Martien Duvall

Deffo Ayagou

Gaurav R Joshi

Thi Tuyet

Mai Tran

Eliane Sutter

Bernard Tribollet

Christophe Mendibide

Claude Duret-Thual

Nicolas Ferrando

Jean Kittel
email: jean.kittel@ifpen.fr.

Corrosion and Hydrogen Permeation in H2S Environments with O2 Contamination, Part 2: Impact of H2S Partial Pressure

Keywords: hydrogen-induced cracking, hydrogen permeation, hydrogen sulfide, sulfide stress cracking

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

INTRODUCTION

T he extent of the degradation of carbon and low-alloy steel components during upstream production varies across a plethora of environmental and operating conditions. Factors such as acid gas concentration, solution chemistry, fluid flow regimes, and oil/water cut influence engineering decisions that range from materials selection to inhibitor dosing. Ferritic materials exposed to environments containing hydrogen sulfide (H 2 S) can suffer hydrogen-induced cracking, sulfide stress cracking (SSC), or stress-oriented hydrogen-induced cracking. This is correctly attributed to the corrosive, and H-entry promoting nature, of dissolved H 2 S.

ISO 15156 guides the selection of carbon and low-alloy steels for use in existing or potential sour operating conditions. [START_REF]Petroleum and Natural Gas Industries-Materials for Use in H 2 S-Containing Environments in Oil and Gas Production -Part 2: Cracking Resistant Carbon and Low Alloy Steels, and the Use of Cast Iron[END_REF] Often, a pH(y)-P H 2 S (x) diagram (see ISO 15156-2) is used as a frame of reference during materials testing; the diagram plots solution acidity (pH) vs. H 2 S partial pressures (kPa) across orders of magnitude and suggests relative "levels of severity." Concerning materials qualification, standards such as NACE TM0177 and TM0284 stipulate materials testing criteria. A critical one is the effective deaeration of H 2 S-containing test solutions and maintaining low dissolved concentrations of oxygen during the testing period (typically 30 d). [START_REF] Nace Tm | Evaluation of Pipeline and Pressure Vessel Steels for Resistance to Hydrogen-Induced Cracking[END_REF][START_REF]Laboratory Testing of Metals for Resistance to Sulfide Stress Cracking and Stress Corrosion Cracking in H 2 S Environments[END_REF] Standards mention that, "O 2 contamination may induce an increase of the corrosion rate and reduce hydrogen evolution and hydrogen entry into the steel," although "systematic studies of the parameters affecting these phenomena have not been reported in the literature." Ambient oxygen has the potential to pollute a test medium through incorrect or insufficient deaeration of the initial test solution, a poor sealing of test reactors, the use of plastic tubing permeable to oxygen, and a multitude of operations during the test (e.g., solution sampling, pH adjustment). From an industrial point of view, oxygen ingress into sour systems through compressors or vapor recovery units is well recognized. [START_REF] Boivin | Sulfur Corrosion due to Oxygen Ingress[END_REF] In the first paper of this series, it was shown how traces of oxygen exacerbate the corrosion of, but diminish the hydrogen permeation across, iron membranes exposed to 35 g/L NaCl saturated with H 2 S at 24°C and atmospheric pressure (partial pressure of hydrogen sulfide gas = 100 kPa) as a function of time over 3 to 4 weeks. 5 In this second paper, with the pH-P H 2 S diagram of ISO 15156 in mind, the study parameters are extended to lower applied P H 2 S , i.e., how the same O 2 contamination influences phenomena in solutions containing lower dissolved H 2 S. As done in the first paper, only model systems are considered here, with pure iron as the test material, the use of unbuffered test solutions, and H 2 S or H 2 S/N 2 mixtures used to control the acid gas concentration (no CO 2 ). As before, this study follows the evolution of the solution chemistries, and the effect on Fe corrosion and hydrogen uptake, as a function of time. Material surface characterization through x-ray diffraction (XRD) further enables the understanding of phenomena that take place in such media.

EXPERIMENTAL PROCEDURES

The relevant preparation and experimental procedures are detailed in Part 1. 5 Pure H 2 S (>99.5% H 2 S) and H 2 S/N 2 gas cylinders (10% H 2 S/balance N 2 ), (1% H 2 S/balance N 2 ), and (0.1% H 2 S/balance N 2 ) were used to establish the different partial pressures of 10 kPa, 1 kPa, and 0.1 kPa, respectively, in the tests. The partial pressure of water vapor at 24°C being 3 kPa, a strict calculation of H 2 S partial pressures leads to a 4% decrease in the reported values. For the sake of simplicity, rounded values were kept for P H 2 S mentioned in the text. In order to determine the impact of O 2 contamination, a number of tests were conducted with pure iron exposed to 35 g/L NaCl solution at various H 2 S partial pressures (P H 2 S ) between 100 kPa and 0.1 kPa. Several types of measurements were performed:

(1) continuous in situ pH measurement and periodic sampling of test solution for chemical analysis; (2) weight-loss and surface analysis of small coupons; (3) electrochemical impedance spectroscopy (EIS) at the H-entry face of the membrane; and (4) hydrogen permeation at the H-exit face of the membrane.

RESULTS AND DISCUSSION

| Impact of Oxygen on Corrosive Medium

The evolution of test solution pH with or without oxygen contamination are shown in Figure 1. The tests at 100 kPa were repeated four times, and the deviation of pH values was always less than ±0.1 (Figure 1[a]). The tests at lower P H 2 S were performed only once, hence no error bars are plotted in the corresponding graphs. Note that as the balance gas is nitrogen, the bulk solution pH is higher at lower P H 2 S .

For exposure times of a few hours, no difference in pH is measured between the test with O In order to measure the O 2 -H 2 S reaction chemistry, which is responsible for solution acidification, 5 periodic sampling of the test solution was performed for the experiments at 100 kPa H 2 S and 10 kPa H 2 S. Analysis by ionic chromatography shows that sulfate (SO 2- 4 ) is the main species formed. Its concentration increased linearly with time in presence of O 2 contamination, while it stayed at minute levels in reference conditions without O 2 (Figure 2). According to the literature, the reaction paths leading to sulfate involves intermediate steps with thiosulfate formation accompanied by the release of H + : [START_REF] Crolet | The Role of Trace Amounts of Oxygen on the Corrosivity of H 2 S Media[END_REF][START_REF] Song | [END_REF]

2H 2 S þ 2O 2 → S 2 O 2- 3 þ H 2 O þ 2H þ (1)
The more stable sulfate species, and elemental sulfur, may then be formed by various paths, e.g., by direct oxidation:

S 2 O 2- 3 þ 1=2O 2 → SO 2- 4 þ S (2)
Thiosulfate is also recognized as a candidate for direct reduction, either as only cathodic reactant, or in combination with atomic hydrogen already present at the surface of the electrode: 8

S 2 O 2- 3 þ 2e -→ S 2-þ SO 2- 3 (3) S 2 O 2- 3 þ H ads þ e -→ S 2-þ SO 2- 3 þ H þ (4)
In this latter case, thiosulfate reduction plays a detrimental role in hydrogen diffusion into the metal, as it reacts with the intermediate species H ads .

Such results of pH change could be indicative of O 2 contamination during SSC qualification tests. Indeed, NACE TM0177 states that "oxygen contamination is evident by a cloudy (opaque) appearance of the test solution when the H 2 S gas enters the test vessel." As soon as steel is present, darkening of the test solution occurs with iron sulfide precipitation, masking any transparency changes that take place in the early stages. The present authors suggest that in situ pH monitoring or chemical detection of sulfates might thus represent more robust methods to detect oxygen contamination in such systems.

| Weight-Loss Corrosion and Surface Analysis

The weight-loss corrosion rates of pure iron (average of two coupons) exposed to solutions at various P H 2 S , with and without O 2 contamination, are compared in Table 2. Four replicates were performed under 100 kPa H 2 S, from which standard deviations were determined. All other tests were performed only once. This table also includes corrosion rates obtained from electrochemical measurements at the entry face of the membrane. The increasing discrepancy between weight-loss and electrochemical measurements, with decreasing P H 2 S , will be discussed in the electrochemical section of this article.

Corrosion rates in reference conditions without O 2 decrease monotonically as P H 2 S decreases. The corrosion rate drops from 0.5 mm/y (100 kPa H 2 S) to 0.1 mm/y at 0.1 kPa H 2 S. This decrease of corrosion rate correlates with the measured increase in pH. This is expected, because dissociated H + aq is one of the cathodic reactants. 9 Indeed, the concentration of dissolved H 2 S is expected to decrease from 1 × 10 -1 M to 1 × 10 -4 M from 100 kPa to 0.1 kPa, which also is recognized as a relevant oxidizing agent that contributes to iron corrosion. 10 It is expected that the water reaction also becomes a relevant contributing cathodic reaction at the highest pH/lowest P H 2 S test solutions.

In the presence of O 2 contamination, a systematic aggravation of corrosion is seen, compared to reference condition without O 2 . At P H 2 S of 100 kPa and 10 kPa, corrosion rates in the presence of oxygen were doubled. Curiously, however, the amplitude of this aggravation increases by more than an order of magnitude (>10×) as P H 2 S decreases even further toward the lowest dissolved H 2 S concentrations and highest pH. To discuss that the measured increase is not a result of "classical" oxygen cathodic corrosion, the concentrations of the dissolved molarities are considered in bulk solution. For the 100 kPa and 10 KPa systems, the [H 2 S] aq are 0.1 M and 0.01 M, respectively (vs. 2 × 10 -5 M of [O 2 ] aq ), i.e., in massive excess. In 1 kPa and 0.1 kPa, the [H 2 S] aq are 0.001 M and 0.0001 M, respectively. The ratios of [H 2 S] aq :[O 2 ] aq , which remain roughly the same throughout the testing period due to a continuous bubbling of both gases, decreased (from 5,000× to 5×), but corrosion rates increased by an order of magnitude for the two lower H 2 S partial pressures. This peculiarity of massively high corrosion rate is not explicable by [O 2 ] aq as a cathodic reactant alone as it is simply not present in high enough concentration to support a high cathodic reduction current (of the order of 100 μA/cm 2 ). The bulk solution acidification effect also contributes, but it is insufficient to cause more than a 10-fold increase in corrosion rates at the lowest P H 2 S .

