Guillermo A A Dosio

Herve  Rey

Je  Re  Mie Lecoeur

Natalia G Izquierdo

Luis A N Aguirreza  Bal

Olivier Turc
email: turc@ensam.inra.fr

A whole-plant analysis of the dynamics of expansion of individual leaves of two sun¯ower hybrids

Keywords: Exponential growth, Helianthus annuus L., leaf expansion, phasic development, phytomer, thermal time, whole-plant-analysis pressure de®cit

Common features in the time-course of expansion of leaves which considerably differed in ®nal area, due to phytomer position, growing conditions and genotype, were identi®ed. Leaf development consisted of two phases of exponential growth, followed by a third phase of continuous decrease of the relative expansion rate. The rate and the duration of the ®rst exponential phase were common to all phytomers, growing conditions and genotypes. Leaves differed in the rate and the duration of the second exponential phase. The decrease of the relative expansion rate during the third phase depended on neither genotype nor growing conditions. It was phytomerdependent and was deduced from the rate of the second phase via a parameter common to all cases studied. Differences in ®nal leaf area among growing conditions were linked to different expansion rates during the second exponential phase. The duration of the phases at any given phytomer position was the same for the two hybrids in different growing conditions. The dates of developmental events (initiation, end of the two exponential phases, full expansion), and the rate of the second exponential phase, were related to phytomer position, de®ning a strict pattern of leaf development at the whole plant level. Using this framework simpli®ed the analysis of the response of leaf expansion to genotype and environment.

Introduction

The next challenge assigned to plant biology, after the complete genome sequencing of model plant species, may well be to simulate the behaviour, in different environments, of genetically characterized phenotypes [START_REF] Chory | National Science Foundation-Sponsored Workshop Report: `The 2010 Project', functional genomics and the virtual plant. A blueprint for understanding how plants are built and how to improve them[END_REF]. To achieve this, analysis of gene expression has to be combined with tools to analyse growth and development at the whole-plant scale and throughout the plant life cycle [START_REF] Boyes | Growth stage±based phenotypic analysis of Arabidopsis: a model for high throughput functional genomics in plants[END_REF].

Descriptors of shoot development, or growth stages, are classically based on the repetitive activity of the shoot apical meristem, which sequentially produces phytomers [START_REF] Lyndon | Plant development. The cellular basis[END_REF]. They consist of counting the number of leaves beyond a given stage (initiation, emergence, unfolding, threshold length) and have been developed for many species [START_REF] Erickson | The plastochron index[END_REF][START_REF] Maurer | Response of peas to environment. III. Assessment of the morphological development of peas[END_REF][START_REF] Haun | Visual quanti®cation of wheat development[END_REF][START_REF] Schneiter | Description of sun¯ower growth stages[END_REF][START_REF] Rickman | The phyllochron: where do we go in the future[END_REF][START_REF] Turc | Leaf primordium initiation and expanded leaf production are co-ordinated through similar response to air temperature in pea (Pisum sativum L.)[END_REF] and, recently, Arabidopsis thaliana [START_REF] Boyes | Growth stage±based phenotypic analysis of Arabidopsis: a model for high throughput functional genomics in plants[END_REF]. The response of growth stages to environmental factors such as temperature, day-length and light intensity has been extensively studied [START_REF] Warrington | Corn growth response to temperature and photoperiod. II. Leaf initiation and leaf appearance rates[END_REF][START_REF] Miglietta | Effect of photoperiod and temperature on leaf initiation rates in wheat (Triticum spp.)[END_REF][START_REF] Jamieson | Prediction of leaf appearance in wheat: a question of temperature[END_REF][START_REF] Kirby | Environmental factors in¯uencing the phyllochron[END_REF]. Such descriptors are very useful in characterizing crop or plant development, but they often mismatch transitional events of individual leaf development (e.g. the beginning of linear growth or the end of leaf expansion) pertinent to the growth response to the environment [START_REF] Turc | Leaf primordium initiation and expanded leaf production are co-ordinated through similar response to air temperature in pea (Pisum sativum L.)[END_REF][START_REF] Lafarge | A model co-ordinating the elongation of all leaves of a sorghum cultivar was applied to both Mediterranean and Sahelian conditions[END_REF].

Analysing the relationships between plant growth and environment is made dif®cult by interactions between them [START_REF] Sinoquet | Short-term interactions between tree foliage and the aerial environment: An overview of modelling approaches available for tree structure±function models[END_REF]. For example, changes in leaf growth, resulting from the environmental conditions prevailing during early leaf development [START_REF] Granier | Leaf expansion and cell division are affected by reducing absorbed light before but not after the decline in cell division rate in the sun¯ower leaf[END_REF] or from genetic variation [START_REF] Dolan | The Okra leaf shape mutation in cotton is active in all cell layers of the leaf[END_REF], modify the micrometeorological conditions within the canopy (e.g. the vertical distribution of light interception), which in turn affect the development of newly initiated leaves and the current microclimate. Threedimensional models of plants, representing the canopy as a set of organs and yielding realistic computer-generated images of plant geometry through time [START_REF] Âaz-Ambrona | Faba bean canopy modelling with a parametric open L-system: a comparison with the Monsi and Saeki model[END_REF][START_REF] Fournier | A 3D architectural and process-based model of maize development[END_REF][START_REF] Rey | Numerical sun¯ower plants responding to light and temperature[END_REF] allow calculation of the radiative environment of each organ at each step of plant development [START_REF] Dauzat | Simulating light regime and intercrop yields in coconut based farming systems[END_REF]. These could, therefore, constitute pertinent tools to analyse ongoing interactions. However, the main dif®culty of building virtual plants seems to deal with the mathematical expression of the genetic variability of plant responses to environmental conditions to include into the computer representation of plants [START_REF] Tardieu | Virtual plants: modelling as a tool for the genomics of tolerance to water de®cit[END_REF].

A framework of analysis of leaf expansion of dicots, based on the time-course of the leaf relative expansion rate, provided a mathematical formalism, together with pertinent stages of leaf development, for the response of leaf expansion to environmental conditions (Granier and Tardieu, 1998a). Two phases were taken into account during leaf development to analyse these responses: during the ®rst one, leaf growth was exponential and, during the second one, leaf relative expansion rate decreased rapidly to reach zero at the end of leaf expansion [START_REF] Denne | Leaf development in Trifolium repens[END_REF][START_REF] Poethig | The developmental morphology and growth dynamics of the tobacco leaf[END_REF]Granier and Tardieu, 1998a). Relative expansion rate was affected by a reduction in absorbed light only if it occurred during the exponential phase [START_REF] Granier | Leaf expansion and cell division are affected by reducing absorbed light before but not after the decline in cell division rate in the sun¯ower leaf[END_REF]. However, the validity of this approach, mainly obtained on leaf 8 of sun¯ower (Helianthus annuus L.) hybrid Albena, remains to be evaluated for all other nodal positions along the stem. Leaf development at other phytomer positions occurs under different environmental and endogenous (e.g. source±sink relationships, reproductive development) conditions that could interfere with leaf expansion and make inadequate a common pattern for all phytomers.

The objective of the present work was to establish a whole-plant pattern of leaf growth and development of sun¯ower consisting of a set of mathematical equations with identi®ed parameters depending either on nodal position, genotype or environment. The dynamics of expansion of all individual leaves were analysed for sun¯ower plants grown under ¯uctuating environmental conditions (light and temperature). Time-courses were expressed on a thermal time basis (Granier and Tardieu, 1998b). The analysis consisted of determining, at all phytomer positions, the dates of occurrence, and the rate and the duration of the successive phases of leaf development relevant for leaf expansion. This was done on two sun¯ower hybrids differing in leaf number, grown at two different locations either as isolated plants, without competition for light, or in a plant canopy.

Materials and methods

Plant culture and growth conditions

Three experiments were performed to investigate the development of leaves originating from all successive phytomers of sun¯ower plants grown in the ®eld under naturally ¯uctuating environmental conditions. Two experiments were carried out near Montpellier, France (43°40¢ N, 3°50¢ E). Sun¯ower (Helianthus annuus L., hybrid Albena) seeds were sown on 5 May 1998 and 11 May 1999 in a deep sandy loam soil (¯uvio-calcaric Cambissol). Seedling emergence occurred a week after sowing in both experiments. Due to a poor soil N content, N fertilization was applied before sowing (80 kg ha ±1 ). Soil water potential was monitored with ®ve tensiometers (DTE 1000 system, Nardeux, Saint-Avertin, France) placed every 0.20 m from 0.30 m to 1.10 m depth, and maintained by irrigation above ±40 kPa in the ®rst 0.5 m of soil during the whole period of leaf area establishment. This practice maintained available soil water content in a range where neither stomatal conductance nor leaf growth were affected [START_REF] Turner | Measurement and in¯uence of environmental and plant factors on stomatal conductance in the ®eld[END_REF]. Two treatments were applied to plants. In the ®rst one, carried out in 1998 and 1999, isolated plants were obtained by thinning to 0.06 plants m ±2 after seedling emergence to avoid light competition between neighbouring plants. In the second treatment, carried out in 1999, a plant canopy was obtained with a ®nal plant density of 5.5 plants m ±2 .

