Extended-Kalman-Filter-like observers for continuous time systems with discrete time measurements
|
|
- Joel Harris
- 5 years ago
- Views:
Transcription
1 Extended-Kalman-Filter-lie observers for continuous time systems with discrete time measurements Vincent Andrieu To cite this version: Vincent Andrieu. Extended-Kalman-Filter-lie observers for continuous time systems with discrete time measurements <hal > HAL Id: hal Submitted on 5 Mar 010 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
2 Extended-Kalman-Filter-lie observers for continuous time systems with discrete time measurements Vincent Andrieu March 5, 010 Abstract In this short note is studied the observer introduced in [1] and [5]. The relationship between the Lipschitz constant and the measurement stepsize is exhibited. For second order system, we evaluate the authorized measurement stepsize. 1 Introduction We consider a continuous time system of the form: ẋ = f(x,u), x = (x 1,...,x n ) (1) with the state x is in R n and the input u is in U R. The solutions of this system are denoted as 1 x(t). The state of this system is accessible via a discrete time measure where (t ) N is a sequence of positive real number defined as: where δ is a positive real number. y = Cx(t ), () t +1 = t +δ Vincent Andrieu is with Université de Lyon, F-696, Lyon, France; Université Lyon 1, Villeurbanne; CNRS, UMR 5007, LAGEP. 43 bd du 11 novembre, Villeurbanne, France vincent.andrieu@gmail.com 1 Solution should be written x(x 0,t,u( )) but to simplify the presentation we prefer this notation. 1
3 The problem under consideration is an observer problem. How can we estimate the state of the system nowing only the measurements. This problem has been addressed in [1] (see also [3] and the related filtering problem in [4]). Following what has been done for continuous time measurement in [], we consider the case in which the system can be written (possibly after a change of coordinates) in the form: with Ax = (x,...,x n 1,0), ẋ = Ax+φ(x,u) (3) y = Cx(t ) = x 1 (t ) and where the function φ : R n U R n satisfies an upper triangular globally Lipschitz condition, i.e. is such that for i in {1,...,n} we have: i φ i (x+e,u) φ i (x,u) c L e j, (x,e) R n R n, u U, (4) j=1 where c L is a positive real number named the Lipschitz constant of the nonlinear system. Inspired by [1] and [5], an estimate of the state can be given as any piecewise continuous function ˆx : R + R n solution of the following continuous-discrete system : ˆx = Aˆx+φ(ˆx,u) Ṡ = θs A, t [δ,( +1)δ) S SA ) ˆx(t ) = ˆx(t ) δs(t ) 1 C (Cˆx(t ) y (5) S(t ) = S(t )+δc C with θ a positive real number. Note that compared to what has been done in [1], the Matrix updated law is a Lyapunov lie one instead of a a Riccati lie one. This one has been used in [5] since it allows to study analytically its limit and to give a better approximation on the bound involved in the highgain approach. Compare to [5] the only difference is the fact that the gain matrix depends on S which is time varying. Main Theorem Inspired by the result of [5], the following theorem can be obtain. For a time function x( ) the notation x(t ) means (when it exists) lim t t,t<t x(t)
4 Theorem 1 There exists a positive real number ζ (which depends only on the dimension of the system), such that, for all δ < ζ c L there exist two positive real numbers θ 1 < θ such that for any θ in [θ 1,θ ], the estimate ˆx converges asymptotically to the state of the system (3). Proof : Let E = Θ 1 (ˆx x) where Θ = diag{1,...,θ n 1 } with θ a positive real number larger then 1 to be specified. The scaled error satisfies for t in [δ,( +1)δ): Ė = Θ 1 A[ˆx x]+θ 1 [φ(ˆx,u) φ(x,u)], and since Θ 1 A = θaθ 1, it yields for t in [δ,( +1)δ): Ė = θae + θ φ(x,e), with θ φ(x,e) = Θ 1 [φ(ˆx,u) φ(x,u)] such that θ φ i (x,e) c i L j=1 E j since θ > 1. Moreover, the scaled error satisfies: [ ) ] E(t ) = Θ 1 ˆx(t ) δs(t ) 1 C (Cˆx(t ) y x(t ) ( ) = I δ S(t ) 1 C C E(t ), where S( ) = ΘS( )Θ. Note that S satisfies: and for t in [t,( +1)δ): S(t ) = S(t )+δc C (6) S = θ S ΘA Θ 1 S SΘ 1 AΘ, = θ S θa S θ SA. (7) Note that the sequence (s ) N such that s = θ S(t ) satisfies where ρ = θδ. Which gives, s +1 = exp( ρ)exp( A ρ)s exp( Aρ)+ρC C, 1 s = exp( ρ)exp( A ρ)s 0 exp( Aρ)+ exp( lρ)exp( A lρ)ρc Cexp( Alρ), l=0 Following what has been done in [5], the following Lemma can be shown (its proof is given in Section 3). 3
5 Lemma 1 There exist two continuous function α 1 and α such that for all ρ > 0, the matrix series converges to a limit denoted s (ρ), formally defined as: such that s = + l=0 exp( lρ)exp( A lρ)ρc Cexp( Alρ), where α 1 and α are two strictly positive continuous functions. α 1 (ρ)i s α (ρ)i, (8) The proof is divided in three steps : Part 1: With this lemma in hand, we will first show that for all ρ > 0, there exists t 0 such that for all t > t 0, the matrix S satisfies: γ 1 (ρ)i < θ S(t) < γ (ρ)i (9) where γ 1 and γ are two continuous positive function defined as, γ 1 (ρ) = exp( ρ)α 1 (ρ)c Int(ρ) 3 c 1, γ (ρ) = α (ρ)c c Int(ρ) 4 (10) where Int( ) denotes the integer part of a positive real number, c 1 and c are two positive real numbers and (c 3,c 4 ) are two positive real numbers such that c 3 < 1 and c 4 > 1. Indeed, for in N and s in [0,δ), S( +s) is solution of (7), hence, θ S( +s) = exp( θs)exp( A θs)s exp( Aθs). With Lemma, for sufficiently large, the matrix s is definite positive, it yields that for all v in R n, v θ S( +s)v = exp( θs) s 1 exp( Aθs)v. Hence, we get for all v in R n : Note that v θ S( +s)v exp( ρ) exp( Aθs)v = exp( Aθs)v (λ min { s 1 }) exp( Aθs)v = exp( AInt(θs))exp( A[θs Int(θs)])v (λ min {exp( A) exp( A)}) Int(θs) exp( A[θs Int(θs)])v (λ min {exp( A) exp( A)}) Int(θs) c 1 v 4