In order to understand this curiousity, reference to surface characterization data is helpful. Figure 3 presents typical XRD patterns taken from Fe coupon surfaces that were exposed to solutions under high partial pressure (100 kPa) and low partial pressure (1 kPa) H 2 S, with and without oxygen contamination. For the 100 kPa case, sample surfaces were comprised exclusively of iron sulfides, in particular mackinawite; under 500 ppb oxygen contamination, additional signatures of mixture of pyrrhotite/pyrite/troillite were observed (it is difficult to distinguish between these as they share many diffraction peaks). At lower P H 2 S of 10 kPa, iron sulfides, predominantly mackinawite, were detected. However, under oxygen contamination, one also can clearly discern the presence of magnetite and elemental sulfur (S 8 ) in addition to the iron sulfides (including greigite). Greigite is an iron sulfide mineral that is considered to form in the presence of traces of oxygen. 11 This is quite significant as the surface products of elemental sulfur and magnetite are never detected at higher partial pressures of H 2 S. Indeed, elemental sulfur corrosion deposits are known to enormously accelerate the corrosion of iron, through galvanic coupling effects. [START_REF] Boivin | Sulfur Corrosion due to Oxygen Ingress[END_REF]12 It is expected to be particularly insidious in field scenarios where an iron sulfide overlayer is exposed to salty brines. [START_REF] Gregg | Corrosion Experiences and Inhibition Practices Managing Wet Sour Salty Gas Pipeline Environments Contaminated with Sulfur Deposits[END_REF] It may well explain why the highest weight-loss corrosion rates are detected at the lowest H 2 S partial pressures. This experimental finding will be further discussed in the electrochemical section of this article.

| Electrochemical Study

In addition to weight-loss coupons, electrochemical measurements were conducted on the iron membrane. Typical EIS measurements of the entry face of the membrane at various times of exposure are presented in Figure 4. In this Figure, the plots labeled (a, c, e, and f) display EIS Nyquist spectra taken from sample surfaces exposed to different P H 2 S solutions without oxygen, whereas the (b, d, f, and h) plots show EIS data taken from surfaces exposed to the same (different) P H 2 S solutions, albeit with oxygen contamination.

Except for the tests conducted at 100 kPa H 2 S without O 2 contamination and at long exposures (Figure 4[a]), all other EIS Nyquist curves are deemed to be quite comparable with each They present a high-frequency capacitive loop, possibly followed by a second semicircle at low frequency. This lowfrequency semicircle is hardly visible at 1 kPa H 2 S and 0.1 kPa H 2 S. A previous article discusses that the impedance behavior of pure iron exposed to 100 kPa H 2 S is characterized by a two-step anodic reaction with an adsorbed intermediate. 14 The charge transfer resistance from which corrosion rate could be calculated is represented by the diameter of the high-frequency time constant observed in Nyquist plots. This interpretation is extended to evaluate the charge transfer resistance for all test conditions as a function of time. A more complex model, including a porous film contribution, had to be used to analyze EIS data obtained at 100 kPa H 2 S, as explained in details in previous articles. 5,14 From the charge transfer resistances determined from the EIS data above, it is then possible to present the evolution of instantaneous corrosion rates, with the application of the Stern and Geary relationship (B = 40 mV/decade used here) and Faraday's law. The results obtained are illustrated in Figure 5.

As expected, without O 2 contamination, corrosion rates decrease sharply in short exposure periods, and then stabilize for the remaining duration of the experiment. The time window of the initial drop of corrosion rate is comparable to pH changes, suggesting that kinetics of corrosion are driven by test solution acidity and FeS formation/solution saturation. With O 2 contamination, the corrosion rates start to decrease at short exposure times, similarly to the case without O 2 . However, rather than stabilizing, they then increase sharply as pH decreases. Looking closely at the profile of the EIS corrosion rates, it is interesting to note that the corrosion rate does not simply increase steadily from the early stages of exposure, but undergoes a sharp transition that imposes a faster rate of corrosion. The trend signifies sudden and enormously increasing rate of reaction that these authors assign to be intimately related to modifications to the protectivity of the sulfide corrosion product films and the deposition of elemental sulfur, as determined by XRD. Field experience suggests that steel corrosion in the presence of elemental sulfur, iron sulfides, and dissolved chlorides can proceed at enormous rates and lead to rapid failure of equipment through deposit galvanic corrosion. [START_REF] Boivin | Sulfur Corrosion due to Oxygen Ingress[END_REF] Furthermore, reference to the geothermal power industry's research efforts on the deliberate oxidation of H 2 S (to eliminate it) through injecting oxygen is highly relevant here. [START_REF] Weres | Reaction of Hydrogen Sulfide with Oxygen in the Presence of Sulfite[END_REF] The reactions of dissolved hydrogen sulfide and oxygen to form thiosulfates, sulfates, and elemental sulfur are suspected to be "autocatalytic," with the possible involvement of cocatalyst radical polysulfido-ions that likely need to build up in concentration. [START_REF] Weres | Reaction of Hydrogen Sulfide with Oxygen in the Presence of Sulfite[END_REF] The H 2 S-O 2 reactions can be catalyzed at active transition metal surfaces (e.g., Ni, Co, Fe), and potentially at iron oxides and iron sulfides. One reproducible experimental feature observed at all P H 2 S to support the "autocatalytic oxidation of H 2 S leading to S 8 deposition" hypothesis is a steady rising corrosion rate followed by a very sudden rise at a given moment during immersion. An induction period prior to an unusually high reaction rate is characteristic of an autocatalytic reaction. [START_REF] Boivin | Sulfur Corrosion due to Oxygen Ingress[END_REF][START_REF] Weres | Reaction of Hydrogen Sulfide with Oxygen in the Presence of Sulfite[END_REF] As noted in Table 2, a significant deviation between weight-loss corrosion rates and electrochemical values is observed at lower P H 2 S , with the former vastly exceeding the latter. The difference is certainly exacerbated in the case of O 2 contamination, reaching a factor of ×20. Upon reflection, the highest corrosion rates of weight-loss coupons, compared to the entry face of permeation membranes, is believed to relate to the different fluid flows that they experience in the experiment cell over the duration of the test (see Figure 2 in Deffo Ayagou, et al. 14 ). Weight-loss coupons were typically suspended just above the purging gas frit. As a consequence, it is expected that these samples were subjected to a quite unpredictable, turbulent flow if compared against the permeation membrane; the latter is situated at the end of a tube (diameter = 40 mm, length = 60 mm) that connects to the main body of the cell. The permeation membrane is concluded to largely experience a laminar, low flow. Considering the distance between the gas 0 200 400 600 0.0 0.5 frit and the permeation membrane (>100 mm), it is quite likely that only the solution reaction O 2 -H 2 S products (thiosulfates and sulfates), rather than dissolved oxygen, interact with the permeation membrane surface. In contrast, the weight-loss coupons were subjected to a direct gas exposure during the first seconds of gas injection. This is suspected to permit the direct oxidation of H 2 S with dissolved oxygen on these catalytic corrosion coupon surfaces, which explains the presence of elemental sulfur at the surface of the coupons for the tests at 1 kPa and 0.1 kPa H 2 S. In these tests, dissolved H 2 S concentration is sufficiently weak to allow aqueous and gaseous O 2 to reach the coupon surfaces, where it can react with sulfides to form elemental sulfur. As a consequence, the electrochemical measurements that are reported in this paper must be considered representative of test solutions containing O 2 -H 2 S reaction products, rather than a vigorous exposure of oxygen gas to the metal surface. Conversely, a relatively good correlation between both methods of corrosion rate evaluation is generally observed at the higher P H 2 S of 100 kPa and 10 kPa. This is considered to be an effect of the vast excess of [H 2 S] aq with respect to [O 2 ] aq , where the low dissolved oxygen concentration is quickly diminished upon reaction with H 2 S to form thiosulfates and sulfates. The corrosion of iron in the relatively higher P H 2 S media appears to be governed by the effect of the pH (3.5 to 5) and dissolved concentration of H 2 S on cathodic reactions, rather than flow effects. 9 The evolution of open-circuit potential values of the pure iron electrode in different partial pressure H 2 S media is presented in Figure 6, (a) without and (b) with oxygen contamination. Visually, it is clear to see that under oxygen contamination, these values increase considerably more if compared against deaerated conditions. Table 3 lists the change in open-circuit potential between the beginning and after 2 to 4 weeks of sample exposure. As the corrosion rate and open-circuit potential are both increasing, mixed potential theory would suggest that it is the increasing cathodic reaction kinetics that play the important role in corrosion acceleration. The opencircuit potential rise in the oxygen polluted conditions also correlates with the increasing solution acidity (Figure 1). It is suspected that the increase in cathodic corrosion reaction kinetics might arise from increasing H + concentration as well as from Reaction (2) (thiosulfate reduction). Indeed, the role of elemental sulfur cannot be ruled out.

