| 1. |
- Bodin, Per, et al.
(författare)
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Selection of best orthonormal rational basis
- 2000
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Ingår i: SIAM Journal on Control and Optimization. - 0363-0129 .- 1095-7138. ; 38:4, s. 995-1032
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Tidskriftsartikel (refereegranskat)abstract
- This contribution deals with the problem of structure determination for generalized orthonormal basis models used in system identification. The model structure is parameterized by a prespecified set of poles representing a finite-dimensional subspace of H2. Given this structure and experimental data, a model can be estimated using linear regression techniques. Since the variance of the estimated model increases with the number of estimated parameters, one objective is to find coordinates, or a basis, for the finite-dimensional subspace giving as compact or parsimonious a system representation as possible. In this paper, a best basis algorithm and a coefficient decomposition scheme are derived for the generalized orthonormal rational bases. Combined with linear regression and thresholding this leads to compact transfer function representations. The methods are demonstrated with several examples.
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| 2. |
- Byrnes, Christopher, et al.
(författare)
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Identifiability and well-posedness of shaping-filter parameterizations : A global analysis approach
- 2002
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Ingår i: SIAM Journal of Control and Optimization. - 0363-0129 .- 1095-7138. ; 41:1, s. 23-59
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we study the well-posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients, covariance lags, or Markov parameters. For example, we determine whether there exists a shaping filter with a prescribed window of Markov parameters and a prescribed window of covariance lags. We show that several such problems are well-posed in the sense of Hadamard; that is, one can prove existence, uniqueness (identifiability), and continuous dependence of the model on the measurements. Our starting point is the global analysis of linear systems, where one studies an entire class of systems or models as a whole, and where one views measurements, such as covariance lags and cepstral coefficients or Markov parameters, from data as functions on the entire class. This enables one to pose such problems in a way that tools from calculus, optimization, geometry, and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm-independent fashion. In this language, we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona de coordinate system on the space of shaping filters, thereby establishing existence, uniqueness, and smooth dependence of the model parameters on the measurements from data.
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| 3. |
- Enqvist, Per
(författare)
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A convex optimization approach to arma(n,m) model design from covariance and cepstral data
- 2004
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Ingår i: SIAM Journal of Control and Optimization. - 0363-0129 .- 1095-7138. ; 43:3, s. 1011-1036
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Tidskriftsartikel (refereegranskat)abstract
- Methods for determining ARMA(n, m) filters from covariance and cepstral estimates are proposed. In [C. I. Byrnes, P. Enqvist, and A. Lindquist, SIAM J. Control Optim., 41 ( 2002), pp. 23-59], we have shown that an ARMA( n, n) model determines and is uniquely determined by a window r(0), r(1),..., r(n) of covariance lags and c(1), c(2),..., c(n) of cepstral lags. This unique model can be determined from a convex optimization problem which was shown to be the dual of a maximum entropy problem. In this paper, generalizations of this problem are analyzed. Problems with covariance lags r(0), r(1),..., r(n) and cepstral lags c(1), c(2),..., c(m) of different lengths are considered, and by considering different combinations of covariances, cepstral parameters, poles, and zeros, it is shown that only zeros and covariances give a parameterization that is consistent with generic data. However, the main contribution of this paper is a regularization of the optimization problems that is proposed in order to handle generic data. For the covariance and cepstral problem, if the data does not correspond to a system of desired order, solutions with zeros on the boundary occur and the cepstral coefficients are not interpolated exactly. In order to achieve strictly minimum phase filters for estimated covariance and cepstral data, a barrier-like term is introduced to the optimization problem. This term is chosen so that convexity is maintained and so that the unique solution will still interpolate the covariances but only approximate the cepstral lags. Furthermore, the solution will depend analytically on the covariance and cepstral data, which provides robustness, and the barrier term increases the entropy of the solution.
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| 4. |
- Heuberger, Peter S. C., et al.
(författare)
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Orthonormal basis functions in time and frequency domain : Hambo transform theory
- 2004
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Ingår i: SIAM Journal of Control and Optimization. - 0363-0129 .- 1095-7138. ; 42:4, s. 1347-1373
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Tidskriftsartikel (refereegranskat)abstract
- The class of finite impulse response (FIR), Laguerre, and Kautz functions can be generalized to a family of rational orthonormal basis functions for the Hardy space H2 of stable linear dynamical systems. These basis functions are useful for constructing efficient parameterizations and coding of linear systems and signals, as required in, e.g., system identification, system approximation, and adaptive filtering. In this paper, the basis functions are derived from a transfer function perspective as well as in a state space setting. It is shown how this approach leads to alternative series expansions of systems and signals in time and frequency domain. The generalized basis functions induce signal and system transforms (Hambo transforms), which have proved to be useful analysis tools in various modelling problems. These transforms are analyzed in detail in this paper, and a large number of their properties are derived. Principally, it is shown how minimal state space realizations of the system transform can be obtained from minimal state space realizations of the original system and vice versa.
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| 5. |
- Lindquist, Anders
(författare)
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On the duality between filtering and Nevanlinna-Pick interpolation
- 2000
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Ingår i: SIAM Journal of Control and Optimization. - 0363-0129 .- 1095-7138. ; 39:3, s. 757-775
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Tidskriftsartikel (refereegranskat)abstract
- Positive real rational functions play a central role in both deterministic and stochastic linear systems theory, as well as in circuit synthesis, spectral analysis, and speech processing. For this reason, results about positive real transfer functions and their realizations typically have many applications and manifestations. In this paper, we study certain manifolds and submanifolds of positive real transfer functions, describing a fundamental geometric duality between filtering and Nevanlinna Pick interpolation. Not surprisingly, then, this duality, while interesting in its own right, has several corollaries which provide solutions and insight into some very interesting and intensely researched problems. One of these is the problem of parameterizing all rational solutions of bounded degree of the Nevanlinna-Pick interpolation problem, which plays a central role in robust control, and for which the duality theorem yields a complete solution. In this paper, we shall describe the duality theorem, which we motivate in terms of both the interpolation problem and a fast algorithm for Kalman filtering, viewed as a nonlinear dynamical system on the space of positive real transfer functions. We also outline a new proof of the recent solution to the rational Nevanlinna Pick interpolation problem, using an algebraic topological generalization of Hadamard's global inverse function theorem.
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| 6. |
- Ghulchak, Andrey, et al.
(författare)
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Duality in $H^infty$ Cone Optimization
- 2002
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Ingår i: SIAM Journal of Control and Optimization. - 1095-7138. ; 41:1, s. 253-277
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Tidskriftsartikel (refereegranskat)abstract
- Positive real cones in the space $H^infty$ appear naturally in many optimization problems of control theory and signal processing. Although such problems can be solved by finite-dimensional approximations (e.g., Ritz projection), all such approximations are conservative, providing one-sided bounds for the optimal value. In order to obtain both upper and lower bounds of the optimal value, a dual problem approach is developed in this paper. A finite-dimensional approximation of the dual problem gives the opposite bound for the optimal value. Thus, by combining the primal and dual problems, a suboptimal solution to the original problem can be found with any required accuracy.
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