Seeds of Dekalb G100 hybrid were sown on 17 November 1997 at the INTA Balcarce Experimental Station, Argentina (37°45¢ S, 58°1 8¢ W), on a loamy clay typic Argiudol. Soil analysis indicated that neither N nor P fertilization were necessary as nutrient availability was adequate to obtain maximum yield of sun¯ower crops under non limiting water conditions [START_REF] Sosa | Evaluacio Ân de la nutricio Ân nitrogenada de girasol cultivado en Balcarce (Buenos Aires, Argentina)[END_REF][START_REF] Andrade | Requerimientos de nutrientes minerales[END_REF]. Soil water content was monitored every 5±7 d using a neutron probe (Troxler 4300, Troxler Electronic Laboratories, INC, Research Triangle Park, NC, USA). Irrigation was supplied to maintain soil water content above 40% of maximum available water content in the ®rst 0.60 m of the soil pro®le during the entire growing season. This management avoided any decrease in stomatal conductance [START_REF] Turner | Measurement and in¯uence of environmental and plant factors on stomatal conductance in the ®eld[END_REF]. After emergence (10 d after sowing), seedlings were thinned to obtain a plant density of 5.5 plants m ±2 .

Incident radiation was measured at 2 m above the soil surface with a PPFD sensor (LI-190SB, Li-Cor, Lincoln, Nebraska, USA). Air temperature and relative humidity were measured at 2 m above the soil surface with a HMP35A capacitive hygrometer (Vaisala Oy, Helsinki, Finland) placed in a ventilated shelter. Leaf temperature was measured with 0.4 mm-diameter copper-constantan thermocouples. Thermocouples were placed at three locations along the stem to characterize the gradient of temperature according to phytomer position. A thermocouple was inserted into the apical bud to measure apex temperature. A second one was pressed to the under-side of the lamina of a growing leaf near the shoot top, and a third one measured the temperature of a mature leaf, near the base of the plant. As the plant developed, thermocouples were progressively moved to upper leaves. All data were transferred every 20 s to a data logger (LTD-CR10 Wiring Panel; Campbell Scienti®c, Shepshed, UK and Delta-T DL2e Logger; Delta-T Devices Ltd, Cambridge, UK at Montpellier and Balcarce, respectively) and averaged every 600 s.

Daily mean values observed between emergence and ®rst anthesis varied from 13.6 to 26.6 °C for air temperature, 12.9 to 25.7 °C for leaf temperature, 14.0 to 64.6 mol m ±2 d ±1 for incident PPFD, while daily maximum VPD varied from 0.05 to 4.6 kPa across dates and locations. Day-to-day meteorological variations in each experiment were greater than variations through year and location (not shown). Periods with low incident PPFD or high VPD were short in all cases: two consecutive days with PPFD lower than 30 mol m ±2 d ±1 never occurred, and periods with VPD higher than 2.0 kPa were limited to a couple of hours around midday, during few consecutive days (not shown). The effect of temperature on leaf expansion was taken into account by the use of thermal time (Granier and Tardieu, 1998b).

Stages of development and thermal time

Seedling emergence was de®ned as the date when the ®rst leaf pair became just visible (3±5 mm length) between open cotyledons. It occurred about 100 °Cd (c. 6 d) after seed imbibition.

A leaf was considered as initiated when its primordium was visible (about 40 mm long) on the apical meristem under a microscope (Leica stereomicroscope, Wild F8Z, Wetzlar, Germany) at magni®cation Q80. Phytomers were numbered from the cotyledonary node. Each of the ®rst four phytomers bear two opposite organs (cotyledons and three leaf pairs). Phyllotaxy is then alternate from phytomer 5 (corresponding to leaf 7) until the involucral bracts surrounding the capitulum. For example, leaf 16 corresponds to phytomer 14. Measurements concerned only the `true' leaves. Cotyledons (phytomer 1) and sessile leaves (intermediate between leaves and bracts), generally present at the last two phytomers under the involucral bracts, were not measured.

The capitulum was considered as initiated when the shoot apical meristem broadened and became dome-shaped. Apex area comprised between 0.1 and 0.2 mm 2 at that stage, equivalent to Floral Stage 1.3 of [START_REF] Marc | Photoperiodic sensitivity of in¯orescence initiation and development in the sun¯ower[END_REF]. First anthesis was reached when stamens became visible on the outer ring of ¯owers on the capitulum.

Thermal time was calculated by daily integration of leaf temperature and a base of 4.8 °C (Granier and Tardieu, 1998b), cumulated either from seedling emergence or from leaf initiation. For a given leaf, the calculation was made successively with apex temperature until the leaf emerged out of the apical bud, with temperature measured in the zone of growing leaves until full expansion and, subsequently, in mature leaves.

Growth measurements

A sampling procedure was designed to reduce variability in plant size due to different dates of seedling emergence [START_REF] Granier | Individual leaf development in Arabidopsis thaliana: a stable thermal-time-based programme[END_REF]. A batch of ten plants close to the median of the distribution of length of the ®rst leaf pair was selected after emergence and considered as reference plants. Length and width of all visible leaves (length >15±20 mm) were non-destructively measured with a rule three times a week on the reference plants until the end of expansion of the last leaf. A highly signi®cant linear relationship (r 2 >0.99, n >300, P <0.0001) was established in each experiment between the product lengthQwidth and leaf area measured using a planimeter (Li-Cor 3100 Area Meter; Li-Cor Inc., Lincoln, Nebraska, USA) in Balcarce and a camera coupled to an interactive image analysis system (Optimas V 6.5; Media Cybernetics, Silver Spring, MD, USA) in Montpellier.

Three times a week in the Montpellier experiments and every 10 d in the Balcarce experiment, 4±6 plants having a developmental stage close to that of the reference plants were harvested to measure leaves enclosed in the apical bud. Leaf area of visible leaves was measured with an image analyser. The apical bud was dissected under a microscope (Leica MZ75, Wetzlar, Germany). Leaves were excised and area was measured through the microscope and a camera coupled with an image analyser.

For each leaf, relative expansion rate (R) was calculated at any given time j from leaf initiation to the end of leaf expansion as the local slope of the relationship between the logarithm of leaf area (A) and thermal time t:

R j =[d(lnA)/dt] j (1)
It was calculated by linear regression on the three values of A and t corresponding to times j±1, j and j+1.

Growth curves ®t

Leaf expansion was analysed as a function of thermal time by considering a succession of leaf developmental phases, characterized by the time-course of the leaf relative expansion rate. Two periods of exponential growth and a third one with an exponential decrease of the relative expansion rate were distinguished (see Results). For each leaf, the thermal-time-course of the logarithm of leaf area from leaf initiation to full expansion was ®tted to a set of three equations. The parameters of the three equations, including the dates of shift between the successive phases, were all estimated simultaneously for each given leaf by least squares ®tting, using an algorithm of generalized reduced gradient (Marquardt±Levenberg algorithm of software SIGMA PLOT 2001 V7.1; SPSS Science, Chicago, IL, USA).

Results

Considerable differences in kinetics of leaf growth and in ®nal leaf area were observed Final leaf area largely differed with phytomer position along the stem (Fig. 1). It varied from 3700 mm 2 for phytomer 2 to 120 000 mm 2 for phytomer 20 of isolated plants of hybrid Albena (Fig. 1A). Kinetics of leaf growth markedly changed from the plant base to the plant tip (Figs 2, 3). In particular, the duration of leaf expansion increased with phytomer number (Figs 2, 3): it equalled 420 °Cd for phytomer 2 and 746 °Cd for phytomer 26 of isolated plants of hybrid Albena (Fig. 2).

Competition with neighbouring plants reduced the ®nal leaf area from phytomer 7 to phytomer 30 of plants grown in a dense canopy compared with isolated plants (Fig. 1A). This reduction increased with phytomer number and exceeded 70% for the highest phytomers (Fig. 1A). As a consequence, leaf area per plant largely differed between isolated plants and plants growing in a canopy (Table 1).

Hybrids Albena and G100 differed in their rates of vegetative development, as well as in reproductive development (Table 1). G100 initiated ®ve leaves fewer than Albena (Fig. 1B): the ®rst involucral bract appeared at phytomer 28 for G100 versus phytomers 33 or 34 for Albena. A lower rate of leaf initiation by about 30% in G100 (Table 1) more than compensated for the difference in leaf number between both genotypes, resulting in a delay of about 60 °Cd of the dates of capitulum initiation and ®rst anthesis in G100 compared to Albena (Table 1). The ®nal leaf area was very similar in hybrids G100 and Albena for phytomers 5 to 21 (Fig. 1B). It was slightly but signi®cantly lower for phytomers 2 to 5, and sharply reduced above phytomer 21 in G100 compared to Albena (Fig. 1B). Final leaf area per plant was consequently lower by about 24% in G100 (Table 1).

The time-course of the relative expansion rate had common features in all cases Nevertheless, the dynamics of growth of all leaves showed strong similarities, based on the kinetics of the relative expansion rate (Figs 2 and 3, bottom graphs). Provided it was expressed per unit thermal time, the time-course of the relative expansion rate had a general shape conserved across years, locations, genotypes, and phytomers (Figs 2,3). In all cases, the relative expansion rate was high for all phytomers during the ®rst 100 °Cd following initiation, rapidly decreased before 150 °Cd (except for phytomer 2, see below), then stabilized for a certain time, and subsequently continuously decreased to reach zero at the end of leaf expansion.

