Djillali Mezhoud

Youcef Bouafia

Mohammed Saad

Jacqueline Saliba

Bond-slip law and short tie behavior without main cracks

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Introduction

During dimensioning reinforced concrete structural elements (Reinforcement and stress testing), the contribution of tensioned concrete is totally neglected, which is sufficient in this case. However, for practical reasons, consideration should be given to:

• The estimation of the crack openings in the service limit state (SLS).

• The calculation of the mean rigidity of the tie rod to correctly evaluate the deformations at the SLS and even at the ultimate limit state (ULS) during the redistribution of forces in the hyperstatic structures (Effects of second order in shape stability and the response in alternating stresses).

CONTACT djillali Mezhoud djillali.mezhoud@hotmail.fr

Numerous studies have been carried out to evaluate the contribution of tensed concrete between cracks. Two types of approach are considered:

• The distribution hypothesis of the adhesion along the tie-rod, without involving the sliding [START_REF] Brice | Idées générales sur la fissuration du béton armé et du béton précontraint [General ideas on the reinforced concrete cracking and prestressed concrete[END_REF]. • Using the equilibrium of forces and the compatibility of deformations from experimental relationships, such as the model of the European Concrete Committee [START_REF]CEB-FIP Model Code. Comité Euro-international du Béton-Fédération Internationale de la Précontrainte[END_REF], between the local stress of steel/concrete adhesion and the corresponding slip, to lead at the resolution of differential equations [START_REF] Balazs | Cracking analysis based on slip and bond stresses[END_REF][START_REF] Chaussin | Bases de la théorie de la fissuration, Fissuration et durabilité du béton [Cracking theory basics, cracking and concrete durability[END_REF][START_REF] Saad | Influence du pourcentage d'acier sur le comportement du béton tendu fissuré: Calcul de l' ouverture des fissures par le biais de "l'acier fictif[END_REF].

To express the solution of the problem, two types of formulation are distinguished:

• Mean fictitious behavioral laws of taut concrete, taking into account the addition of the reinforcement and concrete contributions to the cross section of the 'tie rod' [START_REF] Grelat | Calcul non linéaire des ossatures en béton armé[END_REF][START_REF] Quast | Zur Mitwirkung des Betons in der Zugzone[END_REF][START_REF] Vecchio | The modified compression-field theory for reinforced concrete elements subjected to shear[END_REF][START_REF] Fouré | Déformations limites des armatures tendues et du béton comprimé pour le calcul sismique des structures [Limit deformations of tension reinforcement and compressed concrete for the structures seismic calculation[END_REF]. • Fictitious behavioral laws of tensile steel (concrete-coated steel), implicitly taking into account tensed concrete in the form of a corrective term (Relative elongation deviation Δε, between bare steel and concrete-coated steel for the same tensile force) [START_REF] Espion | Rigidité d'une zone tendue de béton armé [Rigidity of a reinforced concrete tensile zone[END_REF][START_REF] Kwak | Cracking analysis of RC members using polynomial strain distribution function[END_REF][START_REF] Saad | Influence du pourcentage d'acier sur le comportement Post -fissuration du béton armé en traction[Steel percentage influence on the post-cracking behavior of reinforced concrete in tension[END_REF].

Many studies have been carried out on the concrete steel connection, with series of experimental tests [START_REF] Bae | Bond stress between conventional reinforcement and steel fibre reinforced reactive powder concrete[END_REF][START_REF] Metelli | Influence of the relative rib area on bond behaviour[END_REF][START_REF] Casanova | Confinement effects on the steel-concrete bond strength and pull-out failure[END_REF][START_REF] Murcia | bond strength and cyclic bond deterioration of large-diameter bars[END_REF], mostly, the study of the geometry influence of the ribs on the resistance and rigidity [START_REF] Metelli | Influence of the relative rib area on bond behaviour[END_REF] and the effects of passive and active confinement on the maximum stress of the bond [START_REF] Casanova | Confinement effects on the steel-concrete bond strength and pull-out failure[END_REF]. Other studies propose models of local adhesion-slip relationships [START_REF] Bae | Bond stress between conventional reinforcement and steel fibre reinforced reactive powder concrete[END_REF][START_REF] Tastani | Reinforcement and concrete bond: state determination along the development length[END_REF][START_REF] Mang | A new bond slip model for reinforced concrete structures: validation by modelling a reinforced concrete tie[END_REF]. Several authors have investigated the behavior and estimation of crack openings [START_REF] Alam | Measuring crack width and spacing in reinforced concrete members[END_REF][START_REF] Carino | Prediction of cracking in reinforced concrete structures[END_REF][START_REF] Bernardi | Experimental and numerical study on cracking process in RC and R/FRC ties[END_REF] in the tie rods, and especially during the phase of stabilization of these cracks. During this last phase, the tie rod is composed of several 'short tie rods' . The study of these latter makes it possible to understand how the forces redistribution is effected and the evolution of widths of crack openings because limited to the limit state of service by the different regulations around the world evolve [START_REF] Mang | Crack opening estimate in reinforced concrete walls using a steel-concrete bond model[END_REF].

The contribution of concrete, through experimental relations, is expressed here by fictitious behavior laws of steel in tension for short ties without major cracks.

To do this, pull out tests with different reinforcement diameters of high adhesion have been carried out to characterize the first phase of the law of stress-slip behavior (τg) between steel and concrete. Short ties specimens were made with the same concrete, the same cross-section with varying the steel rebar's diameter.

The adhesion-slip curve is obtained through pull-out tests [START_REF] Mang | A new bond slip model for reinforced concrete structures: validation by modelling a reinforced concrete tie[END_REF][START_REF] Zhandarov | Analysis of a pull-out test with real specimen geometry. Part II: the effect of meniscus[END_REF][START_REF] Zhandarov | Analysis of a pull-out test with real specimen geometry. Part I: matrix droplet in the shape of a spherical segment[END_REF][START_REF] Casanova | Bond slip model for the simulation of reinforced concrete structures[END_REF] in order to characterize the curvilinear curve.

Tests on short ties, whose lengths are chosen so that no major cracks appear, will help us to better understand the mode of variation of the mobilization of steel/concrete adhesion and to propose behavior laws of steel in tension for such elements.

The behavior of short ties without major cracks also makes it possible to understand the response of tie rods in general [START_REF] Handika | Influence of interface zone behaviour in reinforced concrete under tension loading: an analysis based on modelling and digital image correlation[END_REF]. In this work, the short ties without cracks highlight according to the loading and characteristics of the two materials the moment when a maximum mobilization of the adhesion is obtained. From this moment, it is interesting to see how this adhesion between steel and concrete decreases. This last observation makes it possible to understand the decrease of the contribution of the concrete after the stabilization of the cracks in a tie rod. ( -g) between steel and concrete adopted by the European

Curves

Concrete Committee

The European Concrete Committee [START_REF]CEB-FIP Model Code. Comité Euro-international du Béton-Fédération Internationale de la Précontrainte[END_REF] adopted the stress-sliding relations 1, 2, 3 and 4 between steel and concrete. These curves (Figure 1) are derived from standardized steel bar coated with concrete pull out tests.

The value of is between 0.25 and 0.40 depending on the confinement of the concrete. The adhesion peak τ 1 is taken as: With a = 2.5 for a confined concrete, otherwise a = 2

The normal compressive stress limit of the concrete f ck is expressed in MPa. The value of the residual stress τ r is taken equal to 0.20 1 .

If the modeled curve (Figure 1) of the different phases seems rather representative, the intervals of variation of the different slips g 1 , g 2 and g 3 are closer to those given by Eligehausen [START_REF] Eligehausen | Local bond stress-slip relationships of deformed bars under generalized excitations[END_REF].

These results (curves until adhesion depletion) are possible only if the percentage of steel in the concrete is very low, generally lower than the minimum ratio allowed by the various regulations around the world. For common cases [START_REF] Saad | Influence du pourcentage d'acier sur le comportement du béton tendu fissuré: Calcul de l' ouverture des fissures par le biais de "l'acier fictif[END_REF][START_REF] Saad | Contribution à l' évaluation d' ouverture des fissures dans les éléments en béton armé [Contribution to evaluation of cracks opening in reinforced concrete elements[END_REF], only the first curve (Figure 1) (relation 1) is effective.

(1) 3. Variation mode of stresses-deformations of short tie rods ( st )

= 1 g g 1 if 0 ≤ g ≤ g 1 (2) = 1 if g 1 ≤ g ≤ g 2 (3) =-1 -3 g 3 -g 2 g + 1 g 3 -3 g 2 g 3 -g 2 if g 2 ≤ g ≤ g 3 (4) = r if g ≥ g 3 (5) 1 = a √ f ck
An analytical method is developed in this paper in order to study the short ties behavior based on the experimental results.

A short tie with a length l t = 2L , a section A t (a × a) constant along the tie and a centered rebars section A s , is considered and subjected to a variable tensile load N t (Figure 2).

The normal stress σ c (x) in the concrete (with a Young Modulus E c ) is considered uniformly distributed and the normal stress in steel σ s (x) (with a modulus of elasticity E s ) remains lower to elastic stress of steel σ e .

Figure 3 shows the normal stress-relative strain diagram (σ -ɛ) of the tie rod where ɛ t represents the relative elongation of the tie rod and Δ s the difference with the bare steel under the same normal stress sn = N t ∕A s .

The rebar is embedded in the concrete and the origin of the abscissa is considered at the point O (Figure 4(a) and (b)). It is clear that, after the application of the load N t , which is relatively small, the adhesion stress increases until it reaches a maximum, then decreases to zero at the abscissa l 1 , called introductory length, (Figure 4(c)). We suppose that the maximum of τ is located at the ends of the tie rod, Figure 4(d).

The normal stress in steel rebar and the relative elongation corresponding to the ends of the tie are denoted respectively σ sn and ɛ sn . The normal stress in steel is variable along l 1 ,σ s1 (ɛ s1 ), and constant along Ll 1 , σ s2 (ɛ s2 ), Figure 4(e). Respectively, we have for concrete along l 1 , σ c1 (ɛ c1 ) and for Ll 1 ,σ c2 (ɛ c2 ).

Here the concrete length is not long enough so that the mobilization of the adhesion allows to create a crack, (relation 6).

With: f ctm : Resistance (average) to effective tensile strength of the concrete; A c : Net section of concrete subjected to tension (A c = A t -A s ).

Variation of the relative elongation of steel over the length l 1 of the tie rod

With (p)the useful perimeter of the steel section and n = E s ∕E c , the variations of the relative elongations of steel and concrete along the length l 1 are written respectively: For the two intervals of length l 1 on either side of the tie rod expressing the equilibrium of strength and the expression of the slip g between the elongations of steel u s and concrete u c , g(x)=u c (x)-u s (x). We can write:

(6) ∫ L (x).p.dx < A c .f ctm s1 (x) = sn - p A s .E s x ∫ 0 ( ).d c1 (x) = p A s .E s . n. (1 -) x ∫ 0 ( ).d
And the differential equation 7 can be obtained: The relation 8 represents the solution of the differential equation [START_REF] Balazs | Cracking analysis based on slip and bond stresses[END_REF][START_REF] Chaussin | Bases de la théorie de la fissuration, Fissuration et durabilité du béton [Cracking theory basics, cracking and concrete durability[END_REF][START_REF] Saad | Influence du pourcentage d'acier sur le comportement Post -fissuration du béton armé en traction[Steel percentage influence on the post-cracking behavior of reinforced concrete in tension[END_REF], and the variation of the bond stress 𝜏 can be deduced

dg dx = du c dx - du s dx = c (x) -s (x) ⇒ d 2 g dx = d c dx - d s dx (7) d 2 g(x) dx 2 - p A s .E s . ̄𝜌 𝜏(x) = 0 Where ̄𝜌 = 1 1 + n.𝜌 (1-𝜌) , 𝜌 = A s A t
The variation of the adhesion along l 1 and its maximum at the extremities τ l1 for x = 0 are written by expressions 9 and 10.

