Amanda M Tadini
email: amandatadini@hotmail.com

Gustavo Nicolodelli

Bruno S Marangoni

Stéphane Mounier

Celia Montes

Débora M B P Milori

Stephane Mounier

Célia R Montes

Evaluation of the roles of metals and humic fractions in the podzolization of soils from the Amazon region using two analytical spectroscopy techniques

Keywords: LIBS, FAAS, Organic matter, Amazon

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

Introduction

The Amazon forest provides important services to humanity and the environment, including high biodiversity, climate regulation, carbon sequestration, and regulation of water and nutrient cycles. Soils play crucial roles in most of the processes that occur on the planet, due to their participation in the main biogeochemical cycles [START_REF] Trumbore | Dinâmica do carbono do solo[END_REF]. These involve pedogenetic processes transforming rocks and sediments, whose characteristics are influenced by combinations of soil formation factors such as the source material, climate, microorganisms, and time [START_REF] Fanning | Soil: morphology, genesis and classification[END_REF].

In the Amazon, Spodosols develop over time from clay soil or sedimentary formations. Organometallic iron (Fe) and aluminum (Al) oxides present in horizons A and E are transported to deeper soil layers, where they accumulate and give rise to horizons rich in organic matter and organometallic species [START_REF] Doupoux | Modelling the genesis of equatorial podzols: age and implications for carbon fluxes[END_REF]. Hence, organic matter in these soils plays a key role in environmental sustainability, since it is related to carbon and nutrient cycling, and is a crucial factor to consider in studies in the fields of global climate change and agronomy. Humic substances (HS) are among the main components of SOM and can be classified, according to their solubility, into humic acid (HA), fulvic acid (FA), and humin (HU) fractions [START_REF] Stevenson | Humus chemistry: genesis, composition and reaction[END_REF].

Determination of the chemical properties of the organic matter, as well as its interaction with metallic species, is essential for understanding the main processes that occur in the soil. The techniques that have been described for this purpose include flame atomic absorption spectrometry (FAAS) and, more recently, laser-induced breakdown spectroscopy (LIBS). The latter is an attractive technique due to its high sensitivity and precision, as well as the requirement for only minimal sample preparation, making it attractive for use in applications that comply with the concepts of Green Chemistry [START_REF] Corrêa | Química Verde: fundamentos e aplicações[END_REF].

This technique is a type of atomic emission spectrometry that uses the generation of plasma by means of high power pulses, resulting in ablation of a small portion of the sample. This material dissociates into ions and excited atoms, emitting a continuum of radiation that enables the measurement of atomic/ionic emission lines and identification of the main elements present in the sample [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), part I: review of basic diagnostics and plasma-particle interactions: still-challenging issues within the analytical plasma community[END_REF].

The main advantages of LIBS are the potential for simultaneous multi-element soil analysis with minimal sample preparation, high processing speed, and low cost of analysis, compared to traditional methods. The LIBS technique has been evaluated as an alternative method for the quantification of contaminants, macro/micronutrients, and carbon (C) in soils [START_REF] Ferreira | Evaluation of laser induced breakdown spectroscopy for multielemental determination in soils under sewage sludge application[END_REF][START_REF] Nicolodelli | Quantification of total carbon in soil using laser-induced breakdown spectroscopy: amethod to correct interference lines[END_REF][START_REF] Nicolodelli | Signal enhancement in collinear double-pulse laser-induced breakdown spectroscopy applied to different soils[END_REF]. It has been used in investigations of texture, pH, and macro-and micronutrients in fertilizers and soil organominerals [START_REF] Villas-Boas | Laser-induced breakdown spectroscopy to determine soil texture: A fast analytical technique[END_REF][START_REF] Marangoni | Phosphorus quantification in fertilizers using laser induced breakdown spectroscopy (LIBS): a methodology of analysis to correct physical matrix effects[END_REF][START_REF] Nicolodelli | Double pulse laser induced breakdown spectroscopy: A potential tool for the analysis of contaminants and macro/micronutrients in organic mineral fertilizers[END_REF]. However, although LIBS has been used for soil analysis, it has not previously been employed to investigate the chemical fractions present in the soil.

The aim of this study was to use the FAAS and LIBS techniques to quantify the metals Fe and Al in Amazonian Spodosols. The metals were quantified in whole soil samples and in the different humic fractions (HA, FA, and HU), in order to understand the role of the SOM in the transport of these metals during the podzolization process.

Materials and Methods

Study area

The soil samples (two Spodosols, denoted P1 and P4) were obtained from a site located in the north of the municipality of Barcelos, near the Demeni River (0°15'18''N; 62°46'36''W) in the middle watershed of the Negro River region in the Amazon plains of Brazil. The regional geology is characterized by sediments of the Içá formation. Sample P1 was from a permanently waterlogged podzol covered by herbaceous meadow of scrubs and grass, locally known as "campina". Sample P4 was from a well-drained podzol under rainforest vegetation, with no influence of groundwater. Further details of the study area can be found in Tadini et al. [START_REF] Tadini | Soil organic matter in podzol horizons of the Amazon region: Humification, recalcitrance, and dating[END_REF].

Preparation of the samples for analysis

The procedures adopted for sampling, preservation, and preparation of the soils followed the recommendations of official methods [START_REF] Boulet | Analyse structurale et cartographie en pédologie: I -Prise en compte de l'organisation bidimensionelle de la couverture pédologique: les études de toposéquences et leurs principaux apports à la connaissance dês sols[END_REF][START_REF] Santos | Manual de descrição e coleta de solo no campo[END_REF][START_REF]Sistema brasileiro de classificação de solos[END_REF]. Extraction and purification of the humic acid (HA), fulvic acid (FA), and humin (HU) fractions followed the procedures proposed by the International Humic Substances Society (IHSS), as used by Rice and MacCarthy [START_REF] Rice | Isolation of humin by liquid-liquid partitioning[END_REF], and Swift [START_REF] Swift | Organic matter characterization[END_REF].

Flame atomic absorption spectrometry (FAAS)

Sample decomposition was performed using 100 mg portions of the soils and the humic fractions (HA, FA, and HU), following the recommendations of the United States Environmental Protection Agency (Method 3052) [START_REF]Method 3052: microwave assisted acid digestion of siliceous and organically based matrices[END_REF]. Quantification of Fe and Al was performed using a PerkinElmer PinAAcle 900T flame atomic absorption spectrometer.

Calibration curves were constructed using metal standard solutions and a blank (water) prepared in 1.0 mol -1 HCl (comparable to the sample preparation).

Laser-induced breakdown spectroscopy (LIBS)

The LIBS system employed a Nd:YAG laser operating at 532 nm in the visible (VIS) region . The VIS pulse had a maximum energy of 180 mJ, a width of 4 ns, and was generated by a Quantel Brilliant Q-Switched Nd:YAG laser coupled to a second harmonic generator module. . An ARYELLE 400-Butterfly system was used to detect and select the wavelengths. The spectrometer was operated in the spectral range 175-330 nm, with resolution of 13-24 pm, and was equipped with an intensified chargecoupled device (ICCD) camera (1024 × 1024 pixels). The beam from the lasers was directed and focused on the sample by means of dichroic mirrors at appropriate wavelength. Two lenses were placed between the sample and the tip of the fiber, for efficient collection of the emitted plasma. The sample support was placed in a microcontrolled x-y stage to enable easy and fast scanning by the laser beam impinging on it.

An eight-channel pulse generator (Model 9618, Quantum Composers) was used to synchronize the delay time between pulse and the detection acquisition during the experiments.

Acquisition of the LIBS spectra was performed using a 532 nm visible laser beam with energy of 30 mJ and accumulation of 5 laser shots. The beam was focused and aligned to hit the sample in overlapping laser shots mode.. The gate width was set at 1.5 μs and the gate time at 1.0μs. Each whole soil and humic fraction sample was submitted to 30 measurements performed in different positions.

The outlier spectra were excluded using a technique called Spectral Angle Mapper (SAM) [START_REF] Keshava | Distance metrics and band selection in hyperspectral processing with application to material identification and spectral libraries[END_REF], involving calculation of a normalized scalar product between the individual spectrum and the average spectrum, returning a value between -1 and 1. The closer the value is to 1, the greater the similarity between the spectra. The imposition of a limit below 1 enables the exclusion of spectra that differ from the average. The use of this procedure resulted in elimination of only around 2% of the spectra, indicative of good stability of the experimental system. Finally, an average spectrum was calculated for each sample.

