Christophe Chesneau

On two simple and sharp lower bounds for exp(x 2 )

Keywords:

In this note, we determine two simple and sharp lower bounds for exp(x 2 ), improving two well-known lower bounds. They are defined by continuous even functions on R using exponential functions for the first lower bound, and polynomial-exponential functions for the second lower bound. The results are illustrated by a numerical study and some graphics.

Introduction

In this note, we propose two sharp lower bounds for the function exp(x 2 ). This can be of interest in many area of mathematics when the function exp(x 2 ) naturally appears. One of the most famous example is the gaussian probability density function; a lower bound for exp(x 2 ) gives an upper bound for this function. The first lower bound proposes an improvement of the well-known exponential (hyperbolic) lower bound: for any x ∈ R, we have exp(x 2 ) ≥ cosh √ 2x , whereas the second lower bound proposes an improvement of the well-known polynomial lower bound: for any x ∈ R, we have exp(x 2 ) ≥ 1 + x 2 + x 4 /2. Both of them can be proved using the exponential series: exp(x 2 ) = +∞ k=0

x 2k /(k!). For other classical inequalities involving exponential functions, we refer to [START_REF] Mitrinović | Elementary Inequalities[END_REF] and [START_REF] Bullen | A Dictionary of Inequalities[END_REF]. The motivation of this note is based on the following remark: the two lower bounds presented above are sharp for small values of |x|, but not when |x| is greater to a certain threshold. We thus propose to improve them by an analytic approach; we split R into two parts, an interval A centered in 0 with a length depending on a tuning parameter a ≥ 0, and its complementary A c , we keep the existing lower bound for exp(x 2 ) on A, and we consider a particular exponentiated warped version of it on A c . The obtained lowers bounds are simple continuous even function on R using exponential functions for the first one and polynomial-exponential functions for the second one; there is no power of x 2 . We thus provide a simple and intuitive alternative to the sharp lower bound proposed by [START_REF] Brenner | Proc. R. Soc. Edinburgh[END_REF], i.e. for any a > 0 and

x ∈ R, m(x; a) = 1 + x 2 /a (a(a+x 2 )) 1/2
. The rest of the note is organized as follows. Section 2 present the first lower bound, supported by a short numerical study and some graphics. Section 3 presents the second lower bound, supported by a short numerical study and some graphics. The proofs of our results are postponed in Section 4.

First lower bound

The following lower bound of exp(x 2 ) is well-known: for any x ∈ R, we have exp(x 2 ) ≥ cosh √ 2x . In the result below, we determine a continuous even function on R denoted by f (x; a) such that, for any a ≥ 0, exp(

x 2 ) ≥ f (x; a) ≥ cosh √ 2x , x ∈ R.
Proposition 1. For any a ≥ 0, we defined the function f (x; a) by

f (x; a) = cosh √ 2x 1 {|x|<a/2} + cosh √ 2(|x| -a) exp(2a|x| -a 2 )1 {|x|≥a/2} , x ∈ R,
where 1 A denotes the indicator function over A, i.e. 1 A = 1 if x ∈ A and 0 elsewhere. Then, for any a ≥ 0 and x ∈ R, we have exp(x 2 ) ≥ f (x; a) ≥ cosh √ 2x .

Note that f (x; a) is a continuous function on R according to x; lim

|x|→a/2 f (x; a) = cosh a/ √ 2 =
f (a/2; a) = f (-a/2; a). Moreover, the lower bound of the second inequality can be attained: f (x; 0) = cosh √ 2x . Several criteria are possible for a suitable choice for a. For instance, one can consider

a * = arg min a≥0 2.5 -2.5 f (x; a) -exp(x 2 ) 2 dx.
Table 1 shows the numerical values of the integral squared error R(a, b) measuring an error between f (x; a) and exp( x Then several figures are proposed. Figure 1 illustrates the well-known inequality: exp(x 2 ) ≥ cosh √ 2x , Figure 2 shows graphically the first inequality in Proposition 1: exp(x 2 ) ≥ f (x; a) with a = 1 and Figure 3 illustrates the second inequality in Proposition 1: f (x; a) ≥ cosh √ 2x with a = 1. Let us mention that the choice of a = 1 is far to be optimal (as we can see in Table 1), it is arbitrary taken. The numerical study and the figures are done with Mathematica (v.11.1). See [START_REF] Wolfram | The Mathematica Book[END_REF].

Second lower bound

Let us now focus on a second well-known lower bound of exp(x 2 ): for any x ∈ R, we have exp(x 2 ) ≥ 1 + x 2 + x 4 /2. In the result below, we determine a continuous even function on R denoted by k(x; a) such that, for any a ≥ 0, exp(x 2 ) ≥ k(x; a) ≥ 1 + x 2 + x 4 /2, x ∈ R.

Proposition 2. For any a ≥ 0, we defined the function k(x; a) by

k(x; a) = 1 + x 2 + x 4 2 1 {|x|<a/2} + 1 + (|x| -a) 2 + (|x| -a) 4 2 exp(2a|x| -a 2 )1 {|x|≥a/2} , x ∈ R.
Then, for any a ≥ 0 and x ∈ R, we have

exp(x 2 ) ≥ k(x; a) ≥ 1 + x 2 + x 4 2 .
Note that k(x; a) is a continuous function according to x; lim |x|→a/2 k(x; a) = 1+a 2 /4+a 4 /32 = k(a/2; a) = k(-a/2; a). The lower bound of the second inequality can be attained: k(x; 0) = 1 + x 2 + x 4 /2. Among the possible criteria to chose a suitable a, one can evaluate a * = arg min a≥0 2.5

-2.5 k(x; a) -exp(x 2 ) 2 dx.
As for Table 1, Table 2 shows the numerical values of error measures: R(a, b) Figure 4 illustrates the well-known lower bound: exp(x 2 ) ≥ 1 + x 2 + x 4 /2, Figure 5 illustrates the first inequality in Proposition 2: exp(x 2 ) ≥ k(x; a) with a = 1 and Figure 6 illustrates the second inequality in Proposition 2: k(x; a) ≥ 1 + x 2 + x 4 /2 with a = 1. Again, let us mention that the choice of a = 1 is just for the illustration; it is far to be optimal.

= b -b k(x; a) -exp(x 2 ) 2 dx and R * (b) = b -b 1 + x 2 + x 4 /2 -exp(x 2 ) 2 dx,
Note that the comparison of the two proposed lower bounds is not immediate; from an analytic point of view, there exists x * > 0 such that cosh

√ 2x < 1 + x 2 + x 4 /2 for x ∈ [-x * ,
x * ] and the reverse for |x| ≥ x * . A fair numerical comparison needs to take the optimal a for each of them with a given criteria, which remains arbitrary. -1.0 -0.5 0.5 1.0

expx 2  1 + x 2 + x 4 2 -3 -2 -1 1
1.5 2.0 2.5 k(x, 1) 1 + x 2 + x 4 2 -3 -2 -1 1

Proofs

Proof of Proposition 1. First of all, let us recall that exp(x 2 ) ≥ cosh √ 2x , x ∈ R. Therefore, using

x 2 = (|x| -a) 2 + 2a|x| -a 2 , we have exp(x 2 ) = exp((|x| -a) 2 ) exp(2a|x| -a 2 ) ≥ cosh √ 2(|x| -a) exp(2a|x| -a 2 ).
Let us set g(x; a) = cosh √ 2(|x| -a) exp(2a|x| -a 2 ). We now propose to study g(x; a) according to the variable a. We have

(x 2 )1 {|x|<a/2} ≥ cosh √ 2x 1 {|x|<a/2} . So, for any x ∈ R, we have exp(x 2 ) ≥ cosh √ 2x 1 {|x|<a/2} + g(x; a)1 {|x|≥a/2} = f (x; a), f (x; a) ≥ cosh √ 2x .
Proposition 1 is proved.

Proof of Proposition 2. First of all, let us recall that exp( Moreover, if a < |x|, we have ∂ ∂a (x; a) > 0, so (x; a) is strictly increasing according to a, and if a > |x|, we have ∂ ∂a (x; a) < 0, so (x; a) is strictly decreasing according to a. Noticing that (x; 0) = (x; 2|x|) = 1 + x 2 + x 4 /2, for any |x| ≥ a/2, we have exp(x 2 ) = (x; |x|) ≥ (x; a) ≥ inf a∈[0,2|x|] (x; a) = inf [ (x; 0), (x; 2|x|)] = 1 + x 2 + x 4 2 .

x 2 ) ≥ 1 + x 2 + x 4 /2, x ∈ R. It follows from x 2 = (|x| -a) 2 + 2a|x| -a 2 that exp(x 2 ) = exp((|x| -a) 2 ) exp(2a|x| -a 2 ) ≥ 1 + (|x| -a) 2 + (|x| -a) 4 2 exp(2a|x| -a 2 ).
This proves that exp(x 2 )1 {|x|≥a/2} ≥ (x; a)1 {|x|≥a/2} ≥ 1 + x 2 + x 4 /2 1 {|x|≥a/2} . For |x| < a/2, we have the reverse of the second inequality: This completes the proof of Proposition 2.

1 + x 2 + x 4 2 = inf

  2 ) over [-b, b]: R(a, b) = b -b f (x; a) -exp(x 2 ) 2 dx and the integral squared error R * (b) measuring an error between cosh √ 2x and exp(x 2 ): R * (b) = b -b cosh √ 2x -exp(x 2 ) 2 dx, for several values of a and b. For the considered values, we see that R(a, b) ≤ R * (b).

  for several values of a and b. For all the considered values, we see that R(a, b) ≤ R * (b).

Table 1 :

 1 Numerical evaluations of R(a, b) and R * (b) for the first lower bound.

Figure 1 :

 1 Figure 1: Superimposed curves of exp(x 2 ) and cosh √ 2x for x ∈ [-b, b] with b = 1 for the first figure, and b = 3 for the second figure.

Figure 2 :

 2 Figure 2: Superimposed curves of exp(x 2 ) and f (x; 1) for x ∈ [-b, b] with b = 1 for the first figure, and b = 3 for the second figure.