Thus, the normal stress exerted at the ends of the tie-rod (nude steel), replaced by its longitudinal relative deformation sn (

dg dx x=0
= sn ) can be written as:

The variation of the longitudinal relative deformation ɛ s1 of the steel along the length l 1 is deduced and after simplification we obtain the expression 12:

Variation of the relative elongation of the steel over the length (Ll 1 ) of the tie rod

At the distance x, comprised in the interval [l 1 ; L] the behavior of the tie-rod is homogeneous, concrete and steel resist together to balance the stress N t . There is compatibility in the deformations ( c2 = s2 )and the sliding between the reinforcement and the concrete is equal to zero (u s = u c ). We may write, in any straight section of this interval of the tie-rod:

(8) g(x) = 1 ∕ (1 -) l 1 -x 2 ∕ (1 -) where = 𝛽 2 1 (1 -𝛼) 2 2(1 + 𝛼) , 𝛽 1 = √ k 1 .p A s .E s . ̄𝜌 , k 1 = 𝜏 1 g 𝛼 1 (9) 𝜏(x) = k 1 ∕ (1 -) l 1 -x 2 ∕ (1 -) (10) 𝜏 l1 = k 1 ∕ (1 -) l 2 ∕ (1 -) 1 (11) sn = 2 (1 -) . 1 ∕ (1 -) l (1 + ) ∕ (1 -) 1 s1 (x) = sn - p A s .E s x ∫ 0 ( ).d = sn - p A s .E s x ∫ 0 k 1 ∕ (1 -) l 1 - 2 ∕ (1 -) .d (12) s1 (x) = ⎡ ⎢ ⎢ ⎣ 1 -̄𝜌 + ̄𝜌 � 1 - x l 1 � (1 + 𝛼) ∕ (1 -𝛼) ⎤ ⎥ ⎥ ⎦ 𝜀 sn for 0 < x ≤ l 1 N t = c2 A t -A s + s2 .A s = c2 A t -A s + n. c2 .A s N t A s = sn = E s . sn = E c . c2 n + 1 - = E c . s2 n + 1 - = E s . s2 1 + 1 - n.
This gives the relation 13 which represents the constant relative elongation of the steel over the length (Ll 1 ).

Proposition of a behavior law of short tie-rods ( -)

With theoretical results above and observations made by several authors on the slip between steel and concrete at the beginning of loading, we propose here laws of behavior σ -ɛ of the short tie rods. This is in agreement with the study conducted by Giuriani [START_REF] Giuriani | On effective axial stiffness of a bar in cracked concrete[END_REF] who showed that the sliding is effective for the spline bars only after a certain pulling force, and thus the τg curve does not pass through the origin of the abscissa g.

Eligehaussen & al [START_REF] Eligehausen | Local bond stress-slip relationships of deformed bars under generalized excitations[END_REF] reported that the slope of the curve is relatively high at the beginning. By increasing the pulling force of the bar, the ribs of the steel rebar abut on the concrete and begin to crush it. At this moment, the first cracks appear and the slip at this stage is relatively small (point A in Figure 5). The partial shearing of concrete rods results in a reduction in the slope between point A and B. The point C represents the moment when the maximum adhesion stress is obtained if the confinement is sufficient. Then the residual stress decreases due to the friction along the sheared concrete zone in the descending branch.

For our study, with relations 12, 13 and Figures 3 and4, assuming a linear variation of σ -ɛ curve of short tie, the diagrams slopes (minimum and maximum) can be deduced for phase 1. With relation 12, when l 1 = L, we obtain the minimum slope:

As a first approximation, we can take the average: s1 = 1 -̄𝜌 2 𝜀 sn and the minimal slope represented by the fictitious module is deduced:

E s 1-̄𝜌 2
With relation 13, when l 1 tends to 0, we obtain the maximum slope either:

E s 1-̄𝜌
So for the first phase, the dummy module E f1 is equal, relation 14 For phase 2, after maximum mobilization of the adhesion between steel and concrete, this bond deteriorates and only a residual adhesion remains, in this case it is possible to take as fictitious module E f2 , relation 15

N t A s = E s .𝜀 sn = E s (1 -̄𝜌 ) 𝜀 s2 (13) 𝜀 s2 = (1 -̄𝜌 ).𝜀 sn for L -l 1 𝜀 s1 = 𝜀 sn if x = 0 𝜀 s1 = (1 -̄𝜌 )𝜀 sn if x = l 1 (14) E f 1 = E s 𝜓 1 with (1 -̄𝜌 ) <𝜓 1 < 1 - ̄𝜌 2 (15) E f 2 = E s 𝜓 2 with 0, 8 <𝜓 2 < 0, 9
The ψ 1 and ψ 2 values are less than unity and will be estimated after carrying out the experimental tests. The curve σ -ɛ limit points of the short tie rods are A and B Figure 3. At point A, at maximum mobilization of adhesion (l 1 = L), Figure 4(e) and using relation 11, the relative elongation of bare steel is written, expression 16.

We have also:

σ snL = E s ɛ snL
In our study, the normal stress in steel is limited to σ e (also ɛ e ) corresponding to the fictional relative elongation of steel ɛ sfe represented by point B, Figure 3 and determined by relation 17. ( 16) In summary, for the short tie-rods, at the elastic limit of concrete, the relation 18 represents the fictitious diagram σ -ɛ of the steel:

𝜀 snL = 2 1 - . 1 ∕ (1 -) L (1 + ) ∕ (1 -) (17) 𝜀 sfe = e -snL E f 2 + sfL with sfL = snL E f 1

Materials

Concrete

The particle size analysis was carried out according to the French standard NF P 18-560 (Table 1), with a maximum diameter, D max = 12.5 mm for gravel and a fineness module for sand, MF = 2.03. A single formulation of an ordinary concrete has been used for all series of tests. The composition of the concrete for one cubic meter determined by Faury's method is given in Table 2. The use of large aggregates (15∕25)was avoided considering the relatively small dimensions of specimens. Prismatic (15 × 15 × 15) cm 3 and cylindrical 11cm × 22.5 cm specimens have been pre- pared for compression and tensile tests respectively. The results of the various properties of the concrete are given in Table 3.

Steel

The used rebars for the pull out and the tie rods tests are steels with high rib adhesion (Figure 6). Tensile tests on nude bars of 18 cm long have been carried out. A yield stress of e = 530 MPa and a longitudinal elastic modulus of E s = 200 GPa have been calculated for the used diameters �=8, 12 and 14 mm.

Note here that the adhesion between steel and concrete depends on the characteristics of the ribs.

The works, of Rehm [START_REF] Rehm | The fundamental law of bond[END_REF], Sorez & Hölzenbein [START_REF] Soretz | Influence of Rib Dimensions of Reinforcing Bars on Bond and Bendability[END_REF] as well as Martin & Noakowski [START_REF] Martin | Verbundverhalten von Betonstählen -Untersuchung auf der Grundlage von Ausziehversuchen[END_REF] showed the geometric importance of the ribs of rebars in the steel-concrete adhesion.

The Studies carried out by Caetano & al [START_REF] Caetano | Bond strength and rib geometry: a comparative study of the influence of deformation patternns on anchorage bond strength[END_REF] and Handika & al [START_REF] Handika | Influence of interface zone behaviour in reinforced concrete under tension loading: an analysis based on modelling and digital image correlation[END_REF] showed the influence of the ribs in considering the height h 1 , the spacing e 1 , the angle of the ribs α 1 (45 to 50 • ) and the angle of inclination of the ribs α 2 (45 to 50 • ) (Figure 7).

The geometrical characteristics of the ribs of the reinforcement used for our tests are shown in Table 4.

Pull out tests

Pull-out tests are carried out with the machine shown in Figure 8. They are conducted on the same bars used for the tie rods. 8 and9)

Geometry of specimens and procedure (Figures

Prismatic specimens with a cross section of 100 mm × 100 mm and a length of 15φ where φ is the rebar diameter were used (Figure 9(a)). Reinforced concrete pull out specimens were designed with a length varying in function of rebars diameter φ. Reinforcing steel rods were placed prior to casting of concrete with a protective layer on the bottom and upper side of embedded steel over a length of 5φ using a PVC hose covering as shown in Figure 9(a) in order to avoid inaccurate stresses in the specimens and friction in the sample. Rebars are embedded into a 15φ concrete cube and the bond length of the steel bar and concrete was equal to5 . As the section of specimens is kept constant, similar to that of RC tensile tests, the percentage of steel is varied = A s ∕A t where A s and A t represent respectively the section of the steel and the section of the specimen.

(18) ⎧ ⎪ ⎨ ⎪ ⎩ = E f 1 . if 0 ≤ ≤ sfL = E f 2 . � -sfL � + snL if sfL ≤ ≤
The pulling out and the withdrawal of the reinforcing element from concrete are performed in the axial tensile mode using a testing machine with a special metal framework. The sample was placed in a holder with a stopper fixed at the lower clamp of the testing machine while the free top part of the rod was fixed by the testing machine (Figure 9(b,c)). The sample was loaded by pulling out the rod at a constant velocity in the vertical direction. The loading force and displacement of the mobile clamp were continuously monitored. In addition, two LVDT sensors were placed in order to measure the slip between the rod and concrete on the bottom and the upper side of the sample in order to study any perturbations as flexion.

Experimental pull out results

Pull out tests were realized on reinforced concrete specimens with high adherence rebars HA8, HA12 and HA14 (Three specimens for each type), and thus different percentage of steel 0.50, 1.13 and 1.54% respectively.

The bond stress slip relationships (τg) are presented in Figure 10 for rebars with different diameters. Tests with the HA14 rebars have broken out and are not shown here.

Characteristic values of the adhesion slip law ( -g)

Based on the bond stress slip curves obtained in Figure 10 and numerical results, the characteristics of the 𝜏 -g law can be deduced. Eligehausen & al [START_REF] Eligehausen | Local bond stress-slip relationships of deformed bars under generalized excitations[END_REF] law, adopted by the European Concrete Committee [START_REF]CEB-FIP Model Code. Comité Euro-international du Béton-Fédération Internationale de la Précontrainte[END_REF] is used for the first phase, with the tests values, we can write the expression 19.

The value of the confinement c for the tie-rods used here is given by the relation 20:

The value of 𝜏 1max is deduced directly from the maximal load obtained with the various pull out tests (Table 5) and the first part of the curvilinear curve at the beginning of loading.

Knowing two states of loading j(g j ;𝜏 j ) and k(g k ;𝜏 k ) with the loading level at k superior to that of j and 𝜏 k >𝜏 j :

Considering the ratio of (b) to (a), expression 21 can be deduced When α is determined, and for the state at the beginning of loading (g;𝜏), g 1max can be deduced by the relation 22:

In our case, α was considered equal to 0.33, the others results are given in Table 5.

Note: The European Concrete Committee [START_REF]CEB-FIP Model Code. Comité Euro-international du Béton-Fédération Internationale de la Précontrainte[END_REF] uses here two coefficients α and a to express the same parameter: the confinement of the concrete. To keep only one, knowing that for = 0.25 and = 0.40 corresponding a = 2 and a = 2.5 respectively, the relation 5 can be written:

(19) 𝜏 = 𝜏 1max g g 1max (20) c = 50 -�∕2
� and c in mm

⎧ ⎪ ⎨ ⎪ ⎩ 𝜏 j = 𝜏 1max � g j g 1max � (a) 𝜏 k = 𝜏 1max � g k g 1max � (b) (21)
= log k j log g k g j = log k j -log g k g j (22) g 1max = g. 𝜏 1max 𝜏 1 ∕ (23) 𝜏 1 = 10 3 + 7 6 √ f ck
Table 5. characteristic values of the τ -g law deduced from the pull out curves.

