Dr Ariana Villarroel Marquez

Dr Gerardo Salinas

Dr Myriam Abarkan

Dr Maël Idir

Prof Cyril Brochon

Georges Hadziioannou

Prof Matthieu Jochen Raoux

Alexander Kuhn

Jochen Lang

Eric Cloutet
email: cloutet@u-bordeaux.fr

Dr Eric Georges Hadziioannou

Cloutet

Lang

Prof A Kuhn

Design of Potassium-Selective Mixed Ion/Electron Conducting Polymers

Keywords: Mixed conducting polymers, PEDOT, crown-ether, ion sensing, coulovoltammetry

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

An approach providing cation selective PEDOT:polyelectrolyte mixed conductors is presented in this communication, based on the structural modification of this ambivalent (ionic and electronic conductive) polymer complex. First, an 18-crown-6 moiety was integrated into the styrene sulfonate monomer structure as a specific metal cation scavenger particularly targeting K + vs Na + detection. This newly functionalized monomer has been characterized by 1 H-NMR titration to evaluate the ion selectivity. Aqueous PEDOT dispersion inks containing the polymeric ion-selective moieties have been designed and their electrical and electrochemical properties have been analysed. These biocompatible inks are the first proof-of-concept step towards ion selectivity in view of their interfacing with biological cells and microorgans of interest in the field of biosensors and physiology.

Mixed conducting polymer films are materials that have gained considerable interest, due to the synergy between the polyelectrolyte, for ionic transport, and the conjugated conducting polymer, for electronic transport. [1][2] Poly-(3,4-ethylenedioxythiophene) (PEDOT), [3] stabilized by poly(styrene sulfonate) (PSS), is one of the most used mixed conducting polymer system for the development of organic electronics. [4][5][6] PSS has a double role in the blend; it works as the counter ion for the conducting doped PEDOT chains and stabilizes the insoluble charged PEDOT in aqueous suspensions. PEDOT:PSS dispersions have been used to obtain thin film electrodes for different applications such as solar cells (OPV), electrochromic electrodes, flexible screen devices (OLED) and organic electrochemical transistors (OECTs). [4], [7][8] In the latter, since mixed conduction occurs in the volume of these devices, the low current resistance of the films allows better electrical recordings with local signal amplification and consequently unprecedented signal detection in biology applications [9][10] pH [11] and ion monitoring. [12][13] An interesting alternative is the enhancement of ion selectivity of these materials by tuning the chemical structures of the conducting polymer or the polyelectrolyte, in order to target an analyte of interest. [14] Complex functionalized polythiophenes, with selective moieties targeting anion and cation analysis have been synthesized. [15][16] Although these mixed conducting polymers present high selectivity, the poor solubility of the monomer in water imposes the use of organic solvents during the electropolymerization. In the frame of a more environmental friendly approach, a promising method to improve solubility may be provided by functionalization of the water soluble polyelectrolyte. In this work, we present an approach to confer cation selectivity to a PEDOT:polyelectrolyte mixed conductor. A specific metal cation scavenger particularly targeting K + detection, the 18-crown-6 moiety, was integrated into the styrene sulfonate monomer through the modification of the corresponding anion. The 18-crown-6 moiety has been chosen due to its well-known affinity towards the target cation K + , based on its size matching, as well as for the commercial availability of its functional derivatives (i.e. aminocrown). [17] Potassium (K + ) was selected as a monovalent target among all the possible choices, due to its importance in the generation and dissipation of electrical cell signals as changes in K + efflux from cells play a major role in cell depolarization and repolarization of membrane potentials. [START_REF] Tien | From Molecules to Networks[END_REF] Incorporation of the ionophore 15-crown-5 in electrodeposited PEDOT for monovalent cations analysis has been previously reported. [START_REF] Jeszke | [END_REF] However, the Synthesis of the newly functionalized monomer (2) was confirmed by 1 H-NMR, 13 C-NMR and EIS-HRMS spectroscopy (see supporting information, monomer synthesis and spectroscopy characterization). This functionalized monomer (2) was characterized by 1 H-NMR titration to evaluate its ion selectivity (i.e. K + vs Na + ). Aqueous PEDOT dispersion inks containing the ion-selective moieties were then designed and their electrical and electrochemical properties have been analyzed. Synthesis of the ion selective monomer, copolymerization of the polyelectrolyte and synthesis of the conductive inks are illustrated in Scheme 1. First, the synthesis of (2) has been achieved in a two-step reaction path. The formation of the styrenesulfonyl chloride (1) was followed by reaction with the functionalized amino phenyl-18-crown-6, to obtain the targeted monomer (2). The 1 H-NMR spectrum of ( 2 Ion selectivity toward K + vs. Na + of newly designed monomer (2) was first investigated via 1 H-NMR titration at various concentrations of potassium and sodium perchlorate (Figure SI-3 and Figure SI-4). Figure 1 shows for instance the chemical shifts obtained during the NMR study of monomer (2) in the presence of 2 molar equivalents in concentration of potassium and sodium perchlorate, respectively. Thus, in the presence of K + , the signals related to the protons in the proximity of the crown ether cavity (signals 9-10 and 2-5) are shifted, as compared to the 1 H-NMR spectrum of the pristine monomer (Figure 1b vs. 1a) in the absence of K + . This behavior can be attributed to the formation of a coordination compound between the K + and the crown ether moiety (signals 2-3 in Fig 1b vs 1a). Furthermore, the chemical shifts related to the vinyl protons (signals 6-7) were not affected by the presence of the cation. This confirms that the binding interaction with the monovalent cations occurs as expected through the crown ether moiety. However, when 1 H-NMR study was performed in the presence of sodium perchlorate at the same concentration as K + , the variation of the chemical shift was negligible (signals 2-3 in Figure 1c vs 1a). From the analysis of the chemical shift as a function of the salt concentration the binding isotherms were obtained (Figure SI-5). Using a 1:1 binding model of the experimental data, the association constants (k a ) of (2) with both cations were evaluated (k a, Na = 121.7  6.9 M -1 and k a, K = 1026.2  118.3 M -1 ). As the K a is one order of magnitude higher for potassium in comparison to sodium, these results confirm the high selectivity of (2) for the K + cation in accordance with the crown ether size-matching rule. In addition, from the dissociation constants (k d = 1/k a ), a stronger binding of K + in this solvent can be inferred. Finally, saturation of the binding sites was obtained only by the addition of the potassium salt (plateau range in figure SI-5). Monomer (2) was then integrated into a polyanion through copolymerization with the monomer 4-styrenesulfonyl (trifluoromethylsulfonyl) imide (STFSI). PSTFSI derivative has been reported by us as an alternative to PSS in PEDOT:PSS materials for further use in transparent organic electrodes. [8], [20] The STFSI fraction provides solubility in water, whilst the functionalized monomer (2) confers the ion-sensitive ability. It appears that the dispersity gets larger as the content in monomer ( 2) is increased, i.e. from 5 to 15% (Table 1) due to the statistical distribution of these moieties in the different polymer chains. Moreover, when the content of the monomer (2) increases up to 40% or 50%, it became complicated to reach the targeted chain length (Mw) (Table 1). A copolymer at a ratio of 85 wt. % of STFSI and 15 wt. % of monomer (2) has been taken as a case study from a larger copolymer family (Table 1). A polyanion P(STFSI-co-S(18-crown-6)) ( 3) with an apparent molar mass of 135 kDa, was for instance obtained (Figure SI-6, PS standard calibration and DMF solvent) with quite broad distribution which can be attributed to specific interactions with SEC columns. The 1 H-NMR of the P(STFSI-co-S(18-crown-6)) (Figure SI-7) clearly shows broadening of signals as compared to the initial monomers' spectra in accordance with the expected expansion of polymer signals in this type of high molar masses polyelectrolytes. [21] The presented copolymer ratio is the theoretical weight percentage (wt%) ratio between STFSI and monomer (2) used for the polymer synthesis. Experimentally it is not possible to quantify the copolymer ratio by the NMR spectra, as the initiator signal overlaps in the aliphatic region with the signals related to the alkyl polymer backbone (δ= 0.75-2.25 ppm). However, the 1 H-NMR chemical shifts from the aromatic rings, polystyrene backbone and crown ether moieties could be assigned at δ = 6.59-7.54ppm, δ = 1.44 ppm and δ=3.53-4 ppm respectively and thus confirmed the copolymer synthesis and correlates with the ratio between comonomers. 21 Table 1. Summary of polyanion (3) synthesized, and used for the stabilization of ion-sensitive PEDOT dispersions. M 0 : STFSI monomer; M: functionalized monomer (2); n M0 / n M : molar ratio between both monomers; Mw: Average of the apparent molecular mass (kDa) of the obtained polyanions copolymers a) and Ð: Dispersity as obtained from SEC a) analysis in DMF using PS standard calibration The ion-sensitive functionalized polyelectrolyte, P(STFSI-co-S(18-crown-6)), was subsequently used as a stabilizer/counterpart of aqueous PEDOT dispersions during the oxidative in-situ polymerization of EDOT [21] in the presence of iron chloride and ammonium persulfate; see Scheme 1c). PEDOT inks were obtained after polymerization for 64h at 10°C Processable and stable aqueous inks were obtained, containing the hydrophobic ion-selective moiety in the polyelectrolyte chains. However, stable PEDOT inks were difficult to obtain when trying to incorporate more than 50% (in weight content) of crown ether modified styrene subunits into the copolymer. In fact, in these cases the dispersion was unstable and the polymer precipitated. Spin-coated PEDOT films presented conductivities in the mS cm -1 range (σ < 1366 mS cm -1 ). Although this is less than what is observed for PEDOT:PSS and PEDOT:PSTFSI (190 S cm -1 and 110 S cm -1 respectively), this conductivity range is sufficient for bioelectronics applications.

In order to evaluate the electrochemical properties of the films and provide more evidence for the occurrence of ion selectivity of these mixed conducting polymers, in-situ electrochemical conductance experiments were carried out. [22] The electrochemical characterization of drop casted films was evaluated in an aqueous 0.1 M tetramethyl ammonium chloride (TMAC) solution (Figure 2a andb). Two separate IDME electrodes, covered with the same conductive ink, were used to analyze their electrochemical response in the presence of potassium and sodium respectively (Figure 2a andb). Since during drop casting, the thickness control of the films is limited, normalized currents are presented in order to allow a reasonable comparison of the samples.

