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## Author manuscript (2009)

### Citations

1278 |
Theory of reproducing kernels
- Aronszajn
- 1950
(Show Context)
Citation Context ...is important because a delay may not in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 10 be tolerable in some real time applications1. We shall also show that for a fixed modelling order N , the abscence of delay undergoes a performance loss. Finally, the question of the “best” choice of the barycentric coordinates in (14) will be approached. To begin, let us recall some well-known facts. Consider the subspace of L2([0, 1]), defined by Hq = span { P {κ,µ} 0 (t), P {κ,µ} 1 (t), · · · , P {κ,µ}q (t) } . (25) Equipped with the inner product (19), Hq is clearly a reproducing kernel Hilbert space [4], [3], [32], with the reproducing kernel Kq(τ, t) = q∑ i=0 P {κ,µ} i (τ)P {κ,µ} i (t) ‖P {κ,µ}i ‖2 . (26) The reproducing property implies that for any function f defined on [0, 1], we have 〈Kq(τ, ·), f(·)〉 = fq(τ), where fq stands for the orthogonal projection of f on Hq. In the rest of the paper, we will set q = N − n and assume that q > 0. Theorem 2 Assume that q 6 κ + n and let x (n) LS,q(t) 4 = q∑ i=0 〈P {κ,µ}i (τ), x(n)(Tτ)〉 ‖P {κ,µ}i ‖2 P {κ,µ} i (t/T ), (27) denote the least-squares qth order polynomial approximation of x(n)(·) in the interval IT0− = [0, T ]. Then x(n)(0; κ, µ;N) is g... |

258 |
Time-delay systems: an overview of some recent advances and open problems
- Richard
(Show Context)
Citation Context ...tion. Remark 1 This is for the anti-causal variant. To obtain the causal counterpart, just replace y(t) for t ∈ [0, T ] by −y(T − t). We immediately deduce from the above theorem that the corresponding causal (κ, µ)-AND is delay-free: x(n)(t; κ, µ;N) ≈ x(n)(t), t > T. (29) Now, since x(n)LS,q(Tt) is the orthogonal projection of x(n)(Tt), t ∈ [0, 1] on Hq, we may rewrite (28) as x(n)(0; κ, µ; N) = 〈Kq(0, t), x(n)(Tt)〉+ e$(0; κ, µ; N) = q∑ `=0 λ` x (n)(0; κ + q − `, µ + `), (30) 1 For instance, it is known that introducing delayed signals in a control loop of a system tends to destabilize it [30], [33]. in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 11 where we recall (14) (for q 6 κ + n). As already suggested, we henceforth allow the barycentric coordinates λ`, ` = 0, . . . , q to belong to R instead of being restricted in Q. We obtain the next result. Proposition 2 Let κ, µ, n and q be four positive integers as above. For any ξ ∈ [0, 1], there exists a unique set of real coordinates λ`(ξ) ∈ R, ` = 0, · · · , q such that q∑ `=0 λ`(ξ) x(n)(0; κ`, µ`) = 〈Kq(ξ, t), x(n)(Tt)〉+ eξ$(0; κ, µ;n + q), (31) for some noise contribution eξ$(0; κ, µ; n + q). Moreover, these coordinates satisf... |

112 |
Concentration Inequalities and Model Selection
- Massart
- 2007
(Show Context)
Citation Context ... case, the approximation (33) with ξ = ξ{q+1}i , 1 6 i 6 q + 1 would be exact for all polynomial signals up to degree N + 1 although this approximation is based on a Taylor expansion of order N . This has an important implication as shown next. in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 12 3.2.4 Delay and precision To continue the discussion, we establish in Proposition 3 below that the quality of the estimation may be significantly improved by admitting a delay. When relating this result with the last remark of the preceding subsection, we somehow recover the paradigm of model section [24]: a simple and approximate model may outperform a precise and more complex one. To begin, let us expend x(n)(t + Tu), for each fixed t, as x(n)(t + Tu) = ∑ ai(t)P {κ,µ} i (u), u ∈ [0, 1], (35) and set ξ{q}i , i = 1, . . . , q for the zeros of P {κ,µ} q , in increasing order. Suppose that q = N − n > 1. For some given i, consider first x(n) ξ {q} i (t; κ, µ; N − 1), the time-delayed τ = Tξ{q}i estimate (34), based on an (N − 1)-order truncated Taylor expansion. Using (35) and (34), the estimation error due to the truncation reads, at each time t, as: e ξ {q} i RN−1(t;κ, µ; n+q−1) = aq+1(t)P {κ... |