-Z Im (Ω•cm 2 ) -Z Im (Ω•cm 2 ) -Z Im (Ω•cm 2 ) -Z Im (Ω•cm 2 ) -Z Im (Ω•cm 2 ) -Z Im (Ω•cm 2 ) -Z Im (Ω•cm 2 ) -Z Im (Ω•cm 2 ) Z Real -R e (Ω•cm 2 ) Z Real -R e (Ω•cm 2 ) Z Real -R e (Ω•cm 2 ) Z Real -R e (Ω•cm 2 ) Z Real -R e (Ω•cm 2 ) Z Real -R e (Ω•cm 2 ) Z Real -R e (Ω•cm 2 ) Z Real -R e (Ω•cm 2 ) (a) (b) (c) (d) (e) (f) (g) (h)
Figures 7(a) through (h) present the results of the hydrogen permeation tests at different partial pressures of H 2 S, with and without oxygen gas contamination of the solutions. As EIS measurements were performed simultaneously with hydrogen permeation, it is possible to compare the evolution of corrosion current density and hydrogen permeation current density as a function of time through evaluating the hydrogen permeation efficiency (E perm ), defined as the ratio between permeation current and corrosion current. The validity of its usage in such experiments is outlined in Part 1. 5 For the studied conditions, it is observed that average permeation currents diminish with decreasing P H 2 S (i.e., lower [H 2 S] aq and [H + ] aq ). Quite remarkably, and as already observed in the previous article, permeation efficiencies are close to 100% for all tests without O 2 contamination, albeit more scattering appears at 0.1 kPa H 2 S. For this test condition (0.1 kPa H 2 S without O 2 ), it is believed that corrosion current densities were overestimated, due to insufficient influence of the charge transfer resistance in the frequency range of impedance measurements (Figure 4[g]). In such conditions, permeation current density might be a more reliable indication of corrosion current density than EIS analysis. Such results with no O 2 contamination illustrate the strong power of H 2 S in hydrogen entry promotion, with 100% efficiency for as low as 0.1 kPa H 2 S. This behavior is observed in very specific hydrogen permeation systems, provided that the entire corrosion current contributes to hydrogen entry (i.e., negligible film protection, hydrogen entry promoters), that there is no impediment to hydrogen entry (i.e., no filming or competitive adsorption), that no hydrogen diffusion limitation in the bulk of the metal occurs (i.e., thin membrane, low trapping), and that the hydrogen extraction is efficient (i.e., Pd coated). [START_REF] Wach | [END_REF] Conversely, under oxygen contamination, the hydrogen permeation current is always found to be weaker relative to the reference solutions. The previous article 5 discusses the continuous decrease of the hydrogen permeation current for the O 2 -polluted 100 kPa system. Interestingly, this is no longer the case for lower P H 2 S solutions. Rather, a lower permeation current (hydrogen diffusion) is simply maintained throughout the experiment duration. As the hydrogen extraction setup is always identical, and identical membranes are used, this is almost certainly a limitation on hydrogen uptake in the oxygen-polluted systems. The impact of O 2 contamination is better illustrated by E perm . Considering this parameter, similar trends are observed whatever the P H 2 S considered in the study. At short immersion times, when O 2 -H 2 S reactions have not yet had time to modify the system, high permeation efficiencies are obtained, typically between 60% and 100%. Then, as O 2 -H 2 S reaction products modify the test solution and the surface scales, a continuous decrease of the permeation efficiency is seen. This decrease is mainly due to the increase of corrosion rate, which takes place at nearly constant J perm , suggesting the participation of an additional cathodic reaction that drives corrosion without promoting hydrogen entry into the metal. It is expected that this contribution comes from thiosulfate reduction (Reactions [3] and [4]) and the subsequent corrosion acceleration effects of elemental sulfur.

CONCLUSIONS

The effect of traces of oxygen on the corrosion of, and hydrogen permeation across, pure iron was studied in a 35 g/L NaCl test solution saturated with H 2 S partial pressures ranging from 100 kPa to 0.1 kPa, as a function of time over 2 to 4 weeks. O 2 contamination was introduced in a continuous manner in the test gas, at a partial pressure of 1.3 kPa corresponding to 500 mass ppb dissolved O 2 at equilibrium. The following conclusions were drawn from this study: In the third part of this series of papers, similar experiments will be performed in experimental conditions more representative of industrial applications. Pure iron membrane will be replaced by low alloy steel, and tests will be conducted in acetate buffered solutions.

(a) (b) (c) (d) (e) (f) (g) (h)

FIGURE 2 .

 2 FIGURE 2. Time evolution of sulfate concentration for the tests at 100 kPa and 10 kPa H 2 S in model systems.

FIGURE 3 .

 3 FIGURE 3. XRD patterns from pure iron surfaces after long-term exposure to H 2 S-containing 35 g/L NaCl (24°C), with and without 500 ppb oxygen contamination, at (a) P H 2 S = 100 kPa and (b) 1 kPa.

FIGURE 4 .

 4 FIGURE 4. Typical impedance of pure iron in 35 g/L NaCl solution saturated with H 2 S at (a, b) 100 kPa, (c, d) 10 kPa, (e, f) 1 kPa, and (g, h) 0.1 kPa; (a, c, e, g) without and (b, d, f, h) with O 2 contamination.

FIGURE 5 .

 5 FIGURE 5. Time evolution of corrosion rates of pure iron determined from EIS analysis.

FIGURE 6 .

 6 Figures 7(a) through (h) present the results of the hydrogen permeation tests at different partial pressures of H 2 S, with and without oxygen gas contamination of the solutions. As EIS measurements were performed simultaneously with hydrogen permeation, it is possible to compare the evolution of corrosion current density and hydrogen permeation current density as a function of time through evaluating the hydrogen permeation efficiency (E perm ), defined as the ratio between permeation current and corrosion current. The validity of its usage in such experiments is outlined in Part 1.5 For the studied conditions, it is observed that average permeation currents diminish with decreasing P H 2 S (i.e., lower [H 2 S] aq and [H + ] aq ). Quite remarkably, and as already observed in the previous article, permeation efficiencies are close to 100% for all tests without O 2 contamination, albeit more scattering appears at 0.1 kPa H 2 S. For this test condition (0.1 kPa H 2 S without O 2 ), it is believed that corrosion current densities were overestimated, due to insufficient influence of the charge transfer resistance in the frequency range of impedance measurements (Figure4[g]). In such conditions, permeation current density might be a more reliable indication of corrosion current density than EIS analysis. Such results with no O 2 contamination illustrate the strong power of H 2 S in hydrogen entry promotion, with 100% efficiency for as low as 0.1 kPa H 2 S. This behavior is observed in very specific hydrogen permeation systems, provided that the entire corrosion current contributes to hydrogen entry (i.e., negligible film protection, hydrogen entry promoters), that there is no impediment to hydrogen entry (i.e., no filming or competitive adsorption), that no hydrogen diffusion limitation in the bulk of the metal occurs (i.e., thin membrane, low trapping), and that the hydrogen extraction is efficient (i.e., Pd coated).[START_REF] Wach | [END_REF][17]

  2 contamination (red symbols) and the reference test without O 2 (black symbols). Initial pH values are determined by acid gas content (i.e., P H 2 S ), while FeS solubility equilibrium controls the final pH, for the test condition without O 2 . For comparison, saturated pH values calculated with Cormed™ † are reported in Table 1, and compared with

								5.5
		4.5
								5.0
	pH	4.0				100 kPa H 2 S 100 kPa H 2 S + O 2	pH
		3.5						4.5
		3.0						4.0
		0	120	240	360	480	600	720	0	120	240	360	480
					Time (h)						Time (h)

Table 1 .

 1 Comparison Between Experimental pH at the Beginning of the Tests and Calculated pH at FeS Saturation in 35 g/L NaCl Solution at 24°C

	P H 2 S (kPa)	100	10	1	0.1
	Exp. pH at 24 h without O 2	4.3±0.1 5.0	5.7	6.8
	Exp. pH at 24 h with O 2	4.2±0.1 5.0	5.5	6.7
	Calculated pH at FeS saturation	4.3	5.0	5.6	6.3

Table 2 .

 2 Impact of O 2 Contamination on Corrosion Rate of Pure Iron Weight-Loss Coupons Exposed to 35 g/L NaCl Solution Saturated with H 2 S at Various Partial Pressures for 2 to 4 Weeks

							Pyrrhotite + Troilite + Pyrite
							Mackinawite
							α-ferrite
		Corrosion Rates	Corrosion Rates
		Without O 2 (μm/y)	with O 2 (μm/y)
	P H 2 S	Weight-Loss EIS	Weight-Loss EIS	100 kPa H 2 S + O 2
	100 kPa	489±68	490	976±248	900
	10 kPa	284	230	523	330
	1 kPa	163	53	2,257	96
	0.1 kPa	101	4	1,111	44
						100 kPa H 2 S
					10	20	30	40	50	60

Table 3 .

 3 Impact of O 2 Contamination on Open-Circuit

	Potential Value Change (from start) of Pure Iron Exposed to
	35 g/L NaCl Solution Saturated with H 2 S at Various Partial
	Pressures for 2 to 4 Weeks
	P H 2 S	100 kPa	10 kPa	1 kPa	0.1 kPa
	H 2 S	+11±5 mV	+17 mV	+21 mV	+9 mV
	H 2 S + O 2	+40±15 mV	+85 mV	+49 mV	+43 mV

µA/cm 2 ) J (µA/cm 2 ) J (µA/cm 2 ) J (µA/cm 2 ) J (µA/cm 2 ) J (µA/cm 2 ) J (µA/cm 2 ) J (µA/cm 2

  Comparisons between J corr and J perm , and evolution of the permeation efficiency, as a function of time. Tests at (a, b) 100 kPa, (c, d) 10 kPa, (e, f) 1 kPa, and (g, h) 0.1 kPa, (a, c, e, g) without and (b, d, f, h) with O 2 contamination.➣ With H 2 S partial pressure from 100 kPa to 0.1 kPa, and in the resulting pH range of 4 to 6, O 2 contamination modifies the test solution chemistry by forming thiosulfates, and the stable sulfate species. These reactions appear to induce a test solution acidification that would extend to over 1 pH unit over 720 h. ➣ O 2 contamination significantly enhances the corrosion of iron in all cases, compared to reference solution without O 2 . At low P H 2 S , as the iron surface is placed close to the source of O 2 entry, the H 2 S-O 2 oxidation reactions take place at the metal/metal sulfide surface, depositing elemental sulfur, and accelerating corrosion rates by an order of magnitude (20×) in the process. When the metal surface is sufficiently far to be exposed only to H 2 S-O 2 reaction products, the factor drops to approximately 2× on average for the time scale of these experiments. ➣ Without O 2 contamination, mackinawite is generally the principal iron sulfide species detected. With O 2 contamination at high P H 2 S , surfaces are revealed to consist of additional iron sulfide (pyrrohtite, troillite) signatures; at lower P H 2 S , clear diffraction peaks for greigite, magnetite, and elemental sulfur are discerned. ➣ Without O 2 contamination, hydrogen permeation density is found to be in the same range as the corrosion current density, irrespective of the P H 2 S , implying a 100% permeation efficiency. This is a clear illustration of the strong hydrogen entry promotion by H 2 S. ➣ Conversely, with O 2 contamination, hydrogen permeation efficiency drops from 60% to 100% initially, to as low as 5% after 2 to 4 weeks. This drop is mainly due to the continuous increase of corrosion rate, indicating that H 2 S-O 2 reactions promote the onset and participation of additional cathodic corrosion reactions.