Initially, during the first potential sweep, peaks of current were not observed for both films (Fig. 2 (a) and (b)). Subsequently, a peak of current starts to appear, between 0.36 V and 0.38 ). This decrease is attributed to a conformational rearrangement of the oligomeric chains of the polymer blend and not to a limitation of charge hopping caused by the trapped positive charges formed during the aniline oxidation. [23] In order to provide more evidence of these conformational changes, the charge-potential plots were obtained by integration of the potentiodynamic plots. [24] From the coulovoltammetric response it is possible to identify changes in structural components driven by electrochemical reactions inside the conducting polymer matrix. [25][26][27][28] These charge-potential plots have been used by Otero et al., by means of the electrochemically stimulated conformational relaxation model (ESCR) to explain the reversible and irreversible structural changes driven by electrochemical reactions of conducting polymers exchanging cations. [29] Since, for mixed conducting materials, composed of conducting polymers and anionic polyelectrolytes, the anion remains inserted in the conducting polymer matrix during the charge/discharge process, exchange of cations occurs. According to the ESCR model, variations in the slope values of the chargepotential plots are attributed to conformational changes of the polymer matrix. [24] The coulovoltammetric response of the films, in a 0.1 M TMAC aqueous solution (Figure SI-11a), presents in the first cycle, starting from the minimum of charge, two oxidation processes: an oxidation/shrinking (m O/S = 0.48 mC V -1 ) and an oxidation/compaction [30] (m O/C = 0.78 mC V -1 ), both caused by the release of cations from the polymer matrix. The latter is followed, at the beginning of the cathodic sweep, by some oxidation inertia and two reduction processes: a reduction/relaxation (m R/R = 0.16 mC V -1 ) and a reduction/swelling (m R/S = 0.50 mC V -1 ), caused by the insertion of tetramethyl ammonium cations. During subsequent scans, the slope value of the oxidation/shrinking and the oxidation/compaction slope decreases

V
(Figure SI-11b, m O/S = 0.26 mC V -1 , m O/C = 0.46 mC V -1
), whilst a new oxidation process appears, between 0.30 V and 0.45 V vs Ag°, with a concomitant increase of slope (m An/S = 0.81 mC V -1 ). Due to the range of potential where it appears, this is attributed to the positive charge formed during the aniline oxidation [31] that, in addition to the oxidation of PEDOT, causes a faster oxidation/shrinking process, hereafter referred to as anilineoxidation/shrinking. Finally, during the cathodic sweep, a faster reduction/swelling process was observed (m R/S = 0.77 mC V -1 ). This may be attributed to a positive "charge trapping", which requires a faster exchange of cations and anions inside the polymer matrix to keep electroneutrality.

The ion selectivity was evaluated in a 0.1 M TMAC aqueous solution in the presence of 10 mM potassium or sodium chloride (Figure 2c andd, respectively). In the presence of KCl, a continuous decrease in current of the aniline oxidation peak (20.5 %) was observed. In contrast, in the NaCl solution, first a small decrease of the aniline oxidation peak was obtained, followed by a slow increase of the peak current (2.4 %). A possible explanation for this observation is that potassium ions provide an electron withdrawing effect on the aniline moiety, which causes an anodic shift of the oxidation potential of the aniline. Thus, the amount of aniline present and able to get oxidized at 0.38 V vs Ag° decreases as the cycling proceeds. The absence of an oxidation peak for the new K + -aniline complex, is due to the low current attributed to such a redox process, which may overlap with the oxidation current of the different oligomeric PEDOT chains as it was stated previously. In the presence of sodium, the cation does not fit into the crown ether cavity and it is easier to oxidize the aniline moiety.

Finally, for both cations, the conductance profile of the films shows the same quasi-linear variation during the potential sweep, whereas a small decrease of conductance was observed as a function of the potential scans (ΔG p-K = 0.17 ± 0.01 mS and ΔG p-Na = 0.28 ± 0.11 mS) (Figure SI-10a and b). Thus, the charge hopping process is not influenced by the presence of different cations. [23] From the charge-potential plots of the films, in the presence of KCl and NaCl (Figure 2e andf, respectively), the change of the maximum oxidation charge (ΔQ) after 10 potentiodynamic cycles was evaluated (ΔQ K + = 88.70  1.59 μC and ΔQ Na + = 48.31  5.65 μC). As the ΔQ for potassium is the double in comparison to sodium, these results confirm the selectivity of the films for the K + cation in accordance with the size-matching rule. In addition, the coulovoltammetric response presents, for all the cycles, the same oxidation/reduction process as in the absence of the monovalent cations (Figure 2e andf). As the experiment proceed, changes on the slope for all the oxidation/reduction process can be observed. From the charge-potential plots (Figure 2e andf, respectively), it can be seen a higher variation of the slope for the aniline-oxidation/shrinking and the reduction/swelling process. In the presence of K + the aniline-oxidation/shrinking and the reduction/swelling present a decrease in the slope value (Δm An/S K = -0.19 mC V -1 , Δm R/S K = -0.03 mC V -1 ). However, in the NaCl solution, both processes show an increase in the slope value (Δm An/S Na = 0.08 mC V -1 , Δm R/S Na = 0.10 mC V -1 ). Since potassium ions, trapped inside the crown ether cavity, are not expelled during the aniline-oxidation/shrinking process, an abrupt rearrangement of the oligomeric chains (between 1 st and 2 nd cycle) followed. This may be due to a slow mass insertion/repulsion of ions during the charge/discharge process. These results correlate with the binding constants obtained by 1 H-NMR (k a, Na = 121.7  6.9 M -1 and k a, K = 1026.2  118.3 M -1 ), where the binding constant for potassium, suggests the formation of a K + -aniline complex inside the polymer matrix. Such a high value of a binding constant is the first indicator of a poor release of K + trapped in the K + -aniline complex. Finally, in order to prepare for a future integration of these new ion-sensitive mixed conducting polymer complexes with biological systems, cell viability was evaluated on such PEDOT films. The quantification of the ratio between dead and total number of cells (dead + alive) grown on these films was comparable to the one in control conditions, suggesting an unchanged growth of β cells on the coverslips covered with films (Figure 3). Thus the polymer material does not exhibit a major cytotoxic effect under the conditions tested. These results open up the possibility of interfacing these ion-sensitive polymers with pancreaticcells, who play a key role in nutrient homeostasis and in diabetes. [START_REF] Tien | From Molecules to Networks[END_REF], [32] In conclusion, a new approach was presented to introduce cation selectivity in a mixed conducting polymer material. A specific metal cation scavenger, particularly targeting K + , was integrated in the PEDOT: PSTFSI complex through the polyanion counterpart by appropriate modification of the styrene subunits. We achieved synthesis and characterization of a functionalized styrene derivative with an ion-selective moiety, i.e. 18-crown-6, monomer

. This monomer was used for the synthesis of a polyanion copolymer, which was then used as a substitute to PSS in the oxidative dispersion polymerization of EDOT in water. All these new compounds have been evaluated for their ability to sense K + metal cations. First, monomer (2) showed a higher affinity for K + compared to Na + . Second, the PEDOT inks composed of such polyanions displayed interesting new electrical and electrochemical characteristics. By in-situ electrochemical conductance measurements and the correspondent charge-potential plots, the higher affinity for K + as compared to Na + of these PEDOT materials has been confirmed. Finally, these materials were not cytotoxic and therefore qualify for applications such as recordings of K + -dependent electrical signals in pancreaticor other cells via the fabrication of ion selective OECTs. were fitted to a 1:1 binding model using Affinimeter [1] software and from these data the global association constants were extracted.

Monomer synthesis and spectroscopy characterization

Step 1-Formation of 4-styrenesulfonyl chloride (1): A round bottom flask, previously flamedried, was charged with dry and degassed acetonitrile (V=60 mL). Oxalyl chloride (5.91 g, 46.6 mmol, 1.2 eq.) and DMF were added. This solution was kept under stirring at room temperature to solubilize the reagents and promote the Vilsmaier-Haack complex formation for 4 or 5 hours. Once the characteristic yellow colour of the solution became stable, 4styrenesulfonic acid sodium salt (8.0 g, 38.8 mmol, 1.0 eq.) was added slowly to the solution under nitrogen atmosphere and at room temperature. After 24h of reaction, the precipitated salt was separated by filtration and the reaction mixture was used in the second step.

Step MgCl 2 1.0) and dried overnight. 50,000 INS-1 clonal -cells were added per well and cultured as described for 3 days. [3] Coverslips were either visualized and documented by phase contrast microscopy or further processed for apoptosis measurements. In the latter case, cell media were replaced by fresh media with or without thapsigargin (0.1 µM, as positive control) for 6h prior to analysis. Imaging was used after de-freeze. The Live/Dead cell kit was used according to the manufacturer and 150 µL of final solution were added per coverslip for 15 min at 37°C followed by imaging. For each data point three coverslips were examined, experiments were repeated independently nine times and areas of fluorescence calculated by

ImageJ. [4] Statistical analysis was performed using GraphPad Prism7.

  Scheme 1. (a) Synthetic route to obtain the 4-styrenesulfonyl(phenyl-18-crown-6) imide

  ) (Figure SI-1) shows the corresponding signals for the styrene aromatic protons at 7.4 ppm and 7.7 ppm. Vinyl protons are evidenced by the presence of the signals at 5.37 ppm and 5.82 ppm. The signal of -CH 2 -O-from the crown moiety is assigned at 3.75 ppm. Finally, the multiplet signal at 6.62 ppm corresponds to the aromatic protons of the crown moiety. The corresponding 13 C-NMR spectrum of (2) also confirms the expected monomer structure (Figure SI-2).

Figure 1 .

 1 Figure 1. 1 H-NMR Spectra (DMSO-d 6 , 400 MHz) obtained for monomer (2) in the absence

  under inert atmosphere, then recovered through purification via ultrafiltration. As illustrative example, an initial copolymer solution (Figure SI-8a) is presented, acting as stabilizer of PEDOT favoring the complex formation and stability of the PEDOT dispersion in water when the polymerization takes place (Figure SI-8b).

  vs Ag°. This additional anodic reaction is attributed to the oxidation of the aniline moiety attached to the crown ether group along the P(STFSI-co-S(18-crown-6)) polyanion. The absence of this oxidation peak, at the first potentiodynamic scan, is due to the low current attributed to such redox process, which overlaps with the oxidation current of the different oligomeric PEDOT chains. It should be noted that PEDOT:PSS and PEDOT:PSTFSI do not show any anodic peak in this range of potentials (Figure SI-9a and b). The conductance profile of the films shows a quasi-linear increase during the potential sweep, caused by the formation of mobile charges in the PEDOT backbone. Whereas, a gradual decrease of conductance was observed as the cycling proceeds (from 5.44 ± 0.75 mS to 4.61 ± 0.37 mS) (Figure SI-10a and b

Figure 2 .

