M Dragon-Louiset
email: dragon@lms.polytechnique.fr

On a predictive macroscopic contact-sliding wear model based on micromechanical considerations

Keywords: Friction, Wear, Energy release rate, Third body, Interface, Integral equations, Asymptotic expansion

A model for the mild wear of two contacting solids and an analytical example are proposed in this article. The model includes the presence of an interface made of damaged materials, ¯uid and wear debris. It consists in a wear criterion, an interface law and complementary relations deduced from the mass conservation. A thermodynamical analysis provides energy-release rates associated with the evolution of the surfaces in contact and the mass ¯uxes due to wear. They are used as characteristic quantities in the formulation of the wear criterion and wear velocities. Given that the physics of the interface modify the global contact conditions, micromechanical considerations are developed and result in an interface law, modeling its evolution with an internal parameter: the volume fraction of wear debris. The relation between this parameter and wear velocities is obtained with the mass conservation equation, which completes the model and allows to apply it in a numerical simulation. As an example, a problem of a rigid punch sliding on an elastic wornout half-plane is treated by means of integral equations, accounting on the presence of an interface according to the previous modeling. Stresses and strains are obtained analytically, as asymptotic expansion ®elds.

Introduction

Although contact-wear phenomena are of frequent occurrence, many diculties appear when engineers attempt to control them. Friction between contacting surfaces induces damage of materials, producing surface and subsurface cracks. As wear occurs, asperities of damaged surfaces are cracked, leading to loss of material and debris appearance in the contact interface. By the way, as wear occurs, geometrical changes take place and contact conditions are signi®cantly modi®ed because of the presence of wear particles. Life expectancy of machines can be reduced seriously, which implies the need for speci®c controls. This phenomenon is observed in nuclear power plants where security must be ensured, despite the wear of some components, therefore implying these components to be changed frequently. For complex structures, a 1 solution to this problem consists in using numerical simulations as a means to predict damage and to select the most resistant materials. This raises the question of a predictive model for contact wear, which could enable the evaluation of damage and loss of material by numerical calculations.

However, during the contact wear process, several mechanisms occur, either simultaneously or not, making an attempt to analyze very dicult. Loss of material produced by asperities cracks can be increased by corrosion; particles can be abrasive, or on the contrary, small enough to cause few eects in the interface. [START_REF] Ko | Wear of power plant components due to impact and sliding[END_REF] precisely describes the dierent wear mechanisms and provides an analysis of the wear of components in nuclear power plants. His article con®rms the complexity of wear problems, making their modeling dicult under conditions considered. Several models are presented in [START_REF] Ko | Wear of power plant components due to impact and sliding[END_REF]). An analysis of more than 5000 papers published by ``Wear'' is proposed by [START_REF] Meng | Wear models and predictive equations; their form and content[END_REF]. In the literature, mostly empirical equations are proposed, established for a particular system using speci®c parameters. These works provide informations about speci®c wear mechanisms and processes, but cannot be applied to other experimental conditions, which means that related equations are no more available. Some authors try to compare mechanical properties of materials, like fatigue limit, with their resistance to wear. Many studies are based on the Archard's equation [START_REF] Archard | Contact and rubbing of ¯at surfaces[END_REF] giving the volume of lost material as proportional to the normal load, to the sliding distance, the coecient of proportionality being called wear coecient. This one depends on operating conditions, therefore experimental data are needed for reliable predictive simulations. [START_REF] Str Omberg | Derivation and analysis of a generalized standard model for contact, friction and wear[END_REF] developed a comprehensive generalized standard model for wear, where Archard's law is introduced and its coecient slightly modi®ed. Few analytical models exist; one can refer to Zmitrowicz (1987a,b,c) for a complete mathematical framework.

More generally, global existing models do not take account of the physics of the interface, being unable, by the way, to provide fair results when dierent mechanisms can occur. As written in [START_REF] Singer | Role of third bodies in friction and wear[END_REF], macroscopic modeling should not ignore the in¯uence of the interface. Moreover, the complexity of a complete micromechanical model discourages its use for macroscopic modeling. Nevertheless, as pointed out by [START_REF] Meng | Wear models and predictive equations; their form and content[END_REF], the translation from microscopic observation to macroscopic laws may be done anyway, as a necessity for a predictive wear model.

The aim of this work is to develop a macroscopic model for mild wear, which includes interface related parameters, complex enough to describe the in¯uence of the preponderant physical phenomena, and simple enough to allow numerical simulations. A basic system of two contacting bodies with an interfacial ¯uid is studied. The ¯uid can be a lubricant or not, compressible or not. The loss of material is characteristic of the wear of one or both solids.

In this paper, a thermodynamical analysis of this system is advanced. Conservation laws are written taking into account the mass ¯uxes and the moving boundaries due to wear. From this, energy-release rates are obtained, depending on the stress and strain state in the sound material and the damaged one. A wear criterion for each solid is formulated, based on the energy-release rates. The interface contains the damaged parts of both materials and the third-body (¯uid and debris). An interface model is necessary to determine the quantities involved in the wear criterion and to calculate the mechanical state of the tribological system.

This macroscopic model is presented in Section 2. In Section 3, this model is built on micromechanical considerations, including an internal variable (volume fraction of wear particles in the interface). It may be interesting to compare the interface law inferred from this study with the Ruina±Kirchho friction law [START_REF] Scholz | Earthquakes and friction laws[END_REF], also expressed with an internal evolutive parameter. The mass conservation provides a relation between the internal variable and the wear criterion presented in Section 2.

These complementary studies allow a wear simulation. The problem of a rigid punch sliding on an elastic half plane submitted to wear and their interface is proposed in Section 4, as an example, treated analytically with integral equations. It may be noted that generally, both contacting solids are not simultaneously losing material; consequently, assuming that the punch is not worn-out is not an important restriction. The problem's statement results in one essential equation (called the wear equation) to solve. Assuming the ¯uid incompressible, asymptotic expansion solutions with respect to the small parameter (fraction of wear particles in the interface) are searched, which allows analytical treatment of the wear equation. The zeroorder solution without wear is given; the ®rst-order solution with wear of the elastic half plane can then be determined.

Macroscopic modeling of contact wear phenomena

In this section, a thermodynamical approach of a system of two contacting solids is developed, in the case of mild wear of one or both solids and loss of material. The interfacial layer of the bodies is more precisely described in the next section; in this one, a global model is built with the assumption of mass ¯uxes from the worn solids to the interface, non-zero ¯uxes being characteristic of wear phenomena. Similar assumptions are made in a complex mathematical framework for wear of two contacting solids proposed by Zmitrowicz (1987a,b,c); however, his study needs further physical interpretation. Before developing the thermodynamical approach, let us consider Stribeck's curves in order to distinguish mild wear from severe wear mechanisms, and to infer some evidence for our study.

