P Grangé

J.-F Mathiot

B Mutet

E Werner

Aspects of Fine-tuning of the Higgs Mass within Finite Field Theories

Keywords: numbers: 11.10.Gh, 11.10.-z, 12.15.Lk PCCF RI 10-05

We re-analyse the perturbative radiative corrections to the Higgs mass within the Standard Model in the light of the Taylor-Lagrange renormalization scheme. This scheme naturally leads to completely finite corrections, depending on an arbitrary scale. The formulation avoids very large individual corrections to the Higgs mass. This illustrates the fact that the so-called fine-tuning problem in the Standard Model is just an artefact of the regularization scheme. It should therefore not lead to any physical interpretation in terms of the energy scale at which new physics should show up, nor in terms of a new symmetry. We analyse the intrinsic physical scales relevant for the description of these radiative corrections.

I. INTRODUCTION

The experimental tests of the Standard Model of particle physics are entering a completely new era with the first pp collisions at LHC (CERN) in the TeV energy range and the discovery of a Higgs-like scalar boson [1]. It is commonly admitted that experimental evidences showing deviations from predictions within the SM should be interpreted as signs of new physics, thereby relegating the SM to the status of an effective field theory (EFT). We know that the renormalizable SM is at most valid up to an energy scale Λ ef f < M P , where M P is the Planck mass (≃ 10 19 GeV ), for quantum gravitational effects become relevant at that scale. However the way to treat the SM as an effective theory at a much lower energy scale of a yet unknown more fundamental theory is rather elusive and demands certain careful guidance from past experiences.

In any physical process, the theoretical consistency requirement of the Standard Model demands that any characteristic intrinsic momentum, denoted by Λ k , which is relevant for the description of this process should be less or equal to Λ ef f . Otherwise, new contributions of order (Λ k /Λ ef f ) n to the Lagrangian of the Standard Model should start to be sizeable. At tree level, this momentum can be defined for instance by any typical kinematical variables of the process, like Q 2 in Deep Inelastic Scattering (DIS). It is thus under complete control. However, beyond tree level, one has to deal with internal momenta in loop contributions, that may be large.

How large are they really? To answer this question, one has to enter into the renormalization procedure in order to define the physical amplitudes in terms of the bare ones calculated from the original Lagrangian of the Standard Model.

While physical observables should be independent of the renormalization scheme which is considered, the use of a particular scheme (together with a regularization procedure) may lead to unphysical interpretations when non-observable bare quantities are concerned. We shall address in this article the particular case of radiative corrections to the Higgs mass.

Beyond tree level, the first radiative corrections to the Higgs mass one has to consider in the Standard Model are radiative corrections from a t t loop as well as Higgs and W, Z bosons loops. Using a naïve cut-off to regularize the bare amplitudes, these corrections immediately lead to a quadratic dependency of the regulated amplitudes on the cut-off scale Λ C [2]. The (square of the) physical mass, M H , defined at the pole of the two-body Green's function, can be schematically written as

M 2 H = M 2 0 + b Λ 2 C + . . . , (1)
where M 0 is the bare mass of the Higgs particle, and b is a combination of the top quark, W, Z bosons and Higgs masses. As it is, this equation has not much physical interest. It is just a definition of the bare mass as a function of the physical mass, in the spirit of the renormalization theory.

The so-called fine-tuning problem arises if one wants to give some kind of physical interpretation to the bare mass M 0 . Since Λ C should be much larger than any characteristic energy scale relevant for the description of the theoretical physical amplitude, a large cancellation between M 2 0 and b Λ 2 C should be enforced by hand -hence the name fine-tuning -unless b is zero (the so-called Veltman condition [3]), or M H is very large, which we now know is not the case.

Our concern in the following is twofold. We shall first illustrate the above general remarks by the explicit calculation of the radiative corrections to the Higgs mass in a finite field theory based on the Taylor-Lagrange Renormalization Scheme (TLRS) [4], in leading order of perturbation theory. Then we compare the results with standard procedures using a naïve cut-off or Dimensional Regularization (DR). Since BPHZ can be recovered from TLRS, our conclusions do also apply to BPHZ [5]. In addition, in analogy with DR, TLRS also contains an explicit arbitrary scale. Secondly, we shall analyse our results in terms of the characteristic scale Λ k , and compare its value for both types, finite or infinite, of renormalization schemes. This may in turn have important consequences for the determination of the relevant momentum and/or energy scales at which new physics should show up. We stick in our study to the Standard Model. The discussion of how the gauge hierachy problem is formulated in TLRS when grand unified theories are concerned is beyond the scope of the present article.

The plan of the article is the following. The overall settings for a consistent formulation of an EFT are first discussed in Sec. II. Then we recall in Sec. III the general features of TLRS. We apply this scheme to radiative corrections to the Higgs mass in the Standard Model in Sec. IV. We discuss our results in the light of the fine-tuning problem in Sec. V, and draw our conclusions in Sec. VI.

II. INTRODUCTORY REMARKS ON THE MEANING OF EFT

We are familiar with the EFT approach from the early days of Quantum Electrodynamics (QED), where a simple example of EFT valid at very low energies, E γ ≪ m e , with m e the electron mass, is provided by the Euler-Heisenberg Lagrangian [6]:

L ef f = 1 4 F µν F µν + a m 4 e (F µν F µν ) 2 + b m 6 e F µν F νσ F σρ F ρµ + O( F 6 m 8 e ). (2)
It is important to emphasize some key features from this EFT analysis: i) the fundamental Gauge, Lorentz, Charge Conjugation and Parity invariance severely constrain the possible forms of EFT terms, ii) the success of the full non-perturbative Euler-Heisenberg expression of L ef f is tied up to the subtraction of the (infinite) free-field effective action. It was further realized that the other (logarithmically divergent) subtraction terms can be seen as an embryonic recognition of charge renormalization. iii) in this low-energy regime, all the information of the full renormalized QED dynamics is embodied in the values of the effective couplings a = α 2 36 and b = 7α 2 90 , with α = e 2 4π . The EFT describes then the low-energy physics, to a given accuracy ǫ, in terms of a finite set of parameters. The successful ab-initio determination of these effective couplings is tied up to the renormalizibility of the full theory. To explore a domain of higher energies still in EFT terms one has first to include operators of higher dimensions and then re-adjust the values of the initial effective couplings through renormalization group (RG)-equations.

The consistency of the procedure is imposed by matching conditions around the energy threshold region where physical predictions should be identical, to a given accuracy, in the full and effective theories. In this way if an energy domain shows up where the matching conditions cannot be fulfilled with the desired accuracy, the problematic is twofold: either the successive addition of operators of higher dimensions is not any more feasible due to fast increasing values of the effective couplings with energy -a verifiable situation-or, more speculative, the full initial theory is itself a renormalizable EFT of a deeper fundamental description of the physics reality involving new degrees of freedom in this energy domain.

Obviously, since the 1940's, the accumulation of experimental evidences in particle physics continuously supported the progression towards the SM formulation. At the same time the seminal works of Euler-Heisenberg heavily influenced similar successful EFT developpments in non-abelian gauge field, SM and string theories, as discussed in [7]. In this respect it is often argued that, although effective field theo-ries contain an infinite number of terms, renormalizability is not an issue since, at a given order in the energy expansion, the low-energy theory is specified by a finite number of couplings and allows then for an order-by-order renormalization. From the works of Euler-Heisenberg and followers and the above discussion this is evidently not the case: an EFT-SM approach cannot be considered without taking into account the full renormalizibility of the underlying theory in the determination of the effective couplings even in the supposed energy threshold region for the onset of new physics.

More recently the necessity for a mass-independent renormalization scheme has clearly been emphasized by A. Pich [START_REF] Pich | Probing the standard model of particle interactions Proceedings of " Summer School in Theoretical Physics[END_REF]: if one cuts without renormalizing loop integrals at a scale Λ where new physics is supposed to become important, one obtains erroneous dependences for mass corrections which are not suppressed by powers of 1 Λ . This makes it impossible to reach converging mass contributions from higher orders of perturbation theory so that the whole approach breaks down. On the contrary, a mass independent renormalization scheme yields results where mass corrections are suppressed by powers of µ Λ where µ is the RG scale parameter. In addition, for a gauge invariant theory like the SM, gauge invariance excludes renormalization schemes violating this invariance, e.g. all cut-off methods.