To account for this, the time-course of expansion of all individual leaves was divided into three periods (Fig. 4). The ®rst two periods were assumed to be exponential, with rates equalling R 1 and R 2 , respectively. During the third period (rapid leaf growth) the relative expansion rate R was ®tted for each leaf to a negative exponential function of thermal time t as follows:

R=R 3 Qexp[±aQ(t±t 2 )] (2)
Change with time of leaf area was therefore ®tted to the following set of three equations:

t 0 <t<t 1 lnA=lnA 0 +R 1 Q(t±t 0 ) (3) t 1 <t<t 2 lnA=lnA 1 +R 2 Q(t±t 1 ) (4) t>t 2 lnA=lnA 2 +R 3 /aQ{1±exp[±aQ(t±t 2 )]} (5)
where A was the leaf area at time t, t 0 the date of leaf initiation, A 0 the leaf area at initiation, R 1 the relative expansion rate during the ®rst exponential period, t 1 the date of the end of the ®rst exponential period, A 1 the leaf area at the end of the ®rst exponential period (calculated with equation 3), R 2 the relative expansion rate during the second exponential period, t 2 the date of end of the second exponential phase, A 2 the leaf area at t 2 (calculated with equation 4), a the parameter characterizing the exponential decrease of the relative exponential rate after t 2 , and R 3 the relative expansion rate at t 2 .

For consistency with data points, ®tted relative expansion rates were averaged on periods of 60 °C (Fig. 4, solid line) and did not show the discontinuities at t 1 and t 2 obtained when calculated by deriving equations 3±5 (Fig. 4, dotted line). The succession of two exponential phases with different growth rates accounted for the decrease in relative expansion rate observed for all leaves between 50 °Cd and 150 °Cd after leaf initiation (Figs 2 and 3, bottom graphs). The results of ®tting are presented as lines on Figs 2 and 3. They accounted for leaf growth in all cases (r 2 >0.999, P <0.0001).

Analysis of the relative expansion rates

The stability of the relative expansion rate during the ®rst 100 °Cd following initiation was not clear for individual leaves (Figs 2 and 3, bottom graphs) because there were, at best, two values per treatment during this period owing to the method of calculation. Conversely, data processing of all leaves studied suggested that relative expansion rate during this ®rst period was common to all phytomers (Fig. 5). Data on leaf area versus thermal time after leaf initiation, gathered from three experiments and including phytomers 3 to 30, ®tted to a common exponential function (r 2 =0.945, n=171, P <0.0001). The slope of the exponential ®t corresponded to the mean relative expansion rate during this ®rst period and equalled 0.060T0.001 °Cd ±1 . The y-intercept of the exponential ®t corresponded to the average area of the leaf primordium when initiated and equalled 0.0050T0.0003 mm 2 . The relative expansion rate during the ®rst phase was then calculated for each individual leaf (Fig. 6A) with the data from Fig. 5 by assuming that the initial area was common to all leaves. Individual rates were scattered around the mean rate (horizontal dotted line) with no signi®cant effect of phytomer position or treatment (Fig. 6A). Therefore, early expansion of leaf primordium, from initiation to c. 100 °Cd later, appeared to be exponential and common to all leaves studied, irrespective of phytomer position, tested growing conditions and genotypes. The relative expansion rate during the second exponential phase was lower than that occurring before 100 °Cd, except for phytomer 2, where the rate was maintained at a similar level (Figs 2 and 3, bottom graphs). It was highest for phytomer 2 (e.g. 0.057 °Cd ±1 for isolated plants) and decreased continuously with phytomer number (e.g. 0.028 °Cd ±1 for phytomer 26 of isolated plants, Figs 2, 6B). A similar trend was observed for isolated plants, plants grown in a dense canopy and for the two hybrids (Fig. 6B). For plants grown in a canopy, competition for light gave a greater decrease in expansion rate with phytomer number (Fig. 6B), without changing the general shape of the decrease. The difference between isolated plants and plants in a canopy increased from phytomer 6 upwards (Fig 6B andinsert). The relative expansion rate during the second exponential phase was similar in both hybrids for phytomers 2 to 21 (Fig. 6B). It sharply decreased for the upper leaves of G100 corresponding to the last initiated Solid arrows indicate the characteristic parameters of any given leaf: t 0 , the date of initiation, R 2 the rate of the second exponential phase, and t 2 , the end of the exponential growth. Dashed arrows indicate the parameters either common to all leaves (A 0 , the leaf area at initiation, t 1 and R 1 , the duration and the rate of the ®rst exponential phase), or deduced from the preceding ones (R 3 , the relative expansion rate at the beginning of the third phase, a, the rate of exponential decrease of the relative expansion rate during the rapid leaf growth, and t f , the date of full expansion). phytomers (phytomers 21 to 25), and, for example, leaf growth of phytomer 24 of G100 did not ®t with that of phytomer 24 of Albena (Figs 1, 3). A similar acceleration of the decrease in expansion rate with phytomer number was also observed for the last initiated phytomers of hybrid Albena (phytomers 25 to 30) either grown as isolated plants or in a dense canopy (Fig. 6B). This accelerated decrease was therefore more related to the proximity of the capitulum than to phytomer position counted from the plant base (Fig. 6B). It was ®tted to the following equation:

R 2(N) =cQ{1-exp[±bQ(N b ±N)]} (6)
where R 2(N) was the second exponential rate for phytomer number N, and N b the phytomer number of the ®rst involucral bract. The same parameter b (b=0.322, r 2 0.966, P <0.001) accounted for the decrease of R 2 on the ®ve upper leaves for both hybrids grown either as isolated plants or in a dense canopy (dotted lines on Fig. 6B).

The relative expansion rate during the third period was characterized with two parameters, R 3 and a. For isolated plants, calculated R 3 did not differ signi®cantly from R 2 , the rate of the second phase (Fig. 6C, insert below main curve, R 3 =R 2 along the dotted line). For plants grown in a canopy, calculated R 3 was very close to that of isolated plants at any phytomer position (Fig. 6C, main graph). Plotting a against R 3 for all leaves studied (Fig. 6C, insert above main curve) resulted in a strong linear relationship with a zero intercept (slope=0.372T0.002, r 2 =0.955, n=123, P <0.0001), common to both hybrids and environmental conditions. So, neither R 3 nor a differed between isolated plants and plants in a canopy at any given phytomer position, showing that the time-course of the relative expansion rate was independent of interplant competition during this last phase. For plants grown in a canopy, R 3 progressively diverged from R 2 as phytomer number increased above 6 (compare Fig. 6B with C).

Stability of the duration of the phases

The duration of the ®rst exponential phase (t 1 ±t 0 ) averaged 92.3T1.7 °Cd, with no signi®cant effect of phytomer position or treatment (Fig. 6D). Its variability was higher for phytomers 2 to 5, probably because the rates of the two successive phases were rather close for such leaves (e.g. phytomer 2, Figs 2 and 3, bottom graphs), making it dif®cult to date precisely a shift between them. For modelling purposes, the duration of the ®rst phase was taken to be common to all leaves studied (Fig. 4).

The duration of the second exponential phase (t 2 ±t 1 ) increased with phytomer number (Fig. 6E) according to a The insert is a zoom of the main curve to make visible the differences between isolated plants and canopy plants for phytomers 5 to 15. Vertical bars correspond to standard errors and are generally smaller than symbol size on main ®gures. Dotted lines correspond to a negative exponential ®t for the ®ve upper `true' leaves (equation 6 in text). The x-intercept corresponded to the phytomer position of the ®rst involucral bract. (C) The main ®gure indicates the relative expansion rate at the beginning of the phase (R 3 ). The relative expansion rate then exponentially decreased with time (rate=parameter a) according to equation 2 (see text). Inserts illustrate the proportionality between R 3 and the parameter a (insert above the main curve) and the equality between R 2 (rate of the second phase) and R 3 for isolated plants (insert below the main curve).

pattern common to plants of hybrids G100 and Albena, either isolated or grown in a canopy. It increased by about 30 °Cd and 7 °Cd per phytomer below and above phytomer 5, respectively. The duration of the second exponential phase was similar in isolated plants and plants grown in a canopy at any given phytomer (Fig. 6E). It was also similar in both hybrids for all alternate leaves (from phytomer 5 upwards, Fig. 6E). Duration was slightly shorter for leaf pairs of G100 at phytomers 2 to 4 (Fig. 6E).

The end of leaf expansion, t f , was calculated with equation 5 as the date when leaf area A equalled 95% of its ®nal value. The duration of the third phase (t f ±t 2 ) increased with phytomer number, but did not signi®cantly differ between isolated plants and plants in a canopy, and between hybrids (Fig. 6F).

A whole-plant representation of leaf development to analyse the response of leaf expansion to environment and genotype Four leaf developmental stages, namely leaf initiation, end of ®rst exponential phase, end of second exponential phase and end of expansion (resp. t 0 , t 1 , t 2 , and t f in Fig. 4) were required to characterize the time-course of leaf expansion. Representing the dates of occurrence of these four events for each phytomer illustrates the co-ordination of the development of the successive leaves (Fig. 7A for hybrid Albena and 7D for hybrid G100). Taken together for a given stage (e.g. initiation), these dates de®ne the progression of this stage along the stem. For each hybrid, the number of initiated phytomers against thermal time was ®tted to three successive linear regressions corresponding to (i) phytomers bearing leaf pairs (phytomer number <5), (ii) intermediate phytomers (phytomers 5 to 25 and 5 to 20 for Albena and G100, respectively), and (iii) the ®ve upper phytomers bearing `true' leaves (Fig. 7A,D). The rates of phytomer initiation of the three successive phytomer classes of Albena plants equalled 0.018, 0.106 and 0.245 phytomer °Cd ±1 (r 2 =0.993, 0.997 and 0.987, respectively, P <0.0001). Initiation rates of G100 were 30± 50% lower: 0.014, 0.071 and 0.125 phytomer °Cd ±1 (r 2 =0.999, 0.997 and 0.997, respectively, P <0.0001). The progression of the other stages (Fig. 7A,D) was deduced from that of initiation and from the duration of the phases according to phytomer number described above (Fig. 6).