Stribeck's curve: regimes I±III

Wear experiments provide x±y plots with x gV =p and y, the friction coecient (g: ¯uid±lubricant viscosity, V: relative velocity of the contacting bodies, p: pressure) called Stribeck's curves, schematically represented in Fig. 1. In the case where gV =p is small, it may be noted that the friction coecient is high (I), which corresponds to dry and severe wear, when surfaces are not entirely protected by the ¯uid. As gV =p increases, the ¯uid is spread over the contacting surfaces and the friction coecient decreases. An unstable phase (II) follows, with an incomplete ¯uid cover: here combined dry and lubricated wear mechanisms occur. Phase (III) corresponds to a stable lubricated hydrodynamic regime, where an interface made of ¯uid is formed. [START_REF] Havet | Caract erisation tribologique de biolubri®ants[END_REF] presents some tribological aspects of lubrication and Stribeck's curves, giving references on this topic.

This con®rms the interaction between the so-called ®rst bodies (contacting solids) and the third body (¯uid and wear products). This evidence results in the necessity to associate a global approach to the wear modeling with the description of the interface evolution, through some internal state variable as will be proposed later. Our purpose in this section is to propose a macroscopic model for the wear of two bodies in

V/p η friction coefficient Ι ΙΙΙ ΙΙ Fig. 1. Stribeck's curve.
contact, the interface being the subject of principal interest in Section 3. Both complementary theoretical approaches, thermodynamical analysis and interface law, can then be used for numerical simulations of the wear of a structure.

The energy release rate approach in wear mechanics

In the present analysis, assuming that wear produces a characteristic dissipation, energy release rates for the phenomena are determined. Energy release rates are used in fracture mechanics to describe cracks propagation. Here, these quantities characterize the production of wear debris by damage of solid and particle detachment. They are used in the formulation of a wear criterion, which is completed in the further study by the interfacial ®lm behavior.

Three-domain model

Let us consider a system of two solids in contact in Fig. 2, one moving with respect to the other. Apparently smooth surfaces are, on a large scale, made of asperities; subsurface materials instead of being unaected, are damaged, cracked because of pressure and friction due to the movement. C 1 and C 2 are boundaries separating sound materials from the damaged ones. In fact, the density of cracks is increasing continuously if we move to the contacting edge of solids 1 and 2. In this model, we will distinguish sound material from the damaged one approximately and separate them by the boundary C i , considering that material becomes damaged and goes through C i , once its macroscopic behavior is no more known and controlled. By the way, we build a three-domain model (third picture in Fig. 2): X 1 and X 2 are, respectively, the sound parts of the solids 1 and 2, X 3 is the interface with a complex behavior containing damaged parts of both solids and the third body made of wear products mixed with the ¯uid, eventually the lubricant.

When the solid i is worn, particles in X 3 are detached and cracks propagate towards sound material; there are mass ¯uxes through C i from X i to X 3 . (The ¯uid and debris ¯ow in X 3 is described in the next section.) Balance equations are written for the system X X 1 X 2 X 3 with moving boundaries C 1 and C 2 . See [START_REF] Pradeilles-Duval | Mechanical transformation and discontinuities along a moving surface[END_REF] for the study of a system with moving boundaries, and Dragon-Louiset and Stolz (1999) for details about this analysis.

Conservation laws and entropy production

Let us introduce the jump or discontinuity of a quantity b through the moving surface C i , i 1; 2: C i . W i is the geometrical celerity of C i , n i is its unit normal vector as shown in Fig. 2. In Eulerian formulation, U de®ning the velocity, time derivative of integrals of b over the domain X is d dt

sbt 3 i b 3i À b i
X b dX i1;2;3 Xi ob i ot divb i U i ! dX i1;2 Ci sbU À W i t 3 i n i dC: 1
Integrals over C 1 and C 2 come therefore from the conservation laws applied to X. Conservation equations are obtained for X i (i 1; 2; 3) and C i (i 1; 2). Mass conservation gives the mass ¯uxes

l i q i U i À W i n i Àq i v i (i 1; 2), with v i W i À U i n i the material velocity of the boundary C i . The equation, l i sw sT t 3 i sU t 3 i r 3i r i =2n i À sqt 3 i n i (i 1;
2), arises from the law of the conservation of energy (w denotes the free energy, s the entropy, T the temperature and q the heat ¯ux). According to the second law of thermodynamics, the internal entropy production W is non-negative; for both boundaries C 1 , C 2 : W Ci l i sst 3 i sq=T t 3 i P 0, and in X 3 , W X 3 ex 0 r : _ e À q _ w s _ T ÀqrT =T 1=T dy P 0 ex is the thickness of the layer. We suppose from now the continuity of the temperature T through both boundaries (T T i on C i ).

Wear criterion

The entropy production due to the wear of X i was expressed above as W Ci , which can be transformed with the equation of conservation of energy and simpli®ed thanks to two assumptions. At ®rst, on the assumption that wear is mild, we presume that normal stresses are continuous through C i . However, owing to the balance equation of momentum: srt 3 i n i l i sU t 3 i ,( i 1; 2); which presupposes that l i sU t 3 i is a second-order term. Secondly, displacements n through C i are supposed to be continuous. Hadamard consistency relations provide sU t 3 i Àv i srnt 3 i n i (i 1; 2). This enables us to write W Ci in the following form:

W Ci v i T i g i À g 3i with g i n i r i rn i n i À q i w i ; g 3i n i r 3i rn 3i n i À q i w 3i ; & 2
where g i and g 3i are energy release rates associated to the wear of solid i. Assuming that there exists a wear threshold g is de®ned for the material of the solid i, we formulate a wear criterion as follows:

if Gg i ; g 3i g i À g 3i À g is < 0; no wear occurs; if Gg i ; g 3i g i À g 3i À g is P 0; wear of i occurs:

& 3
The velocity v i can be inferred from the criterion where wear occurs: v i F hGg i ; g 3i i where F is a function hbi is the positive part of b, i.e. hbi 0i fb 6 0, hbi b if b > 0. Several forms of F can be examined (and confronted with experiments), e.g. linear v i bhGg i ; g 3i i , or polynomial. Another way to determine v i can be to develop a normality law, similar to plasticity; this will be studied in the future. Although g i is easy to determine in the case where both contacting solids are elastic (it seems sucient for mild wear process), g 3i needs further study. The in¯uence of the interface on the ®rst bodies is expressed by g 3i ; it contains the physics of X 3 . We attempt to approach the behavior of the interface in the following section.

A multiscale model of the third body

The three-scale third body

Our aim is not to describe and model the microscopic mechanisms which occur in X 3 , but to use micromechanical considerations in order to describe in a realistic manner the X 3 physics and then model it on a mesoscopic scale. A relevant interface law is inferred from this analysis, for applications on a macroscopic scale. Consider the Fig. 3 where three scales from the microscopic to the macroscopic one via a mesoscopic scale are represented:

(l), microscopic scale: Asperities in the third body on microscopic scale, wear mechanisms. Some authors propose studies of plastic strains, cracks propagation, contact between two asperities on the microscopic scale [START_REF] Barbarin | Instabilit e et frottement en elasticit e: application a un probl eme d'ondes de contrainte[END_REF][START_REF] Stupkiewicz | A model of third body abrasive friction and wear in hot metal forming[END_REF].

