Abstract
Most control engineering exercises begin with the phrase “Given is the system with the transfer function ...”. Alas, in engineering practice, a mathematical description of the plant which is to be controlled is seldom available. The plant first needs to be modeled mathematically. In spite of this fact, control engineering courses generally do not spend much time on modeling. Perhaps this is because modeling can be said to be more of an art than a science. This Chapter begins by defining what is meant by the words model and modeling. It then illustrates various types of models, studies the process of modeling and concludes with the problem of parameter identification and related optimization techniques.
“That’s another thing we’ve learned from your Nation,” said
Mein Herr, “map-making. But we’ve carried it much further
than you. What do you consider the largest map that would
be really useful?”
“About six inches to the mile.”
“Only six inches!” exclaimed Mein Herr. “We very soon got to six
yards to the mile. Then we tried a hundred yards to the mile.
And then came the grandest idea of all! We actually made a
map of the country, on the scale of a mile to the mile!”
“Have you used it much?” I enquired.
“It has never been spread out, yet,” said Mein Herr: “the farmers
objected: they said it would cover the whole country, and shut
out the sunlight! So we now use the country itself, as its own
map, and I assure you it does nearly as well.”
— Lewis Carroll, Sylvie and Bruno Concluded
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Notes
- 1.
This Chapter is an extended version of Chapter 3 in [51].
- 2.
Isaac Newton, English astronomer, physicist, mathematician (1642–1726); widely recognized as one of the most influential scientists of all time and a key figure in the scientific revolution; famous for developing infinitesimal calculus, classical mechanics, a theory of gravitation and a theory of color.
- 3.
Isaac Asimov, American writer (1920–1992).
- 4.
Which science fiction enthusiast is unaffected by the Asimovian character Hari Seldon’s “psychohistory”, which combines history, sociology and statistics to make general predictions about the future behavior of large groups of populations—like the Galactic Empire with a quintillion citizens [7]?
- 5.
Oliver Heaviside, English mathematician and engineer (1850–1925).
- 6.
Albert Einstein, German-American physicist (1879–1955); famous for his explanation of the photoelectric effect and the relativity theory which radically changed the understanding of the universe.
- 7.
Lower case boldface characters are used for vectors and upper case boldface characters are used for matrices throughout the Book unless otherwise specified.
- 8.
Louis Jean-Baptiste Alphonse Bachelier, French mathematician (1870–1946) is credited with being the first person who tried modeling stock prices mathematically [134].
- 9.
Carl Friedrich Gauss, German mathematician (1777–1855); contributed significantly to many fields in mathematics and physics.
- 10.
A unit impulse function contains all frequencies, because its Laplace transform is simply 1. Therefore, theoretically, it is a sufficiently rich input for identification purposes.
- 11.
A minor of a matrix \(\mathbf {A}\) is the determinant of some smaller square matrix, cut down from \(\mathbf {A}\) by removing one or more of its rows or columns. Minors obtained by removing just one row and one column from square matrices are called first minors. First minors that are obtained by successively removing the last row and the last column of a matrix are called leading first minors.
- 12.
A similar test checks the positive or negative definiteness of the Hessian matrix via its eigenvalues to determine whether a critical point (\(f_{x_i}=0\)) is a minimum or a maximum [90].
- 13.
Joseph-Louis Lagrange, French mathematician (1736–1813).
- 14.
Leonardo Fibonacci, Italian mathematician (c.1175–c.1250).
- 15.
Already the Ancient Greek mathematicians, most notably Euclid, were fascinated by the so-called golden ratio which is recurrent in the nature. This ratio is given as \(\frac{a+b}{a}=\frac{a}{b}\).
- 16.
In Germanic literature this method is known as “achsenparallele Suche” or “search parallel to the axes”.
- 17.
Occam’s razor is the law of parsimony, economy or succinctness. It is a principle urging one to select from among competing hypotheses the one which makes the fewest assumptions. Although the principle was known earlier, Occam’s razor is attributed to the 14th-century English Franciscan friar William of Ockham.
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Hacιsalihzade, S.S. (2018). Modeling and Identification. In: Control Engineering and Finance. Lecture Notes in Control and Information Sciences, vol 467. Springer, Cham. https://doi.org/10.1007/978-3-319-64492-9_2
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DOI: https://doi.org/10.1007/978-3-319-64492-9_2
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