 2 Figure 2. Potentiodynamic study of drop casted PEDOT/ P(STFSI 85 -co-S18cr6 15 ) films in a 0.1 M TMAC aqueous solution using E λ = 0.6 V vs Ag°, 10 cycles, v = 10 mV s -1 and ΔV on

Figure 3 .

 3 Figure 3. Cytotoxicity study of the ion sensitive inks: Clonal β cells (INS832-13) cultured for 72 h on (a, c) control coverslips or coverslips covered with the ink studied (PEDOT: PSTFSI-85-co-P(2)-15 with 18-crown-6 moieties). (b, d) Apoptosis induced by thapsigargin (6h, 10 µM) on control coverslips or coverslips covered with the same ink. All micrographs present merged images (green: FYTC filter, live cells; red: YFP filter, dead cells). (e): quantitative assessment of apoptosis. Given are individual data points, means and SEM; N= 9. **, p<0.001; ns, not significant (Kruskal Wallis and Dunne post-hoc test)

2 -

 2 Synthesis of 4-styrenesulfonyl(phenyl-18-crown-6) imide (Sph18cr6SI)(2): In a second flame-dried round bottom flask containing dry and degassed acetonitrile (V=50 mL), triethylamine (4.29 g, 30.2 mmol, 3 eq.) and 4'-aminobenzo-18-crown-6 (3.00 g, 10.1 mmol, 1 eq.) were added and stirred for one hour. During this time, the 4-styrenesulfonyl chloride/acetonitrile solution (formed during Step 1) was cooled at 0 °C. Subsequently, the trifluorosulfonamide solution was added in vacuum atmosphere to the former. Finally, the mixture was allowed to warm up to room temperature and the reaction was kept running under stirring for 16h. Purification was done firstly by removing under low pressure the solvent mixture and solubilizing the monomer in dichloromethane. The product was obtained by extraction with potassium carbonate and a subsequent wash with 1M hydrochloric acid solution. Solvent was removed under low pressure and after drying overnight at 45°C under vacuum, the product was recovered as a brown solid product. (yield=54%

SI- 5 .SI- 6 .SI- 8 . 2 ]SI- 9 .SI- 10 .

 5682910 1 H-NMR (CD 2 Cl 2 , 400 MHz) of 4-styrenesulfonyl(phenyl-18-crown-6) imide(2). Binding isotherms and fitting to a 1:1 binding model of the experimental data from 1 H-NMR titration of the monomer (2) (d-DMSO, 400 MHz). Scattered dots: Experimental data obtained from the difference in NMR shift (ppm) with respect to initial monomer spectra (yaxis -"Signal") vs the increasing concentration of KClO 4 , for a constant monomer concentration, Mt (x-axis -"At/ Mt"). Red curves: Fitting to the different binding isotherms with a 1:1 binding model to obtain the global binding constant (K A ).Radical polymerization and Copolymerization (GENERAL PROCEDURE):The copolymers between STFSI monomer and ion-sensitive styrene monomer (2) were synthesized by reversible addition-fragmentation chain-transfer (RAFT) radical polymerization by mixing, according to the targeted molar mass and composition, the proper quantity of the STFSI/(2) monomers, chain transfer agent and azobisisobutyronitrile (AIBN) in DMF. After several freeze-thaw cycles, the mixture was left to polymerize at 64°C from several hours to few weeks, depending on the molar mass that should be obtained. The polymer was ready after precipitation in tetrahydrofuran (THF) or the adequate solvent mixture, filtration, washing with THF and drying in vacuum oven at 65°C for at least one day. Size exclusion chromatogram (SEC) of a copolymer between STFSI (wt.% = 85) and monomer (2) (wt.% = 15) showing a main population of 135 kDa. Conditions: Polystyrene standard calibration. DMF, LiBr, 70°C. SI-7. 1 H-NMR Spectrum (d 6 -DMSO, 400 MHz) of a copolymer between STFSI (wt.% = 85) and monomer (2) (wt.% = 15). Conductive ion sensitive aqueous PEDOT dispersions: Solutions of iron chloride (III) (0.036 mg, 0.22 mmol), ammonium persulfate (0.161 g, 0.71 mmol) and the styrene-based copolymer (0.210 g) were prepared in individual aqueous solutions and stirred during 5h previous to their use. Afterwards, the polyelectrolyte solution was transferred to the reactor and EDOT monomer was added (0.038 mL, 0.36 mmol) keeping under nitrogen flow at 10°C for 30 min. Addition of the oxidant solutions composed of iron chloride and ammonium persulfate starts the polymerization. The reaction is kept running under static inert atmosphere during 64h. Purification has been done by ultrafiltration, washing firstly with 1M hydrochloric acid (150mL). After stirring for two more hours with a 50% content of the previous used acid solution, a last washing with water was performed. Concentrations were adjusted to about 1 weight % of content after filtration, in correlation with a calibration curve obtained by UV-Vis absorption spectroscopy analysis. Inks were stored protected from light at constant temperature and under stirring. Aqueous dispersion formation of the ion-sensitive PEDOT based ink: a) Before the addition of the oxidants, brownish polyelectrolyte solution; b) After addition of the oxidants, formation of the PEDOT based ink.In-situ electrochemical-conductance measurements: In-situ electrochemical-conductance measurements were carried out using a two potentiostat system, specially built for this application. As working electrodes, DropSens interdigital microarrange electrodes (IDMAE), featuring separated Pt bands (10 μm) deposited on it, were used. A three-electrode electrochemical cell fitted with a Pt foil counter electrode separated aproximatly 0.5 cm from the working electrode (parallel and of the same size as the working electrode), a pseudoreference electrode (Ag wire) and the IDMAE were used for the studies. A constant potential difference of 10 mV provided by one potentiostat was applied between the IDMAE branches to generate the drain current that circulates through the polymer during conductance determinations. The conductance of the dropcasted films was measured in real time during the cyclic voltammetry experiment in an aqueous 0.1 M tetramethyl ammonium chloride (TMAC) solution at 25 °C. Conductance was calculated according to Ohm's Law, which is directly related with specific conductivity. More details about this technique and examples of the use of conductance measurements in conducting polymers can be found elsewhere in the literature [Potentiodynamic response of PEDOT electropolymerized in the presence of (a) PSS and (b) PSTFSI, in a free monomer 0.1 M PSS aqueous solution, v = 50 mV/s, 1st cycle (black line) and 10th cycle (red line). Conductance profile of drop casted PEDOT/ P(STFSI85-co-S18cr615) films in a 0.1 M TMAC aqueous solution using E λ = 0.6 V vs Ag°, v = 10 mV s -1 and ΔV on the WE = 10 mV, WE = Pt IDMAE, CE = Pt foil, in the absence (initial 10 cycles) and the presence of (a) 10 mM KCl and (c) 10 mM NaCl (final 10 cycles). The asterisk indicates the moment when addition of the chloride salts was performed.

  298K): δ(ppm)= 147.9; 145.46; 141.1; 138.4; 135.2; 130.54; 127.9; 127.15; 126.84; 126.63; 117.76; 113.93; 113.14; 107.7; 68.82; 69.74-67.97. m/z (ESI-HRMS) 516.17 ([MNa + ] C 24 H 31 NO 8 SNa requires 516.57).

	). Scheme 1. 1 H-
	NMR (400 MHz; d-DMSO; 298K): δ(ppm)= 7.79 (d, 2H); 7.61 (d, 2H); 6.80 -6.52(q, s, dd,
	4H); 5.99-5.94 (d, 1H); 5.48-5.36 (d, 1H); 3.98 -3.50 (m, 20H). 13 C-NMR (75 MHz; d-
	DMSO;

SI-1.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author. Copyright WILEY-VCH Verlag GmbH & Co. KGaA, 69469 Weinheim, Germany, 2018.

Supporting Information

Design of potassium selective mixed ion/electron conducting polymers Ariana Villarroel Marquez, Gerardo Salinas, Myriam Abarkan, Maël Idir, Cyril Brochon, Georges Hadziioannou, Matthieu Raoux, Alexander Kuhn, Jochen Lang, Eric Cloutet* Chemicals 4'-aminobenzo-18-crown-6; 4-styrene sulfonic acid sodium salt; 4-vinylbenzyl chloride; azobisisobutyronitrile; oxalyl chloride; potassium carbonate; sodium carbonate; triethylamine and all other products were used as received or according to manufacturer's instructions (Sigma Aldrich, France). N,N-dimethylformamide (DMF) was distilled from calcium hydride (CaH 2 ) prior to use. Acetonitrile was distilled prior to use or dried by a solvent dryer machine (MB SPS-800, MBraun). Tetrahydrofurane was distilled from benzophenone/sodium prior to use.

Techniques and instrumentation

Nuclear magnetic resonance (NMR) spectroscopy: 1 H-NMR and 13 C-NMR spectra for structural characterization and titration experiments were recorded on Bruker Avance (300MHz) and Bruker Prodigy (400MHz) spectrometers, respectively, at 300K and the residual proton signal of the solvent was used as the internal standard. Size exclusion chomathography (SEC) was used to determine molar masses and dispersity of the copolymers synthesized. The system (Agilent GPC220HT) is equipped with UV and RI detectors and is equipped with a pre-column GF-1G-7B and two columns Asahipack GF-7MHSi (DMF, 0.1M LiBr, T=75°C, flow rate=0.6mL/min). Polystyrene standards were used for calibration. Film Deposition: A Brewer Science 200CBX spin-coater was used for film deposition. Substrates


Mattia Mattioni
email: mattia.mattioni@uniroma1.it

Alessio Moreschini
email: alessio.moreschini@uniroma1.it

Dorothée Salvatore Monaco
email: salvatore.monaco@uniroma1.it

Dorothée Normand-Cyrot
email: dorothee.normand-cyrot@centralesupelec.fr

Reduction-based stabilization of nonlinear discrete-time systems through delayed state measurements