• X 31 , X 32 : damaged material, cracks, wear particles jammed into the holes between asperities. The ¯uid does not soak completely into the cracks; • X 33 : contact between asperities which are progressively detached. The ¯uid carrying the debris along.

(m), mesoscopic model: Third body on mesoscopic scale: X 3 X 31 X 33 X 32 . This three-area model was proposed by [START_REF] Godet | Extrapolation in tribology[END_REF]1990) and completed by [START_REF] Georges | Comment aborder un probl eme de tribologie[END_REF], [START_REF] Georges | Drainage of thin liquid ®lms between relatively smooth surfaces[END_REF], [START_REF] Berthier | M ecanismes et Tribologie[END_REF]1990), [START_REF] Berthier | Velocity accommodation in fretting[END_REF].

• X 31 , X 32 : porous, damaged media with no shear, no shear-stress but only strains e xx , e yy and stresses r xx , r yy ; • X 33 : solid particle suspension forming a sheared layer and viscous ¯uid ¯ow [START_REF] Dragon-Louiset | Mod elisation microm ecanique de l'interface d'un syst eme tribologique dans une approche thermodynamique de l'usure continue[END_REF].

Strains e xy and e yy , shear-stress r xy and compressive stress r yy .

(M), macroscopic model: Interface on the macroscopic scale: C C 1 X 3 C 2 . There are several models and friction laws in the literature with an internal variable, for example the Ruina-Kircho's friction law [START_REF] Scholz | Earthquakes and friction laws[END_REF]. In this paper, we will propose an interface law adapted to the wear phenomena, giving a relation between the compressive stress r yy and the strain e yy , the shear-stress r xy and _ e xy (because of the viscosity of the solid particle suspension). r yy , e yy , e xy , and e xx are related to the boundary conditions on C 1 and C 2 for a macroscopic point of view. Note: the friction coecient can be inferred from the interface behavior by l r xy =r yy where overlined ®elds mean averages on the layer thickness.

Macroscopic interface law

We will assume, for the application, steady-state wear, 2D, plane strain, incompressible ¯uid, 2 moves at the speed of ÀV x with respect to 1, x and y axis are ®xed in 1.

Internal variable for the layer X 3

Internal variable: u volume fraction of the particles (u u 1 u 2 particles from X 1 and X 2 ) then u f is the volume fraction of ¯uid and u 1 u 2 u f 1. Relations between u, e, v wear velocity and V the relative velocity between 1 and 2 are given by the conservation of mass for the three types of material (debris of 1, debris of 2 and ¯uid).

The balance of mass is oq 3 =otdivq 3 U 3 0 where q 3 q 1 u 1 q 2 u 2 q f u f . In the steady-state case, e ex; y, u i u i x; y for i 1; 2; f. Mean values of volume fractions can also be de®ned by: u i x1=ex ex 0 u i x; ydy for i 1; 2; f. As we described X 3 , X 3i , i 1; 2 are the intermediate areas for the material, which is ®rstly damaged during the process, goes through C i at the wear velocity v i , before being detached and then become a wear debris. Secondly, this material turned into a wear particle is taken away from X 3i to X 33 .InX 3i , u i 9 1. We introduce the parameter a i x, i 1; 2, the volume fraction of material i, which goes from X 3i to X 33 , assuming that only a part of the material in X 3i is removed into the X 33 mixture. (a i can also be considered as a constant parameter, equal to 1 if the last assumption cannot be made.) We can consider that the ¯ow of wear particles goes from X 3i to X 33 at the same velocity as it goes through C i (v i ).

We assume that the ¯uid is incompressible and the particles rigid once detached, and take ÀV =2a s averaged shear velocity for the particles and the ¯uid in the interface, treated as a homogeneous layer. This means that the two-dimensional ¯uid ¯ow near the end points x AEa of the contact interface is disregarded. Taking account of particle incomes a i v i dx due to wear in a section of X 3 between x and x dx, the balance of mass can be written for solid particles and ¯uid, in the steady-state case:

solid a i xv i x V 2 o ox u i xex 0; i 1; 2; 4 fluid o ox 1  À u 1 xÀu 2 xex à 0: 5
If there is no wear particle in the interface, its thickness is the sum of rugosities of both solids equal to e 0 ; Eq. ( 5) becomes 1 À u 1 xÀu 2 xexe 0 on the one hand. On the other hand, the wear criterion gives the velocity v i . Both relations (4) and ( 5) are enough to determine u u 1 u 2 needed to evaluate stresses and strains in the interface, giving by the way the interface law.

Stresses and strains in the interface

Normal compression and viscous shear: r yy jue yy ; 6 r xy gu _ e xy : 7

Strains are given by the boundary conditions:

e yy u 1 y À u 2 y eu ; 8 _ e xy _ u 1 x À _ u 2 x eu : 9
The coecient jufu À1 u 1 =K 1 u 2 =K 2 u f =K f ÂÃ À1 corresponds to Reuss's law of strains additionality or stress homogeneity. In the case of an incompressible ¯uid, jufu À1 u 1 =K 1 Â u 2 =K 2 À1 . The coecient gug 0 1 2:5u designates the viscosity given by Einstein's law for the viscosity of a solid particle suspension [START_REF] Landau | M ecanique des Fluides[END_REF][START_REF] Van Der Wer | Hard-sphere colloidal dispersions: the scaling of rheological properties with particle size, volume fraction, and shear rate[END_REF][START_REF] De Kruif | Hard sphere colloidal dispersions: viscosity as a function of shear rate and volume fraction[END_REF][START_REF] Dragon-Louiset | Mod elisation microm ecanique de l'interface d'un syst eme tribologique dans une approche thermodynamique de l'usure continue[END_REF]. In the case of mild wear, u remains small (u max 9 0:6 for a compact assembly of circles of same radius).

The wear criterion needs to be evaluated near the boundaries C i for both sides X i (g i ) and X 3i (g 3i ). The behavior of X 1 and X 2 is known, so g i can easily be calculated and the behavior of X 3i , for which mean values are no more appropriate, will give g 3i .InX 3i , u i 9 1, r xx T 0 and r yy T 0. r i yy r 3i yy , u i u 3i . Then, we can take r i yy r 3i yy ju i 1e yy , e 3i xx e i xx u i x;x , r 3i xx j H u i 1e 3i xx .

Later on, we will employ the quantities mugu=eu and kuju À1 eufueu, where u was noted u for simplicity. The following resumes the three modeling steps.

