fatty acids FAs, linoleic acid LA, perfluorooctanoic acid PFOA, oleic acid OA, trifluoroethanol TFE

Ammonium perfluorooctanoate (APFOA) was used as surfactant for the separation of free unsaturated C18 fatty acids by micellar electrokinetic chromatography. A simple background electrolyte of 50 mM APFOA water/methanol (90:10, v/v) at pH = 10 enabled repeatable separation of oleic acid, elaidic acid, linoleic acid, and alphalinolenic acid in less than 20 min. Separation conditions were optimized regarding various parameters (organic solvent, counterion, APFOA concentration, and pH).

Because the repulsive interactions between fluorocarbon chains and hydrogenated chains are known to lead to segregation and phase separation, the choice of perfluorinated micelles to separate such perhydrogenated long-chain acids could appear astonishing. Therefore, the cmc, the charge density, and the mobility of the micelles have been determined, resulting in a first description of the separation process.

Introduction

Fatty acids (FAs) represent one of the major lipid classes and are involved in many biological processes. Among the analytical techniques suitable for the separation of FAs, gas chromatography (GC) is the mainly used one. GC-based protocols have been very well optimized especially for the separation of FA methyl esters (FAMEs), and several applications have been described in the literature [START_REF] Eder | Gas chromatographic analysis of fatty acid methyl esters[END_REF][START_REF] Wei | Gas chromatography-mass spectrometry and highperformance liquid chromatography-tandem mass spectrometry in quantifying fatty acids[END_REF][START_REF] Xiang | Application of Derivatization in Fatty Acids and Fatty Acyls Detection: Mass Spectrometry-Based Targeted Lipidomics[END_REF]. Capillary electrophoresis (CE) is another leading technique, now well described for the separation of free FAs [START_REF] Poinsot | A digest of capillary electrophoretic methods applied to lipid analyzes[END_REF][START_REF] De Oliveira | 20 Years of Fatty Acid Analysis by Capillary Electrophoresis[END_REF]. In most of the reported separation conditions, the use of sodium dodecyl sulfate (SDS) is necessary, in combination with organic solvents or cyclodextrins. This ensures a good solubilization of FAs, which are poorly watersoluble and can themselves form micelles.

Micellar electrokinetic Chromatography (MEKC) can be used for the separation of FAs [START_REF] Schmitz | Separation of unsaturated fatty acids and related isomeric hydroperoxides by micellar electrokinetic chromatography[END_REF][START_REF] Melchior | Simultaneous micellar electrokinetic chromatographic determination of isomeric fatty acid hydroperoxides and corresponding hydroxy fatty acids[END_REF][START_REF] Amorim | Screening method for simultaneous detection of elaidic and vaccenic trans fatty acid isomers by capillary zone electrophoresis[END_REF], including detection in real matrixes [START_REF] Sato | A capillary electrophoresis method for free fatty acids screening and acidity determination in biodiesel[END_REF][START_REF] Amorim | Capillary electromigration methods for fatty acids determination in vegetable and marine oils: A review[END_REF][START_REF] Amorim | A fast and validated capillary zone electrophoresis method for the determination of selected fatty acids applied to food and cosmetic purposes[END_REF], as well as non-aqueous capillary electrophoresis (NACE) [START_REF] Azab | Robust Method for High-Throughput Screening of Fatty Acids by Multisegment Injection-Nonaqueous Capillary Electrophoresis-Mass Spectrometry with Stringent Quality Control[END_REF]. In the latter case, the separation is performed using organic solvents (acetonitrile, methanol, isopropanol) in the presence of aqueous ammonia or ammonium acetate. These volatile electrolytes have the advantage to be useable when CE is coupled to electrospray ionization mass spectrometry (in CE/ESI-MS experiments), while MEKC using SDS is not compatible with ESI-MS due to its low volatility. To overcome this major drawback and enable MS detection, volatile surfactants have to be used in replacement of SDS.

Two decades ago, ammonium perfluorooctanoate (APFOA) has been used as surfactant in MEKC experiments [START_REF] Petersson | Direct coupling of micellar electrokinetic chromatography to mass spectrometry using a volatile buffer system based on perfluorooctanoic acid and ammonia[END_REF]. It was found to be a compatible volatile surfactant for MEKC-ESI/MS analysis of polar compounds such as Nmethylcarbamates [START_REF] Van Biesen | Ammonium perfluorooctanoate as a volatile surfactant for the analysis ofN-methylcarbamates by MEKC-ESI-MS[END_REF][START_REF] Moreno-González | Dispersive liquid-liquid microextraction using a low density extraction solvent for the determination of 17 N-methylcarbamates by micellar electrokinetic chromatography-electrospray-mass spectrometry employing a volatile surfactant[END_REF], charged and uncharged phenyl compounds (more or less hydrophobic) [START_REF] Petersson | Direct coupling of micellar electrokinetic chromatography to mass spectrometry using a volatile buffer system based on perfluorooctanoic acid and ammonia[END_REF], pyrene and benzopyrene [START_REF] Takagai | UV-detection capillary electrophoresis for benzo[a]pyrene and pyrene following a two-step concentration system using homogeneous liquid-liquid extraction and a sweeping method[END_REF], benzimidazoles [START_REF] Tejada-Casado | Coupling sweeping-micellar electrokinetic chromatography with tandem mass spectrometry for the therapeutic monitoring of benzimidazoles in animal urine by dilute and shoot[END_REF], amino acids [START_REF] Takagai | Determination of ppb Levels of Tryptophan Derivatives by Capillary Electrophoresis with Homogeneous Liquid-Liquid Extraction and Sweeping Method[END_REF][START_REF] Moreno-González | Micellar electrokinetic chromatography-electrospray ionization mass spectrometry employing a volatile surfactant for the analysis of amino acids in human urine[END_REF], etc. However, its use for FAs separation in MEKC-ESI/MS has not been reported so far. Moreover, the physical and chemical characteristics of the resulting micelles in the BGE have not been characterized neither.

The repulsive interactions between fluorocarbon chains and hydrogenated chains are known to lead to segregation and phase separation [START_REF] Kadi | Demixing of fluorinated and hydrogenated surfactants into distinct aggregates and into distinct regions within aggregates. A combined NMR, fluorescence quenching, and cryo-TEM study[END_REF], due to severe nonidealities.

The mixing of SDS and APFOA was found to result in demixing into two kinds of micelles with either a high-level content of SDS or APFOA [START_REF] Mukerjee | Nonideality of mixing of micelles of fluorocarbon and hydrocarbon surfactants and evidence of partial miscibility from differential conductance data[END_REF]. This is still under debate since another hypothesis about this kind of mixture is that it would form mixed micelles [START_REF] Kadi | Mixed Micelles of Fluorocarbon and Hydrocarbon Surfactants. A Small Angle Neutron Scattering Study[END_REF]. However, it seems that small amounts of hydrogenated surfactants can be incorporated into APFOA-rich micelles, and small amounts of fluorinated surfactants into hydrogenated surfactant-based micelles [START_REF] Nordstierna | Mixed Micelles of Fluorinated and Hydrogenated Surfactants[END_REF]. In this context, could APFOA -used as volatile surfactant -be a good candidate to separate FAs, keeping in mind that it could be later be compatible with MS analysis? As a proof of concept, the latter possibility was positively examined for the separation of low concentrations of fatty acids by APFOA MEKC, keeping simple UV-Vis detection as a first step. Thus, APFOA micelles were used to separate four free unsaturated FAs, namely oleic acid (OA), elaidic acid (EA), linoleic acid (LA), and alpha-linolenic acid (ALA). The interactions underlying the separation of these hydrogenated compounds in the perfluorinated medium were studied, and a first approach of the separation mechanism was proposed. This manuscript will first present the optimization of CE separation. Then, the measurement of the mobility of the APFOA micelles was performed by using a micelle marker we synthesized. The CE instrument was also used to determine the cmc (critical micelle concentration), which was confirmed by NMR experiments. Based on the residence time of FAs and geometric considerations, we propose a preliminary separation mechanism of FAs by APFOA micelles.

Materials and methods

Chemicals and reagents

Perfluorooctanoic acid (96 %, PFOA), FAs (≥ 99 %, including OA, EA, LA, and ALA), 2,2,2-trifluoroethanol (TFE), phenylalanine (99 %), sodium hydroxide, ammonium hydroxide, methanol, ethanol, and acetonitrile (HPLC grade solvents) were purchased from Sigma-Aldrich (Saint-Quentin-Fallavier, France). Perfluorooctanoyl chloride (97 %) was supplied from Fisher Scientific-Alfa Aesar (Illkirch, France). Deuterated solvents were supplied from Eurisotop (Saint-Aubin, France).

