Differential Algebra for Nonlinear Control Theory
This chapter focus is to show the application of the commutative algebra, algebraic geometry and differential algebra concepts to nonlinear control theory, it begins with necessary information of differential algebra, it continues with definitions of sing
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Differential Algebra for Nonlinear Control Theory
Abstract This chapter focus is to show the application of the commutative algebra, algebraic geometry and differential algebra concepts to nonlinear control theory, it begins with necessary information of differential algebra, it continues with definitions of single-input single-output systems, invertible systems, realization and canonical forms, finally we present methods for stabilization of nonlinear systems throughout linearization by dynamical feedback and some examples of such processes.
10.1 Algebraic and Transcendental Field Extensions The notion of field was introduced in the late nineteenth century in Germany, with the purpose of treating equations while avoiding the need of complex calculations, in a way similar to nonlinear equations. Definition 10.1 Let L , K fields such that K ⊂ L. The larger field L is then said to be an extension field of K denoted by L/K [1–4]. On the other hand, consider a ∈ L. 1. a is algebraic over K if there exists a polynomial P(x) with coefficients in K such that P(a) = 0. 2. a is transcendent over K if it is not algebraic, that is, if there is no polynomial P(x) with coefficients in K such that P(a) = 0. √ For example, if K = Q, 2 ∈ R is algebraic over Q since there exists a polyno√ mial, namely p(x) = x 2 − 2, such that p( 2) = 0. If L = R, π, e are transcendental elements over Q. According to the above concepts, the extension L/K is algebraic if and only if every element of L is algebraic over K , it is said to be transcendental if and only if there exist at least one element of L such that it is transcendental over K . © Springer Nature Switzerland AG 2019 R. Martínez-Guerra et al., Algebraic and Differential Methods for Nonlinear Control Theory, Mathematical and Analytical Techniques with Applications to Engineering, https://doi.org/10.1007/978-3-030-12025-2_10
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10 Differential Algebra for Nonlinear Control Theory
Now, consider the algebraic dependence. This idea is similar to linear independence of vectors in vector spaces. Definition 10.2 The elements {ai | i ∈ I } of L are said to be K -algebraically independent if there are no polynomials P(x1 , . . . , xγ ) in any finite number of variables with coefficients in K such that P(a1 , . . . , aγ ) = 0. Elements no K -algebraically independent are said to be K -algebraically dependent [5–7]. It is possible to show the existence of maximal families of elements of L that are K -algebraically independent. Two maximal families have the same cardinality (the same number of elements). Definition 10.3 Let S be a subset of L a maximal family. S is said to be a transcendence basis of L/K if it is algebraically independent over K and if furthermore L is an algebraic extension of the field K(S) (the field obtained by adjoining the elements of S to K). The cardinality of this basis is the transcendence degree of L/K denoted by tr d◦ L/K [8–10]. Example 10.1 1. The extension L/K is algebraic if and only if tr d◦ L/K = 0. 2. Let K = R. We consider the field of rational fun
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