Diameter of the rebar �(mm) HA8 HA12 Percentage of steel (A s /A t = steel section/Tie-rods section) 0.503% Based on the experimental tests, the conventional curve adopted by the European Concrete Committee [START_REF]CEB-FIP Model Code. Comité Euro-international du Béton-Fédération Internationale de la Précontrainte[END_REF], underestimated the value of τ 1 . Therefore, the relation 24 is proposed:

6. Tests on short tie-rods without main cracks

Specimens dimensions and preparation

The section of the tie-rods is the same as those of the pull out tests, Figures 9(a). Figure 11 presents a sketch of the short ties, where rebars are placed in the center with a total length of 480 mm. Rebars are embedded into a length of l t = 190 mm chosen so that no principal cracking can appear during tensile tests. At both end of the RC ties, PVC tubes of 30 mm, are placed in order to avoid inaccurate stresses in the and friction and splitting in the sample. Three specimens were prepared for each type of tie rod (THA8, THA12 and THA14).

Operating mode on the tie-rods

After casting, specimens were covered with a thin sheet of plastic to prevent water loss and were maintained in a climatic chamber at a temperature of 20 °C and a relative humidity (RH) of 95%. Twenty four hours after casting, specimens were stripped off from the molds and kept for curing in lime water, under a temperature condition of 20 °C for 28 days. After 28 days, tensile tests on reinforced concrete ties are carried out with a hydraulic machine with a capacity of 200 kN.

The lower free part of the rods was fixed while the other free part was subjected to a tensile load N t with a constant displacement kinetic of 1 mm/min (Figure 11).

Two cameras (cam 0 and cam 1), were also used in order to measure the displacement of both free parts of the rebar (gau 1, gau 2 and gau 3) (Figure 11). The obtained informations allow us to plot the stress strain diagram (σ -ɛ) of each short tie.

Experimental results and analyzes

The contribution of concrete is studied (in function of the adhesion) in a tie rod by measuring the difference between:

where ɛ sn and ɛ t = Δl t /l t represent, at the same instant, the relative elongations respectively of the steel assumed to be nude and the tie rod under the force N t .

We present in Figure 12, the σ -ɛ curves of tie-rods with diameters equal to HA8 (THA8) and HA12 (THA12).

The relation 25 provides an overview of the variation of adhesion between steel and concrete. The plotting of the curve Δ𝜀 as a function of loading (represented here in Figure 13 by the stress in the steel) shows:

• For THA8 (Δ𝜀8), the 'mobilized' adhesion in the tie rod increases linearly up to the force N t = 16 kN, or a normal stress in the steel rebar of sn = 319 MPa. The relative deformation of the tie rod is then 𝜀 t = 8.5.10 -5 , which corresponds to Δ𝜀8 = 1.51.10 -3 . From this level of loading, the slope decreases considerably (Figures 12 and13). The modulus of elasticity of the tie rod is slightly greater than that of the steel rebar. • For THA12 (Δ𝜀12), the adhesion mobilized between the steel and the concrete in the tie rod increases linearly up to the load N t = 15.7 kN, which corresponds to a normal stress in the steel rebar sn = 139 MPa. At this moment, the relative deformation of the tie rod is equal to 𝜀 t = 3.84.10 -4 , which corresponds to Δ𝜀12 = 3.12.10 -4 . A reduction in the adhesion between the steel and the concrete is then observed (Figures 12 and13) and the modulus of elasticity of the tie rod is close to that of the steel. With the plasticization of the steel ( sn = 450 MPa), the deterioration of the adhesion appears.

Deductions of 1 and 2 values of the proposed laws

At the beginning of loading, phase 1, the experimental curves (Figure 12) are approximated by taking the expression 26 as the value of the coefficient ψ 1 . The other parameters (1α) and γ take into account, respectively, the confinement and the coating.

( As a first approximation for the second phase, after maximum mobilization of the adhesion and its degradation we can take for the coefficient 2 , relation 27:

) Δ𝜀 = 𝜀 sn -𝜀 t (26) 𝜓 1 = 1 - ̄𝜌 𝛾.(1 -) with ⎧ ⎪ ⎨ ⎪ ⎩ 𝛾 = 1 if c ≥ 5.�( or 𝜌<1%) 𝛾 = � � c � (1+ ) if c < 5.�( or 𝜌 ≥ 1%) 25
For the examples treated, THA8 and THA12, with the data of Tables 3 and5 and the relationships 14 to 18, 26 and 27, the equations in Tables 6 and7 are obtained. The representation of these theoretical curves in continuous line is illustrated, Figure 12.

Conclusions

• In the estimation of the adhesion peak τ 1 , of the τg law proposed by the European Concrete Committee, the coefficient 'a' can be replaced by its equivalent as a function of α so as to keep the same confinement parameter (see Relation 24). • The raw (unconventional) characteristic values of the τg law of the first curvilinear curve proposed by the European Concrete Committee can be estimated in the pull out tests at the begining of the curve by the relations proposed here (Relations 21 and 22). • For the pull out tests, under the same conditions, the order of magnitude of the maximum peak of adhesion does not depend on the percentage of steel because the adherent side surface of the bar to the concrete is all the greater as the perimeter is. • The length of the tie-rods here does not allow an adhesion mobilization capable of creating a crack. The introduction length must not exceed half that of the tie-rod. • The fictitious module slope of steel is higher the lower the percentage of steel. After a certain level of loading (maximum mobilization of the adhesion), the slope decreases and approaches that of the bare steel (10 to 20%). At the approach of the elastic normal stress, this slope declines, this is partly explained by the Poisson effect.

( • The proposed laws, here, of simple analytic expressions, for short tie-rods without main cracks, suitably approach the test curves. However, other contributions are needed to improve this approach. • It is clear that each steel-concrete mixture must include, after the granulometric analysis and the concrete formulation, a specific 'pull-out' study to be able to deduce the tie rods behavior. • The results obtained in this study will serve to better understand the behavior of tie-rods in general (Fictitious diagram of steel and concrete, spacing of cracks, crack openings, …). Indeed, during this last phase, the tie rod is composed of 'short tie rods without main cracks' object of this study. Tensile load on the tie rod s = s (x) normal stress of concrete-coated steel at a distance x from the end of the tie rod 𝜀 s = 𝜀 s (x) relative elongation of concrete-coated steel at a distance x from the end of the tie rod c = c (x) normal stress of the concrete at a distance x from the end of the tie 𝜀 c = 𝜀 c (x) relative elongation of the concrete at a distance x from the end of the tie σ sn ( = N t /A s ) normal stress in the bare steel at the ends of the tie rod (x = 0) ɛ sn ( = σ sn /E s ) relative elongation of the steel at the ends of the tie rod (x = 0) s1 , s1 normal stress and corresponding relative elongation in bare steel, along the tie rod on the interval [0;l 1 ] s2 , s2 normal stress and corresponding relative elongation in bare steel, along the tie rod on the interval [l 1 ; L] c1 , c1 normal stress and corresponding relative elongation in concrete, along the tie rod on the interval [0;l 1 ] c2 , c2 normal stress and corresponding relative elongation in concrete, along the tie rod on the interval[l

) 𝜓 2 = 0, 9 if c ≥ 5.�( or 𝜌<1%) 𝜓 2 = 0, 8 if c < 5.�( or 𝜌 ≥ 1%) 27

Nomenclature

Disclosure statement

No potential conflict of interest was reported by the authors.

Figure 1 .

 1 Figure 1. τ -g curves adopted by the ceB.

Figure 2 .

 2 Figure 2. short tie-rod, (a) View full length, (b) straight section.

Figure 3 .

 3 Figure 3. diagram type σ -ɛ of the short tie.

Figure 4 .

 4 Figure 4. slipping between steel and concrete and stresses generated under N t in the tie.

Figure 5 .

 5 Figure 5. adhesion mechanism and diagram from the full bond-sliding relationship according to eligehausen et al. [27].

Figure 6 .

 6 Figure 6. high adhesion steel used.

Figure 7 .

 7 Figure 7. geometric characteristics of the ribs taken into account by caetano and al [33].

Figure 8 .

 8 Figure 8. specimen used and device.

Figure 9 .

 9 Figure 9. specimen and device for the pull-out test, (a) specimen geometry, (b) concrete moving frame, (c) concrete moving frame.

Figure 10 .

 10 Figure 10. stress -slip curves.

Figure 11 .

 11 Figure 11. Tie-rods in tension.

Figure 12 .

 12 Figure 12. diagrams σ -ɛ t of tie-rodsTha8 and Tha12.

Figure 13 .

 13 Figure 13. diagrams Δ𝜀 tof tie-rodsTha8 and Tha12.

Table 1 .

 1 Particle size analysis according to the french standard nf P18-560.

	1.25	32	0.156	90.15	-	-	-	-
	1.6	33	0.250	92.44	0.016	19	-	2.289
	2	34	0.500	95.12	0.08	20	-	2.587
	2.5	35	0.969	97.61	0.1	21	-	2.886
	3.15	36	1.563	98.81	0.125	22	-	3.682
	4	37	2.156	99.20	0.16	23	-	5.174
	5	38	2.906	99.30	0.2	24	0.063	7.761
	6.3	39	4.281	100	0.25	25	0.063	14.73
	8	40	8.94	100	0.315	26	0.063	28.87
	10	41	32.94	100	0.4	27	0.063	52.24
	12.5	42	77.28	100	0.5	28	0.063	69.15
	16	43	99.28	100	0.63	29	0.063	78.71
	20	44	100	100	0.8	30	0.063	84.28
	25	45	100	100	1.0	31	0.063	85.17
	sieve diameter (mm)	sieve module	gravel passing cumulative%	sand passing cumulative %	sieve diameter (mm)	sieve module	gravel passing cumulative%	sand passing cumulative %

sfe

Table 2 .

 2 concrete constituents.

	sand 0/3	629 kg/m 3	26.4%
	rolled gravel 3/15	1148 kg/m 3	48.3%
	cement ceM 32.5 r	400 kg/m 3	16.8%
	Water	202 kg/m 3	8.50%
	Total	2379 kg/m 3	100%

Table 3 .

 3 concrete properties.

	compressive strength at 28 days f c28	33.7 MPa
	average tensile strength at 28 days f ctm 28	2.60 MPa
	Modulus of elasticity of concrete E c cement/Water ratio (C∕E)	35.5 GPa 1.98
	abrams cone subsidence	60 mm

Table 4 .

 4 characteristic values and geometry of the ribs.

	�(mm)	h 1 (mm)	b 1 (mm)	e 1 (mm)	A s mm 2
	8	0.7	0.9	5.3	50.3
	12	1	1.2	7.2	1.13
	14	1.1	1.4	8.4	1.154

Table 6 .

 6 fictitious steel laws for short tie-rods without major cracks Tha8. 4.842.10 6 . if 0 ≤ ≤ 6.470.10 -5 = 2.4.10 5 . + 297.7 if 6.470.10 -5 ≤ ≤ 9.679.10 -4

	rc concrete short ties with ha8 rebar (Tha8) without principal cracking
	E f 1 (MPa)	snL (MPa)	𝜀 sfL 10 -5	E f 2 (MPa)	𝜀 sfe 10 -4
	4.842.10 6	313.5	6.470	2.4.10 5	9.679

=

Table 7 .

 7 fictitious steel laws for short tie-rods without major cracks Tha12. = 2.2.10 5 . + 66.52 if 3.840.10 -4 ≤ ≤ 2.107.10 -3

	RC concrete Short ties with HA12 rebar (THA12) without principal cracking
	E f 1 (MPa)	snL (MPa)	𝜀 sfL 10 -4	E f 2 (MPa)	𝜀 sfe 10 -3
	3.932.10 5	150.9	3.840	2.2.10 5	2.107
		= 3.932.10 5 .	if 0 ≤ ≤ 3.840.10 -4

  1 ;L] ɛ t relative elongation of the tie rod (= Δl t ∕l t ) ɛ sf fictitious elongation of steel Δ𝜀(= 𝜀 sn -𝜀 t ) relative elongation gap between bare steel and tie rod under load N t l t (= 2L) length of tie rod A t straight section of tie rod A s steel section in the tie rod A c net section of concrete subjected to tension (= A t -A s ) ρ Percentage of steel (= A s ∕A t )

	l 1 p	length of introduction useful perimeter of steel (here bar perimeter)
	u s u c lVdT	elongation of steel elongation of concrete linear Variable differential Transformer


Claudia Califano
email: califano@dis.uniroma1.it

Claude H Moog
email: moog@ieee.org

On the Existence of the Normal Form for Nonlinear Delay Systems

The normal form is discussed for nonlinear systems affected by constant commensurate delays. Two different forms are argued. In particular, necessary and sufficient conditions are given under which a nonlinear time-delay system can be decomposed into a (weakly) observable subsystem and a non observable subsystem. Whenever such a decomposition exists, additional conditions are required to ensure the feedback linearization of the weakly observable subsystem. Finally, a full characterization is derived for the nonlinear time delay system to have an unobservable subsystem not directly affected by the input and a weakly observable subsystem which is linearizable by feedback. The performed analysis is carried out within a new geometric framework recently introduced in the literature.