Figure 3 :

 3 Figure 3: Superimposed curves of f (x; 1) and cosh √ 2x for x ∈ [-b, b] with b = 1 for the first figure, and b = 3 for the second figure.

Table 2 :

 2 Numerical evaluations of R(a, b) and R * (b) for the second lower bound.

Figure 4 :

 4 Figure 4: Superimposed curves of exp(x 2 ) and 1 + x 2 + x 4 /2 for x ∈ [-b, b] with b = 1 for the first figure, and b = 3 for the second figure.

Figure 5 :

 5 Figure 5: Superimposed curves of exp(x 2 ) and k(x; 1) for x ∈ [-b, b] with b = 1 for the first figure, and b = 3 for the second figure.

Figure 6 :

 6 Figure 6: Superimposed curves of k(x; 1) and 1 + x 2 + x 4 /2 for x ∈ [-b, b] with b = 1 for the first figure, and b = 3 for the second figure.

  a) = exp 2a|x| -a 2 h(|x| -a),where h(y) is the function defined by h(y) = 2y cosh √ 2y -√ 2 sinh √ 2y , y ∈ R. We have immediately exp 2a|x| -a 2 > 0. Let us investigate the sign of h(|x| -a). Since, for any y ∈ R, is increasing. Thanks to h(0) = 0, if a < |x|, we have h(|x| -a) > 0, implying that ∂ ∂a g(x; a) > 0, so g(x; a) is strictly increasing according to a. If a > |x|, we have h(|x| -a) < 0, implying that ∂ ∂a g(x; a) < 0, so g(x; a) is strictly decreasing according to a. Moreover, ∂ ∂a g(x; a) = 0 if and only if a = |x|. Noticing that g(x; 0) = g(x; 2|x|) = cosh √ 2x , for any |x| ≥ a/2, we haveexp(x 2 ) = g(x; |x|) ≥ g(x; a) ≥ inf a∈[0,2|x|] g(x; a) = inf [g(x; 0), g(x; 2|x|)] = cosh √ 2x .Thus exp(x 2 )1 {|x|≥a/2} ≥ g(x; a)1 {|x|≥a/2} ≥ cosh √ 2x 1 {|x|≥a/2} . For |x| < a/2, we have the reverse of the second inequality: a) ≥ g(x; a), but the well-known lower bound is still valid: exp

  Let us set (x; a) = 1 + (|x| -a) 2 + (|x| -a) 4 /2 exp(2a|x|-a 2 ). We now propose to study (x; a) according to the variable a. Algebraic manipulations and simplifications give a simple expression for ∂ ∂a (x; a):∂ ∂a (x; a) = exp 2a|x| -a 2 (|x| -a) 5 ,Since exp 2a|x| -a 2 > 0, we have ∂ ∂a (x; a) = 0 if and only if a = |x|.

  a) ≥ (x; a), but the well-known lower bound still holds: exp(x 2 )1 {|x|<a/2} ≥ 1 + x 2 + x 4 /2 1 {|x|<a/2} . Hence, for any x ∈ R, we haveexp(x 2 ) ≥ 1 + x 2 + x 4 2 1 {|x|<a/2} + (x; a)1 {|x|≥a/2} = k(x; a), k(x; a) ≥ 1 + x 2 + x 4 2 .

Email: christophe.chesneau@unicaen.fr


Sonia Boscolo

Frédéric Audo

Christophe Finot
email: christophe.finot@u-bourgogne.fr

Impact of initial pulse characteristics on the mitigation of self-phase modulation by sinusoidally time varying phase

Keywords: Self phase modulation, nonlinear fiber optics, pulse shaping

A simple and efficient approach to suppress undesirable self-phase modulation (SPM) of optical pulses propagating in fiber-optic systems is based on imposing a sinusoidal temporal phase modulation on the pulses to offset the chirp generated by SPM (Audo et al. 2017). Here, we present a detailed analysis of this method. We derive an exact formula for the reduction of the SPM-induced rms spectrum broadening of an initially Gaussian pulse enabled by the sinusoidal compensation, and we assess the effects of the initial pulse shape and duration on the effectiveness of the technique by means of numerical simulation. The differences between pre-and postpropagation compensation schemes are also discussed.

Introduction

The propagation of intense ultra-short optical pulses in a Kerr medium such as an optical fiber still remains a critical issue for the performance of many optical systems such as beam delivery, optical communication or pulse amplification systems. This is because the self-phase modulation (SPM) of the propagating pulse usually causes a broadening of the pulse spectrum that is typically accompanied by an oscillatory structure covering the entire frequency range [START_REF] Stolen | Self-phase modulation in silica optical fibers[END_REF][START_REF] Agrawal | Nonlinear Fiber Optics, Fourth Edition[END_REF][START_REF] Wang | High-resolution spectra of self-phase modulation in optical fibers[END_REF]). Several strategies have been proposed and successfully deployed to counteract the deleterious effects of SPM in fiber-optic systems [START_REF] Richardson | High power fiber lasers: current status and future perspectives[END_REF]. The most basic one consists in spectrally filtering the broadened spectrum after nonlinear propagation. However, this approach is power inefficient and may induce large distortions of the pulse temporal profile. More relevant approaches rely on spatial or temporal scaling to reduce the impact of nonlinearity. These include the use of very large mode area fibers or hollow core fibers [START_REF] Ma | Single-mode chirally-coupled-core fibers with larger than 50µm diameter cores[END_REF][START_REF] Michieletto | Hollow-core fibers for high power pulse delivery[END_REF], the well-known technique of chirped pulse amplification (CPA), in which ultra-short laser pulses are stretched in time prior to amplification and recompressed back to ultra-short pulse duration after amplification through the reversal process of stretching [START_REF] Strickland | Compression of amplified chirped optical pulses[END_REF], and new schemes for pulse division amplification [START_REF] Seghilani | Mitigating nonlinear propagation impairments of ultrashort pulses by fractional temporal self-imaging[END_REF]. A different class of approaches tends to exploit the nonlinear properties of the propagation. For example, when the pulse launched in the fiber has an hyperbolic secant shape and its power and duration are correlated, the effects of SPM and anomalous second-order groupvelocity dispersion (GVD) can cooperate in such a way that the pulse either does not change along the fiber length or follows a periodic evolution pattern [START_REF] Hasegawa | Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion[END_REF][START_REF] Agrawal | Nonlinear Fiber Optics, Fourth Edition[END_REF][START_REF] Ouzounov | Generation of Megawatt optical solitons in hollow-core photonic band-gap fibers[END_REF]. A class of optical solitons arising from the interaction of negative fourthorder dispersion and SPM in photonic crystal waveguides has also been recently demonstrated [START_REF] Blanco-Redondo | Husko: Pure-quartic solitons[END_REF]). However, the generation of solitons imposes very stringent conditions on the pulse and fiber/waveguide properties. Other approaches making the most of the Kerr nonlinearity include the exploitation of the peculiar properties of parabolic shaped pulses and self-similar evolution [START_REF] Finot | Optical parabolic pulse generation and applications[END_REF][START_REF] Pierrot | Amplification and compression of temporally shaped picosecond pulses in Ybdoped rod-type fibers[END_REF], the use of other types of pre-shaped input pulses, and the compensation of nonlinear phase shifts with third-order dispersion [START_REF] Shah | High energy femtosecond Yb cubicon fiber amplifier[END_REF][START_REF] Zhou | Compensation of nonlinear phase shifts with thirdorder dispersion in short-pulse fiber amplifiers[END_REF]. However, none of these last techniques preserves the pulse temporal duration.

A simple technique to compensate the nonlinear phase due to SPM and related spectrum broadening of nanosecond or picosecond optical pulses consists in using an electro-optic phase modulator to impart the opposite phase to the pulses. This method, which emulates the use of a material with a negative nonlinear index of refraction, has proved successful in fiber-optic and free-space optical telecommunication applications using phase-shift keying systems [START_REF] Xu | Compensation of nonlinear self-phase modulation with phase modulators[END_REF][START_REF] Ulmer | Compensation of self-phase modulation in fiber-based chirped-pulse amplification systems[END_REF], and in the generation of high-peak-power nanosecond pulses [START_REF] Su | High-peak-power, single-frequency, single-mode, linearly polarized, nanosecond all-fiber laser based on self-phase modulation compensation[END_REF][START_REF] Munroe | Reduction of SPM induced spectral broadening in a high peak power narrow linewidth IR fiber laser using phase modulation[END_REF]. Moreover, the technique can be integrated into existing CPA systems to deal with femtosecond pulses (van Howe et al. 2006;[START_REF] Zhu | Nonlinear distortion free fiber-based chirped pulse amplification with selfphase modulation up to 2π[END_REF]). We have recently demonstrated that for Gaussian shaped input pulses, the use of a simple sinusoidal drive signal for the phase modulator with appropriate amplitude and frequency is sufficient to reduce the nonlinear spectrum broadening to a large degree, and to significantly enhance the spectral quality of the pulses while their temporal duration remains unaffected (Audo et al. 2017). We have also studied the use of a sinusoidal phase compensation for improving the extinction ratio of signals impaired by a coherent background (Audo,Finot 2017).

In this paper, we present a comprehensive analysis of the SPM-mitigation method involving the use of a sinusoidal phase modulation. Most of the previous works are primarily experimental in nature (van Howe et al. 2006;[START_REF] Zhu | Nonlinear distortion free fiber-based chirped pulse amplification with selfphase modulation up to 2π[END_REF][START_REF] Xu | Compensation of nonlinear self-phase modulation with phase modulators[END_REF][START_REF] Ulmer | Compensation of self-phase modulation in fiber-based chirped-pulse amplification systems[END_REF][START_REF] Su | High-peak-power, single-frequency, single-mode, linearly polarized, nanosecond all-fiber laser based on self-phase modulation compensation[END_REF][START_REF] Munroe | Reduction of SPM induced spectral broadening in a high peak power narrow linewidth IR fiber laser using phase modulation[END_REF]) and, for instance, have not discussed the sensitivity of the technique to the initial pulse characteristics in detail. First, we recall the concept of our method and the situation being studied. Next, we present an exact closed formula for the rms spectral width of an initially Gaussian pulse after undergoing SPM and with the corrective phase applied, which confirms the substantial reduction of the SPM-induced spectrum broadening attainable with the phase compensation. Then, we describe the impact of the initial pulse shape and duration on the effectiveness of the technique through numerical simulation of the governing equation. A comparison of pre-and post-propagation compensation schemes is also provided.