86 |
Higher-order sliding modes, differentiation and outputfeedback control
- Levant
- 2003
(Show Context)
Citation Context ... si on 1 - 7 Se p 20 08 Author manuscript, published in "Numerical Algorithms 50, 4 (2009) 439-467" DOI : 10.1007/s11075-008-9236-1 2 1 Introduction Numerical differentiation, i.e., the derivative estimation of noisy time signals, is a longstanding difficult ill-posed problem in numerical analysis and in signal processing and control. It has attracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been ... |

84 |
An algebraic framework for linear identification
- Fliess, Sira-Ramı́rez
(Show Context)
Citation Context ...designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation therefore follows by varying τ . The general ideas leading to this estimation are exposed in section 2. As in the classical approaches, the starting point is an order N > n tr... |

43 | Fliess: A revised look at numerical differentiation with an application to nonlinear feedback control
- Mboup, Join, et al.
- 2007
(Show Context)
Citation Context ...gorithm is already designed for this task. Then, for any τ > 0, it is clear that applying the same algorithm on the new signal observation yτ (t) 4 = Heaviside(t)y(t + τ), will yield an estimate for x(n)(τ). We thus lay the focus on how to estimate x(n)(0) in (4). 2.1.1 Annihilators Of course it is also possible to estimate all the coefficients x(i)(0), i = 0, . . . , N simultaneously. However, not only the coefficients x(i)(0), i 6= n are not necessary for the estimation of x(n)(t) as explaned above, but also simultaneous estimation is more sensitive to noise and numerical computation errors [26]. These drawbacks are avoided in the proposed approach. All the terms sN−ix(i)(0) in (4) with i 6= n, are consequently considered as undesired perturbations which we proceed to annihilate. For this, it suffices to find a linear differential operator, i.e. Π = ∑ finite %`(s) d` ds` , %`(s) ∈ C(s), (5) satisfying ΠxN = %(s)x(n)(0), (6) for some rational function %(s) ∈ C(s). Such a linear differential operator, subsequently called an annihilator for x(n)(0), obviously exists and is not unique. It is also clear that to each annihilator Π, there is a unique %(s) ∈ C(s) such that (6) holds. We sha... |

39 | Questioning some paradigms of signal processing via concrete examples
- Fliess, Mboup, et al.
- 2003
(Show Context)
Citation Context ...cal integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation therefore follows by varying τ . The general ideas leading to this estimation are exposed in section 2. As in the classical approaches, the starting point is an order N > n truncation of the... |

30 |
Bounds on the extreme zeros of orthogonal polynomials
- HIsmail, Li
- 1992
(Show Context)
Citation Context .... (18) In all the sequel, ‖ · ‖ will denote the norm induced by the inner product defined, for two real valued functions f and g, by: 〈f, g〉{κ,µ} = ∫ 1 0 f(t)wκ,µ(t)g(t)dt. (19) The short hand notation 〈·, ·〉 will be used wherever there is no possible confusion. Upon noting that P {κ,µ}0 (t) ≡ 1 and ‖P {κ,µ}0 ‖2 = (µ + n)!(κ + n)! (µ + κ + 2n + 1)! = 1 γκ,µ,n , (20) we have the following: Proposition 1 Let x(n) LS,1(t) denote the first order least-squares polynomial approximation of x(n)(t) in the interval [0, T ]. Then x(n)(0; κ, µ) is given by x(n)(0; κ, µ) = x(n) LS,1(Tξ1) + e$(0; κ, µ), (21) where ξ1 = κ + n + 1 µ + κ + 2(n + 1) (22) is the root of P {κ,µ}1 (t) and e$(0; κ, µ) is the noise contribution as given by the second term in the right hand side of (17). in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 9 3.2.2 Time-delayed derivative estimation For a given τ > 0, let us substitute, in all the preceding developments, y(t) by [Heaviside(τ)y(τ + t)]. This simply amounts to moving the time origin from 0 to τ . As a result, we obtain x(n)(τ ;κ, µ), the estimate of x(n)(τ). Observe however that, the corresponding estimation is anti-causal, since the estimate at time τ is based... |

