Interpolation and identification problems

Abstract

The paper refines the relationship between the method of orthogonal moments for the identification of polynomial systems and the method of Lagrange-and Hermite-type operator interpolation in Hilbert space. The identification accuracy is estimated by the interpolation method and the minimum number of input signals that guarantee the prescribed accuracy is determined.Analysis of the properties and peculiarities of plants using modern information processing methods involves development of a mathematical model of a plant based on observable data -inputs and outputs. Such an approach is called plant identification and consists in determining structure and parameters of a plant. Moreover, many scientific and applied areas (medicine, economy, meteorology, seismology, hydro-and radiolocation, image processing, etc.) deal with various systems whose mathematical models can be represented by operator polynomials of some degree. This explains the importance and necessity of developing both theoretical and practical aspects of the analysis of polynomial plants. Among studies in this field, noteworthy are [1-3] concerned with data interpolation, system identification, and the theory of polynomial systems. Note that the possibility of approximating an arbitrary continuous operator by a polynomial one is justified by theorems similar to the Weierstrass theorem (approximation of a continuous function by polynomials). For example, the Frechet theorem is an analog of the Weierstrass theorem for a continuous functional. Prenter [4] has proved a theorem similar to the Weierstrass theorem for continuous nonlinear operators in a Hilbert space. Istratesku [5] and Daugavet [6] generalized this result to Banach space.The identification problem for plants of unknown structure may be considered as a problem of approximating operators in some functional spaces. Since operator interpolation is one of the methods of solving the approximation problem, it is important to establish relationships between operator interpolation formulas and identification methods for nonlinear systems under deterministic actions. Pupkov, Kapalin, and Yushchenko [7] addressed the identification of polynomial functional systems by the method of orthogonal moments followed by selection of homogeneous functions. Khlobystov and Kashpur [8] showed that this method is equivalent to the interpolation one on the corresponding set of nodes and considered its generalization to operators in a Hilbert space.The present study refines the relationship between the method of orthogonal moments for identification of polynomial systems and the method of Lagrange-type operator interpolation of a special sequence of nodes and Hermitian-type operator interpolation [9] for differentiable operators in a Hilbert space. We will estimate interpolation accuracy using the identification problem in the L T 2 0 ( , )space and the minimum number of input signals providing prescribed accuracy. Let us consider the problem of identifying a nonlinear plant using the method of orthogonal moments fol...

Two interpolation operators on irregularly distributed data in inner product spaces

“…Note that the g i are well defined on the whole X , being bounded as a consequence of the right-hand relation in (6). In fact, this equality holds identically for any z ∈ X , because by (3) it can be interpreted as a ratio of two identical sums.…”

“…However, the norm induced in Y by the inner product is useful to consider error bounds. Moreover, we will adopt the setting where X and Y are Hilbert spaces, according to the existing literature (see, e.g., [6,7,5]). The completeness should be needed to discuss convergence properties, a topic which is beyond the present preliminary investigation.…”

“…1, 1 + 1/20] × [1, 1 + 1/20] (r = 2,3,4,5,6,7,8,9,10) and then constructing Halton points (with generating primes 3 and 7) on the same square. Then, the interpolant has been evaluated at 169 points taking nodes with coefficients (α, β) on a regular 13 × 13 grid over [1, 1 + 1/20] × [1, 1 + 1/20].…”

“…In particular, we can give a rough error estimate by using the properties of the cardinal basis functions; in fact, we have by (6) ‖f…”

Lagrange Interpolation on Arbitrarily Distributed Data in Banach Spaces

“…In Hilbert spaces the interpolation problem can be solved, for instance, by a sort of generalization of the Lagrange formula (see [7]) constructed with scalar products, which is a particular case of polynomial operator interpolant ( [5,6], see [9]). This interpolant can be modified so to get a 112 G. Allasia and C. Bracco cardinal basis solution to the same problem (see [1]), obtaining acceptable approximation performances.…”