Introduction

In this chapter the conditions are investigated under which a given nonlinear single input-single output continuous time system, affected by constant commensurate delays, can be represented up to a feedback as the connection of two subsystems, a weakly observable subsystem, and an unobservable one. The latter naturally yields the notion of zero dynamics of the system. It is also investigated under which conditions, the weakly observable subsystem can be rendered linear. The interest in such a canonical form arises in various control problems including the study of stability and stabilization [START_REF] Pepe | A Lyapunov Krasovskii methodology for ISS and iISS of time-delay systems[END_REF].

As well known, in the continuous delay-free case, under the assumption of defined relative degree, such a decomposition is always possible, and it is referred to as normal form [START_REF] Isidori | Nonlinear Control Systems[END_REF]. In this case the input-output behavior can always be rendered linear through an appropriate choice of the feedback law and can be represented as a chain of integrators. Furthermore for single output systems, it has also been shown that the unobservable dynamics is independent of the control, a property which is in general lost in the multi-output case. This property which has some interesting implications on the study of the zero dynamics of the system is not guaranteed for nonlinear discrete time single-input single-output systems [START_REF] Califano | On the discrete-time normal form[END_REF]. In fact the assumption of defined relative degree allows to achieve the desired decomposition, but in general the unobservable subsystem is influenced by the control.

The case of single-input systems of retarded type is even more involved since the assumption of well defined relative degree does not guarantee anymore the existence of such a decomposition. Furthermore in order to get a linear input-output behavior some additional conditions are needed, and even if these conditions are satisfied, a representation of such a behavior as a chain of integrators can be achieved only under very strong conditions. It is also shown that the unobservable dynamics is in general influenced by the control. Some preliminary results were presented in [START_REF] Califano | De l'existence de la forme normale pour les systemes non lineaires a retards[END_REF].

The outline of this book chapter is as follows. The statements of the problems under investigation are given in Sect. 2, while some recalls on the algebraic and geometric tools used in the chapter for dealing with nonlinear time-delay systems are given in Sect. [START_REF] Califano | On the discrete-time normal form[END_REF]. Section 4 is devoted to the main results. Some concluding remarks are given in Sect. 5.

Preliminaries and Problem Statements

A class of nonlinear single-input single-output systems is considered whose state variables and control are subject to constant and commensurate delays. Without loss of generality, and after an eventual preliminary sampling (as explained in [START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Michiels | Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based Approach[END_REF]), the dynamics reads:

Σ : ẋ[0] = F(x [s] ) + s j=0 G j (x [s] )u [0] (-j) y [0] = H (x [s] ) (1)
where

x T [s] = (x T (t), •••x T (t -s)) ∈ R (s+1)n ,
represents the first (s +1)n components of the state of the infinite dimensional system [START_REF] Califano | Extended lie brackets for nonlinear timedelay systems[END_REF]. The notations x [0] = x(t) ∈ R n , u [0] = u(t) ∈ R and y [0] = y(t) ∈ R stand for the instantaneous values of the state, the input and the output. The functions F(x [s] ), G j (x [s] ) for j ∈[0, s] and H (x [s] ) are assumed analytic in their arguments. Additional assumptions such as dealing with local/global Lipschitz functions or forward completeness may be introduced depending on the requirements on the existence/uniqueness of the solution of [START_REF] Califano | Extended lie brackets for nonlinear timedelay systems[END_REF]. These assumptions play a key role when dealing with stability problems. Denote by

x T [s] (-p) = (x T (t -p), •••x T (t -s -p));
in a similar vein, we define u [s] and u [s] (-p). When there is no confusion, the subscript will be dropped so that x represents x [s] and x(p) represents x [s] (-p).O nt h e contrary if a function τ(•) depends not only on x [s] , but also on some future values of the variable x(t), in order to put in evidence that such a function is not causal, we will say that τ(

•) = τ(x [ * ] ). Given a function τ(x [s]
), its r th derivative with respect to time will be denoted as

τ (r ) (x [s] ) := d r τ(x [s] ) dt r .
The notation x 0

[γ ] represents an equilibrium point. In the following, K will be used to denote the field of meromorphic functions of a finite number of variables in

x [0] (-i), u [0] (-i), ••• , u (k) [0] (-i), k, i ∈ N ,
where N is the set of natural numbers. The notation "d" will represent the differential operator, while "δ" will represent the backward time-shift operator: for a

(x [α] ), f (x [β] ) ∈ K : δ[ a(x [α] ) df(x [β] )]=a(x [α] (-1))δdf(x [β] ) = a(x [α] (-1))df(x [β] (-1)).
The left ring of polynomials in δ with coefficients in K is denoted K (δ].

In this framework a one-form is written as

ω(x,δ)dx [0] = n i=1 ω i (x,δ)dx i,[0] ,ω i (x,δ) ∈ K (δ], i ∈[1, n]. If there exists f (x) such that df(x) = ω(x,δ)dx [0] , then the one-form ω(x,δ)dx [0] is exact. A set of one-forms {ω i (x,δ)dx [0] , i ∈[1, j]} are independent over K (δ] if j i=1 α i (x,δ)ω i (x,δ) = 0 only if α i (x,δ) = 0, ∀i ∈[1, j]. Finally Δ = span K {r 1 (x), ••• , r j (x)
} denotes the distribution spanned by the r i (x)'s. Any element of Δ can be expressed as a linear combination of the r i (x)'s with coefficient in K . Δ will denote its involutive closure. Analogously Δ(x,δ) = span K (δ] {r 1 (x,δ),••• , r j (x,δ)} denotes the submodule spanned by the r i (x,δ)'s. Any element of Δ(x,δ)can be expressed as

r (x,δ) = j i=0 r i (x)α i (x,δ) with α i (x,δ) ∈ K (δ].
Before stating the different addressed problems, we will recall the notions of relative degree of a given output function, observability of the given system, bicausality of a change of coordinates and regular static state feedback, which play a key role. More precisely let us recall that the relative degree r of the output y [0] is defined as the smallest integer r , such that the jth derivative of the output function y ( j) [0] is not influenced by the control, for 0 ≤ j ≤ r -1, while y (r )

[0] is influenced by the (eventually delayed) control. For the class of system considered, algebraic conditions are given in Definition 6, which characterize the relative degree through the use of the extended Lie derivative (Definition 5).

Definition 1 (Observability, [START_REF] Marquez-Martinez | Observability and observers for nonlinear systems with time delay[END_REF]) System Σ with instantaneous state variable x [0] ∈ R n is said to be weakly observable if the observability matrix O(x,δ)satisfying

⎛ ⎜ ⎝ dy [0]
. . .

dy (n-1) [0] ⎞ ⎟ ⎠ = O(x,δ)dx [0] ,
is a full rank matrix.The system is strongly observable if O(x,δ)is unimodular, i.e. it admits a polynomial inverse.

Since we will study the equivalence of systems linked through a change of coordinates, recall the following definition of bicausal change of coordinates adapted from [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF].

Definition 2 (Bicausal change of coordinates) Consider system Σ in the state coor- dinates x. The mapping z [0] = ϕ(x [α]
), where α ∈ N and ϕ ∈ K n , is a local bicausal change of coordinates for Σ if there exists an integer ℓ ∈ N and a function

ψ(z [ℓ] ) ∈ K n such that, assuming z [0] and x [0] defined for t ≥-(α + ℓ), then ψ(ϕ(x [α] ), ••• ,ϕ(x [α] (-ℓ)) = x [0] for t ≥ 0.

Definition 3 (Bicausal regular static state feedback) Consider the input w

[0] ∈ I R m . The feedback w [0] = α(x) + i j=0 β j (x)v [0] (-j)
is a local bicausal regular static state feedback if there exists an integer ℓ ∈ N and functions ᾱ(x) and βμ (x) such that,

α(x) + i j=0 β j (x) ⎛ ⎝ ᾱ(x(-j)) + ℓ μ=0 βμ (x(-j))w [0] (-μ -j) ⎞ ⎠ = w [0] .
According to the definition, the regularity and bicausality of the feedback is equivalent to the unimodularity of the matrix

β(x,δ) = i j=0 β j (x)δ j . (2)
In the single input case, considered in the present chapter, it thus reduces to the independence of δ, that is β(x,δ) = β(x) = 0, or equivalently β 0 (x) = 0 and

β j (x) = 0 for all j ∈[1, i].
The following problems can now be set.

Problem Statement 1.

[Existence of the Generalized Normal Form] Given system (1), with output y [0] characterized by the well defined relative degree r , find, if possible, a regular bicausal static state feedback

u [0] = α(x) + β(x)v [0] ,
and a bicausal change of coordinates

z [0] = z 1,[0] z 2,[0] = ϕ(x)
with z 1,[0] ∈ R r , such that in the new coordinates the closed-loop system reads as:

ż1,[0] = θ 1 (z 1,[ℓ] ) + ℓ j=0 θ 2 j (z 1,[ℓ] )v [0] (-j), ż2,[0] = η 1 (z [ℓ] ) + ℓ j=0 η 2 j (z [ℓ] )v [0] (-j), y [0] = H (z 1,[ℓ] ), (3)
where the subsystem

ż1,[0] = θ 1 (z 1,[ℓ] ) + ℓ j=0 θ 2 j (z 1,[ℓ] )v [0] (-j), y [0] = H (z 1,[ℓ] )
is weakly observable. ⊳ Such a problem can be further developed by requiring that the input-output behavior is described by a linear system, which leads to the second problem examined in this chapter:

Problem Statement 2. [Existence of the Normal Form] Given system (1), with output y [0] characterized by the well defined relative degree r , find, if possible, a regular bicausal static state feedback

u [0] = α(x) + β(x)v [0]
, and a bicausal change of coordinates

z [0] = z 1,[0] z 2,[0] = ϕ(x)
with z 1,[0] ∈ R r , such that in the new coordinates the closed-loop system reads as (3) with a linear (weakly) observable input-output behavior, that is with

ż1,[0] = θ 1 (z 1,[ℓ] ) + ℓ j=0 θ 2 j (z 1,[ℓ] )v [0] (-j) = ℓ j=0 A j z 1,[0] (-j) + ℓ j=0 B j v [0] (-j), y [0] = H (z 1,[ℓ] ) = ℓ j=0 C j z 1,[0] (-j),
where the polynomial matrices

A(δ) = ℓ j=0 A j δ j , B(δ) = ℓ j=0 B j δ j , and C(δ) = ℓ j=0 C j δ j
define a linear (weakly) observable system. ⊳ Finally, the conditions under which the unobservable dynamics defined by the derivative of z 2,[0] in (3) is not affected by the control v are investigated; that is the bicausal change of coordinates

z [0] = z 1,[0] z 2,[0] = ϕ(x)
can be chosen to guarantee that

ż2,[0] = η 1 (z [ℓ] ).
This has interesting consequences when studying the stability properties of the whole system as discussed in Sect. 5.

Remark 1 System (1) is defined for t ≥ 0 once the initial condition x(t) = ϕ(t) is set on the interval [-s, 0]. For the equivalence to the form (3) to hold true for t ≥ 0, it may be necessary to consider the initial condition x(t) = ϕ(t) set on a greater interval [-s, 0], to guarantee that the change of coordinates and the static state feedback are well defined for t ≥ 0, as well as the initial condition z(t) =φ(t) which must be well defined on the interval [-ℓ, 0].