Keywords: Asymptotic stabilization, Delay systems, Lyapunov methods

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION

Throughout the last decades, time-delay systems have been widely investigated due to their involvement in practical scenarios (see [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF]; [START_REF] Valmorbida | Delays and interconnections: Methodology, algorithms and applications[END_REF] for an overview). Among the several strategies, the prediction-based control method represents one of the most appealing approaches from its early introduction by [START_REF] Smith | Posicast control of damped oscillatory systems[END_REF] for systems affected by input or measurement delays (e.g., [START_REF] Krstic | Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch[END_REF]; Karafyllis and Krstic (2013a,b); [START_REF] Gonzalez | Robustness of a discrete-time predictor-based controller for time-varying measurement delay[END_REF]; [START_REF] Zhou | Observer-based output feedback control of discrete-time linear systems with input and output delays[END_REF]; [START_REF] Zhou | Stabilization of linear systems with both input and state delays by observer-predictors[END_REF]). A different and more general approach for input-delayed systems has been initiated by [START_REF] Artstein | Linear systems with delayed controls: A reduction[END_REF]. It relies upon the idea of transporting the control problem over a new delay-free dynamics (the so-called reduced dynamics) which is equivalent to the retarded one in terms of stabilizability. Such an approach has been recently extended to the nonlinear context by Mattioni et al. (2018a,b) for both continuous and discrete-time dynamics affected by input delays. Inspired by input-reduction, the contribution of this paper is twofold: the novel concept of output-reduction is introduced for stabilizing nonlinear discrete-time dynamics when the delay occurs on the measurements of the states; passivity-based control methodologies, developed in [START_REF] Monaco | Nonlinear average passivity and stabilizing controllers in discrete time[END_REF] for nonlinear discretetime dynamics in terms of average passivity, are generalized to the presence of state measurement delays. The reduction approach stands in the possibility of leading the control design over a suitably transformed delay-free dynamics. In addition, it is shown that the construction Supported by Università degli Studi di Roma La Sapienza (Progetti di Ateneo 2018 -Piccoli progetti RP11816436325B63 ) and by Université Franco-Italienne/Università Italo-Francese (Vinci 2019).

of the output-reduction variable (and thus of the stabilizing feedback) requires the knowledge of the instantaneous delayed measurement of the state plus a finite number of past delayed signals over the time-delayed window only. In addition, contrarily to classical prediction, the free evolution of the retarded dynamics is preserved by reduction with a new forced component which accounts for the delay. Moreover, as the delay approaches to zero, the retarded output mapping and damping feedback naturally recover the delay-free ones. The problem is then specified to the case of a linear time-invariant system and the solution validated on an academic example through simulations enlightening the improvements, also in terms of robustness, with respect to classical prediction-based control.

The paper is organized as follows. In Section 2 recalls are provided and the problem is formulated. In Section 3 the main result is enhanced and specified to the case of linear dynamics. In Section 4 an academic example is carried out and simulated whereas Section 5 concludes the paper.

Notations. R and N denote, respectively, the set of real and natural numbers including 0. For any vector v ∈ R n , |v| and v define the norm and transpose of v respectively. Given a square matrix R ≥ 0 and v ∈ R n , the weighted square seminorm is defined as v 2 R := v Rv. I d denotes the identity function or identity matrix while I denotes the identity operator. Given a real-valued function V (•) : R n → R assumed differentiable, ∇V represents its jacobian vector when ∇ denotes the R n vector of partial derivatives. Given a smooth vector field over R n , the Lie operator is defined as

L f = n i=1 f i (x) ∂
∂xi . The arguments of the functions are dropped when clear from the context. A function R(x, δ) = O(δ p ) is said of order δ p , p ≥ 1 if it can be written as R(x, δ) = δ p-1 R(x, δ) and there exist a function θ ∈ K ∞ and δ * > 0 such that ∀δ ≤ δ * , | R(x, δ)| ≤ θ(δ). To simplify the notations, when no confusion arises the time arguments are dropped out.

RECALLS AND PROBLEM STATEMENT

State space representations for discrete-time dynamics

As discussed in [START_REF] Monaco | Discrete-time state representations, a new paradigm[END_REF], a general discrete-time system Σ :

x(k + 1) = F (x(k), u(k)) (1) with x ∈ R n , u ∈ R can be equivalently described through a couple of differential-difference equations x + = F 0 (x) (2a) dx + (u) du = G(x + (u), u) with x + (0) = x + (2b)
where:

F 0 (•) = F (•, 0) is a R n -valued smooth map and G(•, u) = G 1 (•) + i>0 u i i! G i+1 (•) is a vector field on R n assumed complete verifying ∇ u F (x, u) = G(F (x, u), u) for all (x, u) ∈ R n × R. In such a formalism, x + (u) represents a generic curve in R n parameterized by u ∈ R. For all (k, x(k), u(k)) ∈ N × R n × R, integrating (2b) over the interval [0, u(k))
with initial condition fixed through (2a) (i.e. x + (0) = F 0 (x(k))), one recovers the representation (1) in the form of a map

F (x(k), u(k)) = F 0 (x(k)) + u(k) 0 G(x + (v), v)dv. (3)
Hence, x + (u(k)) := x(k + 1) represents the one-step ahead evolution while

x + = x + (0) = F 0 (x(k)) = F (x(k), 0)
defines the one-step free evolution. In this respect, one gets that (1) rewrites for all (x, u) ∈ R n × R as

x + (u) = F (x, u) = e uG(•,u) I d F0(x) (4)
where e uG (•,u) is the flow associated with G(•, u) as the solution to (2b) and characterized by the exponent series uG(•, u) defined as (see [START_REF] Monaco | From chronological calculus to exponential representations of continuous and discrete-time dynamics: A liealgebraic approach[END_REF])

uG(•, u) = uG 1 + u 2 2 G 2 + u 3 3! (G 3 + 1 2 [G 1 , G 2 ]) + O(u 4 ).
(5)

From ( 4) and ( 5), one computes the first terms in the expansion of F (•, u) in u as follows

F (x, u) = e uG1+ u 2 2 G2+ u 3 3! (G3+ 1 2 [G1,G2])+O(u 4 ) I d F0(x) = F 0 (x) + uL G1 I d F0(x) + u 2 2 (L 2 G1 + L G2 )I d F0(x) + O(u 3
). As a consequence of the form (2), given any smooth enough map λ(•) : R n → R, one gets

λ(x + (u)) = h(F 0 (x)) + u 0 L G(•,w) λ(x + (v))dv (6)
or, when exploiting the flow associated with the solution to the differential equation ( 2b)

λ(x + (u)) =e uG(•,u) λ x + = λ(F 0 (x)) + uL G1 λ(F 0 (x)) + u 2 2 (L 2 G1 + L G2 )λ(F 0 (x))) + O(u 3
). For further discussion on the equivalence among (1), ( 4) and (2), the interested reader is referred to [START_REF] Monaco | Discrete-time state representations, a new paradigm[END_REF].

Average passivity and passivation in discrete-time

The concept of average passivity has been introduced in [START_REF] Monaco | Nonlinear average passivity and stabilizing controllers in discrete time[END_REF] to relax the necessity of a direct input/output link that is unavoidable in discrete time to invoke passivity. Denoting by Σ(h) the discretetime system composed with the dynamics (1) and an output map h(•, u) : R n × R → R, the following definition is recalled. Definition 2.1. (u-average passivity). The dynamics (1) with output y = h(x, u) is said to be u-average passive if there exists a positive semi-definite function S(•) :

R n → R ≥0 (the storage function) such that, for all (x, u) ∈ R n ×R S(x + (u)) -S(x) ≤ u 0 h(x + (v), v)dv = uh av (x, u). (7) with h av (x, u) := 1 u u 0 h(x + (v), v)dv (8)
being the u-average output verifying h av (x, 0) = h(F 0 (x), 0).

It is worth mentioning that u-average passivity with respect to h(•, u) is equivalent to usual passivity with respect to the averaged output h av (•, u).

Assuming that the discrete-time dynamics (1) possesses a stable equilibrium (say at the origin) with Lyapunov function

S(•) : R n → R ≥0 verifying, for u = 0, S(F 0 (x)) - S(x) ≤ 0, from Definition 2.1 one can immediately deduce ∆S(x) := S(F (x, u)) -S(x) = S(F 0 (x)) -S(x) + u 0 L G(•,v) S(x + (v))dv ≤ u 0 L G(•,u) S(x + (v))dv, ∀(x, u) ∈ R n × R.
Hence, u-average passivity with respect to the output h(x, u) = L G(•,u) S(x) holds. Since the average output approximately rewrites in O(|u|) as

h av (x, u) = L G(•,0) S(F 0 (x)) + O(|u|) (9)
the passivating output for (1) can be immediately computed starting from real time measures of the state (that is x(k)). The following definition is recalled as instrumental for stabilization purposes. Definition 2.2. (ZSD). Consider the system (1) with output h(x, u) and let

Z ⊂ R n be the largest invariant set contained into {x ∈ R n s.t. h(x, 0) = 0}. Then Σ(h) is zero-state detectable (ZSD) if the origin is asymptotically stable conditionally to Z.
Stabilization via average damping can be deduced as recalled here below. Theorem 2.1. Let the system (1) with output h(x, u) be u-average passive and ZSD. Then, the feedback u = u(x) solution to the damping equality

u + κh av (x, u) = 0, κ > 0 (10)
makes the origin of (1) globally asymptotically stable (GAS).

For the existence of a solution to (10) and computational aspects, the interested reader can refer to [START_REF] Monaco | Nonlinear average passivity and stabilizing controllers in discrete time[END_REF]; [START_REF] Mattioni | Forwarding stabilization in discrete time[END_REF].

Problem statement

In the following, assuming that measurements of the states of (1) are delayed over a time window of length N ≥ 0 (i.e., y x (k) = x(k -N )) we look for an output mapping making the delayed system u-average passive. Standard notations and assumptions for discrete-time delayed systems are adopted. As in [START_REF] Stojanović | A Lyapunov-Krasovskii methodology for asymptotic stability of discrete time delay systems[END_REF]; [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF]; [START_REF] Pepe | On Lyapunov-Krasovskii Characterizations of Stability Notions for Discrete-Time Systems With Uncertain Time-Varying Time Delays[END_REF], C (resp. C u ) denotes the space of functions mapping the set {-N, . . . , -1} into R n (resp. R); x 0 = x 0 (θ) ∈ C represents the initial condition of the system;

u k = u k (θ) = u(k + θ) ∈ C u for θ ∈ {-N, . . . , -1}
denotes the story of the control signal. From now on, for the sake of compactedness, we denote

x k = (x(k -N ), u k ),
with the following standing assumption.

Assumption 2.1. The dynamics (1) possesses a stable equilibrium the origin (that is x = 0) with Lyapunov function S(•) : R n → R ≥0 and is ZSD with respect to the output h(x, u) = L G(•,u) S(x).