6 wherec 1 = min r 1 {λ min {exp( Ar) exp( Ar)}}. Furthermore, notethatdet{exp( A) exp( A)} = 1, hence c 3 := λ min {exp( A) exp( A)} < 1. Consequently, And since, exp( Aθs)v c Int(ρ) 3 c 1 { }) (λ min s 1 λmin {s }, it yields for all sufficiently large and all v in R n and, In a same way, v θ S( +s)v exp( ρ)λ min {s }c Int(ρ) 3 c 1 v v θ S( +s)v (λ max { s 1 }) exp( Aθs)v exp( Aθs)v (λ max {exp( A) exp( A)}) Int(θs) c v where c = max r 1 λ max {exp( Ar) exp( Ar)}. Hence we get v θ S( +s)v λ max {s }c (λ max {exp( A) exp( A)}) Int(θs) v Due to the fact that c 4 := λ max {exp( A) exp( A)} > 1, we get v θ S( +s)v λ max {s }c c Int(ρ) 4 v which, with Lemma implies that equation (9) holds and finishes the first part of the proof. Part : Let now V(E) = E SE. We will now show that it is strictly decreasing providing θ and ρ satisfy an inequality constraint depending on c L the Lipschitz constant of the nonlinear system. The function V satisfies along the trajectories of the system for t in [δ,( +1)δ): V(E) = E [θa S(t)+θ S(t)A+ ] Ṡ E +E S θ φ(x,e) (11) With Schwarz inequality, we have for all t > 0 and we have 3 = θe S(t)E +E S(t) θ φ(x,e). (1) E S(t) θ φ(x,e) E S(t) θ φ(x,e) λ max { S(t)} E θ φ(x,e), (13) 3 Here we have used the inequality: (a a n ) n ( a a ) n 5
7 θ φ(x,e) = n θ φ i (x,e) (14) i=1 Consequently, we get: V(E) n i=1 c L ( i ) ( n ) E j nc L E j n c L E j=1 [ ] λ max { S(t)}nc L θλ min { S(t)} E. (15) Note that, due to equation (9), we get that if we can select θ and δ such that j=1 with ρ = δθ we would get θ > γ (ρ)nc L γ 1 (ρ) λ max { S(t)}nc L θλ min { S(t)} < 0, (16) for all t sufficiently large. Hence we get V(E) < 0 for all t in [δ,(+1)δ), with sufficiently large. Moreover, we have ( ) V(E)(t ) = E(t (I ) δ S 1 C C) S(t ) I δ S 1 C C E(t ) ] = E(t [ S(t ) ) δc C +δ C C S(t ) 1 C C E(t ) where, = E(t ) [ S(t ) δc C +δ C C = V(E)(t ) E(t ) C pce(t ) p = δ δ 1C C ( S(t C) )+δc. Following [4] (see also [1]), note that if we note q = pq = Hence, we get : = 1 ( S(t )+δc C) 1C C [ C S(t ) 1 C + 1 δ ], we have, ] E(t ) ( ) ][ 1C [δ δ C S(t ) 1 I +δc C S(t ) 1 C S(t ) 1 C + 1 ] δ V(E)(t ) = V(E)(t ) E(t ) C q 1 CE(t ) 6
8 with q > 0. Consequently the function V is decreasing along the trajectories of the system for t sufficiently large. Its eigenvalues being lower and upper bounded as shown in the first part of the proof, we get convergence of the estimation error to the origin. Part 3: We now show that the constraint we have on θ and ρ can be given in terms of an upper bound on δ depending on c L. To obtain estimation, we have to find δ and θ such that (16) is satisfied and θ > 1. Note that (16) is equivalent with : δ < ϕ(ρ) c L, with ϕ(ρ) = ργ 1(ρ) nγ (ρ) where γ 1 and γ are defined in (10). Hence, we get: ϕ(ρ) = ρexp( ρ) α 1(ρ) nα (ρ) ( c 1 c3 c c 4 Note that since α 1 (ρ) α (ρ) and c 3 < 1 and c 4 > 1 we get: limϕ(ρ) = 0, lim ϕ(ρ) = 0. ρ 0 ρ + Hence we can define the positive real number κ as : κ = max ρ>0 {ϕ(ρ)}, )Int(ρ) This function being continuous, we get that for all c L δ < κ, there exists ρ 1 (δ) and ρ (δ) such that ϕ(ρ) > c L δ, ρ ( ρ 1 (δ), ρ (δ)), and, ϕ( ρ (δ)) = ϕ( ρ 1 (δ)) = c L δ. The function ϕ being definite positive, we get that lim δ 0 ρ (δ) = +. This implies that there exists 0 < c L ζ κ such that for all 0 < δ < ζ there exist ρ 1 (δ) and ρ (δ) such that we have δ < ρ 1 (δ) < ρ (δ), ϕ(ρ) > c L δ, ρ [ρ 1 (δ),ρ (δ)]. Hence, this implies where θ 1 = ρ 1(δ) δ and θ = ρ (δ) δ ϕ(ρ) c L > δ, θ [θ 1,θ ],1 < θ 1 < θ, which concludes the proof. (17) 7
9 3 Proof of Lemma We want to show that the sequence s = exp( lρ)exp( A lρ)ρc Cexp( Alρ), N, l=0 converges to a limit as goes to infinity denoted s and such that First, note that A being nilpotent, it yields 4 α 1 (ρ)i s α (ρ)i. (18) n 1 C exp( Alρ) = C( Alρ) j = r lρ S n, ( ) 1 where S n = diag 1,..., and, r (n 1)! lρ is a vector defined as : Consequently, we get j=0 r l = ( 1,( ρl),...,( ρl) n 1). s = ρs n M (ρ)s n, where M is the symetric positive (at least) semi-definite matrix defined as ( ) M (ρ) = exp( lρ)r lρr lρ. l=0 Hence to get the result it is sufficient to wor on the sequence M since we have for all in N where M is definite positive: v s v = ρ M 1 (ρ)s n v 4 In fact we have Cexp( Alρ) = r ρl S n O where O is the observability matrix associated to the couple (A, C), i.e., C CA O =. = I. CA n 1 8