This study is performed within a new geometric framework which was recently introduced in [START_REF] Califano | Extended lie brackets for nonlinear timedelay systems[END_REF], and appeared to be successful to derive various canonical forms [START_REF] Califano | Extended lie brackets for nonlinear timedelay systems[END_REF][START_REF] Califano | Linearization of time-delay systems by inputoutput injection and output transformation[END_REF]. More precisely, the conditions are stated in terms of extended Lie brackets of vector fields which are defined on appropriate finite dimensional manifolds.

Recalling Results on Algebraic and Geometric Tools

This Section is devoted to recall some basic tools and results which will be used to address the problems stated in Sect. 2. The approach has shown to be efficient for solving various problems for nonlinear time delay systems, including the identification of the delay [START_REF] Zheng | Identification of the delay parameter for nonlinear time-delay systems with unknown inputs[END_REF].

Differential Representation

Given system Σ defined by (1), its differential representation is

Σ L : d ẋ[0] = f (x [s] , u [s] ,δ)dx [0] + g 1 (x [s] ,δ)du [0] dy [0] = h(x [s] ,δ)dx [0] (4) with f (x [s] , u [s] ,δ) = s i=0 ∂ F(x [s] ) ∂ x [0] (-i) δ i + s j=0 u [0] (-j) s i=0 ∂G j (x [s] ) ∂ x [0] (-i) δ i g 1 (x [s] ,δ) = s j=0 G j (x [s] )δ j = s j=0 g j 1 (x [s] )δ j , h(x [s] ,δ) = s j=0 ∂ H (x [s] ) ∂ x [0] (-j) δ j

Left-and Right-Annihilators

• Let T (x,δ)be a (n × r ) polynomial matrix of rank r . Its left-annihilator Ω(x,δ) consists of all row vectors ω(x,δ) such that ω(x,δ)T (x,δ) = 0. By definition, the submodule Ω(x,δ)is closed in the following sense of left-closure:

P(x,δ)π ∈ Ω ⇒ π ∈ Ω
where P(x,δ)stands for a polynomial operator.

• Let T (x,δ)be a (r ×n) polynomial matrix of rank r . Its right-annihilator Δ(x [ * ] ,δ) consists of all column vectors v(x [ * ] ,δ)such that T (x,δ)v(x [ * ] ,δ) = 0.
By definition, the right-kernel is closed in the following sense of right-closure:

v(x [ * ] ,δ)P(x [ * ] ,δ) ∈ Δ(x [ * ] ,δ) ⇒ v(x [ * ] ,δ) ∈ Δ(x [ * ] ,δ)
• The left closure of the polynomial matrix T (x,δ)is obtained computing the leftannihilator of the right-annihilator of T (x,δ). • The right closure of the polynomial matrix T (x,δ) is obtained computing the right-annihilator of the left-annihilator of T (x,δ). • Note that starting from the causal (n × r ) polynomial matrix T (x,δ) of rank r , its left-annihilator can always be expressed by causal generators. On the contrary starting from the causal (r × n) polynomial matrix T (x,δ)of rank r , it may not be possible to express its right-annihilator through causal generators, that is the entries of the right-annihilator are polynomials in δ whose coefficients are functions which may depend on future time instants t + k for some k > 0. For instance, consider

T (x,δ) = x 1 (-1) + x 1 δ x 2 δ .
Then its left annihilator is

ω = x 2 δ, -x 1 (-2) -x 2 x 1 (-1)
x 2 (-1) δ , which is causal. Now consider the transpose of T (x,δ), namely,

T (x,δ) = x 1 (-1) + x 1 δ, x 2 δ . Its right annihilator is v = x 2 δ -x 1 -x 1 (+1) x 2 (+1) x 2 δ
, which is not causal.

The Action of a Bicausal Change of Coordinates

Given the bicausal change of coordinates z [0] = ϕ(x), let us consider the associated differential representation dz

[0] = T (x,δ)dx [0] . Then, T (x [α] ,δ)is unimodular.
Moreover, its inverse has a polynomial degree γ ≤ α(n -1). Moreover, under such a bicausal change of coordinates, the differential representation ( 4) is transformed into

d ż[0] = f (z, u,δ)dz [0] +g 1 (z,δ)du [0] , dy [0] = h(z,δ)dz [0] , (5)
where

f (z, u,δ) = T (x,δ)f (x, u,δ)+ Ṫ (x,δ) T -1 (x,δ) ϕ -1 (z) g1 (z,δ) = (T (x,δ)g 1 (x,δ)) ϕ -1 (z)
Remark 2 The rank of T (x,δ)may drop around some singular function ϕ(x), which may eventually be a trajectory for the system. In this case the equivalence does not hold true around such a singular trajectory.

Extended Lie Bracket

We next recall the definition of the extended Lie bracket introduced in [START_REF] Califano | Extended lie brackets for nonlinear timedelay systems[END_REF] to tackle nonlinear time-delay systems and which is used to characterize the integrability of 1-forms defined not only from state variables at time t but also from their time shifts.

Definition 4 (Extended Lie Bracket)Let r β (x,δ) = s j=0 r j β (x)δ j , where β = 1, 2. The extended Lie bracket [r k 1 (x), r l 2 (x)] E i is defined on R (i+1)n , i ≥ 0, by r k 1 (•), r l 2 (•) E i = k j=0 [r k-j 1 (•), r l-j 2 (•)] E 0 T |(x(-j),u(-j)) ∂ ∂ x [0] (-j) , (6)
with k = min(k, l, i), and

[r k 1 (•), r l 2 (•)] E 0 = k i=0 ∂r l 2 (x) ∂ x [0] (-i) r k-i 1 (x(-i)) - l i=0 ∂r k 1 (x) ∂ x [0] (-i) r l-i 2 (x(-i)). (7)
The extended Lie bracket makes sense for the infinite dimensional system

ẋ[0] = F(x [s] ) + s j=0 G j (x [s] )u [0] (-j) ẋ[0] (-1) = F(x [s] (-1)) + s j=0 G j (x [s] (-1))u [0] (-j -1) (8)
. . . which is associated to the time-delay system (1). The computation of the extended Lie bracket can be implemented in some computer algebra software as done in [START_REF] Garate-Garcia | A computer algebra system for analysis and control of nonlinear time-delay systems[END_REF].

As for delay-free systems, it is useful to introduce an extended Lie derivative whose definition is given below and differs slightly from the one in [START_REF] Germani | Linearization of input-output mapping for nonlinear delay systems via static state feedback[END_REF][START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF].

Definition 5 (Extended Lie Derivative) Given a function λ(x [s] ) and the vector

r i (x,δ) = s j=0 r j i (x)δ j , the extended Lie derivative L r j i (x) λ(x [s] ) of λ(x [s] ) in the direction r j i is L r j i (x) λ(x [s] ) = j l=0 ∂λ(x [s] ) ∂ x [0] (-l) r j-l i (x(-l)). (9)
One thus gets

r k 1 (•), r l 2 (•) E 0 = L r k 1 (x) r l 2 (x) -L r l 2 (x) r k 1 (x) T ∂ ∂ x [0]
.

This allows to recover standard definitions of Lie derivatives and Lie brackets which are used in the delay free case. From (1), consider now the polynomial vector

F(x [s] ,δ) = ns j=0 F j (x [s] )δ j = ns j=0 F(x [s] )δ j . ( 10
)
The derivative of λ(x [s] ) of order i which is computed for u = 0, is given by

λ (i) (x, 0) = L i F ns (x) λ(x [s] ), for i ∈[1, n].
The definition of relative degree can then be stated as follows.

Definition 6 The function λ(x [s]

) has relative degree k > 0if

L g j 1 L i F ns λ(x) ≡ 0 ∀ j ≥ 0, ∀0 ≤ i < k -1,
and if there exists an integer j ≥ 0 such that

L g j 1 L k-1 F ns λ(x) = 0 (11)
The relative degree is said to be strong if [START_REF] Marquez-Martinez | Observability and observers for nonlinear systems with time delay[END_REF] is fulfilled for j = 0.

Integrability

Next, let us recall a fundamental result on integrability of a submodule, giving conditions under which the left annihilator of the submodule is spanned by exact one-forms.

The following is a more elegant revisited version of a result in [START_REF] Califano | Extended lie brackets for nonlinear timedelay systems[END_REF]. Consider

P j (x,δ) = r 1 (x,δ),••• , r j (x,δ) = s l=0 P jl (x)δ l
where P j0 (x) has full column rank j and

r k (x,δ) = s l=0 r l k (x)δ l , k ∈[1, j],
and let s be the maximum delay in the x variable. Denoting, with some abuse of notations, in compact form

r l = r l 1 , ••• , r l j
and by I n the n × n identity matrix, consider the distributions Δ i , i ≥ 0 which are defined on R (i+s+1)n as follows:

Δ 0 = span K ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ r 0 ••• r s 0 ••• ••• I n ••• . . . . . . ••• 0 I n ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ Δ 1 = span K ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r 0 ••• r s 0 ••• 0 r 0 (-1) ••• r s (-1) 0 ••• I n ••• . . . . . . ••• 0 I n ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (12)
. . .

Δ i = span K ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r 0 ••• r s 0 ••• 0 . . . ••• . . . 0 ••• r 0 (-i) ••• r s (-i) 0 ••• I n ••• . . . . . . ••• 0 I n ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Let ρ i = dim( Δi ) in the neighborhood of x 0 .
Theorem 1 Consider the submodule

Δ(x,δ) = span K (δ] r 1 (x,δ),••• , r j (x,δ) , where r i (x,δ) = s l=0 r l i (x [s] )δ l ,
and such that the matrices

P j (x,δ) = r 1 (x,δ),••• , r j (x,δ) = s l=0
P jl (x)δ l and P j0 (x) have rank j. Let Δ i be the set of distributions defined by [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based Approach[END_REF], which have by assumption constant dimension ρ i = dim ( Δi ) locally around x 0 [i] , with ρ -1 = ns by definition. Then Δ(x,δ) is completely integrable if and only if there exists an integer γ such that ρ γρ γ -1 = j.

If the given submodule Δ(x,δ) is not completely integrable, one may wish to compute the largest number of exact one-forms independent over K (δ] which are in the left-kernel of Δ(x,δ). The following result holds:

Theorem 2 Consider the submodule Δ(x,δ) = span K (δ] r 1 (x,δ),••• , r j (x,δ) with r i (x,δ) = s l=0 r l i (x [s] )δ l ,
and such that matrices

P j (x,δ) = r 1 (x,δ),••• , r j (x,δ) = s l=0
P jl (x)δ l and P j0 (x) have rank j. Let Δ i be the set of distributions defined by [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based Approach[END_REF],w h i c h have by assumption constant dimension

ρ i = dim ( Δi ) locally around x 0 [i]
, where ρ -1 = ns by definition. Let γ be the smallest index such that ρ γ +k -ρ γ +k-1 = ℓ ≥ j, ∀k ≥ 0. The smallest completely integrable submodule Δ(x,δ) ⊃ Δ(x,δ)has rank ℓ. Consequently, there exist nℓ exact one-forms independent over K (δ] which generate the left kernel of Δ(x,δ).

We end this section by stating the following result from [START_REF] Califano | Coordinates transformations in nonlinear time-delay systems[END_REF], which is fundamental for developing the results in the next section.

Theorem 3 Let Δ(x,δ) be a given submodule and let Δ(x,δ) denote the smallest completely integrable submodule such that Δ(x,δ) ⊃ Δ(x,δ). Let j be the rank of Δ(x,δ), then there exist nj exact one forms (dλ 1 , ••• , dλ n-j ) which generate the left annihilator of Δ(x,δ). Furthermore there exist j exact one forms (dλ n-j+1 , ••• , dλ n ), which can be taken to define a basis completion, that is

⎛ ⎜ ⎝ dλ 1 . . . dλ n ⎞ ⎟ ⎠ = T (x,δ)dx [0]
with T (x,δ)unimodular.

