Andrea Fasano

Aurore Bailly

Jeremy Wozniak

Vincent Fourmond

Christophe Léger
email: leger@imm.cnrs.fr

Christophe Léger Catalytic

Catalytic bias and redox-driven inactivation of ancestral FeFe hydrogenases from group B2

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

Introduction

Hydrogenases, the enzymes that oxidize and produce hydrogen, come in three different flavors --FeFe-hydrogenases, NiFe-hydrogenases, and Fe-hydrogenases --named after their metal content. Nitrogenases also produce H 2 as a byproduct of nitrogen reduction.

The active site "H-cluster" of FeFe-hydrogenases (leftmost in fig 1A) consists of a dinuclear Fe 2 cluster that is coordinated by 3 CO and 2 CN ligands, a bridging amine (adt, or azadithiolate), and a cysteine thiolate that covalently attaches the dinuclear site to a [4Fe4S] cluster. The Fe ion that is remote from the cubane is referred to as "distal", or Fe d . The diatomic ligands can be detected by FTIR spectroscopy, and their vibrations have been used to identify various states of the H-cluster, including catalytic intermediates [START_REF] Sommer | Proton Coupled Electronic Rearrangement within the H-Cluster as an Essential Step in the Catalytic Cycle of [FeFe] Hydrogenases[END_REF][START_REF] Laun | Site-Selective Protonation of the One-Electron Reduced Cofactor in [FeFe]-Hydrogenase[END_REF][3][START_REF] Morra | Oxygen Stability in the New [FeFe]-Hydrogenase from Clostridium Beijerinckii SM10 (CbA5H)[END_REF][START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] .

In the catalytic cycle, H 2 binds to the apical, empty coordination site on Fe d when the active site is in the Hox 'resting' state ([4Fe4S] 2+ Fe II Fe I ). H 2 is split into a hydride and a proton which is transferred to the nitrogen of the amine ligand. The H-cluster is connected to the solvent by a series of residues that mediate long range proton transfer (a conserved cysteine, C299 in figure 1A, is in many hydrogenase the 1 st proton acceptor along the chain 3,6 ), elusive gaz channels that guide the diffusion of small molecules, and, in some enzymes, accessory FeS clusters that are used to mediate electron transfer.

FeFe-hydrogenases are present in many microorganisms, both prokaryotes and eukaryotes, and are therefore very diverse [START_REF] Morra | Fantastic [FeFe]-Hydrogenases and Where to Find Them[END_REF] . The environment of the active site is characterized by three polypeptide motifs (P1, [FILT][ST][SCM]C[CS]P[AGSMIV][FWY], P2 and P3) [START_REF] Vignais | Classification, and Biological Function of Hydrogenases: An Overview[END_REF] . Greening et al. have recently defined three phylogenetic groups: A (prototypical and bifurcating), B (ancestral) and C (putative sensory) [START_REF] Greening | Genomic and Metagenomic Surveys of Hydrogenase Distribution Indicate H2 Is a Widely Utilised Energy Source for Microbial Growth and Survival[END_REF] ; Calusisnka et al also defined a group D with putative hydrogenases 10 . An earlier, structural classification is based on the number of accessory FeS clusters: zero (M1), two (M2), or four clusters (M3), as exemplified by the three group A1 enzymes from Chlamydomonas reinhardtii (Cr), Desulfovibrio desulfuricans and Clostridium pasteurianum (CpI), respectively. These three standard enzymes have been the focus of most of the biophysical studies of FeFe hydrogenases, but non-standard hydrogenases have recently been characterized and unexpected features have emerged.

For example, the group A "CbA5H" FeFe hydrogenase from C. beijerinckii (Cb) is protected from oxygen damage by the binding to the distal Fe ion of the conserved cysteine that is the first proton relay; this binding can occur because non-conserved residues that are remote from the active site make the protein loop that bears the cysteine residue more flexible than in other hydrogenases [START_REF] Morra | Oxygen Stability in the New [FeFe]-Hydrogenase from Clostridium Beijerinckii SM10 (CbA5H)[END_REF][START_REF] Winkler | A Safety Cap Protects Hydrogenase from Oxygen Attack[END_REF][START_REF] Rutz | Increasing the O2 Resistance of the [FeFe]-Hydrogenase CbA5H through Enhanced Protein Flexibility[END_REF] . In the group C FeFe hydrogenase from Thermoanaerobacter mathranii, protons are transferred along a pathway that is entirely distinct from that of prototypical hydrogenases [START_REF] Land | Discovery of Novel [FeFe]-Hydrogenases for Biocatalytic H2-Production[END_REF] , and catalysis is 'irreversible', that is H 2 oxidation and production only occur at the price of a large thermodynamic driving force 14 . Related to the above comment about (ir)reversible catalysis, another catalytic property that has attracted much interest is the 'catalytic bias', defined as the ratio of the maximal rates in the two directions of the reaction (H 2 oxidation and production) [START_REF] Fourmond | Reversible Catalysis[END_REF] . Three homologous FeFe hydrogenases from C. pasteurianum illustrate how the protein that embeds the active site can tune the enzyme's catalytic bias by orders of magnitude: the enzyme CpI is equally efficient in the oxidative and reductive directions, whereas the homologous enzymes CpII (group A) and CpIII (group B) are strongly biased for oxidative and reductive catalysis, respectively [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] .

In 2019, the fact that CpIII is nearly inactive for H 2 oxidation in solution assays was related by Peters and coworkers to the ease with which the active site is oxidized above its normal resting state Hox. The overoxidized 'Hox+1' state is reminiscent of the "Hinact" state observed in other FeFe-hydrogenases, but it is observed in CpIII under less oxidizing conditions. These unique properties were tentatively assigned to the more hydrophobic environment of the H-cluster in CpIII, compared to CpI and CpII, which would destabilize the Hox state and thus favor proton reduction [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] .

At that time, the implication of the observation of the Hinact state in CpIII was not discussed. Only shorty after did it become clear that this state, which is also observed when exogenous sulfide binds to the distal Fe in standard FeFe hydrogenases [START_REF] Rodríguez-Maciá | Sulfide Protects [FeFe] Hydrogenases From O2[END_REF][START_REF] Felbek | Mechanism of Hydrogen Sulfide-Dependent Inhibition of FeFe Hydrogenase[END_REF] , reveals the binding to the distal Fe of the intrinsic sulfide of a cysteine residue, as occurs in the enzyme from C. beijerinckii [START_REF] Winkler | A Safety Cap Protects Hydrogenase from Oxygen Attack[END_REF] . The hypothesis that this also occurs in CpIII would be particularly appealing because the environment of its H cluster is cysteine rich: the P1 motif, most comonly xTSCCPxx, is present as ITSCCCPMW. This peculiar TSCCCP motif defines the FeFe hydrogenase subgroups 'M2a' and 'B2' according to Meyer [START_REF] Meyer | FeFe] Hydrogenases and Their Evolution: A Genomic Perspective[END_REF] and Calusinska et al. 10 , respectively. Its functional significance is discussed in this paper, where we examine the catalytic properties of two M2a/B2 enzymes: CpIII (WP_003447632.1) and Megasphaera elsdenii II ("MeII", WP_169013299.1, distinct from the enzyme studied in refs [START_REF] Caserta | FeFe]-Hydrogenase: Do Accessory Clusters Influence O2 Resistance and Catalytic Bias?[END_REF][START_REF] Butt | Direct Electrochemistry of Megasphaera Elsdenii Iron Hydrogenase. Definition of the Enzyme's Catalytic Operating and Quantitation of the Catalytic Behaviour over a Continuous Potential Range[END_REF] ), and a CpIII site directed mutant where the supernumerary cysteine is deleted.

Results

Structural considerations

We examined 274 group B sequences listed on the hydrogenase database website 21 as of may 2023 [START_REF]Hydrogenase Classifier[END_REF] (we removed WP_025640716.1, which lacks the P1 motif, and included the CpIII and MeII sequences). In this group, the xTSCCPxx motif is the most common version of P1 (134 occurrences), followed by TSCCCPxx (75 occurrences). The latter is specific to group B: none of the 955 hydrogenases sequences classified as group A or C listed on the hydrogenase database included a CCC triad.