30 |
Operational Calculus.
- Mikusinski
- 1959
(Show Context)
Citation Context ...in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation therefore follows by varying τ . The general ideas leading to this estimation are exposed in section 2. As in the classical approaches, the starting point is an order N > n truncation of the Taylor series expansion of the signal. One of the key feature of the proposed approach is to operate in the operational calculus domain [36], [27,28]. Therein, an extensive use of differential elimination and a series of algebraic manipulations yield, back in the time domain, an explicit expression for the estimate of d nx(t) dtn |t=τ as an integral operator of the noisy observation within a short time interval [τ, τ +T ]. The differential algebraic manipulations may be chosen in such a way that the time domain integral operators specialise to iterated integrals. This corresponds to a family of pointwise derivative estimators introduced in section 3. This section contains the main contributions of the paper. Therein, we establish a direct ... |

24 |
Compression différentielle de transitoires bruités
- Fliess, Join, et al.
- 2004
(Show Context)
Citation Context ... been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation therefore follows by varying τ . The general ideas leading to this estimation are exposed in section 2. As in the classical approaches, the starting point is an order N > n truncation of the Taylor series expansion of the signal. One of the key feature of the propose... |

21 |
H.K.: Discrete-time implementation of high-gain observers for numerical differentiation.
- Dabroom, Khalil
- 1999
(Show Context)
Citation Context ...n ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 Author manuscript, published in "Numerical Algorithms 50, 4 (2009) 439-467" DOI : 10.1007/s11075-008-9236-1 2 1 Introduction Numerical differentiation, i.e., the derivative estimation of noisy time signals, is a longstanding difficult ill-posed problem in numerical analysis and in signal processing and control. It has attracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these i... |

19 | On Numerical Differentiation Algorithms for Nonlinear Estimation
- Braci, Diop
- 2003
(Show Context)
Citation Context ...esign may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12... |

19 | Control via state estimations of some nonlinear systems,”
- Fliess, Sira-Ramirez
- 2004
(Show Context)
Citation Context ... been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation therefore follows by varying τ . The general ideas leading to this estimation are exposed in section 2. As in the classical approaches, the starting point is an order N > n truncation of the Taylor series expansion of the signal. One of the key feature of the propose... |

18 |
Parameter estimation via differential algebra and operational calculus.
- Mboup
- 2007
(Show Context)
Citation Context ...rator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation therefore follows by varying τ . The general ideas leading to this estimation are exposed in section 2. As in the classical approaches, the starting point is an order N > n truncation of the Taylor s... |

17 |
Robust exponential stabilization for systems with time-varying delays,
- Seuret, Dambrine, et al.
- 2004
(Show Context)
Citation Context ...Remark 1 This is for the anti-causal variant. To obtain the causal counterpart, just replace y(t) for t ∈ [0, T ] by −y(T − t). We immediately deduce from the above theorem that the corresponding causal (κ, µ)-AND is delay-free: x(n)(t; κ, µ;N) ≈ x(n)(t), t > T. (29) Now, since x(n)LS,q(Tt) is the orthogonal projection of x(n)(Tt), t ∈ [0, 1] on Hq, we may rewrite (28) as x(n)(0; κ, µ; N) = 〈Kq(0, t), x(n)(Tt)〉+ e$(0; κ, µ; N) = q∑ `=0 λ` x (n)(0; κ + q − `, µ + `), (30) 1 For instance, it is known that introducing delayed signals in a control loop of a system tends to destabilize it [30], [33]. in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 11 where we recall (14) (for q 6 κ + n). As already suggested, we henceforth allow the barycentric coordinates λ`, ` = 0, . . . , q to belong to R instead of being restricted in Q. We obtain the next result. Proposition 2 Let κ, µ, n and q be four positive integers as above. For any ξ ∈ [0, 1], there exists a unique set of real coordinates λ`(ξ) ∈ R, ` = 0, · · · , q such that q∑ `=0 λ`(ξ) x(n)(0; κ`, µ`) = 〈Kq(ξ, t), x(n)(Tt)〉+ eξ$(0; κ, µ;n + q), (31) for some noise contribution eξ$(0; κ, µ; n + q). Moreover, these coordinates satisfy q∑ `... |