Main Results

In this section we address the problem of characterizing the different geometrical structures analyzed in the Introduction. The conditions are linked to two different properties of the system, basically, what can be achieved under bicausal change of coordinates, and what can be achieved under regular bicausal static state feedback.

Generalized Normal Form

Hereafter we analyze the conditions under which there exist a bicausal change of coordinates and a regular bicausal static state feedback thanks to which the closedloop system can be decomposed in the generalized normal form

ż1,[0] = θ 1 (z 1,[ℓ] ) + ℓ j=0 θ 2 j (z 1,[ℓ] )v [0] (-j), ż2,[0] = η 1 (z [ℓ] ) + ℓ j=0 η 2 j (z [ℓ] )v [0] (-j) (13)
y [0] = H (z 1,[ℓ] ),
with z 1,[0] ∈ R r , r being the relative degree of the output.

Before enouncing the necessary and sufficient conditions, the following result is required, which enlightens under which conditions it is possible to define a bicausal change of coordinates such that the output function and its derivative up to the order r -1 can be expressed as functions of the variable z 1,[0] ∈ R r and its delayed values. This condition, which is necessary to solve Problem 1, does not follow from the assumption of defined relative degree, as it happens in the delay-free case.

The following result holds: Lemma 1 Given system (1), with output y [0] characterized by the well defined relative degree r , there exists a bicausal change of coordinates

z [0] = z 1,[0] z 2,[0] = ϕ 1 (x) ϕ 2 (x) = ϕ(x)
with z 1,[0] ∈ R r , such that in the new coordinates dy (i)

[0] = ϕ i+1 (z 1 ) i ∈[0, r -1] ( 14
)
if and only if setting

⎛ ⎜ ⎝ dy [0] . . . dy (r -1) [0] ⎞ ⎟ ⎠ = Y(x,δ)dx [0] ,
the right annihilator Δ(•,δ)of Y(x,δ)is causal.

Proof Necessity. Assume for instance that there exists a bicausal change of coordinates

z [0] = z 1,[0] z 2,[0] = ϕ 1 (x) ϕ 2 (x) = ϕ(x)
with z 1,[0] ∈ R r , such that in the new coordinates the output and its derivative up to the order r -1 are expressed as functions of the variable z 1 only. Let

dz [0] = dz 1,[0] dz 2,[0] = T 1 (x,δ) T 2 (x,δ) dx [0]
Then one gets immediately that

⎛ ⎜ ⎝ dy [0]
. . .

dy (r -1) [0] ⎞ ⎟ ⎠ = Ψ(z 1 ,δ)dz 1,[0] = Ψ(z 1 ,δ) z 1,[0] = ϕ 1 (x) T 1 (x,δ)dx [0] = Y(x,δ)dx [0]
.

Consider now the matrix T -1 (x,δ). Since T (x,δ) is unimodular by assumption,

T -1 (x,δ) is causal. Furthermore setting T -1 (x,δ) = (L 1 (x,δ),L 2 (x,δ)), by con- struction T 1 (x,δ)L 2 (x,δ) = 0. Consequently, setting Ψ(x,δ) = Ψ(z 1 ,δ) z 1,[0] =ϕ 1 (x) , Y(x,δ)L 2 (x,δ) = Ψ(x,δ)T 1 (x,δ)L 2 (x,δ) = 0.
It follows that the right annihilator Δ(•,δ)of Y(x,δ)is given by Δ(•,δ) = L 2 (x,δ), has rank nr and is causal.

Sufficiency. By assumption Δ(•,δ)is causal, has rank n -r and there exist r exact differentials, independent over K (δ] which lay in the left annihilator. Consequently the rank of the involutive closure Δ(x,δ) of Δ(x,δ) is still nr and there exists r exact differentials dλ 1 (x), ••• , dλ r (x) which generate the left annihilator, so that for i ∈[0, r -1],

dy (i) [0] = r j=1 α ij (x,δ)dλ j (x) (15)
Due to Theorem 3 it is also possible to compute nr functions λ r +1 (x), ••• ,λ n (x), so that

⎛ ⎜ ⎝ dλ 1 (x) . . . dλ n (x) ⎞ ⎟ ⎠ = T (x,δ)dx [0]
with T (x,δ)unimodular. As a consequence, the transformation

z [0] = ⎛ ⎜ ⎝ λ 1 (x) . . . λ n (x) ⎞ ⎟ ⎠
defines a bicausal change of coordinates. Denoting by z 1,[0] the first r -components, and by z 2,[0] the last nr components of z [0] , in these new coordinates, we have that, due to Eq. ( 15), for i ∈[0, r -1]

dy (i) [0] = n j=1 α ij (x,δ)dλ j (x) = r j=1 α ij (x,δ)dλ j (x) = r j=1 ᾱij (z,δ)dz 1,[0]
Since on the left hand-side one has exact differentials, then necessarily ᾱij (z,δ) = ᾱij (z 1 ,δ), which ends the proof.

Theorem 4

Given system (1), with output y [0] characterized by defined relative degree r , there exist a regular bicausal static state feedback

u [0] =α(x) + β(x)v [0] ,
and a bicausal change of coordinates

z [0] = z 1,[0] z 2,[0] = ϕ(x)
with z 1,[0] ∈ R r , such that in the new coordinates the closed-loop system reads as the Generalized Normal Form ( 13), if and only if the following conditions hold:

(i) Let ⎛ ⎜ ⎝ dy [0]
. . .

dy (r -1) [0] ⎞ ⎟ ⎠ = Y(x,δ)dx [0] . (16)
The right annihilator Δ(•,δ)of Y(x,δ)has rank nr and is causal. (ii) dy (r )

[0] , the differential of the r th derivative of the output y (r ) (x [s] , u [s] ) , given by dy (r ) [0] = a(x, u,δ)dx

[0] + b(x,δ)du [0] , (17)
satisfies the following properties: If conditions (i) and (ii) are satisfied, the coordinates z [0] = ϕ(x), which transform the closed-loop system in the form [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF], can be chosen in such a way that θ 2 j (z 1 ) has only the last component different from 0, that is

(iia) b(x,δ)in (17) can be factorized as b(x,δ) = b(x,δ)β(x) (18) with b(x,δ) = ℓ j=0 b j (x)δ ℓ such that d b j (x) ∈ Δ(x,δ) ⊥ , ∀ j ∈[0,ℓ] (19)
θ 2 j (z 1 ) = ⎛ ⎜ ⎜ ⎜ ⎝ 0 . . . 0 ϑ 2 j (z 1 ) ⎞ ⎟ ⎟ ⎟ ⎠ , ∀i ∈[0,ℓ] (21)
Proof Assume that there exist a regular bicausal static state feedback

u [0] =α(x) + β(x)v [0]
and a bicausal change of coordinates,

z [0] = z 1,[0] z 2,[0] = ϕ(x)
with z 1,[0] ∈ R r , such that the closed-loop system in the new coordinates reads [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF].

Then

dy [0] = d H (z 1
) and dy (i)

[0] = dψ i (z 1 )

for i ∈[1, r -1].Let dz [0] = dz 1,[0] dz 2,[0] = T 1 (x,δ) T 2 (x,δ) dx [0] = T (x,δ)dx [0]
be the differential representation of the bicausal change of coordinates. Then, by assumption,

⎛ ⎜ ⎝ dy [0]
. . .

dy (r -1) [0] ⎞ ⎟ ⎠ = Q(z 1 ,δ)dz 1,[0] = Q(z 1 ,δ)| z 1,[0] = ϕ 1 (x) T 1 (x,δ)dx [0] .
so that Y(x,δ)in [START_REF] Zheng | Identification of the delay parameter for nonlinear time-delay systems with unknown inputs[END_REF] is given by Y(x,δ) = Q(x,δ)T 1 (x,δ). It is immediately seen that denoting by Δ(x,δ) the last nr columns of T -1 (x,δ), then Δ(x,δ) is the right annihilator of Y(x,δ), has rank nr and is causal, which proves the necessity of (i).

As for (ii), by assumption in the z-coordinates, and after the regular bicausal static state feedback

u [0] =α(x) + β(x)v [0]
which in the z-coordinates reads

u [0] =α(z) + β(z)v [0] ,
the differential representation of the r th derivative of the output of the closed-loop system is given by

dy (r ) [0] =ã(z 1 , v,δ)dz 1,[0] + b(z 1 ,δ)dv [0] . Let v [0] =ᾱ(z) + β(z)u [0]
be the inverse feedback of

u [0] =α(z) + β(z)v [0] .
Then ᾱ(z) =-β-1 (z) α(z) and β(z) = β-1 (z).

Let dv [0] =γ(z, u,δ)dz [0] + β(z)du [0] be its differential representation and note that d ᾱ(z) =γ(z, 0,δ)dz [0] . Then for the open loop system, one has that

dy (r ) [0] =ā(z, u,δ)dz [0] + b(z,δ)du [0] =ã(z 1 , v,δ)| v [0] =ᾱ(z)+ β(z)u [0] dz 1,[0] + b(z 1 ,δ)γ(z, u,δ)dz [0] + b(z 1 ,δ) β(z)du [0] =â(z, u,δ)dz 1,[0] + b(z 1 ,δ)γ(z, u,δ)dz [0] + b(z 1 ,δ) β(z)du [0] which shows that b(z,δ) = b(z 1 ,δ) β(z) ā(z, u,δ)dz [0] =â(z, u,δ)dz 1,[0] + b(z 1 ,δ)γ(z, u,δ)dz [0] , so that ā(z, 0,δ)dz [0] =â(z, 0,δ)dz 1,[0] + b(z 1 ,δ)γ(z, 0,δ)dz [0] . ( 22
) Recall that b(x,δ) in (17)islinkedto b(z,δ) computed above through the relation b(x,δ) = b(z,δ)| z [0] = ϕ(x)
, so that one gets:

b(x,δ) = b(z,δ)| z [0] = ϕ(x) = b(z 1 ,δ)| z 1,[0] = ϕ 1 (x) β(z)| z [0] = ϕ(x) = b(x,δ)β(x) where b j (x) = b j (z 1 )| z 1,[0] = ϕ 1 (x)
so that for all j ∈[0,ℓ], d b j (x) ∈ Δ(x,δ) ⊥ , thus proving ( 18) and ( 19)in(iia). Analogously a(x, 0,δ)and γ(x,δ)used in (20) are linked respectively to ā(z, 0,δ) and γ(z,δ) defined above through the relations a(x, 0,δ) =ā(z, 0,δ

)| z [0] = ϕ(x) T (x,δ)and γ(x,δ) =γ(z,δ)| z [0] = ϕ(x) T (x,δ). It follows from (22), that a(x, 0,δ)-b(x,δ)γ(x,δ) dx [0] = ( â(z, 0,δ), 0) -b(z 1 ,δ)γ(z,δ) | z [0] = ϕ(x) T (x,δ)dx [0] =ā(z, 0,δ)| z [0] = ϕ(x) dϕ 1 (x)
which proves (20).