In all sequences in group B, the first run of four cysteines, which is Cx2Cx2Cx3C in prototypical hydrogenases, is highly variable (blue dots in SI fig S6), with a consequence on the coordination of the accessory [4Fe4S] clusters that is distal from the active site (SI figure S7), as mentioned before 10,[START_REF] Meyer | FeFe] Hydrogenases and Their Evolution: A Genomic Perspective[END_REF] .

Figure 1A shows the alphafold [START_REF] Jumper | Highly Accurate Protein Structure Prediction with AlphaFold[END_REF] model structure of the WT CpIII, aligned on the structure of CpI (pdb 6N59 [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] ).

The CpI residues are shown in gray. A cluster of red balls indicate the positions of a water molecule that is seen in many structures of group A1 FeFe hydrogenases (see the list in Table S1 in ref [START_REF] Felbek | Mechanism of Hydrogen Sulfide-Dependent Inhibition of FeFe Hydrogenase[END_REF] ). Proton transfer from the active site to the solvent involves C299, this water molecule (HOH 612 in pdb 3C8Y, absent in pdb 6N59), E279, S319, E282 and R286 3,6 (black dots in SI fig S6). According to a ClustalO alignment [START_REF] Sievers | Clustal Omega for Making Accurate Alignments of Many Protein Sequences[END_REF] of 204 group A1 sequences, it is very conserved: the first glutamate is substituted in only 3 sequences, the second glutamate substituted in 4, the serine substituted in only one sequence, and the arginine is substituted (most often with lysine) in only 20 sequences. This series of residues is more variable in group B: the first glutamate is present in all but one group B sequences (WP_013255832.1) out of 274, but the second glutamate is replaced with D or H in about 40 group B sequences (none of which includes the CCC triad) ; the serine is absent from about 30 other group B sequences, and the arginine is rarely present. This shows that there are many small variations on the typical proton transfer pathway in hydrogenases from group B.

The positions of the CpIII residues C223 (equivalent to CpI C300, which coordinates the cubane of the H-cluster), C222, C221, E202, S242, E205 and H209 are shown in green in Figure 1A. The proton transfer pathway is the same in CpI and CpIII, except that a histidine in CpIII replaces R286, and the sulfur of the supernumerary cysteine (C221) is very close to the position of the conserved water molecule. This suggests that C221 mediates proton transfer from C222 to E202 in CpIII. The Alphafold model of MeII (not shown) is similar to that of CpIII, but the putative proton transfer pathway is less well defined ; it would involve C183, C182, E163, E166 but the serine position is occupied by P203 according to the sequence alignment, and the distal arginine by N170 (black dots in SI fig S6). Figure 1B shows the Alphafold model of the structure of the ΔC221 CpIII mutant where one of the three adjacent cysteines is removed from the sequence: the positions of the two cysteines of the TSCCP motif (blue) match those of CpI (gray), suggesting that the environment of the active site in the ΔC mutant is very similar to that of standard hydrogenases.

Electrochemistry

Figure 2A compares the voltammetric signatures of Cr, CpIII and MeII FeFe hydrogenases adsorbed onto a rotating pyrolytic graphite edge electrode in a solution saturated with H 2 . In these experiments, electron transfer between the electrode and the enzyme is direct, the electrode potential is swept up and down at a certain scan rate and the activity is measured as a current, positive for H 2 oxidation, negative for H 2 production. The electrode is rotated to prevent H 2 depletion near the electrode surface.

The CpIII signal in Fig 2A (red) is consistent with that published in ref [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] , identical to that of MeII (orange), and very distinct from that of Cr (green) and other standard hydrogenases [START_REF] Del Barrio | Interaction of the H-Cluster of FeFe Hydrogenase with Halides[END_REF] . The CpIII and MeII enzymes completely inactivate on the scan towards high potentials, above E = -100 mV, but this inactivation is reversible: on the scan towards low potential, a large fraction of the activity is recovered under very reducing conditions (below -400mV): compare the approx. 90% decrease from forward to backward scan at -0.2 V vs the 25% decrease at -0.5 V. SI fig S1 shows the subsequent scan, which demonstrates that the H 2 -oxidation activity is also recovered by the excursion to low potentials. The slight loss of current when the low potential limit of the CV is reached can be due to incomplete reactivation on this particular time scale or film loss.

The CpIII, MeII and Cr CVs shown as plain lines were recorded with the enzyme in a chloride-free buffer at 30°C. Addition of chloride has little effect on the CpIII and MeII CVs (SI fig S2). The reversible oxidative inactivation occurs at a lower potential and is distinct from that of Cr and other group A FeFe hydrogenases : the latter is strictly dependent on the presence of chloride (or bromide) in the buffer [START_REF] Del Barrio | Interaction of the H-Cluster of FeFe Hydrogenase with Halides[END_REF] , as shown by the dashed line in fig 2 . The shape of the CpIII and MeII CVs is reminiscent of that of Cb (gray line in figure 2), although the mechanism appears to be distinct hereafter. Figures 3A andB show a series of CpIII voltammograms recorded at 5°C (A) and 30°C (B) with the same film and increasing values of the high potential limit (where the sweep is reversed), from red to blue. The enzyme was reactivated by a reductive poise at -509 mV for two to three minutes between each CV.

The CVs at 5°C in panel A clearly show that the reactivation consists of two different processes. One occurs around -150 mV and results in a sigmoidal increase in current at about the same potential as the inactivation seen on the upward sweep. But this reactivation is not complete: full activity is only recovered at much lower potential, during the reductive poise. This shows that two distinct inactive species, distinguished by their reactivation kinetics, are produced under oxidizing conditions. In the following we call I 1 and I 2 the species that reactivate at high and low potential, respectively.

The sigmoidal variations of activity seen around -150 mV, and the observation that the potentials where I 1 is formed (on the sweep upward) and reactivates (on the sweep backward) are very similar, suggest that I 1 is in equilibrium with the active form of the enzyme on the time scale of the voltammetry. That this (in)activation is still clearly observed in voltammograms recorded at a much faster scan rate (up to 0.5 V/s in SI fig S5) confirms that it results from a fast reaction. That full reactivation of the enzyme at low potential required a reductive poise (figures 3A and B) shows that the reactivation of the other inactive state, I 2 , is much slower than that of I 1 .

That the sigmoidal reactivation around -150 mV is clearly visible only at 5°C suggests that at high temperature, the formation of I 2 becomes fast enough that the entire sample is converted into I 2 , a species that reactivates only under very reducing conditions. The variable high-potential limit experiments in figures 3A and B suggest that I 1 is not an intermediate between the active form of the enzyme and I 2 . Indeed, if this were to be the case, any increase in the vertex potential (below -150mV) would cause an exponential increase in the rate of formation of I 2 . Instead, the data suggest that the slightly larger amount of I 2 produced when the vertex potential is increased is the mere consequence of time spent at high potential being greater. Similarly, it seems possible to rule out the hypothesis that I 2 is produced from the active form of the enzyme only: in that case, increasing the high potential limit of the sweep would prevent the formation of I 2 , since under these conditions I 1 would accumulate more. The qualitative inspection of the CVs therefore suggests that I 2 is produced upon oxidation of either A or I 1 . This hypothesis is supported by the quantitative analysis of the voltammetry below. • The active enzyme molecules transform into two fully inactive species, I 1 and I 2 .

(

) 1
where Γ is the total enzyme surface coverage, and the square brackets indicate surface concentrations (all in units of mol/cm 2 ).