Principle of operation and situation under investigation

Pulse propagation in an optical fiber is modeled by the standard nonlinear Schrödinger equation (NLSE) [START_REF] Agrawal | Nonlinear Fiber Optics, Fourth Edition[END_REF]:

2 2 2 2 1 || 2 i zt         (1)
where ψ(z,t) is the complex envelope of the pulse, z is the propagation coordinate, t is the retarded time, β2 is the GVD parameter, and γ is the coefficient of Kerr nonlinearity of the fiber. When the effects of the Kerr nonlinearity are considered over those distances and power levels such that we can neglect dispersion, the solution to Eq. ( 1) is simply 𝜓(𝑧, 𝑡) = 𝜓(0, 𝑡) exp 𝑖[𝜑 𝑁𝐿 (𝑧, 𝑡)], where 𝜓(0, 𝑡) is the field amplitude at z = 0, and 𝜑 𝑁𝐿 (𝑧, 𝑡) = 𝐵𝐼(𝑡) is the SPM-induced nonlinear phase shift. Here,

2 0 ( ) (0, ) / It ψ t P 
is the (longitudinally unchanging) intensity profile of the pulse, P0

is the peak pulse power, and B = P0 z is the maximum phase shift that occurs at the pulse center located at t = 0 and is widely known as the B-integral. In the presence of loss or gain, the physical propagated length z is replaced with an effective length defined as Leff = [1 -exp(-z)]/, where α>0 (<0) accounts for loss (gain) [START_REF] Agrawal | Nonlinear Fiber Optics, Fourth Edition[END_REF]. For an initially transform-limited pulse, the presence of a temporal variation of the instantaneous frequency (chirp) NL(t) = -dNL/dt causes broadening of the pulse spectrum over its initial width at z = 0, while the nonlinear nature of the chirp leads to spectral oscillations.

In order to illustrate our idea and provide some simple analytic design rules, we consider an initial chirp-free pulse with a Gaussian temporal intensity profile: IG(t) = exp(¬t 2 /T0 2 ), where T0 is the 1/e intensity half-duration, propagating in a highly nonlinear fiber. The initial pulse and fiber parameters used in the numerical simulation of Eq. ( 1) are: full-width at half-maximum (fwhm) pulse duration = 100 ps, P0 in the Watt range, fiber length = 500 m,  = 10/W/km and 2 = 3•10 ¬3 ps 2 /m (Audo et al. 2017). With such parameters, the nonlinearity-dominant regime of propagation is applicable. Figure 1 shows the pulse characterizations at the fiber output for a Bintegral accumulated over the fiber length of B = 2 rad. We can see that the SPM-induced frequency chirp NL has a non-monotonic temporal variation across the pulse with two extrema around the pulse center (Fig. 1(a), red curve). The time separation between these extrema and the corresponding frequency excursion can be computed by  = 2 •T0 and  = B 2 exp(¬1/2) / T0, respectively (Agrawal 2006). The pulse spectrum has been significantly broadened over its initial width by the SPM-generated frequency components and has developed a multi-peak structure as a result of interference between distinct points in the pulse having the same instantaneous frequency (Fig. 1(b)).

A common approach to transform a chirped pulse into a close-to-Fourier-transform-limited pulse consists in customizing the pulse spectral phase by applying a parabolic phase profile or more complex profiles in the frequency domain. The temporal intensity profile of the pulse is significantly modified with this approach and features a much shorter duration whereas the pulse spectrum remains unaffected. The SPM-compensation method that we study here is radically different as the phase of the pulse is modified directly in the time domain. By imparting to the pulse a phase shift that is opposite to the nonlinear phase shift, the spectrum broadening caused by SPM can be completely reversed [START_REF] Xu | Compensation of nonlinear self-phase modulation with phase modulators[END_REF][START_REF] Munroe | Reduction of SPM induced spectral broadening in a high peak power narrow linewidth IR fiber laser using phase modulation[END_REF][START_REF] Su | High-peak-power, single-frequency, single-mode, linearly polarized, nanosecond all-fiber laser based on self-phase modulation compensation[END_REF]. In principle, if the pulse shape is known, the appropriate drive signal for the phase modulator can be synthesized. However, because this approach entails high optoelectronics bandwidth requirements it is of significant practical importance to minimize the optoelectronics bandwidth that is involved.

In the context of the transmission of telecommunication signals, it has been shown that approximating the signal waveform with just one or two harmonics suffices to achieve a significant amount of SPM compensation [START_REF] Xu | Compensation of nonlinear self-phase modulation with phase modulators[END_REF][START_REF] Ulmer | Compensation of self-phase modulation in fiber-based chirped-pulse amplification systems[END_REF][START_REF] Zhu | Nonlinear distortion free fiber-based chirped pulse amplification with selfphase modulation up to 2π[END_REF]. This can also be expected from our results in Fig. 1(a): the observed time dependence of the SPM-induced frequency chirp of the Gaussian pulse indeed suggests that this chirp can be offset by driving the phase modulator with the simplest waveform that can be synthetized electronically, i.e. a sinusoid: C(t) = AC cos(C t+0), producing the chirp c = AC C sin(C t+0). An effective cancellation of the nonlinear phase imposes the phase shift 0 = Inspired by a previous numerical work relating to the optimization of the SPM-driven spectral compression process [START_REF] Boscolo | Enhanced nonlinear spectral compression in fibre by external sinusoidal phase modulation[END_REF], we choose the angular frequency and amplitude of the modulation for the Gaussian pulse as (Audo et al. 2017):

  00 ( // / 2 1 2 ) 2.221 / 0.386 / 2 / C CTG C CTG CTG TT A A B exp B                            (2)
With this parameter choice, the external phase modulation indeed compensates the nonlinear phase accumulated by the pulse to a large degree: the total chirp, c + NL, is now canceled almost completely over the central region of the pulse (Fig. 1(a), blue curve). This entails a significantly narrower spectrum that is now close to its original width (Fig. 1(b1)). Small pedestals are still present in the spectrum, but their intensity level is below 20 dB. A measure of how close a pulse is to its Fourier transform limit is provided by the spectral Strehl ratio, defined as the ratio of the maximum spectral brilliance of the actual pulse to the brilliance obtained assuming a transformlimited pulse. While after nonlinear propagation in the fiber, the maximum relative brilliance of the pulse spectrum has dropped to 0.22 the application of the external phase modulation reestablishes a Strehl ratio of 0.73. It is also interesting to represent the pulse electric field on the complex plane (I(t); (t)) (Fig. 2).

While the pulse affected by SPM traces a curve similar to an Archimedean spiral (I(t); B I(t)) (red curve), we can readily see from the graph that after phase compensation the phase is nearly constant over most of the pulse (blue solid curve), indicating a close-to-Fourier-transform-limited profile. On the contrary, higher values of the accumulated B-integral entail stronger deviations from the ideal profile (blue dashed curve).

SPM-compensation capability and analytical results

We have numerically assessed the ability of our method to counteract SPM over a wide range of values of the B-integral accumulated in the fiber. The results obtained for the initially 100-ps-wide Gaussian pulse and the corrective phase modulation specified in the previous section are summarized in Fig. 3, and highlight that the corrective approach can provide mitigation of SPM up to very high B values: the fwhm of the pulse spectrum is largely preserved and no spectral ripple develops. However, as already seen in Fig. 2, the cancellation of the nonlinear phase is not complete and the intensity level of spectral side lobes tends to grow with increasing accumulated B-integral. This is also reflected in the evolution of the root-mean-square (rms) spectral width of the pulse with the B-integral (Fig. 3(b)): the rms spectrum broadening is reduced by a factor close to 4 relative to the case when no phase correction is used but it experiences a continuous increase with B due to growing spectral side-lobes. The same trend is observed in the evolution of the Strehl ratio of the pulse: while this remains well above 0.5, it slowly deteriorates with the increase of B.

An exact formula for the rms spectral width of the pulse is derived hereafter. (5)and Eq. ( 9), respectively.

Spectral width calculation

Following the calculation described by S. C. Pinault and M. J. Potasek in [START_REF] Pinault | Frequency broadening by self-phase modulation in optical fibers[END_REF] and used by us in (Boscolo et al. 2017a)

dd dd                             (3)
where ()  is the Fourier transform of the pulse envelope () t  . Using Fourier-transform and convolution theorems in a combination with integration by parts, 2   can be expressed in terms of

() t  : 2 2 2 22 '( ) '( ) *( ) ( ) ( ) t dt t t dt t dt t dt              (4)
where ' and * denote derivation and complex conjugation, respectively.