Hereinafter, we address the problem of computing a stabilizing passivity-based controller for discrete-time dynamics of the form (1) based on delayed measurements of the state; namely, we seek a class of outputs y r (k) = h r (x k , u(k)) (depending on x(k -N ) and a finite buffer of past values of the control signal) that makes the system (1) u-average passive with a suitable storage functional S r (•) : 11) and in such a way that the corresponding damping feedback makes the origin of (1) GAS. The solution we propose relies upon the definition of the so-called output reduction variable ν(•) : C × C u → R n (as the counterpart of input reduction proposed by Mattioni et al. (2018a)) whose passivity properties will be shown to be equivalent to the ones of (1) in presence of delayed state measurements. Remark 2.1. All the results do apply to nonlinear systems issued from sampling along the lines of [START_REF] Mattioni | Sampled-data reduction of nonlinear input-delayed dynamics[END_REF] also covering the case of non-entire delays. Remark 2.2. Assumption 2.1 is applicable to a large varieties of discrete-time dynamics as, for instance, issued from sampling of passive systems.

C → R ≥0 verifying ∆S r (x k ) = S r (x k+1 ) -S r (x k ) ≤ u(k)h av r (x k , u(k)) (

DAMPING THROUGH OUTPUT REDUCTION

Starting from the dynamics (1) and assuming

F 0 (•) = F (•, 0) invertible 1 , we introduce the output reduction variable ν(k) = r(x k ) as, for all k ≥ 0, ν(k) =r(x k ) =F -N 0 (x(k)) = F -N 0 (•) • F N (x(k -N ), u k ) (12) with F -N 0 (•) = F -1 0 (•) • • • • • F -1 0 (•) N times F N (•, u k ) = F (•, u(k -1)) • • • • • F (•, u(k -N )) for initial condition u 0 ≡ 0, r(x 0 , u 0 ) = F -N 0 (•) • F N (x(-N ), 0) = x(-N ) and verifying r(x(k -N ), 0) = x(k -N ) when u ≡ 0.
It is a matter of computations to prove the following result.

1 There exists F -1

0 : R n → R n s.t. F -1 0 (F 0 (x)) = F 0 (F -1 0 (x)) = x.
Proposition 3.1. Consider the dynamics (1) under Assumption 2.1 and state measurements affected by a delay of length N ≥ 0. Let the output reduction variable be (12), then the output reduced dynamics (or simply reduced dynamics) is free of delay and evolves as

ν(k + 1) = F -N (ν(k), u(k)) (13)
with

F -N (ν, u) := F -N 0 (•) • F (•, u) • F N 0 (ν).
Additionally, (13) admits the differential-difference representation

ν + = F 0 (ν) (14a) dν + (u) du = G -N (ν + (u), u) with ν + (0) = ν + (14b)
with, by definition

G -N (•, u) := Ad F -N 0 G(•, u). ( 15
)
The operator Ad 15) denotes the transport of a vector field G(•, u) along the mapping F -N 0 (•) (see [START_REF] Monaco | From chronological calculus to exponential representations of continuous and discrete-time dynamics: A liealgebraic approach[END_REF]) so getting

F -N 0 G(•, u) in (
G -N (•, u) := L G(•,u) F -N 0 (•) • F N 0 (•) = ∇[F -N 0 (•)]G(•, u) x=F N 0 (•) = i≥0 u i i! G [-N ],i+1 (•) with analogously G [-N ],i (•) := Ad F -N 0 G i (•) for i ≥ 1.
Exploiting the exponential representation in (4), one gets through composition that

ν(k) =e u(k-1)G -N (•,u(k-1)) • e u(k-2)G -N +1 (•,u(k-2)) • • • • • e u(k-N )G-1(•,u(k-N )) I d x(k-N )
with according to ( 5)

uG (•, u) =uG [ ],1 + u 2 2 G [ ],2 + u 3 3! (G [ ],3 + 1 2 [G [ ],1 , G [ ],2 ]) + O(u 4 )
for all ∈ Z. Hence, (12) approximately rewrites as

ν(k) =x(k -N ) + N i=1 u(k -i)G [-N +i-1],1 (x(k -N )) + O(|u k | 2 ). Remark 3.1.
For the construction of the reduction (12), invertibility of F 0 (•) = F (•, 0) can be weakened to require submersivity only [START_REF] Monaco | From chronological calculus to exponential representations of continuous and discrete-time dynamics: A liealgebraic approach[END_REF]). Remark 3.2. Differently from the continuous-time case [START_REF] Artstein | Linear systems with delayed controls: A reduction[END_REF]; [START_REF] Mattioni | Reduction-based stabilization of time-delay nonlinear dynamics[END_REF]), the reduction variable and the reduced model can be exactly computed through successive composition of the involved mappings. Remark 3.3. The reduced dynamics (13) is free of delays and the reduction ( 12) is computed as a prediction of the state at time k starting from the measures at k -N brought backwards through the inverse drift of the system. Moreover, the computation of (12) requires only the knowledge of the state at time k-N (that is measured) plus a finite number of past control signals over the time window [k -N, k) which can be stored into a finitedimensional buffer. As a consequence, the initial condition problem (that is typical of prediction-based control) is overcome by reduction as at the initial step k = 0 the state x 0 = x(-N ) is measured and the control past story can be fixed by the designer (e.g., at u 0 = 0). Moreover, As N → 0, the reduction variable recovers the state measure that is ν(k) = x(k). Remark 3.4. The reduction (12) and the reduced dynamics (13) are different from their input-delay counterparts (Mattioni et al. (2018a)). In the present case, ( 12) is capturing the effect of the delay over the state measurements and not directly onto the evolution of the dynamics (1) which is free of delay in open loop. Indeed, the delayed measures affect the system evolutions once a statefeedback is applied.

As the reduced dynamics (13) preserves the same controlfree component as (1) the following result can be proven. Proposition 3.2. Consider the dynamics (1) under Assumption 2.1 and state measurements affected by a delay of length N ≥ 0. Consider the output reduction variable (12) evolving according to (13). Then, the reduced dynamics ( 13) is u-average passive with respect to the output

y(k) = ĥr (ν(k), u(k)) with ĥr (ν, u) = L G -N (•,u) S(ν) (16a) ĥav r (ν, u) = 1 u u 0 ĥr (ν + (v), v)dv (16b)
and the same storage function S(•) : R n → R ≥0 .

Proof. Using the Lyapunov function in Assumption 2.1 as storage function for ( 13), one has that

∆S(ν) = S(F -N (ν, u)) -S(ν) = S(F 0 (ν)) -S(ν) ≤0 + u 0 L G -N (•,v) S(ν + (v))dv ≤ u 0 L G -N (•,u) S(ν + (v))dv = u ĥav r (ν, u)
so getting the result. Remark 3.5. The average output (16b) can be approxi-

mated in O(|u| 2 ) as ĥav r (ν, u) =L G [-N ],1 S(F 0 (ν))+ u 2 L 2 G [-N ],1 + L G [-N ],2 S(F 0 (ν)) + O(|u| 2 ).
From Proposition 3.2, because (13) evolves with the same drift as (1), average passivation of the reduced dynamics can be inferred from the delay-free case. Moreover, the new output (16a) making the reduced system (13) average passive is explicitily parameterized by the delay N through the new transported vector field G -N (•, u) in ( 15). It turns out, as proved in the following result, that u-average passivation of (13) allows to infer average passivation of the original system (1) with the new reduction output (16a). Corollary 3.1. Consider the dynamics (1) under Assumption 2.1 and state measurements affected by a delay of length N ≥ 0. Then, the dynamics (1) is u-average passive with output functional y r (k) = h r (x k , u(k)) where

h r (x k , u(k)) = ĥr (ν(k), u(k)) ν(k)=r(x k ) =L G -N (•,u(k)) S r (x k ) (17)
and storage functional S r (x k ) := S(r(x k )) > 0 verifying the dissipativity inequality ( 11) for all k ≥ 0.

Proof. The average passivating output ( 17) rewrites as

L G -N (•,u(k)) S r (x k ) = L G(•,u(k)) S(F -N 0 (x(k))
) that is, the delay-free passivating output computed over the reduction ν(k) = F -N 0 (x(k)) with x(k) being the predicted state brought N -step backward through the drift of (1) only. The average reduced output reads

h av r (x k , u(k)) = 1 u(k) u(k) 0 L G(•,v) S(F -N 0 (x + (v)))dv so yielding ∆ k S r (x k ) =∆ k S(ν(k)) ≤ u(k) 0 L G -N (•,v) S(ν + (v))dv = u(k) 0 L G(•,v) S(F -N 0 (x + (v)))dv.
Accordingly, average passivity of the reduced dynamics (13) with output (16a) implies passivity of the original dynamics (1) with output (17). Remark 3.6. As one might have expected, Corollary 3.1 states that for passivation (1) to be achieved despite delayed state measurements, the output and the storage need to be functional of the state x k = (x(k -N ), u k ) of the overall time-delay system. Remark 3.7. When N → 0 one recovers the delay-free passivating output. Remark 3.8. Based on Remark 3.5 one has

h av r (x k , u(k)) = L G [-N ],1 S(F 0 (x(k -N ))) + N i=1 u(k-i)L G [-N +i-1],1 S(F 0 (x(k-N )) + u(k) 2 × L 2 G [-N ],1 +L G [-N ],2 S(F 0 (x(k-N )))+O(|(u k , u(k))| 2 ).
From the arguments above, stabilization through delayed averaged output damping can be achieved provided that ZSD holds. Theorem 3.1. Consider the dynamics (1) under Assumption 2.1 and state measurements affected by a delay of length N ≥ 0. Then, the damping reduction-based feedback u = u r (x k ) solution to the damping equality 12) and (16a) makes the origin GAS for (1).

u + κ r h av r (x k , u) = 0, κ r > 0 (18) with h av r (x k , u(k)) = ĥav r (ν(k), u(k)) and ν(k) = r(x k ) as in (
Proof. For proving the result it is enough to show that ZSD of the output delay-free system (1) with h(x, u) = L G(•,u) S(x) implies ZSD of (1) under the delayed output (17). To this end, Let Z r be the largest invariant set contained in

{x k ∈ C s.t. L G -N (•,0) S r (x k ) = L G(•,0) S(x(k - N )) = 0}. As F 0 (•) is invertible (with non singular ja- cobian by definition) and for u ≡ 0, ν(k) = x(k - N ) and G -N (•, 0) = Ad F -N 0 G(•, 0), one gets {x k ∈ C s.t. L G -N (•,0) S r (x k ) = 0} ≡ {x ∈ R n s.t. L G(•,0) S(x) =
0} and thus the result. Remark 3.9. Exploiting the definition of ( 12), the averaged passivating output rewrites as

h av r (x k , u(k)) = 1 u(k) u(k) 0 L G(•,v) S(F -N 0 (x + (v)))dv
computed on the basis of the reduced dynamics (13). By approximating the expression above as

h av r (x k , u(k)) = L G(•,0) S(F 0 (x(k -N ))) + O(|(u k , u(k))|
) one gets that (3.2) gets the form an expansion around the delay-free (9) computed at the measured retarded state x(k -N ) plus higher terms taking into account the effect of the delay and vanishing as u → 0. This is not the same as in the prediction approach where the passivating output

h av p (x k , u(k)) = 1 u(k) u(k) 0 L G(•,v) S(x + (v))dv
is the prediction of the delay-free (8) N -step forward (i.e.,

h av p (x k ) = h av (F N (x(k -N ), u k ) and x(k) = F N (x(k - N ), u k )))
. By approximating the above expression one gets

h av p (x k , u(k)) = L G(•,0) S(F N +1 0 (x(k-N )))+O(|(u k , u(k))|)
that is the delay-free output map (9) computed over the predicted state x(k) = F N 0 (x(k -N )). Remark 3.10. On the basis of Remark 3.8, an approximate solution to (18) rendering the origin of (1) locally asymptotically stable is given by

u ap r (x k ) = -κr (x(k -N )) (x k ) (x k ) =L G [-N ],1 S(F 0 (x(k -N ))) + N i=1 u(k -i)L G [-N +i-1],1 S(F 0 (x(k -N )) κr (•) =2κ r 2 + κ r L 2 G [-N ],1 + L G [-N ],2 S(F 0 (•)) -1
.