As for the sufficiency, assume that the conditions are satisfied and let the r th derivative of the output be

y (r ) [0] =ā(x) + ℓ j=0 b j (x)u [0] (-j).
Consider the regular bicausal static state feedback

u [0] =α(x) + β(x)v [0] = β -1 (x) -α(x) + v [0] ,
where β(x) is computed from (18), while α(x) is computed starting from γ(x,δ) satisfying (20), with

dα(x) = γ(x,δ)dx [0] .
For the closed-loop system, the r th derivative of the output becomes

y (r ) [0] =ā(x) + ℓ j=0 b j (x)β -1 (x(-j))(-α(x(-j)) + v [0] (-j))
Due to (18), each coefficient b j (x) = b j (x)β(x(-j)), so that

y (r ) [0] =ā(x) + ℓ j=0 b j (x)(-α(x(-j)) + v [0] (-j))
From (i) choose a bicausal change of coordinates z [0] = ϕ(x), such that dz 1,0 ⊥ Δ(x,δ)with z 1,[0] ∈ I R r . Then, by Lemma 1,

dy (i) [0] ∈ span K (δ] {dz 1,[0] }, for i ∈[ 0, r -1]
. Furthermore in these new coordinates the r th derivative of the output of the closed-loop system is

y (r ) [0] =ā(x)| x [0] = ϕ -1 (z) - ℓ j=0 ( b j (x)α(x(-j)))| x [0] = ϕ -1 (z) + ℓ j=0 b j (x)| x [0] = ϕ -1 (z) v [0] (-j) Due to (19), d b j (x) ∈ Δ ⊥ (x,δ), so that b j (x)| x [0] = ϕ -1 (z) = b j (z 1 )
. Finally due to (20), we get

d ā(x) - ℓ j=0 ( b j (x)δ j dα(x)) ∈ Δ(x,δ) ⊥ ,
so that also

d ā(x) - ℓ j=0 d( b j (x)α(x(-j))) ∈ Δ(x,δ) ⊥ ,
which proves that for the closed-loop system in the new coordinates

y (r ) [0] =ã(z 1 ) + ℓ j=0 b j (z 1 )v [0] (-j) Due to Lemma 1,inthez-coordinates one has that ⎛ ⎜ ⎝ dy [0] . . . dy (r -1) [0] ⎞ ⎟ ⎠ = Q(z 1 ,δ)dz 1,[0] (23)
Consider the derivative of both sides of (23). Then one gets that for the closed-loop system

⎛ ⎜ ⎝ d ẏ[0] . . . dy (r ) [0] ⎞ ⎟ ⎠ = Q(z 1 ,δ)dz 1,[0] + Q(z 1 ,δ)d ż1,[0] = P 1 (z 1 , v,δ)dz 1,[0] + P 2 (z 1 ,δ)dv [0] , (24)
that is

Q(z 1 ,δ)d ż1,[0] = P1 (z, v,δ)dz 1,[0] + P 2 (z 1 ,δ)dv [0] . (25)
The proof is carried out by contradiction, by showing that d ż1,[0] cannot depend on dz 2,[0] . Assume in fact that this is not the case. Then,

d ż1,[0] = P11 (z,δ)dz 1,[0] + P12 (z,δ)dz 2,[0] + P13 (z,δ)dv [0]
Thus, premultiplying both sides by Q(z 1 ,δ), one gets that

Q(z 1 ,δ)d ż1,[0] = Q(z 1 ,δ) P11 (z,δ)dz 1,[0] + Q(z 1 ,δ) P12 (z,δ)dz 2,[0] + Q(z 1 ,δ) P13 (z,δ)dv [0]
which compared with (25) implies that Q(z 1 ,δ) P12 (z,δ) = 0. Since by assumption Q(z 1 ,δ) is a full rank matrix, this can only happen if P12 (z,δ) = 0, which thus proves the thesis. As a consequence the system can be put in the form [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF]. To end the proof we have to show that the z 1,[0] coordinates can always be chosen in order to guarantee that the θ 2 j 's in [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF] are of the form (21). This follows immediately by considering the subsystem

ż1,[0] = θ 1 (z 1,[ℓ] ) + ℓ j=0 θ 2 j (z 1,[ℓ] )v [0] (-j), y [0] = H (z 1,[ℓ] ),
whose associated differential representation ( 4) is characterized by

g 11 (z 1,[ℓ] ,δ) = ℓ j=0 θ 2 j (z 1,[ℓ] )δ j
Since dy

(i) [0] g 11 (z 1 ,δ) = 0fori ∈[0, r -2], then Δ(z 1 ,δ) = span K (δ] {g 11 (z 1 ,δ)}
is completely integrable and since rank Δ(x,δ) = 1, there exists r -1 exact differentials which lay in the left-annihilator. Due to Lemma 1 there exists a bicausal change of coordinates z1,[0] =φ 1 (z 1 ), such that in the new coordinates

g11 (z 1,[ l] ,δ) = ⎛ ⎜ ⎜ ⎜ ⎝ 0 . . . 0 * ⎞ ⎟ ⎟ ⎟ ⎠ that is ż1,[0] = θ1 (z 1,[ l] ) + l j=0 θ2 j (z 1,[ l] )v [0] (-j), y [0] = H (z 1,[ l] ), with θ2 j (z 1,[ l] ) = ⎛ ⎜ ⎜ ⎜ ⎝ 0 . . . 0 θ2 j (z 1 ) ⎞ ⎟ ⎟ ⎟ ⎠ , ∀ j ∈[0, l]
which ends the proof.

We next give two examples which show how to apply the previous result.

Example 1 Consider the dynamics

ẋ1,[0] = x 1,[0] x 1,[0] (-1) + x 2,[0] -x 1,[0] + v [0] ẋ2,[0] =-(x 1,[0] x 1,[0] (-1) + x 2,[0] ) 3 -(x 1,[0] x 1,[0] (-1) + x 2,[0] -x 1,[0] + v [0] )x 1,[0] (-1) -(x 1,[0] (-1)x 1,[0] (-2) + x 2,[0] (-1) -x 1,[0] (-1) + v [0] (-1))x 1,[0] + v [0] (-1)
with output function

y [0] = x 1,[0] x 1,[0] (-1) + x 2,[0]
Then we get that the output has relative degree r = 1 and we have that

dy [0] = x 1,[0] (-1) + x 1,[0] δ, 1 dx [0] .
Accordingly, the right annihilator is

Δ(x,δ) = -1 x 1,[0] (-1) + x 1,[0] δ which is causal. Since dy [0] is closed we have that dλ 1 (x) = dy [0]
dz [0] = x 1,[0] (-1) + x 1,[0] δ 1 10 dx [0] → z [0] = x 1,[0] x 1,[0] (-1) + x 2,[0] x 1,[0]
the system reads

ż1,[0] =-z 3 1,[0] + v [0] (-1), ż2,[0] = z 1,[0] -z 2,[0] + v [0] , y [0] = z 1,[0] .
Example 2 Consider again the dynamics (1), but with output function

y [0] = x 1,[0] x 1,[0] (-1) + x 2,[0] x 2,[0] (-1)
In this case,

dy [0] = x 1,[0] (-1) + x 1,[0] δ x 2,[0] (-1) + x 2,[0] δ ,
which is still closed. However the right annihilator in this case is

Δ(x,δ) = - x 2,[0] (-1) x 1,[0] (-1) b 0 -x 2,[0] (+1)δ b 0 + x 1,[0] (+1)δ , with b 0 = x 1,[0] x 2,[0] (+2) -x 2,[0] x 1,[0] (+2) x 2,[0] (+1)x 1,[0] (-1) -x 1,[0] (+1)x 2,[0] (-1)
x 1,[0] (-1).

Obviously the generator of Δ(x,δ) can never be chosen causal, so that there is no causal function λ(x) such that dx [0] ∈ span K (δ] {dy [0] , dλ}.

We end this section by giving the conditions under which the unobservable dynamics in [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF] can be chosen in order to be independent of the control. The following result holds true. Theorem 5 Given system (1), with output y [0] characterized by the well defined relative degree r , there exist a regular bicausal static state feedback u [0] = α(x) + β(x)v [0] , and a bicausal change of coordinates

z [0] = z 1,[0] z 2,[0] = ϕ(x)
with z 1,[0] ∈ R r , such that in the new coordinates the closed-loop system reads as [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF] with η 2 j = 0,j∈[0,ℓ], if and only if the conditions of Theorem 4 are satisfied, and additionally, (a) given the differential representation (4), associated with the system,

Δ(x,δ) = span K (δ] {g 1 (x,δ)}
is completely integrable, that is, there exists an index γ such that ρ γρ γ -1 = 1;

(b) in the z-coordinates in which the system reads (13) with θ 2 j of the form (21), denoting by η 2 (z,δ) = ℓ j=0 η 2 j (z)δ j and by ϑ 2 (z 1 ,δ) = ℓ j=0 ϑ 2 j (z 1 )δ j , then

η 2 (z,δ) = ⎛ ⎜ ⎝ m 1 (z,δ) . . . m n-r (z,δ) ⎞ ⎟ ⎠ ϑ 2 (z 1 ,δ)
Proof The proof of the necessity is immediate if one considers that, condition (a) ensures that there exist n -1 exact differentials in the left kernel of g 1 (x,δ), which are the candidates for the bicausal change of coordinates. The second condition is necessary to guarantee that one can find a bicausal change of coordinates which achieves the result by transforming only the last nr components, since the first r ones are related to the input output behavior.

As for the sufficiency, suppose for simplicity that the system has already been transformed in the form [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF], with θ 2 j of the form (21). Since (a) holds true one can find n -1 exact differentials which lie in the left kernel of

Δ(x,δ) = span K (δ] {g 1 (z,δ)}.
Starting with the computation of such exact differentials at the first step one gets

dz 11,[0] , ••• , dz 1r -1,[0] .
We now have to show the form of the last nr exact differentials. To this end note that, due to b), a basis for Δ(x,δ) ⊥ is

Δ(x,δ) ⊥ = span K (δ] {dz 1i,[0] , i ∈[1, r -1], dz 2 j,[0] -m j (z,δ)dz 1 r,[0] , j ∈[1, n -r ]}
Since the previous collection of one-forms is closed, then any other collection which generates Δ(x,δ) ⊥ must be obtained through a unimodular matrix. Let dϕ 2 j (z), j ∈[1, ••• , nr ], be the set of exact differentials such that

Δ(x,δ) ⊥ = span K (δ] {dz 1i,[0] , i ∈[1, r -1], dϕ 2 j (z), j ∈[1, n -r ]} then ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ dz 11,[0]
. . .

dz 1 r -1,[0]
dϕ 21 (z) . . .

dϕ 2 n-r (z) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = I 0 T 21 (z,δ) T 22 (z,δ) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ dz 11,[0]
. . .

dz 1 r -1,[0] dz 21,[0] -m 1 (z,δ)dz 1 r,[0]
. . .

dz 2 n-r,[0] -m n-r (z,δ)dz 1 r,[0] ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
where T 22 (z,δ)is unimodular. Consequently the change of coordinates

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ dz 11,[0] . . . dz 1 r,[0] dϕ 21 (z) . . . dϕ 2 n-r (z) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ is bicausal.

Normal Form

In this section we analyze the conditions under which there exist a bicausal change of coordinates and a regular bicausal static state feedback thanks to which the closedloop system can be decomposed in the form [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF], with a linear input output behavior, that is the closed-loop system displays a linear input-output relation and reads in the new coordinates

ż1,[0] = ℓ j=0 A j z 1,[0] (-j) + ℓ j=0 B j v [0] (-j), ż2,[0] = η 1 (z [ℓ] ) + ℓ j=0 η 2 j (z [ℓ] )v [0] (-j) (26) y [0] = ℓ j=0 C j z 1,[0] (-j),
with z 1,[0] ∈ R r , r being the relative degree of the output. Of course, in this case the conditions of the previous Theorem 4 must be satisfied together with some new additional ones. The following result holds true. Theorem 6 Given system (1), with output y [0] characterized by the well defined relative degree r , there exist a regular bicausal static state feedback

u [0] =α(x) + β(x)v [0] ,
and a bicausal change of coordinates

z [0] = z 1,[0] z 2,[0] = ϕ(x) with z 1,[0] ∈ R r ,
(x,δ)dx [0] is left-closed. (ii') b(x,δ)in (18) is independent of x that is b(x,δ) = b(δ). Accordingly, (19) is always satisfied. (iii') the exact differential γ(x, 0,δ)dx [0] = dα(x) must satisfy a(x, 0,δ)-b(δ)γ (x,δ) dx [0] ∈ span R(δ] {dλ 1 , ••• , dλ r } (27)
The coordinates z 1 = λ 1 , ••• ,λ r T can be chosen in such a way that the B j 's in the closed-loop system (26), only have the last component different from 0, that is,

B j = ⎛ ⎜ ⎜ ⎜ ⎝ 0 . . . 0 b jr ⎞ ⎟ ⎟ ⎟ ⎠ , j ∈[0,ℓ]. (28)
Proof Of course the conditions of Theorem 4 must be satisfied, since the structure (26) represents a special case of the structure [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF]. The necessity and sufficiency of the additional conditions are proved hereafter. As for the necessity, if the system can be put in the form (26), then one gets that in the good coordinates

⎛ ⎜ ⎝ dy [0] . . . dy (r -1) [0] ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ C(δ) . . . C(δ)A r -1 (δ) ⎞ ⎟ ⎠ dz 1,[0] = Q(δ)T 1 (x,δ)dx [0] ,
which proves the necessity of (i'). Furthermore, for the closed-loop system one must have that

dy (r ) [0] =ã(δ)dz 1,[0] + b(δ)dv [0]
For any regular bicausal static state feedback v [0] =ᾱ(z) + β(z)u [0] , one has that

dv [0] =γ(z, u,δ)dz [0] + β(z)du [0] , so that dy (r ) [0] =ã(δ)dz 1,[0] + b(δ) γ(z, u,δ)dz [0] + b(δ) β(z)du [0] =ã(z, u,δ)dz [0] + b(δ) β(z)du [0]
which immediately proves (ii') and (iii').