10 It yields, ( I ( I ρ λ min {M 1 (ρ)}λ min {S n }) s ρ λ max {M 1 (ρ)}λ max {S n }) or, Consequently ρλ min {M (ρ)}(λ min {S n }) I s ρλ max {M (ρ)}λ max ({S n }) I ρ (n 1)! λ min{m (ρ)}i s ρλ max {M (ρ)}i So, provided its limit exists, (18) is satisfied with: α 1 (ρ) = ρ (n 1)! λ min{m (ρ)}, α (ρ) = ρλ max {M (ρ)} (19) where, M (ρ) = lim + M (ρ) 1. We first show that the for all sufficiently large, the matrix M is definite positive. First note that: ( n 1 ) M (ρ) exp( (n 1)ρ) r lρ r lρ, n. Note that for all v in R n, ( n 1 ) v r lρ r n 1 lρ v = r lρ v = R(ρ)v λ min {R(ρ) R(ρ)} v, l=0 l=0 where R is a Vandermonde matrix 5 R = (r 0,...,r n 1 ). Hence, we get M (ρ) λ min {R(ρ) R(ρ)}exp( (n 1)ρ)I, n. which shows that M is definite positive for all n.. We now show that M exists and is a continuous matrix function of ρ. Note that l=0 (M (ρ)) i,j = l=0 exp( lρ)( lρ) i+j = ( ρ) i+j i+j π (ρ), ρi+j 5 This implies that it is a full ran matrix. 9
11 where π (φ) = l=0exp( lρ). Note that we have π (ρ) = lim + π (ρ) = 1 1 exp( ρ) Consequently, (M (ρ)) i,j = ( ρ) i+j i+j π (ρ), ρi+j Note that this function is continuous 6 for all ρ > 0. 4 Numerical evaluation of ζ In this paragraph, we try to give the value of ζ in the simple case where the dimension of the system is n =. Computation of M, α 1 and α : We have, M (ρ) = Note that when n =, we have, 1 1 exp( ρ) ρ exp( ρ) (1 exp( ρ)) ρ exp( ρ) (1 exp( ρ)) ρ [exp( ρ)+exp( ρ)] (1 exp( ρ)) 3 α 1 (ρ) = λ min {M (ρ)}, α (ρ) = λ max {M (ρ)} Employing WolframAlpha website, we are able to give explicitly this eigenvalues as a function of ρ. Computation of c 1, c, c 3 and c 4 : We have also, ( ) 1 r exp( A r)exp(ar) = r 1+r, 6 Moreover, it satisfies: Hence, we have: lim [ρ(m (ρ)) i,j ] = ( 1) (i+j ) (i+j )!, ρ 0 lim α 1(ρ) > 0 ρ 0 10
12 consequently, and, λ min {exp( A r)exp(ar)} = +r r r +4 λ max {exp( A r)exp(ar)} = +r +r r +4. Note that r λ min {exp( A r)exp(ar)} is a strictly decreasing function on R + and that r λ max {exp( A r)exp(ar)} is a strictly increasing function on R +. Consequently, we obtain c 1 = c = 3 5,c 3 = c 4 = 3+ 5 Note that with these data, the function ϕ defined in (17) can be computed threw matlab. Its maximal value is reached for κ = max ρ>0 {ϕ(ρ)} = 0.00, ρ opt = Argmax ρ>0 {ϕ(ρ)} = > κ. Consequently, in this case, ζ = Better approximation of γ 1 and γ : In the particular case of n = a better approximation can be given. Indeed, we have, min s<δ {λ min{exp( Aθs) exp( Aθs)}} = +ρ Hence, we can tae: (+ρ ) 4, ρ = θδ. γ 1 (ρ) = exp( ρ)α 1 (ρ) +ρ (+ρ ) 4, γ (ρ) = α (ρ) +ρ + (+ρ ) 4. for The function ϕ(ρ) = ργ 1(ρ) nγ (ρ) can be computed threw matlab. Its maximal value is reached κ = max ρ>0 {ϕ(ρ)} = , ρ opt = Argmax ρ>0 {ϕ(ρ)} = > κ. Consequently, in this case, ζ = This implies that given the Lipschitz constant of the system c L, the observer which maximizes the time between two measures can be tuned with δ = c L, θ = 35.9 c L. 11
13 5 Conclusion In this short note, the continuous discrete observer presented in [5] has been studied. The maximal time between two measurement has been exhibited as a function of the Lipschitz constant. Even for small dimension, the obtained value seems to be small. References [1] F. Deza, E. Busvelle, JP Gauthier, and D. Raotopara. High gain estimation for nonlinear systems. Systems & control letters, 18(4):95 99, 199. [] J. P. Gauthier, H. Hammouri, and S. Othman. A simple observer for nonlinear systems applications to bioreactors. IEEE Transactions on Automatic Control, 37(6): , 199. [3] H. Hammouri, M. Nadri, and R. Mota. Constant gain observer for continuous-discrete time uniformly observable systems. In Proc. 45th IEEE Conference on Decision and Control, pages , 006. [4] A.H. Jazwinsi. Stochastic processes and filtering theory. Mathematics in Science and Engineering, [5] M. Nadri and H. Hammouri. Constant gain observer for continuous-discrete time uniformly observable systems. Submitted to System & Control letters,
A new simple recursive algorithm for finding prime numbers using Rosser s theorem
A new simple recursive algorithm for finding prime numbers using Rosser s theorem Rédoane Daoudi To cite this version: Rédoane Daoudi. A new simple recursive algorithm for finding prime numbers using Rosser
More informationFull-order observers for linear systems with unknown inputs
Full-order observers for linear systems with unknown inputs Mohamed Darouach, Michel Zasadzinski, Shi Jie Xu To cite this version: Mohamed Darouach, Michel Zasadzinski, Shi Jie Xu. Full-order observers
More informationSolution to Sylvester equation associated to linear descriptor systems
Solution to Sylvester equation associated to linear descriptor systems Mohamed Darouach To cite this version: Mohamed Darouach. Solution to Sylvester equation associated to linear descriptor systems. Systems
More informationSmart Bolometer: Toward Monolithic Bolometer with Smart Functions
Smart Bolometer: Toward Monolithic Bolometer with Smart Functions Matthieu Denoual, Gilles Allègre, Patrick Attia, Olivier De Sagazan To cite this version: Matthieu Denoual, Gilles Allègre, Patrick Attia,
More informationCase report on the article Water nanoelectrolysis: A simple model, Journal of Applied Physics (2017) 122,