• The observed catalytic current is the product of the steady-state response of the active enzyme times the time-dependent fraction of active enzyme [START_REF] Barrio | Redox (in)activations of Metalloenzymes: A Protein Film Voltammetry Approach[END_REF][START_REF] Fourmond | Modelling the Voltammetry of Adsorbed Enzymes and Molecular Catalysts[END_REF] :

(2)

• The steady-state catalytic response of the active enzyme is modeled by the generic "EECr" equation [START_REF] Fourmond | Modelling the Voltammetry of Adsorbed Enzymes and Molecular Catalysts[END_REF][START_REF] Fourmond | Understanding and Design of Bidirectional and Reversible Catalysts of Multielectron, Multistep Reactions[END_REF] . • I 1 is in redox equilibrium with the active enzyme.

(3) (we expect E°1 around -150 mV).

• I 2 is produced slowly (on the time scale of the voltammetry) either from I 1 (hypothesis 1), from A (hypothesis 2), or indistinctly from either A or I 1 (hypothesis 3). If we note k i the pseudo 1st-order rate constant of production of I 2 (inactivation) and k a the rate of disappearance of I 2 (activation), the hypotheses translate into:

(4) (5)
(6)

• The rate constants k i and k a depend on potential in a sigmoidal manner:

Note that k i tends to k i max at high potential, and k a tends to k a max at low potential. This (in)activation kinetics is expected if the formation of I 2 follows an "ECE" mechanism (using the electrochemical terminology), with an oxidation (e.g. the formation of Hox, the most oxidized catalytic intermediate) that precedes the inactivation, a chemical transformation, and a final oxidation that locks the inactive state. But we make no a priori assumption on the values of E 0 2 and E 0 3 . We used the QSoas software [START_REF] Fourmond | QSoas: A Versatile Software for Data Analysis[END_REF] to fit the model to two successive scans to accurately determine the kinetics of slow reactivation at low potential. Figure 4 shows the best fit of the 3 rd model to three selected voltammograms of WT CpIII at pH 7, 30°C (the model was fitted to all the CVs shown in fig 3B globally, we show only 3 CVs in figure 4 for clarity). The other two hypotheses gave less satisfactory modeling (SI figures S3 and S4), as expected from the above qualitative discussion of figure 3. That the 3 rd hypothesis gives the best fit of the voltammetry implies that the two inactivation processes are independent of one another. It may be, for example, that one corresponds to a transformation of the active site, and the other results from a remote process.

Panel D shows the dependence of k i and k a on E that is deduced from the global fitting procedure. We obtained k i max =0.08 s -1 , k a max =0.08 s -1 , E 0 1 =-139mV, E 0 2 =-317mV, and E 0 3 =-384mV for CpIII at 30°C, pH 7. We estimated that the errors on E 2 and E 0 3 are around ± 50 mV. The value of E 0 1 defines the equilibrium between the active enzyme and I 1 at high potential. The value of E 0 2 is that of the redox step that precedes the oxidative formation of I 1 ; here it is close to the potential of Hred/Hox measured in IR titrations of standard hydrogenases (this parameter has not been measured for CpIII), and the value of E 0 3 is close to the value of the reduction potential of the Hox/Hinact transition in CpIII (-380mV at pH 8 in ref [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] ), suggesting that the I 2 inactive species is Hinact. In an effort to identify the functional significance of the supernumerary cysteine in the TSCCCP motif, we produced the ΔC221 CpIII variant, by deleting one of the three adjacent cysteine residues. The deletion does not impair activity in either direction, as shown by the CV of the ΔC221 variant in figure 3C. However, the shape of the CV is very different from that of WT CpIII and MeII. In particular, the inactivation seems to involve only the "I 1 " species (which is produced and reactivated at high potential). Indeed, the reactivation at high potential is more pronounced than with the WT and complete. A small hysteresis is observed at high potential; it may also be present in the WT voltammetry but hidden by the second inactivation process (the formation of I 2 ), which is not detected in the variant.

In addition to the (in)activation kinetics, the outcome of the fitting procedure is the steady-state response of the fully active enzyme (green in figure 4C) [START_REF] Fourmond | Modelling the Voltammetry of Adsorbed Enzymes and Molecular Catalysts[END_REF][START_REF] Fourmond | Understanding and Design of Bidirectional and Reversible Catalysts of Multielectron, Multistep Reactions[END_REF] . The latter is defined by two catalytic potentials, E cat ox and E cat red . The difference between (E cat ox + E cat red )/2 and E eq , the H + /H 2 Nernst potential, gives the catalytic bias of the enzyme, defined as the ratio of the oxidative and reductive limiting currents, which would be observed in the absence of any inactivation process.

Table 1 shows the values of the catalytic bias of CpIII and Cr, based on the analysis of the voltammograms shown in figures 2A and 4. We conclude that the intrinsic catalytic bias is about the same in all cases, with the enzymes being equally active in both directions of the reaction.

Discussion

There has been much interest recently in the reaction of the H-cluster of standard [START_REF] Rodríguez-Maciá | Sulfide Protects [FeFe] Hydrogenases From O2[END_REF][START_REF] Felbek | Mechanism of Hydrogen Sulfide-Dependent Inhibition of FeFe Hydrogenase[END_REF][START_REF] Rodríguez-Maciá | Caught in the Hinact : Crystal Structure and Spectroscopy Reveal a Sulfur Bound to the Active Site of an O2 -Stable State of [FeFe] Hydrogenase[END_REF] and non-standard hydrogenases [START_REF] Winkler | A Safety Cap Protects Hydrogenase from Oxygen Attack[END_REF][START_REF] Corrigan | Evidence of Atypical Structural Flexibility of the Active Site Surrounding of an [FeFe] Hydrogenase from Clostridium Beijerinkii[END_REF] from group A (Cr and Cb, respectively) with extrinsic or intrinsic sulfide. The formation of the sulfide-bound oxidized inactive state is revealed by a particular FTIR signature called H inact reversed upon reduction. CpIII is another example of hydrogenase that can be observed in the Hinact state [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] , whereas it belongs to the phylogenetically distinct group B. The relations between the formation of Hinact in CpIII, inactivation and protection from O 2 have not been discussed. The structural determinants of this reaction in CpIII were also unknown, but we have been intrigued by the cysteine-rich TSCCCP motif present in CpIII and other group B FeFe hydrogenases, such as MeII. This motif defines the B2 subgroup.

In the absence of a crystallographic structure of any FeFe hydrogenase from group B, we examined an AlphaFold prediction of the structure of CpIII hydrogenase (Figure 1). The overall fold was already predicted to be very similar to that of standard hydrogenases (SI fig S6 in ref [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] ). The AlphaFold model shown here in figure 1A suggests that the sulfur of the supernumerary cysteine (C222) is involved in proton transfer.

Redox-driven inactivation is a usual feature of FeFe hydrogenases, but significant variations have recently been observed. Oxidative, reversible inactivation is observed with CpI and other standard FeFe hydrogenases from group A, but it occurs only under very oxidizing conditions and only in the presence of chloride, bromide [START_REF] Del Barrio | Interaction of the H-Cluster of FeFe Hydrogenase with Halides[END_REF] or sulfide [START_REF] Felbek | Mechanism of Hydrogen Sulfide-Dependent Inhibition of FeFe Hydrogenase[END_REF] . All experiments herein were performed in a chlorideand sulfide-free buffer, which implies that the mechanism of inactivation is distinct.

Reversible oxidative inactivation occurs with Clostridium beijerinckii FeFe hydrogenase (Cb, also from group A, gray in fig 2) in the absence of chloride or exogenous sulfide, and it results from the formation of the Hinact state upon binding to the distal Fe of the H-cluster of the sulfur of the cysteine that is equivalent to CpI C299 [START_REF] Winkler | A Safety Cap Protects Hydrogenase from Oxygen Attack[END_REF][START_REF] Corrigan | Evidence of Atypical Structural Flexibility of the Active Site Surrounding of an [FeFe] Hydrogenase from Clostridium Beijerinkii[END_REF][START_REF] Heghmanns | The Oxygen-Resistant [FeFe]-Hydrogenase CbA5H Harbors an Unknown Radical Signal[END_REF] (see figure 1A). The observation that CpIII is also oxidized to the Hinact state in the absence of sulfide (Table S3 in ref ) strongly suggests that the binding of the cysteine to Fe d occurs in CpIII, as it does in Cb. This is probably one of the two reasons CpIII inactivates at high potential. The slow reduction of Hinact may be the cause of the lag observed in solution assays of CpIII (SI section 1.2 in ref [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] ). The voltammetry of CpIII and MeII being very similar, it is tempting to suggest that this bond formation also occurs in MeII and other hydrogenases from group B. The oxidative formation of a vicinal disulfide bridge [START_REF] Richardson | Broad Analysis of Vicinal Disulfides: Occurrences, Conformations with Cis or with Trans Peptides, and Functional Roles Including Sugar Binding[END_REF] between C222 and C221 may also result in enzyme inactivation, but this alternative hypothesis does not explain the observation of the Hinact state, so we considered it unlikely.