In the case of a Gaussian pulse affected by SPM, the closed-form evaluation of the integrals above yields [START_REF] Pinault | Frequency broadening by self-phase modulation in optical fibers[END_REF] :

  2 2 0 4 1 1 0.8774 33 B B           (5)
where 0   is the initial rms spectral width of the pulse. Note that 0 0.8774

B      for 1 B .
When a sinusoidal temporal phase modulation is imposed on the pulse, we obtain:

    2 2 2 2 00 1 22 2 2 22 0 00 0 2 2 2 22 0 0 4 1 2 exp cos 8 33 1 1 cos 2 CC C CC C TT CC T B A B T A T e e                                                             (6)
This, with the choice 0   , reduces to:

  22 0 1 22 2 2 2 2 2 2 2 0 00 0 4 1 2 exp 1 8 33 C CC C C C C T T B A B T A T e                            (7) It is notable that if we set   0 /2 C CTG T    
in Eq. ( 7), we can find the optimum amplitude of the modulation A'CTG that minimizes the rms spectrum broadening generated by SPM by a derivative test:

' 2 2 0 0 C CTG C C AA A            , yielding 2 2 /16 /2 e ' 0.384 2 1e CTG B AB         (8)
This amplitude value is in excellent agreement (less than 0.5 percent change) with the value ACTG given in Eq. ( 2), which was obtained through a more natural analysis. Equation ( 8) also confirms that as long as the propagation regime in the fiber is purely nonlinear, the duration of the initial pulse does not affect the amplitude of the corrective phase. Now, when we substitute Eq. (2) into Eq. ( 7), we obtain:

  2 2 1 2 22 2 16 2 1/2 1/2 0 4 2 2 1 1 1 0.2146 3 3 2 C BB e e B ee                               (9)
The rms spectral width evolutions with the accumulated B-integral predicted by Eqs. ( 5), ( 9) are in perfect agreement with the results of numerical simulations (Fig. 3(b)), thus confirming that for a 100 ps pulse the propagation is dominated by SPM. These predictions can be used to draw some interesting conclusions: while both evolutions follow the same law (1+(k B) 2 ) 1/2 , the growth factor k is approximately four times smaller when the phase correction is applied. This brings about a close to fourfold reduction in spectrum broadening at large B:

Impact of initial pulse shape

In the previous sections, we have illustrated our proposed concept with a Gaussian input pulse and showed that an additional sinusoidal modulation of the temporal phase is remarkably effective at lessening the spectrum broadening induced by SPM. However, it is well known that SPM is highly sensitive to the initial pulse shape, and one may wonder whether our approach works equally well with other widely used pulse shapes. Our phase compensation method is in fact capable of effectively counteracting the unwanted effects of SPM on the propagation of other pulse waveforms, provided that these can be fitted by a parabolic profile in their central part (Boscolo et al. 2017a). To corroborate this statement, here we illustrate the case of a hyperbolic secant pulse like those delivered by soliton lasers. Following the same idea of reducing the chirp in the central region of the pulse and a similar calculation, one may derive the analogue of the formulae presented in Eq. ( 2) for the pulse intensity profile IS(t) = sech 2 (T/T0):

00 3/2 31 ln 31 4 3 3 1 ln 3 2.385 / 0.322 1 CTS CTS TT B B A                                  (10)
Figure 4 illustrates the impact of a sinusoidal phase modulation with the frequency and amplitude chosen based on Eq. ( 10) on a 100 ps hyperbolic secant pulse after it undergoes SPM in the fiber as obtained by numerical integration of Eq. (1). We note that in the case of a purely nonlinear propagation, an analytic expression for the SPM-broadened spectrum of an hyperbolic secant pulse can be obtained [START_REF] Phan Huy | Characterization of self-phase modulation in liquid filled hollow core photonic bandgap fibers[END_REF]. We can see in Fig. 4 that the parameter choice of Eq. ( 10) achieves fairly good cancellation of the chirp of the pulse near the pulse center. The flat phase achieved around the peak intensity is also visible in the polar graph of Fig. 5. However, the deviations from perfect phase compensation on the pulse wings are rather strong, and these result into strong side lobes in the spectrum. Accordingly, despite the considerable enhancement contributed by the phase correction, the Strehl ratio of the pulse remains below 0.5. 10) and from the optimized modulation parameters are plotted with blue and green dotted lines, respectively.

In order to improve our results, we have used an iterative procedure to find the best parameters of the modulating sinusoid: starting from the values CTSand ACTS provided by Eq. ( 10), we have scanned the values close to this initial guess and selected the combination that yielded the highest Strehl ratio. The results obtained with the optimized parameters are plotted in Fig. 4 with green lines. With this strategy, the Strehl ratio has increased up to 0.74 and the frequency extent of the spectral side lobes has lessened. As we have observed in our previous work relating to nonlinear spectral compression [START_REF] Boscolo | Enhanced nonlinear spectral compression in fibre by external sinusoidal phase modulation[END_REF], a constructive interference between different pulse parts having the same instantaneous frequency is favorable for attaining high Strehl ratios. This however occurs at the expense of a slight increase in the intensity level of the spectral side lobes. We would like to emphasize that, contrary to the parameter set of Eq. ( 10), the optimized modulation parameters do not achieve cancellation of the chirp near the pulse center. This is also readily seen in Fig. 5: the phase of the pulse is far from the Fourier transform limit.

Hence, such choice of modulation parameters would make the phase correction method unsuitable for application in the context of phase regeneration of phase-encoded telecommunication signals. Figure 6 and 7 summarize the performance of the phase compensation method using optimized parameters across a wide range of B values, and highlight an excellent mitigation of SPM. We can see in Fig. 6 that with the application of the corrective phase modulation, there is no noticeable spectrum broadening in terms of fwhm extension with increasing nonlinear phase shift accumulated in propagation while only a moderate broadening is observed in terms of rms width.

However, the presence of side lobes in the spectrum is apparent. Figure 7 shows that the optimized sinusoidal modulation triggers a threefold decrease in the rms spectrum broadening generated from SPM compared to the modulation based on Eq. ( 10), and the Strehl ratio remains above 0.5 for B values as high as 17 rad. The frequency and amplitude of the modulating sinusoid are plotted as functions of the B-integral in panels (c) and (d) of Fig. 7. We can note that the optimum frequency is significantly lower than the frequency CTS defined in Eq. ( 10) and, contrary to CTS, it evolves with the accumulated nonlinear phase shift. It is noteworthy that with the range of modulation frequencies being studied, it is possible to drive the phase modulator with the master clock that has been used in [START_REF] Ulmer | Compensation of self-phase modulation in fiber-based chirped-pulse amplification systems[END_REF] to generate RZ50-like signals. The optimum amplitude of the phase modulation is slightly higher than the value ACTS from Eq. ( 10), while it features a similar linear variation with the B-integral.

Impact of initial temporal pulse duration

So far we have described the application of our SPM-mitigation method to optical pulses propagating through an optical fiber in the nonlinearity-dominant regime. However, it is well known that for long fiber lengths and/or short pulse durations, the effects of dispersion on propagation become important. The aim of this section is to assess the impact of the temporal duration of the initial pulses on the performance of our technique. To this end, we have varied the fwhm duration of a Gaussian input pulse from a few to a hundred picoseconds, and studied the compensation of the SPM acquired by the pulse upon propagation in a highly nonlinear fiber with the same parameters as those given in Sec. 2. Figure 8 gives a characterization of the pulse after accumulating a B-integral of 2 rad in the fiber as a function of its initial temporal duration in both the frequency domainthrough the evaluation of the rms spectral broadening factor and Strehl ratio -and the time domainthrough the evaluation of the rms pulse width and excess kurtosis parameter [START_REF] Decarlo | On the meaning and use of kurtosis[END_REF]. Firstly, we observe that initial temporal durations above 30 ps do not affect either the spectral or the temporal metrics of the pulse, which pinpoints unaltered quality of the spectrum and characteristics of the temporal intensity profile for these pulse durations. In particular, the excess kurtosis is null and the duration of the output pulse equals the initial duration. This proves that the regime of propagation is dominated by Kerr nonlinearity with no influence of dispersion, and hence nonlinearity can be treated as being a lumped temporal phase modulator.

The picture is rather different for shorter initial pulse durations: the width of the pulse spectrum at the exit of the fiber increases significantly while the SPM-induced spectrum broadening diminishes slightly with decreasing initial duration. Besides, we can note a fall in effectiveness of the corrective phase modulation chosen based on Eq. ( 2) when it is applied as a post-compensation (blue solid line). For pulse durations shorter than 5 ps, the spectrum after phase compensation is even wider than the uncompensated case, highlighting the ineffectiveness of the design rules of Eq. ( 2) when the fiber dispersion comes into play. Similarly, the Strehl ratio of the pulse after phase compensation drops with decreasing initial duration. These features are ascribed to the changes experienced by the temporal intensity profile of the pulse: the combination of normal GVD and SPM makes the pulse temporally broaden and change shape toward a parabolic [START_REF] Anderson | Wave-breaking in nonlinear optical fibers[END_REF][START_REF] Finot | Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device[END_REF]) and then almost rectangular pulse form. This is evidenced by the excess kurtosis parameter approaching -1. We note that accurate analytical formulae describing the evolution of the temporal and spectral rms widths and of the kurtosis of a pulse propagating in a nonlinear dispersive fiber are available [START_REF] Burgoyne | Nonlinear pulse propagation in optical fibers using second order moments[END_REF][START_REF] Rosenberg | Evolution of optical pulses towards wave breaking in highly nonlinear fibres[END_REF]) and may form the basis for the further development of design guidelines for our SPMcompensation scheme in the presence of dispersion. It is also worth mentioning that in the propagation regime being studied the nonlinearity still prevails over dispersion. Conversely, for longer propagation distances in the fiber, the broadening of the pulse spectrum may saturate and the pulse may enter spectronic dynamics [START_REF] Zeytunyan | Nonlinear-dispersive similariton of passive fiber[END_REF][START_REF] Castelló-Lurbe | Dispersion-to-spectrum mapping in nonlinear fibers based on optical wave-breaking[END_REF][START_REF] Parriaux | Spectral broadening of picosecond pulses forming dispersive shock waves in optical fibers[END_REF].

We have optimized the parameters of the sinusoidal phase modulation applied as postcompensation for each value of the initial pulse duration using the same iterative procedure as described in the previous section. The results are plotted in Fig. 8 with green lines, and show that an almost complete compensation of the SPM at any pulse duration is achievable by use of this approach. We will return to this point below. Furthermore, the effectiveness of the approach becomes more pronounced when shorter durations are involved. 2) and applied as a post-compensation, with the phase modulation chosen based on Eq. ( 2) and applied as a pre-compensation, and with the optimized phase modulation applied as a post-compensation are plotted with red, blue, yellow and green lines, respectively. The dotted blue lines recall the results obtained with a Gaussian pulse propagating in the quasi-pure nonlinear regime and a postcompensation based on Eq. ( 2). The vertical dotted lines indicate the temporal durations used in Figs. 1 and 7 and in Figs. 9 and 11.

