Frequency domain curve fitting with maximum amplitude criterion and guaranteed stability

Hakvoort, R.G.; Hof, Paul M.J. Van den

doi:10.1080/00207179408921496

Cited by 30 publications

(15 citation statements)

References 9 publications

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“…Second, it is explicitly indicated in earlier overviews of the motion control field, e.g., already in Steinbuch and Norg (1998, Section 4.2, limitations of tools), that modeling tools are not readily available. Third, besides the fact that many system identification approaches with different criteria have been developed, a significant number of results addressing issues with the numerical implementation of these methods have been developed, including frequency scaling (Pintelon and Kollár, 2005), amplitude scaling (Hakvoort and Van den Hof, 1994), the use of classical orthonormal polynomials and orthogonal rational functions (Heuberger et al, 2005, Section 3.1), (Ninness et al, 2000), (Ninness and Hjalmarsson, 2001), and more recently the use of orthonormal basis functions with respect to a discrete databased inner product (Bultheel et al, 2005), (Van Herpen et al, 2014). Related numerical issues are also seen in subspace identification, see e.g., Verdult et al (2002) and Chiuso and Giorgio (2004).…”

Section: Introductionmentioning

confidence: 99%

Identification of High-Tech Motion Systems: An Active Vibration Isolation Benchmark

et al. 2015

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The benchmark Active Vibration Isolation System (AVIS) in this paper is a complex high-tech industrial system used in vibration and motion control applications. The system is complex in the sense of high order flexible dynamics and multiple inputs and outputs. The aim of this benchmark is to compare different black box, linear time invariant system identification algorithms. Different large data sets are provided, enabling the use of both frequency domain and time domain identification approaches. The idea of the benchmark is to investigate both model accuracy and numerical reliability of the computational steps. Several reference solutions are provided, which demonstrate the challenging aspects of this benchmark already in the singleinput single-output case. The benchmark data and additional info are available on the website of Tom Oomen 1 .

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Section: Introductionmentioning

confidence: 99%

Identification of High-Tech Motion Systems: An Active Vibration Isolation Benchmark

et al. 2015

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“…The related theory, as presented in [12], shows that for any scalar stable all-pass transfer function with balanced realization the sequence of functions (7) generates an orthonormal basis for the space of stable systems , with the property that (8) As a result there exist unique and such that (9) Note that with the McMillan degree of (dimension of ), and . The advantage of this generalized basis is that if an appropriate choice of dynamics (set of poles) is incorporated into , and thus into the basis functions , then the series expansion (9) shows an increasing rate of convergence. Consequently, the accuracy of a finite expansion model will substantially increase; for more details on the use of these basis functions see [12], [34], and [25].…”

Section: Model Parameterization With Orthonormal Basis Functionsmentioning

confidence: 99%

“…The estimated parameter vector is denoted by (11) IV. PARAMETER ESTIMATE Building upon (9) for the data generating system, we will write (12) with and (13) Using the system's equations, similar as in [17], we now can express the ETFE by (14) where (15) and where is a term due to the past of the input signal, . Written in vector notation we can rewrite this into…”

Section: Model Parameterization With Orthonormal Basis Functionsmentioning

confidence: 99%

“…For this situation a large number of identification methods exist, mostly dealing with least squares criteria [14], [29], [38], [31]. Maximum amplitude criteria are considered in [32] and [9], while special purpose multivariable algorithms are discussed, e.g., in [15] and [1]. Recently subspace algorithms also have been analyzed for frequency domain identification [21].…”

Section: Introductionmentioning

confidence: 99%

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Frequency domain identification with generalized orthonormal basis functions

Vries

Hof

1998

IEEE Trans. Automat. Contr.

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Abstract-A method is considered for the identification of linear parametric models based on a least squares identification criterion that is formulated in the frequency domain. To this end, use is made of the empirical transfer function estimate (ETFE), identified from time-domain data. As a parametric model structure use is made of a finite expansion sequence in terms of recently introduced generalized basis functions, being generalizations of the classical pulse and Laguerre and Kautz types of bases. An asymptotic analysis of the estimated models is provided and conditions for consistency are formulated. Explicit and transparent bias and variance expressions are established, the latter ones also valid in a situation of undermodeling.

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“…Established identification algorithms include [26], the SK-iteration [34], and the Gauss-Newton iteration [1], which (iteratively) compute the least-squares solution to a linear systems of equations, for polynomial models or rational parametrizations. Although conceptually straightforward, the associated numerical conditioning is often extremely poor, as is evidenced by the developments in [1], [18], [28], [41], [44], which provide partial solutions for ill-conditioning. In [30], [37], [21], a fundamentally different solution strategy is pursued.…”

Section: Introductionmentioning

confidence: 99%

Bi-Orthonormal Polynomial Basis Function Framework With Applications in System Identification

Herpen

Bosgra

Oomen

2016

IEEE Trans. Automat. Contr.

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Abstract-Numerical aspects are of central importance in identification and control. Many computations in these fields involve approximations using polynomial or rational functions that are obtained using orthogonal or oblique projections. The aim of this paper is to develop a new and general theoretical framework to solve a large class of relevant problems. The proposed method is built on the introduction of bi-orthonormal polynomials with respect to a data-dependent bi-linear form. This bi-linear form generalises the conventional inner product and allows for asymmetric and indefinite problems. The proposed approach is shown to lead to optimal numerical conditioning (κ = 1) in a recent frequency-domain instrumental variable system identification algorithm. In comparison, it is shown that these recent algorithms exhibit extremely poor numerical properties when solved using traditional approaches. I. INTRODUCTIONApplications of system identification and control often involve numerical computations. The accuracy of these computations determines the quality of the resulting model or controller. This has led to considerable research to develop numerically reliable algorithms, see, e.g., [38] and [8], [4] for a general overview. Many computations involve weighted least-squares type problems for systems that are parametrized in terms of (vector) polynomials or rational forms, where orthogonal or oblique projections play a central role.In the field of system identification, weighted nonlinear least-squares criteria are particularly common, see [29]. Established identification algorithms include [26], the SK-iteration [34], and the Gauss-Newton iteration [1], which (iteratively) compute the least-squares solution to a linear systems of equations, for polynomial models or rational parametrizations. Although conceptually straightforward, the associated numerical conditioning is often extremely poor, as is evidenced by the developments in [1 , a fundamentally different solution strategy is pursued. The strategy is based upon the construction of orthogonal polynomials with respect to a data-dependent inner product, which directly provides the solution to the approximation problem in terms of polynomials. Essentially, this yields optimal conditioning of the associated linear system of equations, i.e., κ = 1.Besides developments in view of reliable algorithms that involve least-squares type solutions, recently, more general

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Frequency domain curve fitting with maximum amplitude criterion and guaranteed stability

Cited by 30 publications

References 9 publications

Identification of High-Tech Motion Systems: An Active Vibration Isolation Benchmark

Identification of High-Tech Motion Systems: An Active Vibration Isolation Benchmark

Frequency domain identification with generalized orthonormal basis functions

Bi-Orthonormal Polynomial Basis Function Framework With Applications in System Identification

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