The voltammetric signatures of CpIII and MeII in figure 2 may look similar to that of Cb at first (figure 2). However we observed that the (in)activation kinetics of CpIII and MeII is different from that of Cb. In particular two distinct inactive species, referred to here as I 1 and I 2 , are produced independently.

We consider most likely that the 2 nd inactive species (I 2 ) is the Hinact state. Indeed, the reduction of Hinact occurs at low potential (-400mV at pH 8 according to the results in ref [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] ), which matches the reactivation kinetics of I 2 (E 0 3 =-384mV at pH 7, according to the above described modeling of the voltammetry).

The observation that the binding of the cysteine and the formation of Hinact occur in these group B enzymes, which are phylogenetically distinct from Cb, is intriguing. The three residues that have been identified as allowing the formation of Hinact in Cb by making the TSCCP loop flexible (L364, P386 and A561) [START_REF] Winkler | A Safety Cap Protects Hydrogenase from Oxygen Attack[END_REF] are not conserved in CpIII and MeII (see the orange frames in SI fig S6), but we observed that in CpIII, this inactivation disappears when the supernumerary cysteine is deleted. This suggests that in WT CpIII, C221 pushes the proximal cysteine (C222) closer to Fed and hence favors the formation of the Fed-Scys bond. It therefore appears that different peculiar structural features in Cb on one side, and group B2 hydrogenases on the other, give the protein the ability to allow the proximal cysteine to bind Fe d .

Binding of a nearby residue to Fe d should protect the H-cluster from O 2 . This is so in Cb, but the same protection mechanism operates in a certain type of NiFe hydrogenases, where the binding to a metal ion of the active site of a nearby aspartate protects the enzyme from oxygen attack [START_REF] Kulka-Peschke | Reversible Glutamate Coordination to High-Valent Nickel Protects the Active Site of a [NiFe] Hydrogenase from Oxygen[END_REF][START_REF] Kumar | The Fully Oxidized State of the Glutamate Coordinated O2-Tolerant [NiFe]-Hydrogenase Shows a Ni(III)/Fe(III) Open-Shell Singlet Ground State[END_REF] However, we have found no evidence that CpIII hydrogenase is resistant to O 2 . This may be because the formation of Hinact is too slow (0.08s -1 at 30°C in CpIII, compared to 1s -1 at 5°C in Cb [START_REF] Winkler | A Safety Cap Protects Hydrogenase from Oxygen Attack[END_REF] ), so that the disruption of the active site by O 2 occurs faster than the enzyme can protect itself by forming the H inact state. The between fast formation of Hinact and greater resistance was demonstrated in a series of Cb site directed mutants [START_REF] Winkler | A Safety Cap Protects Hydrogenase from Oxygen Attack[END_REF][START_REF] Rutz | Increasing the O2 Resistance of the [FeFe]-Hydrogenase CbA5H through Enhanced Protein Flexibility[END_REF] .

The other inactive species, I 1 , is produced and reduced more quickly and at much higher potential (-150mV) than I 2 irrespective of the presence of the third cysteine in the P1 motif (that is, in WT CpIII and in the ΔC variant, fig 3C). Since the formation of I 2 occurs independently from the production of I 1 at the active site (hypothesis 3 in the modeling of the voltammetry), this inactivation, which is not affected by the change in coordination of Fe , must be the consequence of a remote transformation; we speculate that this is the overoxidation of an accessory cluster, maybe the distal cluster whose coordination pattern is very peculiar in group B hydrogenases (SI figure S7). This hypothesis will have to be tested by mutagenesis.

To discuss the catalytic bias and the shape of the voltammograms, one needs to acknowledge that the catalytic response of the enzyme is modulated by the inactivation processes: the observed voltammogram is actually the product of the steady-state response of the active enzyme (an intrinsic property of the active site) times the time-dependent fraction of enzyme that is active (considering the two redox-dependent, reversible inactivation processes, eq 2). The quantitative modeling of the voltammetry (figure 4) allows us to untangle the two contributions, to obtain the steady-state response (green in figure 4C) and the kinetics of inactivation. The shape of the former tells whether the enzyme is an equally good catalyst in the two directions of the reaction, or if it is biased in one particular direction. This is quantified by the difference between the average catalytic potential and the Nernst potential of the H + /H 2 couple (eq 9). Table 1 shows that CpIII hydrogenase is actually not intrinsically biased in any direction (the ratio of the limiting currents is close to one), just like Cr. This contrasts with the observation that in solution assays [START_REF] Artz | Tuning Catalytic Bias of Hydrogen Gas Producing Hydrogenases[END_REF] , CpIII is much more active in the reductive than in the oxidative direction. The reason for this apparent discrepancy is that in solution assays, the enzyme inactivates under the oxidative conditions that are required to drive H 2 oxidation, hence the very small oxidative activity. The same is true for Cb, which is very efficient at oxidizing H 2 , but whose strong H 2 -oxidation activity is hidden by the oxidative inactivation, except in some variants where this inactivation is slowed (figure 3d in ref [START_REF] Winkler | A Safety Cap Protects Hydrogenase from Oxygen Attack[END_REF] ).

If one is interested in using the enzyme to catalyze H 2 oxidation, that it inactivates under oxidative condition is clearly an issue that has to be considered. However, if one's goal is to elucidate how the residues that surround the H-cluster may tune its catalytic properties and the catalytic bias, then any redox driven inactivation must be factored out. Here, we conclude that the properties of the active site of CpIII hydrogenase are not very different from those of group A hydrogenases (e.g. Cb or Cr), in that the enzyme is intrinsically equally active in both directions. That Cb and CpIII hydrogenase inactivate at high potential does not reveal any difference in terms of H-cluster redox and catalytic properties.

Figure 1 .

 1 Figure 1. The model structure of CpIII (A) and the ΔC variant (B), aligned on the structure of CpI. Gray sticks indicate the positions of selected CpI residues (light gray, pdb 6N59 5 ), the red spheres show the positions of a conserved water molecule in various structures of group A1 FeFe hydrogenases (see the list in Table S1 in ref 17 ). Green sticks show the CCC triad of CpIII and four other residues that are probably involved in proton transfer (2 nd rank relaxed AlphaFold model). B: The position of the CC diad in the model structure of the ΔC mutant of CpIII (blue), compared to the position of the CpI residues (gray). SI figure S7 compares the environments of the accessory FeS clusters in CpI and CpIII.

Figure 2 :

 2 Figure 2: Cyclic voltammetry of CpIII (red), MeII (orange), Cb (gray) and Cr (green), all at pH 7 under one atm. of H 2 . The signals shown using a solid line were recorded without chloride, dashed line in the presence of 0.1M chloride. The Cb voltammogram was scaled down three-fold to ease the comparison. The Cr voltammogram with NaCl was scaled up 1.3 times to compensate for film-loss and allow a proper comparison with the voltammogram with Na 2 SO 4 . Arrows show the directions of the sweeps. A capacitive current recorded in an experiment without enzyme was subtracted from each voltammogram. Conditions: 30°C (except Cb, 5°C); 20 mV/s; 3000 rpm.