Case report on the article Water nanoelectrolysis: A simple model, Journal of Applied Physics (2017) 122, 244902 Juan Olives, Zoubida Hammadi, Roger Morin, Laurent Lapena To cite this version: Juan Olives,
More informationMethylation-associated PHOX2B gene silencing is a rare event in human neuroblastoma.
Methylation-associated PHOX2B gene silencing is a rare event in human neuroblastoma. Loïc De Pontual, Delphine Trochet, Franck Bourdeaut, Sophie Thomas, Heather Etchevers, Agnes Chompret, Véronique Minard,
More informationOn path partitions of the divisor graph
On path partitions of the divisor graph Paul Melotti, Eric Saias To cite this version: Paul Melotti, Eric Saias On path partitions of the divisor graph 018 HAL Id: hal-0184801 https://halarchives-ouvertesfr/hal-0184801
More informationThe FLRW cosmological model revisited: relation of the local time with th e local curvature and consequences on the Heisenberg uncertainty principle
The FLRW cosmological model revisited: relation of the local time with th e local curvature and consequences on the Heisenberg uncertainty principle Nathalie Olivi-Tran, Paul M Gauthier To cite this version:
More informationCan we reduce health inequalities? An analysis of the English strategy ( )
Can we reduce health inequalities? An analysis of the English strategy (1997-2010) Johan P Mackenbach To cite this version: Johan P Mackenbach. Can we reduce health inequalities? An analysis of the English
More informationb-chromatic number of cacti
b-chromatic number of cacti Victor Campos, Claudia Linhares Sales, Frédéric Maffray, Ana Silva To cite this version: Victor Campos, Claudia Linhares Sales, Frédéric Maffray, Ana Silva. b-chromatic number
More informationEaster bracelets for years
Easter bracelets for 5700000 years Denis Roegel To cite this version: Denis Roegel. Easter bracelets for 5700000 years. [Research Report] 2014. HAL Id: hal-01009457 https://hal.inria.fr/hal-01009457
More informationUnbiased minimum variance estimation for systems with unknown exogenous inputs
Unbiased minimum variance estimation for systems with unknown exogenous inputs Mohamed Darouach, Michel Zasadzinski To cite this version: Mohamed Darouach, Michel Zasadzinski. Unbiased minimum variance
More informationVibro-acoustic simulation of a car window
Vibro-acoustic simulation of a car window Christophe Barras To cite this version: Christophe Barras. Vibro-acoustic simulation of a car window. Société Française d Acoustique. Acoustics 12, Apr 12, Nantes,
More informationLinear Quadratic Zero-Sum Two-Person Differential Games
Linear Quadratic Zero-Sum Two-Person Differential Games Pierre Bernhard To cite this version: Pierre Bernhard. Linear Quadratic Zero-Sum Two-Person Differential Games. Encyclopaedia of Systems and Control,
More informationDispersion relation results for VCS at JLab
Dispersion relation results for VCS at JLab G. Laveissiere To cite this version: G. Laveissiere. Dispersion relation results for VCS at JLab. Compton Scattering from Low to High Momentum Transfer, Mar
More informationStickelberger s congruences for absolute norms of relative discriminants
Stickelberger s congruences for absolute norms of relative discriminants Georges Gras To cite this version: Georges Gras. Stickelberger s congruences for absolute norms of relative discriminants. Journal
More informationExact Comparison of Quadratic Irrationals
Exact Comparison of Quadratic Irrationals Phuc Ngo To cite this version: Phuc Ngo. Exact Comparison of Quadratic Irrationals. [Research Report] LIGM. 20. HAL Id: hal-0069762 https://hal.archives-ouvertes.fr/hal-0069762
More informationPasserelle entre les arts : la sculpture sonore
Passerelle entre les arts : la sculpture sonore Anaïs Rolez To cite this version: Anaïs Rolez. Passerelle entre les arts : la sculpture sonore. Article destiné à l origine à la Revue de l Institut National
More informationExogenous input estimation in Electronic Power Steering (EPS) systems
Exogenous input estimation in Electronic Power Steering (EPS) systems Valentina Ciarla, Carlos Canudas de Wit, Franck Quaine, Violaine Cahouet To cite this version: Valentina Ciarla, Carlos Canudas de
More informationNew estimates for the div-curl-grad operators and elliptic problems with L1-data in the half-space
New estimates for the div-curl-grad operators and elliptic problems with L1-data in the half-space Chérif Amrouche, Huy Hoang Nguyen To cite this version: Chérif Amrouche, Huy Hoang Nguyen. New estimates
More informationOn Symmetric Norm Inequalities And Hermitian Block-Matrices
On Symmetric Norm Inequalities And Hermitian lock-matrices Antoine Mhanna To cite this version: Antoine Mhanna On Symmetric Norm Inequalities And Hermitian lock-matrices 015 HAL Id: hal-0131860
More informationEvolution of the cooperation and consequences of a decrease in plant diversity on the root symbiont diversity
Evolution of the cooperation and consequences of a decrease in plant diversity on the root symbiont diversity Marie Duhamel To cite this version: Marie Duhamel. Evolution of the cooperation and consequences
More informationA remark on a theorem of A. E. Ingham.