Transitions identification

Identification of the Fe and Al transitions was performed using pixel correlation [START_REF] Marangoni | Phosphorus quantification in fertilizers using laser induced breakdown spectroscopy (LIBS): a methodology of analysis to correct physical matrix effects[END_REF] and the NIST database [START_REF]NIST Atomic Spectra Database[END_REF]. Briefly, calculation was made of the linear Pearson correlation between the intensity of a specific pixel and the variation of the concentration. This process was repeated for all pixels in the spectrum. The reference elemental concentration value was measured by FAAS. In this way, the pixels with higher correlation values were more likely to be representative of a transition line. This analysis assisted in eliminating saturated transitions and those with interference. As a result, three transition lines were selected for each metal: Al I (237.20 nm), Al II (265.25 nm), Al (281.62 nm), Fe I (248.33 nm), Fe II (273.95 nm), and Fe (302.05 and 302.06 nm). The last Fe transition is a superposition of two Fe lines, so the integrated area of the entire region was used as a single intensity value.

Multivariate calibration

The LIBS intensity was linearly correlated with the Al and Fe concentrations of the samples, resulting in a calibration curve. The LIBS intensity was obtained by means of a multivariate calculation involving the areas for each transition and the area for the background, as shown in Equation 1:

(1)
where A i is the area of each atomic transition, C is the background area, and α i and β are positive parameters fitted to the best linear correlation. The A i values were calculated by summing the intensities of pixels inside a transition. The value of C was obtained as the sum of the intensities in a region with no apparent transitions.

Results and Discussion

Table 1 shows the values for carbon, Fe, and Al in the whole soil samples and in the humic fractions extracted from the Amazonian Spodosols, obtained using FAAS. The Al concentration profile showed an increase with greater depth in the horizons of soils P1 and P4 (Table 1). In the case of Fe, the highest values were found for the subsurface horizons (A and A-E) and the deep Bh horizons: Bh-C 240 (P1) and Bh 370-380 (P4). A possible explanation for this behavior was that in the case of the hydromorphic Spodosol (P1), there was mobilization of suspended organometallic complexes and metal oxides, controlled by the oscillating groundwater level. In flooded soils, anaerobic environments are created, with the organic matter acting as an electron receptor and enabling reduction of the macro-and micronutrients present [START_REF] Camargo | Electrochemical Changes in wetland soils[END_REF][START_REF] Sousa | Solos alagados: reações de redox[END_REF].

Hence, the results suggested that there was greater mobilization of the organometallic complexes and the reduced iron in soil P1, compared to the well-drained Spodosol (P4), with the precipitation of Fe and decreasing mobilization of the organometallic complexes along the profile. Consequently, there was the accumulation of Fe in horizons with greater amounts of organic matter, resulting in spodic Bh horizons in this system.

The results obtained in this study were in agreement with the findings of Ishida [START_REF] Ishida | Caracterização e Gênese de Solos e de Depósito de Caulim Associado[END_REF] and Santos [START_REF] Santos | Estudo da matéria orgânica e composição elementar de solos arenosos de regiões próximas a São Gabriel da Cachoeira no Amazonas[END_REF] for a Latosol-Spodosol system in the Amazon region. It was reported that there was accumulation of Al and Fe in the Bh horizons, with the highest value obtained for Al. This accumulation was suggested to be due to translocation of the metals, especially Al, to the spodic Bh horizons in the form of organic complexes.

In previous work by Tadini et al. [START_REF] Tadini | Soil organic matter in podzol horizons of the Amazon region: Humification, recalcitrance, and dating[END_REF], involving the dating of these Amazonian Spodosols and determination of the humification indices, it was observed that soil P1 presented an accumulation of young organic material with a low degree of humification, which was displaced with depth, due to physical and chemical processes. The highest values for humification of the whole soil and for the humic acid fraction were obtained for the Bh-C horizon, at 240 cm. The data obtained in the present study (Table 1)

showed that this horizon presented the highest concentrations of Fe and Al in the whole soil and the humic fractions (HA, FA, and HU).

The well-drained Spodosol (soil P4) showed the contribution of old and poorly humified organic matter in horizons Bh 170-180 and Bh 370-380. Therefore, the results indicated that the transfer of iron and aluminum to deeper profiles of soil P1 involved the participation of young and less humified organic matter. Transfer of the metals in soil P4 involved older organic matter with a low degree of humification, which could be attributed to the presence of more recalcitrant compounds derived from lignin [START_REF] Tadini | Soil organic matter in podzol horizons of the Amazon region: Humification, recalcitrance, and dating[END_REF].

These results indicated that the three humic fractions were involved in the process of podzolization of the Amazonian Spodosols. The FA fraction played a predominant role in the transport of Al (8.8 to 27.2 g/kg), since this fraction had the highest amount of this metal in its structure, while the HA fraction was responsible for the transport of Fe (0.2 to 2.3 g/kg).

The LIBS technique was also used for analysis of the whole soil samples and the humic fractions extracted from the Amazonian Spodosols. Figure 1 shows typical LIBS spectra obtained for horizon A of the hydromorphic Spodosol (P1). 0,0 0,5 1,0 the organic matter of these Amazonian Spodosols (the fulvic acids) was most important for the transport of Al. The results shown in Figures 2 and3 were in agreement with those presented in Figure 1. Consequently, in the case of the hydromorphic Spodosol (P1), the presence of groundwater outcrops favored the movement of this metal towards the deeper horizons, as observed by the high amounts of this metal in the whole soil (Table 1), compared to the well-drained Spodosol (P4). The results demonstrated that use of the LIBS technique enabled acquisition of important information concerning the humic fractions of the soil organic matter. In particular, it was possible to quantify the contributions of the different humic fractions to the processes of podzolization in these soils. The LIBS technique is faster than conventional techniques and was able to provide important information about key elements present in the humic fractions of organic matter in Amazonian soils.

Conclusions

The results showed that the fulvic acid fraction was associated with high amounts of Al, while the humic acid fraction was more selective towards Fe. The transfer of Fe and Al to deeper profiles involved young and less humified organic matter (soil P1), or older organic matter with a low degree of humification (soil P4). It could therefore be concluded that the three humic fractions (HA, FA, and HU) were involved in the soil podzolization process, with the FA and HA fractions playing predominant roles in the transport of Al and Fe, respectively.

The potential of LIBS as a technique capable of assisting in investigation of the processes of formation of Amazonian Spodosols was also evaluated. The data obtained showed strong correlations with the results of analyses using the reference method (FAAS), for Fe in the humic acid and humin fractions (R = 0.95 and 0.97, respectively), and for Al in the fulvic acid fraction (R = 0.97). This is the first study using LIBS applied to samples of environmental humic fractions. In combination with FAAS, it enabled understanding of the roles of the humic fractions in the transport of nutrients and in the process of formation of Amazonian Spodosols.

Figure 1 .

 1 Figure 1. Typical LIBS spectra for the hydromorphic spodosol (P1 A 0-15 cm): (a) whole soil, (b) humic acids, (c) fulvic acids, and (d) humin.

Figure 2

 2 Figure2shows the graphs for the correlations between the LIBS intensities and the Fe concentrations (g/kg) obtained by FAAS analysis of the whole soil samples and the humic fractions. Very strong Pearson correlations were obtained for the humic acid and humin fractions, with R values of 0.95 and 0.97, respectively (Figures2(a) and 2(b)), while weaker correlations were found for the whole soils and the fulvic acid fractions. This suggested that the former two fractions, especially the humic acids, strongly participated in the transport of Fe to other horizons of the Amazonian Spodosols.

Figure 2 .

 2 Figure 2. Correlations between the LIBS intensities and the Fe concentrations (g/kg) in the Amazonian Spodosol samples: (a) humic acids; (b) humin.

Figure 3

 3 Figure 3 shows the correlation graphs for the LIBS intensity and [Al] (g/kg) determined by FAAS, for the whole soil samples and the fulvic acid fractions. Strong

Figure 3 .

 3 Figure 3. Correlations between the LIBS intensities and the Al concentrations (g/kg) in the Amazonian Spodosol samples: (a) whole soil; (b) fulvic acids.

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Typical LIBS spectra for the hydromorphic spodosol (P1 A 0-15 cm): (a) whole soil, (b) humic acids, (c) fulvic acids, and (d) humin.

Table 1 .

 1 Concentrations of carbon (%) and the metals Fe and Al (g/kg) in the whole soil samples and in the humic fractions (HA, FA, and HU) extracted from the Amazonian Spodosols.