16 |
Time-Varying High-Gain Observer for Numerical Differentiations,”
- Chitour, Y
- 2002
(Show Context)
Citation Context ... -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 Author manuscript, published in "Numerical Algorithms 50, 4 (2009) 439-467" DOI : 10.1007/s11075-008-9236-1 2 1 Introduction Numerical differentiation, i.e., the derivative estimation of noisy time signals, is a longstanding difficult ill-posed problem in numerical analysis and in signal processing and control. It has attracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these intere... |

15 |
Analyse et representation de signaux transitoires : application a la compression, au debruitage et a la detection de ruptures. In:
- Fliess, Join, et al.
- 2005
(Show Context)
Citation Context ...by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation therefore follows by varying τ . The general ideas leading to this estimation are exposed in section 2. As in the classical approaches, the starting point is an order N > n truncation of the Taylor series expansion of the signal. One of the key feature of the proposed approach is to op... |

15 | A numerical procedure for filtering and efficient high-order signal differentiation
- Ibrir, Diop
- 2004
(Show Context)
Citation Context ...th order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estima... |

13 |
B.Y.: A simple improved velocity estimation for low-speed regions based on position measurements only.
- Su, Zheng, et al.
- 2006
(Show Context)
Citation Context ... si on 1 - 7 Se p 20 08 Author manuscript, published in "Numerical Algorithms 50, 4 (2009) 439-467" DOI : 10.1007/s11075-008-9236-1 2 1 Introduction Numerical differentiation, i.e., the derivative estimation of noisy time signals, is a longstanding difficult ill-posed problem in numerical analysis and in signal processing and control. It has attracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been ... |

12 |
Algorithme de Schur, espaces à noyau reproduisant et théorie des systèmes
- Alpay
- 1998
(Show Context)
Citation Context ...portant because a delay may not in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 10 be tolerable in some real time applications1. We shall also show that for a fixed modelling order N , the abscence of delay undergoes a performance loss. Finally, the question of the “best” choice of the barycentric coordinates in (14) will be approached. To begin, let us recall some well-known facts. Consider the subspace of L2([0, 1]), defined by Hq = span { P {κ,µ} 0 (t), P {κ,µ} 1 (t), · · · , P {κ,µ}q (t) } . (25) Equipped with the inner product (19), Hq is clearly a reproducing kernel Hilbert space [4], [3], [32], with the reproducing kernel Kq(τ, t) = q∑ i=0 P {κ,µ} i (τ)P {κ,µ} i (t) ‖P {κ,µ}i ‖2 . (26) The reproducing property implies that for any function f defined on [0, 1], we have 〈Kq(τ, ·), f(·)〉 = fq(τ), where fq stands for the orthogonal projection of f on Hq. In the rest of the paper, we will set q = N − n and assume that q > 0. Theorem 2 Assume that q 6 κ + n and let x (n) LS,q(t) 4 = q∑ i=0 〈P {κ,µ}i (τ), x(n)(Tτ)〉 ‖P {κ,µ}i ‖2 P {κ,µ} i (t/T ), (27) denote the least-squares qth order polynomial approximation of x(n)(·) in the interval IT0− = [0, T ]. Then x(n)(0; κ, µ;N) is given ... |

12 |
A.: Estimation des derivees d’un signal multidimensionnel avec applications aux images et aux videos. In:
- Fliess, Join, et al.
- 2005
(Show Context)
Citation Context ...by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation therefore follows by varying τ . The general ideas leading to this estimation are exposed in section 2. As in the classical approaches, the starting point is an order N > n truncation of the Taylor series expansion of the signal. One of the key feature of the proposed approach is to op... |

11 |
Online exact differentiation and notion of asymptotic algebraic observers.
- Ibrir
- 2003
(Show Context)
Citation Context ...3 19 24 0, v er si on 1 - 7 Se p 20 08 Author manuscript, published in "Numerical Algorithms 50, 4 (2009) 439-467" DOI : 10.1007/s11075-008-9236-1 2 1 Introduction Numerical differentiation, i.e., the derivative estimation of noisy time signals, is a longstanding difficult ill-posed problem in numerical analysis and in signal processing and control. It has attracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting ... |

10 |
A class of second-order integrators and low-pass differentiators.
- Al-Alaoui
- 1995
(Show Context)
Citation Context ...ttracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised throu... |

9 |
Linear time-derivatives trackers.
- Ibrir
- 2004
(Show Context)
Citation Context ... si on 1 - 7 Se p 20 08 Author manuscript, published in "Numerical Algorithms 50, 4 (2009) 439-467" DOI : 10.1007/s11075-008-9236-1 2 1 Introduction Numerical differentiation, i.e., the derivative estimation of noisy time signals, is a longstanding difficult ill-posed problem in numerical analysis and in signal processing and control. It has attracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been ... |