	J perm
	J corr
	Time (h) J perm J corr E perm J () Time (h) E perm J perm J corr E perm J perm J corr E perm E perm (%) E perm (%) E perm (%)	J perm J corr E perm	J perm J corr E perm	E perm (%) E perm (%)	Time (h) J perm J corr E perm J perm J corr E perm E perm (%)	E perm (%) E perm (%)
	Time (h)			Time (h)
	FIGURE 7.

APRIL 2019 • Vol. 75 • Issue 4 397

ACKNOWLEDGMENTS

The authors express their gratitude to Alexandre Bonneau for his active participation in the implementation of the experimental setup and Remy Mingant, for EIS analysis. The contribution of IFPEN analysis department in the surface characterization and solution analysis is also greatly acknowledged.


Jian Wang

Denis Efimov

S Aranovskiy

A A Bobtsov

Fixed-time estimation of parameters for non-persistent excitation

The problem of estimation in the linear regression model is studied under the hypothesis that the regressor may be excited on a limited initial interval of time only. Then the estimation solution is searched on a finite interval of time also based on the framework of finite-time or fixed-time converging dynamical systems. The robustness issue is analyzed and a short-time input-to-state stability property is introduced for fixed-time converging time-varying systems with a sufficient condition, which is formulated with the use of a Lyapunov function. Several estimation algorithms are proposed and compared with existing solutions. The performance of the estimators is demonstrated in numerical experiments.

I. INTRODUCTION

One of the basic and the most popular problems in the theory of identification and estimation is the parameter estimation in a linear regression model:

y(t) = ω (t)θ + w(t), t ∈ R,
where θ ∈ R n is the vector of unknown constant parameters that is necessary to find, ω : R → R n is the regressor function (usually assumed to be bounded and known), y(t) ∈ R is the signal available for measurements with a measurement noise w : R → R (here R denotes the set of real numbers). There are plenty of methods to solve this problem that need a complete statistics on the process (in other words these tools are mainly oriented on estimation off-line): the linear least squares, the maximum-likelihood estimation, the Bayesian linear regression, the principal component regression [START_REF] Yan | Linear Regression Analysis: Theory and Computing[END_REF], [START_REF] Rencher | ser. Wiley Series in Probability and Statistics[END_REF], to mention a few. In the theory of adaptive control and identification there exist also many methods for adaptive and on-line estimation [START_REF] Ioannou | Adaptive Systems with Reduced Models[END_REF], [START_REF] Ljung | System Identification: Theory for the User[END_REF], [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF], and applicability of many of them is based on the condition of persistence of excitation [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF], [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF], thus these approaches are also implicitly based on the asymptotic statistics. Recently, several concepts have been proposed to relax the requirement on the excitation [START_REF] Praly | Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases[END_REF], [START_REF] Barabanov | On global asymptotic stability of ẋ = -φ(t)φ (t)x with φ(t) bounded and not persistently exciting[END_REF], [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF], with improved estimation algorithms [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF], [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF], [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF], [START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF], which require only an interval estimation of the regressor ω(t).

Considering convergence on a finite interval, the amplitude of the initial error becomes of great importance, since if this deviation is not bounded, then it is complicated to ensure global convergence of the estimates to their ideal values in a limited time. A notion that overcomes this drawback has been proposed recently, and it is called fixed-time or predefined-time stability or convergence [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]. The objective of the present work is to design several fixed-time convergent algorithms solving the parameter estimation problem in a linear regression model, which are independent in the initial guesses for the values of parameters and regressor excitation 1 , and whose robustness against the measurement noise is assessed using the input-to-state stability theory. To this end, a new notion of fixed-time input-to-state stability on a short interval of time is proposed together with a Lyapunov function analysis. The introduced approach is based on the dynamic regressor extension and mixing (DREM) method [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF], [START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF], which allow the vector estimation problem to be decoupled on a series of scalar ones. The obtained solutions are compared with ones from [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF], [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF], [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF].

The outline of this note is as follows. Some preliminary results are introduced in section II. The problem statement is given in Section III. The estimation algorithms are designed in Section IV, where also the convergence and robustness conditions are established. Simple illustrating examples are considered in Section V.

Notation

• R+ = {x ∈ R : x ≥ 0}, where R is the set of real number.

• |x| denotes the absolute value for x ∈ R or a vector norm for x ∈ R n , and the corresponding induced matrix norm for a matrix A ∈ R n×n is denoted by A .

• For a Lebesgue measurable and essentially bounded function x : R → R n denote x ∞ = ess sup t∈R |x(t)|, and define by L∞(R, R n ) the set of all such functions with finite norms

• ∞; if +∞ -∞ |x(t)| 2 dt < +∞
then this class of functions is denoted by L2(R, R n ). • The identity matrix of dimension n × n is denoted as In.

• A sequence of integers 1, 2, ..., n is denoted by 1, n.

• Define e = exp(1).

• Define the Lambert function Lambert : R → R, also called the omega function or product logarithm, as the branches of the inverse relation of the function f (z) = ze z for z ∈ R.

• Denote s α = |s| α sign(s) for any s ∈ R and α ∈ R+.

II. PRELIMINARIES

Consider a time-dependent differential equation:

dx(t)/dt = f (t, x(t), d(t)), t ≥ t0, t0 ∈ R, (1)
where x(t) ∈ R n is the state vector, d(t) ∈ R m is the vector of external disturbances and d ∈ L∞(R, R m ); f : R n+m+1 → R n is a continuous function with respect to x, d and piecewise continuous with respect to t, f (t, 0, 0) = 0 for all t ∈ R. A solution of the system (1) for an initial condition x0 ∈ R n at time instant t0 ∈ R and some d ∈ L∞(R, R m ) is denoted as X(t, t0, x0, d), and we assume that f ensures definiteness and uniqueness of solutions X(t, t0, x0, d) in forward time at least on some finite time interval [t0, t0 + T ), where T > 0 may be dependent on the initial condition x0, the input d and the initial time t0.

A. Stability definitions

Let Ω, Ξ be open neighborhoods of the origin in R n , 0 ∈ Ω ⊂ Ξ, then following [START_REF] Ríos | Homogeneous time-varying systems: Robustness analysis[END_REF], [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF] introduce several stability notions:

Definition 1. At the steady state x = 0 the system (1) with d = 0 is said to be

(a) short-time stable with respect to (Ω, Ξ, T 0 , T f ) if for any x0 ∈ Ω and t0 ∈ [-T 0 , T 0 ], X(t, t0, x0, 0) ∈ Ξ for all t ∈ [t0, t0 +T f ];
(b) short-finite-time stable with respect to (Ω, Ξ, T 0 , T f ) if it is short-time stable with respect to (Ω, Ξ, T 0 , T f ) and finite-time converging from Ω with the convergence time T t 0 ,x 0 ≤ t0 + T f for all x0 ∈ Ω and t0 ∈ [-T 0 , T 0 ]; (c) globally short-finite-time stable for T 0 > 0 if for any bounded set Ω ⊂ R n containing the origin there exist a bounded set Ξ ⊂ R n , Ω ⊂ Ξ and T f > 0 such that the system is short-finite-time stable with respect to (Ω, Ξ, T 0 , T f );

(d) short-fixed-time stable for T 0 > 0 and T f > 0, if for any bounded set Ω ⊂ R n containing the origin there exists a bounded set Ξ ⊂ R n , Ω ⊂ Ξ such that the system is short-finite-time stable with respect to (Ω, Ξ, T 0 , T f ).

The notions of Definition 1 can be equivalently formulated using the functions from the classes K and GKL, e.g. the system (1) with d = 0 is globally short-fixed-time stable for T 0 > 0 and T f > 0 at the origin if there exists β ∈ GKL such that for all x0 ∈ R n and t0 ∈ [-T 0 , T 0 ]:

|X(t, t0, x0, 0)| ≤ β(|x0|, t -t0) ∀t ∈ [t0, t0 + T f ]
and β(|x0|, T f ) = 0. In addition, if (1) is short-fixed-time stable, then it is also globally short-finite-time stable. In [START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF], [START_REF] Lebedev | The problem of stability in a finite interval of time[END_REF], [START_REF] Dorato | Short-time stability in linear time-varying systems[END_REF], [START_REF] Weiss | On the stability of systems defined over a finite time interval[END_REF] the short-time stability is considered for a fixed initial time instant t0 only. The notion of short-fixed-time stability has not been defined in [START_REF] Ríos | Homogeneous time-varying systems: Robustness analysis[END_REF], [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF].