	Samples Metals Area Horizon Depth (cm)	Soils	Carbon (%) HA FA	HU	Soils	Concentration of metals (g/kg) HA FA	HU
		A	0-15	22.5 ± 0.9	45 ± 3	18.2 ± 0.5	52.03 ± 0.08	1.20 ± 0.01	0.4 ± 0.2	0.07 ± 0.01	0.3 ± 0.1
		A-E	15-30	1.01 ± 0.09	51 ± 1	26.9 ± 0.2	24.38 ± 0.01	0.95 ± 0.01	0.3 ± 0.1	*	0.2 ± 0.1
	P1	Bh	40-50	2.1 ± 0.2	56.2 ± 0.2	36.8 ± 0.8	36.67 ± 0.03	0.77 ± 0.01	0.3 ± 0.1	0.12 ± 0.01	0.2 ± 0.1
		Bh-C	240	3.2 ± 0.2	46 ± 2	47.9 ± 0.1	*	6.44 ± 0.01	0.6 ± 0.2	*	0.4 ± 0.1
		C	350	1.2 ± 0.2	51 ± 3	38.7 ± 0.9	5.17 ± 0.00	2.13 ± 0.01	0.2 ± 0.1	0.07 ± 0.01	1.0 ± 0.2
	Fe	A A	0-20 20-30	3.9 ± 0.2 3.5 ± 0.2	45.2 ± 0.3 49.4 ± 0.01	34 ± 3 21.6 ± 0.2	41.43 ± 0.07 *	2.29 ± 0.01 2.28 ± 0.01	1.8 ± 0.5 1.9 ± 0.4	0.2 ± 0.1 *	0.2 ± 0.1 0.2 ± 0.1
		A	30-40	0.89 ± 0.01	50 ± 2	10.8 ± 0.1	5.76 ± 0.02	2.60 ± 0.01	2.3 ± 0.3	*	0.3 ± 0.1
	P4	Bh	110-120	0.42 ± 0.02	50.0 ± 0.1	39.8 ± 0.6	*	1.81 ± 0.01	1.1 ± 0.2	*	0.4 ± 0.1
		Bh	170-180	0.38 ± 0.01	52 ± 6	3.1 ± 0.0	5.33 ± 0.02	1.72 ± 0.01	1.4 ± 0.2	*	0.3 ± 0.1
		Bh	370-380	0.48 ± 0.01	52.8 ± 0.3	32.8 ± 0.2	2.26 ± 0.01	2.49 ± 0.01	1.2 ± 0.6	*	0.4 ± 0.1
		Bh	380-390	0.78 ± 0.01	54.5 ± 0.5	34.5 ± 0.3	*	1.66 ± 0.01	1.3 ± 0.2	*	0.3 ± 0.1
		A	0-15	22.5 ± 0.9	45 ± 3	18.2 ± 0.5	52.03 ± 0.08	43 ± 1	*	16.8 ± 0.2	*
		A-E	15-30	1.01 ± 0.09	51 ± 1	26.9 ± 0.2	24.38 ± 0.01	39 ± 1	*	13.8 ± 0.5	*
	P1	Bh	40-50	2.1 ± 0.2	56.2 ± 0.2	36.8 ± 0.8	36.67 ± 0.03	57 ± 1	*	8.8 ± 0.3	*
		Bh-C	240	3.2 ± 0.2	46 ± 2	47.9 ± 0.1	*	92 ± 1	*	*	*
		C	350	1.2 ± 0.2	51 ± 3	38.7 ± 0.9	5.17 ± 0.00	52 ± 1	*	20.0 ± 0.1	5.0 ± 0.1
	Al	A A	0-20 20-30	3.9 ± 0.2 3.5 ± 0.2	45.2 ± 0.3 49.4 ± 0.01	34 ± 3 21.6 ± 0.2	41.43 ± 0.07 *	48 ± 1 47 ± 1	6.4 ± 0.4 *	9.5 ± 0.1 27.2 ± 0.2	* *
		A	30-40	0.89 ± 0.01	50 ± 2	10.8 ± 0.1	5.76 ± 0.02	46 ± 1	*	17.5 ± 0.1	*
	P4	Bh	110-120	0.42 ± 0.02	50.0 ± 0.1	39.8 ± 0.6	*	39 ± 1	*	13.7 ± 0.4	*
		Bh	170-180	0.38 ± 0.01	52 ± 6	3.1 ± 0.0	5.33 ± 0.02	49 ± 1	*	16.5 ± 0.3	*
		Bh	370-380	0.48 ± 0.01	52.8 ± 0.3	32.8 ± 0.2	2.26 ± 0.01	41 ± 1	*	20.5 ± 0.2	*
		Bh	380-390	0.78 ± 0.01	54.5 ± 0.5	34.5 ± 0.3	*	51 ± 1	*	21 ± 1	*
	P1: hydromorphic spodosol; P4: well drained spodosol; * below the detection limit of the calibration curve (%C = 0.04; Fe: 0.10 mg/kg and Al: 2.5 mg/kg).

Table 1 .

 1 Concentrations of carbon (%) and the metals Fe and Al (g/kg) in the whole soil samples and in the humic fractions (HA, FA, and HU) extracted from the Amazonian Spodosols. P4: well drained spodosol; * below the detection limit of the calibration curve (%C = 0.04; Fe: 0.10 mg/kg and Al: 2.5 mg/kg).

	Samples Metals Area Horizon Depth (cm)	Soils	Carbon (%) HA FA	HU	Soils	Concentration of metals (g/kg) HA FA	HU
		A	0-15	22.5 ± 0.9	45 ± 3	18.2 ± 0.5	52.03 ± 0.08	1.20 ± 0.01	0.4 ± 0.2	0.07 ± 0.01	0.3 ± 0.1
		A-E	15-30	1.01 ± 0.09	51 ± 1	26.9 ± 0.2	24.38 ± 0.01	0.95 ± 0.01	0.3 ± 0.1	*	0.2 ± 0.1
	P1	Bh	40-50	2.1 ± 0.2	56.2 ± 0.2	36.8 ± 0.8	36.67 ± 0.03	0.77 ± 0.01	0.3 ± 0.1	0.12 ± 0.01	0.2 ± 0.1
		Bh-C	240	3.2 ± 0.2	46 ± 2	47.9 ± 0.1	*	6.44 ± 0.01	0.6 ± 0.2	*	0.4 ± 0.1
		C	350	1.2 ± 0.2	51 ± 3	38.7 ± 0.9	5.17 ± 0.00	2.13 ± 0.01	0.2 ± 0.1	0.07 ± 0.01	1.0 ± 0.2
	Fe	A A	0-20 20-30	3.9 ± 0.2 3.5 ± 0.2	45.2 ± 0.3 49.4 ± 0.01	34 ± 3 21.6 ± 0.2	41.43 ± 0.07 *	2.29 ± 0.01 2.28 ± 0.01	1.8 ± 0.5 1.9 ± 0.4	0.2 ± 0.1 *	0.2 ± 0.1 0.2 ± 0.1
		A	30-40	0.89 ± 0.01	50 ± 2	10.8 ± 0.1	5.76 ± 0.02	2.60 ± 0.01	2.3 ± 0.3	*	0.3 ± 0.1
	P4	Bh	110-120	0.42 ± 0.02	50.0 ± 0.1	39.8 ± 0.6	*	1.81 ± 0.01	1.1 ± 0.2	*	0.4 ± 0.1
		Bh	170-180	0.38 ± 0.01	52 ± 6	3.1 ± 0.0	5.33 ± 0.02	1.72 ± 0.01	1.4 ± 0.2	*	0.3 ± 0.1
		Bh	370-380	0.48 ± 0.01	52.8 ± 0.3	32.8 ± 0.2	2.26 ± 0.01	2.49 ± 0.01	1.2 ± 0.6	*	0.4 ± 0.1
		Bh	380-390	0.78 ± 0.01	54.5 ± 0.5	34.5 ± 0.3	*	1.66 ± 0.01	1.3 ± 0.2	*	0.3 ± 0.1
		A	0-15	22.5 ± 0.9	45 ± 3	18.2 ± 0.5	52.03 ± 0.08	43 ± 1	*	16.8 ± 0.2	*
		A-E	15-30	1.01 ± 0.09	51 ± 1	26.9 ± 0.2	24.38 ± 0.01	39 ± 1	*	13.8 ± 0.5	*
	P1	Bh	40-50	2.1 ± 0.2	56.2 ± 0.2	36.8 ± 0.8	36.67 ± 0.03	57 ± 1	*	8.8 ± 0.3	*
		Bh-C	240	3.2 ± 0.2	46 ± 2	47.9 ± 0.1	*	92 ± 1	*	*	*
		C	350	1.2 ± 0.2	51 ± 3	38.7 ± 0.9	5.17 ± 0.00	52 ± 1	*	20.0 ± 0.1	5.0 ± 0.1
	Al	A A	0-20 20-30	3.9 ± 0.2 3.5 ± 0.2	45.2 ± 0.3 49.4 ± 0.01	34 ± 3 21.6 ± 0.2	41.43 ± 0.07 *	48 ± 1 47 ± 1	6.4 ± 0.4 *	9.5 ± 0.1 27.2 ± 0.2	* *
		A	30-40	0.89 ± 0.01	50 ± 2	10.8 ± 0.1	5.76 ± 0.02	46 ± 1	*	17.5 ± 0.1	*
	P4	Bh	110-120	0.42 ± 0.02	50.0 ± 0.1	39.8 ± 0.6	*	39 ± 1	*	13.7 ± 0.4	*
		Bh	170-180	0.38 ± 0.01	52 ± 6	3.1 ± 0.0	5.33 ± 0.02	49 ± 1	*	16.5 ± 0.3	*
		Bh	370-380	0.48 ± 0.01	52.8 ± 0.3	32.8 ± 0.2	2.26 ± 0.01	41 ± 1	*	20.5 ± 0.2	*
		Bh	380-390	0.78 ± 0.01	54.5 ± 0.5	34.5 ± 0.3	*	51 ± 1	*	21 ± 1	*
	P1: hydromorphic spodosol;