9 |
Orthogonal polynomials, 3rd edn.
- Szego
- 1967
(Show Context)
Citation Context ...sing a smaller estimation time, T , will tend to reduce |eRN (0)|. But at the same time, a large T is required to filter out the noise showing that the choice for T should obey a compromise. in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 6 3 Pointwise derivative estimation Now that we have fixed the notations and the guidelines of our methodology, we investigate in this section some detailed properties and performance of a class of pointwise derivative estimators. These will be derived from a particular family of annihilators. And, as we shall shortly see, the Jacobi orthogonal polynomials [35] are inherently connected with these estimators. A least squares interpretation then naturally follows [25], [26] and this leads to one of the main contribution of the paper: the numerical differentiation is as efficient as an appropriately chosen time delay is tolerated. 3.1 Purely integral estimators A linear differential operator Π is said to be in finite-integral form if, and only if, each %`(s) in (5) is in the form %`(s) = 1sH`( 1 s ) for some polynomial H`. We consider the family of annihilators defined below. Lemma 2 For any κ, µ ∈ N, the differential operator ΠN,nκ,µ = 1 sN+µ+1 dn+κ d... |

8 |
Signals and Systems, 2nd edn
- Haykin, Veen
- 2002
(Show Context)
Citation Context ...formance significantly degrades as compared to the preceding results although the signal model is more precise. If we now relax the delay-free constraint, it becomes possible to take advantage of the more flexible second order model for the signal. This is illustrated in the following simulation (see Figure 4), where we keep the same settings for T , but with κ = µ = 1 now. The solid line curve in Figure 4 represents the estimates obtained as in Proposition 2, with the barycentric coordinates given by equation (68) where ξ = ξ{2}2 is the largest root of P {κ,µ} 2 . For comparison 4 See, e.g., [17] for this well known concept in signal processing in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 17 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 −8 −6 −4 −2 0 2 4 6 Time Fig. 2 Estimation of the signal derivative: minimal (0, 0)-AND. purpose, the curve has been shifted by a number of samples corresponding to the delay ξ{2}2 T . The same type of behaviors are also observed for the esti0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 −8 −6 −4 −2 0 2 4 6 8 Time Fig. 3 Non minimal, delay-free derivative estimation. mation of the higher order derivatives. As an illustration, we give in figure 5 and fi... |

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L.B.: Approximating noncausal IIR digital filters having arbitrary poles, including new Hilbert transformer designs, via forward/backward block recursion.
- Rader, Jackson
- 2006
(Show Context)
Citation Context ...., the derivative estimation of noisy time signals, is a longstanding difficult ill-posed problem in numerical analysis and in signal processing and control. It has attracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization ste... |

7 |
Numerical differentiation and parameter estimation in higher-order linear stochastic systems.
- Duncan, Mandl, et al.
- 1996
(Show Context)
Citation Context ...er differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation ... |

7 |
Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics. Longman Scientic & Technical,
- Saitoh
- 1988
(Show Context)
Citation Context ...nt because a delay may not in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 10 be tolerable in some real time applications1. We shall also show that for a fixed modelling order N , the abscence of delay undergoes a performance loss. Finally, the question of the “best” choice of the barycentric coordinates in (14) will be approached. To begin, let us recall some well-known facts. Consider the subspace of L2([0, 1]), defined by Hq = span { P {κ,µ} 0 (t), P {κ,µ} 1 (t), · · · , P {κ,µ}q (t) } . (25) Equipped with the inner product (19), Hq is clearly a reproducing kernel Hilbert space [4], [3], [32], with the reproducing kernel Kq(τ, t) = q∑ i=0 P {κ,µ} i (τ)P {κ,µ} i (t) ‖P {κ,µ}i ‖2 . (26) The reproducing property implies that for any function f defined on [0, 1], we have 〈Kq(τ, ·), f(·)〉 = fq(τ), where fq stands for the orthogonal projection of f on Hq. In the rest of the paper, we will set q = N − n and assume that q > 0. Theorem 2 Assume that q 6 κ + n and let x (n) LS,q(t) 4 = q∑ i=0 〈P {κ,µ}i (τ), x(n)(Tτ)〉 ‖P {κ,µ}i ‖2 P {κ,µ} i (t/T ), (27) denote the least-squares qth order polynomial approximation of x(n)(·) in the interval IT0− = [0, T ]. Then x(n)(0; κ, µ;N) is given by x(... |