A remark on a theorem of A. E. Ingham. K G Bhat, K Ramachandra To cite this version: K G Bhat, K Ramachandra. A remark on a theorem of A. E. Ingham.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2006,
More informationHook lengths and shifted parts of partitions
Hook lengths and shifted parts of partitions Guo-Niu Han To cite this version: Guo-Niu Han Hook lengths and shifted parts of partitions The Ramanujan Journal, 009, 9 p HAL Id: hal-00395690
More informationA new approach of the concept of prime number
A new approach of the concept of prime number Jamel Ghannouchi To cite this version: Jamel Ghannouchi. A new approach of the concept of prime number. 4 pages. 24. HAL Id: hal-3943 https://hal.archives-ouvertes.fr/hal-3943
More informationThomas Lugand. To cite this version: HAL Id: tel
Contribution à la Modélisation et à l Optimisation de la Machine Asynchrone Double Alimentation pour des Applications Hydrauliques de Pompage Turbinage Thomas Lugand To cite this version: Thomas Lugand.
More informationapproximation results for the Traveling Salesman and related Problems
approximation results for the Traveling Salesman and related Problems Jérôme Monnot To cite this version: Jérôme Monnot. approximation results for the Traveling Salesman and related Problems. Information
More informationInfluence of a Rough Thin Layer on the Potential
Influence of a Rough Thin Layer on the Potential Ionel Ciuperca, Ronan Perrussel, Clair Poignard To cite this version: Ionel Ciuperca, Ronan Perrussel, Clair Poignard. Influence of a Rough Thin Layer on
More informationOn size, radius and minimum degree
On size, radius and minimum degree Simon Mukwembi To cite this version: Simon Mukwembi. On size, radius and minimum degree. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2014, Vol. 16 no.
More informationCompleteness of the Tree System for Propositional Classical Logic
Completeness of the Tree System for Propositional Classical Logic Shahid Rahman To cite this version: Shahid Rahman. Completeness of the Tree System for Propositional Classical Logic. Licence. France.
More informationL institution sportive : rêve et illusion
L institution sportive : rêve et illusion Hafsi Bedhioufi, Sida Ayachi, Imen Ben Amar To cite this version: Hafsi Bedhioufi, Sida Ayachi, Imen Ben Amar. L institution sportive : rêve et illusion. Revue
More informationA Slice Based 3-D Schur-Cohn Stability Criterion
A Slice Based 3-D Schur-Cohn Stability Criterion Ioana Serban, Mohamed Najim To cite this version: Ioana Serban, Mohamed Najim. A Slice Based 3-D Schur-Cohn Stability Criterion. ICASSP 007, Apr 007, Honolulu,
More informationSoundness of the System of Semantic Trees for Classical Logic based on Fitting and Smullyan
Soundness of the System of Semantic Trees for Classical Logic based on Fitting and Smullyan Shahid Rahman To cite this version: Shahid Rahman. Soundness of the System of Semantic Trees for Classical Logic
More informationOn Newton-Raphson iteration for multiplicative inverses modulo prime powers
On Newton-Raphson iteration for multiplicative inverses modulo prime powers Jean-Guillaume Dumas To cite this version: Jean-Guillaume Dumas. On Newton-Raphson iteration for multiplicative inverses modulo
More informationAnalysis of Boyer and Moore s MJRTY algorithm
Analysis of Boyer and Moore s MJRTY algorithm Laurent Alonso, Edward M. Reingold To cite this version: Laurent Alonso, Edward M. Reingold. Analysis of Boyer and Moore s MJRTY algorithm. Information Processing
More informationOn infinite permutations
On infinite permutations Dmitri G. Fon-Der-Flaass, Anna E. Frid To cite this version: Dmitri G. Fon-Der-Flaass, Anna E. Frid. On infinite permutations. Stefan Felsner. 2005 European Conference on Combinatorics,
More informationThe magnetic field diffusion equation including dynamic, hysteresis: A linear formulation of the problem
The magnetic field diffusion equation including dynamic, hysteresis: A linear formulation of the problem Marie-Ange Raulet, Benjamin Ducharne, Jean-Pierre Masson, G. Bayada To cite this version: Marie-Ange
More informationA Simple Proof of P versus NP
A Simple Proof of P versus NP Frank Vega To cite this version: Frank Vega. A Simple Proof of P versus NP. 2016. HAL Id: hal-01281254 https://hal.archives-ouvertes.fr/hal-01281254 Submitted
More informationUnfolding the Skorohod reflection of a semimartingale
Unfolding the Skorohod reflection of a semimartingale Vilmos Prokaj To cite this version: Vilmos Prokaj. Unfolding the Skorohod reflection of a semimartingale. Statistics and Probability Letters, Elsevier,
More informationSome explanations about the IWLS algorithm to fit generalized linear models
Some explanations about the IWLS algorithm to fit generalized linear models Christophe Dutang To cite this version: Christophe Dutang. Some explanations about the IWLS algorithm to fit generalized linear
More informationQuestion order experimental constraints on quantum-like models of judgement
Question order experimental constraints on quantum-like models of judgement Patrick Cassam-Chenaï To cite this version: Patrick Cassam-Chenaï. Question order experimental constraints on quantum-like models
More informationComputable priors sharpened into Occam s razors
Computable priors sharpened into Occam s razors David R. Bickel To cite this version: David R. Bickel. Computable priors sharpened into Occam s razors. 2016. HAL Id: hal-01423673 https://hal.archives-ouvertes.fr/hal-01423673v2
More informationTropical Graph Signal Processing
Tropical Graph Signal Processing Vincent Gripon To cite this version: Vincent Gripon. Tropical Graph Signal Processing. 2017. HAL Id: hal-01527695 https://hal.archives-ouvertes.fr/hal-01527695v2
More informationOn the link between finite differences and derivatives of polynomials
On the lin between finite differences and derivatives of polynomials Kolosov Petro To cite this version: Kolosov Petro. On the lin between finite differences and derivatives of polynomials. 13 pages, 1
More informationThermodynamic form of the equation of motion for perfect fluids of grade n
Thermodynamic form of the equation of motion for perfect fluids of grade n Henri Gouin To cite this version: Henri Gouin. Thermodynamic form of the equation of motion for perfect fluids of grade n. Comptes
More informationChebyshev polynomials, quadratic surds and a variation of Pascal s triangle
Chebyshev polynomials, quadratic surds and a variation of Pascal s triangle Roland Bacher To cite this version: Roland Bacher. Chebyshev polynomials, quadratic surds and a variation of Pascal s triangle.