Starting from the above approximations, a bounded globally asymptotically stabilizing feedback can be computed along the lines of [START_REF] Mattioni | Forwarding stabilization in discrete time[END_REF].

The LTI case

Consider the case in which the discrete-time dynamics (1), is represented as a LTI system, x(k + 1) = Ax(k) + Bu(k) (19) with A marginally Schur stable and |A| = 0 (that is no eigenvalues in zero), with Lyapunov function S(x) = 1 2 x P x and state measurements affected by a delay of length N ≥ 0. By applying Proposition 3.2, the output reduction variable (12) specifies as

ν(k) = x(k -N ) + k-1 i=k-N A k-1-N -i Bu(i) (20)
and the reduced dynamics (13) as

ν(k + 1) = Aν(k) + B -N u(k) (21)
with B -N := A -N B. As a consequence of Proposition 3.2, the reduced dynamics ( 21) is u-average passive with respect to y(k) = B -N P ν(k) with storage function S(ν) = 1 2 ν P ν and average output ĥav r (ν, u) = B -N P Aν+

1 2 B -N P B -N u. Accordingly, one rewrites S r (x k ) = x(k -N ) + k-1 i=k-N A k-1-N -i Bu(i) 2 P > 0 h r (x k , u) = B -N P (x(k -N ) + k-1 i=k-N A k-1-i-N Bu(i)) h av r (x k , u) = B -N P A(x(k -N ) + k-1 i=k-N A k-1-i-N Bu(i)) + 1 2 B -N P B -N u(k)
so that the stabilizing feedback solution to ( 18) is given by

u(k) =- 2κ r P Aν(k) 2 + κ r B -N P B -N = - 2κ r P A -(N -1) x(k) 2 + κ r B -N P B -N .

AN ACADEMIC EXAMPLE

Consider a discrete-time bilinear system described as

x(k + 1) = Ax(k) + (M x(k) + B)u(k) (22) with matrices A = 0 1 - 1 2 3 2 , M = 0 0 0 1 , B = 1 0 .
The system (22) satisfies Assumption 2.1 with associated maps

F 0 (x) = Ax, G(x, u) = (1, x 1 -u) , x = (x 1 , x 2 ) ,
and storage function

S(x) = 1 2 x P x with P = 2 -3 -3 5 .
When a delay of length N ≥ 0 affects the state measurement, the reduction variable for (22) reads

ν(k) =A -N Ψ(k -1, k -N )x(k -N ) + k-1 j=k-N A -N Ψ(k -1, j)Bu(j), with Ψ(k, k 0 ) = k i=k0+1 (A + M u(i)) and Ψ(k, k) = I. Accordingly, one defines the reduced dynamics ν(k + 1) = Aν(k) + (M -N ν(k) + B -N )u(k), with M -N = A -N M A N , B -N = A -N B and, moreover, G -N (ν, u) = M -N (A + M -N u) -1 (ν -B -N u) + B -N .
Thus, the damping feedback in Theorem 3.1 gets the form

u(k) = -κ r (B -N + M -N ν(k)) P Aν(k) 1+ κr 2 (B -N +M -N ν(k)) P (B -N +M -N ν(k)) (23)
making the origin of (22) GAS with κ r > 0.

Simulation. To validate the effectiveness of the proposed method, let us compare the reduction-based controller ( 23) with the prediction-based control, namely

u p (k) = -κ p (B + M x p (k)) P Ax p (k) 1 + κp 2 (B + M x p (k)) P (B + M x p (k)) , (24)
with κ p > 0, x p (k) = x(k) = A N ν(k) both applied to the system (22) (Remark 3.9). In Figure 1, simulations of the closed-loop system are reported when considering both the reduction-based feedback (23) and the predictionbased feedback (24) under condition x 0 (-N ) = (-1, 1) and a delay of length N = 1. From those figures, it is shown that, although comparable, the reduction-based controller provides smoother trajectories and zero decay faster than the prediction one so highlighting the anticipating effect of the reduction. We note that, as N increases significantly, reduction still ensures closed-loop stabilization albeit with degrading performances in the first N time steps. In Figure 2, robustness with respect to delay mismatch is tested. Indeed, the design (for both prediction and reduction) is based on a nominal value of the delay N = 2 whereas the actual delay is set to N r = 4 > N . In this case, the reduction-based controller still guarantees stabilization of the closed loop with acceptable performances whereas the prediction-based one fails.

CONCLUSIONS AND PERSPECTIVES

A new passivity-based stabilizing control strategy is proposed for discrete-time nonlinear systems affected by delayed state measurements. Based on the concept of outputreduction, a new variable is computed from the delayed state measure at time k ≥ 0 and the story of the control signal over the delayed time window with no need of prediction. The MIMO case follows the same lines as Mattioni et al. (2018a). The results are promising also in terms of robustness with respect to uncertainties on the delay for which a formal study is undergoing. Current work is addressing the case of stabilization through reduction for nonlinear systems with general delayed output measurements (that may not be the state) also under large delays.

Fig. 1 .Fig. 2 .

 12 Fig. 1. N = 2 and κ r = κ p = 10.


Mohamed Elobaid
email: mohamed.elobaid@uniroma1.it

Mattia Mattioni
email: mattia.mattioni@uniroma1.it

Dorothée Salvatore Monaco
email: salvatore.monaco@uniroma1.it

Dorothée Normand-Cyrot
email: dorothee.normand-cyrot@centralesupelec.fr

Sampled-data tracking under model predictive control and multi-rate planning

Keywords: Digital implementation, Nonlinear predictive control, Tracking

INTRODUCTION

Several control problems involve the cancellation of the so called zero-dynamics. This issue is even more evident in the nonlinear context when solving input-output feedback linearization or tracking problems (see [START_REF] Isidori | Nonlinear control systems[END_REF]). The problem becomes more delicate under sampling and, in particular, when direct digital control approaches are employed to design the controller. In that case in fact, the minimum-phase property of the system can even be lost for small sampling periods, due to the appearance of the unstable so-called sampling zeros ( Åström et al. (1984); [START_REF] Monaco | Zero dynamics of sampled nonlinear systems[END_REF] for linear and nonlinear cases). The idea of making use of a piecewise constant control over sub-intervals of the sampling interval, that is multi-rate control, has been properly introduced in the nonlinear context to overcome the aforementioned pathologies [START_REF] Monaco | An introduction to motion planning using multirate digital control[END_REF]). The possibility of varying the control several times in the sampling interval confers more degrees of freedom to the control action (Monaco andNormand-Cyrot (1991, 2001)). A major limitation in the use of multi-rate control stands in its intrinsic dependence on the model so implying lack of robustness. Moreover, since the input signal changes over sub-intervals of the sampling period, the control works in open loop at all those sub-intervals since the corresponding measures of the states are not available. On the other hand, model predictive control (MPC) represents a powerful and effective design technique that relies upon the solution of a constrained optimization problem subject to the system dynamics plus possible additional requirements over a finite time horizon. Indeed, it has now become the mainstay for regulation and tracking in the industry for constrained multi-input/multi-output (MIMO) systems (see [START_REF] Camacho | Model predictive control[END_REF] 1989)). Additionally, under suitable assumptions, it has been proved that the feedback generated by a nominal nonlinear model predictive problem is inherently robust with respect to different perturbing actions and model uncertainties (e.g., [START_REF] Picasso | Robustness analysis of nominal model predictive control for nonlinear discrete-time systems[END_REF]; [START_REF] Grimm | Examples when model predictive control is nonrobust[END_REF]; [START_REF] Magni | Nonlinear model predictive control[END_REF]). Still, as the intuition suggests, some difficulties may arise when discrete-time models are used by the MPC designer to ensure a prefixed reference profile for the output. As sampling induces unstable zerodynamics, a naive implementation of MPC feedback might yield unboundedness of the internal trajectories and thus of the feedback. To overcome this issue, standard MPC implementations introduce further penalizing weights on the objective function and/or constraints (typically LMI) over the optimization problem (e.g., [START_REF] Bemporad | Robust model predictive control: A survey[END_REF]; [START_REF] Byun | Predictive control: A review and some new stability results[END_REF]). As a drawback, such a stratagem is not based on the understanding of the source of unstability so that ad-hoc tuning is needed. Motivated by these reasons, in Elobaid et al. (2019a) a first attempt to handle those aspects has been proposed by directly designing MPC utilizing multi-rate inputs and the equivalent multi-rate sampled-data model for prediction. Under small penalties on the input, this approach has been shown to be effective as no further modifications of the optimization problem are required for preserving boundedness of the closed loop; however it requires huger capabilities of the sample and hold devices. In this respect, this work is aimed to weaken this demand by leaving the holding and sampling devices (i.e., actuators and sensors) synchronous. The new proposed control scheme combines a classical single-rate MPC controller with a multi-rate planner that computes, starting from samples of the desired output profile, a suitable admissible reference trajectory to be fed to the MPC. The inner MPC controller working on the fast sampling rate will now guarantee the prefixed boundedness of the internal behaviour of the overall system with no need of introducing further constraints. Roughly speaking, in the proposed control scheme, MPC is used to robustify multi-rate design, and multi-rate planning is used to improve MPC. This work is organized as follows. In Section 2 recalls on single and multi-rate sampling and model predictive control are given. In Section 3 the main ideas and motivations behind multi-rate planning for MPC control are introduced. Section 4 is devoted to the proposed MPC-MR control scheme which is applied to the simplified model of a planar vertical take-off and landing in Section 5. Concluding remarks end the paper.