We shall consider in what follows two kinds of renormalization schemes, depending on whether the renormalization of the bare amplitudes is finite or infinite. The latter is the one which is widely used in standard perturbation theory à la Feynman. In this scheme, the choice of a regularization procedure is a necessary prerequisite to give a mathematical sense to apriori divergent bare amplitudes. In the literature two regularization methods are mainly used: i)The first one exhibits a very large mass scale, denoted by Λ C . This mass scale is either a naïve cut-off in (Euclidean) four-momentum space, or the mass of Pauli-Villars (PV) particles in the PV regularization scheme. This explicit mass scale should be much larger then any characteristic energy, or momentum, scale relevant in the calculation of the theoretical physical amplitude.

ii)The second one, the so-called dimensional regularization (DR) procedure, amounts to extending the space-time dimension D away from 4. The divergences of the original amplitudes show up as singularities in ε = 4 -D, with ε > 0. In this case, the bare amplitudes depend on a finite, and arbitrary, mass scale µ (the so-called "unit of mass" [START_REF] Hooft | [END_REF]).

Using these regularization procedures, any bare amplitude is thus made finite. However, the original divergences are recovered in the limiting process Λ C → ∞ or ǫ → 0. In this sense the renormalization schemes using these regularizations procedures are called "infinite". Of course, all fully renormalized amplitudes are indeed equivalently finite irrespective of the regularization method which has been used to derive them, provided the symmetries of the system are conserved.

The prototype of a finite renormalization scheme is the well-known BPHZ procedure. In this scheme, any bare amplitude is made finite by subtracting as many terms as necessary from the Taylor expansion at zero external momenta of the integrand. All Feynman integrals being convergent, no further regularization is required. In this context we shall focus on the relevance of the TLRS. This scheme originates from the well known observation that the divergences of bare amplitudes can be traced back to the violation of causality, originating from ill-defined products of distributions at the same point [START_REF] Scharf | Finite QED: the causal approach[END_REF][START_REF] Aste | Finite field theories and causality, Proceedings of the International Workshop "LC2008 Relativistic nuclear and particle physics[END_REF]. The correct mathematical treatment, known since a long time, is to consider covariant fields as Operator Valued Distribution (OPVD), these distributions being applied on test functions with well-defined properties [START_REF] Schwartz | Théorie des distributions[END_REF][START_REF] Epstein | [END_REF][14]. These considerations led to the development and applications of TLRS [15,16].

III. THE TAYLOR-LAGRANGE RENORMALIZATION SCHEME

Any quantum field φ(x) -taken here as a scalar field for simplicity -should be considered as an OPVD [START_REF] Schweber | An Introduction to Relativistic Quantum Field Theory[END_REF][START_REF] Collins | Renormalization[END_REF][START_REF] Haag | Local Quantum Physics: Fields, Particules, Algebras[END_REF][START_REF] Stora | Lagrangian field theory[END_REF]. This has been known for a long time. However, its full significance for practical calculations was not fully recognized until recently [4,[START_REF] Scharf | Finite QED: the causal approach[END_REF][START_REF] Aste | Finite field theories and causality, Proceedings of the International Workshop "LC2008 Relativistic nuclear and particle physics[END_REF][START_REF] Epstein | [END_REF][14][15].

As any distribution, quantum fields should be defined by their application on test functions, denoted by ρ, with well identified mathematical properties [START_REF] Schwartz | Théorie des distributions[END_REF]. The physical field ϕ(x) is thus defined by [4] ϕ(x) ≡ d 4 y φ(y)ρ(x -y) .

(

) 3
If we denote by f the Fourier transform of the test function, we can further write ϕ(x) in terms of creation and destruction operators, leading to

ϕ(x) = d D-1 p (2π) D-1 f (ε 2 p , p 2 ) 2ε p a † p e ip.x + a p e -ip.x , (4)
with ε 2 p = p 2 + m 2 . From this decomposition, it is apparent that test functions should be attached to each fermion and boson fields. Each propagator being the contraction of two fields should be proportional to f 2 . In order to have a dimensionless argument for f , we shall introduce an arbitrary scale Λ to "measure" all momenta. Λ can be any of the masses of the constituents. To deal with massless theories, we shall consider some a-priori arbitrary value. The final expression of any amplitude should be independent of Λ.

As recalled in Ref. [4], the test function f should have peculiar properties. It is chosen as a super regular partition of unity (see [4] for more details on PU's), i.e. a function of finite support which is 1 everywhere except at the boundaries. As a super regular test function (SRTF), it vanishes as well as all its derivatives, at this boundaries, in the UV and in the IR domains.

The boundary conditions of the test functionwhich in this study is assumed to depend on a one dimensional variable X -should embody a scale invariance inherent, in the UV domain for instance, to the limit X → ∞ since in this limit η 2 X also goes to ∞, where η 2 is an arbitrary dimensionless scale. This can be done by considering a running boundary condition for the test function, i.e. a boundary condition which depends on the variable X according to

f (X ≥ H(X)) = 0 for H(X) ≡ η 2 Xg(X) . (5)
This condition defines a maximal value, X max , with f (X max ) = 0 1 . In order to extend the test function to 1 over the whole space, we shall consider a set of functions g(X), denoted by g α (X), where by construction α is a real positive number smaller than 1. A typical example of g α (X) is given in Ref. [4], where it is shown that in the limit α → 1 -, with η 2 > 1, the running support of the PU test function then stretches over the whole integration domain, X max → ∞ and f → 1. In this limit g α (X) → 1 -. This running condition is equivalent to having an ultra-soft cut-off [15], i.e. an infinitesimal drop-off of the test function in the asymptotic limit, the rate of drop-off being governed by the arbitrary scale η 2 . A similar scale invariance is also present in the IR domain, when X → 0.

With these properties, the TLRS can be summarized as follows, first in the UV domain. Starting from a general amplitude A written for simplicity in a one 1 The square η 2 in ( 5) is only here for later convenience dimensional space as

A = ∞ 0 dX T > (X) f (X) , (6)
where T > (X) is a singular distribution, we apply the following general Lagrange formula to f (X), after separating out an intrinsic scale a from the (running) dynamical variable

X f > (aX) = - X a k k! ∞ a dt t (a -t) k ∂ k+1 X X k f > (Xt) .
(7) This Lagrange formula is valid for any order k, with k > 0, since f is chosen as a SRTF. It is therefore equal to its Taylor remainder for any k. After integration by part in (6), and using (7), we can thus express the amplitude A as

A = ∞ 0 dX T > (X) f (X) , (8)
where T > (X) is the so-called extension of the singular distribution T > (X). In the limit f → 1, it is given by [4]

T > (X) ≡ (-X) k a k k! ∂ k+1 X XT > (X) η 2 a dt t (a -t) k . ( 9
)
The value of k in (9) corresponds to the order of singularity of the original distribution T > (X) [4]. In practice, it can be chosen as the smallest integer, positive or null, which leads to a non singular extension T > (X). If in the absence of the test function T > (X) leads to a logarithmic divergence in (6), k is 0. It is 1 if the divergence is quadratic. With this choice for k, the extension of T > (X) is no longer singular due to the derivative in [START_REF] Scharf | Finite QED: the causal approach[END_REF], so that we can safely perform the limit f → 1 in [START_REF] Pich | Probing the standard model of particle interactions Proceedings of " Summer School in Theoretical Physics[END_REF], and obtain

A = ∞ 0 dX T > (X) , (10)
which is well defined but depends on the arbitrary dimensionless scale η. This scale is the only remnant of the presence of the test function.