More informationAxiom of infinity and construction of N
Axiom of infinity and construction of N F Portal To cite this version: F Portal. Axiom of infinity and construction of N. 2015. HAL Id: hal-01162075 https://hal.archives-ouvertes.fr/hal-01162075 Submitted
More informationLow frequency resolvent estimates for long range perturbations of the Euclidean Laplacian
Low frequency resolvent estimates for long range perturbations of the Euclidean Laplacian Jean-Francois Bony, Dietrich Häfner To cite this version: Jean-Francois Bony, Dietrich Häfner. Low frequency resolvent
More informationThere are infinitely many twin primes 30n+11 and 30n+13, 30n+17 and 30n+19, 30n+29 and 30n+31
There are infinitely many twin primes 30n+11 and 30n+13, 30n+17 and 30n+19, 30n+29 and 30n+31 Sibiri Christian Bandre To cite this version: Sibiri Christian Bandre. There are infinitely many twin primes
More informationDissipative Systems Analysis and Control, Theory and Applications: Addendum/Erratum
Dissipative Systems Analysis and Control, Theory and Applications: Addendum/Erratum Bernard Brogliato To cite this version: Bernard Brogliato. Dissipative Systems Analysis and Control, Theory and Applications:
More informationFrom Unstructured 3D Point Clouds to Structured Knowledge - A Semantics Approach
From Unstructured 3D Point Clouds to Structured Knowledge - A Semantics Approach Christophe Cruz, Helmi Ben Hmida, Frank Boochs, Christophe Nicolle To cite this version: Christophe Cruz, Helmi Ben Hmida,
More informationCutwidth and degeneracy of graphs
Cutwidth and degeneracy of graphs Benoit Kloeckner To cite this version: Benoit Kloeckner. Cutwidth and degeneracy of graphs. IF_PREPUB. 2009. HAL Id: hal-00408210 https://hal.archives-ouvertes.fr/hal-00408210v1
More informationOn Symmetric Norm Inequalities And Hermitian Block-Matrices
On Symmetric Norm Inequalities And Hermitian lock-matrices Antoine Mhanna To cite this version: Antoine Mhanna On Symmetric Norm Inequalities And Hermitian lock-matrices 016 HAL Id: hal-0131860
More informationOn the longest path in a recursively partitionable graph
On the longest path in a recursively partitionable graph Julien Bensmail To cite this version: Julien Bensmail. On the longest path in a recursively partitionable graph. 2012. HAL Id:
More informationOn Poincare-Wirtinger inequalities in spaces of functions of bounded variation
On Poincare-Wirtinger inequalities in spaces of functions of bounded variation Maïtine Bergounioux To cite this version: Maïtine Bergounioux. On Poincare-Wirtinger inequalities in spaces of functions of
More informationA non-linear simulator written in C for orbital spacecraft rendezvous applications.
A non-linear simulator written in C for orbital spacecraft rendezvous applications. Paulo Ricardo Arantes Gilz To cite this version: Paulo Ricardo Arantes Gilz. A non-linear simulator written in C for
More informationTowards an active anechoic room
Towards an active anechoic room Dominique Habault, Philippe Herzog, Emmanuel Friot, Cédric Pinhède To cite this version: Dominique Habault, Philippe Herzog, Emmanuel Friot, Cédric Pinhède. Towards an active
More informationNorm Inequalities of Positive Semi-Definite Matrices
Norm Inequalities of Positive Semi-Definite Matrices Antoine Mhanna To cite this version: Antoine Mhanna Norm Inequalities of Positive Semi-Definite Matrices 15 HAL Id: hal-11844 https://halinriafr/hal-11844v1
More informationOn The Exact Solution of Newell-Whitehead-Segel Equation Using the Homotopy Perturbation Method
On The Exact Solution of Newell-Whitehead-Segel Equation Using the Homotopy Perturbation Method S. Salman Nourazar, Mohsen Soori, Akbar Nazari-Golshan To cite this version: S. Salman Nourazar, Mohsen Soori,
More informationOptimal operation of sublimation time of the freeze drying process by predictive control: Application of the Software
Optimal operation of sublimation time of the freeze drying process by predictive control: Application of the MPC@CB Software Nawal Daraoui, Pascal Dufour, Hassan Hammouri, Aurélie Hottot To cite this version:
More informationSelf-dual skew codes and factorization of skew polynomials
Self-dual skew codes and factorization of skew polynomials Delphine Boucher, Félix Ulmer To cite this version: Delphine Boucher, Félix Ulmer. Self-dual skew codes and factorization of skew polynomials.