The interface model

(1) wear criterion and velocity Gg i ; g 3i g i À g 3i À g is P 0 v i F hGg i ; g 3i i ; i 1; 2:

(2) conservation of mass

solid a i xv i x V 2 o ox u i xex 0 i 1; 2; fluid 1 À u 1 xÀu 2 xexe 0 :
(3) viscous shear and pressure

r xy xmux _ u 1 x À _ u 2 x ; r yy xkuxu 1 y xÀu 2 y x:

The wear equation for incompressible ¯uid

The case of an elastic half-space (solid 2) and a cylindrical rigid punch (solid 1) is examined with an incompressible ¯uid. Only one of the solids is worn, the half-space, it follows that there is only one kind of debris in the interface: u u 2 and 1 u u f . In order to obtain displacements, stresses and strains at the surface of the half-space, which is covered in the contact area by the interface made of ¯uid and wear debris, we will use the Galin's integral equations [START_REF] Galin | Contact Problems of the Theory of Elasticity[END_REF]. The study of an elastic half-space submitted to contact wear due to a rigid punch was done by [START_REF] Galin | Contact problems of the theory of elasticity in the presence of wear[END_REF] and [START_REF] Galin | Axisymmetric contact problem of the theory of elasticity in the presence of wear[END_REF]. In those papers, there is nevertheless no wear debris between the punch and the half-space, therefore wear phenomena in those studies has no in¯uence on the contact pressure and displacements. Our purpose is to take into account the presence and the in¯uence of the third body using the interface model.

Problem's statement

The x-and y-axes are moving with the punch, whose symmetry axis is o-center and has abscissa x 0 . The displacements of the punch u 1 denoted u p are u p

x , with the following assumption _ u p x 9 0, and u p y x d x À x 0 2 =2R. From now on, v 2 , a 2 , u 2 , u, r yy , K 2 and r xy are, respectively, denoted as v, a, u, u, r yy , K and r xy in order to simplify.

The contact area is bounded by x a and x Àa. Ahead of the punch, no debris is present in the interface: ux a0. Let us take a a constant parameter; the balance of mass Eq. ( 4) becomes avxe 0 V =2ou=oxx0. By the way, once v is induced by the wear criterion, u is given by uxÀ 2a Ve 0 x a vt dt; x 6 a: 10

In the case of mild wear, keeping ®rst-order terms for u in the model of the interface is enough. Relations between stresses and displacements described before are r xy gu=eu _ u p x À _ u x and r yy fueu u p y À u y . The ¯uid being incompressible, jufu À1 u=K À1 . The layer's thickness eue 0 =1 À u is given by Eq. ( 5 Because of the assumptions made on _ u p x and _ u x , Eq. ( 12) is similar to a plasticity threshold, which depends here on the internal variable u and on V. It is consistent with phase III of Stribeck's curve, where s=p G gV =p i.e. s G gV .

The wear equation

Galin's equations [START_REF] Galin | Contact Problems of the Theory of Elasticity[END_REF] are

E 21 À m 2 du x dx x 1 À 2m 21 À m r yy xpv 1 p a Àa r xy s ds s À x ; 14 E 21 À m 2 du y dx xÀ 1 À 2m 21 À m r xy xpv 1 p a Àa r yy s ds s À x : 15
Reciprocal relations can be found in [START_REF] Bui | Inverse Problems in the Mechanics of Materials[END_REF]. The principal value is denoted as pv, de®ned by

pv b a f s ds s À x lim 30 xÀ a f s ds s À x b x f s ds s À x ! : 16
Solutions f of pv b a f s ds=s À x gx where g is a given function satisfying some regularity conditions, can be found in [START_REF] Muskhelishvili | Singular Integral Equations[END_REF][START_REF] Muskhelishvili | Some Basic Problems of the Mathematical Theory of Elasticity[END_REF]. With c 1 E=21 À m 2 and c 2 1 À 2m=21 À m.B y replacing u y using Eq. ( 13) in Eq. ( 15), and r xy using Eq. ( 12) in both relations ( 14) and ( 15), we obtain

c 1 du x dx xc 2 r yy x g 0 V e 0 pv 1 p a Àa 1 1:5us ds s À x ; 17 c 1 e 0 K d dx r yy xux ÂÃ pv 1 p a Àa r yy s ds s À x c 1 du p y dx xc 2 g 0 V e 0 1 1:5ux: 18
The criterion gives v F g 2 ; g 32 , u is determined by the balance of mass equation ( 10) and r xy by the viscous interface law equation ( 12). We have to solve Eq. ( 18) to obtain r yy (where r yy appears as well in the integral pv as outside this integral), u y being then given by Eq. ( 13) and u x by Eq. ( 17). Finally, the wear equation ( 18) for r yy is the one to solve in order to obtain stresses, displacements and strains at the surface of the half-space. Its solution can be computed, or evaluated approximately for a given type of function (for example if r yy is developed into a Fourier series or into Chebyshev's polynomial). Another way to solve this equation is to search asymptotic expansion solutions, with respect to a small parameter related to ux. This is discussed in the following sections.

The normal stress r yy can be computed. An iterative algorithm can determine approximated ®elds r, e, then calculate v, u, solve the wear equation ( 18) and update u, repeating this until convergence. This convergence is brie¯y discussed here.

Considering that the normal stress r 1 is obtained knowing r 0 , after the evaluation of v 0 and u 0 , all mappings are linear except the ®rst one v 0 F r 0 , which is generally nonlinear, depending on the physics and experimental conclusions. Assuming that F is a Lipschitzian function and the mapping M giving the solution r n1 r 0 Mr n can be proved to be a contraction for small values of u, the problem solution is the ®xed point of the nonlinear mapping: r r 0 Mr [START_REF] Courant | Methods of Mathematical Physics[END_REF]. By the way, the convergence of the algorithm can be proved.

Asymptotic expansion solutions for a mixture with incompressible ¯uid

In the case of mild wear, u is assumed to be small. We will consider that the zero-order solution is obtained for u 0 (no wear occurs). Higher-order solutions are deduced from the zero-order one and other lower-order solutions. The small parameter used for the expansion can be x sup jxj a juxj; the asymptotic expansion ®elds take the form: The wear rate v is function of the energy release rates g 2 and g 32 , which depend on r xx and r yy . Let v i at the i-order be obtained with only the i À 1-order stresses r iÀ1 yy , r iÀ1 xx :v i F g 2 ; g 32 r iÀ1 yy ; r iÀ1 xx . This assumption is not necessary to solve the problem and has no important consequence on its solution, but it simpli®es the equations for analytical study. Notice that r iÀ1 1:5g 0 V e 0 u i x for i P 1:

v xv 1 x 2 v 2 ÁÁÁ; u xu 1 x 2 u 2 ÁÁÁ; u x u 0 x xu 1 x x 2 u
V X 28
Because u p y is given by Eq. ( 20), h 0 is a regular function and satis®es a H older condition on Àa; a [START_REF] Muskhelishvili | Singular Integral Equations[END_REF]. Zero-order solution can easily be determined, as shown hereafter. This is no more valid for higher orders. However, on the assumption that ®rst-order terms for u provide precise enough results (Section 4.1), there is no sense in searching higher-order solutions; thus, only zero-and ®rst-order problems are solved. Total pressure in the contact area is inferred from both zero-and ®rst-order solutions:

P P 0 xP 1 À a Àa r 0 yy x h xr 1 yy x i dx: 29

Zero-order solution

At ®rst, let us consider the contact without wear to determine the zero-order solution. According to Eqs. ( 23) and ( 24), respectively, r 0 xy g 0 V =e 0 , u 0 y xu p y x. The stress r 0 yy is given by Eq. ( 25), where u 0 and the punch displacement was replaced by Eq. ( 20):

pv 1 p a Àa r 0 yy s ds s À x c 1 R x c 2 g 0 V e 0 À c 1 x 0 R ! h 0 x: 30
(See Appendix A for the solution.) The consistency condition a Àa h 0 sa 2 À s 2 À1=2 ds 0 concerning the right-hand side of Eq. ( 30) yields h 0 xc 1 x=R and x 0 is obtained by setting the bracketed term of Eq. ( 30) to zero. The zero-order solution r 0 yy is, therefore, the Hertzian contact pressure. We ®nally determine du 0

x =dxx using Eq. ( 26). The zero-order solution is given by v 0 0;

u 0 0; r 0 xy g 0 V e 0 ; r 0 yy xÀ c 1 R a 2 À x 2 p ; 31 du 0 x dx xÀ c 2 R a 2 À x 2 p g 0 V pc 1 e 0 ln a À x a x ; x 0 c 2 c 1 g 0 VR e 0 ;
u 0 y xu p y xd x À x 0 2 =2R:

First-order solution

Assuming that v 1 F g 2 ; g 32 r 0 yy ; r 0 xx , u 1 x is determined by Eq. ( 22), r 1 xy and u 1 y are given by Eqs. ( 23) and ( 24), with r 1 yy solution of pv 1

p a Àa r 1 yy s ds s À x Àc 1 e 0 K d dx r 0 yy xu 1 x c 2 1:5g 0 V e 0 u 1 xh 1 x: 32
u 1 a0 but uÀaT 0 Eq. ( 22). The derivative h 1 in Eq. ( 32) becomes therefore singular for x Àa, because r 0 yy Àc 1 =R a 2 À x 2 p . (For the criterion presented before and in the case of a linear or polynomial function F Eq. ( 21), taking into account that v depends on r yy squared, u 1 x (22) is a polynomial with no singularity.)

It implies that h 1 does not satisfy a H older condition in x Àa. Let us express h 1 as a sum of four terms; its regular part h r 1 for x PÀa; a, its singular part h s 1 , xCa x and ÀxCa x with C a constant to determine: h 1 xh r 1 xh s 1 xxCa xÀxCa x. Instead of solving Eq. ( 32), we will move forward h s 1 xÀxCa x to the next order equation. Thus, Eq. ( 32) is replaced by

pv 1 p a Àa r 1 yy s ds s À x h r 1 xxCa x; 33
where the right-hand member satis®es a H older condition on Àa; a. The term 1=xh s 1 xÀCa x is added to h 2 x for second-order equation; if further orders solutions are searched, a similar regularization is needed for each step. The constant C is given by the consistency condition

a Àa h r 1 s  xCa s à a 2 À s 2 À1=2 ds 0: 34
The ®rst-order solution is then v 1 xF g 2 ; g 32 r 0 yy ; r 0 xx ;

u 1 xÀ 2a Ve 0 x a v 1 t dt; r 1 xy x 1:5g 0 V e 0 u 1 x;
C given by Eq: 34; 35 r 1 yy xsolution of Eq: 33;

du 1 x dx xÀ c 2 c 1 r 1 yy x 1:5g 0 V c 1 e 0 pv 1 p a Àa u 1 s ds s À x ; u 1 y xÀ e 0 K r 0 yy xu 1 x:

Analytical example

The ®rst-order term r 1 yy can be determined analytically for a simpli®ed wear criterion. Taking for example Gg 2 ; g 32 Hr yy 2 ,a n dv bhGg 2 ; g 32 i , the ®rst-order velocity is given by v 1 bHr 0 yy 2 , according to the assumptions made above. Using Eq. ( 35), the volume fraction u 1 , the constant C and the ®rst-order normal stress r 1 yy are calculated. The parameter abH is expressed as v. The constant C is evaluated so that h r 1 xxCa x satis®es the consistency condition Eq. ( 34), which implies C vc 2 1 =R 2 2a 2 4=3pc 2 1 =KVRÀc 2 g 0 =e 2 0 ÂÃ . Finally, using the reciprocal formula (A.7) (Appendix A), it follows from Eq. (33) that

r 1 yy r 0 yy vc 1 R & À 2 p c 2 1 KVR a 2 À x 2 ln a À x a x À 2ax ! c 2 g 0 e 2 0 x 2 À 5 2 a 2 ! 2a 2 4 3p c 2 1 KVR À c 2 g 0 e 2 0 !' : 36
In Fig. 4 are drawn r 0 yy as a dashed line and r yy r 0 yy xr 1 yy as a solid line (with negative units). (The following parameters was chosen: E 2 Â 10 À11 , m 0:34, g 0 3:5 Â 10 À4 , e 0 1 Â 10 À6 , R 0:02, V 1, x 0:1, v 30 Â 10 À12 .) The ®rst-order normal stress xr 1 yy appears as a correction for r 0 yy corresponding to the in¯uence of the wear process. For a fair estimate of this correction, some parameters set here to unity (e.g. b necessary to evaluate the wear velocity) must be experimentally determined, as material properties.

Conclusion

A wear criterion derived from the second law of thermodynamics is proposed in this paper, taking into account the mass ¯uxes due to the production of the wear debris. Speci®c energy release rates arise in this analysis and can be interpreted as the energy dissipated during the process of asperities cracks. The wear velocities of the contacting bodies are inferred from the criterion. This global study is coupled to the interface law which describes, in the case of mild wear, the average behavior of this complex area made of damaged subsurfaces and third body. The volume fraction of particles in this interface is introduced as the internal parameter. An equation of mass conservation completes the study, providing a relation between the internal parameter and the wear velocity deduced from the wear criterion.

This model can be applied to two contacting structures losing material, and solved by the ®nite element method, but also, as it is proposed in this article, by integral equations. The problem of an elastic half space and a rigid punch with their interface leads to a single wear equation to solve. Its solution, the normal stresses in the contact area, can be determined as an asymptotic expansion with respect to the small parameter: the volume fraction of debris. An analytical example is proposed, for a simpli®ed wear criterion.

The loss of material and the changes of surface geometry can be evaluated easily with this model, where the wear velocity and the volume fraction of debris are the essential parameters. An important quantity in tribology is the friction coecient, which sometimes varies making modeling dicult; it can be inferred here from the shear and the normal stress in the interface and is not a data.

Fig. 2 .

 2 Fig. 2. Three-domain model.

Fig. 3 .

 3 Fig. 3. From the microscopic scale to the macroscopic one.

Fig

  Fig. 4. r 0 yy (---) and r yy r 0 yy xr 1 yy .