Figure 3 :

 3 Figure 3: Cyclic voltammograms of WT CpIII recorded by changing the upper potential limits (red to blue from the lower to the greater vertex potential) at 5°C (panel A) and 30°C (panel B). Cyclic voltammogram of the CpIII mutant, missing the supernumerary cysteine of the TSCCCP motif, at 30°C (panel C). The capacitive current recorded in experiments carried out without enzyme was subtracted from all the voltammograms. Other conditions: pH 7; 20 mV/s; 3000 rpm. We used the following assumptions to fit a model to the CVs in figs 3A and B.

Figure 4 :

 4 Figure 4: fit of the 3 rd model to three selected voltammograms of WT CpIII among those of figure 3B. Panels A, B and C show the experimental data in black and the fits of the 3 rd model in dash red lines. Panel D shows the potential dependence of k i (solid line) and k a (dash line) obtained from the global fit 29 . The green dashed line in panel C shows the steady state catalytic response simulated from the parameters of the best by forcing the inactivation rate constants to be zero. Vertical lines in panel C indicate the two catalytic potentials E cat ox and E cat red and the equilibrium potential E eq . Solid and dashed vertical lines in panel D note the values of E 0 2 and E 0 3 , respectively.

Table 1 .

 1 The catalytic potentials (in mV vs SHE) and catalytic bias under one atm. of H 2 of Cr and CpIII at 30°C, pH 7, 1 bar H 2 , E eq = -419 ± 2 mV.

		E cat	ox	E cat	red	(E cat	ox +E cat	red )/2 i lim ox /i lim red	figure
	CpIII	-390	-455	-423	0.85	4
	Cr	-362	-448	-405	2.5	2A

Acknowledgements

This research was funded by the Centre National de la Recherche Scientifique, Aix Marseille Université, Agence Nationale de la Recherche (ANR-21-21-CE50-0041), Région Sud. This work received support from the french government under the France 2030 investment plan, as part of the Initiative d'Excellence d'Aix-Marseille Université -A*MIDEX, AMX-22-RE-AB-097. The authors are very grateful to Frédérique Berger, glass blower at Aix Marseille University and the proteomic facility of the Institut de Microbiologie de la Méditerranée (IMM, CNRS-AMU, FR3479), Marseille Proteomique (MaP), for performing proteomic analyses by mass spectrometry.

Supporting Information

Molecular biology and biochemistry procedures, additional electrochemical experiments and data analyses, sequence alignments. The SI file is available free of charge at https://pubs.acs.org


Koen Scheres
email: k.j.a.scheres@tue.nl

Romain Postoyan
email: romain.postoyan@univ-lorraine.fr

Dragan Nesic
email: dnesic@unimelb.edu.au

W P Maurice

H Heemels

Output-Based

Maurice Heemels
email: m.heemels@tue.nl

Output-based event-holding control in presence of measurement noise

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

Output-based Event-holding Control in Presence of Measurement Noise K.J.A. Scheres, R. Postoyan, D. Nešić, W.P.M.H. Heemels

Abstract-We present rules to stabilize the origin of a networked system, where data exchanges between the plant and the controller only occur when an output-dependent inequality has been satisfied for a given amount of time. This strategy, called Event-Holding Control (EHC), differs from timeregularized event-triggered control (ETC) techniques, which generate transmissions as soon as a triggering condition is verified and the time elapsed since the last transmission is larger than a given bound. Indeed, the clock involved in EHC is not running continuously after each transmission instant, but only when a criterion is verified. We propose an output-based design of these triggering mechanisms that are robust to additive measurement noise and ensure an input-to-state stability (ISS) property. This EHC scheme naturally has a positive lower bound on the transmission interval. Additionally, we show via an example that, in presence of measurement noise, Zeno-like behavior, where events are generated near the minimum interevent time consistently, may occur when the system is close to the attractor. We introduce space-regularization to mitigate this issue, resulting in an input-to-state practical stability (ISpS) property rather than ISS.

I. INTRODUCTION

In digital control systems with feedback loops, it is well known that the maximum achievable system performance depends on the rate at which information is exchanged between the plant and the controller. In some cases, this communication rate may be limited due to physical constraints, such as energy usage in battery-powered electronics or communication bandwidth in systems that are physically distributed and thus rely on low bandwidth communication to exchange information. In such cases, Event-triggered Control (ETC), see, e.g., [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF], [START_REF] Årzén | A simple event-based pid controller[END_REF], has been proposed as an alternative sampling strategy, whereby the communication times are not determined by a clock, but instead by a certain event occurring in the system. As these events depend on the state of the system, the underlying philosophy is that communication resources are only used when necessary. When designing the triggering mechanisms, one has to be careful to avoid so-called Zeno behavior, referring to the situation where an infinite number of transmissions are generated in finite time, which is clearly undesirable. In fact, it is desirable that the time between two consecutive events is lower-bounded by a positive number, which is often called the minimum inter-event time (MIET).

Many ETC designs are available in the literature with these properties, although, surprisingly, few dealing formally with the presence of measurement noise [START_REF] Hardouin | Distributed eventtriggered consensus of multi-agent systems with measurement noise and guaranteed interval bounds[END_REF]- [START_REF]Robustifying event-triggered control to measurement noise[END_REF]. It is known that Zeno behavior may occur for many existing triggering mechanisms, as the inter-event times are not robust to measurement errors. A famous example is the relative triggering mechanisms [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] for which Zeno behavior is excluded in the nominal case, but, it may occur when (arbitrary small) measurement noise is introduced, as shown in [START_REF] Borgers | Event-separation properties of event-triggered control systems[END_REF], [START_REF] Casau | On the robustness of nominally well-posed event-triggered controllers[END_REF]. To address this issue, in the previous works [START_REF] Scheres | Eventtriggered control in presence of measurement noise: A spaceregularization approach[END_REF], [START_REF]Robustifying event-triggered control to measurement noise[END_REF], space-regularization was introduced, whereby the triggering condition is regularized using a tuning parameter. By selecting this tuning parameter sufficiently large, a MIET is guaranteed even in presence of noise, and the resulting stability property is an Input-to-State practical Stability (ISpS) property instead of an Input-to-State Stability (ISS) one.

It is also possible, as was shown in [START_REF] Scheres | Eventtriggered control in presence of measurement noise: A spaceregularization approach[END_REF], [START_REF]Robustifying event-triggered control to measurement noise[END_REF], to design relative triggering mechanisms that do not exhibit Zeno in the typical sense by exploiting time-regularization [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF] under extra conditions, whereby a MIET is enforced by design. However, in presence of measurement noise, the system may display Zeno-like [START_REF] Yang | Periodic event/self-triggered consensus for general continuous-time linear multi-agent systems under general directed graphs[END_REF] behavior when close to the attractor, whereby the triggering times are consistently close to or at the MIET, thereby reducing to (almost) periodic time-triggered control.

In [START_REF] Wang | State-feedback event-holding control for nonlinear systems[END_REF], state feedback Event-holding Control (EHC) was proposed as an alternative to the time-regularized static event-triggered control, whereby the local clock is only run if a certain condition is violated. This strategy is motivated by several beneficial features. First, compared to time-regularized ETC, which also enforces a strictly positive minimum inter-event time, a transmission is not triggered as soon as a given amount of time has passed since the last transmission and the triggering rule is satisfied, but only when the latter has been verified for a given amount of time. This is essentially different and may help to decrease the amount of transmissions, as we will show on a numerical example. Second, it is well-suited for practical setups such as those operated by supervisory control and data acquisition (SCADA) system for instance, where a "hold time parameter" is used to adjust the maximal period that a local device holds its event before reporting to the master system. In this context, an event refers to the detection of a condition, see [START_REF]DNP3: User and Reference Manual[END_REF] for more details.