As for short pulse durations the interaction between nonlinearity and dispersion in the fiber is no longer negligible and so the pulse dynamics becomes more complex than the development of a simple temporal phase term, it is relevant here to compare the results that can be obtained by imparting the sinusoidal phase modulation before or after propagation in the fiber. Indeed, while for a purely nonlinear propagation the disposition of the nonlinear element and the external phase modulator does not impact the overall SPM-mitigation process, this is no longer the case when dispersion becomes important. In practice, while post-compensation has shown to be effective when it is done in the receiver of a telecommunication system [START_REF] Xu | Compensation of nonlinear self-phase modulation with phase modulators[END_REF], applications dealing with high-power pulses may favor a pre-compensation scheme to avoid damages to the optoelectronic component due to excessive optical power and mitigate the deleterious Brillouin backscattering [START_REF] Ulmer | Compensation of self-phase modulation in fiber-based chirped-pulse amplification systems[END_REF][START_REF] Su | High-peak-power, single-frequency, single-mode, linearly polarized, nanosecond all-fiber laser based on self-phase modulation compensation[END_REF]. It is noteworthy that with the phase modulation applied as a pre-compensation, the process can be reinterpreted as spectral focusing with the nonlinear phase accumulated by the pulse in propagation compensating an initial and opposite phase modulation [START_REF] Zohrabian | Compression of the spectrum of picosecond ultrashort pulses[END_REF][START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF]: the spectrum is broadened by the pre-compensation, and the SPM process then compresses it back to its original shape. These pulse dynamics are in some ways analogous to the dynamics experienced by a pulse while it undergoes SPM-based spectral compression in a fiber [START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF][START_REF] Markaryan | Spectral compression of ultrashort laser pulses[END_REF][START_REF] Planas | Spectral narrowing in the propagation of chirped pulses in single-mode fibers[END_REF]Finot,Boscolo 2016). However, in the case of a phase pre-compensation, the temporal intensity profile of the pulse does not evolve towards a parabolic shape; on the contrary, it experiences an increase of its kurtosis and becomes leptokurtic of a similar kind as a triangular profile [START_REF] Boscolo | Passive nonlinear pulse shaping in normally dispersive fiber systems[END_REF]. The results that we have obtained by applying the sinusoidal modulation chosen based on Eq. ( 2) as a pre-compensation (yellow lines in Fig. 8) are even worse than those obtained within the post-compensation scheme, thus emphasizing that the design rules of Eq. ( 2) are not appropriate for an efficient mitigation of the SPM of short pulses. Now we focus on initial pulses having a fwhm duration of 14 ps, for which a sinusoidal modulation based on Eq. ( 2) can be generated by a 40-GHz phase modulator; such devices are widely commercially available. The pulse characterization showed in Fig. 9 obtained for an accumulated B-integral of 2 rad substantiates the trends already captured by Fig. 8: for the pulse duration being considered, the temporal intensity profile of the pulse is affected by propagation in the fiber and the changes depend on whether the phase correction is applied before or after propagation. The quality of the SPM-mitigation highly depends on the compensation scheme being used. A pre-compensation with parameters chosen according to Eq. ( 2) yields poor mitigation.

Optimizing the parameters of the sinusoidal modulation may help increase the spectral peak brightness and approach performances similar to those achieved with a post-compensation based on the parameters of Eq. ( 2). However, much better results both in terms of Strehl ratio enhancement and suppression of the residual side lobes in the spectrum can be obtained using a post-compensation with optimized parameters. The remarkable SPM-mitigation afforded by this approach is also readily apparent from the polar graph of Fig. 10, which highlights a notable reduction of the maximum phase of the pulse. The results obtained with the sinusoidal modulation chosen based on Eq. (2) (solid) are compared with those obtained with the optimized sinusoidal modulation (dashed). The accumulated B-integral is B = 2 Further, we have tested the compliance of these observations with higher levels of SPM experienced by the 14 ps input pulse upon propagation in the fiber. The results are summarized in Fig. 11 and indeed confirm the trends discussed above: the evolutions of the Strehl ratio and the spectral broadening factor evidence that the use of Eq. ( 2) within a pre-compensation scheme is inefficient. The use of optimized parameters for the sinusoidal modulation applied as a precompensation achieves performances similar to those of a post-compensation based on Eq. (2).

Applying the sinusoidal modulation with optimized parameters as a post-compensation provides a nearly perfect SPM cancellation with a Strehl ratio that approaches 1 and an rms spectral width after fiber propagation that nearly equals the initial width. It is remarkable that the effectiveness of this approach does not seem to be impaired by the level of SPM accumulated by the pulse and the compensation process appears to work perfectly for B-integral values as high as 16 rad. This quite counterintuitive behavior can in fact be explained by the linearization of the temporal phase accumulated by the pulse upon propagation that is contributed by normal GVD [START_REF] Anderson | Wave-breaking in nonlinear optical fibers[END_REF][START_REF] Rosenberg | Evolution of optical pulses towards wave breaking in highly nonlinear fibres[END_REF]. We can also see in Fig. 11(c) that the frequency of the sinusoidal modulation does not change significantly with the level of accumulated SPM. The optimized frequency is threefold smaller than the frequency chosen based on Eq. ( 2), thereby being more easily attainable with current optoelectronic components. The optimized amplitude of the 20 modulation is 20 percent larger than the amplitude given by Eq. (2) (Fig. 11(d)) but yet remains fully accessible with current technology [START_REF] Godil | Picosecond time-lenses[END_REF][START_REF] Plansinis | Spectral changes induced by a phase modulator acting as a time lens[END_REF].

Conclusion

We have presented an in-depth characterization of the mitigation of the impact of SPM on optical pulses propagating through an optical fiber relying on imparting a sinusoidal phase modulation to the pulses in the time domain. We have presented simple analytic guidelines for the choice of the frequency and amplitude of the modulation, and described a calculation that gives an exact closed formula for the rms spectral width of an initially Gaussian pulse after undergoing SPM and with the corrective phase applied, which confirms the remarkable reduction of the SPM-induced spectrum broadening attainable with the compensation approach. We have then extended our analysis to other commonly used pulse shapes such as hyperbolic secant pulses, and shown that for such pulses optimization of the parameters of the modulating sinusoid through a scan of the amplitude-frequency space outperforms the parameter choice based on the analytic guidelines.

Further, we have assessed the impact of the temporal duration of the initial pulses on the effectiveness of the compensation technique, and emphasized the important role played by the fiber dispersion in the compensation process for short pulses: the interaction between SPM and GVD engenders significantly different performances depending on whether the sinusoidal modulation is applied as a pre-or post-compensation. Specifically, we have shown that remarkable SPM mitigation is attainable by use of a post-compensation based on optimized parameters.

Handling pulses as short as 10 ps with our SPM-mitigation scheme is fully realistic by means of existing optoelectronic components, while the use of cross-phase modulation in the fiber could also be a solution [START_REF] Mouradian | Spectro-temporal imaging of femtosecond events[END_REF]. In this paper, we have focused on the regime of normal dispersion that is typical of fiber amplifiers, yet the use of an anomalously dispersive fiber may significantly change the conclusions drawn [START_REF] Paré | Split compensation of dispersion and self-phase modulation in optical communication systems[END_REF].

We believe that the present study can provide a deeper insight into the association of the SPM effects with an external sinusoidal phase modulation and can be of interest for multistage fiber amplifier architectures. We also expect that some features of the spectral compression process driven by SPM discussed in our previous work (Boscolo et al. 2017b) can be readily extended to the SPM-mitigation scheme described in this paper so that the present scheme should be rather stable against amplitude fluctuations and signal-to-noise ratio degradation of the input pulses. We also believe that, similarly to a nonlinear spectral compression stage, the present technique could be combined with fiber laser architectures into new cavity designs [START_REF] Ilday | Nonlinearity management: a route to high-energy soliton fiber lasers[END_REF][START_REF] Boscolo | Amplifier similariton fiber laser with nonlinear spectral compression[END_REF]. Other prospected applications of the proposed approach include the reduction of different unwanted temporal phase defects such as those developed by pulses upon propagation in gain switched fibers or semiconductors devices.

Figure 1 :

 1 Figure 1: Impact of phase correction on a 100-ps Gaussian pulse after undergoing SPM in a fiber as obtained by numerical integration of Eq. (1): (a) temporal intensity and chirp profiles, and (b) spectral intensity profile on linear and logarithmic scales at the entrance (black) and the exit of the fiber before (red) and after (blue) phase compensation by a sinusoidal signal with frequency and amplitude chosen based on Eq. (2). The B-integral accumulated by the pulse is B = 2rad. The corrective chirp δc is plotted with a green dotted line.

Figure 2 :

 2 Figure 2: Polar graph (I(t);(t)) of the electric field of a 100 ps Gaussian pulse after undergoing SPM in a fiber before (red) and after (blue) phase compensation by a sinusoidal signal with frequency and amplitude chosen based on Eq. (2). The results obtained for the B-integral values B = 2 rad and B = 4 rad are plotted with thick solid lines and thin dashed lines, respectively.

Figure 3 :

 3 Figure 3: (a) Numerically computed evolution of the spectrum (logarithmic scale, 20 dB dynamics) of an initial 100 ps Gaussian pulse after undergoing SPM in a fiber with the accumulated B-integral, before (subplot 1) and after (subplot 2) phase compensation by a sinusoidal signal with frequency and amplitude chosen based on Eq. (2). (b)-(c) Evolutions of the rms spectral broadening factor and the Strehl ratio of the pulse at the exit of the fiber. The results obtained without and with phase correction are plotted with red and blue lines, respectively. The vertical dotted lines indicate the B-integral values used in Figs. 1 and 2. The red and blue circles represent the analytical predictions from Eq.

B

  >> 1, which explains the numerically observed trend in Fig.3(b).