Acknowledgments

The authors are grateful for the financial support provided by the São Paulo State Research Foundation (FAPESP, grants 2011/03250-2, 2012/51469-6, 2013/07276-1, and 2013/13013-3) and the Brazilian National Council for Technological and Scientific Development (CNPq, grants 303478/2011-0, 306674/2014-9, 150087/2017-9 and 232225/2014-1-SWE). The authors would like to thank Dr. Célia Regina Montes (ESALQ/ USP -Piracicaba, SP) for having given samples for analysis.


H Ríos

D Efimov

W Perruquetti

An Adaptive Sliding-Mode Observer for a Class of Uncertain Nonlinear Systems

Keywords: Adaptive observer, Sliding-modes, Nonlinear systems

In this paper the problem of simultaneous state and parameter estimation is studied for a class of uncertain nonlinear systems. A nonlinear adaptive sliding-mode observer is proposed based on a nonlinear parameter estimation algorithm. It is shown that such a nonlinear algorithm provides a rate of convergence faster than exponential, i.e. faster than the classic linear algorithm. Then, the proposed parameter estimation algorithm is included in the structure of a sliding-mode state observer providing an ultimate bound for the full estimation error attenuating the effects of the external disturbances. Moreover, the synthesis of the observer is given in terms of linear matrix inequalities. The corresponding proofs of convergence are developed based on Lyapunov function approach and input-to-state stability theory. Some simulation results illustrate the efficiency of the proposed adaptive sliding-mode observer.

INTRODUCTION

Despite a significant advance in development of identification tools for nonlinear systems, the controllers have to cope with a high level of uncertainty, mainly due to lack of knowledge on the system parameters and the fact that the whole system state is not available to measure. This situation justifies the need for the adaptive control design that has received a great deal of attention in control theory during the last decades. Due to this attention, this area has grown to turn into one of the widest in terms of algorithms, techniques for design, analytical tools, and so on (see, for instance [START_REF] Ioannou | Robust Adaptive Control[END_REF] and [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF]). One important problem in the adaptive control area is the design of adaptive observers, i.e. the design of observers estimating simultaneously the whole state and the parameters of the system by some on-line adaptation law [START_REF] Besanc ¸on | Nonlinear Observers and Application[END_REF].

In this context, there exist a lot of literature related to the adaptive observers design for linear systems (see, for instance [START_REF] Sastry | Adaptive Control. Stability, Convergence and Robustness[END_REF], [START_REF] Lüders | An adaptive observer and identifier for linear systems[END_REF], [START_REF] Carroll | An adaptive observer for single input single output linear systems[END_REF], and [START_REF] Narendra | Stable Adaptive Systems[END_REF]). Most of these results are based on appropriated change of state coordinates to some canonical form in order to provide a state estimation together with persistence of excitation conditions to ensure the parameter estimation.

For nonlinear systems, one of the first results was proposed by [START_REF] Bastin | Stable adaptive observers for nonlinear time-varying systems[END_REF] extending some linear results. In the same vein, several results based on output injection transformations are given for nonlinear systems that are equivalent to linear observable systems in the Brunovsky observer form (see, for example [START_REF] Marino | Global adaptive observers for nonlinear systems via filteres transformations[END_REF] and [START_REF] Marino | Nonlinear Control Design: Geometric, Adaptive & Robust[END_REF]). Afterwards, in [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF] a unifying adaptive observer form is proposed for nonlinear systems providing asymptotic state estimation as well as parameter estimation under some passivity-like conditions. For multiple-input multiple-output (MIMO) linear time-varying systems, an adaptive observer is proposed by [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (MIMO) linear time varying systems[END_REF] and it is also valid for affine state nonlinear systems. In [START_REF] Xu | Nonlinear system fault diagnosis based on adaptive estimation[END_REF], a more general design of adaptive observers is proposed for a class of single-output uniformly observable nonlinear systems. For the case of uniformly observable multiple-input multiple-output nonlinear systems, in [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF] an adaptive observer is proposed to exponentially estimate the state and the unknown parameters under a persistent excitation condition. The structure of this observer gives some flexibility to obtain high-gain-like observers and adaptive sliding-mode-like observers. In [START_REF] Liu | Robust adaptive observer for nonlinear systems with unmodeled dynamic[END_REF] a robust adaptive observer is provided for nonlinear systems with disturbances and unmodeled dynamics based on adaptive nonlinear damping. Nevertheless, such an observer is just able to estimate the state. In [START_REF] Stamnes | Redesign of adaptive observers for improved parameter identification in nonlinear systems[END_REF], a redesign of adaptive observers is proposed for nonlinear systems based on adaptive laws that use delayed measurements. These delayed observers improve the performance of the parameter estimation but increase the computational load. More recently, in [START_REF] Tyukin | Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters[END_REF], based on the concepts of weakly attracting sets and non-uniform convergence, an adaptive observer is proposed to asymptotic reconstruction of the state and parameter values in a particular class of forwardcomplete single input-single-output nonlinear systems. Adaptive impulsive observers are proposed in [START_REF] Chen | Adaptive impulsive observers for nonlinear systems: Revisited[END_REF] and [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF] for a class of nonlinear systems. Particularly, in [START_REF] Chen | Adaptive impulsive observers for nonlinear systems: Revisited[END_REF], the proposed observer is modelled by an impulsive differential equation, the asymptotic convergence is analyzed based on a time-varying Lyapunov function approach and some sufficient conditions are formulated in terms of linear matrix inequalities. On the other hand, in [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF], a new impulsive adaptive observer shows that an impulsive feedback can improve the convergence rate or relax the requirement on persistency of excitation. For a class of nonlinear sampled-data systems, in [START_REF] Folin | Sampled-data adaptive observer for a class of state-affine output-injection nonlinear systems[END_REF] a new hybrid adaptive observer is designed and shown to be exponentially convergent under some common conditions. In the same vein, in [START_REF] Ahmed-Ali | Adaptive observer design with heat pde sensor[END_REF] an adaptive observer is proposed for systems with cascade structure including finitedimensional dynamics followed by infinite-dimensional dynamics. The given adaptive observer is based on a combination of backstepping method and the extended Kalman observer approach, and it is shown to be exponentially convergent under a persistent excitation condition. However, most of the works previously mentioned do not consider external disturbances, in addition, the convergence rate of the parameter and state estimation errors is exponential or asymptotic. In addition, it has been shown in [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF] that an improvement of convergence rate of parameter estimation error cannot be achieved by a simple increasing the observer gains. Some other solutions have been proposed in the sliding-mode area where the presented solutions demonstrate a remarkable insensitivity with respect to certain class of external disturbances [START_REF] Shtessel | Sliding Mode Control and Observation[END_REF]. In the context of fault detection, in [START_REF] Yan | Adaptive sliding-mode-observer-based fault reconstruction for nonlineary systems with parametric uncertainties[END_REF] an adaptive sliding-mode observer is provided for a class of nonlinear systems with unknown parameters and faults. Using the inherent features of the slidingmode observers, a fault reconstruction is given under relative degree of the output with respect to the fault equal to one. Finally, it is worth saying that most of the mentioned adaptive observers propose linear parameter estimation algorithms.