Thus, in this paper, we extend the ideas of event-holding control [START_REF] Wang | State-feedback event-holding control for nonlinear systems[END_REF] to the output-feedback case, and we design triggers that are robust to measurement noise. As we will see in Section VII, EHC may still lead to Zeno-like behavior in presence of measurement noise when close to the attractor. We thus introduce space-regularization to improve the "steady-state" behavior, at the price of a practical stability property. As the time between consecutive events is lower bounded by design, the space-regularization parameter can be selected arbitrarily, contrary to [START_REF] Scheres | Eventtriggered control in presence of measurement noise: A spaceregularization approach[END_REF], [START_REF]Robustifying event-triggered control to measurement noise[END_REF], where this parameter should be sufficiently large to ensure non-Zenoness and requires the knowledge of an upper-bound of the noise. Although a practical stability property is ensured, the added benefit of using EHC is that, by properly tuning this parameter, the average asymptotic closeness to the attractor may remain unaffected while the "steady-state" behavior in terms of average inter-event times improves significantly. This is not possible using [START_REF] Scheres | Eventtriggered control in presence of measurement noise: A spaceregularization approach[END_REF], [START_REF]Robustifying event-triggered control to measurement noise[END_REF], due to the (positive) lower bound on the space-regularization parameter, which is used to avoid Zeno. Our contributions are hence as follows:

1) we generalize EHC to the output-feedback case;

2) we show that, with the proposed event-holding function, the system is ISS with respect to measurement noise; 3) by applying space-regularization, the behavior close to the attractor may be improved in terms of average transmission rates, however, the resulting triggering mechanism will lead to ISpS instead of ISS; 4) using numerical simulations, we show that, by tuning the space-regularization parameter, the asymptotic closeness to the attractor may remain unaffected, while the average inter-event times improve significantly.

II. PRELIMINARIES

A. Notation

The sets of all non-negative integers are denoted N. The field of all reals and all non-negative reals are indicated by R and R ⩾0 , respectively. The vector in R N whose elements are all zeros is denoted by 0 N . For any vector u ∈ R m , v ∈ R n , the stacked vector u ⊤ v ⊤ ⊤ is denoted by (u, v). By ⟨•, •⟩ and | • | we denote the usual inner product of real vectors and the Euclidean norm, respectively. For any x ∈ R N , the distance to a closed non-empty set A ⊆ R N is denoted by |x| A := min y∈A |x -y|. We use U • (x; v) to denote Clarke's generalized directional derivative of a locally Lipschitz function U at x in the direction v, i.e., U • (x; v) := lim sup h→0 + , y→x (U (y + hv) -U (y))/h, which reduces to the standard directional derivative when U is continuously differentiable; see [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] for more details. Given a vector x ∈ R nx and a set

A ⊆ R nx ×R ny , Π x (A) denotes the projection of A onto the x-plane R nx , i.e., Π x (A) = {z ∈ R nx | ∃y ∈ R ny s.t. (z, y) ∈ A}.
By ∧ and ∨ we denote the logical and and or operators, respectively. For a measurable signal w : R ⩾0 → R nw , we denote by ∥w∥ ∞ := ess sup t∈R ⩾0 |w(t)| its essential L ∞ -norm provided it is finite, in which case we write w ∈ L ∞ . We use the usual definitions for comparison functions K, K ∞ and KL, see [START_REF] Goebel | Hybrid dynamical systems: modeling, stability, and robustness[END_REF].

B. Hybrid systems

Based on an extension [START_REF] Heemels | Hybrid systems with continuous-time inputs: subtleties in solution concepts and existence results[END_REF] of the formalism of [START_REF] Goebel | Hybrid dynamical systems: modeling, stability, and robustness[END_REF], we model hybrid systems

H(F, C, G, D, X, V) as ξ ∈ F (ξ, ν) ξ + ∈ G(ξ, ν) (ξ, ν) ∈ C, (ξ, ν) ∈ D, (1)
where ξ ∈ X ⊆ R n ξ denotes the state, ν an external input taking values in

V ⊆ R nν , C ⊆ X × V the flow set, D ⊆ X × V the jump set, F : X × V ⇒ R n ξ the (set-valued) flow map and G : X × V ⇒ R n ξ the (set- valued) jump map.
Sets C and D are assumed to be closed. We refer to [START_REF] Goebel | Hybrid dynamical systems: modeling, stability, and robustness[END_REF] for notions related to (1) such as hybrid time domains or hybrid arcs. For a hybrid time domain E, sup t E := sup {t ∈ R ⩾0 : ∃j ∈ N such that (t, j) ∈ E}.

Note that the disturbances ν are defined on the non-negative real line R ⩾0 , not on a hybrid time domain, see [START_REF] Heemels | Hybrid systems with continuous-time inputs: subtleties in solution concepts and existence results[END_REF] for a detailed motivation of this choice.

Definition 1. Given a set of inputs V ⊆ L ∞ , a hybrid system H is persistently flowing if all maximal solution pairs (ϕ, v) ∈ S H with v ∈ V are complete in the t-direction, i.e., sup t dom ϕ = ∞.

In this paper, we consider systems H that are persistently flowing, i.e., systems whose maximal solutions are t-complete, and thus we focus on the stability notions below. Definition 2. When H is persistently flowing with V ⊆ L ∞ , we say that a closed, non-empty set A ⊂ R nx is input-tostate practically stable (ISpS), if there exist β ∈ KL, γ ∈ K and d ⩾ 0 such that for any solution pair

(x, v) with v ∈ V |x(t, j)| A ⩽ β(|x(0, 0)| A , t) + γ(∥v∥ ∞ ) + d, (2)
for all (t, j) ∈ dom x. We say that A is input-to-state stable (ISS) when (2) holds with d = 0.

To prove that a given non-empty closed set A is IS(p)S, we will use the following Lyapunov conditions. Proposition 1. Consider a persistently flowing system H with a set of inputs V ⊆ L ∞ and let A ⊂ R n ξ be a non-empty closed set. If there exist a locally Lipschitz

V : dom V → R ⩾0 , α, α, α ∈ K ∞ , γ ∈ K and d ∈ R ⩾0 such that 1) dom V ⊃ Π ξ (C ∪ D), 2) for any (ξ, ν) ∈ C ∪ D, α(|ξ| A ) ⩽ V (ξ) ⩽ α(|ξ| A ), 3) for all (ξ, ν) ∈ C and f ∈ F (ξ, ν), V • (ξ; f ) ⩽ -α(V (ξ)) + γ(|ν|) + d, 4) for all (ξ, ν) ∈ D and any g ∈ G(ξ, ν), V (g) -V (ξ) ⩽ 0,
then A is ISpS, and A is ISS when item 3) holds with d = 0.

III. PROBLEM FORMULATION

We consider a continuous-time plant P of the form

P :      ẋp = f p (x p , u, v), y = g p (x p ), y = y + w, (3)
where x p ∈ R nx p is the plant state, u ∈ R nu the control input, v ∈ R nv a process disturbance, y ∈ R ny the output of the system unaffected by measurement noise, y ∈ R ny the output of the system including measurement noise and w ∈ R ny the additive measurement noise. The system is controlled by a dynamic controller C with dynamics

C : ẋc = f c (x c , y ), u = g c (x c , y ), (4)
where x c ∈ R nx c is the controller state, y ∈ R ny the most recently received noisy output of the system P, and u ∈ R nu the output of the controller. The functions f p and f c are assumed to be continuous, and g p and g c are assumed to be continuously differentiable and zero at zero. We proceed by emulation and assume that the controller C has been designed such that the origin of P is robustly stabilized, in the sense that the closed-loop dynamics satisfy certain properties that will be formalized in the following. Networked control setup where the controller and plant are (physically) separated and communicate via a packet-based network.

Remark 1. We do not impose restrictions on the existence or boundedness of the derivative of w, as was required in, e.g., [START_REF] Mousavi | On integral input-to-state stability of event-triggered control systems[END_REF], [START_REF] Abdelrahim | Robust event-triggered output feedback controllers for nonlinear systems[END_REF]. Moreover, we also do not need to know an upper bound on the L ∞ -norm as in [START_REF] Scheres | Eventtriggered control in presence of measurement noise: A spaceregularization approach[END_REF], [START_REF]Robustifying event-triggered control to measurement noise[END_REF]. Indeed, the only requirement on the signal w is that it is measurable in the mathematical sense.