Remark 1. In the literature, short-time stability [START_REF] Dorato | Short-time stability in linear time-varying systems[END_REF] is frequently called stability over a finite interval of time [START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF], [START_REF] Lebedev | The problem of stability in a finite interval of time[END_REF], [START_REF] Weiss | On the stability of systems defined over a finite time interval[END_REF], but following [START_REF] Ríos | Homogeneous time-varying systems: Robustness analysis[END_REF], we prefer here the former notion to avoid a confusion with finite-time stability from [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], since both concepts of stability are used in the paper.

B. Robust stability definitions

Consider the following definition of robust stability for (1) with d = 0:

Definition 2.
[14] The system (1) is said to be (a) short-finite-time ISS with respect to (Ω, T 0 , T f , D) if there exist β ∈ GKL and γ ∈ K such that for all x0 ∈ Ω, all d ∈ L∞(R, R m ) with d ∞ < D and t0 ∈ [-T 0 , T 0 ]:

|X(t, t0, x0, d)| ≤ β(|x0|, t -t0) + γ( d ∞) ∀t ∈ [t0, t0 + T f ]
and β(|x0|, T f ) = 0;

(b) globally short-finite-time ISS for T 0 > 0 if there exist β ∈ GKL and γ ∈ K such that for any bounded set Ω ⊂ R n containing the origin there is T f > 0 such that for all x0 ∈ Ω, all d ∈ L∞(R, R m ) and t0 ∈ [-T 0 , T 0 ]:

|X(t, t0, x0, d)| ≤ β(|x0|, t -t0) + γ( d ∞) ∀t ∈ [t0, t0 + T f ]
and β(|x0|, T f ) = 0 (the system is short-finite-time ISS with respect to (Ω, T 0 , T f , +∞));

(c) short-fixed-time ISS for T 0 > 0 and T f > 0, if there exist β ∈ GKL and γ ∈ K such that for all x0 ∈ R n , all d ∈ L∞(R, R m ) and t0 ∈ [-T 0 , T 0 ]:

|X(t, t0, x0, d)| ≤ β(|x0|, t -t0) + γ( d ∞) ∀t ∈ [t0, t0 + T f ]
and β(|x0|, T f ) = 0 (for any bounded set Ω ⊂ R n containing the origin the system is short-finite-time ISS with respect to

(Ω, T 0 , T f , +∞)).

As we can conclude from Definition 2, if the system (1) is short-finite-time ISS, then for d = 0 there exists Ξ ⊂ R n , Ω ⊂ Ξ such that the system is short-finite-time stable with respect to (Ω, Ξ, T 0 , T f ). Again, if (1) is short-fixed-time ISS, then it is also globally short-finite-time ISS. The difference of global short-finite-time or short-fixed-time ISS and a conventional (finite-time or fixed-time) ISS [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] is that in the former case the stability property is considered on a finite interval of time [t0, t0 + T f ] only.

Theorem 1. Let the constants T 0 ≥ 0 and T f > 0 be given. Let the system in (1) possess a Lyapunov function

V : R × R n → R+ such that for all x ∈ R n , d ∈ R m and t ∈ [-T 0 , T 0 + T f ]: α1(|x|) ≤ V (t, x) ≤ α2(|x|) α1, α2 ∈ K∞; (2)
V (t, x) ≤ -u(t) V 1-η (t, x) + V 1+η (t, x) + κ(|d|)
for κ ∈ K, η ∈ (0, 1) and a function u : R → R+ satisfying t+ t u(s)ds ≥ υ > 0, > 0 for all t ∈ [-T 0 , T 0 + T f ] . Then the system (1) is short-fixed-time ISS for T 0 and T f with

β(s, t) = α -1 1 (2 η -1 min{max{0, η α η 2 (s) -η υ 4 (t -)}, 1 η α -η 2 (s) + η υ 4 (t -) }), γ(s) = α -1 1 [2 η -1 ( κ(s) + max{( 4 υ κ(s)) 1 1-η , ( 4 υ κ(s)) 1 1+η })]
provided that

T f ≥ 2(1 + 4 ηυ ) .
Proof. First of all note that for any t ∈ (0, T f ] such that t = ν + µ with an integer ν ≥ 1 and µ ∈ [0, ) we have:

t 0 +t t 0 u(t)dt = ν i=1 t 0 +i t 0 +(i-1) u(t)dt + t 0 +t t 0 +ν u(t)dt ≥ ν i=1 t 0 +i t 0 +(i-1) u(t)dt ≥ νυ = νυ t t = υ ν ν + µ t ≥ υ ν ν + 1 t ≥ υ 2 t. Denote V (t) = V (t, X(t, t0, x0, d))
and define two sets of instants of time:

T1 = {t ∈ [t0, t0 + T f ) : 4 υ κ( d ∞) < V 1-η (t) + V 1+η (t)}, T2 = {t ∈ [t0, t0 + T f ) : 4 υ κ( d ∞) ≥ V 1-η (t) + V 1+η (t)}, then obviously [t0, t0 + T f ) = T1 ∪ T2. Next, consider t ∈ [t1, t2) ⊂ T1, then V (t) ≤ -u(t) - υ 4 V 1-η (t) + V 1+η (t) ≤ -u(t) - υ 4 max{V 1-η (t), V 1+η (t)}
and integrating this inequality we obtain:

V η (t) ≤ min{max{0, V η (t1) -η t t 1 u(s) - υ 4 ds}, 1 V -η (t1) + η t t 1 u(s) -υ 4 ds } for all t ∈ [t1, t2). Thus, V η (t) ≤ min{max{0, V η (t1) -η υ 4 (t -t1)}, 1 V -η (t1) + η υ 4 (t -t1) } for all t ∈ [t1 + , t2). From the inequality V (t) ≤ κ(|d|)
for all t ∈ [t1, t1 + ) we get:

V (t) ≤ V (t1) + κ( d ∞)(t -t1) or V η (t) ≤ V η (t1) + η κ η ( d ∞).
Therefore,

V η (t) ≤ min{max{0, V η (t1) -η υ 4 (t -t1 -)}, 1 V -η (t1) + η υ 4 (t -t1 -) } + η κ η ( d ∞)
for all t ∈ [t1, t2) ⊂ T1, then the estimate that is satisfied for all t ∈ [t0, t0

+ T f ) = T1 ∪ T2 is V η (t) ≤ min{max{0, V η (t0) -η υ 4 (t -t0 -)}, 1 V -η (t0) + η υ 4 (t -t0 -) } + η κ η ( d ∞) + max 4 υ κ( d ∞) η 1-η , 4 υ κ( d ∞) η 1+η
and the system is ISS as needed for the given β and γ. The constraint on T f follows from the estimate on the fixed time of convergence for the case d ∞ = 0. Indeed, V η (t) ≤ 1 for all t ≥ t0 + T1 with T1 = (1 + 4 ηυ ) using the term with the power 1 + η, and starting from the unit value V (t) = 0 for all t ≥ t0 + T1 + T2 with T2 = (1 + 4 ηυ ) using the term with the power 1 -η. Then T f ≤ T1 + T2.

The case without the second term V 1+η has been analyzed in [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF]:

Theorem 2.
[14] Let the constants > 0, D > 0, T 0 , T f ∈ R+ and > 0 be given. Let the system in (1) possess a Lyapunov function

V : R × Ω → R+, where Ω = {x ∈ R n : |x| ≤ }, such that for all x ∈ Ω, all |d| ≤ D and all t ∈ [-T 0 , T 0 + T f ]: α1(|x|) ≤ V (t, x) ≤ α2(|x|) α1, α2 ∈ K∞; V (t, x) ≤ -u(t)V 1-η (t, x) + κ(|d|) κ ∈ K, η ∈ (0, 1)
for a function u : R → R+ satisfying

t+ t u(s)ds ≥ υ > 0 for all t ∈ [-T 0 , T 0 + T f ] . Then the system (1) is short-finite-time ISS with respect to (Ω , T 0 , T f , D) for Ω = {x ∈ Ω : β(|x|, 0) + γ(D) ≤ }, β(s, t) = α -1 1 3 1-η η 4 α η 2 (s) - η 2 η υ 4 (t -) 1 η , γ(s) = α -1 1 3 1-η η 4 4 υ -1 κ(s) 1 1-η + κ(s)
provided that

T f ≥ [1 + 2 η η 4 υ sup x 0 ∈Ω α η 2 (|x0|)].
In both theorems, the condition imposed on u is a version of PE for a finite interval of time. Finally, let us formulate a useful lemma:

Lemma 1.
[32] Let x, y ∈ R and p > 0, then for any κ1 ∈ (0, 1) there exists κ2 > 0 such that

x x + y p ≥ κ1|x| p+1 -κ2|y| p+1 .
In particular, κ2 = max{1 + κ1,

κ 1 (1-κ 1/p 1
) p }.

C. Dynamic regressor extension and mixing method

Consider the estimation problem:

x(t) = ω (t)θ, t ∈ R, (3)
y(t) = x(t) + w(t),
where x(t) ∈ R is the model output, θ ∈ R n is the vector of unknown constant parameters that is necessary to estimate, ω : R → R n is the regressor function (usually assumed to be bounded and known), y(t) ∈ R is the signal available for measurements with a measurement noise w : R → R. Introduce the following hypothesis:

Assumption 1. Let ω ∈ L∞(R, R n ) and w ∈ L∞(R, R).
As it has been proposed in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF], in order to overcome the limitations imposed by the condition that ω is PE and also to improve the transient performance, the DREM procedure transforms (3) to n new one-dimensional regression models, which allows the decoupled estimates of θi, i = 1, n to be computed under a condition on the regressor ω that differs from the persistent excitation.