Figure 4 :

 4 Figure 4: Impact of phase correction on a 100 ps hyperbolic secant pulse after undergoing SPM in a fiber as obtained by numerical integration of Eq. (1): (a) temporal intensity and chirp profiles, and (b) spectral intensity profile on linear and logarithmic scales at the entrance (black) and the exit of the fiber before (red) and after phase compensation by a sinusoidal signal with frequency and amplitude chosen based on Eq. (10) (blue) and with optimized frequency and amplitude (green). The B-integral accumulated by the pulse is B = 2 rad. The corrective chirp δc resulting from Eq. (10) and from the optimized modulation parameters are plotted with blue and green dotted lines, respectively.

Figure 5 :

 5 Figure 5: Polar graph (I(t);(t)) of the electric field of a 100 ps hyperbolic secant pulse after undergoing SPM in a fiber before (red) and after phase compensation by a sinusoidal signal with frequency and amplitude chosen based on Eq. (10) (blue) and with optimized frequency and amplitude (green). The accumulated B-integral is B = 2

Figure 6 :

 6 Figure 6: Numerically computed evolution of the spectrum (logarithmic scale, 20 dB dynamics) of an initial 100 ps hyperbolic secant pulse after undergoing SPM in a fiber with the accumulated B-integral, before (subplot a) and after phase compensation by a sinusoidal signal with frequency and amplitude chosen based on Eq. (10) (subplot b) and with optimized frequency and amplitude (subplot c). The vertical dotted line indicates the B-integral value used in Figs. 4 and 5.

Figure 7 :

 7 Figure 7: Numerically computed evolutions of (a) the rms spectral broadening factor and (b) the Strehl ratio an initial 100 ps hyperbolic secant pulse after undergoing SPM in a fiber with the accumulated B-integral. (c)-(d) Evolutions of the frequency and amplitude of the sinusoidal phase modulation. The results obtained without phase modulation, with the phase modulation chosen based on Eq. (10), and with the optimized phase modulation are plotted with red, blue and green lines, respectively. The dotted blue lines in subplots (c) and (d) represent the modulation frequency and amplitude used for a Gaussian pulse based on Eq. (2). The vertical dotted lines indicate the B-integral value used in Figs. 4 and 5.

Figure 8 :

 8 Figure 8: Numerically computed evolutions of (a) the rms spectral broadening factor, (b) the Strehl ratio, (c) the rms temporal broadening factor, and (d) the excess kurtosis parameter of an initial Gaussian pulse after undergoing SPM in a fiber with the input pulse duration. The accumulated B-integral is B = 2 The results obtained without phase modulation, with the phase modulation chosen based on Eq. (2) and applied as a post-compensation, with the phase

Figure 9 :

 9 Figure 9: Impact of phase correction on a 14 ps Gaussian pulse after undergoing SPM in a fiber as obtained by numerical integration of Eq. (1): (a) temporal intensity and chirp profiles, and (b) spectral intensity profile on linear and logarithmic scales at the entrance (black) and the exit of the fiber without (red) and with phase compensation by a sinusoidal signal with frequency and amplitude chosen based on Eq. (2) (solid) and with optimized frequency and amplitude (dashed). The results obtained with a pre-compensation scheme (yellow) are compared with those obtained with a post-compensation scheme (blue). The B-integral accumulated by the pulse is B = 2 rad.

Figure 10 :

 10 Figure 10: Polar graph (I(t);(t)) of the electric field of a 14 ps Gaussian pulse after undergoing SPM in a fiber without phase compensation (red), with phase pre-compensation (yellow) and with phase post-compensation (blue).

Figure 11 :

 11 Figure 11: Numerically computed evolutions of (a) the rms spectral broadening factor and (b) the Strehl ratio of an initial 14 ps Gaussian pulse after undergoing SPM in a fiber with the accumulated B-integral. (c)-(d) Evolutions of the frequency and amplitude of the sinusoidal phase modulation. The results obtained without phase modulation, with the phase modulation applied as a pre-compensation and with the phase modulation applied as a post-compensation are plotted with red, yellow and blue lines, respectively. The results obtained with the sinusoidal modulation chosen based on Eq. (2) (solid) are compared with those obtained with the optimized sinusoidal modulation (dashed). The vertical dotted lines indicate the B-integral value used in Figs. 9 and 10.

  , we compute the rms spectral width 

							 by computing the
	variance
	22	2	2	( )	22 ( )	2
				( )	22 ( )

Acknowledgements

We thank Bertrand Kibler and Julien Fatome for very stimulating discussions about the initial concept. We also thank Todd Ulmer for bringing to our knowledge several important previous publications on the method being studied. We acknowledge financial support by the Région Bourgogne (Pari Photcom) and the Agence Nationale de la Recherche (Labex Action ANR-11-LABX-01-01). This work was also supported by the European Union within the framework of the operational Program FEDER-FSE Bourgogne 2014-2020 and by the Institut Universitaire de France.


Sébastien Canitrot
email: sebastien.canitrot@cran.uhp-nancy.fr

Taha Boukhobza

Frédéric Hamelin

Sinuhé Martinez-Martinez

Control/fault diagnosis interactions analysis for structured bilinear systems

Keywords: Structural properties, bilinear systems, fault diagnosis, graph theory

INTRODUCTION

Fault detection and isolation () problem has received considerable attention in the two last decades [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF][START_REF] Frank | Analytical and qualitative modelbased fault diagnosis -a survey and some new results[END_REF][START_REF] Gertler | Observer based approach to fault detection and isolation for non linear systems[END_REF]. Detection and isolation of incipient faults is important for safety critical systems where a malfunction can cause human and material damages. Bilinear systems () are amongst the simplest nonlinear systems and therefore are particularly adapted to analysis compared to more complicated nonlinear systems. They can be used to represent a wide range of physical, chemical, biological and social systems, as well as manufacturing processes which cannot be satisfactorily controlled under the linearity assumption. The properties and behavior of bilinear systems have been investigated and a number of useful results have been derived. Up to now, the problem of residual generation for  is mainly treated using algebraic or geometric tools [START_REF] Yang | State observation, failure detection and isolation (FDI) in bilinear systems[END_REF][START_REF] Hammouri | A geometric approach to fault detection and isolation for bilinear systems[END_REF]. One of the most important results about  of  is given in [START_REF] Hammouri | A geometric approach to fault detection and isolation for bilinear systems[END_REF]. Using a geometric approach, the authors provide necessary and sufficient conditions for the solvability of the so-called Bilinear Fundamental Problem of Residual Generation (). This problem extends the Fundamental Problem of Residual Generation () defined and solved for linear systems in [START_REF] Massoumnia | Failure detection and identification[END_REF]. The solvability conditions of such a problem for nonlinear systems, which are affine in the control inputs and in the failure modes, have been also given in (Hammouri et al., 1999;[START_REF] De Persis | A geometric approach to nonlinear fault detection and isolation[END_REF]. Many studies on structured systems are related to the graph-theoretic approach. Until now, this approach is mainly dedicated to linear systems. [START_REF] Dion | Generic properties and control of linear structured systems: A survey[END_REF] reviews the most significant results in this area. From these works, it results that the graph-theoretic approach provides simple and elegant solutions and so is very well suited to analyze large scale or/and uncertain systems. The quality of the results obtained for linear systems by using a graphic approach has motivated us to apply the latter to the problem of residual generation for structured bilinear systems (). The main contribution of this note is the study of the so-called "active"-. This problem, as we define it more precisely later, is related to the existence of particular control inputs which make the  solvable. Indeed, reducing the  problem to the usual  is quite restrictive, since the control input can be used to achieve the detection and the isolation of faults. Effectively, in practice, many  strategies use inputs, or more generally, some specific functioning modes to detect and to isolate the faults occurring on the system. In this context, it seems interesting to study the input influence on the solvability of the . More precisely, we study two kinds of control inputs. Firstly, we search to solve the active- considering all control inputs as generic constant. Next, we set to zero some inputs considering the others constant. All the results provided in this paper are based on the graph-theoretic approach. The solvability conditions for residual generation problem are related to comparison of expressions and to computation of paths in a digraph. This makes our approach very well suited to study large scale and/or uncertain systems. The paper is organized as follows: after section 2, which is devoted to the active- formulation, a digraph representation of structured bilinear systems is defined in section 3. Some sufficient conditions for the solvability of active-, are enounced in section 4. Finally, some concluding remarks are made.

PROBLEM STATEMENT

We consider  of the form:

Σ Λ :            ẋ(t) = A 0 x(t) + m ℓ=1 u ℓ (t)A ℓ x(t) + E f 1 (t) + Hw(t) y(t) = Cx(t) (1) where x(t) ∈ R n , u(t) ∈ R m , f 1 (t) ∈ R, w(t) ∈ R d and y(t) ∈ R p
are respectively the state, the input, the fault, the disturbance and the output vectors. In this study, we treat the case of system with only one fault f 1 . However, if the system is corrupted by several faults, the study can also be made to each fault considering other faults as disturbances. A ℓ , for ℓ = 0, . . . , m, C, E and H represent matrices of appropriate dimensions, which elements are either fixed to zero or assumed free nonzero parameters. We can parameterize these nonzero entries by real (nonzero) parameters λ i , i = 1, . . . , h forming a parameter vector Λ = (λ 1 , . . . , λ h ) T ∈ R h . If all the nonzeros parameters λ i are fixed, we obtain an admissible realization of structured system Σ Λ . Theoretic properties of each realization can be studied according to the values of λ i . We say that a property is true generically if it is true for almost all the realizations of structured system Σ Λ . Here, "for almost all the realizations" is to be understood as "for all parameter values (Λ ∈ R h ) except for those in some proper algebraic variety in the parameter space". The proper algebraic variety for which the property is not true is the zero set of some nontrivial polynomial with real coefficients in the system parameters λ 1 , λ 2 , . . . , λ h or equivalently it is an algebraic variety which has Lebesgue measure zero.