The output y is broadcast over the digital network to the controller, see Fig. 1. The corresponding transmissions occur at time instants t k , k ∈ N, which are generated by an Event-Triggering Mechanism (ETM), to be designed. Because of the packet-based communication over the network, the controller does not have continuous access to y, but only to its latest transmitted value y . When the ETM transmits the measured output of the plant over the network, y is updated according to

y ((t k ) + ) = y(t k ), ∀t ∈ [t k , t k+1 ] (5)
In between transmissions the estimate evolves in a Zero-Order-Hold (ZOH) fashion, i.e., ˙ y = 0.

For modeling purposes, we define y and w, where

y((t k ) + ) = y(t k ), ˙ y = 0, ∀t ∈ [t k , t k+1 ], w((t k ) + ) = w(t k ), ˙ w = 0, ∀t ∈ [t k , t k+1 ]. (7)
Hence, w is the value of w at the last transmission instant of the ETM. Due to the aforementioned definitions, we obtain that y = y + w.

We define the true network-induced error e as the difference between the sampled output y without measurement noise and the current output y without measurement noise:

e := y -y. (8)
Note that e is not known by the ETM, and therefore, cannot be used by the corresponding local triggering condition for determining t k , k ∈ N. Hence, we also define the measured network-induced error e as the difference between the estimated output y and the current measured output y, which are both affected by noise, i.e., e := y -y = e + w -w.

The ETM does have access to e.

As mentioned above, the transmission instants are generated by the ETM, which we describe in the next section.

IV. EVENT-HOLDING CONTROL

As the ETM requires a clock, we introduce the timer variable τ ∈ R ⩾0 . During flow, the timer τ satisfies the differential inclusion

τ ∈ σ(Γ( y, e)), (10)
where σ : R ⇒ {0, 1} is a set-valued map given by

σ(s) :=      {1}, if s > 0, [0, 1], if s = 0, {0}, if s < 0. (11)
The function Γ determines when the clock will run (i.e., when τ = 1), and it will be constructed in the following. The transmission instants t k , k ∈ N, are defined by

t 0 = 0, t k+1 = inf{t > t k | Γ( y(t), e(t)) ⩾ 0 ∧ τ (t) ⩾ τ H }, (12)
where τ H denotes the holding time, which is designed below.

When y is broadcast over the network, the timer τ is reset according to τ + = 0.

V. HYBRID MODEL

We model the overall system as a hybrid system H as in Section II-B, for which a jump corresponds to the broadcasting of the noisy output y over the network. The full state for H becomes ξ := (x, e, w, τ ) ∈ X, where X := R nx × R ny × R ny × R ⩾0 , with n x := n xp + n xc . We define the concatenated exogenous inputs ν := (v, w) ∈ V, where V := R nv × R ny . The flow map F : X × V ⇒ X can then be written as F (ξ, ν) := f (x, e, w, v), g(x, e, w, v), 0 ny , σ(Γ( y, e))).

(

) 14
Based on (3), ( 4) and ( 9), we obtain

f (x, e, w, v) := f p (x p , g c (x c , g p (x p ) + e + w), v) f c (x c , g p (x p ) + e + w) , (15)
where based on (3), ( 6) and ( 9), we obtain g(x, e, w, v) := -∂g p ∂x p f p (x p , g c (x c , g p (x p ) + e + w), v) .

(16) According to [START_REF] Scheres | Eventtriggered control in presence of measurement noise: A spaceregularization approach[END_REF] and ( 13), the jump map G :

X × V → X is G(ξ, ν) := (x, 0, w, 0). (17)
Using [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF], the flow set C ⊆ X × V and jump set D ⊆ X × V are given by

C := {(ξ, ν) ∈ X × V | τ ⩽ τ H ∨ Γ( y, e) ⩽ 0} , ( 18
) D := {(ξ, ν) ∈ X × V | τ ⩾ τ H } . ( 19
)

VI. MAIN RESULTS

To ensure that the hybrid system has the desired stability properties, several conditions have to be satisfied. To that end, we require the following conditions to hold. Condition 1. There exist a locally Lipschitz function W : R ny → R ⩾0 , a continuous function

H : R nx × R ny × W → R ⩾0 , α W , α W > 0, L W ⩾ 0 and ϑ W ∈ K such that (i) for any e ∈ R ny , α W |e| ⩽ W (e) ⩽ α W |e|, (ii) for all e ∈ R ny , x ∈ R nx , w, w ∈ R ny and v ∈ R nv , W • (e; g(x, e, w, v)) ⩽ L W W (e) + H(x, e, w, v).
Item (ii) of Condition 1 imposes an upper bound on the growth of e between successive transmission instants along the solutions to H. This condition can always be satisfied by taking H(x, e, w, v) ⩾ |W • (e; g(x, e, w, v)| with H continuous and L W = 0. Next to Condition 1, we also require that the closed-loop system is robustly stable in the following sense.

Condition 2. There exist a locally Lipschitz function

V : R n → R ⩾0 , K ∞ -functions α V , α V , α V , α W , ϑ V , ϑ ϱ ∈ K, a function J : R nx × R ny × R ny × R nv → R, a locally Lipschitz function ϱ ∈ K ∞ , L ϱ ∈ R and γ > 0 such that (i) for all x ∈ R nx , α V (|x|) ⩽ V (x) ⩽ α V (|x|), (20)
(ii) for all x ∈ R nx , ν ∈ V and e, w ∈ R ny ,

V • (x; f (x, e, w, v)) ⩽ -α V (V (x)) -α W (W (e)) -ϱ(y) -J(x, e, w, v)-H 2 (x, e, w, v)+γ 2 W 2 (e)+ϑ V (|ν|),
where ϱ : R ny → R ⩾0 is defined as ϱ(y) := ϱ(|y|), (iii) for all x ∈ R nx , ν ∈ V and e, w ∈ R ny , ϱ • (y; -g(x, e, w, v)) ⩽ L ϱ ϱ(y) + H 2 (x, e, w, v) + J(x, e, w, v) + ϑ ϱ (|ν|).

Function V is an ISS-Lyapunov function used to prove the stability of the closed-loop system H. It is important to note that Condition 2 (and 1) is a property of the closed-loop system H and is independent of the communication network, as e is interpreted as a disturbance acting on the output y in Condition 2. Items (i) and (ii) of Condition 2 imply that system ẋ = f (x, e, w, v) is L 2 -stable from (W, √ ϑ V ) to H. This type of condition is often used in the literature on NCS, see, e.g., [START_REF] Abdelrahim | Robust event-triggered output feedback controllers for nonlinear systems[END_REF], [START_REF] Dolk | Output-based and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF], [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF], where examples of systems satisfying these conditions are provided. Such a robustness assumption is natural as we proceed by emulation; the continuous-time controller (4) must satisfy some robustness property with respect to the perturbations induced by the network. The function ϱ in item (iii) will be involved in the design of the local event triggering conditions. Item (iii) of Condition 2 is a growth condition on ϱ. Similar assumptions are made in the context of periodic event-triggered control [START_REF] Wang | Periodic event-triggered control for nonlinear networked control systems[END_REF], where relevant case studies are provided to illustrate how Conditions 1 and 2 are verified. However, note that the scope, the model and the design are essentially different here.

Given Conditions 1 and 2, we define δ : [0, 1] → R as

δ(λ) := γλ 1 -λL ϱ (21)
and

λ ⋆ := 1, if L ϱ ⩽ -γ, min 1, 1 Lϱ+γ , if L ϱ > -γ. (22)
The next lemma states an important property of δ; its proof is given in the appendix.

Lemma 1. For any λ ∈ (0, λ ⋆ ), δ(λ) ∈ (0, 1) holds.

The following condition is required to ensure that the resulting closed-loop system is IS(p)S. Condition 3. There exist ϵ ∈ (0, 1) and λ ∈ 0, min λ ⋆ , λ such that δ(λ) < ϵ with

λ := 1-ϵ 2 2L W ϵ+γ , if L ϱ ⩾ 0, 1-ϵ 2 2L W ϵ+γ-ϵ 2 Lϱ , if L ϱ < 0. (23)
Note that it is always possible to satisfy Condition 3 by selecting λ sufficiently small, as δ(λ) → 0 when λ → 0.