Preparation of samples and buffers

Stock solutions of separation buffer -the ammonium perfluorooctanoate (APFOA) micellar solutions -were prepared by weighting and dissolving the appropriate amounts of perfluorooctanoic acid (PFOA) in aqueous ammonium hydroxide (NH4OH).

CE separation parameters

CE analysis was performed on an Agilent Technologies CE7100 system (Waldbronn, Germany). Separation of FAs was carried out on a 50 µm id x 365 µm od x 51 cm effective length (59 cm total length) bare-fused silica capillary (Picometrics -Adelis Technologies, Labège, France).

New capillaries were rinsed with 1 M NaOH for 60 min, water (HPLC grade) for 30 min, and background electrolyte (BGE) for 15 min. Between two analyses, the capillary was rinsed with methanol for 3 min, 1 M NaOH for 3 min, water for 2 min, and BGE for 3 min. At the end of each day, the capillary was rinsed with methanol for 3 min, 1 M NaOH for 15 min, and water for 5 min. The pressure applied for the rinsing procedures was 950 mbar. Samples were hydrodynamically injected at 50 mbar for 5 s. All analyses were carried out at 25 °C and at 25 kV for 30 min; the wavelength of UV direct detection was set at 195 nm.

Cmc determination of APFOA

CE method

Solutions containing PFOA at [START_REF] De Oliveira | 20 Years of Fatty Acid Analysis by Capillary Electrophoresis[END_REF][START_REF] Amorim | Capillary electromigration methods for fatty acids determination in vegetable and marine oils: A review[END_REF][START_REF] Moreno-González | Dispersive liquid-liquid microextraction using a low density extraction solvent for the determination of 17 N-methylcarbamates by micellar electrokinetic chromatography-electrospray-mass spectrometry employing a volatile surfactant[END_REF][START_REF] Kadi | Demixing of fluorinated and hydrogenated surfactants into distinct aggregates and into distinct regions within aggregates. A combined NMR, fluorescence quenching, and cryo-TEM study[END_REF][START_REF] Liu | Predictions of the Fluorine NMR Chemical Shifts of Perfluorinated Carboxylic Acids, C n F 2 n +1 COOH ( n = 6-8)[END_REF][START_REF] Kay | Solvent structure in aqueous mixtures. III. Ionic conductances in ethanol-water mixtures at 10 and 25�C[END_REF][START_REF] Terabe | Electrokinetic separations with micellar solutions and open-tubular capillaries[END_REF]40,45,50,60,70,80, 90, 100 mM, and 5.1-fold more ammonia were flushed for 10 min in the separation capillary, then a voltage of +25 kV was applied. The currents were recorded and the curve I c = f([APFOA]), which represents the variation of I c as a function of APFOA concentration, was drawn as reported in [START_REF] Cifuentes | Determination of Critical Micelle Concentration Values Using Capillary Electrophoresis Instrumentation[END_REF]. We obtained a curve which allowed the calculation of the cmc. This curve shows two linear portions, the first one at low APFOA concentrations as described in equation 1:

𝐼 𝑐 = 𝑎 1 • [𝐴𝑃𝐹𝑂𝐴] (1)
with a1 = 6.67•10 -7 A•mM -1 and the second straight line at high APFOA concentrations as described below in equation 2:

𝐼 𝑐 = 𝑎 2 • [𝐴𝑃𝐹𝑂𝐴] + 𝑏 2 (2)
with a2 = 4.5•10 -7 A•mM -1 , b2 = 4.10•10 -6 A

[APFOA] is expressed in mM, and I c in A. At the intersection of these two lines,

[APFOA] is equal to the cmc, which can then be obtained by combining equations 1 and 2 as follows:

𝑐𝑚𝑐 = 𝑏 2 /(𝑎 1 -𝑎 2 ) (3)

NMR method

All solutions for NMR experiments were prepared with deuterated water as solvent. The determination of the critical micelle concentration (cmc) was performed as described by Xing et al. [START_REF] Xing | NMR investigation on micellization of ammonium/tetraalkylammonium perfluorooctanoates[END_REF] using the following equation based on the theory of pseudo-phase transition model:

∆𝛿 = 𝑐𝑚𝑐 𝐶 (𝛿 𝑚𝑜𝑛 -𝛿 𝑚𝑖𝑐 ) + 𝛿 𝑚𝑖𝑐 -𝛿 𝑚𝑜𝑛 ( 4
)
Where Δδ is the chemical shift variation of selected 19 The target compound was prepared via a classical Schotten-Baumann acylation.

Phenylalanine (0.215 g, 1.3 mmol) was dissolved in 2 mL aqueous NaOH (0.2 M) and placed at 0 °C. Perfluorooctanoyl chloride (0.433 g, 1 mmol) was added dropwise followed by additional 2 mL aqueous NaOH (0.2 M). The mixture was stirred for 1 h at 0 °C then for 4 h at room temperature. The white suspension was acidified till pH = 1 using aqueous 6 M HCl. After filtration and thorough washes with water then methanol, the resulting solid was dried under vacuum affording a white powder (0.157 g, 28 % yield). NMR characterizations were in accordance with literature [START_REF] Ma | Dual-Functional Small Molecules for Generating an Efficient Cytochrome P450BM3 Peroxygenase[END_REF][START_REF] Cong | Activation of Wild-Type Cytochrome P450BM3 by the Next Generation of Decoy Molecules: Enhanced Hydroxylation of Gaseous Alkanes and Crystallographic Evidence[END_REF].

High-resolution mass spectrometry (HRMS) analysis was performed by electrospray

Influence of the organic solvent added in the BGE

The presence of organic solvent usually helps dissolving FAs and often allows better separation. It also has an influence on the viscosity of the BGEs and consequently affects the separation times. The addition of 10 % (v/v) acetonitrile (ACN), methanol, or ethanol in the BGE was tested. The corresponding viscosities were reported to be 0.96, 1.07, and 1.20 cP for water/ACN, water/MeOH, and water/EtOH, respectively [START_REF] Thompson | Viscosity measurements of methanol-water and acetonitrile-water mixtures at pressures up to 3500bar using a novel capillary time-of-flight viscometer[END_REF][START_REF] Kay | Solvent structure in aqueous mixtures. III. Ionic conductances in ethanol-water mixtures at 10 and 25�C[END_REF]. The obtained electropherograms (Figure 2) showed that acetonitrile led to the fastest separation of FAs but also to the noisiest baseline. Electropherograms with more stable baselines were obtained with methanol and ethanol, and the three FAs were well separated. However, the presence of ethanol in the BGE lead to the longest migration times. Thus, methanol was selected as it allows obtaining an efficient separation in the shortest migration time. The methanol content in the BGE was then optimized. By increasing it from 10 %, 15 % to 20 % (v/v), the viscosity of the separation medium was increased, leading to longer migration times and peak broadening. At 20 % (v/v) methanol in the BGE, we did not observe any peak for OA (the most slowly migrating FA) after 70 min of analysis (data not shown). Thus, the methanol content was set at 10% (v/v) for the next optimizations.

Effect of the counter ion

It is well-known that the micelle properties not only depend on the nature of the surfactant but also on the type of counterion present at the micellar surface. In this study, we used ammonia or trimethylamine to adjust the pH of perfluorooctanoic acid solutions, resulting in the presence of NH4 + or (CH3)3NH + as counterions, respectively.

The results showed that, as compared to NH4 + , longer migration times were obtained for ALA, LA, and OA in the presence of (CH3)3NH + and that the corresponding electropherogram presented a very noisy baseline (Figure 3), while the peak symmetries were not improved. Consequently, the ammonium ion NH4 + was selected as counterion in the separation electrolyte.

Effects of APFOA concentration and NH4OH/PFOA ratio

The influence of APFOA concentration on the separation of OA, LA, and ALA was also studied (Figure 4). APFOA concentrations ranging from 20 mM to 60 mM were tested ([NH4OH] = 68-204 mM at R = 3.4), in the presence of 10% MeOH in the BGE. At 20 mM APFOA, the three FAs cannot be separated, while the separation was achieved at 25 mM APFOA but the peak resolutions were too low (data are not shown for those two concentrations). At such low concentrations, just above the cmc, only very small quantities of APFOA micelles are formed in the BGE. In contrast, the three FAs were well separated at 30 mM APFOA and above (Figure 4). Increasing APFOA concentration improved peak resolutions but also led to increased migration times.