This whole-plant diagram (Fig. 7A,D) constitutes a framework of analysis of leaf expansion by positioning for each leaf the dates of development pertinent for its expansion. Differences in ®nal leaf area between isolated plants and plants in a canopy concerned phytomers 6 and upwards (Fig. 7B), and were linked to different expansion rates during the second exponential phase (Fig. 6B). Leaf expansion was therefore affected by interplant competition for light between the end of exponential growth at phytomer 5 and that at phytomer 6. This occurred around 300 °Cd after plant emergence (dotted vertical line on Fig. 7A), long before any detectable effect on visible leaves (Fig. 7C). The growth rate of the second exponential phase of subsequent leaves was more and more affected (Fig. 6B), due to an increasing part of this exponential phase occurring during the constraint (Fig. 7A).

Differences between the two genotypes in the timecourse of leaf area per plant (Fig. 7C) were mainly due to the slower initiation rate and to the lower ®nal number of leaves of hybrid G100 compared to Albena (Fig. 7A,D). The delayed growth curve of G100 (Fig. 7C) was associated to delayed development of successive individual leaves. For example, leaf of phytomer 15 of G100 was initiated 172 °Cd after seedling emergence, i.e. 60 °Cd later than the equivalent leaf of Albena, and this delay was conserved through the successive stages of development of that leaf (vertical arrows on Fig. 7A,D) until full expansion (765 versus 705 °Cd). The lower ®nal leaf area per plant of G100 (Fig. 7C) was mainly explained by the lower ®nal leaf number (Fig. 7A,D). Changing the phytomer number of the ®rst involucral bract (N b ) in equation 6 (28 instead of 33.5) was enough to account for the ®nal area of leaves of G100 compared with Albena.

Discussion

A common pattern of development for all leaves of two sun¯ower hybrids grown under ¯uctuating conditions The results suggest that, irrespective of phytomer position, tested environmental conditions and genotype, leaf development consists of the same succession of three phases characterized by the thermal-time-course of leaf relative expansion rate (Fig. 4 and Equations 3±5). Internode extension in maize was described with a similar threephase framework (Fournier and Andrieu, 2000a).

The relative expansion rate was high (0.06 °Cd ±1 ) and common to all phytomers during a ®rst short period (92 °Cd). It was followed by a second exponential phase with a different rate, while the relative expansion rate decreased continuously during the third phase. The earliest period was not mentioned in previous works on phytomer 6 (Granier and Tardieu, 1998a), probably due to a limited number of measurements during the few days following primordium initiation, and because it is not easily distinguishable from the next period for phytomers close to the plant base (due to similar relative expansion rates during the two phases), as is the case for phytomer 2 in the present work (Figs 2,3). It was only evident when examining higher phytomers (Figs 2, 3), and gathering data from 30 phytomers (Fig. 5). Adding this ®rst exponential period to the two-phase framework previously presented (Granier and Tardieu, 1998a) is not problematic because the parameters of this early phase (i.e. primordium area at initiation, and rate and duration of the phase) were common to all leaves, and seemed to be insensitive to the environmental conditions tested.

This breakdown of leaf development allows a simple analysis of the effects of genotypes and growing conditions on leaf expansion. Leaf area strongly differed among the different situations described here. Nevertheless, these differences were accounted for by a limited number of variables of leaf growth and development. As shown for phytomer 6 (Granier and Tardieu, 1998b), the duration of the phases was stable for any given phytomer through different years and locations if expressed on a thermal time basis. This stability was not observed for elongation of internodes in maize submitted to more severe light de®cit (Fournier and Andrieu, 2000b). Irrespective of phytomer position along the stem, the response to a limitation of intercepted light per plant affected leaf expansion only during the exponential phase via a reduction of the relative expansion rate, as observed for phytomer 6 [START_REF] Granier | Leaf expansion and cell division are affected by reducing absorbed light before but not after the decline in cell division rate in the sun¯ower leaf[END_REF]. The relative expansion rate was affected neither during the ®rst exponential phase, nor during the third phase (rapid growth phase). As a consequence, in the studied cases, the ®nal leaf area was determined as early as the end of the second exponential phase, approximately half-time between initiation and full expansion.

Relative expansion rate decreased and exponential growth duration increased with phytomer number according to a pattern common to different genotypes and environments Differences between phytomers concerned only the rate and the duration of the second exponential phase. These differences ranged according to a strict pattern common to the two studied hybrids and the different environmental conditions: the duration increased linearly, while the rate decreased continuously with phytomer position. This pattern resulted in the typical bell-shaped distribution of ®nal leaf area along the stem [START_REF] Kobayashi | Growth analysis of plants as an assemblage of internodal segments: a case of sun¯ower plants in pure stands[END_REF]. Decreasing relative expansion rates versus phytomer number were observed on other species, including dicots [START_REF] Cordero | Plastochron index for juvenile and mature forms of Hedera helix L. (Araliaceae)[END_REF][START_REF] Granier | Individual leaf development in Arabidopsis thaliana: a stable thermal-time-based programme[END_REF] and monocots [START_REF] Gallagher | Field studies of cereal leaf growth[END_REF][START_REF] Lafarge | A model co-ordinating the elongation of all leaves of a sorghum cultivar was applied to both Mediterranean and Sahelian conditions[END_REF]. A linear increase in the duration of expansion with phytomer number was also described in pea [START_REF] Turc | Leaf primordium initiation and expanded leaf production are co-ordinated through similar response to air temperature in pea (Pisum sativum L.)[END_REF] and sorghum [START_REF] Lafarge | A model co-ordinating the elongation of all leaves of a sorghum cultivar was applied to both Mediterranean and Sahelian conditions[END_REF]. Conversely, [START_REF] Kobayashi | Growth analysis of plants as an assemblage of internodal segments: a case of sun¯ower plants in pure stands[END_REF] considered that the duration from leaf emergence to full expansion was common to all phytomers, but its analysis was based on calendar time and not thermal time.

A ®rst consequence of these results is that the notion of plastochron index, a continuous developmental scale based on leaf number [START_REF] Erickson | The plastochron index[END_REF], largely used in physiological studies in a wide range of species [START_REF] Lamoreaux | The plastochron index: a review after two decades of use[END_REF][START_REF] Gagnon | Establishment of a plastochron index and analysis of the sink-to-source transition in leaves of Moricandia arvensis (L.) DC. (Brassicaceae)[END_REF] should be used with great caution in sun¯ower. One of the three underlying assumptions, i.e. similar relative growth rates between successive leaves [START_REF] Erickson | The plastochron index[END_REF][START_REF] Lamoreaux | The plastochron index: a review after two decades of use[END_REF] is not ful®lled, except during the earliest phase. Nevertheless, a leaf plastochron index could be acceptable if limited to a few consecutive leaves or to a relatively short period of plant development [START_REF] Russin | Modeling nonexponential growth of a short-cycling plant, Tagetes erecta (Asteraceae)[END_REF][START_REF] Gagnon | Establishment of a plastochron index and analysis of the sink-to-source transition in leaves of Moricandia arvensis (L.) DC. (Brassicaceae)[END_REF].

The involvement of reproductive development in leaf expansion rates has been shown, for example, on leaves of Hedera helix which exhibit speci®c morphology and anatomy depending on whether they were initiated before (juvenile) or after (mature) ¯ower induction [START_REF] Stein | Comparative developmental anatomy of shoots of juvenile and adult Hedera helix[END_REF]. The relative expansion rate remains stable on successive juvenile leaves, whereas it decreases with phytomer number for mature leaves [START_REF] Cordero | Plastochron index for juvenile and mature forms of Hedera helix L. (Araliaceae)[END_REF]. Similarly, relative expansion rate remains stable until leaf 37 for short-day-grown Arabidopsis thaliana producing more than 40 leaves [START_REF] Groot | Short-day-grown Arabidopsis thaliana satis®es the assumptions of the plastochron index as a time variable in development[END_REF], while it decreases as early as leaf 6 for long day-grown plants producing 9 leaves [START_REF] Granier | Individual leaf development in Arabidopsis thaliana: a stable thermal-time-based programme[END_REF]. In the present study, the effect of the reproductive development on the dynamics of the relative expansion rate seemed to be limited to the ®ve last initiated leaves. For those leaves, the same pattern, based on phytomer distance to the capitulum, was obtained for two hybrids differing in ®nal leaf number, irrespective of absolute phytomer number (Fig. 6B). On the other hand, the continuous decrease in exponential rate observed for the two hybrids from plant base (phytomers 2 to 20) preceded reproductive events. It seemed to be ontogenetic, but not directly driven by reproductive development.

Other patterns of leaf development associated with phytomer position are frequently mentioned in the literature. For example, many plants produce different types of leaves during their developement [START_REF] Allsopp | Heteroblastic development in vascular plants[END_REF][START_REF] Poethig | Phase change and the regulation of shoot morphogenesis in plants[END_REF], and some of these differences, concerning leaf shape and anatomy, are regular features of shoot development and are considered as components of a genetically regulated programme of shoot maturation [START_REF] Poethig | Leaf morphogenesis in ¯owering plants[END_REF][START_REF] Kerstetter | The speci®cation of leaf identity during shoot development[END_REF]. The regular change with phytomer position of duration and rate of expansion observed here could be dependent on such a programme, modulated by reproductive events and environmental conditions.