The extension of singular distributions in the IR domain can be done similarly [4,15]. For an homogeneous distribution in one dimension, with T < [X/t] = t k+1 T < (X), the extension of the distribution in the IR domain is given by

T < (X) = (-1) k ∂ k+1 X X k+1 k! T < (X)ln (ηX) , (11)
with η = η 2 -1. The usual singular distributions in the IR domain are of the form T < (X) = 1/X k+1 . In that case T < (X) reads

T < (X) = (-1) k k! ∂ k+1 X ln (ηX) , (12)
where the derivative should be understood in the sense of distributions. Doing this, the extension T < (X) is nothing else than the pseudo-function (Pf) of 1/X k+1 [4, 12, 15]

T < (X) = Pf 1 X k+1 . ( 13
)
The extension T < (X) differs from the original distribution T < (X) only at the X = 0 singularity. For massive theories with a mass scale M , it is easy to translate this arbitrary dimensionless scale η to an arbitrary "unit of mass" µ = ηM . For massless theories, one can identify similarly an arbitrary unit of mass µ = ηΛ, as we shall see in the next section. This unit of mass, analogous to the unit of mass of DR [START_REF] Hooft | [END_REF], should be kept arbitrary in order to check that physical observables are indeed independent of µ (or η) after proper renormalization, order by order in perturbation theory.

Note that we do not need to know the explicit form of the test function in the derivation of the extended distribution T > (X). We only rely on its mathematical properties and on the running construction of the boundary conditions.

IV. APPLICATION TO THE FINE-TUNING PROBLEM

In leading order of perturbation theory, the radiative corrections to the Higgs mass in the Standard Model are shown in Fig. 1. We have left out, for simplicity, all contributions coming from ghosts and Goldstone bosons. Each diagram in this figure gives a contribution to the self-energy -iΣ(p 2 ), where p is the four-momentum of the external particle, and we have

M 2 H = M 2 0 + Σ(M 2 H ) . (14)
Using a naïve cut-off to regularize the amplitudes, these radiative corrections lead to the well known mass

M 2 H = M 2 0 + 3Λ 2 C 8π 2 v 2 M 2 H + 2M 2 W + M 2 Z -4m 2 t +. . . , (15)
where m t , M W,Z and M H are the masses of the top quark, W, Z and Higgs bosons respectively, and v is the vacuum expectation value of the Higgs potential in the Standard Model. The dots include logarithmic corrections in Λ C as well as contributions independent of Λ C in the large Λ C limit.

The calculation of the four different types of contributions shown in Fig. 1 is very easy in TLRS. Let us first illustrate the calculation of the simple Higgs loop contribution in Fig. 1.b. In Euclidean space one has

-iΣ 1b,H = - 3iM 2 H 2v 2 ∞ 0 d 4 k E (2π) 4 1 k 2 E + M 2 H f k 2 E Λ 2 , ( 16
)
where k 2 E is the square of the four-momentum k. As already mentioned in Sec. III, Λ is an arbitrary momentum scale. The test function f provides the necessary (ultra-soft) cut-off in the calculation of the integral.

After an evident change of variable, we get

Σ 1b,H = 3M 4 H 32π 2 v 2 ∞ 0 dX X X + 1 f M 2 H Λ 2 X (17) = 3M 4 H 32π 2 v 2 ∞ 0 dX 1 - 1 X + 1 f M 2 H Λ 2 X .
The first term under the integral, with no intrinsic scale, can be reduced to a pseudo-function, using [START_REF] Schwartz | Théorie des distributions[END_REF]. Indeed, with Z = 1/X, we have

∞ 0 dXf (X) = ∞ 0 dZ Z 2 f M 2 H Λ 2 1 Z (18) = ∞ 0 dZ Pf 1 Z 2 = - 1 Z ∞ = 0 .
The notation f (u)| a simply indicates that f (u) should be taken at the value u = a, the lower limit of integration being taken care of by the definition of the pseudo-function. This result is reminiscent of the property d D p(p2 ) n = 0, for any n, in DR [START_REF] Collins | Renormalization[END_REF]. The self-energy thus writes

Σ 1b,H = - 3M 4 H 32π 2 v 2 ∞ 0 dX 1 X + 1 f M 2 H Λ 2 X . ( 19
)
The constant factor M 2 H /Λ 2 in the argument of the test function has no physical meaning since it can be absorbed by a rescaling of the arbitrary dimensionless scale η. This can be easily seen by applying the Lagrange formula (7) with the intrinsic scale a = M 2 H /Λ 2 and k = 0. It can thus safely be removed 2 .

We can now apply the Lagrange formula for k = 0. Using the boundary condition on the support of the test function

Xt ≤ H(X) = η 2 Xg α (X) , (20)
we finally get, in the limit f → 1

Σ 1b = - 3M 4 H 32π 2 v 2 ∞ 0 dX∂ X X X + 1 η 2 1 dt t = - 3M 4 H 32π 2 v 2 ln η 2 . ( 21
)
It is easy to see that using a naïve cut-off on k 2 E one would have obtained, in the large Λ C limit

Σ C 1b,H = 3M 2 H 32π 2 v 2 Λ 2 C -M 2 H ln Λ 2 C M 2 H . (22)
For completeness, we recall below the result of the direct calculation of ( 16) in DR

Σ DR 1b,H = 3M 4 H 32π 2 v 2 - 2 ε + c -ln µ 2 M 2 H , (23)
where c = γ E -1 -ln4π and γ E is the Euler constant. We can proceed further to the calculation of the tt polarization correction indicated in Fig. 1.a (left diagram). This one depends explicitly on the (square of the) momentum p of the external particle. Similarly to (16), in the Euclidean space, and using Feynman parametrization, one gets

-iΣ 1a (p 2 ) = 12im 2 t v 2 1 0 dx ∞ 0 d 4 k E (2π) 4 k 2 E -x(1 -x)p 2 E -m 2 t [k 2 E + x(1 -x)p 2 E + m 2 t ] 2 f k 2 E Λ 2 , ( 24
)
We thus have,

Σ 1a (p 2 ) = - 3m 2 t 4π 2 v 2 1 0 dx M 2 (x, p 2 ) ∞ 0 dX X X -1 (X + 1) 2 f X M 2 (x, p 2 ) Λ 2 (25) with M 2 (x, p 2 ) = x(1 -x)p 2 E + m 2 t and X = k 2 E /M 2 (x, p 2
). The integral over X, which we shall call I, can be decomposed into three parts, I = I 1 + I 2 + I 3 , with

I 1 = ∞ 0 dXf X M 2 (x, p 2 ) Λ 2 , I 2 = -3 ∞ 0 dX 1 X + 1 f X M 2 (x, p 2 ) Λ 2 , ( 26
)
I 3 = 2 ∞ 0 dX 1 (X + 1) 2 f X M 2 (x, p 2 ) Λ 2 . ( 27
)
The first one is zero according to [START_REF] Collins | Renormalization[END_REF]. The second one can be calculated following the derivation of ( 19), with an intrinsic scale M 2 (x, p 2 )/Λ 2 in the Lagrange formula (7) for k = 0. This gives, using Λ = M H ,

I 2 = ln η 2 M 2 H M 2 (x, p 2 ) , (28)
while the third one is trivial and gives I 3 = 2 in the limit f → 1. The self-energy correction from the tt polarization diagram is thus

Σ 1a (p 2 ) = - 3m 2 t 4π 2 v 2 1 0 dx M 2 (x, p 2 ) -3 ln η 2 M 2 H M 2 (x, p 2 ) + 2 . ( 29
)
It is interesting to calculate directly (25) without decomposing the integral I. To do that, we should apply the Lagrange formula for k = 1. We thus get for this integral, with an intrinsic scale a =

M 2 (x, p 2 )/M 2 H , I = - ∞ 0 dX X∂ 2 X X 2 (X -1) (X + 1) 2 1 a η 2 a dt t (a -t) .
(30) The self energy calculated in this way, and denoted by Σ 1a (p 2 ), is given by

Σ 1a (p 2 ) = - 3m 2 t 4π 2 v 2 1 0 dx M 2 (x, p 2 ) 3η 2 M 2 H M 2 (x, p 2 ) -3 ln η 2 M 2 H M 2 (x, p 2 ) -3 . (31)
Comparing ( 29) and (31), we see that the calculation of the extension of a singular distribution is not unique. However, the self-energies differ either by a true constant (which depends on the arbitrary scale η, and is thus irrelevant in the calculation of the physical mass of the Higgs particle and more generally of any physical observable), or by a redefinition of η. They are thus said to be almost equivalent in the sense that they give identical physical, i.e. fully renormalized, amplitudes.