Moreover, for the highest surfactant concentration, some problems appeared such as unstable current or clogging of the capillary. With a view to analyzing more complex samples containing ALA, LA, and OA, the quality of separation was considered as the most relevant criterion to choose the BGE composition. The quality of separation was evaluated considering peak visibility and separation as well as baseline quality.

Compared to 40 mM, the baseline is more stable at 50 mM and the peak 3 is more visible. Therefore, 50 mM APFOA along with 10% MeOH was selected as BGE composition. The pH also plays an important role in the separation buffer because it defines the ionization forms of species in solution and thus affects their electrophoretic mobilities.

The separation of FAs often requires a buffer at pH > 9 to ensure a strong electroosmotic flow (EOF). In our case, working at high pH also helped to maintain the buffering efficiency of the ammonia solution (pKa ~ 9.2). The influence of the ammonia concentration or the pH was then evaluated. Therefore, the initial ratio R =

Optimization of the methanol content in the sample (injection buffer)

The methanol content in the sample was optimized to enable analysis of both cis-and trans-isomers of FAs. In a first approach, the sample preparation of FAs consisted in their dilution in the BGE. The three cis-isomers ALA, LA, and OA were found to easily dissolve in the BGE consisting of 50 mM APFOA with 10 % (v/v) methanol. However, in the same conditions, the trans-isomer of OA, namely elaidic acid (EA), was found to form an opaque solution, resulting in an electropherogram with unstable baseline.

Therefore, the BGE was gradually diluted with methanol to get an injection buffer containing EA giving satisfying CE analysis. Considering solubility, peak widths, and intensities, the optimal amount was found to be 53 % (v/v) methanol. This composition was then used for all samples. Cyclodextrins and derivatives are often added in the BGE to improve the separation of isomer mixtures. Thus, we tried adding 5 to 15 mM β-cyclodextrin in the BGE to get a better OA vs EA separation. However, in these conditions, the resolution of those two latter peaks was not improved.

Despite the optimization of all above-mentioned parameters, we always observed asymmetric peaks. Only the dilution of FA samples improved peak symmetry.

However, APFOA has a high UV absorption which severely limits FA detection upon dilution and imposes working at high analyte concentration (1mM) in sample. The mechanism allowing the separation between FAs using APFOA micelles remains unclear: how the poor interactions between perfluorinated chains and perhydrogenated chains could assist the separation of these unsaturated C18 acids?

To understand this point, we initiated the characterization of APFOA micelles. We first conducted experiments using the CE instrument to determine the cmc and the charge density of the micelles. Then, we also determined the mobility of the perfluorinated micelles by introducing a PFOA analogue marker, namely the N-perfluorooctanoyl phenylalanine.

Determination of the cmc of APFOA

The cmc of APFOA in the presence of 10 % (v/v) methanol and at R = 5.1 As proposed by Cifuentes et al [START_REF] Cifuentes | Determination of Critical Micelle Concentration Values Using Capillary Electrophoresis Instrumentation[END_REF] we can measure the cmc of the APFOA in such solution (Figure 7A). Here, the cmc was determined to be 19 +/-0.9 mM. The confirmation of the cmc value was performed by NMR experiments by following the chemical shift variations of 19 F, and by plotting them as a function of the reciprocal PFOA concentration (Figure 7B). The resulting curve consisted in two linear parts, the second having an almost zero slope. The cmc was obtained at the intersection of these two parts, and found to be 19 +/-0.9 mM. Thus, the NMR data also confirmed this value.

Degree of counterion dissociation of the perfluorinated micelles

The degree of counterion dissociation can be determined as the ratio of the slopes above cmc (a2) and below cmc (a1) in the I = f(C t ) representation (Figure 7A) [START_REF] Junior | Theoretical and experimental studies of the aggregation number in the ammonium perfluorooctanoate/water binary mixture[END_REF], as described in the experimental section. This ratio gives the degree of counterion dissociation, β, and subsequently the degree of counterion association α: α = (1-β). In our case β = 0.68, which is a very high value as compared to the one obtained for SDS micelles (β=0.34 [START_REF] Ma | Dual-Functional Small Molecules for Generating an Efficient Cytochrome P450BM3 Peroxygenase[END_REF], or β=0.28 [START_REF] Bales | A Definition of the Degree of Ionization of a Micelle Based on Its Aggregation Number[END_REF]). In another study, Kancharla et al. have found a β value of 0.47, in the absence of MeOH in the BGE. The value was found to greatly increase in the presence of urea [START_REF] Kancharla | Perfluorooctanoate in Aqueous Urea Solutions: Micelle Formation, Structure, and Microenvironment[END_REF]. Short-chain alcohols and urea have similar effects on the cmc of APFOA as its value decreases when these compounds are added to the APFOA solutions [START_REF] Kancharla | Perfluorooctanoate in Aqueous Urea Solutions: Micelle Formation, Structure, and Microenvironment[END_REF][START_REF] Dong | Controlling the self-assembly of perfluorinated surfactants in aqueous environments[END_REF]. Therefore, both MeOH and urea could have a similar effect on the counterion dissociation: the presence of MeOH in the BGE could explain the high β-value observed in our experimental conditions. Such a significant value makes the micelles clearly negatively charged and not neutral.

Measurements of the mobilities of the micelles and the free APFOA

To measure the mobility of the APFOA micelles (µ mic ), an aromatic analog of PFOA was synthesized by grafting PFOA on phenylalanine via an amide link. The obtained compound, which will have an important affinity for APFOA micelles, will allow measuring the migration time of these aggregates by UV-detection at 260 nm (absorption of phenylalanine). We obtained a migration time of 146 ± 5 min (n = 3), which is much higher than the one of the measured EOF, tEOF = 3.25 min (injection of water/MeOH 47/53, v/v). In MEKC, the negatively charged micelles normally have a negative mobility compared to the EOF. Moreover, the velocity of the micelles in absolute value is very close to the one of the EOF: the calculated values are µ EOF = 61.7•10 -9 m 2 •V -1 •s -1 and µ mic = -60.4•10 -9 m 2 •V -1 •s -1 . Thus, there is almost no movement of the micelles inside the capillary and the separation of FAs in these conditions depends on their interaction and residence time in the micelles.

Capacity factor (k') and residence time (RT) of the FAs in the micelles

To better understand the mechanism underlying the separation of FAs in our experimental conditions, their capacity factor and their residence time in the micelles have been studied. Terabe et al [START_REF] Terabe | Electrokinetic separations with micellar solutions and open-tubular capillaries[END_REF][START_REF] Terabe | Electrokinetic chromatography with micellar solution and open-tubular capillary[END_REF] proposed the measurement of k' to appreciate the interaction between the solutes (here the FAs) and the micelles, for neutral species. Khaledi et al completed this study for acidic solutes [START_REF] Khaledi | Micellar electrokinetic capillary chromatography of acidic solutes: migration behavior and optimization strategies[END_REF]. k' is defined as:

𝑘 ′ = 𝑛 𝑚𝑖𝑐 /𝑛 𝑎𝑞 = (µ 𝑎𝑔 -µ 𝐹𝐴 )/(µ 𝑚𝑖𝑐 -µ 𝑎𝑔 ) (5)
where nmic and naq are the molar quantities of the solute in the micelles and in the aqueous media, respectively, µ ag is the mobility of the FAs using a micellar medium (i.e. corresponding to the fatty acid migration time), while µ FA is their mobility in the absence of micelles. µ FA is not easy to determine because FAs are not soluble in 10% methanol and 255 mM ammonia. µ FA was thus estimated at the cmc of APFOA, considering that, at the cmc, the concentration of micelles is very low and can be neglected. In these conditions, we found an average value of µ FA = -20.8•10 -9 m 2 •V -

1 •s -1 .
The residence time RTmic of the solute in the micelle is defined as:

𝑅𝑇 𝑚𝑖𝑐 = 𝑡 𝑟 • 𝑘 ′ /(1 + 𝑘′) (6)
where tr and k' are the migration time and the capacity factor, respectively. Table 1 reports the values obtained for the migration time, capacity factor, and residence time for the three FAs ALA, LA, and OA. The residence times of FAs are ranging from 6.1 min to 12.4 min in the micelles, suggesting that FAs spend 56 % to 73 % of their migration time into the micelles. Such high values were not expected since interactions are usually not favored between perhydrogenated molecules and perfluorinated micelles. In addition, the residence time decreases as the number of unsaturations of the FAs increases: OA (1 unsaturation) is more prone to insert into micelles than ALA (3 unsaturations). As a consequence, reminding that the micelles almost do not move inside the capillary, FAs that reside less in these micelles will migrate first in the order ALA>LA>OA. In this system, the separation mechanism originates directly from the interactions between the analytes and the micelles. From these observations, we can suggest a separation process similar to what is usually observed in the case of SDS used in MEKC, which remains quite surprising considering the repulsive interactions between perhydrogenated and perfluorinated chains. Complementary structural analysis at the meso scale will provide more in depth characterization of the separation mechanism [START_REF] Ta | Separation of unsaturated C18 fatty acids using perfluorinated-micellar electrokinetic chromatography: II) Role of nano-structuration[END_REF].