Conclusion

The whole-plant framework established in this paper is a tool to structure the analysis of the response of leaf expansion to genotypes and environment. It identi®es a set of relevant parameters and variables (rates and duration, and their pattern of change with phytomer number), the variability of which has now to be analysed and quanti®ed in response to a wider range of genotypes and environmental conditions. Because it locates spatially and temporally the growth of the organs and the susceptibility of the expansion to environmental constraints (depending on organ developmental stage), this framework could constitute a pertinent phenological module for a computergenerated 3-dimensional representation of virtual plants responding to environment, thus simulating the phenotypes [START_REF] Rey | Numerical sun¯ower plants responding to light and temperature[END_REF].

fellowship of the Comision de Investigaciones Cientõ ®cas (CIC, Pcia de Buenos Aires, Argentina).

Fig. 1 .

 1 Fig. 1. Final leaf area as a function of phytomer number for isolated plants (open circles, 1998 and ®lled squares, 1999) or plants in a dense canopy (open squares) of ®eld-grown hybrid Albena (A), and for plants of hybrids Albena (open squares) and G100 (®lled triangles) grown in a dense ®eld canopy (B). Vertical bars correspond to standard deviation.

Fig. 2 .

 2 Fig. 2. Thermal-time-course of leaf area (linear and logarithm scale) and leaf relative expansion rate at phytomers 2, 10, 18, and 26 of isolated plants of hybrid Albena grown in 1998 (open circles) and 1999 (®lled squares). Vertical bars correspond to standard deviation. The three successive vertical dashed lines on each graph correspond, respectively, to the end of the ®rst exponential phase, the end of the second exponential phase and the leaf full expansion. Solid line corresponds to the three-phase ®t (see text and Fig. 4).

Fig. 3 .

 3 Fig. 3. Thermal-time-course of leaf area (linear and logarithmic scale) and leaf relative expansion rate at phytomers 2, 10, 18, and 24 for plants of hybrids Albena (open squares) and G100 (®lled triangles) grown in a dense canopy. Vertical bars correspond to standard deviation. The three successive vertical dashed lines on each graph correspond, respectively, to the end of the ®rst exponential phase, the end of the second exponential phase and the leaf full expansion. The dotted line corresponds to the end of the second exponential phase at phytomer 2 for Albena. All other dates were common to both hybrids. Solid line corresponds to the three-phase ®t (see text and Fig. 4).

Fig. 4 .

 4 Fig. 4. A three-phase ®t to describe the time-course of expansion of all leaves studied. The change with thermal-time of the logarithm of the leaf area (B) was ®tted to two successive linear functions (®rst and second exponential phases) and to an exponentially decreasing relative expansion rate during the third phase (phase with rapid leaf growth on a linear scale (A)). Corresponding change with time of the relative expansion rate is indicated in C (dotted line). The solid line in C corresponds to the time-course of the relative expansion rate when averaged on periods of 60 °Cd to be consistent with experimental data (Figs 2 and 3, bottom graphs). Vertical dashed lines indicate the end of the phases.Solid arrows indicate the characteristic parameters of any given leaf: t 0 , the date of initiation, R 2 the rate of the second exponential phase, and t 2 , the end of the exponential growth. Dashed arrows indicate the parameters either common to all leaves (A 0 , the leaf area at initiation, t 1 and R 1 , the duration and the rate of the ®rst exponential phase), or deduced from the preceding ones (R 3 , the relative expansion rate at the beginning of the third phase, a, the rate of exponential decrease of the relative expansion rate during the rapid leaf growth, and t f , the date of full expansion).

Fig. 5 .

 5 Fig. 5. Thermal-time-course of leaf area (logarithmic scale) between 0 and 100 °Cd after leaf initiation for phytomers 3 to 30 for isolated plants of hybrid Albena grown in 1998 (open circles) and 1999 (®lled squares), and plants of hybrids Albena (open squares) and G100 (®lled triangles) grown in a dense ®eld canopy.

Fig. 6 .

 6 Fig. 6. Relative expansion rate (A, B, C) and duration (D, E, F) of each of the three phases of leaf development as a function of phytomer position for isolated plants of hybrid Albena grown in 1998 (open circles) and 1999 (®lled squares), and plants of hybrids Albena (open squares) and G100 (®lled triangles) grown in a dense ®eld canopy. (B) The insert is a zoom of the main curve to make visible the differences between isolated plants and canopy plants for phytomers 5 to 15. Vertical bars correspond to standard errors and are generally smaller than symbol size on main ®gures. Dotted lines correspond to a negative exponential ®t for the ®ve upper `true' leaves (equation 6 in text). The x-intercept corresponded to the phytomer position of the ®rst involucral bract. (C) The main ®gure indicates the relative expansion rate at the beginning of the phase (R 3 ). The relative expansion rate then exponentially decreased with time (rate=parameter a) according to equation 2 (see text). Inserts illustrate the proportionality between R 3 and the parameter a (insert above the main curve) and the equality between R 2 (rate of the second phase) and R 3 for isolated plants (insert below the main curve).

Fig. 7 .

 7 Fig. 7. A whole-plant framework to analyse leaf expansion in sun¯ower. Number of initiated phytomers, number of phytomers beyond ®rst exponential phase, number of phytomers beyond exponential growth and number of phytomers with full expanded leaves as a function of thermal time after plant emergence in sun¯ower hybrids Albena (A) and G100 (D). Seed imbibition (imb.) occurred about 100 °Cd before emergence. Final leaf area according to phytomer number (B, E), and thermal-time-course of plant leaf area (C) for Albena isolated plants (®lled squares) and plants of hybrids Albena (open squares) and G100 (®lled triangles) grown in a dense canopy. For a given phytomer, for example, phytomer 15 in (A) (Albena) and (D) (G100), the diagram indicates the dates of occurrence of the four developmental stages characterizing leaf expansion (vertical arrows). At a given date, e.g. 300 °Cd (vertical dotted line), it indicates the number and the nodal position of the phytomers in each of the phases of development.

Table 1 .

 1 Final leaf area per plant, rates of leaf development and dates of reproductive development for isolated plants of hybrid Albena and plants of hybrids Albena and G100 grown in a dense ®eld canopy Values are means TSD.

	Hybrid	Density	Final leaf area per plant (m 2 )	Rate of phytomer initiation ( °Cd ±1 )	Rate of leaf appearance ( °Cd ±1 )	Capitulum initiation ( °Cd)	Anthesis (°Cd)
	Albena	Isolated	2.34T0.09	0.106T0.002	0.094T0.002	236T13	824T14
	Albena	Canopy	1.06T0.06	0.106T0.002	0.090T0.002	236T13	832T16
	G100	Canopy	0.82T0.06	0.071T0.002	0.058T0.003	297T14	902T15

Acknowledgements

Guillermo Dosio thanks the Ministerio de Cultura y Educacio Ân de la Nacio Ân Argentina, the French Embassy in Argentina, the De Âpartement Environnement et Agronomie (INRA, France), and the Universidad Nacional de Mar del Plata (Argentina) for grants supporting his PhD thesis in France. We thank Benoõ Ãt Suard for daily technical assistance from sowing to data analysis, Thierry Lefeuvre for data collection and analysis in 1998, Guillaume Sinniger for data collection in 1999, Philippe Naudin for irrigation management in 1998, and Cristina Larraburu for technical assistance in Argentina experiments. This work was partly supported by the Convention de coope Âration scienti®que INRA (France)±INTA (Argentina). Luis Aguirrezabal is member of the Consejo Nacional de Investigaciones Cientõ ®cas y Te Âcnicas (CONICET, Argentina). Natalia Izquierdo is


José Geraldo Telles

Julio Cesar De Castro Basilio

Americo Cunha
email: americo@ime.uerj.br

Tiago Roux Oliveira
email: tiagoroux@uerj.br

On the classical and fractional control of a nonlinear inverted cart-pendulum system: a comparative analysis

Keywords: fractional calculus, state-derivative feedback, fractional-order control, cart-pendulum nonlinear system

The use of fractional-order controllers to drive dynamical systems to a desired/target configuration became extremely popular in the last decade, with many studies stating that they present superior performance when compared to the integer-order counterparts, especially for nonlinear systems. Following this trend, the purpose of this chapter is to verify the possibility of improving the performance of the control of an inverted cart-pendulum system using fractional-order integrators. The strategy is to employ the classical pole location linear method to calculate the gains of the controller and then to compare the performance between integer-order and fractional-order integrators, the last one that are calculated using an optimization method.

Introduction

The idea of the Fractional Calculus was introduced in 1695 when Bernoulli, Leibniz, and L'Hôpital exchanged letters about the possibility of a non-integer derivative order. This discussion involved many mathematicians as Euler, Fourier, and Laplace, among others [START_REF] Monje | Fractional-order systems and controls: fundamentals and applications[END_REF][START_REF] Ortigueira | Fractional calculus for scientists and engineers[END_REF][START_REF] Ortigueira | What is a fractional derivative?[END_REF]. During the following centuries, many pure and applied mathematicians contributed to the development of the fractional calculus theory and many different fractional operators have been proposed, among them the Grünwald-Letnikov, Riemann-Liouville and Caputo fractional derivatives (Li and Dend, 2007).