Using a naïve cut-off to calculate the self-energy (24), one would have obtained

Σ C 1a (p 2 ) = - 3m 2 t 4π 2 v 2 1 0 dx M 2 (x, p 2 ) Λ 2 C M 2 (x, p 2 ) -3 ln Λ 2 C M 2 (x, p 2 ) + 2 . ( 32
)
For completeness, we recall below the result in DR

Σ DR 1a (p 2 ) = - 3m 2 t 4π 2 v 2 1 0 dx M 2 (x, p 2 ) - 6 ε + 3c -3 ln µ 2 M 2 (x, p 2 ) + 1 . (33)
We can already see from these results that TLRS and DR lead to a similar p 2 -dependent logarithmic term, with the identification η 2 = µ 2 /M 2 H . They both depend on a completely arbitrary constant. The quadratic and logarithmic divergent terms using a cutoff procedure are transmuted in TLRS into contributions depending on the arbitrary dimensionless scale η.

The other contributions to the radiative corrections to the Higgs mass indicated in Fig. 1 can be calculated similarly. The final correction to the bare mass can thus be written schematically as

M 2 H = M 2 0 + ā η 2 + b ln[η 2 ] + cte . (34)
This should be compared with the corrections indicated in Eqs. (1,15) when using a naïve cut-off. We emphasize again that η in (34) has no reasons whatsoever to be very large. It is a completely arbitrary real dimensionless parameter bigger than one. The exact expressions of ā and b are of no physical interest since physical observables should be independent of η. They are completely finite and depend only on the masses of the top quark, W, Z and Higgs bosons and on the vacuum expectation value v of the Higgs field.

Away from the on mass shell condition p 2 = M 2 H , the constant term includes p 2 -dependent logarithmic corrections which give rise to the well-known running of the mass. These corrections are identical in all three schemes, as expected.

V. DISCUSSION

We shall discuss our results in terms of the various scales appearing in the calculation of the radiative corrections to the Higgs mass, and more generally, to any physical observable. Three of them are of physical origin, and depend on the dynamical content of the underlying theory and on the kinematical conditions of the physical process under consideration, one is of mathematical origin, and two are completely arbitrary and are linked to the renormalization procedure.

The first physical scale is of course Λ ef f which defines the domain of validity of the underlying theory. It fixes, in the Standard Model, the energy scale above which new physics should show up. In that case, the Standard Model Lagrangian, L SM , should be supplemented by effective operators, and one should work with the following effective Lagrangian

L ef f = L SM + i>0 O i (Λ ef f ) i ≡ L SM + ∆L , (35)
where O i is a set of local operators of dimension (mass) i+4 , compatible with the symmetries of the system. This has been examplified in Eq.( 2) for the case of QED. For a given physical process, these new contributions are proportional to (Λ k /Λ ef f )

i . These operators originate from integrating out from the action the degrees of freedom of mass greater than Λ ef f .

The second one corresponds to the kinematical scale defined by the physical process under consideration. It can be for instance Q 2 in DIS. We shall call it Λ Q for simplicity. From a phenomenological point of view, in a bottom-up approach, Λ ef f should correspond to the value of Λ Q for which theoretical predictions within the Standard Model are not corroborated by experimental results. The last physical scale we have already mentioned is the characteristic momentum relevant for the calculation of a given amplitude, called Λ k . As we shall see below, it is intimately linked to the regularization procedure which are used. Generally speaking, we should expect Λ

Q < Λ k < Λ ef f .
The mathematical scale is simply the cut-off Λ C used in the calculation of any integral over momenta in internal loops. As mentioned above, it can not have any physical interpretation. It should be chosen large enough, and one should check that any physical observable is independent -within a given accuracy ǫ -of the exact value of Λ C . In the literature, this mathematical scale is often taken equal to Λ ef f . We prefer here to separate clearly both scales since one has a physical interpretation while the other has not. As we shall explain below, this distinction is of particular interest for finite renormalization schemes like TLRS.

The last two scales are related to the renormalization procedure. The first one is the arbitrary scale η introduced in (5) in TLRS. It is the analog of the arbitrary mass scale µ of DR, with µ ≃ ηΛ. The second one is the mass scale, called R, which is chosen to fix the bare parameters of the original Lagrangian in terms of physical measurable quantities 3 . This is the so-called renormalization point. It appears of course in both finite or infinite renormalization schemes. These two scales are closely related to the RG analysis, in the sense that all physical observables should be independent of both η (or µ) and R.

In order to determine Λ k from a quantitative point of view, we shall proceed in the following way. Writing the self-energy as

Σ(p 2 ) = Λ 2 C 0 dk 2 E σ(k 2 E , p 2 ) , (36)
we shall define the characteristic momentum Λ k by requiring that the reduced self-energy defined by

Σ(p 2 ) = Λ 2 k 0 dk 2 E σ(k 2 E , p 2 ) ( 37
)
differs from Σ(p 2 ) by ǫ in relative value, i.e. with the constraint

Σ(p 2 ) Σ(p 2 ) = 1 -ǫ , (38)
provided we have | Σ(p 2 )| < |Σ(p 2 )|. In the Standard Model, ǫ can be taken of the order of 1%. We show in Fig. 2 the characteristic scale Λ k calculated for two typical expressions of the self-energy of the Higgs particle, as a function of Λ C . The first expression is the bare one given by Σ(M 2 H ) in (14), while the second one is the fully (on-shell) renormalized amplitude, i.e. with both mass and wave function renormalization, defined by [START_REF] Cheng | Gauge theory of elementary particle physics[END_REF]

Σ R (p 2 ) = Σ(p 2 ) -Σ(M 2 H ) -(p 2 -M 2 H ) dΣ(p 2 ) dp 2 p 2 =M 2 H (39) and calculated at two different values of p 2 , p 2 = -10 M 2
H and p 2 = -100 M 2 H . The calculation is done using a typical contribution to the self energy, namely (32) for the cut-off regularization scheme, while (30) is taken for the calculation in TLRS, with an upper limit for the X integral given by Λ 2 C /M 2 (x, p 2 ). Note that the derivation summarized in Sec. III to calculate the extension of the distribution T (X) in the UV domain is valid for any test function with finite support. Since T (X) in ( 8) is not singular anymore, all corrections from the finite support of f in the upper limits of the integrals over X or t give a correction of order 1/X max = M 2 H /Λ 2 C . The results indicated in Fig. 2 exhibit two very different behaviors. If one considers first the calculation of the bare amplitude, the use of a naïve cut-off regularization scheme does not allow to identify any characteristic momentum Λ k . Since Λ k is always very close to Λ C , all momentum scales up to Λ C are involved in the calculation of the bare self-energy. This is indeed a trivial consequence of the fact that the renormalization of the bare amplitude is infinite in that case. This would also be the case in any infinite renormalization scheme using DR (in the limit of dimension 4). However, using TLRS, we can clearly identify a characteristic momentum Λ k , since it reaches a constant value for Λ C large enough. Note also that in this renormalization scheme, we can choose a value of Λ C which is rather arbitrary, as soon as it is much larger than any mass or external momentum of the constituents. It can even be infinite, since it does not have any physical meaning, the only requirement being that physical amplitudes should be independent, within an accuracy ǫ, of the precise value of Λ C . This behavior is typical of finite renormalization schemes. ) defined from the condition (38), in two different schemes: with a naïve cutoff (solid line) and using TLRS (dashed line). The calculation is done for MH = 125 GeV, with η 2 = 100. We also show on this figure Λ k calculated with the fully renormalized self-energy (39) for p 2 = -10 M 2 H (dotted line) and p 2 = -100 M 2 H (dash-dotted line).