Concluding remarks

In this article we have shown that perfluorinated micelles of APFOA in the presence of 10% methanol and a ratio [NH4OH]/[PFOA] = 5.1 allow the separation of four unsaturated C18 FAs by MEKC. The relatively high electrophoretic mobility (µ mic =-60.4•10 -9 m 2 •V -1 •s -1 ) of the micelles is certainly due to the high degree of counterion dissociation of the perfluorinated micelles (β=0.68). This mobility is much higher than the micelles of SDS (-22.2 m 2 •V -1 •s -1 ) [START_REF] Terabe | Electrokinetic separations with micellar solutions and open-tubular capillaries[END_REF]. In the literature it is reported that the mixture of perfluorinated micelles with perhydrogenated ones exhibits severe non idealities [START_REF] Mukerjee | Nonideality of mixing of micelles of fluorocarbon and hydrocarbon surfactants and evidence of partial miscibility from differential conductance data[END_REF]. However, the successful separation of the different FAs indicates that there is a specific interaction with the micelles, which constitute a pseudo-chromatographic phase, because their displacements are very slow, i.e. their electrophoretic mobilities are almost compensated by the EOF. Indeed, the residence times of FAs in the micelles are long, more than 56% of their full migration times. In this study we show that it is possible to separate FAs thanks to their number of double bounds or the double bond position in the carbon chain. However, the characterization of the interactions between the FAs and the micelles is not accessible with CE. Therefore we conducted other complementary studies involving scattering experiments which are described in a complementary article [START_REF] Ta | Separation of unsaturated C18 fatty acids using perfluorinated-micellar electrokinetic chromatography: II) Role of nano-structuration[END_REF].

  The concentration ratios R, defined as R = [NH4OH]/[PFOA], were R = 3.4 or R = 5.1. As a consequence, [NH4OH] was always proportionally related to [PFOA], depending on the R value. The addition of NH4OH to increase pH almost did not change the ionic strength because excess NH4OH remained mostly in a neutral form close to its pKa (9.2). The formation of NH4 + cations only came from the reaction between NH4OH and PFOA, maintaining [NH4 + ] equal to the initial [PFOA]. The pH of these stock solutions were measured using a Seven Compact S220 pH-meter (Mettler-Toledo, Viroflay, France). The pH values were 9.5 and 10 for R = 3.4 and R = 5.1, respectively. The working BGEs used for optimizing the method were prepared by diluting 100 mM APFOA aqueous stock solutions with water and organic modifiers (i.e. MeOH, EtOH, or ACN) to get the desired final APFOA concentration and organic solvent content. As an illustrative example, the BGE composed of 50 mM APFOA along with 10% MeOH at pH 10 was prepared by mixing the 100 mM APFOA aqueous stock solution (pH=10, R=5.1) with water and MeOH in the ratio 50-40-10 (volume fractions). 30 mM FAs stock solutions were prepared in methanol and kept at 5 °C before use. The injection buffers containing the samples were prepared in 23 mM APFOA water/MeOH (47:53, v/v), to allow solubilization of the four FAs (1 mM).

Fluorine- 19

 19 NMR experiments at the different concentrations of APFOA[START_REF] De Oliveira | 20 Years of Fatty Acid Analysis by Capillary Electrophoresis[END_REF][START_REF] Melchior | Simultaneous micellar electrokinetic chromatographic determination of isomeric fatty acid hydroperoxides and corresponding hydroxy fatty acids[END_REF][START_REF] Amorim | Capillary electromigration methods for fatty acids determination in vegetable and marine oils: A review[END_REF][START_REF] Moreno-González | Dispersive liquid-liquid microextraction using a low density extraction solvent for the determination of 17 N-methylcarbamates by micellar electrokinetic chromatography-electrospray-mass spectrometry employing a volatile surfactant[END_REF][START_REF] Kadi | Demixing of fluorinated and hydrogenated surfactants into distinct aggregates and into distinct regions within aggregates. A combined NMR, fluorescence quenching, and cryo-TEM study[END_REF][START_REF] Liu | Predictions of the Fluorine NMR Chemical Shifts of Perfluorinated Carboxylic Acids, C n F 2 n +1 COOH ( n = 6-8)[END_REF][START_REF] Kay | Solvent structure in aqueous mixtures. III. Ionic conductances in ethanol-water mixtures at 10 and 25�C[END_REF] 40, 50, 60, 80, and 92 mM) including TFE as chemical shift reference (2 mM) were acquired on a Bruker Avance 400 spectrometer (Bruker Biospin AG, Fallanden, Switzerland) equipped with a 5 mm triple resonance probe (TXO) with19 F direct detection (376 MHz).One dimensional (1D)19 F pulse sequence with power-gated decoupling for 1 H (zgpg in Bruker library) was used at 25 °C with the following parameters: acquisition time of 1.74 s with 256K data points and a spectral width of 200 ppm, a relaxation delay of 2 s and 16 scans.Diffusion NMR spectra were recorded with a classical stimulated echo NMR experiment using bipolar gradients (stebpgp1s in Bruker library). The acquisition parameters were as follows: 64K data points, acquisition time 1.19 s, spectral width 80 ppm, number of scans 16, relaxation delay 3 s, pulse field gradient length 3.6 ms and recovery delay 1 ms, diffusion time 200 ms. The spoiler gradients were 0.6 ms long. Sixteen experiments were recorded with a linear gradient sampling from 5 to 95% of the maximum gradient strength.All NMR data were processed using the TOPSPIN 4.0.8 software with one level of zero-filling and an exponential line-broadening function of 0.5 Hz. Spectra were calibrated at -76.96 ppm on the CF3 signal of TFE. The data sets of diffusion NMR experiments were processed with TOPSPIN software and transferred to the Origin 9.1 software. The exponential fit tool was then used to determine the values of the selfdiffusion coefficient (D).19 F NMR assignments of APFOA were given at 50 mM (CE conditions) as previously reported by Liu and Goddard[START_REF] Liu | Predictions of the Fluorine NMR Chemical Shifts of Perfluorinated Carboxylic Acids, C n F 2 n +1 COOH ( n = 6-8)[END_REF]: F2 (-117.91 ppm, tt, 3.0 and 12.8 Hz, -CF2-COOH), F3 (-122.58 ppm, m, -CF2-CF2-COOH), F4 (-122.86 ppm, m, -CF2-CF2-CF2-COOH), F5 (-123.49 ppm, m, -CF2-CF2-CF2-CF3), F6 (-123.65 ppm, m, -CF2-CF2-CF3), F7 (-127.23 ppm, m, -CF2-CF3), F8 (-82.34 ppm, tt, 2.2 and 10.1 Hz, -CF2-CF3).

F 1 .

 1 Scheme 1. Synthesis of the micelle marker by direct acylation of phenylalanine

3 Results and discussion 3 . 1

 31 Optimization of MEKC conditions for the separation of mono-, di-and tri-unsaturated free C18 FAsA standard mixture of three cis-isomers free C18 FAs (OA, LA, and ALA) was first analyzed by using a BGE of 50 mM APFOA at pH 9.5. At this initial chosen pH, the ratio between concentrations of ammonium hydroxide and perfluorooctanoic acid ([NH4OH]/[PFOA]) was R = 3.4. Peak assignation was realized by comparison to reference migration time of each acid, as individually injected. The obtained electropherogram exhibited a noisy baseline due to a strong absorption of the APFOA medium at 195 nm (Figure1). In addition, all FAs peaks were asymmetric. In order to gain in separation performance, the composition of the BGE was optimized (nature and amount of the added organic solvent, APFOA concentration, [NH4OH]/[PFOA] ratio), along with the presence of methanol in the sample composition.