The Riemann-Liouville fractional integral [START_REF] Li | Remarks on fractional derivatives[END_REF] of order 𝛼, where 𝛼 ∈ ℝ and 𝛼 > 0, is defined in terms of a convolution type operation between the real-valued function 𝑦(𝑡) and the kernel 𝑡 !"# 𝐼 ! %,' 𝑦(𝑡) = 1 𝛤(𝛼) / (𝑡 -𝜏) !"# 𝑦(𝜏) 𝑑𝜏

' % , 𝑡 > 𝑎 (1)
being 𝛤 the Gamma function, and the Riemann-Liouville fractional derivative (Li and Dend, 2007) is defined in terms of the classical derivative of order 𝑛 ∈ ℤ ( of this fractional-order integral

𝐷

)* ! %,' 𝑦(𝑡) = 1 𝛤(𝑛 -𝛼) 𝑑 + 𝑑𝑥 + / (𝑡 -𝜏) +"!"# 𝑦(𝜏) 𝑑𝜏 ' % 𝑡 > 𝑎, 𝑛 -1 < 𝛼 < 𝑛 ,

(2) which is a global operator (not local as in classical calculus) that presents "memory".

However, other definitions for a fractional derivative are also possible, like one by Grünwald-Letnikov [START_REF] Li | Remarks on fractional derivatives[END_REF]) / (𝑡 -𝜏) +"!"# 𝑦 (+) (𝜏) 𝑑𝜏 ' % 𝑡 > 𝑎, 𝑛 -1 < 𝛼 < 𝑛 .

(4)

The development of new definitions is an active field of research in mathematics [START_REF] Caputo | A new definition of fractional derivative without singular kernel[END_REF]Khalil el al. 2014;Zheng andZhao 2019 Ortigueira andTrujillo 1012;[START_REF] Katugampola | New approach to a generalized fractional integral[END_REF], but in the last decade, mainly thanks to the development of numerical methods to simulate fractional systems [START_REF] Li | Numerical approaches to fractional calculus and fractional ordinary differential equation[END_REF][START_REF] Deng | Efficient algorithms for solving the fractional ordinary differential equations[END_REF][START_REF] Tepljakov | FOMCOM: a MATLAB toolbox for fractional-order system identification and control[END_REF], fractional operators started to be used in engineering analysis [START_REF] Katsikadelis | Generalized fractional derivatives and their applications to mechanical systems[END_REF][START_REF] Caputo | Hysteresis cycles and fatigue criteria using anelastic models based on fractional derivatives[END_REF]Lewandoswski and Pawlak 2018;[START_REF] Lin | Eigenvalue and eigenvector derivatives of fractional vibration systems[END_REF][START_REF] Dai | On generalized fractional vibration equation[END_REF] and control theory [START_REF] Tepljakov | Fractional-order modeling and control of dynamic systems[END_REF][START_REF] Shah | Review of fractional PID controller[END_REF][START_REF] Chen | Fractional-order PIλ controller synthesis for steam turbine speed governing systems[END_REF][START_REF] Bingul | Comparison of PID and FOPID controllers tuned by PSO and ABC algorithms for unstable and integrating systems with time delay[END_REF][START_REF] Li | Fractional active disturbance rejection control[END_REF][START_REF] Balachandran | Controllability of fractional damped dynamical systems[END_REF][START_REF] Barbosa | Effect of fractional-orders in the velocity control of a servo system[END_REF]Wang et al. 2009 Xue and[START_REF] Xue | Fractional-order PID controller design for fractionalorder system[END_REF][START_REF] Delavari | Fractional-order control of a coupled tank[END_REF]), especially when dealing with time-delay [START_REF] Martelli | Stability of PID-controlled second-order time-delay feedback systems[END_REF] or nonlinear and chaotic systems [START_REF] Azar | Fractional-order control and synchronization of chaotic systems[END_REF][START_REF] Niu | Analysis of Duffing oscillator with time-delayed fractionalorder PID controller[END_REF]Shen et al. 2014;Shen et al. 2014).

Among the favorable characteristics these fractional-order operators offer, two of them can be highlighted: (i) their global application provide a natural framework for describing phenomena with memory; and (ii) the fractional exponent offers a kind of additional degree of freedom to tune a controller, opening opportunities for additional improvements of performance in the controller design.

Trying to explore this second feature, in this manuscript a state-feedback control system is proposed to stabilize an inverted cart-pendulum system (Figure 1) and then, the possibility of improving the performance of the control system using fractional integrators is analyzed. The inverted pendulum has been chosen as a benchmark since it is a classical control problem a nonlinear system widely studied using integer-order controllers [START_REF] Kharola | A comparison study for control and stabilisation of inverted pendulum on inclined surface (IPIS) using PID and fuzzy controllers[END_REF][START_REF] Wang | MBPOA-based LQR controller and its application to the doubleparallel inverted pendulum system[END_REF][START_REF] Prasad | Optimal control of nonlinear inverted pendulum system using PID controller and LQR: performance analysis without and with disturbance input[END_REF][START_REF] Wang | Simulation studies of inverted pendulum based on PID controllers[END_REF], which started to be tested in fractionalcontrol literature as well [START_REF] Mousa | Optimal fractionalorder proportional-integral-differential controller for inverted pendulum with reduced order linear quadratic regulator[END_REF].

Proposed control system

Figure 2 shows the state-space feedback control system proposed to control the inverted cart-pendulum system. It can be noted that the angular velocity of the pendulum and the velocity of the mass are chosen to be the observed states and they must be integrated 𝑰 𝜶 , as defined by the Riemann-Liouville fractional integral in Equation 1, to obtain the angular position of the pendulum and the position of the mass, respectively. The pole location is a classical and widely method used to calculate the values of each gain 𝑘 7 and that has been developed for linear systems using integerorder controllers. The locations of the poles are chosen based in the performance desired for the closed-loop system. Since this method is based on linear system hypothesis, the linearized cart-pendulum model is employed

𝑚𝑙 𝑥̈+ (𝐼 + 𝑚𝑙 ! )𝜃 ̈-𝑚𝑔𝑙𝜃 = 0 (𝑀 + 𝑚)𝑥̈+ 𝑚𝑙𝜃 ̈-𝑢 = 0, ( 5
)
where M is the cart mass; 𝑚 is the inverted pendulum mass; I is pendulum moment of inertia; 𝑙 is the distance from the center of pendulum's mass to the fixation point; 𝜃 is the angle between the pendulum and a perpendicular axis through the cart's centroid; 𝑥 is the cart's horizontal displacement; and 𝑢 is the force applied to the cart to control the system. Now define the state variables 𝑥 # , 𝑥 8 , 𝑥 9 and 𝑥 : as 𝑥 " = 𝜃, 𝑥 ! = 𝜃 ̇, 𝑥 3 = 𝑥, 𝑥 4 = 𝑥̇. Then considering the angular position 𝜃 and cart position 𝑥 as the outputs of the system, obtained the equations for the system as follows:

𝐱̇= 𝐀𝐱 + 𝐁 𝑢, 𝑦 = 𝐂𝐱, 𝑢 = -89𝑘 1 𝐼 𝛼 1 + 𝑘 2 :𝑥 ! + 9𝑘 3 𝐼 𝛼 2 + 𝑘 4 :𝑥 # +; + 𝑘 𝑙 𝜉, 𝐷 𝛼 𝑙 𝜉 = 𝑟 -𝑦 = 𝑟 -𝐼 𝛼 3 𝑥 # , ( 7
)
where 𝐱 is the state vector; 𝑢 is the control signal; 𝑦 is the output signal; 𝜉 is the output of the integrator; 𝑟 is the reference input signal; 𝐀 is the matrix of states; 𝐁 is the input matrix; 𝐂 is the output matrix; and 𝐊 is the gain matrix (𝑘 # , 𝑘 8 , 𝑘 9 and 𝑘 : ).

For this cart-pendulum, the matrices are:

𝐀 = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 0 1 0 0 𝑚𝑔𝑙(𝑀+𝑚) 𝐼(𝑀+𝑚)+𝑀𝑚𝑙 2 1 0 0 0 0 0 1 - 𝑚 2 𝑔𝑙 2 𝐼(𝑀+𝑚)+𝑀𝑚𝑙 2 0 0 0 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ , 𝐁 = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 0 - 𝑚𝑙 𝐼(𝑀+𝑚)+𝑀𝑚𝑙 2 0 𝐼+𝑚𝑙 2 𝐼(𝑀+𝑚)+𝑀𝑚𝑙 2 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ , 𝐂 = [ 1 0 0 0 ]. (8)
The following values are adopted for the system parameters:

𝑚 = 0.1 𝑘𝑔, 𝑀 = 2 𝑘𝑔, 𝑙 = 0.5 𝑚 and 𝐼 = 0.006 𝑘𝑔 • 𝑚 8 .
The linear model using classical control will be used to obtain values for the 𝑘′𝑠. It will be used the pole location method and the closed-loop poles are chosen to be located at:

𝜇 # = -1 + 𝑗√3 𝜇 8 = -1 -𝑗√3 𝜇 9 = -5 𝜇 : = -5 𝜇 ? = -5,
where 𝑗 = √-1.