Main Contribution

This paper contributes with a nonlinear adaptive sliding-mode observer based on a nonlinear parameter identification algorithm for uncertain nonlinear systems. The proposed nonlinear adaptive sliding-mode observer is a modified version of that one proposed in [START_REF] Efimov | Enhancement of adaptive observer robustness applying sliding mode techniques[END_REF]. Such a modification lies in the inclusion of a nonlinear parameter identification algorithm that provides a rate of convergence faster than exponential, i.e. faster than classic linear algorithms. Then, the proposed parameter identification algorithm is included in the structure of a sliding-mode state observer providing an ultimate bound for the state and parameter estimation error attenuating the effects of the external disturbances. This solution ensures that the level of observer robustness with respect to some matched disturbances is improved as well as the rate of convergence. Moreover, the synthesis of the () observer is given in terms of linear matrix inequalities. The corresponding proofs of convergence are developed based on Lyapunov function approach and input-to-state stability theory. Some simulation results illustrate the efficiency of the proposed nonlinear adaptive sliding-mode observer.

Structure of the paper

The problem statement is presented in the Section 2. The proposed adaptive sliding-mode observer as well as the convergence proof are given in Section 3. The simulation results are illustrated by Section 4. Some concluding remarks are discussed in Section 5. Finally, the proofs of all proposed results are postponed to the Appendix.

Notation

Let q denote the Euclidean norm of a vector q ∈ R n ; 1, n a sequence of integers 1, ..., n; and I n an identity matrix of dimension n × n. For a matrix Q ∈ R m×n , denote its smallest singular value σ min (Q) = λ min (Q T Q) and its induced norm as

Q := λ max (Q T Q) = σ max (Q),
where λ max (respectively, λ min ) is the maximum (respectively, the minimum) eigenvalue, and σ max is the largest singular value. For a Lebesgue measurable function u : R ≥0 → R m define the norm u (t0,t1) := ess sup t∈(t0,t1) u(t) , then u ∞ := u (0,+∞) and the set of functions u with the property u ∞ < +∞ is denoted as L ∞ . For a matrix Q : R ≥0 → R m×n denote Q ∞ := Q (0,+∞) . A continuous function α : R ≥0 → R ≥0 belongs to class K if it is strictly increasing and α(0) = 0; it belongs to class K ∞ if it is also unbounded. A continuous function β : R ≥0 × R ≥0 → R ≥0 belongs to class KL if for each fixed s, β(r, s) ∈ K with respect to r, and for each fixed r, β(r, s) is decreasing to zero with respect to s. The notation ∇V (x)f (x) denote the directional derivative of a continuously differentiable function V with respect to the vector field f evaluated at any point x.

PROBLEM STATEMENT

Consider a class of uncertain nonlinear systems that can be written, essentially after a change of coordinates, as follows

ẋ = Ax + φ(y, u) + G(t, y, u)θ + Dw, (1)
y = Cx, (2)
where x ∈ R n is the state vector, y ∈ R p is the measurable output vector, u ∈ R m is the control input vector, θ ∈ R q is a vector of unknown constant parameters, and w ∈ R l is a vector of external disturbances. The matrices A, C and D are known, they have corresponding dimensions, and the pair (A, C) is detectable. The functions φ : R p × R m → R n and G : R ≥0 × R p × R m → R n×q are also known and they ensure uniqueness and existence of solutions for system (1) for all admissible disturbances. The aim of this paper is to provide estimations of the state and parameter vectors, i.e. x and θ, respectively; only using the information of the output y and attenuating as much as possible the effects of the external disturbances w.

The following assumptions are introduced for the system (1)-( 2).

Assumption 1

The trajectories of the system, the control input, and the external disturbances belong to L ∞ , i.e.

x ∞ < +∞, u ∞ < +∞, and w ∞ < +∞, respectively; and G(t, y(t), u(t)) ∞ < +∞ for all t ≥ 0.

Despite the fact that boundedness is assumed, the disturbance w may have a large magnitude; thus, special attenuation techniques have to be applied to provide an acceptable state estimation for system [START_REF] Ioannou | Robust Adaptive Control[END_REF]. Regarding boundedness of the state x, an exact estimate on the state maximum amplitude is not necessary, and since estimation of all variables is achieved in a finite time, the boundedness is also compulsory on a finite interval only.

()

Preliminaries

Consider the following nonlinear system

ẋ = f (x, w), (3)
where x ∈ R n is the state, w ∈ R l is the external disturbances, and f : R n × R l → R n is a locally Lipschitz function. For an initial condition x 0 ∈ R n and an external disturbance w ∈ L ∞ , denote the solution by x(t, x 0 , w) for any t ≥ 0 for which the solution exists.

The following stability properties for system (3) are introduced (for more details see [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF], [START_REF] Dashkovskiy | Input to state estability and allied system properties[END_REF] and [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]).

Definition 1

The system (3) is said to be Input-to-State practically Stable (ISpS) if for any w ∈ L ∞ and any x 0 ∈ R n there exist some functions β ∈ KL, γ ∈ K and a constant κ ∈ R ≥0 such that

x(t, x 0 , w) ≤ β( x 0 , t) + γ( w ∞ ) + κ, ∀t ≥ 0. The system (3) is said to be Input-to-State Stable (ISS) if κ = 0.

These properties also have a Lyapunov function characterization.

Definition 2 A smooth function V : R n → R ≥0 is said to be an ISpS Lyapunov function for system (3) if for all x ∈ R n and any w ∈ L ∞ there exist some functions

ψ 1 , ψ 2 , ψ 3 ∈ K ∞ , χ ∈ K, and a constant κ ∈ R ≥0 such that ψ 1 ( x ) ≤ V ≤ ψ 2 ( x ), x ≥ χ( w ∞ ) + κ ⇒ ∇V (x)f (x, w) ≤ -ψ 3 ( x ).
The function V is said to be an ISS Lyapunov function for system (3) if κ = 0.

Theorem 1 [START_REF] Dashkovskiy | Input to state estability and allied system properties[END_REF] The system (3) is ISpS (ISS) if and only if it admits an ISpS (ISS) Lyapunov function.

Let us consider the following interconnected nonlinear system

ẋ1 = f 1 (x 1 , x 2 , w), ( 4
) ẋ2 = f 2 (x 1 , x 2 , w), (5)
where

x i ∈ R ni , w ∈ R l , and f i : R n1 × R n2 × R l → R ni
ensures existence of the system solutions at least locally, for i = 1, 2. Suppose that there exist ISpS Lyapunov functions V 1 and V 2 , for (4) and [START_REF] Lüders | An adaptive observer and identifier for linear systems[END_REF], respectively; such that, for all x i ∈ R ni and any w ∈ L ∞ there exist some functions

ψ i1 , ψ i2 , ψ i3 ∈ K ∞ , γ i , χ i ∈ K and some constants κ i ∈ R ≥0 with i = 1, 2, the following hold ψ i1 ( x i ) ≤ V i (x i ) ≤ ψ i2 ( x i ), i = 1, 2, ( 6
) V 1 (x 1 ) ≥ max[χ 1 (V 2 (x 2 )), γ 1 ( w ) + κ 1 ] ⇒ ∇V 1 (x 1 )f 1 (x 1 , x 2 , w) ≤ -ψ 13 (V 1 (x 1 )), (7) V 2 (x 2 ) ≥ max[χ 2 (V 1 (x 1 )), γ 2 ( w ) + κ 2 ] ⇒ ∇V 2 (x 2 )f 2 (x 1 , x 2 , w) ≤ -ψ 23 (V 2 (x 2 )). (8)
Then, the following nonlinear small-gain result is introduced for the interconnected system (4)-( 5) in terms of ISpS Lyapunov functions.