Figure 1 .

 1 Figure 1. Separation of the three cis-isomers free FAs C18: (1) alpha linolenic acid (ALA), (2) linoleic acid (LA), and (3) oleic acid (OA). CE conditions: fused silica capillary (id: 50 µm, Ltot: 59 cm, Leff: 51 cm), BGE consisting of 50 mM APFOA at pH 9.5, [NH4OH]/[PFOA] = 3.4; samples injected at 50 mbar for 5s, +25 kV, 25 °C, detection at 195 nm. FA injected concentration: 1 mM.

Figure 2 .

 2 Figure 2. Effect of organic solvents on the separation of three unsaturated free FAs C18: (1) alpha linolenic acid ALA, (2) linoleic acid LA, and (3) oleic acid OA. ACN: acetonitrile, EtOH: ethanol, MeOH: methanol. CE conditions: fused silica capillary (id: 50 µm, Ltot: 59 cm, Leff: 51 cm). BGE: 50 mM APFOA at pH 9.5 water/organic solvent (90:10, v/v), [NH4OH]/[PFOA] = 3.4. Samples injected at 50 mbar for 5 s. +25 kV, 25 °C, λ: 195 nm. FAs injected concentration: 1 mM.

Figure 3 .

 3 Figure 3. Effect of counterion on the separation of three unsaturated free FAs C18: (1) alpha linolenic acid ALA, (2) linoleic acid LA, and (3) oleic acid OA. CE conditions: fused silica capillary (id: 50 µm, Ltot: 59 cm, Leff: 51 cm). a) BGE: 50 mM ammonium perfluorooctanoate pH 9.5/MeOH (90:10, v/v) and b) BGE: 50 mM trimethylammonium perfluorooctanoate pH 9.5 water/MeOH (90:10, v/v). Samples injected at 50 mbar for 5 s. +25 kV, 25 °C, λ: 195 nm. FAs injected concentration: 1 mM.

Figure 4 .= 3 . 4 .

 434 Figure 4. Effect of APFOA concentration on the separation of three unsaturated free FAs C18: (1) alpha linolenic acid ALA, (2) linoleic acid LA, and (3) oleic acid OA. CE conditions: fused silica capillary (id: 50 µm, Ltot: 59 cm, Leff: 51 cm). BGE: 50 mM APFOA at pH 9.5 water/methanol (90:10, v/v), [NH4OH]/[PFOA] = 3.4. Samples injected at 50 mbar for 5 s. + 25 kV, 25 °C, λ: 195 nm. FAs injected concentration: 1 mM.

[

  NH4OH]/[PFOA] = 3.4 (pH = 9.5) was then increased to 5.1 (pH = 10). The separation performances of the three FAs were compared in these two conditions. Different parameters including migration times, peak widths, and intensities where under focus, and the repeatability of the analysis was evaluated. The results in Figure5A (left)showed that increasing 1.5 times the ammonia concentration in the BGE led to slightly longer migration times of FAs (2 % for ALA, 4 % for LA, and 6 % for OA). At R = 3.4, FA peak widths were reduced, and peak intensities were slightly higher than at R = 5.1 (Figure5A, center and right). In contrast, higher ammonia concentration (R = 5.1) led to better repeatabilities of peak widths and intensities, in particular for OA, for which RSDs were 7 % and 5 %, respectively (Figure5B), instead of 23.5 % and 15.5 %, respectively, at R = 3.4. Therefore, the [NH4OH]/[PFOA] ratio at 5.1 has finally been preferred and selected for the separation of these three FAs.

Figure 5 .

 5 Figure 5. A) Effects of R=[NH4OH]/[PFOA] on the migration times, peak widths, and intensities of three FAs: alpha linolenic acid (ALA), linoleic acid (LA), and oleic acid (LA). A triplicate injection has been performed for each sample. All reported values were averaged. B) Effect of the R ratio on the interdays repeatability (n = 4) of the previous parameters, reported as relative standard deviation (RSD, %).

Figure 6

 6 presents the CE analysis a sample containing a mixture of ALA, LA, OA, and EA (1 mM each), using the simple (previously optimized) BGE consisting of 50 mM APFOA ([NH4OH]/[PFOA] = 5.1, pH = 10) with 10 % (v/v) methanol. It shows the effective separation of these FAs differing by the number of unsaturations (ALA, LA, and OA) and the cis/trans configuration (OA and EA, with resolution peak = 1.6).

Figure 6 .

 6 Figure 6. Separation of four unsaturated free FAs C18: (1) alpha linolenic acid ALA, (2) linoleic acid LA, (3) oleic acid OA, and (4) elaidic acid EA. CE conditions: fused silica capillary (id: 50 µm, Ltot: 59 cm, Leff: 51 cm). BGE: 50 mM APFOA at pH=10 water/MeOH (90:10, v/v), [NH4OH]/[PFOA] = 5.1. Samples

(

  [NH4OH]/[PFOA]) was determined. The cmc is the minimal concentration at which APFOA micelles start to form, and above which the concentration of free APFOA (C RCOO-) remains constant and equal to the cmc. The formation of micelles induces a change in the conductivity of the medium. The conductivities of APFOA solutions were obtainable via the current intensities in the capillary, which were plotted as a function of PFOA concentration (Figure7A). The curve consisted in two linear parts with different slopes: the cmc was obtained at the intersection of these two parts.

Figure 7 . 19 F

 719 Figure 7. Determination of PFOA critical micellar concentration in the presence of 10 % (v/v) methanol by (A) CE and (B) NMR. (A) Current intensity in the capillary as a function of PFOA

  H NMR (DMSO-d6, 400 MHz): (9.76 ppm, d, 3 JHH = 8.5 Hz, 1H, NH), (7.34-7.16 ppm, m, 5H, CHarom), (4.57 ppm, ddd, 3 JHH = 8.5 Hz, 3 JHH = 4 Hz, 3 JHH = 12 Hz, 1H, α CH),

	(Xevo™ G2 QTOF, Waters). Mass obtained for C17H9NO3F15 (M-H) -are: m/z
	560.0343 (calculated) and m/z 560.0347 (experimental) (+0.7 ppm).
	(3.23 ppm, dd, 2 JHH = 14 Hz, 3 JHH = 4 Hz, 1H, 1/2xCH2), (3.01 ppm, dd, 2 JHH = 14 Hz,
	3 JHH = 12 Hz, 1H, 1/2xCH2).
	13 C NMR (DMSO-d6, 100 MHz): (171.3 ppm, -COOH), (156.6 ppm, -CONH), (137.3
	ppm, Cq,arom), (128.9 ppm, CHarom), (128.1 ppm, CHarom), (126.4 ppm, CHarom), (116.6
	ppm, -CF3), (110.2 ppm, -CF2), (110.2 ppm, -CF2), (110.0 ppm, -CF2), (109.7 ppm, -
	CF2), (108.5 ppm, -CF2), (107.8 ppm, -CF2), (54.2 ppm, -CH), (35.4 ppm, -CH2)

ionization (ESI) coupled to a Time of Flight (TOF) analyzer, in the negative mode 1 19 F NMR (DMSO-d6, 376 MHz): (-80.3 ppm, 3F), (-118.6 ppm, 2F), (-121.6 ppm, 2F), (-122.0 ppm, 2F), (-122.6 ppm, 4F), (-125.9 ppm, 2F).

Table 1 .

 1 MEKC parameters of the perfluorinated micelles. Migration time tr, capacity factor k', residence time in the micelles RTmic, of the three FAs in the perfluorinated micelles. tmc 146.0 min, t0 3.25 min, [APFOA] 50 mM

		tr (min)	k'	RTmic (min)
	ALA	10.8	1.3	6.1
	LA	13.2	1.9	8.6
	OA	16.9	2.8	12.4

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest

The authors have declared no conflict of interest.


Daniel Cunico
email: daniel.cunico@phd.unipd.itangelo.cenedese@unipd.it

Angelo Cenedese

Luca Zaccarian
email: zaccarian@laas.fr

Mauro Borgo
email: mauro.borgo@bft-automation.com

Fig. 1. The electric gate used for the experimental tests. Two-degree-of-freedom Robust Feedback Control of a Sliding Gate Automation

Keywords: Industrial Motion Control, Trajectory Tracking, Robust Control, Linear Matrix Inequalities

A control strategy consisting of a feedforward action and a robust feedback for a gate automation is presented, where a low-cost and non-regenerative motor drive is used. A model of the system is developed and feedback linearization is used to compensate for the highly nonlinear dynamics of the electric drive. To achieve good motion tracking performance we design a smooth reference associated with a feedforward action, based on the nominal model of the system. In addition, based on a model of the uncertainties a robust feedback controller is tuned by solving a set of linear matrix inequalities, combining the optimization of a LQR cost with some pole placement constraints. Finally, we test the proposed control strategy on an experimental device, obtaining satisfactory results.