Using the Matlab to calculate the 𝑘′𝑠, the following values are obtained:

𝑘 # = -200.6 𝑘 8 = -50.3 𝑘 9 = -70.1 𝑘 : = -46.8 𝑘 @ = -63.8.
The next step is to verify if the performance of the system can be improved changing the orders of the integrators by simulations using the FOMCOM toolbox [START_REF] Tepljakov | FOMCOM: a MATLAB toolbox for fractional-order system identification and control[END_REF]. The simulations are done considering the following initial conditions:

𝜃 2 = 10° 𝜃 ̇2 = 0 𝑥 2 = 0 𝑥̇2 = 0.
Since it is desired to verify the basins of attraction of the closed loop system, the simulations also consider a nonlinear model of the inverted cart-pendulum, given by 𝑚𝑙 𝑐𝑜𝑠 𝜃 𝑥̈+ (𝐼 + 𝑚𝑙 ! )𝜃 ̈-𝑚𝑔𝑙 𝑠𝑖𝑛 𝜃 = 0

(𝑀 + 𝑚)𝑥̈+ 𝑚𝑙 𝑐𝑜𝑠 𝜃 𝜃 ̈-𝑚𝑙 𝑠𝑖𝑛 𝜃 𝜃 ̇! -𝑢 = 0. ( 9
)
Another decision is to maintain the values of 𝛼 9 and 𝛼 A equal to 1. This decision is based in the fact that lower values would introduce an offset in the mass position and higher values would make the position of the mass more oscillatory.

Fractional-controller optimization

Simulations are performed to evaluate the behavior of the control-system integrators with the combination of different orders, integer and fractional, and so find the point or region where the performance criteria are minimal.

These simulations are divided into two stages: global analysis and detailed analysis. In the first stage, the global analysis is performed observing the performance in a larger range of integrators, where the integrators had their order varied by 0.05 between: 0.1 ≤ 𝛼 # ≤ 2.0 and 0.1 ≤ 𝛼 8 ≤ 2.0.

Based on the results of this simulation, the second stage have been performed, reducing the interval of the order of the integrators according to the region that has the lowest ISE criterion values and settling time. Therefore, for detailed analysis, several systems are simulated with variations of 0.01 within the ranges:

0.6 ≤ 𝛼 # ≤ 1.2 and 0.7 ≤ 𝛼 8 ≤ 1.3.
The performance of this system is evaluated using the following criteria:

• Integral of the Squared Error (ISE) of the angular position of the pendulum:

𝐽 B = / 𝜃 8 𝑑𝑡 C 2
• Settling time of the pendulum 𝑡 B .

• Integral of the Squared Error (ISE) of the cart position:

On the classical and fractional control of a nonlinear inverted cart-pendulum system …

𝐽 D = / 𝑥 8 𝑑𝑡 C 2
• Settling time of the cart 𝑡 D . These criteria are imposed to verify the best combination of the orders 𝛼 # and 𝛼 8 .

Angular position of the pendulum

In the problem of an inverted cart-pendulum, the angular position of the pendulum is the most important output of the system for dynamic control, especially when the purpose of the control is to keep the pendulum in equilibrium in a near upright position. Therefore, the shorter the time and the oscillation for the system to reach this equilibrium, the better its performance. Therefore, during the simulations below, the setting time 𝑡 B and the ISE criterion 𝐽 B of the angular position are observed.

Global analysis

The first simulation to understand the behavior of the angular position varies 𝛼 # and 𝛼 8 in the range 0.1 to 2. In this way, the ISE criterion for different combinations of integrators produce the results shown in Figures 3 and4. This first result already allows one to conclude that a system with low amplitudes and small oscillations (smaller 𝐽 B ) can be obtained through fractional integrators, as they present better results than integer integrators (𝛼 # = 𝛼 8 = 1). Figure 4 shows the region of integrators that have the lowest ISE value in blue, reaching minimum values close to 𝐽 B =0.01.

Observing the results of another evaluation criterion, the numbers of combinations of integrators can be reduced, further detailing the analysis. Figures 5 and6 show the results of the settling time of the first simulation. As in the results of the ISE criterion, the shortest settling time of the system does not correspond to the integers, but the combination of the fractional integrators 𝛼 # = 0.85 and 𝛼 8 = 0.90, as shown in Figure 5. These results also contribute to the details of the study, since it reduces the number of combinations of integrators with satisfactory results. If a controller with integer integrators (𝛼 #,8 = 1) has a settling time of 4.8 seconds, then the satisfactory results will be those below that. Therefore, for the next simulation the interval is reduced to:

0.6 ≤ 𝛼 # ≤ 1.2 and 0.7 ≤ 𝛼 8 ≤ 1.3.
This range is represented by the central region of the settling time of the contour plot (Figure 6), bounded by dashed lines, where this minimum point and the other points satisfactory.

Detailed analysis

In this second simulation, the intention is to find the minimum ISE criterion value of the angular position (

𝐽 E = ∫ θ 8 𝑑𝑡 C 2
) together with the settling time 𝑡 B less than 4.8 seconds.

Thus, from the range defined in items 3.2 and 4.1.1, the ISE criteria and the settling time of the different combinations of integrators presented the results shown in Figures 7,8, 9 and 10. The results of the graph in Figures 7 and8 indicate to the minimum value of 𝐽 E equal to 0.014, with the fractional integrators approximately 0.94 ≤ 𝛼 # ≤ 1.00 and 0.70 ≤ 𝛼 8 ≤ 0.74, reaffirming that the use of fractional-order in the integrator is more advantageous.

As shown in Figure 9, the minimum settling time of the angular position is 2.315 and 3.182 seconds, with fractional integrators equal to 𝛼 # = 1.20 / 𝛼 8 = 0.70 and 𝛼 # = 0.86 / 𝛼 8 = 0.90 , respectively. However, the region around 𝛼 # = 1.20 / 𝛼 8 = 0.70 has high values of 𝐽 B (see figure 7). Figure 10 shows that there is a large region with different combinations of integrators with 𝑡 B below 4.8 seconds, even so for the two criteria these regions are different. Therefore, it is necessary to find an intersection of these criteria.

Therefore, to find the minimum value, the following optimization criterion is used Minimize 𝐽 B = ∫ 𝜃 8 𝑑𝑡.

C

Imposing a settling time for the pendulum 𝑡 B ≤ 4.8𝑠 and for the cart position 𝑡 D ≤ 5𝑠, the values obtained for the control system are 𝛼 # = 1.08 and 𝛼 8 = 0.95.

Position of the cart

The dynamic control of the cart's position output is important when the system's objective is to reduce the cart's travel as much as possible while maintaining the pendulum in equilibrium in the vertical position, i.e. the final angle 𝜃 F near zero. Therefore, the shorter the settling time for the system to reach this equilibrium with lower amplitudes and oscillations, the better its performance. Therefore, during the simulations below, the settling time 𝑡 D and the ISE criterion 𝐽 D of the cart position are observed.

Global analysis

The simulation is performed to comprehend the behavior of the cart position in a global way, varying 𝛼 # and 𝛼 8 in the range of 0.1 to 2. Thus, the ISE criterion of the different combinations of integrators presented the results shown in Figure 11 and 12. To reduce the number of combinations of integrators and to further detailed analysis, the results of settling time are observed, as shown in Figures 13 and14.

Figures 13 and14 shows that the shortest settling time in the system does not correspond to the integer integrators, but the combination of the fractional integrators 𝛼 # = 0.95 and 𝛼 8 = 0.95, with settling time equal to 4.0 seconds. These results also contribute to the details of the study, since it reduces the number of combinations of integrators with satisfactory results. If a controller with integer integrators (𝛼 #,8 = 1) has a settling time of the cart position equal to 5.0 seconds, then the satisfactory results are be those below that. Therefore, the same simulation of the detailed analysis of the angular position is conducted with intervals equal to:

0.6 ≤ 𝛼 # ≤ 1.2 and 0.7 ≤ 𝛼 8 ≤ 1.3.
This range is represented by the central region of the settling time of the contour plot (Figure 14), bounded by dashed lines, where this minimum point and the other points satisfactory.

Detailed analysis

In the detailed analysis the objective is to find the minimum ISE criterion value of the cart position (𝐽

G = ∫ x 8 𝑑𝑡 C 2
) together with the 𝑡 D settling time less than 5 seconds. Thus, from the interval defined in items 3. Figure 18 shows that there is a small region with different combinations of integrators with 𝑡 D below 5 seconds, even so for the two criteria these regions are different. Therefore, it is necessary to find an intersection of these criteria.

Therefore, to find the minimum value, the following optimization criterion is used

Minimize 𝐽 D = ∫ 𝑥 8 𝑑𝑡 C 2 .
Imposing a settling time for the pendulum 𝑡 B ≤ 4.5𝑠 and for the cart position 𝑡 D ≤ 5𝑠, the values obtained for the control system are also 𝛼 # = 1.10 and 𝛼 8 = 0.95.

Optimized controller

The previous items indicate that for both, the control of the angular position and the control of the position of the cart, the ideal value for the order of the integrators is equal to 𝛼 # = 1.08/1,10 and 𝛼 8 = 0.95. Thus, it is concluded that the fractionals result in a better performance. Otherwise, the optimization should have indicated the order of the integrators equal to 1.

To reinforce the advantages of fractional controllers, graphical and performance comparisons are made, based on the criteria: settling time and ISE criteria. The fractional integrators 𝛼 # = 1.09 (middle term between the optimization of 𝐽 B and 𝐽 D ) and 𝛼 8 = 0.95 were used. The graphical comparisons of system responses are shown in Figure 19 and Figure 20.

The graphs with the comparison between the control systems with integer and fractional integrators show better performance in both criteria. The values of the performance criteria of both systems are presented in the Table 1. The results in Table 1 refer to the optimization of the ISE criterion with the settling time, where the ISE angular position presenting the best performance among the other criteria (approximately 23%). There is also an improvement of the 14% in the ISE criteria of the cart position and the settling time of the angular position. The only one that does not have a considerable improvement is the settling time of the cart position. However, it is concluded that there is a possibility of improving the controllers using fractional integrators.