If we now consider the characteristic momentum scale relevant for the description of the fully renormalized amplitude Σ R , we can also identify a finite value for Λ k since it saturates at sufficiently large values of Λ C compared to the typical masses and external momenta of the system. This behavior is extremely similar to the result obtained in the above analysis of the bare amplitude Σ using TLRS. This is indeed not surprising since the fully renormalized amplitude is also completely finite. It depends only slightly on the external kinematical condition Λ Q (given here by -p 2 ). In any case, the characteristic momentum scale is of the order of 10 times Λ Q , and, what is more important, it is independent of Λ C . One can check that Σ R is of course identical in all renormalization schemes.

VI. CONCLUSIONS

We have analyzed in this article the fine-tuning problem in the Standard Model of particle physics in the light of the recently proposed Taylor-Lagrange renormalization scheme. Since this scheme leads naturally to completely finite bare amplitudes -in contrast to a naïve cut-off regularization scheme which leads to quadratic divergences -the so-called fine-tuning problem can not have any physical reality in this scheme.

In order to understand in more quantitative details the differences between the various schemes, we have analysed the bare amplitudes, as well as the fully renormalized ones, in terms of the characteristic momentum scale relevant for the description of radiative corrections to the Higgs mass. In the case of the bare amplitudes, we find that this characteristic momentum scale is finite and independent of the cut-off Λ C , provided it is large enough, when using TLRS, while it is as large as the cut-off scale in a cut-off regularization scheme. This forcludes any physical analysis in terms of a characteristic momentum scale when one uses a naïve cut-off -or using DR near four-dimensional space-time -on the bare amplitudes. The reason resides in an infinitely large bare amplitude for both regularization schemes.

On the contrary, we can clearly identify a characteristic relevant momentum scale when using TLRS. This scale is finite, while the value of Λ C can be very large, independently of the precise value of Λ ef f . In that case Λ k is completely determined by the dynamics of the underlying theory, as it should, and not by the mathematical properties of an ill-defined integral. It should only satisfy the consistency condition Λ k < Λ ef f . This condition can be interpreted in two different ways. If Λ ef f is known (top-down approach), one should verify this condition in order to check the consistency of the effective theory. If Λ ef f is not known (bottom-up approach), one should use this condition to induce a lower limit on Λ ef f , given by Λ k . This is the case of the Standard Model.

As expected, the equivalence between the use of different schemes preserving gauge symmetry is restored if one analyses the fully renormalized amplitudes. In that case, one finds that the characteristic momentum scale is equivalent for all such schemes, and it is of the order of the masses, or external momenta, of the constituents of the system.

A remarkable feature of TLRS is that the identification of this characteristic momentum scale in the calculation of any amplitude can be done already at the level of the bare amplitude, in four physical spacetime dimensions. This is at variance with both the usual DR or cut-off regularization procedures.

Our analysis of radiative corrections to the Higgs mass in the Standard Model has shown that the characteristic momentum scale is of the order of the typi-cal mass scale given, on the mass shell, by the physical Higgs mass. Moreover, once the various relevant scales have been clearly identified, one can not give, on the sole consideration of radiative corrections to the Higgs mass, any information on the energy/momentum scale at which new physics should show up since this characteristic momentum is independent of Λ ef f . It also does not rely on any new symmetry.

FIG. 1 .

 1 FIG. 1. Radiative corrections to the Higgs mass the Standard Model. For simplicity, we have not shown contributions from ghosts or Goldstone bosons.

  correction

This could also be done more directly by choosing a particular value for Λ.

The scale R may in fact correspond to a set of many different scales, if one chooses for instance to fix the coupling constants at different momentum scales for all external particles[START_REF] Cheng | Gauge theory of elementary particle physics[END_REF].

ACKNOWLEDGEMENTS

We acknowledge financial support from CNRS/IN2P3-Department during the course of this work. E. Werner is grateful to Alain Falvard and Fabrice Feinstein for their kind hospitality at the LUPM.


Tushar Jain
email: tushar.jain@aalto.fi

Joseph Julien

Keywords: fault-tolerant control, finite-time stability, frequency spectrum, behavioral theory

come

A note on frequency spectrum analysis and finite-time stability of the time trajectory-based active fault-tolerant control Tushar Jain * Joseph J. Yamé * *

INTRODUCTION

A fault, in general, is defined as un-permitted dynamics that changes the dynamics of a closed-loop system in such a way it no longer satisfies the desired specifications. Thus, the aim of fault tolerant control (FTC) is to counteract those altered dynamics by applying a suitable control law such that the system encore achieves the prescribed specifications. Predominately, a traditional process to reestablish the desired specifications undergoes the following two cascade stages: Fault Detection and Diagnoses (FDD), and controller reconfiguration (CR). The purpose of FDD is to use available signals to detect, identify, and isolate possibly the sensor faults, actuator faults, and any other system faults. Subsequently, the CR module reckons to-berequired actions so the system can still continue to operate safely even under faulty conditions [START_REF] Blanke | Diagnosis and Fault tolerant control[END_REF].

Focusing on some of the disadvantages, described in [START_REF] Jain | Synergy of canonical control and unfalsified control concept to achieve fault tolerance[END_REF], of the above classical approach to address an FTC problem, the authors have proposed a novel technique in Jain et al. [2012a]. This approach is developed by taking the time-trajectory viewpoint of behavioral system theory [START_REF] Polderman | Introduction to Mathematical Systems Theory: A Behavioral Approach[END_REF]. We term this FTC approach as time trajectory-based active fault-tolerant control (TAFTC). In this approach, no a priori information about the plant is used in real-time. In addition, neither the model of the plant is determined. Hence, no use of FDD module is seen in this approach. Instead, TAFTC strategy uses only the real-time measurements generated by the system. Finally, based on the desired specifications, the controller is reconfigured online such that an occurred fault can be accommodated.

While putting forward this novel approach, the stability issues of the overall scheme were not addressed. A preliminary work on the stability issue was addressed in Jain et al. [2012c]. In this paper, we use the concept of finite-time stability (FTS) that dates back to the sixties [START_REF] Weiss | Finite time stability under perturbing forces and on product spaces[END_REF]. Since then various version of FTS were introduced in the literature [START_REF] Dorato | An overview of finite-time stability[END_REF], e.g. in the context of state-space equations, in the context of transfer functions, for time-varying systems, etc. Basically, the FTS concept differs from classical stability in two important ways. First, it deals with systems whose operation is limited to a fixed finite interval of time. Second, FTS requires prescribed bounds on system variables. Most of the results related to stability and performance in the AFTC research community are defined over an infinite time interval. This could only be feasible if a model of the plant is available in real-time. Unlikely, the TAFTC strategy deals with the behavior of the system over a fixed finite time interval. Therefore, FTS offers practical advantages. In addition, the concept of FTS has never been presented or dealt within the literature of fault-tolerant systems.

The second contribution of this paper is to present the spectral analysis of a TAFTC system. The key concept within the TFTC strategy contains only the timedomain specifications. At the outset, it is difficult to realize that which desired specifications in frequency-domain are achieved by the closed-loop system. Consequently, in addition to proving the FTS, another contribution is to investigate the frequency-domain specifications achieved by this novel TAFTC scheme. This helps one to gain more insights and to infer explicit advantages offered by this scheme.

TIME TRAJECTORY-BASED ACTIVE FAULT-TOLERANT CONTROL

We view a dynamical system as an exclusion law that indicates which trajectories are admissible for the system. A trajectory is a vector-valued function s : T → S, t → s(t) that take its values in the signal space S where T ⊆ R, S ⊆ R s with s denoting the dimension of s(t). The behavior of such systems can be expressed as the set of solutions of a system of linear, constant-coefficient differential equations.

The set of all linear differential systems with s variables will be denoted by L s . The system is defined by a linear differential equation

R 0 s + R 1 d dt s + R 2 d 2 dt 2 s + . . . + R n d n dt n s = 0, (1)
where R i , i = 0, 1, 2, . . . , n are real constant matrices belonging to R •×s with finite number of rows and s columns. Equation ( 1) can be compactly written as

R d dt s = 0, R(ξ) ∈ R •×s [ξ], (2)
with

R(ξ) = R 0 + R 1 ξ + R 2 ξ 2 + . . . + R n ξ n where R •×s [ξ]
denotes the set of • × s polynomial matrices with real coefficients and indeterminate ξ. Then the behavior B is given by the set B = {s ∈ (R s ) R |s satisfies (2)}.