  ), thus by linearization, we denote

	mu	gu eu	g 0 e 0	1 1:5u and kufueu	e 0 K	u:	11
	Assuming that _ u p x 9 0 for the punch and _ u x ÀV ou x =ox9ÀV for the half-space, we obtain r xy xmux _ u p x À _ u x Ar xy x g 0 e 0 1 1:5uxV ;	12
	r yy xkuxu p y xÀu y xAr yy x	e 0 K	uxu p y xÀu y x:	13

  -order terms for v and u; v 1 , u 1 being the normalized wear rate and volume fraction, respectively. The following equations given in Section 4

	2 x ÁÁÁ; y x 2 u 2 y xu 1 u y u 0 y ÁÁÁ; r xy r 0 xy xr 1 r yy r 0 yy xr 1 yy x 2 r 2 yy ÁÁÁ xy ÁÁÁ; xy x 2 r 2	19
	with no zerou p y xd	x À x 0 2 2R	;				20
	v F g 2 ; g 32 ;							21
	uxÀ	2a Ve 0	a	x	vt dt; x 6 a;	22
	r xy x	g 0 V e 0	1 1:5ux;		23
	r yy x	e 0 K	uxu p y xÀu y x 24
	will now be used to ®nd zero-order and higher-order solutions together with the wear equation
	c 1	e 0 K	d dx	r yy xux ÂÃ		pv	1 p	a Àa	r yy sds s À x	c 1	du p y dx	xc 2	g 0 V e 0	1 1:5ux;	25
	c 1	du x dx	xc 2 r yy x	g 0 V e 0	pv	1 p	a Àa	1 1:5usds s À x	:	26

Acknowledgements

The author wishes to thank M. Zbinden of Electricit e de France for his support through an EDF Grant. Many thanks also are addressed to Professors H.D. Bui and Z. Mr oz for their encouragement and comments on this work.

Appendix A

Let us solve the Hilbert equation (A.1), where f is the unknown bounded function at x AEa and g is a given function, satisfying a H older condition for x PÀa; a:

As will be seen by using the reasoning presented below, whose details can be found in [START_REF] Muskhelishvili | Singular Integral Equations[END_REF], f can be determined by a reciprocal relation. (See also [START_REF] Bui | M ecanique de la Rupture Fragile[END_REF] and [START_REF] Muskhelishvili | Singular Integral Equations[END_REF] for the complete mathematical background.) Introducing the holomorphic function Uzpv1=2ip a Àa f t dt=t À z vanishing at in®nity, we can write the following Plemelj formulae valid for x PÀa; a

where U x is the value of Ux on the y > 0 side of the line x PÀa; a and U À x is its value on the y < 0 side. Let Z be a second holomorphic function de®ned by Zza À z 1=2 a z 1=2 . It can easily be noted that Z satis®es Z xÀZ À x, thus dividing the second Plemelj formula (A.3) by Z x, and similarly the ®rst one Eq. (A.2), we obtain,

Eqs. (A.4) and (A.5) are the two Plemelj formulae where f x=Z xpv 1 pi a Àa gt=iZ tdt=t À x,on condition that Uz=Zz vanishes at in®nity, with Uz=Zzpv1=2ip a Àa gt=iZ tdt=t À z. (for x PÀa; a, Z xa 2 À x 2 1=2 ). Because lim z3I Uz0 and lim z3I ZzÀijzj, this condition is satis-®ed, provided that a Àa gt dt a 2 À t 2 1=2 0: A:6

Under this condition, f is given by

In Section 5.1, the solution to Eq. ( 30) can be obtained knowing that pv1=p 1 À1 tdt= 1 À t 2 p t À x 1 for x PÀ1; 1, or using expansion of h 0 x=a 2 À x 2 1=2 , where h 0 is polynomial [START_REF] Muskhelishvili | Singular Integral Equations[END_REF].


Deividas Eringis

John Leth

Zheng-Hua Tan

Rafal Wisniewski

Mihaly Petreczky
email: mihaly.petreczky@centralelille.fr

Explicit construction of the minimum error variance estimator for stochastic LTI state-space systems

Keywords:

In this short article, we showcase the derivation of the optimal (minimum error variance) estimator, when one part of the stochastic LTI system output is not measured but is able to be predicted from the measured system outputs. Similar derivations have been done before but not using state-space representation.

Introduction

Realization theory of stochastic LTI state-space representations with exogenous inputs is a mature theory [START_REF] Lindquist | Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification[END_REF]; [START_REF] Katayama | Subspace Methods for System Identification[END_REF]. In particular, there is a constructive theory for constructing a minimal stochastic LTI state-space representation of a process with exogenous input . The construction uses geometric ideas, and it is based on oblique projection of future outputs onto past inputs and outputs.

Note that it in system identification it is often assumed that ( , ) has a realization by an autonomous stochastic LTI system driven by white noise. Indeed, if has a realization by a stochastic LTI state-space representation, and has a realization by a LTI state-space representation with exogenous input , then under some mild assumptions (absence of feedback from to ) ( , ) will be the output of an autonomous stochastic LTI state-space representation. It is then natural to ask the question how to construct a minimal stochastic LTI realization of with input , from an LTI realization of the joint process ( , ), instead of computing a realization of using oblique projections.

In this paper we present an explicit construction of a minimal stochastic LTI space-representation of with an exogenous input from an autonomous stochastic LTI staterepresentation of the joint process ( , ). The basic idea is as follows: we will assume that there is no feedback from to and then use the result of Jozsa, Petreczky and Camlibel (2018) stating that there exists a minimal LTI realization of ( , ) matrices of which admit a An explicit construction of a realization of with input could useful for several reasons: it could be useful in proofs and it has the potential to provide an alternative to existing system identification algorithms. There are many subtleties in the consistency analysis of subspace identification algorithms with inputs [START_REF] Lindquist | Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification[END_REF]; [START_REF] Katayama | Subspace Methods for System Identification[END_REF]; [START_REF] Chiuso | On the ill-conditioning of subspace identification with inputs[END_REF], so an alternative approach involving the identification of an autonomous model of ( , ) could be advantageous in some cases.

Our motivation for developing an explicit construction of an LTI realization of with input from a LTI realization of ( , ) was that this construction turned out to be useful in providing non-asymptotic error bounds of PAC-Bayesian type [START_REF] Alquier | On the properties of variational approximations of Gibbs posteriors[END_REF] for LTI systems Eringis, Leth, Tan, Wisniewski, Esfahan and Petreczky (2021). The latter could be a first step towards extending the PAC-Bayesian framework for stochastic state-space representations, in particular, to recurrent neural networks.

More precisely, one of the byproducts of the construction of this paper is a one-to-one relationship between LTI systems which generate ( , ) and optimal linear predictors of future values of based on past values of . The latter predictor is just the deterministic part of a realization of with input in forward innovation form. That is, finding a realization of ( , ) boils down to finding the best predictor in a parametrized family of predictors. This is done by minimizing the empirical prediction error. In ? PAC-Bayesian error bounds are formulated relating the expected prediction error and the empricial prediction error for each predictor. In turn, the derivation of these error bounds require the use of LTI state-space representations of ( , ) and of minimal LTI realizations of with input in forward innovation form.