Theorem 2 [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF] Suppose that the interconnected system (4)-( 5) has ISpS Lyapunov functions V 1 and V 2 satisfying the condition (6)- [START_REF] Bastin | Stable adaptive observers for nonlinear time-varying systems[END_REF]. If there exists some constant κ 0 ∈ R ≥0 such that

χ 1 • χ 2 (r) < r, ∀r > κ 0 , (9)
then the interconnected system (4)-( 5) is ISpS. The system (4)-( 5) is

ISS if κ 0 = κ 1 = κ 2 = 0. ()

ADAPTIVE SLIDING-MODE OBSERVER

Let us introduce the following adaptive observer

Ω = (A -LC)Ω + G(t, y, u), ( 10
) θ = ΓΩ T C T y -C x α , (11)
ẋ = Ax + φ(y, u) + G(t, y, u) θ + L(y -C x) + kDsign[F (y -C x)] + Ω θ, (12)
where Ω ∈ R n×q represents an auxiliary variable, θ ∈ R q is the estimation of θ while x ∈ R n is the estimation of x. The function

• α := | • | α sign(•), with | • | and sign(•)
understood in the component-wise sense, the function sign[q] := q/ q for any vector q ∈ R m , the observer matrix gain L ∈ R n×p has to be selected such that (A -LC) is Hurwitz and

0 < Γ T = Γ ∈ R q×q , while F ∈ R l×p , k, α, ∈ R ≥0 are designed later. Note that if the signal G(y, u
) is persistently exciting (PE), then due to the filtering property of the variable Ω, the variable CΩ is also PE. The adaptive sliding-mode observer ( 10)-( 12) represents a modified version of the one proposed in [START_REF] Efimov | Enhancement of adaptive observer robustness applying sliding mode techniques[END_REF]. Such a modification lies in the nonlinear parameter identification algorithm [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF] (in [START_REF] Efimov | Enhancement of adaptive observer robustness applying sliding mode techniques[END_REF] just the case when α = 1 was studied, i.e. the linear case). In this paper, it will be shown that the nonlinear algorithm [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF] may improve the rate of convergence and the accuracy of the given estimation. However, from another side, the nonlinearity in (11) also complicates the proof drastically with respect to [START_REF] Efimov | Enhancement of adaptive observer robustness applying sliding mode techniques[END_REF].

In the following, some properties of the nonlinear parameter estimation algorithm [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF] are presented but before let us introduce the following assumption.

Assumption 2 Let 0 < min ≤ σ min (CΩ(t)) for all t ≥ 0 and ||CΩ|| ∞ ≤ max < +∞.
The existence of max follows from Assumption 1 for the Hurwitz matrix A -LC and it cannot be computed a priori in a generic case. The existence of min can be assured if the matrix CΩ has a full column rank, and it is also hard to estimate the value of min in advance. However, the values of min and max can be evaluated numerically during experiments.

Nonlinear Parameter Estimation Algorithm

Let us define the errors θ := θ -θ and δ := x -x + Ω θ. Hence, taking into account (10)- [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (MIMO) linear time varying systems[END_REF], the error dynamics are given by

θ = -ΓΩ T C T CΩ θ -Cδ α , ( 13
) δ = (A -LC)δ + D(w -ksign[F (y -C x)]). ( 14
)
The following lemma shows that the system ( 13) is ISS with respect to the input δ for any α ∈ [0, 1).

Lemma 3 Let Assumption 2 be satisfied. Then, the system (13), with α ∈ [0, 1) and Γ = Γ T > 0, is ISS with respect to the input δ. Moreover, its trajectories satisfy the following bounds: [START_REF] Liu | Robust adaptive observer for nonlinear systems with unmodeled dynamic[END_REF]

θ(t) ≤ 2λmax(Γ) (2λ min (Γ)) α-1 2 θ(0) 1-α - 2 α-1 α α+1 min (2λ min (Γ)) α+1 2 2(1 + α) t 1 1-α , ∀t ≤ T θ ( θ(0)),
θ(t) ≤ λmax(Γ) λ min (Γ) µ θ , ∀t > T θ ( θ(0)), (16) ()
with

µ θ := p 1 α+1 (1 + α) 2 + 2 2-α α α-1 1 α+1 min (1 + α) α α+1 C δ ∞ T θ ( θ(0)) ≤ max   0, 2(1 + α) (2λ min (Γ)) α-1 2 θ(0) 1-α -(2λ max (Γ)) α-1 2 µ 1-α θ 2 α-1 α α+1 min (2λ min (Γ)) α+1 2   ,
and any θ(0) ∈ R q .

The proofs of all results are postponed to the Appendix. Hence, it is concluded that the solutions of system ( 13) are ultimately bounded with its trajectories satisfying the bounds given by ( 15) and ( 16) for any α ∈ [0, 1). Moreover, some important ISS properties with respect to the input δ are provided for the system [START_REF] Xu | Nonlinear system fault diagnosis based on adaptive estimation[END_REF].

Remark 1 Lemma 3 shows that the solutions of the system (13) enter into the bound (16) at most in a finite time T θ ( θ(0)) for any α ∈ [0, 1). In addition, the parameter estimation error may be reduced according to the choice of the parameter α ∈ [0, 1) since the size of µ θ depends on this value. Now, the following lemma shows that system ( 14) is ISS with respect to the inputs θ and w.

Lemma 4

Let Assumption 1 be satisfied. Suppose that the following matrix inequalities

(A -LC) T P + P (A -LC) + β -1 P + (βr + 2 )C T C ≤ 0, (17)
P D -C T F T ≤ 0, (18)
are feasible for a matrix 0 < P T = P ∈ R n×n , matrices F ∈ R l×p , L ∈ R n×p , and constants β, r, > 0, then the system [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], with k = w ∞ , is ISS with respect to the inputs θ and w.

Moreover, its trajectories satisfy the following bounds:

δ(t) ≤ e -ζ 1 2 t λ max (P ) λ min (P ) δ(0) , ∀t ≤ T δ (δ(0)), (19)
δ(t) ≤ λ max (P ) λ min (P ) µ δ , ∀t > T δ ( θ(0)), (20)
with

ζ 1 := (1 -ρ)(β -1 λ min (P ) + C 2 ) λmax(P ) , µ δ := √ max ρ(β -1 λ min (P ) + C 2 ) θ ∞ + µ δw , µ δw := 2 F ρ(β -1 λ min (P ) + C 2 ) w ∞ , T δ (δ(0)) ≤ max   0, 2 ln ( δ(0) ) -ln λmin(P ) λmax(P ) µ δ ζ 1    ,
ρ ∈ (0, 1), and any δ(0) ∈ R n .

Note that in order to attenuate the effects of the external disturbances w, the size of the parameter µ δw could be minimized.

Convergence of the Adaptive Observer

In the following, based on the statements given by Lemmas 3, 4, and small-gain arguments, it will be shown that the interconnected error system (13)-( 14) is ISS with respect to the external disturbance w, for any α ∈ [0, 1).

Theorem 5

Let Assumptions 1 and 2 be satisfied. Suppose that the matrix inequalities (17)-( 18) and

  p 2 α+1 (1 + α) 2 + 2 2-α α α-1 2 α+1 C 2 √ β max λ min (Γ) 2 min (1 + α) 2α α+1 λ max (P )λ max (Γ) ρλ min (P )   < 1, (21)
are feasible for matrices ), and ρ ∈ (0, 1), then the interconnected error system (13)-( 14) is ISS with respect to the input w. Moreover, with no external disturbances, i.e. w = 0, the system (13)-( 14) is globally asymptotically stable.

0 < P T = P ∈ R n×n , 0 < Γ T = Γ ∈ R q×q , F ∈ R l×p , L ∈ R n×p , constants β, r, , p > 0, α ∈ [0, 1
Note that Theorem 5 implies that the estimation error e := x -x = δ + Ω θ is also ISS since

e(t) ≤ (1 + ||Ω||) θ(t) δ(t) , ∀t ≥ 0, (22)
for any α ∈ [0, 1).

After the statements given by Lemmas 3 and 4; and Theorem 5, one can highlight the following points:

1. For the ideal case, i.e. w = 0 with k = 0, the estimations θ and x converge to the real values θ and x, respectively; and the rate of convergence for θ is faster than exponential, i.e. faster than the classic linear algorithm, for any α ∈ [0, 1), which is our motivation for designing of such a nonlinear estimation scheme. 2. For the perturbed case, i.e. w = 0, one can show by taking V e = e T P e that

Ve ≤ e T (A -LC) T P + P (A -LC) e + 2e T P (ΩΓΩ T C T Ce α + G(t, y, y) θ)

+ 2e T C T F T (w -ksign[F Ce]),
and therefore, if one fixes k = w ∞ , the effect of the external disturbance w is completely attenuated. 3. The condition [START_REF] Chen | Adaptive impulsive observers for nonlinear systems: Revisited[END_REF] introduces structural restrictions over the triple (A, D, C); specifically, it must not have invariant zeros, and the relative degree of the output y with respect to the input w must be equal to one. In order to avoid these restrictions some approaches are proposed in [START_REF] Efimov | Adaptive tunning to bifurcation for time-varying nonlinear systems[END_REF] and [START_REF] Edwards | On the solvability of the constrained Lyapunov problem[END_REF]. 4. To find a solution of the matrix inequality [START_REF] Tyukin | Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters[END_REF], one can rewrite it as follows

(A -LC) T P + P (A -LC) + τ 1 P + τ 2 C T C ≤ 0, (23)
where τ 1 = β -1 and τ 2 = βr + 2 are new variables. Then, using the fact that XY T + Y X T ≤ XΛ -1 X T + Y ΛY T holds for every X ∈ R k×n , Y ∈ R k×n , and 0 < Λ = Λ T ∈ R n×n (see, for instance [START_REF] Poznyak | Advanced Mathematical Tools for Control Engineers: Volume 1: Deterministic Techniques[END_REF]); and applying Schur's complement, it follows that  

A T P + P A -C T Y T -Y C + τ 2 C T C τ 1 2 I n P τ 1 2 I n -Λ -1 0 P 0 -Λ   ≤ 0, (24)
is equivalent to [START_REF] Yan | Adaptive sliding-mode-observer-based fault reconstruction for nonlineary systems with parametric uncertainties[END_REF] for any fixed 0 < Λ T = Λ ∈ R n×n and Y = P L ∈ R n×p . Note that ( 24) is now a linear matrix inequality with respect to matrices P and Y , and parameters τ 1 and τ 2 ; respectively. One can compute the design matrix L = P -1 Y . Then, the matrix inequality ( 21) can be numerically verified with the corresponding values of the solution of ( 24) and fixing α and ρ.

