A philosophy for the modelling of realistic nonlinear systems

Howlett, Phil; Torokhti, Anatoli; Pearce, Charles

doi:10.1090/S0002-9939-03-07164-8

A philosophy for the modelling of realistic nonlinear systems
HTML articles powered by AMS MathViewer

by Phil Howlett, Anatoli Torokhti and Charles Pearce PDF

Proc. Amer. Math. Soc. 132 (2004), 353-363 Request permission

Abstract:

A nonlinear dynamical system is modelled as a nonlinear mapping from a set of input signals into a corresponding set of output signals. Each signal is specified by a set of real number parameters, but such sets may be uncountably infinite. For numerical simulation of the system each signal must be represented by a finite parameter set and the mapping must be defined by a finite arithmetical process. Nevertheless the numerical simulation should be a good approximation to the mathematical model. We discuss the representation of realistic dynamical systems and establish a stable approximation theorem for numerical simulation of such systems.

References

B. Russell, On the notion of cause, Proc. Aristotelian Soc. 13 (1913), 1–25.
Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
P. L. Falb and M. I. Freedman, A generalized transform theory for causal operators, SIAM J. Control 7 (1969), 452–471. MR 0682782, DOI 10.1137/0307033
Jan C. Willems, Stability, instability, invertibility and causality, SIAM J. Control 7 (1969), 645–671. MR 0275936, DOI 10.1137/0307047
I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. Translated from the Russian by A. Feinstein. MR 0264447
Irwin W. Sandberg and Lilian Xu, Uniform approximation of multidimensional myopic maps, IEEE Trans. Circuits Systems I Fund. Theory Appl. 44 (1997), no. 6, 477–485. MR 1452221, DOI 10.1109/81.585959
A. P. Torokhti and P. G. Howlett, On the constructive approximation of non-linear operators in the modelling of dynamical systems, J. Austral. Math. Soc. Ser. B 39 (1997), no. 1, 1–27. MR 1462806, DOI 10.1017/S0334270000009188
P. G. Howlett and A. P. Torokhti, A methodology for the constructive approximation of nonlinear operators defined on noncompact sets, Numer. Funct. Anal. Optim. 18 (1997), no. 3-4, 343–365. MR 1448895, DOI 10.1080/01630569708816764
P. G. Howlett and A. P. Torokhti, Weak interpolation and approximation of non-linear operators on the space $\scr C([0,1])$, Numer. Funct. Anal. Optim. 19 (1998), no. 9-10, 1025–1043. MR 1656408, DOI 10.1080/01630569808816872
P. M. Prenter, A Weierstrass theorem for real, separable Hilbert spaces, J. Approximation Theory 3 (1970), 341–351. MR 433214, DOI 10.1016/0021-9045(70)90039-0
Vincent J. Bruno, A Weierstrass approximation theorem for topological vector spaces, J. Approx. Theory 42 (1984), no. 1, 1–3. MR 757098, DOI 10.1016/0021-9045(84)90049-2
G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems 2 (1989), no. 4, 303–314. MR 1015670, DOI 10.1007/BF02551274
I. K. Daugavet, On operator approximation by causal operators and their generalizations. II: Nonlinear case (Russian), Methods of Optimiz. and their Applic., Irkutsk. Sib. Energ. Institut (1988), 166–178.
Anatoli P. Torokhti and Phil G. Howlett, On the best quadratic approximation of nonlinear systems, IEEE Trans. Circuits Systems I Fund. Theory Appl. 48 (2001), no. 5, 595–602. MR 1854954, DOI 10.1109/81.922461
A. Torokhti and P. Howlett, Optimal Fixed Rank Transform of the Second Degree, IEEE Trans. on Circuits and Systems. Part II, Analog and Digital Signal Processing, 48, No. 3 (2001), 309–315.