Usually, the fault diagnosis problem and the control problem are separated but there exist some control/diagnosis interactions studies as those concerning  (Fault Tolerant Control). In this note we make a complementary study of these interactions. Our aim is to provide sufficient conditions for the generic solvability of the following problem. Consider  Σ Λ , we define the active- as the problem of finding filter of the form:

ż(t) = φ(z(t), y(t)) + ϕ(z(t), y(t))u(t) δ(t) = h(z(t), y(t)) (2)
with z(t) ∈ R r and δ(t) ∈ R, such that it generically exists an input u(t) so that

• if f 1 (t) = 0 then δ(t) → 0 for all initial condi- tions. • else, δ(t) 0 for all initial conditions That is to say, δ(t) is insensitive to w(t).
The "classical"  is to find a filter (2) such that δ(t) enables the detection of f 1 (t) for any admissible inputs u(t). It is well-known that in bilinear systems, the fault detection is affected by the control input. In fact, a specific control value can make possible the detection of a fault, which is not detectable for all other values of u(t). In another way, a specific control value can make impossible the detection of a fault, which is detectable for all other values of u(t). Like many structural properties, we think that the interactions between control and fault diagnosis can be more easily handled using a graph-theoretic approach defined in the next section.

GRAPHICAL REPRESENTATION OF STRUCTURED BILINEAR SYSTEMS

The graph, directly obtained from the system structure, is interesting for several reasons. In particular, the graph-based results, given by weak burden calculation, are intuitive and are easily applicable to large scale systems.

Digraph definition for 

The digraph associated to bilinear system Σ Λ is noted G(Σ Λ ) and is constituted by a vertex set V and an edge set E : G(Σ Λ ) = (V, E). The vertices are associated to the variables of Σ Λ and the directed edges represent links between these variables. To differentiate variables from their corresponding vertices, the latter are written in bold fonts.

The vertex set is V = X ∪ Y ∪ W ∪ {f 1 } where X = {x 1 , . . . , x n } is the set of state vertices, Y = {y 1 , . . . , y p } is the set of output vertices, W = {w 1 , . . . , w d } is the set of disturbance vertices and f 1 is the fault vertex. The edge set is E = m ℓ=0 A ℓ -edges∪C-edges∪E-edges∪ H-edges where A ℓ -edges = {(x i , x j )|A ℓ ( j, i) 0} for ℓ = 0, . . . , m, C-edges = {(x i , y j )|C( j, i) 0}, E-edges = {(f 1 , x j )|E( j) 0} and H-edges = {(w i , x j )|H( j, i) 0}. (v i , v j ) denotes a directed edge from begin vertex v i ∈ V to end vertex v j ∈ V.
We take the following notation : Ā0 -edges = A 0 -edges ∪ C-edges ∪ E-edges ∪ H-edges and for ℓ = 1, . . . , m, Āℓ -edges = A ℓ -edges. To each edge e ∈ Āℓ -edges is associated an unique indice u ℓ and we indicate u ℓ over each edge e in the digraph representation. If an edge e 1 ∈ Āℓ 1 -edges and an edge e 2 ∈ Āℓ 2 -edges have the same begin and end vertices, only one edge is represented and the information over the edge will be u ℓ 1 , u ℓ 2 with ℓ 1 < ℓ 2 .

Example 1. Consider  defined by:

A 0 =                     0 0 0 0 0 0 0 0 0 0 0 0 0 λ 1 0 λ 2 λ 3 0 0 0 0 0 λ 4 0 0                     , A 1 =                     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ 5 0 0 0 0 0 0 λ 6 0                     , A 2 =                     0 0 0 0 0 0 0 0 0 0 λ 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0                     , E =                     0 λ 8 0 0 0                     , H =                     λ 9 0 0 0 0                     and C = 0 0 0 0 λ 10 .
This model is associated to digraph 1.

We can see that the graph is easily obtained from the system structure.

Notations and definitions

• P = v r 0 u i 1 ---→ v r 1 . . . v r k-1 u i k ---→ v r k
denotes a path P, which contains vertices v r 0 , . . . , v r k and where (v r j , v r j+1 ) ∈ Āi j+1 -edges for j = 0, . . . , k -1. v r 0 and v r k are respectively the begin vertex and the end vertex of P. We associate to P an unique indice noted σ(P) and defined by the ordered monomial

u i 1 • u i 2 • . . . • u i k .
A path P is uniquely characterized by its indice and the vertices it covers. P is an Y-topped path if its end vertex v r k ∈ Y.

• Some paths are disjoint if they have no common vertices.

• A set of k l disjoint paths from V 0 ∈ V to V 1 ∈ V is called a linking of dimension k l from V 0 to V 1 . A linking i composed of paths P i 1 , P i 2 , . . . , P i k is denoted 1 w 1 f 1 x 2 x 4 x 5 x 3 x 1 y u 0 u 0 u 0 ,u 1 u 1 u 0 u 2 u 0 u 0 u 0 1 w 1 f 1 x 2 x 4 x 5 x 3 x 1 y u 0 u 0 u 0 ,u 1 u 1 u 0 u 2 u 0 u 0 u 0 Figure 1. Digraph 1 associated to Example 1 m i = k j=1 P i j . A linking from V 0 to V 1 is maximum when k l is maximum. We denote by ρ(V 0 , V 1 ) the maximum number of disjoint paths from V 0 to V 1 . In example 1, ρ(W ∪ {f 1 }, Y) = 1.
• The set of all maximum linkings m j 1 , m j 2 , . . . , m j q

from V 0 to V 1 is denoted by M(V 0 , V 1 ) = q ℓ=1 m j ℓ .

ACTIVE-BFPRG

The active- is a quite vast problem because of the influence of the control input on the solution.

Practically, it is tedious to solve this problem whatever the control, this is why in this paper we focus only on the simplest cases.

Case of constant input

In a first time, we consider that each control input is defined constant (u(t) = u), this restriction can be justified by the fact that control inputs are constant in the steady state. In this case, structured bilinear system Σ Λ is equivalent to structured linear system Σ 1 Λ with system parameters depending on the control value.

Σ 1 Λ : ẋ(t) = Ã0 x(t) + E f 1 (t) + Hw(t) y(t) = Cx(t) (3) where Ã0 = A 0 + m ℓ=1 u ℓ A ℓ .
The  solvability can then be studied using the geometrical conditions given by [START_REF] Hammouri | A geometric approach to fault detection and isolation for bilinear systems[END_REF]. This approach requires the determination of R * , the minimal (C, A)-invariant subspace containing H, and T * , the minimal (C, A)-unobservability subspace containing H. The problem is solvable if E T * . This study does not take part of the control input influence on fault detection as shown by example 2.

Example 2. Consider  defined by :

A 0 =                0 0 0 0 0 0 0 0 λ 1 0 0 0 0 λ 2 0 0                , A 1 =                0 0 0 0 0 0 0 0 0 λ 3 0 0 0 0 0 0                , H =                0 λ 4 0 0                , E =                λ 5 0 0 0               
and C = 0 0 λ 6 0 0 0 0 λ 7 .

This system is represented by digraph 2. For this example, it can be shown that

R * = span                               0 λ 4 0 0                ,                0 0 λ 3 λ 4 0                ,                0 0 0 λ 2 λ 4                               and T * = R 4 .
Then, since E ∈ T * , the  has no solution.

If we consider that the control input is generically constant (u 1 (t) = u 1 ), the system matrices become:

Ã0 =                0 0 0 0 0 0 0 0 λ 1 λ 3 u 1 0 0 0 λ 2 0 0                , H =                0 λ 4 0 0                , E =                λ 5 0 0 0               
and C = 0 0 λ 6 0 0 0 0 λ 7 .

In this case, the  is solvable using the geometrical conditions [START_REF] Hammouri | A geometric approach to fault detection and isolation for bilinear systems[END_REF]. Indeed,

R * = span                               0 λ 4 0 0                ,                0 0 λ 3 λ 4 u 1 λ 2 λ 4                               and T * = span                               0 1 0 0                ,                0 0 λ 3 λ 4 u 1 λ 2 λ 4                               . Since E T * , the  is solvable.
In a practical point of view, the following filter makes possible the detection of fault f 1 :

ẋ(t) = Ax(t) + K(y(t) -C x(t)) δ(t) = Q(y(t) -C x(t)) (4)
with matrices K and Q chosen such that QC(sI -A + KC) -1 H = 0 and QC(sI

-A + KC) -1 E 0.
For the example, we have

K =                k 11 k 12 k 21 k 22 k 31 k 32 k 41 k 42                and Q = q 1 q 2
under the constraints q 2 = -q 1 λ 3 u 1 6 λ 2 λ 7 , k 11 = -k 12 λ 2 λ 7 λ 3 u 1 λ 6 and k 32 = λ 3 (k 42 λ 2 λ 7 -k 31 λ 2 λ 6 +k 41 λ 3 λ 6 )

λ 2 2 λ 7 (subject to stability constraints).

Consequently, it appears for this example that the active- has a solution whereas the "classical"  is not solvable. The following proposition which is based on the results of (Commault et al., 2002) gives a sufficient condition to the active- solvability. This condition is necessary and sufficient in the case of control inputs as generic constants.

Proposition 1 : Consider  Σ Λ . In the case where all the control input components are constant,  Σ Λ is equivalent to structured linear system Σ 1 Λ . The active- is then solvable if :

ρ(W ∪ {f 1 }, Y) = ρ(W, Y) + 1 (5)
If the condition of proposition 1 is not satisfied then there are two possibilities for detecting fault f 1 . The first one consists in increasing the value of ρ(W ∪ {f 1 }, Y) (by addition of particular sensors to the system) and the second one consists in decreasing the value of ρ(W, Y) (using the influence of the control). We are interested in the second case. The parameters of the obtained linear system are dependent of the control. In fact, setting some control components to zero results in edges removal in the Example 3. Consider the system represented by digraph 3.