() 5. The feasibility of ( 24) is ensured for sufficiently small τ 1 and τ 2 , due to the fact that the pair (A, C) is observable. 6. The parameter µ δw can be written, in terms of τ 1 and τ 2 , as follows

µ δw = √ 8 F √ ρ 2τ 1 λ min (P ) + τ 2 C 2 -r τ 1 C 2 1 2
, with the constraint r < τ 1 τ 2 to ensure that > 0. Then, in order to minimize the size of µ δw , i.e. attenuate the effects of the external disturbances w, it is possible to propose de following optimization problem.

Optimization Problem. The minimization of µ δw , for a fixed ρ ∈ (0, 1) and a matrix F satisfying [START_REF] Chen | Adaptive impulsive observers for nonlinear systems: Revisited[END_REF], is equivalent to maximize λ min (P ), τ 1 and τ 2 , i.e.

γ, τ 1 , τ 2 → max P,F,Y,γ,τ 1 ,τ 2
subject to ( 18), ( 24) and γI ≤ P.

(25)

SIMULATION RESULTS

Duffing's Oscillator

Let us consider the following exited Duffing system

ẋ = 0 1 1 -µ x + 0 u + 0 -y 3 θ + 1 0 w, y = x 1 ,
where u = 0.3 cos(t), w = 0.5 sin(2t), µ = 0.2, θ = 3 and x(0) = (2, 1) T . For these parameters, the system develops a chaotic behavior and Assumption 1 is clearly satisfied. Let us apply the statements given by Theorem 5 for both cases, i.e. the ideal and the perturbed case.

Let us fix the matrix Γ = 100I 2 and the gain k = 1. Then, SeDuMi solver among YALMIP in Matlab is used to find a solution for the LMIs ( 18), [START_REF] Efimov | Enhancement of adaptive observer robustness applying sliding mode techniques[END_REF], and ( 21 For this matrix L and the given G(t, y, u), it is possible to show, after some simulations, that Assumption 2 is satisfied. All the simulations have been done in Matlab with the Euler discretization method, sample time equal to 0.001, and initial conditions Ω(0) = (0, 0) T , θ(0) = 0 and x(0) = (0, 0) T . The results for the ideal case, i.e. w = 0, with α = 0.0, 0.5 (nonlinear algorithm) and α = 1.0 (linear algorithm) for the parameter estimation algorithm, are depicted by Figure 1. One can see that the estimations θ and x, given by the nonlinear algorithm, converge to the real values θ and x, respectively; faster than the linear algorithm.

The results for the perturbed case, i.e. w = 0.5 sin(2t), with α = 0.0, 0.5 (nonlinear algorithm) and α = 1.0 (linear algorithm) for the parameter estimation algorithm, are depicted by Figure 2. In this case, the estimations θ and x converge to a neighborhood of the real values θ and x, respectively. One may see that the nonlinear algorithm still converges, to a neighborhood of the real value, faster than the linear algorithm. Now, some results with different values of α are presented in order to show how the estimation error ( θ, e T ) T is improved with respect to the choice of α. The results, for the ideal and perturbed case, are depicted by Figure 3. It is possible to see that there exists a value of α, in this case α = 0.5, such that the size of the estimation error bound ( 22) is minimized. One can also see that the nonlinear algorithm, i.e. α ∈ [0, 1), may increase the precision of the estimation with respect to the linear algorithm, i.e. α = 1.

Chua's Oscillator

Consider the following Chua system

ẋ =   0 0 0 1 -1 1 0 0 0   x +   cy 1 + y 2 -y 3 1 0 0 0 0 -y 2   θ +   0 1 0   w, y = x 1 x 2 T ,
where w = 0.5(sin(2t) + 2 sin(0.2t) + 0.1), c = 0.5, θ = (16, 10) T and x(0) = (3.9, -3.2, 0.03) T . For these parameters, the system develops a chaotic behavior and Assumption 1 is also satisfied. Let us apply again the statements given by Theorem 5 for both cases, i.e. the ideal and the perturbed case.

() Fixing the matrix Γ = 100I 3 and the gain k = 1. Then, SeDuMi solver among YALMIP in Matlab is used to find a solution for the LMIs ( 18), [START_REF] Efimov | Enhancement of adaptive observer robustness applying sliding mode techniques[END_REF], and ( 21 For this matrix L and the given G(t, y, u), one can show, after some simulations, that Assumption 2 is also satisfied. The simulations have been done in Matlab with initial conditions Ω(0) = 0 3×2 , θ(0) = (0, 0) T and x(0) = (0, 0, 0) T . The results for the ideal case, i.e. w = 0, with α = 0.0, 0.5 (nonlinear algorithm) and α = 1.0 (linear algorithm) for the parameter estimation algorithm, are depicted by Figure 4. One can see that the estimations θ and x, given by the nonlinear algorithm, converge to the real values θ and x, respectively; faster than the linear algorithm.

Time [sec]
The results for the perturbed case, i.e. w = 0.5(sin(2t) + 2 sin(0.2t) + 0.1), with α = 0.0, 0.5 (nonlinear algorithm) and α = 1.0 (linear algorithm) for the parameter estimation algorithm, are depicted by Figure 5. In this case, the estimations θ and x converge to a neighborhood of the real values θ and x, respectively. One may see that the nonlinear algorithm still converges, to a neighborhood of the real value, faster than the linear algorithm. Some results with different values of α are also presented in order to show how the estimation error ( θ, e T ) T is improved with respect to the choice of α. The results, for the ideal and perturbed case, are depicted by Figure 6. It is possible to see that there exists a value of α, in this case α = 0.5, such that the size of the estimation error bound [START_REF] Shtessel | Sliding Mode Control and Observation[END_REF] is minimized. One can also see that the nonlinear algorithm, i.e. α ∈ [0, 1), may increase the precision of the estimation with respect to the linear algorithm, i.e. α = 1.

CONCLUSIONS

In this paper an adaptive sliding-mode observer based on a nonlinear parameter estimation algorithm is proposed for uncertain nonlinear systems. The given adaptive sliding-mode observer is a modified version of that one proposed by [START_REF] Efimov | Enhancement of adaptive observer robustness applying sliding mode techniques[END_REF]. Such a modification lies in the inclusion of a nonlinear parameter estimation algorithm that provides a rate of convergence faster than exponential, i.e. faster than the classic linear algorithm. Then, the proposed parameter estimation algorithm is included in the structure of a sliding-mode state observer providing an ultimate bound for the state and parameter estimation error. This solution ensures that the level of observer robustness with respect to some matched disturbances is improved as well as the rate of convergence. The corresponding proofs of convergence are developed based on Lyapunov function approach and input-to-state stability theory. Simulation results illustrate the efficiency of the proposed adaptive sliding-mode observer. where x ∈ R n is the state, w ∈ R l is the external disturbances, and f : R n × R l → R n ensures existence of the system solutions at least locally. The following preliminary result describes some ISS properties for system (3) and quadratic Lyapunov functions.