I. INTRODUCTION

Access automation systems are widely used to restrict the entry of vehicles into a property or to manage the access of people in a building. In this sector, the industries need to provide products with excellent performance, but at the same time they must have low production costs. These devices can be considered as typical electromechanical motion systems [START_REF] Lyshevski | Electromechanical systems, electric machines, and applied mechatronics[END_REF] consisting of an electric motor that moves a load at the desired speed through a mechanical transmission, while their control system must be simple enough to be implemented on embedded hardware with limited computational capacity. The control architecture is usually divided in two hierarchical levels [START_REF] Crowder | Electric drives and electromechanical systems: applications and control[END_REF], [START_REF] Lambrechts | Trajectory planning and feedforward design for electromechanical motion systems[END_REF]. A motion planner generates the desired reference based on the parameters set by the user, while a feedback controller minimizes the deviation of the actual trajectory from the desired reference.

In this paper, we consider the control design of an automatic electric gate, a common access automation system shown in Fig. 1. Due to manufacturing costs, non-regenerative DC drives [START_REF] Hughes | Electric motors and drives: fundamentals, types and applications[END_REF], [START_REF] Ioannidis | AC-DC & DC-DC converters for DC motor drives[END_REF] are usually used to command the motor, which cannot generate a controlled braking torque because they only operate in two quadrants. Furthermore, the dynamics of these drives can be highly nonlinear, increasing the difficulties in the design of the control system. Following [START_REF] Cunico | Nonlinear modeling and feedback control of boom barrier automation[END_REF], we illustrate the dynamics of the electric drive and we employ feedback linearization to compensate its nonlinear effects. We design a polynomial motion reference to be tracked by the feedback controller and defined as a function of the user requirements. Jerk limitation in the trajectory planning is introduced since it provides a means to limit the variation rate of the actuator torques [START_REF] Gasparetto | A technique for time-jerk optimal planning of robot trajectories[END_REF] and is of great relevance in reducing the structure vibrations, wear on the machinery and positioning error during the movement [START_REF] Fang | Smooth and time-optimal S-curve trajectory planning for automated robots and machines[END_REF]- [START_REF] Nguyen | On algorithms for planning s-curve motion profiles[END_REF]. The main control goal is to obtain good tracking performance and a smooth maneuver, considering that the controller should be robust with respect to possible slow unmodeled dynamics, small delays in the feedback loop and quantization effects. The control law is defined as the sum of a feedforward action, computed on the basis of a nominal model, and a robust feedback term that takes into account model uncertainties, e.g. coming from the load inertia. The feedback controller synthesis is formulated as a linear matrix inequality (LMI) optimization problem [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF]. This numerical approach has several advantages. Indeed it is possible to take into account a wide range of plant parameters occurring at different operating points, resulting in robust stability of the gate automation. In addition, the LMI constraints allow us to solve a linear quadratic regulator (LQR) problem [START_REF] Olalla | Robust LQR control for PWM converters: An LMI approach[END_REF], [START_REF] Ge | Robust PID controller design via LMI approach[END_REF] and to shape the performance specifications constraining the eigenvalues of the closed-loop system in a desired region of the left half-plane [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF].

The paper is organized as follows. Section II introduces the mechanical system model, the identification procedure and the electrical drive with the feedback linearization technique. Section III presents the proposed control method together with the reference generation and feedforward action, while the robust feedback control is design in Section IV. Experimental results are shown in Section V, followed by conclusions in Section VI. For reasons of confidentiality, all the units of measure are normalized and the technological details are omitted. However, this does not affect the contribution of the work, which is fully parametric and can be implemented on any similar electromechanical system, thereby making our control technique broadly applicable and easily reproducible.

II. SYSTEM MODEL

The electric gate application can be schematized as shown in Fig. 2. The input of the system is the duty cycle of a PWM signal δ that controls the voltage u a supplied to the motor. The DC motor exerts the torque τ m on the mechanical subsystem through a gearbox, allowing the gate to move with a desired linear speed v(t). The mechanical system is illustrated in Fig. 3. In the following, a model of the system is presented. The electrical equation of a DC motor [START_REF] Krishnan | Electric motor drives: modeling, analysis and control[END_REF] is:

u a (t) = R a i a (t) + L a di a (t) dt + e a (t), (1)
where u a (t) is the terminal voltage, i a (t) is the armature current, R a is the winding resistance, L a is the inductance and e a (t) is the back electromotive force (BEMF). Denoting by ω m the mechanical speed of the motor, the BEMF and the torque exerted at the motor shaft correspond to

e a (t) = k t ω m (t), τ m (t) = k t i a (t), (2)
where k t is the torque constant. Because of the power balance between the input electrical power and the output mechanical power, the two constants in eq. ( 2) coincide. The motor is connected to the load by a two-stage transmission, considered to be rigid: a worm gear reducer connected to a pinion and a rack actuator. The mechanical equation of the system is

τ m (t) = J m dω m (t) dt + b m ω m (t) + τ p (t) + τ d (t). (3)
where τ p is the pinion-related torque, τ d accounts for the Coulomb friction affecting of the mechanical transmission and J m , b m are the motor inertia and viscous friction coefficients. Denoting by ω p the speed of the pinion, we have that:

ω p = N g N w ω m = n 1 ω m , τ p = n 1 η 1 τ p , τ p (t) = J p dω p (t) dt + b p ω p (t) + τ (t), (4)
where N g is the number of teeth in the gear, N w is the number of threads in the worm, n 1 is the worm gear ratio, η 1 < 1 is the first gear transmission efficiency, τ is the load side torque, J p and b p are the pinion inertia and friction, respectively. In a similar way, denoting by v(t) the gate linear speed, we have

v = r p ω p = n 2 ω p =⇒ n 2 = v ω p = r p , τ = n 2 η 2 τ , τ (t) = m dv(t) dt + bv(t), (5)
where r p is the pinion radius, n 2 , η 2 are the ratio and the efficiency of the rack and pinion transmission, m and b are the load mass and viscous friction. Note that the rack and pinion gears convert the rotational motion into a linear one. By combining (3)-( 5) the overall mechanical system equation is obtained as

τ m (t) = J tot dω m (t) dt + b tot ω m (t) + τ c sign(ω m (t)), J tot = J m + J p n 2 1 η 1 + m n 2 1 n 2 2 η 1 η 2 , b tot = b m + b p n 2 1 η 1 + b n 2 1 n 2 2 η 1 η 2 , (6)
with the non-smooth term τ d = τ c sign(ω m ). Fig. 3. Scheme of the mechanical system of the gate automation.

To estimate the model parameters we perform a set of experiments with an independent laboratory acquisition system to collect data under sufficient excitation conditions. For identification purposes, a National Instruments DAQ (USB-6216) has been used to acquire the data with a 16-bit ADC resolution and with a 12-bit resolution encoder (Eltra ER38F) mounted on the motor. The torque constant k t can be estimated from the first equation in ( 2) by connecting the motor to a differential voltage probe and by mechanically coupling the motor to another one. Controlling the motor at different speeds, the value of k t is then obtained from u a = k t ω m (from ( 1) and ( 2)) with a linear least squares estimate of the collected BEMF and speed data. The remaining parameters are estimated following the procedure used in [6, eq. ( 11)-( 12)] by acquiring voltage, current and speed signals after imposing constant acceleration/deceleration phases to the motor during the data collection experiment.