As for the sufficiency, since the conditions of Theorem 4 are satisfied, in the new coordinates the system reads as [START_REF] Oguchi | A finite spectrum assignment for retarded non-linear systems and its solvability condition[END_REF].

Due to (i'), we can take as z

1,[0] coordinates z 1,[0] = (λ 1 (x), ••• ,λ r (x)) T . Taking u [0] = β -1 (x)(-α(x) + v [0]
) as the feedback law, where β(x) and α(x) are chosen as in Theorem 4 one gets that for the closed loop system, the r th derivative of the output becomes Consequently, its differential representation is dy (r ) [0] = d ā(x) -ℓ j=0 b j dα(x(-j)) + ℓ j=0 b j dv [0] (-j) =ã(δ)dz as well as that dy [0] = C(δ)dz 1,[0] . This ends the proof.

In this case, one may also investigate under which conditions the unobservable dynamics is independent of the control variable. The generalization of Theorem 5 is straightforward. The following result, whose proof is omitted since it follows the same lines as the proof of Theorem 5, holds:

Some Concluding Remarks

Various normal forms have been derived for single-output time-delay systems. Some of those normal forms include candidates for defining the so-called zero dynamics. However, as already underlined in [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF], the extension of the concept of zero dynamics to nonlinear systems with delays is not straightforward. As a matter of fact, in the delay-free case, zeroing the output, which has by assumption relative degree r , leads to the computation of a residual dynamics, called zero dynamics, which has dimension n -r , and in the case of single-input single-output systems, is independent of the control u.Asshownin [START_REF] Isidori | Nonlinear Control Systems[END_REF], this property has some important consequences on the stabilization procedure, since if the zero dynamics is locally asymptotically stable, then one may choose the input in order to satisfy some requirements on the input-output behavior. In the case of delay systems, instead, the zero dynamics may have a dimension greater than nr as shown in [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF]. However if the conditions of Theorem 4 (respectively Theorem 6) hold, then the system can be split, as already noted, into two subsystems, the first one representing the input-output behavior, and the second one influenced by the control and the state variable of the first subsystem.

The stability properties of this last subsystem become important when designing feedback laws, to guarantee a certain required input-output behavior.

(

  iib) there exists a function α(x) such that denoting by γ(x,δ)dx [0] = dα(x), then, a(x, u,δ)in (17) computed for u = 0 and b(x,δ)in (18) satisfy a(x, 0,δ)-b(x,δ)γ(x,δ) dx [0] ∈ Δ(x,δ) ⊥ (20)

  and a possible completion to get a bicausal change of coordinates is dz 2,[0] = dx 1,[0] . One thus has that in the coordinates

  x)β -1 (x(-j))(-α(x(-j)) + v [0] (-j))Due to (ii') and (iii'), each coefficient b j (x) = b j β(x(-j)), so thaty (r ) [0] =ā(x) + ℓ j=0 b j (-α(x(-j)) + v [0] (-j))

  1,[0] + b(δ)dv [0] Furthermore, from (i'), (δ)T 1 (x,δ)dx [0] = Q(δ)dz 1,[0] . (δ)d ż1,[0] = P 1 (δ)dz 1,[0] + P 2 (δ)du [0] .Multiply both sides on the left by the adjugate matrix[Q(δ)] (a) of Q(δ). Since [Q(δ)] (a) Q(δ) = q o (δ)I,where q 0 is the determinant of Q(δ) and I is the identity matrix, we getq o (δ)d ż1,[0] =[Q(δ)] (a) P 1 (δ)dz 1,[0] +[Q(δ)] (a) P 2 (δ)du [0]which immediately implies (from the identity of polynomials) that d ż1,[0] = A(δ)dz 1,[0] + B(δ)du[0]

Theorem 7 η 2

 72 Given system[START_REF] Califano | Extended lie brackets for nonlinear timedelay systems[END_REF], with output y [0] characterized by the well defined relative degree r , there exist a regular bicausal static state feedbacku [0] =α(x) + β(x)v [0] ,and a bicausal change of coordinatesz [0] = z 1,[0] z 2,[0] = ϕ(x)with z 1,[0] ∈ R r , such that in the new coordinates the closed-loop system reads as (26) with η 2 j = 0,j∈[0,ℓ], if and only if the conditions of Theorem 6 are satisfied and additionally (a) given the differential representation (4), associated to the system, Δ(x,δ) = span K (δ] {g 1 (x,δ)} is completely integrable, that is there exists an index γ such that ρ γρ γ -1 = 1; and (b) in the z coordinates in which the system is (26) with B j of the form (28), we have η 2 (z,δ) (z,δ) = ℓ j=0 η 2 j δ j and b(δ) = ℓ j=0 b j δ j .

  such that in the new coordinates the closed-loop system is in the Normal Form (26) if and only if the conditions of Theorem 4 are satisfied and additionally

	(i') Y(x,δ) in (16) can be factorized as Y(x,δ) = Q(δ)T 1 (x,δ), with Q(δ) a
	matrix with coefficients in R(δ], and the submodule generated by T 1


Majda Lafhel

Youssef Mourchid

Benjamin Renoust

Hocine Cherifi

Mohammed El

Assessing movie similarity using a multilayer network model

Keywords: Multilayer networks, Movies, Network portrait, Network similarity. Locations

This work investigates the movie similarity issue. A multilayer network model capturing various aspects of the story is built from movie scripts. Based on this representation, movies are compared not from summary or actors but using their storyboard. We rely on the "Portrait divergence" to quantify distances between graph characteristics. We illustrate the effectiveness of this approach in comparing movie series.

Introduction

In the computer vision literature similarity is a hot research topic mainly concerned with quality [START_REF] Pastrana-Vidal | Predicting subjective video quality from separated spatial and temporal assessment[END_REF][START_REF] Pastrana-Vidal | Frame dropping effects on user quality perception[END_REF][START_REF] Abouelaziz | Blind 3d mesh visual quality assessment using support vector regression[END_REF], classification [START_REF] Lasfar | Content-based retrieval in fractal coded image databases[END_REF][START_REF] Demirkesen | A comparison of multiclass svm methods for real world natural scenes[END_REF][START_REF] Hassouni | Hos-based image sequence noise removal[END_REF][START_REF] Lasfar | Content-based retrieval in fractal coded image databases[END_REF] or low-level analysis [START_REF] Rital | A combinatorial edge detection algorithm on noisy images[END_REF][START_REF] Rital | Weighted adaptive neighborhood hypergraph partitioning for image segmentation[END_REF][START_REF] Hassouni | Hos-based image sequence noise removal[END_REF][START_REF] Mourchid | An image segmentation algorithm based on community detection[END_REF] issues. Recently, network models have been increasingly used to support movie understanding. Usually, they concentrate on character interactions to bring out a global picture of the story [START_REF] Labatut | Extraction and analysis of fictional character networks: A survey[END_REF][START_REF] Viard | Movie rating prediction using content-based and link stream features[END_REF][START_REF] He | Srn: The movie character relationship analysis via social network[END_REF].

In a previous work we proposed a multilayer network model that captures more elements of the movie story including character, dialogues, and locations [START_REF] Mourchid | Multilayer network model of movie script[END_REF][START_REF] Mourchid | Movienet: a movie multilayer network model using visual and textual semantic cues[END_REF]. This model initially focused on the movie script has been extended recently to include visual content [START_REF] Lafhel | Movie script similarity using multilayer network portrait divergence[END_REF]. Network extracted from the movie are exploited to compute similarity. Therefore, the visual content similarity issue reduces to computing similarities between network layers. Plenty of methods are proposed in the literature for comparing monolayer networks [START_REF] Tantardini | Comparing methods for comparing networks[END_REF]. Here, we restrict our attention to a method recently introduced by Bagrow et al. [START_REF] Bagrow | An information-theoretic, all-scales approach to comparing networks[END_REF]. The so-called Network Portrait Divergence is based on the network portrait [START_REF] Bagrow | Portraits of complex networks[END_REF]. It characterizes a network by its degree distribution at various distances. Experiments performed on the movie series show encouraging results that corroborates user perception.

Majda Lafhel

Mohammed V University E-mail: majdalafhel1@gmail.com Youssef Mourchid L@bISEN, LSL Team, Yncréa Ouest Brest, France

Multilayer Movie Model

In the context of movie understanding the four Ws are defined as follows: Who? Refers to the characters; Where? Refers to the locations the action takes place; What? Refers to the subjects the characters talk about; When? Refers to the time guiding the succession of actions. We proposed a multilayer graph model that put these entities (characters, locations, and subjects) together in relation to form a story. As shown in Figure 1, this graph has three layers representing characters, locations, and topics (represented by keywords). Two types of relations are modeled: intra-layer relations between nodes of the same category (e.g. two characters conversing, the temporal transition between two locations, two Keywords are in the same conversation); and inter-layer relations (e.g. a character being at a specific location, a keyword is pronounced by a character, a keyword is mentioned in a conversation taking place at a specific location). We Rely mainly on the movie script to extract these entities and their relationships. Indeed, it is usually very well structured, and it contains all the key components to automatically analyze a movie. Investigation of this model have been performed on the whole 6-movie cycle of Star Wars saga. For each movie of the saga, we extract its corresponding multilayer network. Then, for each layer, we compute its corresponding portrait which encodes many structural informations of the graph. The portrait of a network can be computed using the B-matrix which denotes the number of nodes that have exactly k members in their respective l-shells (l-shells: the subsets of nodes at distance l from a given node v i ). Portraits are useful for showing an intuitive picture of a network. They provide a useful visualization and allows for rigorous statistical comparison between networks. Finally to compute the similarity between layers, for each couple of layer we compute their divergence.

Results

In the first experiment, we compute for each layer of each movie its corresponding portrait as shown in Figure 2. Due to the lack of space, we only present the result of Episode I, II and III. One can notice that layers of the same category have almost the same portrait. This proves that portrait is a useful property which can characterize networks of the same category. To compute the similarity between network we use the portrait divergence 0 ≤ DJS ≤ 1. It quantifies the dissimilarity of the two networks, with smaller DJS for more similar networks and larger DJS for less similar networks. One can notice in Figure 3, that character layers of Episode II and III are the more similar. In the Star Wars story, these three episodes compose the prequel trilogy that contains almost the same characters. We can also observe that Episode IV, V, and VI are much similar compared to other episodes (I, II, III). Indeed, these episodes constitute the story of the original trilogy, which similar cooccurrence of almost all the characters in the three episodes. A little similarity is observed between Episode III and IV. Indeed, episode IV is the bridge of Saga story between the prequel and the original trilogy. Similar observations can be made with the other layers (locations and keywords).

Conclusion

To summarize, extensive investigations with multiple movie series have shown that, computing the similarity between networks, especially for networks which encompass the movie story, the proposed multilayer model, can be a good choice to uncover similar movies. This methodology can be incorporated into recommender systems in order to increase their efficiency.

Fig. 1

 1 Fig. 1 Movie Multilayer Network

Fig. 2

 2 Fig. 2 Portraits of each layer of the movie for the three episodes.

Fig. 3

 3 Fig. 3 Divergence between layers for 6-movie episodes of Star Wars saga.