The contribution of the paper can also be viewed as as follows. We wish to construct an estimator of ( ) given past ( < ) and present ( = ) measurements of ( ). We consider a specific class of relationships, specifically when the two processes are related by a common stochastic linear time invariant(LTI) state-space system, i.e. [ ( ) ( )] is an output of an LTI system. The problem of find this estimator can also be thought of as trying to estimate non-measurable quantities of a system from measurable quantities.

Related work

As it was pointed out above, stochastic realization theory with inputs is a mature topic with several publications, see the monographs [START_REF] Lindquist | Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification[END_REF]; [START_REF] Katayama | Subspace Methods for System Identification[END_REF]; [START_REF] Caines | Linear Stochastic Systems[END_REF] and the references therein. However, we have not found in the literature an explicit procedure for constructing a stochastic LTI statespace realization in forward innovation form of with input from the joint stochastic LTI state-space realization of ( , ). The current note is intended to fill this gap.

We will need to further analyse the relationship between and by feedback-free assumption. In [START_REF] Granger | Economic processes involving feedback[END_REF], the author defines what it means for one process to cause another, a similar notion to feedback. In [START_REF] Caines | Weak and strong feedback free processes[END_REF], the authors further extend the notion and define weak and strong feedback free processes. As strong feedback free condition implies weak feedback free, we consider the relaxed case of weak feedback free throughout the paper. Using causal real rational transfer function matrices to describe processes and , and analysing these processes with feedback free assumption, yields a straightforward construction of estimator of given in frequency domain, see [START_REF] Caines | Feedback between stationary stochastic processes[END_REF] and Gevers and Anderson (1982). In this paper we study this problem in time domain, using LTI state-space representations.

Outline This paper is organised as follows. Below we start by defining the notation and terminology used in this paper, then in Section 2 we reformulate the state-space system driven by innovation of [ ( )

( )] into a state-space system, which yields a realisation of , driven by and the innovation of a purely non-deterministic part of . Afterwards in Section 2.1, given this new realisation we provide the optimal (in the sense of minimum error variance) predictor of . And finally we summarise by stating an algorithm to compute such predictors in Section 3

Notation and terminology Let denote a -algebra on the set Ω and be a probability measure on . Unless otherwise stated all probabilistic considerations will be with respect to the probability space (Ω, , ). In this paragraph let denote some euclidean space. We associate with the topology generated by the 2-norm || ⋅ || 2 , and the Borelalgebra generated by the open sets of . The closure of a set is denoted . For ⊆ ℕ and stochastic variables , 1 , 2 , … with values in ℝ we denote by ( | { } ∈ ) the conditional expectation of with respect to the -algebra ({ }) generated by the family { } ∈ . Recall that ( ) define an inner product in 2 (Ω, , ) and that ( | { } ∈ ) can be interpreted as the orthogonal projection onto the closed subspace 2 (Ω, ({ } ∈ ), ) which also can be identified with the closure of the subspace generated by { } ∈ . That is,

2 (Ω, ({ } ∈ ), ) = ∑ ∈ | ∈ ℝ (1)
with only a finite number of summands in (1) being nonzero when = ℕ. Moreover, for a closed subspace of 2 (Ω, , ) and a stochastic variable with values in and (|| || 2 2 ) < ∞, we let ( | ) denote the dim( )dimensional vector with th coordinate equal to ( | ) with denoting the th coordinate of .

There are two closed subspaces of particular importance. Following [START_REF] Lindquist | Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification[END_REF], for a discrete time stochastic process ( ) with values in and (|| ( )|| 2 2 ) < ∞, we write -( ) for the closure of the subspace in 2 (Ω, , ) generated by the coordinate functions ( ) of ( ) for all < . That is,

-( ) = ∑ -1 =-∞ ( ) | ∈ (2)
with indicating transpose and only a finite number of summands in (2) being nonzero. In a similar manner we define

+ ( ) = ∑ ∞ = ( ) | ∈ , ( 3
) ( ) = ∑ ∞ =-∞ ( ) | ∈ . ( 4
)
Let , and be closed subspaces of 2 (Ω, , ). We then define

∨ = { + | ∈ , ∈ } (5)
and say that and are orthogonal given , denoted ⟂

| , if -( | ) -( | ) = 0 (6)
for all ∈ and ∈ . We use the following notation,  = ℝ ,  = ℝ and for the disjoint union

 * = ⨆ ∞ =1  we write = ( 1 , … , ) in place of the more correct ( , ) = (( 1 , … , ), ) for an element in  * .

Formulating a Realisation

Suppose we want to construct an estimator of the output stochastic process ( ) ∶ Ω →  given a sequence of measurements as inputs obtained from the stochastic process ( ) ∶ Ω → . In order to narrow down and formally describe the estimation problem, we assume that the processes ( ) and ( ) can be represented as outputs of an LTI system in forward innovation form: Assumption 1. The processes ( ) and ( ) can be generated by a stochastic discrete-time minimal LTI system on the form

( + 1) = ( ) + ( ) (7a) ( ) ( ) = ( ) + ( ) (7b)
where

∈ ℝ × , ∈ ℝ × , ∈ ℝ ( + )× for ≥ 0, , > 0 and ∈ ℝ , ∈ ℝ , ∈ ℝ and
are stationary, square-integrable, zero-mean, and jointly Gaussian stochastic processes. The processes and are called state and noise process, respectively. Recall, that stationarity and square-integrability imply constant expectation and that the covariance matrix ( ( ), ( )) = [( ( ) -[ ( )])( ( ) -[ ( )]) ] only depends on time lag ( -). Furthermore, we require that is stable (all its eigenvalues are inside the open unit circle) and that for any , ∈ ℤ, ≥ 0, [ ( ) ( --1)] = 0, [ ( ) ( -)] = 0, i.e., the stationary Gaussian process ( ) is white noise and uncorrelated with ( -). We identify the system (7) with the tuple ( , , , , ); note that the state process is uniquely defined by the infinite sum

( ) = ∑ ∞ =1 -1 ( -).
Before we can continue we have to consider the relationship between and . For technical reasons we can not have feedback from to , as would then be determined by a dynamical relation involving the past of the process . As such we have Assumption 2

Assumption 2. There is no feedback from to , following definition 17.1.1. from [START_REF] Lindquist | Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification[END_REF], i.e.,

-( ) ⟂ + ( ) | -( )
holds, i.e., the future of is conditionally uncorrelated with the past of , given the past of .

As a passing remark, the no feedback assumption is equivalent to weak feedback free assumption [START_REF] Caines | Weak and strong feedback free processes[END_REF] or Granger non-causality [START_REF] Granger | Economic processes involving feedback[END_REF]. Thus the no feedback assumption can be stated as does not Granger cause .