For this purpose n -1 linear operators Hj : L∞(R, R) → L∞(R, R) are introduced for j = 1, n -1 (for instance an operator Hj can be chosen as a stable linear time-invariant filter with the transfer function Wj(s) = α j s+β j , where s ∈ C is a complex variable and αj = 0, βj > 0 are selected to filter the noise w in (3); or it can realize the delay operation with the transfer function Wj(s) = e -τ j s for τj > 0). Note that y ∈ L∞(R, R) under Assumption 1, then these operators are applied to the measured output y(t) of (3), and using the superposition principles (the operators Hj are linear) we obtain:

ỹj(t) = Hj(y(t)) = ω j (t)θ + wj(t), j = 1, n -1,
where ỹj(t) ∈ R is the j th operator output, ωj : R → R n is the j th filtered regressor function and wj(t) : R → R is the new j th noise signal, which is composed by the transformation of the noise w(t) by Hj and other exponentially converging components related to the initial conditions of the filters. By construction ωj ∈ L∞(R, R n ) and wj ∈ L∞(R, R) for all j = 1, n -1. Define new vector variables

Ỹ (t) = [y(t) ỹ1(t) . . . ỹn-1(t)] ∈ R n , W (t) = [w(t) w1(t) . . . wn-1(t)] ∈ R n
and a time-varying matrix

M (t) = [ω(t) ω1(t) . . . ωn-1(t)] ∈ R n×n ,
then stacking the original equation ( 3) with the n -1 filtered regressor models we construct an extended regressor system:

Ỹ (t) = M (t)θ + W (t).
For any matrix M (t) ∈ R n×n the following equality is true [START_REF] Gantmacher | The Theory of Matrices[END_REF]:

adj (M (t)) M (t) = det (M (t)) In, even if M (t)
is singular, where adj (M (t)) is the adjugate matrix of M (t) and det (M (t)) is its determinant. Recall that each element of the matrix adj (M (t)),

adj (M (t)) k,s = (-1) k+s M k,s (t)
for all k, s = 1, n, where M k,s (t) is the (k, s) minor of M (t), i.e. it is the determinant of the (n -1) × (n -1) matrix that results from deleting the k th row and the s th column of M (t). Define

Y (t) = adj (M (t)) Ỹ (t), W (t) = adj (M (t)) W (t), φ(t) = det (M (t)) ,
then multiplying from the left the extended regressor system by the adjugate matrix adj (M (t)) we get n scalar regressor models of the form:

Yi(t) = φ(t)θi + Wi(t) (4)
for i = 1, n. Again, by construction Y ∈ L∞(R, R n ), W ∈ L∞(R, R n ) and φ ∈ L∞(R, R).
For the scalar linear regression model [START_REF] Ljung | System Identification: Theory for the User[END_REF] the conventional gradient estimation algorithm takes the form:

θi(t) = γiφ(t) Yi(t) -φ(t) θi(t) , γi > 0 (5)
for all i = 1, n, where now the estimation processes for all components of θ are decoupled, and the adaptation gain γi can be adjusted separately for each element of θ. However, all these estimation algorithms are dependent on the same regressor φ(t) (determinant of

M (t)).
Define the parameter estimation error as e(t) = θ -θ(t), then its dynamics admits the differential equation:

ėi(t) = -γiφ(t) (φ(t)ei(t) + Wi(t)) , i = 1, n (6)
and the following result can be proven for the DREM method: Theorem 3.

[21], [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF] Consider the linear regression system (3) under Assumption 1. Assume that for the selected operators Hj :

L∞(R, R) → L∞(R, R), j = 1, n -1: +∞ t 0 φ 2 (t)dt = +∞ (7)
for any t0 ∈ R, then the estimation algorithm (5) has the following properties:

(A) If W ∞ = 0, then the system (6) is globally asymptotically stable at the origin iff (7) is valid.

(B) For all W ∈ L2(R, R n ) we have e ∈ L∞(R, R n ), in addition, for all t0 ∈ R and ei(t0) ∈ R:

|ei(t)| ≤ e -γ i t t 0 φ 2 (τ )dτ |ei(t0)| + γi 2 1 -e -2γ i t t 0 φ 2 (τ )dτ t t 0 W 2 i (s)ds
for all t ≥ t0 and i = 1, n.

Obviously, if the signal φ(t) is PE, then the error dynamics is ISS with respect to W ∈ L∞(R, R n ) and an exponential convergence rate can be guaranteed [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF], [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF], [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF].

III. PROBLEM STATEMENT

Consider the static linear regression model (3) under Assumption 1, and assume that the DREM method has been applied in order reduce the initial problem of vector estimation to n scalar regressor models in the form [START_REF] Ljung | System Identification: Theory for the User[END_REF]. Note that

Y ∈ L∞(R, R n ), W ∈ L∞(R, R n )
and φ ∈ L∞(R, R) under Assumption 1 and due to properties of the DREM approach. Relaxing the condition [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF], which is also imposed on an infinite interval of time, assume that ω(t) may not admit PE, and that φ(t) is a converging function of time (more precise properties will be formulated separately for the designed algorithms).

It is necessary to propose an algorithm generating an estimate θ(t) ∈ R n of the vector of unknown parameters θ ∈ R n , and for W ∞ = 0 providing the property of short-fixed-time stability (see Definition 1) of the estimation error e(t) = θ -θ(t) dynamics under assumptions 1 for some given T 0 and T f . If W ∞ = 0 then short-fixed-time ISS for T 0 and T f (see Definition 2) has to be guaranteed.

Since by applying DREM method the problem is decoupled on n independent ones, for brevity of notation, we will further omit the index i in (4) by assuming that n = 1:

Y (t) = φ(t)θ + W (t), (8)
then θ ∈ R, Y ∈ L∞(R, R), W ∈ L∞(R, R) and φ ∈ L∞(R, R).

IV. DESIGN OF ESTIMATION ALGORITHMS CONVERGING IN SHORT-FINITE-TIME

Two different solutions to the posed estimation problem are proposed in this section, whose difference consists in the requirements imposed on excitation of φ(t) and on the guaranteed robustness abilities with respect to W (t).

A. Algorithm 1

Consider an adaptive estimation algorithm proposed in [START_REF] Ríos | Time-varying parameter identification algorithms: Finite and fixed-time convergence[END_REF], [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF]:

θ(t) = φ(t){γ1 Y (t) -φ(t) θ(t) 1-α (9) +γ2 Y (t) -φ(t) θ(t) 1+α }
for γ1 > 0, γ2 > 0 and α ∈ [0, 1), with θ(t0) ∈ R, which admits the following properties:

Proposition 1. Let Assumption 1 be satisfied, and for given T 0 > 0 and T f > 0,

t+ t min{|φ(s)| 2-α , |φ(s)| 2+α }ds ≥ υ > 0 ( 10
)
for all t ∈ [-T 0 , T 0 + T f ] and some ∈ 0,

T f 2 . Take min{γ1, γ2} > 2 2+ α 2 αυ T f 2 -1 ,
then the estimation error e(t) = θ -θ(t) dynamics of (9) is short-fixed-time ISS for T 0 and T f .

Proof. The error dynamics for the estimation algorithm (9) can be written as follows:

ė(t) = -φ(t){γ1 φ(t)e(t) + W (t) 1-α +γ2 φ(t)e(t) + W (t) 1+α }.
Consider a Lyapunov function candidate V (e) = 0.5e 2 , whose derivative has an upper estimate for some κ11, κ12 ∈ (0, 1) and κ21, κ22 > 0 coming from Lemma 1:

V (t) = -γ1e(t)φ(t) φ(t)e(t) + W (t) 1-α -γ2e(t)φ(t) φ(t)e(t) + W (t) 1+α ≤ -γ1κ11|e(t)φ(t)| 2-α -γ2κ12|e(t)φ(t)| 2+α +γ1κ21|W (t)| 2-α + γ2κ22|W (t)| 2+α ≤ -u(t) V 1-α 2 (t) + V 1+ α 2 (t) + σ(|W (t)|)
for any e(t) ∈ R and W (t) ∈ R, where

u(t) = min{2 1-α 2 γ1κ11|φ(t)| 2-α , 2 1+ α 2 γ2κ12|φ(t)| 2+α }, σ(s) = γ1κ21s 2-α + γ2κ22s 2+α
is a function from class K∞. Note that

u(t) ≥ 2 1-α 2 min{γ1κ11, γ2κ12} min{|φ(t)| 2-α , |φ(t)| 2+α },
then under the imposed restrictions for φ, the system is short-fixed-time ISS for T 0 and T f due to Theorem 1 provided that the constraint

T f ≥ 2(1 + 2 2+ α 2 α min{γ1κ11, γ2κ12}υ )
is satisfied. The imposed restriction on min{γ1, γ2} guarantees that there exist κ11, κ12 ∈ (0, 1) such that all trajectories converge to the origin faster than

T f if W ∞ = 0.
It is important to clarify the differences between the restrictions imposed on φ(t) in Theorem 3 and Proposition 1 for the case W ∞ = 0.

Note that (7) allows us to establish the fact of asymptotic convergence of the estimation error e(t), but it does not permit to evaluate the rate of convergence. Of course, the condition (7) can be formulated on a finite interval of time:

t 0 +T t 0 φ 2 (t)dt = +∞
that implicitly implies unboundedness of φ(t) contrarily to its admissible convergence in [START_REF] Barabanov | On global asymptotic stability of ẋ = -φ(t)φ (t)x with φ(t) bounded and not persistently exciting[END_REF].

B. Algorithm 2

And, finally, let us introduce a version of the algorithm [START_REF] Praly | Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases[END_REF], which extends the nonlinear paradigm of the former by time-varying powers:

θ(t) = sign(φ(t)){γ1 Y (t) -φ(t) θ(t) β(t) +γ2 Y (t) -φ(t) θ(t) ς+β(t) } (11) γ1 > 0, γ2 > 0, ς > 1, β(t) = |φ(t)| 1 + |φ(t)| ,
where θ(t0) ∈ R. The idea of this design is that the power β(t) is approaching zero together with the regressor φ(t), then the contribution of the regressor in the adaptation rate is proportional to |φ(t)| β(t) , which is strictly separated with zero even for a convergent regressor, the detailed proof of this fact is given below.