Notations: All functions and vector fields defining the dynamics are assumed smooth and complete over the respective definition spaces. In the paper T will be the length of the sampling period, and any suitable subinterval will be denoted by δ. M U denotes the space of measurable and locally bounded functions u : R → U with U ⊆ R. U s ⊆ M U denotes the set of piecewise constant functions over time intervals of fixed length s ∈]0, T * [ and T * small enough; i.e.

U s = {u ∈ M U s.t. u(t) = u k , ∀t ∈ [ks, (k + 1)s[; k ≥ 0}. Given a vector field f , L f denotes the Lie derivative operator, L f = n i=1 f i (•) ∂ ∂xi .
The Lie exponential operator is denoted as e L f and defined as e L f := I + i≥1 L i f i! with I being the identity operator. Finally, x P = x P x denotes the seminorm of x ∈ R n for some

P ≥ 0. A function β(•) : [0, ∞) → [0, ∞) that is zero in zero and strictly increasing and unbounded is said to be of class κ ∞ . A function R(x, δ) = O(δ p ) is said to be of order δ p , p ≥ 1 if, whenever it is defined, it can be written as R(x, δ) = δ p-1 R(x, δ) and there exists a function β(δ) ∈ κ ∞ and δ > 0 s.t. ∀δ ≤ δ , | R(x, δ)| ≤ β(δ)

PRELIMINARIES AND RECALLS

Sampled-data systems

Single-rate (SR) and multi-rate (MR) sampled-data (SD) equivalent models of a continuous-time process are recalled in the sequel [START_REF] Monaco | Issues on nonlinear digital systems[END_REF]). Consider a nonlinear continuous-time input-affine system ẋ = f (x) + g(x)u, y = h(x) (1) and let u(t) ∈ U T and y(t) = y k for t ∈ [kT, (k + 1)T [ (with T ≥ 0 being the sampling period). Then, denoting x k := x(kT ), y k := y(kT ), u k := u(kT ) for k ≥ 0, the evolutions of ( 1) at the sampling instants t = kT with T ≥ 0, are described by its single-rate (SR) sampled-data equivalent model

x k+1 = F T (x k , u k ), y k =h(x k ) (2)
where the mapping

F T (•, •) : R n × R → R n admits the following series expansion in powers of T F T (x k , u k ) =e T (L f +u k Lg) x x k = x k + i>0 T i i! (L f + u k L g ) i x k
.

(3) Since a closed-form expression for the state and output evolutions of ( 1) does not exist in general, only approximated expansions in power of T can be computed. In this respect, if the series expansion (3) is characterized by a finite number of terms, then system ( 1) is said to be finitely discretizable [START_REF] Monaco | Issues on nonlinear digital systems[END_REF]). It is a matter of computations to verify that if (1) has well defined relative degree, say r ≤ n, the relative degree of the sampled-data equivalent model always falls to r d = 1; namely, one has

y k+1 =h(x k ) + r i=1 T i i! L i f h(x) x k + T r r! u k LgL r-1 f h(x) x k + O(T r+1 )
so that ∂y k+1 ∂u k = 0. As a consequence, whenever r > 1, the sampling process induces a further zero-dynamics of dimension r -1 (the sampling zero-dynamics, [START_REF] Monaco | Zero dynamics of sampled nonlinear systems[END_REF]) that is in general unstable for r > 1. In that case inversion-like techniques via single-rate sampling cannot be achieved while guaranteeing stability of the internal dynamics.

To overcome this pathology multi-rate (MR) sampling, corresponding to sample the state and output at lower frequency with respect to changes of the piecewise constant control, has been introduced in Monaco and Normand-Cyrot (1991). Accordingly, denoting by u i k (t), x i k (t) and y i k (t) the input, state and output variables at any t = kT + (i -1)δ for i = 1, . . . , m (with

u k = u 1 k , x k = x 1 k and y k = y 1 k
), the multi-rate equivalent model of order m of (1) gets the form

x k+1 =F T m (x k , u 1 k , . . . , u m k ) (4) with T = mδ, u(t) ∈ U δ and F T m (x k , u 1 k , . . . , u m k ) =e δ(L f +u 1 k Lg) . . . e δ(L f +u m k Lg) x x k = F δ (•, u m k ) • • • • • F δ (x k , u 1 k ).
It has been proved by the authors that, with m = r, the MR sampled-data model has vector relative degree r δ = (1, . . . , 1) under the choice of a suitable output vector specified by the output itself and its first (r-1) derivatives. Accordingly, the corresponding zero-dynamics inherits the zero-dynamics stability properties of (1) (see [START_REF] Monaco | Zero dynamics of sampled nonlinear systems[END_REF]). In addition in [START_REF] Mattioni | On partially minimum phase systems and nonlinear sampled-data control[END_REF], it has been shown that the minimum-phase condition can be relaxed by increasing the MR order. This MR-SD equivalent model can then be used to design tracking controllers over the sampling interval T as developed in Section 3.

Sampled-data unconstrained MPC

The problem of tracking a reference v(t) at the sampling instants is typically addressed within the framework of nonlinear model predictive control. Roughly speaking, the feedback is computed to minimize a quadratic cost function of the form

J = np i=1 e k+i Q + u k+i-1 R (5)
with e k (y k , v k ) a suitable tracking error, Q > 0, R ≥ 0 being appropriate penalizing weights and n p , n c the prediction and control horizons respectively. The functional cost (5) is repeatedly optimized at each sampling instant t = kδ over a finite horizon while the feedback is applied via a receding horizon implementation.

As well know, tracking control implicitly requires zerodynamics cancellation. In this respect, as the above problem is formulated in the sampled-data context, instability of the closed loop system unavoidably arises due to the sampling zero-dynamics. To overcome this, the uncon-strained MPC problem is typically enriched by redefining the cost function as

J = np-1 i=1 e k+i Q + u k+i-1 R + V np s.t. x k+1 = F T (x k , u k ),
x np ∈ X f with V np and X f being the terminal cost and constraints set suitably constructed for keeping the state bounded [START_REF] Camacho | Model predictive control[END_REF]. This new formulation gives rise to possible feasibility issues so that solution might not exist for a given set of initial conditions. On the other hand, reformulation of the tracking MPC problem under multi-rate sampling allows to overcome those unstability and feasibility issues while avoiding unnecessary complications. Indeed, it was proved in Elobaid et al. (2019a) that, given (1) with well-defined relative degree and the MPC problem over (5) subject to the MR-SD model (4), the optimal SD feedback always exists and is uniquely defined as a formal series in powers of δ for all n p = n c ≥ 1. Thus using MR control at the implementation level of MPC allows to handle the issues arising due to sampling, without resorting to penalizing terminal costs and constraints. However, it places a greater burden on the sampling and hold devices, assumes a cheap control, and does not address the issue of working in open loop over intervals of length T ; this motivates the proposed control scheme.

MULTI-RATE PLANNING FOR MPC CONTROL

For system [START_REF]Nominal tracking, R = 0, Q = I Fig. 2. Perturbed tracking, = 0.5, R = 0.5I, Q = I and white actuation noise with mean 10 -3[END_REF], it is required to design a single rate piecewise constant state feedback (with sampling period δ) that tracks samples of suitable reference v(•) at prefixed sampling instants (that is v k = v(kT ), T = mδ) by minimizing the cost functional (5) at all t = kδ, k ≥ 0 and ensuring boundedness of the closed loop trajectories. In the nominal multi-rate scenario and when R = 0 and Q = I, this can be achieved by considering (4) with m ≥ 1 and solving in (u 1 k , u 2 k , .., u m k ) the following system of nonlinear algebraic equations

H(F T m (x k , u 1 k , . . . , u m k )) = V k+1 (6
) where H : R n → R m and the vector V are respectively suitable augmented output and reference vectors so as to guarantee that, by invoking the implicit function theorem, the solution exists. As already commented, major limits of this stand in its lack of robustness with respect to model uncertainties and sampling approximations and to the fact it works in open loop over time intervals of length T . How to improve its effectiveness? We propose to design a sampleddata single-rate control law, acting at all t = kδ and based on the corresponding sampled measures of the state, through a sampled-data MPC procedure that makes use of the intermediate reference output values y i k resulting from the application of the nominal control sequence computed from (6) to the MR-SD model (4). This reference is said to be admissible in the sense of the definition below. Definition 3.1. A sampled-data reference sequence {v k , k ≥ 0} is said to be admissible for (1) from x 0 ∈ R n if, for a suitable integer m ≤ n, equality (6) has a solution {u i k , i = 1, . . . , m} which remains bounded. It will be said SR or MR admissible if m = 1 or m > 1, respectively.

Note that, from Definition 3.1 a MR admissible sequence can be suitably enriched to be SR admissible, as suggested by the following result. Theorem 3.1. Consider the system (1) and let v(t) be a reference signal to be tracked at t = kT for k → ∞. Denote by {v k = v(kT ), k ≥ 0} the sequence of samples of the reference that is assumed to be MR admissible for a suitable m > 1 under the input sequence {û i k , k ≥ 0, i = 1, . . . , m} solution to (6) for all k ≥ 0. Let {ŷ i k , k ≥ 0, i = 1, . . . , m} be the augmented reference generated by ŷ1 k = v k , and for, i = 2, . . . , m, ŷi

k = h(x i k ) with x1 k = x(kT ) xi k = F δ (x i-1 k , ûi-1 k ).
Then, the unconstrained MPC problem defined via the cost ( 5) with e k+i = ŷi k -y i k subject to the SR-SD model ( 2) admits a solution which is bounded for n p = n c ≥ m and R = 0.

Proof: Whenever {v k , k ≥ 0} is admissible, then there exists a solution sequence {û i k , k ≥ 0, i = 1, . . . , m} such that ( 6) is solved and s.t y k+1 = v k+1 , k ≥ 0. Then applying the solution to (2), one gets xi

k = F δ (x i-1 k , ûi-1 k ) and correspondingly the intermediate output values ŷi k = h(x i k ).
Setting δ = T m this sequence is SR admissible by construction, and by Definition 3.1 there exists a sequence of controls such that y(kT ) = vk , so implying feasibility. To show that the MPC optimization problem recovers this solution, one sets n p = n c and the proof proceeds along the lines of (Elobaid et al., 2019a, Th.2, Prop.7).