Several results can now be deduced from Assumption 2. First, by (Lindquist and Picci, 2015, Proposition 2.4.2), we obtain the following relation between projections

[ ( )| ( )] = [ ( )| - +1 ( )], (8)
[ ( )| -( ) ∨ -( )] = [ ( )| -( )]. (9)
Secondly, from (Lindquist and Picci, 2015, Ch. 17) it follows that the process can then be decomposed into a deterministic part and a stochastic part , as follows

( ) = ( ) + ( ), (10)
( ) = [ ( )| ( )] = [ ( )| - +1 ( )], (11)
( ) = ( ) -( ). (12)
Note that, as a consequence of ( 11) and ( 12)

[ ( ) ( )] = 0 ∀ , ,
i.e., and are uncorrelated. Moreover, the process can be realised by a state-space system in forward innovation form

( + 1) = ( ) + ( ), (13a)
( ) = ( ) + ( ), (13b)
= ( ) -[ ( )| -( )]. (13c)
Finally, from (Lindquist and Picci, 2015, Proposition 17.1.3.) we get

-( ) ∨ - +1 ( ) = -( ) ⊕ - +1 ( ), (14)
where ⊕ denotes orthogonal sum, and

( ) = ( ) -[ ( )| -( ) ∨ - +1 ( )]. (15)
Now consider a similarity transformation of ( 7) such that ̄ = -1 , ̄ = and ̄ = -1 are upper block triangular (see also Section 3)

̄ ,1 ( + 1) ̄ ,2 ( + 1) = 1,1 1,2 0 2,2 ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ̄ ̄ ,1 ( ) ̄ ,2 ( ) + 1,1 1,2 0 2,2 ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ̄ 1 ( ) 2 ( ) (16a) ( ) ( ) = 1,1 1,2 0 2,2 ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ̄ ̄ ,1 ( ) ̄ ,2 ( ) + 1 ( ) 2 ( ) (16b)
where [ 1 ( ) 2 ( )] = ( ), and such that ( 3 , 2,2 ) is observable. Moreover, , ∈ ℝ × , , ∈ ℝ × , , ∈ ℝ × , with 1 = and 2 = . From [START_REF] Jozsa | Relationship between granger noncausality and network graph of state-space representations[END_REF] it then follows that ( 2,2 , 2,2 , 2,2 , 2 ) is a minimal Kalman representation of , and hence 2 ( ) is the innovation process of i.e.,

2 ( ) = ( ) -[ ( ) | -( )] = ( ) -[ ( ) | -( ) ∨ -( )]. (17)
Moreover, the transformed system ( 16) induce a relation between the output and input . In detail, from (16b) we also have

2 ( ) = ( ) -2,2 ̄ ,2 ( ). (18)
Hence, substituting ( 18) in ( 16) yields the following realisation of

̄ ,1 ( + 1) ̄ ,2 ( + 1) = 1,1 1,2 -1,2 2,2 0 2,2 -2,2 2,2 ̄ ,1 ( ) ̄ ,2 ( ) (19a)
+ 1,2 2,2 ( ) + 1,1 0 1 ( ) (19b)
( ) = 1,1 1,2 ̄ ,1 ( ) ̄ ,2 ( ) + 1 ( ) (19c)
Note that 1 ( ) is the innovation process of (with respect to ), i.e., (20)

Optimal predictor

The goal in this section is to derive an optimal predictor (in the sense of minimum variance). Firstly, we claim that ( ) = 1 ( ) -( ( )| 2 ( )) = 1 ( ) -0 2 ( ) (21) where 0 = ( [ ( ) 2 ( )]) ( [ 2 ( ) 2 ( )]) -1 is the minimum variance linear estimator of ( ) given 2 ( ), see (Lindquist and Picci, 2015, Proposition 2.2.3.). In order to show (21), we first show that

Figure 1 :

 1 Figure 1: process and prediction ̂

However, using definition of 2 ( ) from ( 17) we have

which equals -( ) ∨ - +1 ( ) and therefore

Again from ( 17) it follows that 2 ( ) ⟂ -( )∨ -( ), thus ( 22) holds.

The relation ( 21) now follows since

and therefore

by using (20). Now from ( 21) and ( 18) we get

which can be applied to (19) to obtain the following realization of

Finally we are in a position to derive a formula for the minimum variance predictor [ ( ) | - +1 ( )]. That is, a formula for the orthogonal projection of ( ) given past and present values of . First define ̂ ( ) = [ ̄ ( ) | - +1 ( )], then from (30b) we get

where (33) follows from (15). Now (30a) can be used to derive a dynamical expression for ̂ as follows

Clearly [ ( )| - +2 ( )] = ( ). For the projection in (35) we have

since the state vector ̄ ( ) can be expressed as an infinite sum using (30a), where ( 15) is used for ( )

Finally, from (13c) we observe that

= 0 -0 since ( ) ⟂ ( ) by ( 12). In summary we obtain the following formula for the minimum variance predictor

In order to predict ( ) based on - +1 ( ), the system (38) would yield the minimum prediction error variance

Summary

Given ( , , , ), which describe the system

and assume that there is no feedback from to .

Step 1 Compute forward innovation representation (Lindquist and Picci, 2015, Section 6.9), by first solving the discrete Lyapunov equation

then compute

Then solve a discrete algebraic Riccati equation

Finally compute the covariance of innovation process ( ) and the Kalman gain

Then we obtain a realisation in forward innovation form

Step 2 Find a similarity transformation to obtain the system in upper block triangular form ( 16).

First compute the svd of the observability matrix related to .

where is diagonal matrix with entries [ 1 , 2 , … , ] and 1 ≤ 2 ≤ ⋯ ≤ . Note that the order of the singular values has been reversed compared to standard practice. The nonsingular matrix , which performs the desired similarity transformation is given by = . Moreover, 1 (from ( 16)) is given by the number of singular values which are zero, i.e. the nullity of the observability matrix in (48), and 2 = -1 . Now compute the upper block triangular representation by

and partition

Step 3 Compute the minimum variance predictor (38).

Illustrative Example

To illustrate the findings consider the system

From system (53) we can see that is feedback free from .

To make the example interesting we will not work with the system (53) but instead apply a similarity transformation to the system such that it is not in a nice and clean form. We use = 0.5 0.9 0.5 0.1

and obtain ( + 1) = 1.08 -0.23 0.58 0.26 ( ) + 0.5 1.4 0.5 0.6 ( ),

Note that if we had started with system (55), it would not be clear if was feedback free of . In this example we know that is feedback free, therefore we proceed by computing the forward innovation form as in Step 1 we get ( + 1) = 1.08 -0.23 0.58 0.26 ( ) + 0.5 0.9 0.5 0.1 ( ),

The triangular form is obtained by applying SVD to the observability matrix 0 0 0 1.9764 -0.7 -0.707 -0.7 0.7 = 1.25 -1.25 0.62 -0.62 (57)

Note that there is one singular value equal to zero, therefore there is one unobservable state and as such 1 = 1. Taking = we obtain a system in triangular form