6 |
Design of high-order digital differentiators using L1 error criteria.
- Chen, Lee
- 1995
(Show Context)
Citation Context ... derivative estimation of noisy time signals, is a longstanding difficult ill-posed problem in numerical analysis and in signal processing and control. It has attracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, th... |

4 |
C.T.: Digital signal processing.
- Roberts, Mullis
- 1987
(Show Context)
Citation Context ...n, i.e., the derivative estimation of noisy time signals, is a longstanding difficult ill-posed problem in numerical analysis and in signal processing and control. It has attracted a lot of attention due to its importance in many fields of engineering and applied mathematics. A number of different approaches have been proposed. Methods based on observer design may be found in the control literature [7], [6], [18]. See also [19,22,34] for other approaches from the control litterature. In signal processing, it is very common to cast the problem in terms of frequency domain digital filter design [31], [29], [5]. This is motivated by the observation that an ideal nth order differentiator has a frequency response of magnitude ωn. Another interesting approach, due to [2], consists in inverting the minimum phase transfer function of a properly designed numerical integrator to obtain a digital differentiator. Similar ideas have also been presented by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularizati... |

2 |
Sira Ramırez, H.: Closed-loop fault-tolerant control for uncertain nonlinear systems.
- Fliess, J
- 2005
(Show Context)
Citation Context ...by [8], in the continuous-time context. All these interesting approaches have been developped as candidate alternatives to the very classical one, based on least-squares polynomial fitting or (spline) interpolation. When completed with a regularization step, this classical approach may be very efficient (see i.e. [20], [9]) in off-line applications. In this paper, the numerical differentiation problem is revised through the algebraic framework of parameter estimation initially presented in [15] (see also [14] and [25]). Our purpose is to improve a new approach which started in [11,16], and in [13,12,10], for solving various questions in control and in signal and image processing. Given a smooth signal x and an order n, a key point of our approach is to consider d nx(t) dtn |t=τ , for each fixed τ > 0, as a single parameter to be estimated from a noisy obervation of the signal. A pointwise derivative estimation therefore follows by varying τ . The general ideas leading to this estimation are exposed in section 2. As in the classical approaches, the starting point is an order N > n truncation of the Taylor series expansion of the signal. One of the key feature of the proposed approach is to op... |

2 |
Polynomials, 2nd edn.
- Lorentz
- 1986
(Show Context)
Citation Context ... , q This shows that if the relation (28) is valid for q > 1, then it must also be valid for q + 1. Since it has been established for q = 1, we conclude that (28) holds for all q > 1. ut Proof (Proof of Proposition 2) As for the equation (58) in the previous proof, it is easy to see that the linear combination Pq `=0 λ`x (n)(0; κ`, µ`) is given by qX `=0 λ`x (n)(0; κ`, µ`) = Z 1 0 B(τ)w{κ,µ}(τ)y(n)(Tτ)dτ = 〈B(τ), y(n)(Tτ)〉. (64) with B(τ) = qX `=0 λ` γκ`,µ`,n τ q−`(1− τ)` = qX `=0 λ` φ` b`,q(τ), (65) where b`,q(τ) 4 = “q ` ” τq−`(1− τ)`, ` = 0, . . . , q, are the Bernstein basis polynomials [23] of degree q and φ` 4 = γκ`,µ`,n“ q ` ” . Since the set ˘ b`,q(τ), ` = 0, . . . , q ¯ also forms a basis for the subspace Hq (see (25)), we may equate Kq(ξ, τ) = qX `=0 qX m=0 bm,q(ξ) (B−1)m,` b`,q(τ). (66) in ria -0 03 19 24 0, v er si on 1 - 7 Se p 20 08 24 Here, the matrix B stands for the Gramian of the Bernstein polynomials b`,q(·), with respect to the inner product of Hq . Now, observe that the first part of the proposition claims that, for each ξ ∈ [0, 1], the equation B(τ) = Kq(ξ, τ) (67) admits a unique solution for (λ0, . . . , λq). With the notations λ = 2 64 λ0 . .. λq 3 75 ; bq(τ)... |