The ideal order values found can be different according to the priority that is given to the dynamic control of the system, that is, if the objective is to improve the settling time of the cart independent of the settling time pendulum, the order of the integrators will be different of the optimum found.

Basins of attraction

Optimized controller (𝜶 𝟏 = 𝟏 and 𝜶 𝟐 = 𝟎. 𝟗𝟓)

The basins of the attraction of the system are compared with integer and fractional integrators. Figure 21 show the result of the attraction basins in terms of angular position and angular velocity. For the fractional controller, 𝛼 # = 1.09 and 𝛼 8 = 0.95 are considered.

The graph shown in Figures 21 show that the use of fractional-order integrators reduces the basin of attraction, despite improving performance. With the integer-order the angular position of the attraction basin is close to 50 degrees, while the fractional-order is below 40 degrees. The reduction in the number of controllable cases with different initial conditions using the fractional-order was 36% compared to the integer-order. Controller with wider attraction basin (𝜶 𝟏 = 𝟎. 𝟗𝟎 and 𝜶 𝟐 = 𝟎. 𝟖𝟓)

From the previous results, where the basin of attraction was reduced using an optimal controller, seek to find combinations of fractional orders that resulted result the opposite, i.e. an extension of the attraction basin. Therefore, the fractional integrators 𝛼 # = 0.90 and 𝛼 8 = 0.85 are used and Figure 22 shows the attraction basins of the three cases (integer-order, optimal fractional-order and fractional-order with a wider attraction basin). The basin of attraction of the new integrator with fractional orders 𝛼 # = 0.90 and 𝛼 8 = 0.85 had an increase of 11% compared to the whole order integrator basin. Thus, it can be said that the use of fractional orders reduces and enlarges the attraction basin, being related to the combination of orders used. In addition, this reduction or expansion is also related to the performance of the controller (settling time and ISE criteria). To assess the impact on the system's behavior, performance and graphical comparisons were made. The graphical comparisons of system responses are shown in Figure 23 and Figure 24.

The graphs show that performance is impaired, with fractional order resulting in higher amplitudes. For a better analysis, the settling time and the ISE criteria are shown in Table 2. Among the results of the criteria presented in Table 2, only the setting time of the angular position of the pendulum has the best performance in comparison with the integer-order controller, with a time reduction of 34%, even better than the optimal controller. The most significant worsening in performance was in the settling time of the cart position, with an increase in time of 18%. The other criteria had no significant variations.

Conclusions

Based on the results shown in this chapter, it can be concluded that the using of fractional integrators improves the performance of the control of a cart-pendulum system reducing the ISE of both the angular position and the position of the cart and the settling time either for the same combinations of 𝑘 7 , but, as expected, the basin of attraction is reduced.

However, disregarding some performance criteria, fractional integrators can expand the attraction basin and improve a certain performance criterion at the same time, as shown in Table 2, where even with the 11% increase in the attraction basin, there is an 34% reduction in settling time compared to integer-order integrators.

The proposed approach seems to be very appealing for control systems in which state-derivatives are easier to be obtained or measured than the state signals [START_REF] Assunção | Robust state-derivative feedback LMI-based designs for multivariable linear systems[END_REF]).

Figure 1 :

 1 Figure 1: Schematic illustration of the cart-pendulum system.

Figure 2 :

 2 Figure 2: Illustration of the proposed controller for the inverted cart-pendulum system.

Figure 3 :

 3 Figure 3: ISE criterion of the angular position for 𝟎. 𝟏 ≤ 𝜶 𝟏,𝟐 ≤ 𝟐.

Figure 4 :

 4 Figure 4: ISE criterion of the angular position for 𝟎. 𝟏 ≤ 𝜶 𝟏,𝟐 ≤ 𝟐.

Figure 5

 5 Figure 5 shows regions with extremely low levels of settling time, however these regions have very oscillation values and high 𝐽 & values. Therefore, the point where the settling time and little oscillation is lower has been indicated in the graph.

Figure 5 :

 5 Figure 5: Settling time of the angular position for 𝟎. 𝟏 ≤ 𝜶 𝟏,𝟐 ≤ 𝟐.

Figure 7 :

 7 Figure 7: ISE criterion of the angular position for 𝟎. 𝟔 ≤ 𝜶 𝟏 ≤ 𝟏. 𝟐 and 𝟎. 𝟕 ≤ 𝜶 𝟐 ≤ 𝟏. 𝟑 .

Figure 8 :

 8 Figure 8: ISE criterion of the angular position for 𝟎. 𝟔 ≤ 𝜶 𝟏 ≤ 𝟏. 𝟐 and 𝟎. 𝟕 ≤ 𝜶 𝟐 ≤ 𝟏. 𝟑.

Figure 9 :

 9 Figure 9: Settling time of the angular position for 𝟎. 𝟔 ≤ 𝜶 𝟏 ≤ 𝟏. 𝟐 and 𝟎. 𝟕 ≤ 𝜶 𝟐 ≤ 𝟏. 𝟑.

Figure 10 :

 10 Figure 10: Settling time of the angular position for 𝟎. 𝟔 ≤ 𝜶 𝟏 ≤ 𝟏. 𝟐 and 𝟎. 𝟕 ≤ 𝜶 𝟐 ≤ 𝟏. 𝟑.

Figure 11 :

 11 Figure 11: ISE criterion of the cart position 𝟎. 𝟏 ≤ 𝜶 𝟏,𝟐 ≤ 𝟐.

Figure 12 :

 12 Figure 12: ISE criterion of the cart position for 𝟎. 𝟏 ≤ 𝜶 𝟏,𝟐 ≤ 𝟐.

Figure 14 :

 14 Figure 14: Settling time of the cart position for 𝟎. 𝟏 ≤ 𝜶 𝟏,𝟐 ≤ 𝟐.

  2 and 4.2.1, the ISE criterion and the settling time of the different combinations of integrators presented the results shown in Figures 15, 16, 17 and 18.

Figure 15 :

 15 Figure 15: ISE criterion of the cart position for 𝟎. 𝟔 ≤ 𝜶 𝟏 ≤ 𝟏. 𝟐 and 𝟎. 𝟕 ≤ 𝜶 𝟐 ≤ 𝟏. 𝟑 .

Figure 16 :

 16 Figure 16: ISE criterion of the cart position for 𝟎. 𝟔 ≤ 𝜶 𝟏 ≤ 𝟏. 𝟐 and 𝟎. 𝟕 ≤ 𝜶 𝟐 ≤ 𝟏. 𝟑. The results of the graph in Figures 15 and 16 indicate to the minimum value of 𝐽 G equal to 0.22, with the fractional integrators approximately 1.10 ≤ 𝛼 # ≤ 1.15 and 0.95 ≤ 𝛼 8 ≤ 1.05, again reaffirming that the use of fractional-order in the integrator offers better results.

Figure 17 :

 17 Figure 17: Settling time of the car position for 𝟎. 𝟔 ≤ 𝜶 𝟏 ≤ 𝟏. 𝟐 and 𝟎. 𝟕 ≤ 𝜶 𝟐 ≤ 𝟏. 𝟑.

Figure 18 :

 18 Figure 18: Settling time of the cart position for 𝟎. 𝟔 ≤ 𝜶 𝟏 ≤ 𝟏. 𝟐 and 𝟎. 𝟕 ≤ 𝜶 𝟐 ≤ 𝟏. 𝟑.

Figure 19 :

 19 Figure 19: Time series of the angular position of the pendulum when ISE criteria is minimized imposing 𝒕 𝜽 ≤ 𝟒. 𝟖𝒔 and 𝒕 𝑿 ≤ 𝟓𝒔.

Figure 20 :

 20 Figure 20: Time series of the position of the cart when ISE criteria is minimized imposing 𝒕 𝜽 ≤ 𝟒. 𝟖𝒔 and 𝒕 𝑿 ≤ 𝟓𝒔.

Figure 21 :

 21 Figure 21: Basin of attraction in the angular position vs angular velocity plane for the integrator with integer-order (𝜶 𝟏 = 𝟏 and 𝜶 𝟐 = 𝟏) and fractional-order (𝜶 𝟏 = 𝟏. 𝟎𝟗 and 𝜶 𝟐 = 𝟎. 𝟗𝟓).

Figure 22 :

 22 Figure22: Basins of attraction in the angular position vs angular velocity plane for the three cases of the integrators: integer-order, optimal fractional-order and fractional-order with a wider attraction basin.

Figure 23 :

 23 Figure 23: Comparison of time series of the angular position of the pendulum between integer-order and fractional-order with wider attraction basins.

Figure 24 :

 24 Figure 24: Comparison of time series of the position of the cart between integer-order and fractional-order with wider attraction basins.

Table 1 -Performance values of systems with integer and fractional integrators after opti- mization of the ISE criterion of the angular position

 1

			ISE			Settling Time
	Parameter
		IO	FO	%	IO	FO	%
	Angular position	68.63	53.04	-23	4.87	4.19	-14
	Cart position	0.26	0.22	-14	5.07	4.91	-3

Table 2 -Performance values of systems with integer and fractional integrators with wider attraction basins.

 2

			ISE			Settling Time
	Parameter
		IO	FO	%	IO	FO	%
	Angular position	68.63	70.70	+3	4.87	3.20	-34
	Cart position	0.26	0.27	+4	5.07	6.00	+18

Acknowledgments

The authors are grateful for the financial support received from the Brazilian agencies CAPES, CNPq and FAPERJ.