More informationCharacterization of Equilibrium Paths in a Two-Sector Economy with CES Production Functions and Sector-Specific Externality
Characterization of Equilibrium Paths in a Two-Sector Economy with CES Production Functions and Sector-Specific Externality Miki Matsuo, Kazuo Nishimura, Tomoya Sakagami, Alain Venditti To cite this version:
More informationSome tight polynomial-exponential lower bounds for an exponential function
Some tight polynomial-exponential lower bounds for an exponential function Christophe Chesneau To cite this version: Christophe Chesneau. Some tight polynomial-exponential lower bounds for an exponential
More informationNumerical Exploration of the Compacted Associated Stirling Numbers
Numerical Exploration of the Compacted Associated Stirling Numbers Khaled Ben Letaïef To cite this version: Khaled Ben Letaïef. Numerical Exploration of the Compacted Associated Stirling Numbers. 2017.
More informationThe Mahler measure of trinomials of height 1
The Mahler measure of trinomials of height 1 Valérie Flammang To cite this version: Valérie Flammang. The Mahler measure of trinomials of height 1. Journal of the Australian Mathematical Society 14 9 pp.1-4.
More informationOn the Griesmer bound for nonlinear codes
On the Griesmer bound for nonlinear codes Emanuele Bellini, Alessio Meneghetti To cite this version: Emanuele Bellini, Alessio Meneghetti. On the Griesmer bound for nonlinear codes. Pascale Charpin, Nicolas
More informationNodal and divergence-conforming boundary-element methods applied to electromagnetic scattering problems
Nodal and divergence-conforming boundary-element methods applied to electromagnetic scattering problems M. Afonso, Joao Vasconcelos, Renato Mesquita, Christian Vollaire, Laurent Nicolas To cite this version:
More informationThe core of voting games: a partition approach
The core of voting games: a partition approach Aymeric Lardon To cite this version: Aymeric Lardon. The core of voting games: a partition approach. International Game Theory Review, World Scientific Publishing,
More informationVoltage Stability of Multiple Distributed Generators in Distribution Networks
oltage Stability of Multiple Distributed Generators in Distribution Networks Andi Wang, Chongxin Liu, Hervé Guéguen, Zhenquan Sun To cite this version: Andi Wang, Chongxin Liu, Hervé Guéguen, Zhenquan
More informationTheoretical calculation of the power of wind turbine or tidal turbine
Theoretical calculation of the power of wind turbine or tidal turbine Pierre Lecanu, Joel Breard, Dominique Mouazé To cite this version: Pierre Lecanu, Joel Breard, Dominique Mouazé. Theoretical calculation
More informationOn Politics and Argumentation
On Politics and Argumentation Maria Boritchev To cite this version: Maria Boritchev. On Politics and Argumentation. MALOTEC, Mar 2017, Nancy, France. 2017. HAL Id: hal-01666416 https://hal.archives-ouvertes.fr/hal-01666416
More informationA non-commutative algorithm for multiplying (7 7) matrices using 250 multiplications
A non-commutative algorithm for multiplying (7 7) matrices using 250 multiplications Alexandre Sedoglavic To cite this version: Alexandre Sedoglavic. A non-commutative algorithm for multiplying (7 7) matrices
More informationNumerical Modeling of Eddy Current Nondestructive Evaluation of Ferromagnetic Tubes via an Integral. Equation Approach
Numerical Modeling of Eddy Current Nondestructive Evaluation of Ferromagnetic Tubes via an Integral Equation Approach Anastassios Skarlatos, Grégoire Pichenot, Dominique Lesselier, Marc Lambert, Bernard
More informationContinuous-discrete time observer design for Lipschitz systems with sampled measurements
JOURNAL OF IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1 Continuous-discrete time observer design for Lipschitz systems with sampled measurements Thach Ngoc Dinh, Vincent Andrieu, Madiha Nadri, Ulysse Serres
More informationBERGE VAISMAN AND NASH EQUILIBRIA: TRANSFORMATION OF GAMES
BERGE VAISMAN AND NASH EQUILIBRIA: TRANSFORMATION OF GAMES Antonin Pottier, Rabia Nessah To cite this version: Antonin Pottier, Rabia Nessah. BERGE VAISMAN AND NASH EQUILIBRIA: TRANS- FORMATION OF GAMES.
More informationA simple kinetic equation of swarm formation: blow up and global existence
A simple kinetic equation of swarm formation: blow up and global existence Miroslaw Lachowicz, Henryk Leszczyński, Martin Parisot To cite this version: Miroslaw Lachowicz, Henryk Leszczyński, Martin Parisot.
More informationThe beam-gas method for luminosity measurement at LHCb
The beam-gas method for luminosity measurement at LHCb P. Hopchev To cite this version: P. Hopchev. The beam-gas method for luminosity measurement at LHCb. XLVth Rencontres de Moriond: Electroweak Interactions
More informationClimbing discrepancy search for flowshop and jobshop scheduling with time-lags
Climbing discrepancy search for flowshop and jobshop scheduling with time-lags Wafa Karoui, Marie-José Huguet, Pierre Lopez, Mohamed Haouari To cite this version: Wafa Karoui, Marie-José Huguet, Pierre
More informationParticle-in-cell simulations of high energy electron production by intense laser pulses in underdense plasmas
Particle-in-cell simulations of high energy electron production by intense laser pulses in underdense plasmas Susumu Kato, Eisuke Miura, Mitsumori Tanimoto, Masahiro Adachi, Kazuyoshi Koyama To cite this
More informationON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS
ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS Abdelhafid Younsi To cite this version: Abdelhafid Younsi. ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS. 4 pages. 212. HAL Id:
More informationAll Associated Stirling Numbers are Arithmetical Triangles
All Associated Stirling Numbers are Arithmetical Triangles Khaled Ben Letaïef To cite this version: Khaled Ben Letaïef. All Associated Stirling Numbers are Arithmetical Triangles. 2017.