For this example, in the case where control inputs are considered to be generically constant, we have

ρ(W, Y) = 2 and ρ(W∪{f 1 }, Y) = 2 ρ(W, Y)+1.
We can then conclude that the detection of fault f 1 with u 1 (t) = u 1 (generic) is impossible. But, if we consider the system with u 1 (t) = 0, then we obtain ρ(W, Y) = 1 and ρ(W

∪ {f 1 }, Y) = 2 = ρ(W, Y) + 1.
Condition of proposition 1 is satisfied and so it is possible to detect fault f 1 . The next part is dedicated to the formulation of this particular case in general contest.

Case of some cancelled inputs

Our objective hereafter is to characterize which control inputs have to be set to zero or not in order to detect f 1 . At this aim let us denote by T = {i | u i 0} the set of control inputs not fixed to zero. In this case, structured linear system Σ 1 Λ is equivalent to a second structured linear system Σ 1 Λ (T ) given by :

Σ 1 Λ (T ) : ẋ(t) = Â0 (T )x(t) + E f 1 (t) + Hw(t) y(t) = Cx(t) (6)
where Â0 (T )

= A 0 + i∈T u i A i .
The digraph associated to system

Σ 1 Λ (T ) is now defined by G(Σ 1 Λ (T )) = (V, E(T )) with E(T ) = Ā0 -edges i∈T Āi -edges.
We denote by ρ u (V 0 , V 1 , T ) the maximum number of disjoint paths from V 0 to V 1 assuming all the control inputs generically constant except for u j with j T such that u j = 0. From the results of (Commault et al., 2002), the following proposition is given.

Proposition 2 : Consider  Σ Λ . In the case where all the control input components are constant,  Σ Λ is equivalent to structured linear system Σ 1 Λ . The active- is then solvable if :

∃T such that ρ u (W ∪ {f 1 }, Y, T ) > ρ u (W, Y, T ) (7)
Since proposition 2 can be generally verified for different subsets T , it is interesting to set a procedure in order to obtain all subsets T which enable the fault detection.

Characterization of all solutions

If we consider that ρ(W ∪ {f 1 }, Y) = ρ(W, Y), we can deduce that linkings belonging to M(W, Y) of cardinality l 1 belong necessarily to M(W ∪ {f 1 }, Y) of cardinality l 2 then M(W, Y) ⊆ M(W ∪ {f 1 }, Y) and l 2 ≥ l 1 . Let us assume that M(W, Y) = m 1 ∪ . . . ∪ m l 1 and M(W∪{f 1 }, Y) = m 1 ∪. . .∪m l 1 ∪. . .∪m l 2 . Linkings m i for i = 1, . . . , l 1 do not contain any path from f 1 to Y. By the way, linkings m i for i = l 1 + 1, . . . , l 2 contain necessarily a path from f 1 to Y. If l 2 = l 1 , we can conclude that there is no path from f 1 to Y and consequently, fault f 1 cannot be detected without adding a new sensor.

Considering now a maximum linking containing a path from f 1 , the proposed approach consists in comparing this linking with all other maximum linkings (with no path from f 1 ). This comparison needs some new definitions.

• Three operations ⊙, ⊕ and ⊘ on set U = {u 0 , . . . , u m } are defined. ⊙ and ⊕ are commutative and associative.

• To path P = v r 0 u i 1 ---→ v r 1 . . . v r k-1 u i k ---→ v r k , we associate the expression ψ(P) = u i 1 ⊙ u i 2 ⊙ . . . ⊙ u i k .
• Considering u 1 = u, the following relations are used in order to simplify expressions.

• u k 1 ⊙ u k 2 = u k 1 +k 2 with k 1 ≥ 1 and k 2 ≥ 1. • (u k 1 i 1 ⊙ ψ 1 ) ⊘ (u k 3 i 1 ⊙ ψ 2 ) =              (u k 1 -k 3 i 1 ⊙ ψ 1 ) ⊘ ψ 2 if k 1 > k 3 . ψ 1 ⊘ (u k 3 -k 1 i 1 ⊙ ψ 2 ) if k 1 < k 3 . ψ 1 ⊘ ψ 2 if k 1 = k 3 .
• The complement operation on expression ψ is denoted by ψ with the following properties :

u 1 ⊙ u 2 ⊙ . . . ⊙ u k = ū1 ⊕ ū2 ⊕ . . . ⊕ ūk and u 1 ⊕ u 2 ⊕ . . . ⊕ u k = ū1 ⊙ ū2 ⊙ . . . ⊙ ūk .
• To each maximum linking m i ∈ M(V 0 , V 1 ), we associate the expression ψ(m i ) = P j ∈m i ψ(P j ). The latter enables to know which control inputs appear in the maximum linking m i . The interest is that we can deduce which control inputs could be set to zero in order to decrease the size of m i .

• Function cp(m i , m ℓ ) makes the comparison between linkings m i and m ℓ It is defined such that: If ψ 1 ⊘ ψ 2 is the expression of ψ(m i ) ⊘ ψ(m ℓ ) after all simplifications then cp(m i , m ℓ ) = ψ 1 .

• P f (ℓ) denotes the path from f 1 belonging to m ℓ . ψ(P f (ℓ)) enables to know which control inputs appear in the path from f 1 to Y. The interest is that we can deduce which control inputs could be different from zero in order to keep path P f (ℓ) in m ℓ .

More precisely, we will be able to find which control inputs have to be equal to zero and which control inputs have to be different from zero in order to obtain all the possible solutions by computing the set S as below.

N(ℓ) indicates the control inputs belonging to ψ(m i ) for i = 1, . . . , l 1 and not to ψ(m ℓ ). The complement of N(ℓ) correspond to the control inputs which can be set to zero without deleting the link between f 1 and Y. Furthermore, D(ℓ) indicates which control inputs have to be different from zero in order to keep the path from f 1 in the linking m ℓ .

From a computational point of view, N(ℓ) and D(ℓ) can be determined by mean of the following algorithm. For ℓ = l 1 + 1 to l 2 N(ℓ) = According to the general form of N(ℓ) which is (ū l 1 k 1 ⊙ ūl 2 k 2 ⊙ . . . ⊙ ūl s k s ) ⊕ (ū l 11 k 11 ⊙ . . . ⊙ ūl t k t ) ⊕ . . ., we create the set S(ℓ) = {{ū k 1 , ūk 2 , . . . , ūk s }, {ū k 11 , ūk 12 , . . .}, . . .}.

Finally, if D(ℓ) is on the form

u d 1 c 1 ⊙ u d 2 c 2 ⊙ . . . ⊙ u d v
c v , we modify set S(ℓ) by adding u c 1 , u c 2 , . . . , u c v to all elements of S(ℓ) in order to obtain S(ℓ) = {{ū k 1 , ūk 2 , . . . , ūk s , u c 1 , u c 2 , . . . , u c v }, . . .}. Each element of S(ℓ) is composed of elements ūd and u e . Elements ūd give control inputs having to be set to zero and u e gives control inputs having to be different from zero. From a practical point of view, it is important to remove some elements of S(ℓ) according to the following rules :

• if an element s g of S(ℓ) contains ū0 then S(ℓ) = S(ℓ)\{s g }, since u 0 is generically different from zero because it depends on the system parameters.

• if an element s h of S(ℓ) contains ūi and u i then S(ℓ) = S(ℓ)\{s h }, since control inputs cannot be set to zero and different from zero at the same time.

Proposition 3

Consider  Σ Λ . In the case where all the control input components are constant,  Σ Λ is equivalent to structured linear system Σ 1 Λ . The active- is then solvable if :

S = l 2 ℓ=l 1 +1 S(ℓ) ∅ (8)
Moreover, all the solutions are given by set S. Let s 1 = {ū i , . . . , u j , . . .} be an element of S. If ūi ∈ s 1 then u i have to be set to zero. If u j ∈ s 1 then u j have to be different from zero. If u k s 1 and ūk s 1 then u k can be indifferently set to zero or different from zero.

Figure 2 .

 2 Figure 2. Digraph 2 associated to Example 2

Figure 3 .

 3 Figure 3. Digraph 3 associated to Example 3 digraph representation and then, the fault detection may be possible.

  m i ), ψ(m ℓ )) D(ℓ) = ψ(P f (ℓ)) Endwhere maximum linkings m i do not contain any path from f 1 and maximum linkings m ℓ contain a path from f 1 .

In example 3, there are 4 paths from W ∪ {f 1 } to Y.

0 ⊙ u 2 . Now, the method consists in compare m 2 and m 3 with m 1 by the computation of S.

We conclude that the active- is solvable if u 1 = 0 and u 2 = constant 0 (since u 0 is generically different from zero).

Remark

There also exists the case of particular constant inputs (but not equal to zero) making the fault detection possible. For example, from the system represented in figure 4, we can write a relation on the form ÿ1 (t) = α 1 f 1 (t) + α 2 w 1 (t) + α 3 u 1 (t)w 1 (t) + α 4 u1 (t)x 1 (t) (with α i depending on the system parameters) and then if u 1 (t) is equal to the appropriate constant (u 1 (t) = -α 2 α 3 ) we obtain ÿ1 (t) = α 1 f 1 (t). We see that a very particular value for the control input makes the fault detectable but this particular value can only be obtained from the knowledge of the parameters. Consequently, this case is not in adequation with the fact that we consider structured systems, which means that we do not know the parameters value.

CONCLUSION

In this paper, we propose a new analysis tool to check the solvability of the  considering the control input influence. For this purpose, sufficient conditions to the existence of input values making the so-called active- solvable have been provided on the basis of graph-theoretic approach. All the conditions we have enounced in this paper do not depend on the parameters of the system and have an intuitive interpretation. Moreover, they are very easy to check by means of well-known combinatorial techniques and simply by hand for small systems. From a computational point of view, note that our approach is particularly suited for large-scale systems and is free from numerical difficulties. Indeed, the proposed algorithm and conditions can be easily implemented because they require simple computations on expressions and are based on finding paths in graphs. Furthermore, the graphical approach makes easy the visualization of the system structure.

In fact, we have seen that among the possibilities to make fault detectable, there are the control inputs influence and the sensor placement. In further work, we will investigate the sensor placement to fulfill the fault detection of the system when this present study do not give any solution.