Lemma 6

Let V : R n → R ≥0 be a smooth function. If there exist some positive constants ψ 1 , ψ 2 , ψ 3 , ψ 4 > 0 and γ ∈ (0, 1] such that then the system (3) is ISS with respect to the input w. Moreover, its trajectories satisfy the following bounds

ψ 1 x 2 ≤ V (x) ≤ ψ 2 x 2 , ( 26
) ∇V (x)f (x, w) ≤ -ψ 3 V γ (x), ∀ x ≥ µ := ψ 4 w ∞ , (27)
x(t, x 0 , w) ≤      1 √ ψ 1 ψ 1-γ 2 x 0 2(1-γ) -ψ 3 (1 -γ)t 1 2(1-γ) , for γ ∈ (0, 1), e -ψ 3 2 t ψ 2 ψ 1 x 0 , for γ = 1, ∀t ≤ T (x 0 ), ( 28
) x(t, x 0 , w) ≤ ψ 2 ψ 1 µ, ∀t > T (x 0 ), (29)
where

T (x 0 ) ≤          max 0, ψ 1-γ 2 x0 2(1-γ) -ψ 1-γ 1 µ 2(1-γ) ψ 3 (1-γ)
, for γ ∈ (0, 1), max 0,

2 ln x0 -ln ψ 1 ψ 2 µ ψ 3 , for γ = 1.

Proof

According to Definition 2, the function V is an ISS Lyapunov function. Therefore, from Theorem 1, the () system ( 3) is ISS with respect to the input w. Then, from [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] and the comparison principle (see, e.g. [START_REF] Khalil | Nonlinear Systems[END_REF]), it follows that (32)

where c 1 := 2λmax(Γ) and c 2 := 2λ min (Γ). The time derivative of V θ along the trajectories of ( 13) is given by

Vθ = θT Γ -1 -ΓΩ T C T CΩ θ -Cδ α , = - θT Ω T C T CΩ θ -Cδ α . (33)
Using Young's inequality it is derived:

θT Ω T C T CΩ θ -Cδ α ≥ p i=1 |(CΩ θ) i ||(CΩ θ) i -(Cδ) i | α - 2α α (1 + α) 1+α Cδ α+1 α+1 ,
and applying Jensen's and Young's inequalities one gets:

θT Ω T C T CΩ θ -Cδ α ≥ 2 α-1 p i=1 |(CΩ θ) i | |(CΩ θ) i | α -|(Cδ) i | α - 2α α (1 + α) 1+α Cδ α+1 α+1 ≥ 2 α-1 α 1 + α p i=1 |(CΩ θ) i | α+1 - 2α 1 + α 2 α-2 + α α-1 (1 + α) α Cδ α+1 α+1 = 2 α-1 α 1 + α CΩ θ α+1 α+1 - 2α 1 + α 2 α-2 + α α-1 (1 + α) α Cδ α+1 α+1 .
By relations between vector norms, CΩ θ ≤ CΩ θ α+1 holds since 2 > α + 1, and Cδ α+1 ≤ p 1 α+1 Cδ ∞. Therefore, one finally obtains:

Vθ ≤ -2 α-1 α 1 + α CΩ θ α+1 + 2pα 1 + α 2 α-2 + α α-1 (1 + α) α Cδ α+1 ∞ ,
and taking into account Assumption 2, one obtains that Vθ is upper bounded as follows:

Vθ ≤ -2 α-1 α 1 + α α+1 min

θ α+1 + 2pα 1 + α 2 α-2 + α α-1 (1 + α) α C α+1 δ α+1 ∞ , or θ ≥ p 1 α+1 (1 + α) 2 + 2 2-α α α-1 1 α+1 min (1 + α) α α+1 C δ ∞ =⇒ Vθ ≤ - 2 α-1 α 1 + α α+1 min θ α+1 . ( 34
) ()
where α δ = β -1 λ min (P ) + C 2 . Then, for all ρ ∈ (0, 1), it follows that

Vδ ≤ -(1 -ρ)α δ δ 2 , ∀ δ ≥ 2 max ρα δ θ ∞ + 2 F w ∞ √ ρα δ .
(37)

Then, from (37) and (36), it is given that

Vδ ≤ - (1 -ρ)α δ λmax(P ) V δ , ∀ δ ≥ 2 max ρα δ θ ∞ + 2 F w ∞ √ ρα δ .
Therefore, based on Lemma 6, V δ is an ISS Lyapunov function for system [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], and the system ( 14) is ISS with respect to the inputs θ and w, and hence its trajectories satisfy the bounds given by ( 19) and [START_REF] Folin | Sampled-data adaptive observer for a class of state-affine output-injection nonlinear systems[END_REF].

Proof of Theorem 5: Let us consider the following functions

V θ = 1 2 θT Γ -1 θ, V δ = δ T P δ.
Based on Lemmas 3 and 4, one can conclude that ()

(2λmax(Γ)) -1 θ 2 ≤ V θ ≤ (2λ min (Γ)) -1 θ 2 , V θ ≥   p 2 α+1 (1 + α) 2 + 2 2-α α α-1 2 α+1 C 2

Figure 1 .

 1 Figure 1. The estimations of θ and x -Ideal Case (α = 0.0, α = 0.5 and α = 1.0)

Figure 2 .

 2 Figure 2. The estimations of θ and x -Perturbed Case (α = 0.0, α = 0.5 and α = 1)

  ), respectively. The following feasible solution, with Λ = I 3 , is found = (0.3386 18.4631) , τ 1 = 0.1478, τ 2 = 294.9623, β = 6.7652, r = 39.2397, = 14.7481.

Figure 3 .

 3 Figure 3. Estimation Error for different values of α. The top graph shows the estimation error for the ideal case, i.e. w = 0; while the bottom graph illustrates the perturbed case, i.e. w = 0.5 sin(2t).

TimeFigure 4 .

 4 Figure 4. The estimations of θ and x -Ideal Case (α = 0.0, α = 0.5 and α = 1.0)

ACKNOWLEDGEMENTH.TimeFigure 5 .

 5 Figure 5. The estimations of θ and x -Perturbed Case (α = 0.0, α = 0.5 and α = 1)

Figure 6 .

 6 Figure 6. Estimation Error for different values of α. The top graph shows the estimation error for the ideal case, i.e. w = 0; while the bottom graph illustrates the perturbed case, i.e. w = 0.5(sin(2t) + 2 sin(0.2t) + 0.1).

⇒λ 2 ,⇒ 21 ( 2 α+1 ( 1 +

 22121 Vθ ≤ -2 α-1 α α+1 min (2λ min (Γ)) min (P ) δ 2 ≤ V δ ≤ λmax(P ) δ Vδ ≤ -(1 -ρ)α δ λmax(P ) V δ .Therefore, the conditions (6)-(8) are satisfied withψ 11 ( θ ) = (2λmax(Γ)) -1 θ 2 , ψ 12 ( θ ) = (2λ min (Γ)) -1 θ 2 , ψ 13 (V θ ) = 2 α-1 α α+1 min (2λ min (Γ)) δ ) = λ min (P ) δ 2 , ψ 22 ( δ ) = λmax(P ) δ 2 , ψ 23 (V δ ) = (1 -ρ)α δ λmax(P ) V δ , χ 1 (V δ (δ)) =and then, it follows thatχ 1 • χ 2 (r) =   p α) 2 + 2 2-α α α-1 2 α+1 C 2 √ β max λ min (Γ)2 min (1 + α) 2α α+1 λmax(P )λmax(Γ) ρλ min (P )   r.

  ), respectively. The following feasible solution, with Λ = I 2 , is found

	P =	28.6444 0	0 0.4455	, L =	30.5749 64.3579	, F = 28.6444,
	τ 1 = 0.0281, τ 2 = 604.7644, β = 35.6235, r = 15.2789,	= 30.2382.

Thus, from (34) and (32), it follows that

Consequently, based on Lemma 6, V θ is an ISS Lyapunov function for system [START_REF] Xu | Nonlinear system fault diagnosis based on adaptive estimation[END_REF], and the system [START_REF] Xu | Nonlinear system fault diagnosis based on adaptive estimation[END_REF] with α ∈ [0, 1) is ISS with respect to the input δ, and hence its trajectories also satisfy the bounds given by ( 15) and ( 16) for any α ∈ [0, 1).

Proof of Lemma 4: Consider the candidate Lyapunov function

which satisfies the following inequality

(36)

The time derivative of V δ along the trajectories of ( 14) is given by

then, if ( 17)-( 18) are satisfied it follows that

and, if one fixes k = w ∞ it is given that

Recalling that Cδ = y -C x + CΩ θ, then

Using the fact that

it is obtained

Hence, if the parameters P , Γ, α, β, and ρ are such that the matrix inequality ( 21) holds, then one obtains that

Therefore, all the conditions of Theorem 2 are satisfied, and thus, one concludes that the interconnected error system (13)-( 14) is ISS with respect to the external input w for any α ∈ [0, 1). Moreover, with w = 0, the system (13)-( 14) is globally asymptotically stable.