Proposition 2. Let Assumption 1 be satisfied, and ϑ ∈ L∞(R, R+) where ϑ(t) = W (t) φ(t) . Then the estimation error e(t) = θ -θ(t) dynamics of (11) is short-finite-time ISS for any T 0 ≥ 0 with the input ϑ.

Let additionally for given T 0 > 0 and T f > 0,

t+ t |φ(s)| ς ds ≥ υ > 0 ( 12
)
for all t ∈ [-T 0 , T 0 + T f ] and some ∈ (0, T f ), and

min{γ1, γ2} > √ 2 1 + φmax + 4 (ς-1)υ (T f -)g(xmin)
,

where φmax = max t∈[-T 0 ,T 0 +T f ] |φ(t)|, g(x) = x x 1+x
and xmin = Lambert(e -1 ), then the estimation error e(t) = θ -θ(t) dynamics of ( 11) is short-fixed-time ISS for T 0 and T f with the input ϑ.

Proof. The error dynamics for the estimation algorithm [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF] can be written as follows:

ė(t) = -sign(φ(t)){γ1 φ(t)e(t) + W (t) β(t) +γ2 φ(t)e(t) + W (t) ς+β(t) }.
Consider a Lyapunov function V (e) = 0.5e 2 and observe that

r a(t) (t) ≥    r a min (t) r(t) ≥ 1 r amax (t) r(t) < 1 ≥ min{1, r amax min }, r a(t) (t) ≤    r amax (t) r(t) ≥ 1 r a min (t) r(t) < 1 ≤ max{1, r amax max }
for any r : R → R+ and a : R → R+ such that rmin = inf t∈R r(t), rmax = sup t∈R r(t), amin = inf t∈R a(t) and amax = sup t∈R a(t)

for some rmin, rmax, amin, amax ∈ R+, then the time derivative of V for the estimation error dynamics admits an upper estimate:

V (t) = -sign(φ(t))e(t){γ1 φ(t)e(t) + W (t) β(t) +γ2 φ(t)e(t) + W (t) ς+β(t) } = -γ1|φ(t)| β(t) e(t) e(t) + ϑ(t) β(t) -γ2|φ(t)| ς+β(t) e(t) e(t) + ϑ(t) ς+β(t) ≤ -γ1κ11|φ(t)| β(t) |e(t)| 1+β(t) -γ2κ12|φ(t)| ς+β(t) |e(t)| 1+ς+β(t) +γ1κ21|φ(t)| β(t) |ϑ(t)| 1+β(t) + γ2κ22|φ(t)| ς+β(t) |ϑ(t)| 1+ς+β(t) ≤ - √ 2 min{γ1κ11, γ2κ12}g(|φ(t)|)[ξ1 (V (t)) +|φ(t)| ς ξ2 (V (t))] + σ( ϑ ∞), where
ξ1(s) =    s 0.5 s ≥ 1 s 0.5+φmax 1+φmax s < 1 , ξ2(s) =      s 1+ς 2 s ≥ 1 s 1+ς+ φmax 1+φmax 2 s < 1 , σ(s) = [γ1κ21 max{1, φ φmax 1+φmax max } +γ2κ22 max{1, φ ς+ φmax 1+φmax max }s ς ] ×    s 1+2φmax 1+φmax s ≥ 1 s s < 1
and κ11, κ12 ∈ (0, 1), κ21, κ22 > 0 are from Lemma 1 (since φ(t) is upper bounded in amplitude due to Assumption 1 such constants exist). Note that g(0) = g(1) = 1 and

∂g(x) ∂x = 1 + ln(x) 1 1 + x g(x) 1 + x , ∂ 2 g(x) ∂x 2 = x 2 ln 2 (x) + x(x + 1) 2 xg(x)(x + 1) 4 , then ∂ 2 g(x)
∂x 2

≥ 0 for x ≥ 0, and the function g(x) has the only minimum xmin ∈ [0, 1] that satisfies the equality:

1 + ln(xmin) 1 1 + xmin = 0.
Solving this equation we obtain xmin = Lambert(e -1 ), then g(xmin) 0.757. Therefore, the estimate for the Lyapunov function can be represented as follows:

V (t) ≤ -[ξ1 (V (t)) + |φ(t)| ς ξ2 (V (t))] + σ( ϑ ∞) ≤ -min{1, |φ(t)| ς }ξ (V (t)) + σ( ϑ ∞),
where = √ 2 min{γ1κ11, γ2κ12}g(xmin) and

ξ(s) =      s 0.5 + s 1+ς 2 s ≥ 1 s 0.5+φmax 1+φmax + s 1+ς+ φmax 1+φmax 2 s < 1 .
By repeating the arguments of theorems 1 and 2, the short-fixed-time and the short-finite-time ISS can be established, respectively (in the latter case there is no restriction on excitation of φ(t) and even the conventional results [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] can be used).

In order to check the restrictions on T f for the case W ∞ = 0, assume that V (t0) > 1, denote by T1, T2 > 0 the instants of time such that V (t0 + T1) = 1 and V (t0 + T1 + T2) = 0, and use the estimate

V (t) ≤ -max{V 0.5+φmax 1+φmax (t), |φ(t)| ς V 1+ς 2 (t)}, then since 0.5+φmax
1+φmax ∈ (0, 1) and 1+ς 2 > 1 we obtain

V ς-1 2 (t) ≤ 1 V -ς-1 2 (t0) + ς-1 2 υ 4 (t -t0 -) for all t ∈ [t0, t0 + T1], and V 0.5 1+φmax (t) ≤ 1 - 0.5 1 + φmax (t -t0 -T1) for all t ∈ [t0 + T1, t0 + T1 + T2].
Hence, the upper bounds on T1 and T2 follow (due to the properties of β(t) the time T2 is independent in the excitation):

T1 ≤ [1 + 8 (ς -1) υ ], T2 ≤ 2 1 + φmax ,
which give by resolving the inequality T1 + T2 ≤ T f the required restriction on γ1 and γ2.

Thus, the idea of the algorithm [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF] consists in the utilization of a nonlinearity such that the function g(|φ(t)|) becomes separated with zero overcoming the absence of excitation in the system. The price for that is the robustness with respect to a noise W with a well-defined ratio W (t) φ(t) . Remark 2. One of the most important features of estimation algorithms, after estimation error convergence in the ideal case, is their robustness with respect to measurement noises. In our case, since the regressor φ(t) may converge to zero, the appearance of W (t) = 0 additionally limit the time of convergence, since it is reasonable to use the output Y (t) for estimation with t ∈ [t0, t0 + T ] only while

|Y (t)| > |W (t)| + ε for some ε > 0. If |Y (t)| ≤ |W (t)| + ε (or |Y (t)| is almost equal to |W (t)
| for a sufficiently small ε), then the measured output mainly contains the measurement noise, and it is ambiguous to ask an algorithm to estimate θ due to a bad ratio between the signal and the noise. In this sense the requirement, that the signal W (t) φ(t) is well-defined, is not much restrictive (roughly speaking it just assumes that the ratio between the useful signal and the noise lies in reasonable limits).

Remark 3. If instead of [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF], for any t0 ∈ [-T 0 , T 0 + T f ] there exist 0 > 0 and υ0 > 0 such that

t 0 + 0 t 0 |φ(s)| ς ds ≥ υ0,
i.e. the regressor is always excited on an initial interval of time [t0, t0 + 0] only, then using the same arguments as in the proof of Proposition 2 it is possible to show that there exists T f > 0 for which the estimation error dynamics of ( 11) is short-fixed-time ISS for T 0 and T f with the input ϑ.

If the regressor φ(t) is just asymptotically converging without crossing zero, then the algorithms ( 9) and ( 11) can be applied for any finite T > 0 and t0 ∈ R.

C. Comparison with other approaches

Two approaches will be considered in the next section for a numeric comparison.

1) Concurrent learning approach:

The concurrent learning (CL) approach [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF], [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF] is based on combination of a conventional gradient algorithm with a static update law, whose application in our case leads to the following estimator:

θ(t) = γ1φ(t)(Y (t) -φ(t) θ(t)) (13)
+γ2 p r=0 φ(tr)(Y (tr) -φ(tr) θ(t)), γ1 > 0, γ2 > 0,
where tr ∈ [t0, t0 + T f ] for all r = 0, . . . , p and p ≥ 0 is the number of stored points (in the above sum and in this subsection it is assumed that if tr ≤ t for r = 0, . . . , h with h < p only, then the sum on the first h items is taken). The part proportional to γ1

is the conventional linear algorithm, the part proportional to γ2 represents an additional static feedback, which allows the requirement on persistence of excitation to be relaxed by keeping in memory p successful measurements. Indeed, considering the dynamics of the estimation error e(t) = θ -θ(t) for ( 13) we obtain: Thus, independently on the excitation properties of φ(t) on the interval t ∈ [-T 0 , T 0 + T f ], if the instants tr, r = 0, . . . , p have been properly selected such that p r=0 φ 2 (tr) > 0 (which is always realizable since φ(t) is assumed to be available for measurement), then the asymptotic convergence of e(t) with an exponential rate to the origin in the absence of W and ISS for W ∞ = 0 are obviously guaranteed. In such a case the term proportional to γ2 introduces a kind of σ modification [START_REF] Ioannou | Adaptive Systems with Reduced Models[END_REF], and γ1 = 0 can be imposed.

ė(t) = -γ1φ 2 (t) + γ2
In our simple scalar context the algorithm (13) can be further modified. Indeed, we do not need to wait for convergence of (13) since for the case with γ1 = 0 the final estimate of (13) can be calculated directly: Obviously, in the noise-free scenario the ideal value is recovered by ( 14) at t = t0 if φ(t0) = 0. In the presence of noise, application of [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF] at t = t0 eliminates dependence on the amplitude of initial error |e(t0)| = |θ -θ(t0)|. The algorithm [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF] needs minimal computational power, and the instants tr can be optimized in order to ensure a better identification of θ.