A complete and detailed description of the motor drive dynamics is given in [6, Section V], from which the main result concerning the feedback linearization of the electrical dynamics is briefly explained in the following. The driver is composed by a Graetz bridge that rectifies the alternating voltage source V ac , and the motor is controlled by chopping the non-negative semi-sinusoids v + and thus modifying the average voltage output depending on the firing angle of the switch. The equation that describes the average voltage ūa during a period T of the drive is given by

ūa (δ, e a ) = e a (1 -δ) + √ 2V ac π (1 -cos(πδ)), (7)
where the first term is the weighted contribution of ūa (δ, e a ) during the "off" phase, while the weighted contribution in the "on" phase is given by the semi-sinusoid v + with amplitude √ 2V ac and period T , as per the second term. By adopting a feedback linearization approach, the objective is to select the duty cycle δ so that the plant seen by the controller is linear. However, function [START_REF] Gasparetto | A technique for time-jerk optimal planning of robot trajectories[END_REF] is only invertible in a range depending on e a and, specifically, it depends on the minimum ūa,m (e a ) and maximum ūa,M (e a ) values of ūa attainable by choosing δ ∈ [0, 1]. From [6, Proposition 1] by using a feedback linearization approach, we have that for any u ∈ [ū a,m (e a ), ūa,M (e a )], selecting the duty cycle δ as

δ = δ m (e a )+ 1-2δ m (e a ) π acos 1- 2(u -ūa,m (e a )) ūa,M (e a ) -ūa,m (e a ) (8)
the resulting average input obtained from ( 7) is ūa (δ, e a ) = u + ψ with |ψ| ≤ 0.01001(ū a,M (e a ) -ūa,m (e a )). The values of δ producing the minimum and maximum voltage in [START_REF] Gasparetto | A technique for time-jerk optimal planning of robot trajectories[END_REF] are given by [6, eq. ( 15)]

δ m (e a ) = 1 π arcsin e a √ 2V ac ∈ 0, 1 2 , δ M (e a ) = 1 -δ m (e a ) ∈ 1 2 , 1 , (9)
and the corresponding voltages are, respectively, [6, eq. ( 16)] Therefore, by selecting δ as in [START_REF] Fang | Smooth and time-optimal S-curve trajectory planning for automated robots and machines[END_REF], it results that the error between the requested input u and the applied input ūa is only about one percent of the input range.

ūa,m (e a ) = √ 2V ac (1 -δ m )sin(πδ m ) + 1 π - cos(πδ m ) π , ūa,M (e a ) = √ 2V ac δ m sin(πδ m ) + 1 π + cos(πδ m ) π . (10)

III. TWO-DEGREE-OF-FREEDOM CONTROL DESIGN

In this section we describe a two degree-of-freedom control design that combines a feedforward control and a robust feedback to achieve the desired tracking performance of the automatic gate. The control system architecture is shown in Fig. 4. The reference is generated by polynomial functions in order to obtain a smooth trajectory. The feedback block in Fig. 4 consists of a PI controller tuned with an LMI optimization approach discussed in Section IV. The overall control law u, used to compute δ from the feedback linearizing law [START_REF] Fang | Smooth and time-optimal S-curve trajectory planning for automated robots and machines[END_REF], is given by the sum of two terms u(t) = u ff (t)+u fb (t), where u ff is the feedforward input associated with the reference motion r and u fb is the error feedback stabilizer exploiting the plant measurement y = [y θ y ω ] corresponding to the mechanical variables. The linearized plant model, given by eqs. ( 1), [START_REF] Cunico | Nonlinear modeling and feedback control of boom barrier automation[END_REF], has been identified in Section II. Due to eq. ( 8), the dynamics from u to ūa is almost an identity (with a 1% error) when u ∈ [ū a,m (e a ), ūa,M (e a )]. In the following the reference and feedforward blocks of Fig. 4 are discussed.

A polynomial motion reference is designed where jerk constraints are introduced in the trajectory planning to limit the variation rate of the actuator torques. In access automation applications, the reference is usually customized by the user according to the different gate installations that may arise, and consists of four main sections: acceleration, constant high speed, deceleration and constant low speed phases. The last section is needed to comply with safety regulations, requiring that a certain final space θ f of the total displacement is traveled at low speed, to avoid dangerous impacts. As shown in Fig. 5, the motion profile r(t) consists of seven segments, among which the first three constitute the acceleration stage, while the last three represent the deceleration and low speed phases, respectively. The reference must be parametric and computed online based on the following values set by the user: the high ω 1 and low ω 2 velocities, the acceleration time t a and the deceleration time t d . Recall that the motor torque in (6) cannot be negative, namely τ m ≥ 0, since the drive is non-regenerative and it cannot generate a controlled braking torque. Therefore a long enough deceleration phase is needed. The corresponding time evolution of the variables, depicted in Fig. 5, is described for each segment by a set of polynomials and can be deduced from the differential relationship among position θ m , velocity ω m , acceleration a m and jerk j m of the motor:

θ m,i (t) = θ m,i-1 + ω m,i-1 t + 1 2 a m,i-1 t 2 + 1 6 j m,i t 3 , (11a)
ω m,i (t) = ω m,i-1 + a m,i-1 t + 1 2 j m,i t 2 , ( 11b
)
a m,i (t) = a m,i-1 + j m,i t, (11c)
where i = {1, 2, . . . , 7} denotes the specific phase under consideration. With reference to Fig. 5, based on the four parameters t a , t d , ω 1 , ω 2 mentioned above, we first introduce the following "acceleration" and "deceleration" jerk values

j a = ω 1 2/9t 2 a , j d = ω 1 -ω 2 10/21t 2 d , (12)
and then the reference is generated as follows:

• The acceleration phase is divided in three intervals

I 01 = [t 0 , t 1 ], I 12 = [t 1 , t 2 ], I 23 = [t 2 , t 3 ] of equal lengths |I 01 | = |I 12 | = |I 23 | = 1
3 t a with constant jerks j a , 0, -j a as in [START_REF] Olalla | Robust LQR control for PWM converters: An LMI approach[END_REF], so that one obtains from [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF] 11) ω m (t 6 ) = ω 2 and a m (t 6 ) = 0.

• The final approaching phase I 67 = [t 6 , t 7 ] has length |I 67 | = T end (defined below) that ensures θ m (t 7 ) -θ m (t 6 ) = θ f . In summary, the following jerk values are imposed, with times t 0 < t 1 < • • • < t 7 as shown in Fig. 5:

j(t) =                j a if t 0 < t ≤ t 1 , 0 if t 1 < t ≤ t 2 , t 3 < t ≤ t 4 , t 6 < t ≤ t 7 , -j a if t 2 < t ≤ t 3 , -20j d if t 4 < t ≤ t 5 , j d if t 5 < t ≤ t 6 ,
Then, the reference values r(t) = (r θ (t), r ω (t), r a (t)) = (θ * m (t), ω * m (t), a * m (t)) can be computed from [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF] after imposing the continuity conditions. The duration of the constant speed sections I 34 and I 67 can be obtained as follows

T end = θ f ω 2 , T slide = θ tot -θ a -θ d -θ f ω 1 , (13)
where θ tot is the total displacement of the gate, θ a and θ d denote the space covered in the acceleration and deceleration phases, respectively. The feedforward term u ff is defined as a function of the reference r = (r θ , r ω , r a ) and is introduced to improve the control accuracy and the response speed. To select u ff , we first consider the linearized plant model given by eqs. ( 1), [START_REF] Cunico | Nonlinear modeling and feedback control of boom barrier automation[END_REF].

Since the electrical time constant of the motor is much smaller than the mechanical time constant, the dynamics of the current can be ignored for the feedforward generation. Therefore, by imposing L a dia(t) dt = 0, along the reference motion r, equation ( 1) becomes

0 = -R a i a -k t r ω + u ff , (14)
which provides i a = u ff -ktrω

Ra

. Substituting this expression in [START_REF] Cunico | Nonlinear modeling and feedback control of boom barrier automation[END_REF], we obtain the simplified mechanical model

ṙω = - k 2 t + R a b tot R a J tot r ω + k t R a J tot u ff - τ c J tot sign(r ω ), (15)
where r ω denotes the reference speed. It is readily seen that a suitable feedforward action (a right inverse of ( 15)) is

u ff = R a J tot k t r a + k 2 t + R a b tot k t r ω + R a k t τ c sign(r ω ). ( 16
)
Indeed replacing ( 16) in [START_REF] Krishnan | Electric motor drives: modeling, analysis and control[END_REF] one obtains ṙω = r a , as desired.