Note that the statement above does not assume that the reference can be tracked in continuous time, but merely that it is MR admissible. Given a reference v(t) that system (1) can exactly track (in the sense of [START_REF] Isidori | Nonlinear control systems[END_REF][Chapter 4]), then a fast sampling of this reference v k is MR admissible, namely one can define H(x) = (h(x), L f h(x), . . . , L r-1 f h(x)) , V = (v, v, . . . , v (r-1) ) as in [START_REF] Monaco | Multirate sampling and zero dynamics: from linear to nonlinear[END_REF] corresponding to which which (6) admits a solution. The following result can be hence given. Corollary 3.1. Suppose system (1) is minimum-phase and has a well defined relative degree r ≤ n, then equality (6) always admits a solution with a multi-rate of order m ≥ r.

Proof: Since (1) has a well defined relative degree, then the input-output feedback linearization problem is solvable so that, under feedback and change of coordinates, [START_REF]Nominal tracking, R = 0, Q = I Fig. 2. Perturbed tracking, = 0.5, R = 0.5I, Q = I and white actuation noise with mean 10 -3[END_REF] Isidori (2013)[Chapter 4]. Accordingly, the MR equivalent model of order m ≥ r is given by

reads ż = Az + Bv, η = q(z, η) + p(z, η)v with A, B, q(•), p(•) as in
z k+1 = A T m z k + B T m v k , η k+1 = F T m (z k , η k , v k ) (7) with A T m = (e Aδ ) m , B T m = A m-1 sr B sr . . . B sr and A sr = e Aδ , B sr = δ 0 e Aδ u(τ )dτ and F T m (z k , η k , v k ) = e δ(Lq+v 1 k Lp) . . . e δ(Lq+v m k Lp) x (z k ,η k ) .
In this setting, z k corresponds to H so that (6) reduces to a linear map on the multi-rate inputs v k . Because B δ m is full rank by construction and because the relative degree is invariant under feedback, one gets the solution vk = (B T m ) -1 (V k+1 -A δ m z k ). Moreover, by the fact that η = q(z, η) + p(z, η)v is stable, and combined with Theorem 3.1 and the arguments in [START_REF] Monaco | Zero dynamics of sampled nonlinear systems[END_REF], one gets that the planned trajectories are admissible for the overall dynamics over the small interval δ = T m . Theorem 3.1 applies to finitely discritizable systems.

PLANNING AND CONTROL ALGORITHM

In this section, and referring to the discussion above, we present in a detailed manner the control scheme for designing a sampled-data feedback u k = u(x k ) ensuring tracking of a given output profile at the sampling instants t = kT for all k ≥ 0 by exploiting a planned admissible trajectory generated via the MR model (4). We assume that the dynamics ( 1) is finitely descritizable with F T m (•, û1 k , . . . , ûr k ) denoting the corresponding multirate finite model of order m (possibily computed under coordinate change and preliminary feedback). The following algorithm is proposed by using the admissible sequence {ŷ i k , i = 1, . . . , m, k ≥ 0} defined in Theorem 3.1 as a reference trajectory for the MPC with n p = m. Such a trajectory is computed and updated at all t = kT based on the nominal multi-rate solution defined through (4). Thus, for all t ∈ {kT, kT + δ, . . . , kT + (m -1)δ} the planned reference sequence is fed to the MPC for computing the optimizing controller which is guaranteed to exist for R ≥ 0 small enough by virtue of Theorem 3.1. Specifically, the algorithm works over the steps depicted in Algorithm 1.

Algorithm 1 Planning and control algorithm

1: Initialization: V a ← (v k+1 , v k+2 ) x k ← x(kT ), Q ← Q, R ← R, m ← m 2: while t ≥ 0 do 3: if t = (k + j)T, j ∈ Z ≥0 then 4: k ← k + j 5: (ŷ k , ŷ) = Planning(x k ) 6: u k = Control(m, Q, R) 7:
else 8:

for t = kT + iδ, i = 1, . . . , m -1 do 9:

u i k = Control(m, Q, R) 10: procedure (ŷ k , ŷk+1 ) = Planning(x k , V a ) 11: x1 k ← x k        û1 k . . . ûm k . . . ûm k+1        = (h • F T m ) -1 (x k , v k+1 ) (h • F T m ) -1 (•, v k+2 ) • (F T m ) -1 (x k , v k+1 ) (8)
12:

for j = 0 : 1 do 13:

for i = 1 : m do xi+1 k+j = F δ (x i k+j , ûi k+j ), ŷi k+j =h(x i k+j ). 14: procedure u k = Control(m, Q, R, ŷ1 k , . . . , ŷnp k ) 15: n p ← m 16: u k = argmin u k np i=1 ŷi k -y i k Q + u i-1 k R 17: u k = u 1 k Remark 4.1.
From the previous arguments, we only exploit the samples of the reference over two big steps (that is v k+1 , v k+2 and correspondingly setting in the MR planner (6) an augmented output vector H a (x) = (y k , y k+1 ) ). This is due to the fact that in the implementation of a receding horizon algorithm, we will need explicitly the values of the desired reference sequence over n p = m steps, and writing the second iteration of the MPC (i.e., at time t = kT + δ), one notes the explicit dependence of the (optimal) control on values of the desired output trajectories at ŷm+1 k = ŷ1 k+1 . Remark 4.2. The design the planner can be worked out on a simplified sampled-data model so to reduce the computational burden related to solve equality (8). When the conditions of Corollary 3.1 are met, the computations associated with the planner are simply the inversion of a matrix B T m which is full rank by construction. Consequently, as shown in the case study, one in principle uses a simplified finite model for the planner, while a more exhaustive one is employed by MPC for prediction. Remark 4.3. The proposed control scheme inherits the nominal robustness properties of MPC (e.g. [START_REF] Grimm | Nominally robust model predictive control with state constraints[END_REF]; [START_REF] Picasso | Robustness analysis of nominal model predictive control for nonlinear discrete-time systems[END_REF]; [START_REF] Grimm | Examples when model predictive control is nonrobust[END_REF]). This will be further demonstrated in the following case study.

THE PVTOL AS A CASE STUDY

Let the model of the PVTOL (planar vertical take-off and landing) aircraft take the form: ẍ = -sin(θ

)v 1 + cos(θ)v 2 z = cos(θ)v 1 -1 + sin(θ)v 2 θ = v 2 (9)
with output y = h(x, ẋ, z, ż, θ, θ) = (x, z) .

Construction of the simplified MR planner model

To use our control scheme, we first define the multi-rate sampled-data model of the PVTOL. To this end, it is known (e.g. Di Giamberardino and Djemai (1994)) that ( 9) is feedback equivalent to a finitely discretizable system, by setting

v = 1 cosθ + ε θ2

  coordinates change ζ = ϕ(x, ẋ, z, ż, θ, θ) = ϕ 1 (•) + ϕ 2 (•) with ϕ 1 (•) = cos θ -θ sin θ 0 -sin θ 0 -θ cos θ , ϕ 2 (rate of order 2 on u 1 and 4 on u 2 can be employed to ensure the invertibility of the sampled-data dynamics. Explicitly writingu 1 (t) = u j 1 (k), t ∈ [(k + j -1 2 )T, (k + j 2 )T [, j = 1, 2 u 2 (t) = u j 2 (k), t ∈ [(k + j -1 4 )T, (k + j 4 )T [, j =1, 2, 3, 4.

and developing the calculations and rearranging the terms, we obtain the MR SD equivalent model of the PVTOL as:

with δ ≥ 0 being the sampling period and T = 4δ. The model (10) will be used for planning the admissible reference trajectories.

Planning and control

For all t = kT , planning of the intermediate output references is made on the basis of the simplified equivalent model ( 10). Hence one gets for all t = kT + (i -1)δ, an ad-

, and thus the intermediate reference output values {(x i k+j , ẑi k+j ), i = 1, 2, 3, 4 and j = 0, 1} for system (9). Consequently, for all t = kT + (i -1)δ, the MPC computes the feedback u i k (for i = 1, . . . , 4) with the sampleddata SR model of the PVTOL (in the form ( 2)) used for prediction. This feedback is then applied to the simulation model of system (9) while recomputing the reference for all t = kT .

Simulations

In the following we will compare the proposed control algorithm (denoted MR-MPC) to the stand-alone MPC with the trajectory planner proposed by Luigi Biagiotti (2019) (denoted FIR-MPC). Figure ?? depicts the nominal scenario (with no external perturbations nor model uncertainties), while in Figure ?? actuator disturbances as well as uncertainties on the parameter are considered. In all simulations δ = 1 seconds while n p = n c = 4 seconds and Q = I. Both schemes will utilize a sequential quadratic programming SQP based optimization solver. Time varying references are fixed on both the lateral and vertical displacements (that is x, z respectively) as a ramp signal (shown in black) with velocity v 0 = 1 m/s to be tracked at t = kT, T = 4δ. It results that the proposed control algorithm favourably compares to the FIR-MPC. In the nominal case (Figure ??), we set R = 0. Contrarily to the MR-MPC scheme, the FIR-MPC is unable to follow the reference over the sampling steps δ and an off-set is evident. While the number of iterations in each instant to compute the minimizer is slightly more in the MR-MPC case, the minimum obtained is lower, namely it is V np = 0 compared to 0.5 obtained by the FIR-MPC. In Figure ??, R > 0 and a white noise is added on the actuation signal. In addition the parameter = 0.5 in the simulation model, while the nominal value = 0.8 is used to construct the planner and the prediction model for the MPC. In this case as well, the proposed algorithm is able to track the reference over the big intervals T , despite the effect of uncertainties being evident (although acceptable i.e. |θ| ≤ 0.1 rads) in the internal dynamics. Note also that while the number of iterations required to obtain the minimum is comparable between the two algorithms, the value function was still lower V np = 0.08 for MR-MPC compared to 0.1 for FIR-MPC. Further simulations are reported in [START_REF] Elobaid | Sampled-data tracking under model predictive control and multirate planning: further simulations and remarks[END_REF] comparing the MR-MPC with FIR-MPC and stand-alone MR control in different situations.

CONCLUSIONS

We establish an intuitive implementation of sampled-data tracking control through multi-rate planning utilizing discrete time nonlinear single-rate MPC for reference tracking. This is done by highlighting the role played by ensuring the availability of good and admissible references to the MPC controller so to improve the effectiveness of the two stand-alone design methodologies. Future works concern the study of the effect of incorporating constraints on the control, on the benefits incurred by the proposed scheme.