We now define the maximum allowable event-holding time (MAET) T H : R ⩾0 × R ⩾0 → R ⩾0 . Fix λ, ϵ such that Condition 3 is satisfied. Select any µ ∈ (δ(λ), ϵ/δ(λ)]. The MAET is then given by

T H (λ, µ) :=      1 L W r arctan(θ), γ > L W , 1 L W µ-δ(λ) (δ(λ)+1)(µ+1) , γ = L W , 1 L W r arctanh(θ), γ < L W , (24)
where

θ := r(µ -δ(λ)) γ L W (1 + µδ(λ)) + µ + δ(λ) , r := γ L W 2 -1 ,
and L W , L δ and γ come from Conditions 1 and 2.

We are now ready to state the main result of this paper. Its proof is omitted for space reasons.

Theorem 1. Consider the hybrid system H and suppose Conditions 1-3 hold. We define for all ξ ∈ X and ν ∈ V, Γ : R ny × R ny → R as

Γ( y, e) := (1 + ζ)γα W | e| 2 -λδ(λ)ϱ 1 1+κ | y| -d, (25)

VII. NUMERICAL CASE STUDY

We consider the controlled Lorenz model of fluid convection in [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback controllers[END_REF] affected by measurement noise, ẋ1 = -ax 1 + ax 2 , ẋ2 = bx 1 -x 2 -x 1 x 3 +u, ẋ3 = x 1 x 2 -cx 3 and y = x 1 , where a, b, c > 0 are related to some physical constants. The static output feedback controller u = -p1 p2 a + b y, where p 1 , p 2 > 0, globally stabilizes the origin. The proposition below ensures the satisfaction of Conditions 1 and 2 for this system.

Proposition 2. Let a > 1, p 1 > 2a 2 +1 2a and p 2 > 2a 2 . The controlled Lorenz system satisfies Conditions 1 and 2 with

α W , α W = 1, W (e) = |e|, L W = 0, L ϱ = 0, H(x) := a(|x 1 | + |x 2 |), J = 0, α W (s) = α W s 2 , γ 2 = 2p 2 p1 p2 a + b 2 + α W and ϱ(s) = 2as 2 .
Proof. We take W (e) = |e|. By definition, α W = α W = 1.

Moreover, for all e ∈ R ny , W • (e; g(x, e, w))

⩽ |a(x 1 - x 2 )| ⩽ a(|x 1 | + |x 2 |) =: H(x)
. By Young's inequality, we find that H 2 (x) ⩽ 2a 2 x 2 1 +2a 2 x 2 2 . We introduce the quadratic Lyapunov function V (x) = p 1 x 2 1 + p 2 x 2 2 + p 2 x 2 3 . We have for any x ∈ R nx and e, w ∈ R ny Let ϱ(|y|) = 2ay 2 such that ⟨∇ϱ(y), -g(x, e, w)⟩

⟨∇V (x), f (x, e, w)⟩ = -2ap 1 x 2 1 -2p 2 x 2 2 -2cp 2 x 2 3 - 2p
⩽ 2a(-2ax 2 1 + 2ax 1 x 2 ) ⩽ 2a(ax 2 1 + ax 2 2 ) ⩽ 2a 2 x 2 1 + 2a 2 x 2 2 = H 2 (x)
. Thus, L ϱ = 0 and J = 0. Hence, we find that For the parameter values we select a = 10, b = 28, c = 8/3 and we set p 1 = a + 1.1, p 2 = 2a 2 + 1, α W = 0.05 and we obtain γ = 147.4331. We select λ = (2γ) -1 and ϵ = 0.6, such that λ < λ < λ ⋆ and Condition 3 is satisfied. We select µ = δ(λ), µ = ϵ/δ(λ), τ H = T H (λ, µ) = 2.7973 • 10 -3 and ζ = κ = 0.01.

⟨∇V (x), f (x, e, w)⟩ ⩽ -(2ap 1 -2a 2 -2a)x 2 1 -(p 2 -2a 2 )x 2 2 -2cp 2 x 2 3 + (γ 2 -α W )e 2 + γ 2 w 2 -ϱ(|y|) -H 2 (x) (26
Figures 2 and3 depict the trajectories and the inter-event times in absence of noise, respectively. The average interevent times are generally significantly above the minimum holding time τ H when no noise is present. We now add noise to the output signal. The noise is generated as a piecewise constant signal, where a new value of w is taken uniformly in the interval [-10 -3 , 10 -3 ] every 10 -6 seconds. In Figure 4, the inter-event times are portrayed for the case where no space-regularization is used, i.e., when d = 0. Observe that Zeno-like behavior is present due to the noise. For systems where the noise is consistently present, i.e., when w(t) does not vanish as t → ∞, the space-regularization parameter d can be used to attain improved "steady-state" behavior in terms of average inter-event times when close to the attractor, as depicted in Figure 5, where we selected d = 2.5 • 10 -4 . By carefully selecting d, the average distance to the attractor in "steady-state" can be maintained, as depicted in Figure 6. Moreover, adding space-regularization does not significantly impact the transient response, thereby not degrading performance.

To highlight the benefits of EHC, we also implement a time-regularized static ETC algorithm for the controlled Lorenz equations, see, e.g., [START_REF] Abdelrahim | Robust event-triggered output feedback controllers for nonlinear systems[END_REF]. We tune the triggering condition in ETC such that the MIET of the ETC algorithm is approximately equal to the MAET τ H , while keeping γ constant. We run 50 simulations of each triggering mechanism using random initial conditions, i.e., for each pair of triggering mechanisms we select some x(0, 0) ∈ [0, 1] 3 according to a uniform distribution. The average inter-event times and number of events for a 20 second simulation can be found in Table I. Clearly, EHC outperforms static ETC with time regularization, which underlines the benefit of EHC with respect to time-regularized static ETC in this example.

VIII. CONCLUSIONS

We presented an output-based event-holding strategy that is robust to measurement noise. The proposed strategy guarantees a lower bound on the inter-event times by design. We obtain ISS for the closed-loop system, if no space regularization is used, however, in presence of measurement noise, it may lead to Zeno-like behavior when close to the attractor. By applying space-regularization, the interevent times may be significantly improved as illustrated in simulations. Due to the system being Zeno-free by design, the space-regularization parameter can be tuned such that the resulting trajectories maintain similar properties in terms of (average) asymptotic closeness to the attractor, while not impacting the performance in the transient response.

We showed these beneficial properties on the controlled Lorenz equations to demonstrate the efficacy of the proposed approach.

  Fig. 1.Networked control setup where the controller and plant are (physically) separated and communicate via a packet-based network.

Fig. 2 .Fig. 3 .

 23 Fig.2. Trajectories of controlled Lorenz equations under the proposed strategy with initial condition x 0 = (-20, -20, 30) and no measurement noise, i.e., w(t) = 0 for all t ∈ R ⩾0 .

2 x 2

 2 p1 p2 a + b (e + w).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Inter-event times for the event-holding control strategy and bounded noise, i.e., |w|∞ ≈ 10 -3 , when d = 0.

) with γ 2 = 2p 2 p1 p2 a + b 2 +

 22 α W , α W > 0 and ϱ(s) = 2as 2 , which satisfies Condition 2 when p 1 > a + 1 and p 2 > 2a 2 .

= 2 . 5 • 10 - 4 Fig. 6 .

 251046 Fig.6. Difference of |x| for the event-holding control strategy and bounded noise, i.e., |w|∞ ≈ 10 -3 , when applying space-regularization with constant d = 0 (in red) or d = 10 -4 (in blue).

TABLE I AVERAGE

 I INTER-EVENT TIMES AND NUMBER OF EVENTS FOR EHC AND ETC OVER 50 SIMULATIONS.

		Average IET Total # events
	EHC	0.01097	1822.78
	ETC	0.00286	6976.92

His work is supported by the ANR under grant HANDY ANR-18-CE40-0010, and the IRP ARS from CNRS.