More informationSTATISTICAL ENERGY ANALYSIS: CORRELATION BETWEEN DIFFUSE FIELD AND ENERGY EQUIPARTITION
STATISTICAL ENERGY ANALYSIS: CORRELATION BETWEEN DIFFUSE FIELD AND ENERGY EQUIPARTITION Thibault Lafont, Alain Le Bot, Nicolas Totaro To cite this version: Thibault Lafont, Alain Le Bot, Nicolas Totaro.
More informationOptimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients
Optimized Schwarz Methods for Maxwell Equations with Discontinuous Coefficients Victorita Dolean, Martin Gander, Erwin Veneros To cite this version: Victorita Dolean, Martin Gander, Erwin Veneros. Optimized
More informationA proximal approach to the inversion of ill-conditioned matrices
A proximal approach to the inversion of ill-conditioned matrices Pierre Maréchal, Aude Rondepierre To cite this version: Pierre Maréchal, Aude Rondepierre. A proximal approach to the inversion of ill-conditioned
More informationQuantum efficiency and metastable lifetime measurements in ruby ( Cr 3+ : Al2O3) via lock-in rate-window photothermal radiometry
Quantum efficiency and metastable lifetime measurements in ruby ( Cr 3+ : Al2O3) via lock-in rate-window photothermal radiometry A. Mandelis, Z. Chen, R. Bleiss To cite this version: A. Mandelis, Z. Chen,
More informationA Simple Model for Cavitation with Non-condensable Gases
A Simple Model for Cavitation with Non-condensable Gases Mathieu Bachmann, Siegfried Müller, Philippe Helluy, Hélène Mathis To cite this version: Mathieu Bachmann, Siegfried Müller, Philippe Helluy, Hélène
More informationUnderstanding SVM (and associated kernel machines) through the development of a Matlab toolbox
Understanding SVM (and associated kernel machines) through the development of a Matlab toolbox Stephane Canu To cite this version: Stephane Canu. Understanding SVM (and associated kernel machines) through
More informationBasic concepts and models in continuum damage mechanics
Basic concepts and models in continuum damage mechanics Djimedo Kondo, Hélène Welemane, Fabrice Cormery To cite this version: Djimedo Kondo, Hélène Welemane, Fabrice Cormery. Basic concepts and models
More informationInteractions of an eddy current sensor and a multilayered structure
Interactions of an eddy current sensor and a multilayered structure Thanh Long Cung, Pierre-Yves Joubert, Eric Vourc H, Pascal Larzabal To cite this version: Thanh Long Cung, Pierre-Yves Joubert, Eric
More informationMultiple sensor fault detection in heat exchanger system
Multiple sensor fault detection in heat exchanger system Abdel Aïtouche, Didier Maquin, Frédéric Busson To cite this version: Abdel Aïtouche, Didier Maquin, Frédéric Busson. Multiple sensor fault detection
More informationComparison of Harmonic, Geometric and Arithmetic means for change detection in SAR time series
Comparison of Harmonic, Geometric and Arithmetic means for change detection in SAR time series Guillaume Quin, Béatrice Pinel-Puysségur, Jean-Marie Nicolas To cite this version: Guillaume Quin, Béatrice
More informationThe Learner s Dictionary and the Sciences:
The Learner s Dictionary and the Sciences: Geoffrey Williams To cite this version: Geoffrey Williams. The Learner s Dictionary and the Sciences:: Mismatch or no match?. Corpora, Language, Teaching, and
More informationEddy-Current Effects in Circuit Breakers During Arc Displacement Phase
Eddy-Current Effects in Circuit Breakers During Arc Displacement Phase Olivier Chadebec, Gerard Meunier, V. Mazauric, Yann Le Floch, Patrice Labie To cite this version: Olivier Chadebec, Gerard Meunier,
More informationWidely Linear Estimation with Complex Data
Widely Linear Estimation with Complex Data Bernard Picinbono, Pascal Chevalier To cite this version: Bernard Picinbono, Pascal Chevalier. Widely Linear Estimation with Complex Data. IEEE Transactions on
More informationA numerical analysis of chaos in the double pendulum
A numerical analysis of chaos in the double pendulum Tomasz Stachowiak, Toshio Okada To cite this version: Tomasz Stachowiak, Toshio Okada. A numerical analysis of chaos in the double pendulum. Chaos,
More informationReduced Models (and control) of in-situ decontamination of large water resources
Reduced Models (and control) of in-situ decontamination of large water resources Antoine Rousseau, Alain Rapaport To cite this version: Antoine Rousseau, Alain Rapaport. Reduced Models (and control) of
More informationA Study of the Regular Pentagon with a Classic Geometric Approach
A Study of the Regular Pentagon with a Classic Geometric Approach Amelia Carolina Sparavigna, Mauro Maria Baldi To cite this version: Amelia Carolina Sparavigna, Mauro Maria Baldi. A Study of the Regular
More informationEFFECT OF THE ONE-DIMENSIONAL STRUCTURE ON THE ENERGY TRANSFER IN Li6Gd (BO3)3
EFFECT OF THE ONE-DIMENSIONAL STRUCTURE ON THE ENERGY TRANSFER IN Li6Gd (BO3)3 C. Garapon, B. Jacquier, Y. Salem, R. Moncorge To cite this version: C. Garapon, B. Jacquier, Y. Salem, R. Moncorge. EFFECT
More informationTrajectory Optimization for Differential Flat Systems
Trajectory Optimization for Differential Flat Systems Kahina Louadj, Benjamas Panomruttanarug, Alexre Carlos Brao-Ramos, Felix Antonio Claudio Mora-Camino To cite this version: Kahina Louadj, Benjamas
More informationWater Vapour Effects in Mass Measurement
Water Vapour Effects in Mass Measurement N.-E. Khélifa To cite this version: N.-E. Khélifa. Water Vapour Effects in Mass Measurement. Measurement. Water Vapour Effects in Mass Measurement, May 2007, Smolenice,
More information