(3) The representation used in (2) is called the kernel representation of B, and we often write it as B = ker(R( d dt )). From the above, clearly a dynamical system is now represented by a set of operating signals given in (3). The shift from representing a dynamical system as an input/output processor standpoint to an equivalent set of solutions is the key idea within the novel time trajectory-based FTC approach.

Feedback Interconnection of Dynamical Systems

The concept of interconnection plays the central role in modeling and control of system in the behavioral framework. By an interconnected system, we mean a system that consists of interacting subsystems. Here, we deal within a special type of interconnection, termed as the feedback interconnection. Let P ∈ L s denotes the behavior of the plant and C ∈ L s denotes the behavior of the controller, where s = r T y T u T T , whose values lie in the signal space S having the dimension s = r + y + u. In the sequel, we denote this column vector by s = col(r, y, u). See Fig. 1 for a pictorial description of P and C. Taking the behavioral point of view, we can now define the trajectorybased behaviors of the plant as

P = s = col(r, y, u) ∈ S T R d dt s = 0 , (4)
where

R(ξ) = [0 r D p (ξ) -N p (ξ)] with D p (ξ) ∈ R •×y [ξ], N p (ξ) ∈ R •×u [ξ]
being co-prime polynomials, and 0 r representing the zero matrix of dimension r. From the classical input/output point of view, y is considered as the output of the plant and u as the input. With this partition of inputs and outputs, evidently D p (ξ) -1 N p (ξ) = G(ξ) defines a proper rational matrix with D p (ξ) = 0 [Willems 1991, Section VIII]. In a similar way, the behavior of the controller is given by

C = s = col(r, y, u) ∈ S T C d dt s = 0 , (5)
where

C(ξ) = [N c (ξ) -N c (ξ) -D c (ξ)] with D c (ξ) ∈ R •×u [ξ], N c (ξ) ∈ R •×y [ξ]
being co-prime polynomial, and

D c (ξ) -1 N c (ξ) = H(ξ)
representing a proper rational matrix with D c (ξ) = 0. In this controller configuration, u is the output of the controller, and (r, y) are the inputs. The interconnection of P and C through the shared variable s results in a system in which this shared variable satisfies the dynamics of both P and C. The behavior of this interconnection system is termed as the controlled behavior or the implemented behavior K, and this is given by the set K = P ∩ C, which is equivalent to

K = {s = col(r, y, u) |s ∈ P and s ∈ C } , (6)
where the symbol '∩' denotes the interconnection operation. Note that the interconnection symbol '∩' is akin to the interconnection of sets. The following definition formalizes the above concept of implementability. Definition 1. (Implementability) Let P ∈ L s be a linear differential system, C ∈ L s be a controller, and K ∈ L s . Whenever K is obtained by interconnecting P and C, we say that "C implements K". In addition, for a given K ∈ L s whenever there exists C ∈ L s such that C implements K, K is said to be implementable by the interconnection.

The above definition implies that K is the restricted behavior satisfying the dynamics of both P and C. Accordingly, a given K ∈ L s is implementable by an interconnection with respect to P if and only if K ⊆ P.

Effect of Faults

The real-time notion of controlling a faulty system is that the operating plant must achieve the control objectives at anytime, i.e. regardless of any occurrence of a fault. In this respect, we can single out a subset of plants' behavior as desirable. We call it the desired behavior, denoted by D ∈ L s . The underlying principle is that the set of solutions belonging to the desired behavior satisfy the control objectives. We define the desired behavior as

D = s = col(r, y, u) ∈ S T |J(s) ≤ λ , (7)
where J : (R s ) R → R, s → J(s) defines the control performance functional with λ ∈ R denoting the threshold limit below which the performance is considered satisfactory.

Faults affect the dynamics of the system in a way that the control specifications are not satisfied. However, in some cases, the operating controller in the feedback control loop is extremely robust making a fault tolerable within an FTC system. Hence, no change in the control law would Definition 2. (Minor Faults). A fault is said to be a minor fault whenever there is no need of reconfiguring the controller in the closed-loop. Definition 3. (Major Faults). A fault is said to be a major fault whenever K D.

y(t) r(t) u(t) Controller Plant y(t) (a) In classical sense r(t) y(t) u(t) P C s K (b) In behavioral sense
The real-time FTC problem we are dealing with is posed in the following way. Given the desired behavior D, the problem is to find an appropriate controller C, without using any a priori knowledge about the system in "realtime", which have the suitable control actions such that the controlled behavior K satisfy the desired behavior D at anytime.

Design and Implementation of TAFTC

In the traditional structure of AFTC systems, it is a fact that a precise knowledge of running plant is required during the fault diagnosis operation. On the other side, the novelty of utilizing the behavioral approach lies in its time-trajectories outlook of approaching an FTC problem, where no such knowledge is required. Nevertheless, the first stage for developing a fault-tolerant system requires Failure Mode and Effective Analysis (FMEA) [START_REF] Blanke | Diagnosis and Fault tolerant control[END_REF]. FMEA's objective is to forecast systematically how fault effects on elements relate to faults at inputs, or outputs within the elements, and what reactions should be imposed on the system when a certain fault appears. Therefore, a mandatory prerequisite for achieving faulttolerance is to have an effective FMEA of the system. We termed this phase as the Analysis & Development (AD) phase, which aims to provide a complete coverage of possible occurring faults in the closed-loop as well as the corresponding remedial measures. From the AD phase, it is assumed that a finite set of controllers

C = {C 1 , C 2 , . . . , C N } (8
) is constructed, which makes the desired behavior D implementable. An approach to perform this analysis procedure is discussed in [START_REF] Wu | Coverage in fault-tolerant control[END_REF] and the references therein.

The Behaviors

All modeling assumptions about the operating plant are embedded within the R(ξ) matrix given in (4). At a run time, this matrix will not be determined. We form a measurement set M, which is nonempty subset of S T . From the real-time viewpoint, this is formalized in the following definition. Definition 4. (Experimental plant's behavior). Given a vector space of time-dependent signals S T , a dynamical sys-

P C 1 s C 2 C N C Supervisor faults σ(t)
[O τ (x)](t) = x(t), t n -τ ≤ t < t n ; 0, otherwise.
where t n = nτ, ∀n = 1, 2, . . .. The role of introducing the time-truncation operator is to produce time-dependent subsets M τ of M for the interval of length τ ∈ R. From the above definition, for any controller C together with the behavior of the plant, we have

O τ (O -1 τ (M τ ) ∩ C) ⊆ O τ (K), ( 10
) where O -1 τ (M τ ) denotes the pre-image of M τ .
It is interesting to note from (10) that the controlled behavior, by construction, is formulated independently of the matrix R(ξ).

Set-up of the TAFTC Architecture The structure of the real-time fault-tolerant control in the behavioral context is provided in Fig. 2. Indeed, this structure is similar to what is usually used in the traditional projection-based approach. However, the novelty lies in the construction of the supervisor. Note that the plant in the figure is shaded as we do not have any a priori knowledge of it in real-time. Conversely, in the traditional approach, the supervisor is constructed by assuming the plant's knowledge which performs the "when-which" task, i.e. when to change the control law, and which controller should be switched in the closed-loop. To perform this task without the aforesaid assumptions, we construct a supervisor as illustrated in Fig. 3, whose job is to select and switch in run time the correct controller within the closed-loop in one-shot, i.e. in a single switch. This supervisor is constructed using a bank of filters, a Performance Index Generator (PIG) block and a controller selector block.