IV. ROBUST FEEDBACK CONTROL DESIGN We design a robust feedback control action using an optimization algorithm that aims to guarantee the stability, to shape the transient performance by constraining the closed-loop eigenvalues in a region of the left half-plane and to optimize a linear quadratic performance index. Considering the optimized solutions of the simplified model [START_REF] Krishnan | Electric motor drives: modeling, analysis and control[END_REF], and defining the error vector e = [e θ e ω ] = [r θ -y θ r ω -y ω ] ∈ R 2 , the error dynamics follows:

ė = Ae + Bu fb = 0 1 0 - k 2 t RaJtot + btot Jtot e + 0 -kt RaJtot u fb . ( 17
)
Some elements involved in the system matrices A and B may be uncertain or time varying. Then, we can express (17) as a function of the uncertain terms that are grouped in a vector p

ė = A(p)e + B(p)u fb . ( 18
)
We consider that each uncertain parameter p i is bounded between a minimum and a maximum value p i and p i , namely p i ∈ [p i , p i ], and we select p 1 = 1 Jtot and p 2 = 1 RaJtot . In this case study, we consider that the resistance R a and the total inertia J tot are uncertain parameters. The components of the parameter vector p are restricted inside the following intervals

p 1 ∈ [1/J tot,M , 1/J tot,m ], (19a)
p 2 ∈ [1/(R a,M J tot,M ), 1/(R a,m J tot,m )], (19b)
where R a,m , J tot,m , R a,M and J tot,M are the minimum and maximum values of the model parameters. Since A and B depend linearly on the uncertain vector p, a polytope of n = 4 vertices that contains all the possible values of the uncertain matrices can be defined characterized by matrices A j and B j , with j = 1, . . . , n. For an extensive explanation of polytopic models of uncertainty see [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF].

The feedback control law is given by

u fb = Ke = k p k d e, (20)
where K is selected to place closed-loop poles in the complex plane in the intersection of three elementary LMI regions: an α-stability region, a disk of radius ρ, and a conic sector determined by ϑ. Also, the optimal LQR problem consists in finding a control input for (17) that minimizes the integral cost

J = ∞ 0 e Q e + u fb R u fb , (21)
where Q and R are symmetric positive-definite matrices. Fixing parameters α ≥ 0, ρ > α, ϑ ∈ [0, π/2], the LQR controller is found by solving the optimization problem

max β∈R, W ∈R 2×2 , X∈R 1×2
β subject to:

W = W > 0 (22a) βI -W < 0 (22b) M + M + 2αW < 0 (22c) (M + M )sin(ϑ) (M -M )cos(ϑ) (M -M )cos(ϑ) (M + M )sin(ϑ) ≤ 0 (22d) -ρW M M -ρW ≤ 0 (22e)
  M + M W X W -Q -1 0 X 0 -R -1   < 0, (22f)
M j + M j < 0, j = 1, . . . , n, (22g)
where M := A 0 W + B 0 X is defined by the nominal model A 0 and B 0 , while M j := A j W + B j X represent the vertices A j and B j of the uncertainty polytope. Constraints (22c), (22d), (22e) shape the region of the complex plane, modifying the dynamical properties of the system, while constraints (22b), (22f) optimize the linear quadratic index (21). Constraint (22g) guarantees stability of the error dynamics in the presence of model uncertainty.

Proposition 1. Selecting α ≥ 0, ϑ ∈ [0, π/2] and ρ > α, for any feasible solution to (22), selecting K = XW -1 the following properties hold: i) the closed-loop matrix (A j + B j K) has eigenvalues with absolute value less than ρ, ii) the damping factor of the poles is larger than cos(ϑ), iii) for all p in the uncertainty polytope, (A j (p) + B j (p)K) has eigenvalues with real part smaller than -α, iv) the LQR cost satisfies J ≤ β -1 |e(0)| 2 along all solutions.

Proof. i-iii) See the results in [6, Proposition 3]. iv) Performing a Schur-complement twice on (22f) we obtain

M + M + W QW + X RX < 0. (23)
where A cl = A+BK and u = Ke. Defining V (e) = e P e with P = W -1 and noting that (24) implies that A cl W +W A cl < 0 and that e(t) converges to zero as t → ∞, we may integrate (24) between 0 and T and take the limit as T → ∞ to obtain For this application, we choose a matrix Q to penalize the speed error, while R is used to avoid large sizes of K. This choice induces desirable results, as illustrated in the next Section. It is worth to point out that while solving the LMI optimization problem (22) requires the use of convex optimization tools, once the controller gains are determined, the implementation of the proposed solution in a real embedded control system does not require more computational effort than a classical control scheme. Moreover, one of the main advantages with this formulation is to include uncertainties in the feedback gain selection, while classical LQR control is only valid for systems without uncertainty.

V. EXPERIMENTAL RESULTS

In order to verify the results derived in the previous sections we implemented the proposed control strategy in an industrial electric gate. Specifically, the controller has been written in C language and implemented on a 8-bit microcontroller (Microchip PIC18F device) mounted on a control board provided by the company developing the application. In Fig. 6, we show the set of responses of the closed loop system obtained by varying the tuning parameter α of the LMI optimization. Increasing the values of α results in better tracking of the reference, especially in the acceleration phase. However, a larger overshoot can be observed at the end of the acceleration phase, due to the larger feedback gains. Note also that the reference is badly followed at the start of the maneuver, probably due to the significant mechanical backlash of the transmission system. The reference motion was validated by testing different user settable parameter values. Compared to the trapezoidal trajectory characterized by an impulsive jerk profile, the adopted motion profile has limited jerk and attenuates the excessive stresses generated by the step variations of acceleration on the actuators and on the mechanical structure, thus extending the useful life of the device. In Fig. 7 the robustness of the proposed controller (with α = 1.5) is tested for a model parameter uncertainty. Specifically, the mass of the physical gate has been changed to have a variation of the total inertia of ±25 percent with respect to the nominal inertia J tot . It can be noted that, while the input actions δ and u increase or decrease to compensate for the different inertia, the outputs remain similar thereby showing successful feedback action. Finally, the response of the proposed controller (with α = 1.5) is compared with the one of the production standard controller in Fig. 8, resulting in a significant improvement of the tracking performance. The production controller is a PI speed controller, manually tuned by adjusting the gains until acceptable results are obtained. In Fig. 8(c) it can be also observed that the applied voltage for the proposed controller does not saturate as it remains always within its minimum and maximum values, computed using [START_REF] Nguyen | On algorithms for planning s-curve motion profiles[END_REF].

VI. CONCLUSIONS

In this paper, a control architecture for an electric gate is proposed. The nonlinear dynamics of the motor drive is compensated by a feedback linearization action, the feedforward input is computed as a function of a smooth speed reference signal, and a robust feedback controller is tuned using LMI optimization. The effectiveness of this approach has been illustrated with experimental tests carried out on a prototype device. Future work may include explicit consideration of the saturation in the control algorithm and the implementation of adaptive control techniques to eliminate systematic errors in the trajectory tracking task, in addition to alternative feedback control architectures, such as a conventional P-PI cascade with a P position controller and an underlying PI speed controller.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Two-degree-of-freedom control system diagram.

Substituting M =

 = AW + BX, X = KW and left-right multiplying by e we get e (A cl W + W A cl )e + e Qe + u Ru < 0 ∀e = 0.

  τ ))dτ = V (e(0)) = e(0) P e(0). (25) Hence, by using constraint (22b) we can ensure that J = e(0) P e(0) = e(0) W -1 e(0) ≤ β -1 |e(0)|

Fig. 6 .

 6 Fig. 6. Experimental responses of the gate automation using the proposed two-degree-of-freedom control for different values of the tuning parameter α. (a) motion reference ω * m and speed ωm; (b) motor current ia; (c) control input u and (d) duty cycle δ obtained from the feedback linearization. Constants ω m,M , i a,M , u M and t M are normalization factors.

  Fig. 2. Electric gate system blocks scheme.

	δ(t)	Electric Drive	ua(t)	DC motor	τm(t)	system Mechanical	v(t)
				Drive
					Gear
		Motor	ωp(t)	Pinion
		ωm(t)
		Worm			Gate
			v(t)

  ω m (t a ) = ω m (t 3 ) = ω 1 .• The "sliding" interval I 34 = [t 3 , t 4 ] is executed with constant speed ω 1 and length |I 34 | = T slide , defined below. • The deceleration phase is characterized by a short interval I 45 = [t 4 , t 5 ] with |I 45 | = 1 21 t d having a constant negative jerk -20j d followed by a longer phase I 56 = [t 5 , t 6 ] with |I 56 | = 20 21 t d and positive jerk j d ; with the choice in (12), one gets from (

  Fig. 7. Experimental output (a) and input (b) in the presence of a load inertia perturbation of ±25%.Fig. 8. Comparison between the performance of the proposed controller and the production standard controller.

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