Optimally conditioned instrumental variable approach for frequency-domain system identification

Herpen, Robbert van; Oomen, Tom; Steinbuch, M Maarten

doi:10.1016/j.automatica.2014.07.002

Cited by 30 publications

(30 citation statements)

References 34 publications

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“…A 16 th order successful industrial case study is presented in [22]. Furthermore, an experimental example and comparison with pre-existing approaches is presented in [39].…”

Section: Discussionmentioning

confidence: 99%

“…. , m, two distinct polynomial bases are required in order to achieve bi-orthogonality with respect to the general bi-linear form (21), whereas ii) for symmetric positive definite weights w T 1k w 1k , as encountered in the inner product (22), it is possible to achieve orthogonality with a single polynomial basis, since in that case S ij = 0 ∀ i, j = 0, 1, . .…”

Section: B Achieving Optimal Numerical Conditioning Through the Use mentioning

confidence: 99%

“…Further applications to specific identification problems are developed in [19,Chap. 3], [22], where a vector polynomial framework is employed.…”

Section: Introductionmentioning

confidence: 99%

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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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“…A 16 th order successful industrial case study is presented in [22]. Furthermore, an experimental example and comparison with pre-existing approaches is presented in [39].…”

Section: Discussionmentioning

confidence: 99%

Section: B Achieving Optimal Numerical Conditioning Through the Use mentioning

confidence: 99%

See 1 more Smart Citation

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

Herpen

Bosgra

Oomen

2016

IEEE Trans. Automat. Contr.

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“…The obtained frequency resolution is 0.5 Hz. The parametric model

bold-italicĴ (z)

is estimated using an iterative identification procedure in . The resulting underlying state‐space model for

bold-italicĴ (z)

is of 44th order.…”

Section: Experimental Validationmentioning

confidence: 99%

Data‐driven multivariable ILC: enhanced performance by eliminatingLandQfilters

Bolder

Kleinendorst

Oomen

2016

Intl J Robust & Nonlinear

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Summary Iterative learning control (ILC) algorithms enable high‐performance control design using only approximate models of the system. To deal with severe modeling errors, a robustness filter Q is typically employed. Irrespective of the large performance enhancement, these approaches thus require a modeling effort and are subject to a performance/robustness tradeoff. The aim of this paper is to develop a fully data‐driven ILC approach that does not require a modeling effort and mitigates the performance/robustness tradeoff. The main idea is to replace the use of a model by dedicated experiments on the system. Convergence conditions are developed in a finite‐time framework, and insight in the convergence aspects is presented using a frequency‐domain analysis. Extensions to increase the convergence speed are proposed. The developed framework is validated through experiments on a multivariable industrial flatbed printer. Both increased performance and robustness are demonstrated in a comparison with closely related model‐based ILC algorithms. Copyright © 2016 John Wiley & Sons, Ltd.

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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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Optimally conditioned instrumental variable approach for frequency-domain system identification

Cited by 30 publications

References 34 publications

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

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

Data‐driven multivariable ILC: enhanced performance by eliminatingLandQfilters

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

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