Bank of Filters

Since no knowledge of the plant's model is available, we only obtain a measurement set M τ during the interval of length τ , which is composed of the trajectories u(t) and y(t) produced by the plant. If a controller C were in the loop when the plant produced the trajectories col(y, u), then the restrictions imposed by the controller behavior (5) would be

D c (ξ)u(t) = N c (ξ)r(t) -N c (ξ)y(t), t n -τ ≤ t < n for some r ∈ (R r ) R or equivalently, N c (ξ)r(t) = D c (ξ)u(t) + N c t n -τ ≤ t < t n . (11)
Assume that all controllers in the controller's bank (8) are stable causally left invertible, then based on the observed set M τ the corresponding reference trajectory r(t) can be evaluated as ri (t) = (N ci (ξ)) -1 (D ci (ξ)u(t)+N ci (ξ)y(t)), t n -τ ≤ t < t n , (12) where i ∈ {1, 2, . . . , N }. For the controller C in the actual loop, the trajectory r(t) asymptotically converges to r(t), since subtracting (11) from (12) yields N (ξ)(r(t) -r(t)) = 0, where N (ξ) is the stable polynomial. Equation (11), in fact, yields the controlled behavior K as defined in (6), since here (y, u) ∈ P and (r, y, u) ∈ C. Consequently, for a measurement set M τ ⊆ O τ (P), if there exists a trajectory ri (t) corresponding to the i th controller C i , i = 1, 2, . . . , N , then it would yield the corresponding controlled behavior 12) defines a filter which reconstructs the virtual reference signal r(t) from the measurement set M τ [START_REF] Safonov | The unfalsified control concept and learning[END_REF]. From this, we have now determined the controlled behavior of all controllers with respect to P at run time, however, no knowledge of the plant's model is used here.

O τ (K i ) ⊇ O τ (O -1 τ (M τ ) ∩ C i ), ∀i ∈ {1, 2, . . . , N }. Equation (

PIG block

: The measurements generated by the plant together with the virtual reference, i.e. ŝ = col(r, y, u) ∈ S T , are then fed to the PIG block. This block yields N performance indices {J(ŝ i ), i = 1, 2, . . . , N }, (13) for the corresponding N controllers, which are evaluated by considering the signal ŝ during the interval of length τ .

Controller Selection

: The controller selector block subsequently produces a piecewise constant signal (the switching signal) σ(t) based on {J(ŝ i )} N i=1 whose job is to select the controller that implements D from the bank of controllers. The switching signal is a map from the time axis T to the controllers index set {1, 2, ..., N }, i.e. σ : T → {1, 2, ..., N }. The control performance is evaluated during the interval of length τ , and if it requires switching of the controller, the switch will occur after time τ exclusively. Therefore, it imposes a lower bound on the length of intervals between successive switches. This minimum length of time in which a controller is active in the loop is known as the dwell time [START_REF] Morse | Lectures notes on logically switched dynamical systems[END_REF]. The control selection logic is then realized through σ(t) = σ(t n ) for t n ≤ t < t n+1 with the updating rule

σ(t n+1 ) = σ(t n ), if K ⊆ D; argmin{J(ŝ i )} i =σ(tn) , if K D. ( 14
)
The controller selector block contains the control selection algorithm given in ( 14).

FINITE-TIME STABILITY OF TAFTC

In the behavioral system theory, the stability is not considered as a property of a dynamical system, but of a trajectory generated by the system. However, for linear systems, the stability can be viewed as a property of the itself. The following definition formalizes the idea of stability within the class of linear systems. Definition 5. A linear dynamical system is said to be stable whenever all elements of its behavior B, i.e. s = col(r, y, u) ∈ B are bounded on the half-line [0, ∞).

Otherwise, it is said to be unstable.

One can see that the above definition can be put in line to the classical concept of input/output stability, especially, bounded-input bounded-output stability. Indeed, this concept of stability considers either the trajectories characterized over an infinite time interval or the mathematical model of the system. Since, we do not have any a priori mathematical model of the system in real-time and we do not have trajectories up to time t → ∞, we use the concept of finite-time stability to deal with AFTC systems. Definition 6. Given positive scalars T 1 , T 2 ∈ R, (T 2 > T 1 ), a linear dynamical system is said to be finite-time stable whenever all elements of its behavior B, i.e. s = col(r, y, u) ∈ B are bounded by a prespecified limit over the interval t ∈ [T 1 , T 2 [.

One can easily infer that the scalars defined above are in line with time limits specified in Definition 4. Note the difference between Definitions 5 and 6. First, the latter involves trajectories characterized over a finite time interval. Secondly, quantitative bounds are prespecified on trajectories unlike to the former definition. We want to guarantee that the novel TAFTC system is a finite-time stable system. To prove this, the desired behavior D that captures the dynamics of the closed-loop, should be finitetime stable. Otherwise, no controller can guarantee FTS of the closed-loop. First, we shall show when a closed-loop can be certified as a finite-time stable. Proposition 1. Given a dynamical system Σ C = (T, S, C), the desired behavior D, and a measurement set M τ ⊆ O τ (P), the interconnection between P and C yields a 'stand-alone' finite-time stable system if

O τ (O -1 τ (M τ ) ∩ C) ⊆ O τ (D). ( 15
)
Proof Suppose the performance functional in the desired behavior D is chosen as

J(s = col(r, y, u)) = u + y r + α 0 ,
where α 0 is a non-zero constant and • denotes the Euclidean norm. Within a finite-time interval of length τ , the stable desired behavior would be defined by

O τ (D) = s = col(r, y, u)|J(s τ ) = u τ + y τ r τ + α 0 ≤ λ .
(16) Whenever the interconnection between the controller C and the unknown plant P is made, then (r, y, u) τ ∈ O τ (C) and (y, u) τ ∈ O τ (P) ⊇ M τ . Note that no knowledge about the plant is used; instead we use the experimental plant's behavior. If the above interconnected system is included in the desired behavior then trajectories satisfy the inequality (16), which is equivalent

u τ ≤ λ 1 r τ + δ 1 , y τ ≤ λ 2 r τ + δ 2 .
with Σ 2 i=1 i = λ and δ i = λ i α 0 , i = 1, 2. Hence, according to the Definition 6, the closed-loop is a 'stand-alone' finitetime stable system. Indeed, at a runtime the FTS of the closed-loop system cannot be guaranteed in the event of fault occurrence. In the sequel, we shall show when an operating closed-loop system can be certified as a finite-time stable system at a run-time. As shown earlier, there exists a controller in the bank that can achieve the desired behavior, and hence the FTS. A trivial solution is that one can test every controller sequentially in the closed-loop such that the Proposition 1 can be applied. However, at a runtime illustrating FTS in such a manner can lead the system to an unrecoverable mode, i.e. the mode from where an occurred fault cannot be accommodated. Therefore, to guarantee the FTS of the overall TAFTC scheme, it is sufficient showing that the controller that can stabilize the system, in the sense of FTS, switches in one-shot into the closed-loop. Proposition 2. Given the stable desired behavior D, and a measurement set M τ ⊆ O τ (P). For any occurrence of a fault, the time trajectory-based fault-tolerant control system is finite-time stable.

Proof Without any loss of generality, consider a bank of three controllers C = {C 1 , C 2 , C 3 } is constructed in the AD phase. The real-time operation is initiated with an unknown P interconnected with the

C 1 . Thus, O τ (O -1 τ (M τ )∩ C 1 ) = O τ (K 1 ) ⊆ O τ (D)
. Suppose, a minor fault occurs into the system. Since the occurred minor fault does not change the behavior of the system, we would still have O τ (K 1 ) ⊆ O τ (D). This implies that there is no need to change the controller C 1 . Hence, the closed-loop system is finite-time stable. Consider now that a major fault occurs into the system. Indeed, this will change the behavior of the plant, and suppose, this new behavior is then given by P f . From the Definition 3, it implies that an occurrence of major fault causes K 1 D. Therefore, whenever the last inclusion satisfies, the operating controller in the loop is invalidated implying that a fault has occurred.

Again, without any loss of generality, suppose, that controller is C 2 and not C 3 . Using the measurement set M τ generated by the plant, two virtual reference signals are evaluated by (11), which gives two sets of trajectories: ŝ2 = col(r 2 , y, u), and ŝ3 = col(r 3 , y, u). As described in Section 2.3, for these set of trajectories, we would have two corresponding virtual interconnected system, namely K2 and K3 , defined as K2 = {(ŝ 2 = col(r 2 , y, u)|(y, u) ∈ P f and (r 2 , y, u) ∈ C 2 }, K3 = {(ŝ 3 = col(r 3 , y, u)|(y, u) ∈ P f and (r 3 , y, u) ∈ C 3 }.