The drawback of the CL approach is that the result of its application is very sensitive to selection of the instants tr, r = 0, . . . , p and the values of the noise W (tr), which are unknown.

2) Linear fixed-time converging observers: A very elegant estimation solution has been proposed in [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF], following it the linear regression problem can be rewritten as follows:

θ = 0 = γiφ(t)[Y (t) -φ(t)θ -W (t)],
where γi ∈ R with i = 1, 2 are design parameters defined later. Then, estimators can be introduced as usual linear observers (in the form of ( 5)):

θi(t) = γiφ(t)[Y (t) -φ(t) θi(t)],
where θi(t) ∈ R are the conventional estimates of θ for i = 1, 2. Next, for any delay τ > 0 we obtain:

θ -θi(t) = e -γ i t t-τ φ 2 (s)ds (θ -θi(t -τ )) -γi t t-τ
e -γ i t σ φ 2 (s)ds φ(σ)W (σ)dσ for i = 1, 2, and subtracting these equations we get:

e -γ 1 t t-τ φ 2 (s)ds -e -γ 2 t t-τ φ 2 (s)ds θ = θ2(t) -θ1(t) +e -γ 1 t t-τ φ 2 (s)ds θ1(t -τ ) -e -γ 2 t t-τ φ 2 (s)ds θ2(t -τ ) + t t-τ γ1e -γ 1 t σ φ 2 (s)ds -γ2e -γ 2 t σ φ 2 (s)ds φ(σ)W (σ)dσ,
from which the following fixed-time estimator can be designed:

θ(t) = e -γ 1 t t-τ φ 2 (s)ds -e -γ 2 t t-τ φ 2 (s)ds -1 [ θ2(t) -θ1(t) +e -γ 1 t t-τ φ 2 (s)ds θ1(t -τ ) -e -γ 2 t t-τ φ 2 (s)ds θ2(t -τ )], (15)
under the hypothesis that the gains γ1, γ2 and the delay τ are selected such that e -γ 1 t t-τ φ 2 (s)ds = e -γ 2 t t-τ φ 2 (s)ds for all t ∈ [t0 + τ, t0 + T f ] (see [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF] for the details). Obviously, the estimation error e(t) = θ -θ(t) for ( 15) satisfies:

|e(t)| ≤ Kt 0 W ∞ ∀t ∈ [t0 + τ, t0 + T f ],
where

Kt 0 = sup t∈[t 0 +τ,t 0 +T f ] e -γ 1 t t-τ φ 2 (s)ds -e -γ 2 t t-τ φ 2 (s)ds -1 × t t-τ γ1e -γ 1 t σ φ 2 (s)ds -γ2e -γ 2 t σ φ 2 (s)ds |φ(σ)|dσ,
and the upper bound on the error is undefined for t ∈ [t0, t0 + τ ). Therefore, an issue of this approach is its robustness. Indeed, if the regressor signal φ(t) is approaching zero, then the gain e -γ 1 t t-τ φ 2 (s)dse -γ 2 t t-τ φ 2 (s)ds -1

becomes large, while the integral may be continuously growing, which leads to the growth of Kt 0 . Since here the estimation is performed on a bounded interval of time, then some of the gains γ1 and γ2 can be selected to be slightly negative, to improve the accuracy of estimation and noise sensitivity.

V. EXAMPLE Consider a measured scalar signal y(t) that consists of two terms. The first term is a harmonic oscillation, and the second term is a decaying process:

y(t) = θ1 sin(t) + θ2 1 t + 1 + w(t),
where w(t) is a measurement noise, and θ1, θ2 are unknown constant parameters. Since the second term decays, the parameter θ1 can be estimated asymptotically with a standard gradient estimator; however, estimation of θ2 is more challenging. The signal y(t) can be rewritten as [START_REF] Ioannou | Adaptive Systems with Reduced Models[END_REF], where the regressor ω(t) = [sin(t) 1 t+1 ] , obviously, is not persistently exciting. Thus, standard gradient or least-square approaches do not guarantee asymptotic estimation. This section illustrates how θ2 can be estimated by the means of the DREM procedure and the proposed fixed-time parameter estimation algorithms.

First, the DREM procedure is applied to get a scalar linear regression equation for θ2. To this end, choose the delay operator H(u(t)) = u(t -τ d ), where τ d > 0 is the tuning parameter, and define Then the procedure described in subsection II-C yields

ω1(t) = H(ω(t)) = ω(t -τ d ), ỹ1(t) = H(y(t)) = y(t -τ d ).
Y2(t) = φ(t)θ2 + W2,
where Y2, W2, and φ are computed as it has been explained above, and

φ(t) = sin(t) 1 + t -τ d - sin(t -τ d ) 1 + t . It is worth noting that ∞ τ d φ 2 (s)ds = C ≤ ∞,
and the requirement (7) of Theorem 3 is not satisfied; thus, the algorithm (5) cannot be applied. However, since φ(t) decays only asymptotically, there exist and υ such that ( 10) and ( 12) are satisfied on any finite interval of time.

For simulations, the algorithms ( 9), ( 11), ( 14) and ( 15) are tuned as follows. For algorithms ( 9) and ( 11 For the algorithm [START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF] the values τ = 0.5, γ1 = 5 and γ2 = 15 are chosen; note that for this algorithm it is required that γ1 = γ2. To avoid singularities in [START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF], no division is performed when the absolute value of the denominator is less than 10 -4 . All algorithms are initialized with θ2(0) = 0, while the true value is θ2 = -2. For all algorithms no estimation is performed for t ∈ [0, τ d ], τ d = 1 since φ(t) is identically zero on this interval due to the DREM procedure.

Simulation results for the algorithms ( 9), ( 11), ( 14) and ( 15) for the noise-free case w(t) ≡ 0 are given in Fig. 1. In the left part the plot with transients for the nominal values of γ1 and γ2 is given, while in the right part, in order to illustrate the ability for acceleration of convergence, the estimates are shown for the values of γ1 and γ2 multiplied by 4 and τ d = 0.5 (note that in our case, decreasing the value of τ d leads to a weaker excitation of φ(t), see the expression above). Simulation results for the same algorithms for the case when w(t) is a uniformly distributed noise, |w(t)| ≤ 0.2, are given in Fig. 2. The mean values and root-mean-squared deviations computed over the interval t ∈ [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF][START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF] are given in Tab. I.

As expected, all algorithms provide finite-time parameter estimation. In the noise-free scenario, the algebraic algorithms ( 14) and [START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF] provide one-step estimation. Since the algorithm [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF] does not gather new measurements as φ(t) decays, it has the smallest steady-state error deviations but suffers from the largest mean error. The algorithms [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF] and [START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF] show high noise sensitivity, while the algorithm [START_REF] Praly | Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases[END_REF] has both small mean error and small error oscillations, where the trade-off is the largest transient time for similar tuning parameters.

All these results confirm the theoretical findings of this note.

VI. CONCLUSIONS

The problem of estimation in the linear regression model has been considered on a bounded interval of time. Several estimators are designed, which are based on the framework of finite-time or fixed-time converging dynamical systems. In order to analyze the robustness of these estimation algorithms, a short-time fixed-time input-to-state stability property has been introduced for time-varying systems and its sufficient Lyapunov condition is given. The synthesized estimation algorithms are compared with existing solutions in numerical experiments. 9), ( 11), ( 14) and ( 15), estimation error θ 2 -θ2 (t) vs. time, seconds. The case of no measurement noise (to the left with the nominal values of γ 1 and γ 2 , to the right with four times higher values) 9), ( 11), ( 14) and ( 15), estimation error θ 2 -θ2 (t) vs. time, seconds. The case of uniformly distributed measurement noise

p r=0 φ 2

 2 (tr) e(t) -γ1φ(t)W (t) -γ2 p r=0 φ(tr)W (tr).

,

  the noise-free hypothesis: Y (tr) = φ(tr) θ(t), φ(tr) = 0 ∀r = 0, . . . , p, the following simple fixed-time converging algorithm can be proposed:θ(t) = p r=0 φ(tr)Y (tr)whose estimation error e(t) = θ -θ(t) admits an upper bound for all t ∈ [t0, t0 + T f ]:|e(t)| ≤ W ∞ max1≤h≤p h r=0 |φ(tr)| h r=0 φ 2 (tr) .

  ) the parameters γ1 = γ2 = 10, α = 0.25, and ς = 1.25 are chosen. For the algorithm (14), a point φ(tr) is stored if it is sufficiently large and sufficient amount of time has passed: tr = arg inf t≥t r-1 +0.1 {|φ(t)| ≥ 0.1}.

Figure 1 .

 1 Figure 1. Simulation results for the algorithms (9), (11), (14) and (15), estimation error θ 2 -θ2 (t) vs. time, seconds. The case of no measurement noise (to the left with the nominal values of γ 1 and γ 2 , to the right with four times higher values)

Figure 2 .

 2 Figure 2. Simulation results for the algorithms (9), (11), (14) and (15), estimation error θ 2 -θ2 (t) vs. time, seconds. The case of uniformly distributed measurement noise

Table I THE

 I MEAN VALUES (MV) AND ROOT-MEAN-SQUARED DEVIATIONS (RMS) OF ESTIMATION ERRORS FOR UNIFORMLY DISTRIBUTED MEASUREMENT

			NOISE.
		Algorithm 1 Algorithm 2 Algorithm 3 Algorithm 4
	MV•10 2	0.1	4.0	6.1	3.8
	RMS•10 2	0.7	20.6	0.1	23.5

This work was partially supported by 111 project No. D17019 (China) and by the Government of Russian Federation (Grant 08-08).