Interestingly, we have obtained the virtual closed-loop behaviors with every controller without putting actually the controllers into the loop. From the above, C 2 is supposed to be that right controller, not C 3 . This, indeed, satisfies O τ ( K2 ) ⊆ O τ (D). Consequently, the controller C 2 will be switched in to the closed-loop by the switching logic ( 14), instead of the controller C 3 . As shown earlier, r = r, which implies that K2 = K 2 and hence O τ (K 2 ) ⊆ O τ (D). Therefore, the time trajectory-based fault tolerant strategy is finite-time stable.

Above proposition demonstrates that the stability is guaranteed for the overall TAFTC scheme. However, a source of stability violation could be because of the switching between the controllers. The lack of dynamical consistency between the "state trajectories" of the to-be-switched control and the unknown plant is shown as the prime cause of stability violation [START_REF] Yamé | On bumps and reduction of switching transients in multicontroller systems[END_REF]. Due to this inconsistency, bump appears whenever a new controller is introduced in the closed-loop at a run-time. To address this issue, we have used the algorithm proposed in [START_REF] Jain | On implementing on-line designed controller for smooth interconnection in the behavioral framework[END_REF], which guarantees the smooth interconnection between the controller and the unknown plant. The concept of smooth interconnection is demonstrated by using the behavioral framework. The main idea is to enforce the dynamical consistency between the controller and the plant such that whenever a controller is introduced in the loop, no bumps would appear. In other words, we reset the "state" of the to-be-switched controller without using any knowledge about the plant in real-time.

FREQUENCY SPECTRUM ANALYSIS OF TAFTC

From Fig. 3, it can be seen that the performance evaluator block plays a significant role in the selection of one of the controllers from the bank. Generally, the performance functional in the PIG block captures the control objective or the desired specification. No doubt this functional is composed of the plant trajectories together with the virtual reference signal. In the existing literature, with respect to this latter signal, various types of performance functional exist, see Jain et al. [2012c] for a brief overview. In this section, we shall see which properties are depicted by this functional from the viewpoint of frequency-domain.

First, we shall discuss the integral squared error (ISE) as the performance functional chosen in the desired behavior. Indeed, ISE is a usual and common performance index evaluator in practical control engineering problems [START_REF] Jain | A real-time projection-based approach for fault accommodation in nrels 5mw wind turbine systems[END_REF]. The desired behavior is then given by

D = {s = col(r, y, u)|J(s) = t êi (ς) 2 dς ≤ λ} (17)
where êi = ri -y, i = 1, 2, . . . , N is the virtual error. Taking the spectral viewpoint, the virtual reference signal is given by ri (f ) = H -1 i (f )u(f ) + y(f ), where H -1 i (f ) is the inverse spectrum of the i-th controller. Above equations yield êi = H -1 i (f )u(f ). One should keep in mind that the plant is operating in closed-loop, i.e. the signal u(f ) depends on the plant output signal y(f ). The behavior of the running plant G is captured by the signals (u, y), and in the classical input/output approach, this is expressed as y(f ) = G(f )u(f ), where G(f ) is the frequency spectrum of the plant. On the other hand, we can also formally write u(f ) = G -1 (f )y(f ), where G -1 (f ) is the inverse spectrum of the plant. The above equations yield virtual error signal as êi = H -1 i (f )G -1 (f )y(f ), i = 1, 2, . . . , N . Since the signal y is same for all virtual closed-loops being tested, the performance formally depends on H -1 i (f )G -1 (f ), and this can be written as

J = f H -1 i (ς)G -1 (ς)y(ς) 2 dς (18)
It is interesting to note that the plant G in the last expression is the actual plant operating mode, and any occurrence of a fault results in a change of the operating mode. One would have certainly recognized that

L Hi (f ) = G(f )H i (f ) (19
) is the loop transfer function of the feedback system (H i , G), specifically with respect to H i since the plant is same for all the virtual loops. Clearly, the switching logic made the comparison on the basis of the inverse of the loop transfer L -1 Hi (f ). Consequently, the virtual feedback loop having the highest loop transfer gain (integral (18) will be minimal in that case) will have its controller switched in the actual real-time closed-loop by the supervisor. In fact, recalling that sensitivity is defined by S = (1 + L -1 ) where L is the loop transfer, one can clearly see that the supervisor seeks for the feedback loop which gives a better sensitivity function.

Since any other functional that capturing the control objectives can be chosen in the desired behavior. Now we will carry out the similar analysis with another performance functional used in [START_REF] Safonov | The unfalsified control concept and learning[END_REF], i.e. J(s) = w 1 * (y -ri ) 2 + w 2 * u) 2 -ri 2 (20) where * is the convolution operator, signals (r(t), y(t), u(t)) have their usual meanings and w 1 , w 2 are dynamical weights. In this case, the desired behavior would be then expressed as D = {s = col(r, y, u)|J(s) ≤ 0} (21) Taking the spectral viewpoint, and putting ri (f ) =

H -1 i (f )u(f ) + y(f ), u(f ) = G -1 (f )y(f ) in (20) we get w 1 (f )H -1 i (f )G -1 (f ) 2 H -1 i (f )G -1 (f ) + 1 2 + w 2 (f )G -1 (f ) 2 H -1 i (f )G -1 (f ) + 1 2 ≤ 1 w 1 (f )(GH i ) -1 (f ) 2 (GH i ) -1 (f ) + 1 2 + w 2 (f )G -1 (f ) 2 (GH i ) -1 (f ) + 1 2 ≤ 1 w 1 (f ) 2 (GH i )(f ) + 1 2 + w 2 (f )H i (f ) 2 (GH i )(f ) + 1 2 ≤ 1 w 1 (f )S i (f ) 2 + w 2 (f )H i (f )S i (f ) 2 ≤ 1
where S i (f ) = 1 1+G(f )Hi(f ) is the sensitivity function with respect to H i (f ), i = 1, 2, . . . , N . Again, one could have readily recognized that the supervisor selects a controller that satisfies an upper bound w -1 1 (f ) on S i (f ). Specifically, it is desired that |S i (f )| ≤ |w 1 (f )| -1 for f ≥ 0, while w 2 (f ) aims to limit the control effort. The weighting function w 1 (f ) gathers the desired specifications to be achieved.

From the above analysis, we conclude that without using any a priori model of the plant the supervisor selects one of the controllers that minimizes the sensitivity function. Roughly speaking, the performance depends on the controller-plant pairing (H i , G) and the well -performing controller-plant pairings are pairings (H j , G j ) where j ∈ {1, 2, ..., N } is the index of the actual plant mode, that is, those pairings in which the controller matches the actual running mode for which it has been designed in the AD phase.

CONCLUSION

In this paper, we have gained some insights of the time trajectory-based fault-tolerant control. First, we showed the finite-time stability of an active fault-tolerant system. In the TAFTC approach, no a priori information about the plant is used in real-time. The desired behavior in this approach is generally defined in terms of time trajectories. We have showed what properties are actually satisfied by this novel FTC mechanism in the frequency-domain.

Fig. 1 .

 1 Fig.1. Feedback interconnection : shaded behavior indicates that no a priori information is available in real-time be required. With the above considerations, we define two classes of faults, namely minor faults, and major faults[START_REF] Jain | Behavioral system theoretic approach to fault tolerant control[END_REF]. Definition 2. (Minor Faults). A fault is said to be a minor fault whenever there is no need of reconfiguring the controller in the closed-loop. Definition 3. (Major Faults). A fault is said to be a major fault whenever K D.

Fig. 2 .

 2 Fig. 2. System Architecture of time trajectory-based Active FTC system in the behavioral context tem Σ P = (T, S, P), and a measurement set M ⊆ S T , the behavior of the plant P is a superset of the (experimental) measurement on time intervals, i.e. M τ ⊆ O τ (P) (9) where O τ is the time truncation operator given as

Fig. 3 .

 3 Fig. 3. Internal structure of the supervisor within the time trajectory-based FTC scheme of Fig. 2