In order to substantiate the main aim of the dissertation i.e. the “systematic use of geometric way of thinking” in control technology first I would like to give a very brief historical survey to show how fruitful and profitable it was in the field of the natural sciences. Since the historical background of these methods normally are not mentioned (neither in the standard university-education of Mathematics nor in the more specific scientific papers), for collecting this information (rigorously only for this purpose) I intensively used the materials available on the Web at the pages of Wikipedia, the free encyclopedia [R26]. The result of this brief historical research was quite surprising and shocking for me because it revealed that Mankind has clear, precise, and well generalized concepts of this subject area practically only from the middle of the 19th Century.
3.1. Certain Representative Examples of Uneven Development
From a historical point of view it can be stated that the main concepts had crystallized only “recently” that has the interesting consequences that certain fundamental mathematical methods widely used in Technical Sciences obtained rigorous mathematical explanation only after their invention. To mention only a few significant examples: when Euler invented one of the fundamental equations of Fluid Dynamics in 1755 no systematic concepts of vectors, tensors, or other directed quantities were available [R27]. When Maxwell published his famous “Treatise on Electricity and Magnetism” in 1892 [R28] both Hamilton’s “quaternions” [R29] as well as Grassmann’s “vectors” already existed [R30] (he worked on this idea from 1832), however, the latter concept became widely available only a few years after issuing the “Treatise”, therefore Maxwell used quaternions for the quantitative description of electromagnetic phenomena. This observation highlights the
“incidental nature” of the development in sciences. As is well known the later issues of the “Treatise” already used the concept of vectors and tensors instead of quaternions. It was an interesting and inspiring question to look after what kind of Electrodynamics we could have now if the “custom” of using quaternion prevailed.
For instance, in a common work with Iván Abonyi and János F. Bitó we found that the two invariants in Electrodynamics could be more easily explored by using the complex extension of Quaternion Algebra than by using tensors. It was also found that the significant components of the relativistic tensor formulation of Electrodynamics could be also identified in the quaternion representation [R31]. On this reason in the next part I present a very brief historical summary of the fundamental concepts.
3.2. Historical Antecedents of Geometric Way of Thinking
Until the 1st half of the 20th Century the development of Mathematics aimed at serving the needs of natural and technical sciences. In the history of the
"quantitative sciences" geometric way of thinking always played a pioneering role.
The principles of geometry first were reduced to a small set of axioms by Euclid of Alexandria, a Greek mathematician who worked during the reign of Ptolemy I (323-283 BC) in Egypt. His method of proving mathematical theorems by logical reasoning from accepted first principles remained the backbone of mathematics even in our days, and is responsible for that field's characteristic rigor [R32].
Following the pioneering work clarifying the phenomenology of Classical Mechanics by Galilei and Newton, in his fundamental work entitled "Mécanique Analytique" [R33] Joseph-Louis Lagrange (1736-1813) solved various optimization problems under constraints, introduced the concept of “Reduced Gradient” and that of what we refer to nowadays as “Lagrange Multipliers” [R34]. It has to be noted that at that time the concept of "linear vector spaces" was not clarified at all.
The first mathematical means of describing quantities with direction, i.e. the quaternions introduced by Sir William Rowan Hamilton (1805-1865) appeared not very long time after Lagrange's death [R29]. In the 19th Century quaternions were generally used for such purposes. For instance, in the first edition of Maxwell's famous “Treatise on Electricity and Magnetism” quaternions were used for describing the "directed" magnetic and electric fields.
The first known appearance of what are now called “linear algebra” and the notion of a “vector space” is related to Hermann Günther Grassmann (1809-1877), who started to work on the concept from 1832. In 1844, Grassmann published his masterpiece [R30] that commonly is referred to as the "Ausdehnungslehre", ("theory of extension" or “theory of extensive magnitudes”). This work was mainly inspired by Lagrange's "Mécanique analytique" [R33]. Grassmann showed that once geometry is put into the algebraic form he advocated, then the number three has no privileged role as the number of spatial dimensions: the number of possible dimensions is in fact unbounded [R35].
The close relationship between geometry and algebra was realized and strongly utilized by William Kingdon Clifford (1845-1879) who introduced various
“associative algebras”, the so called "Clifford Algebras" [R36]. As special cases Clifford Algebras contain the algebra of the real, the complex, the dual numbers, the quaternion algebra, and the algebra of octonions (biquaternions) [R37]. His
“Geometric Algebra” is widely used in technical sciences as e.g. in computer graphics, robotics, etc.
Equipped with the concepts of linear vector spaces Marius Sophus Lie (1842-1899) in his PhD dissertation studied the properties of geometric symmetry transformations [R38]. One of his greatest achievements was the discovery that continuous transformation groups (now called after him Lie groups) could be better understood by studying the properties of the tangent space of the group elements, that form linear vector spaces (the vector space of the so-called infinitesimal generators), and with the commutator as multiplication also form algebras, the so called “Lie Algebras”.
In the very fertile period of Mathematics, in the 19th Century Georg Friedrich Bernhard Riemann (1826-1866) elaborated the geometry of curved spaces in a special form that made it possible to study physical quantities as tensors even if the geometry of the space differs from the Euclidean Geometry [R39]. This concept was very fruitfully used in the General Theory of Relativity.
David Hilbert (1862-1943) [R40] extended the concept of the Euclidean Geometry to linear, normed, complete metric spaces in which the norm originates from a scalar product.
Stefan Banach (1892-1945) [R41] introduced the more general concept, the concept of Banach Spaces that are linear, normed, complete metric spaces in which the norm not necessarily originates from a scalar product. The great practical advantage of Banach's invention is that by adding various norms to the same mathematical set various complete, linear, normed metric spaces can be obtained that
offer a wide basis for elaborating diverse practical variants and solutions pertaining to the essentially same basic idea.
Vladimir Igorevich Arnold (1937-) [R42] studied the Symplectic Geometry and Symplectic Topology that are extremely useful means of studying the behavior of various mechanical and other physical systems.
The geometric way of thinking outlined above appeared in one of the best textbooks used for teaching functional analysis, too (the excellent book by László Máté [R43]).
By the middle of the eighties of the past century certain elements of the sophisticated geometric concepts were systematically utilized in control technology.
The first edition of Isidori’s book in 1985 [R44] contained cahpeters as “Geometric Theory of State Feedback” and “Geometric Theory of Nonlinear Systems”. An even more systematic surveay and application of Group Theory and Differentiable Manifolds can be found in Jurdjevic’s book from 1997 [R45].
Another, very important mathematical tool that makes it easy to apply geometric way of thinking is the Singular Value Decomposition (SVD). The history of matrix decomposition goes back to the 1850s. During the last 150 years several mathematicians — Eugenio Beltrami (1835–1899), Camille Jordan (1838–1921), James Joseph Sylvester (1814–1897), Erhard Schmidt (1876–1959), and Hermann Weyl (1885–1955), who were perhaps the most important ones, contributed to establishing the existence of the singular value decomposition and developing its theory [R46]. Thanks to the pioneering efforts of Gene Golub, there exist efficient, stable algorithms to compute the singular value decomposition [R47]. Certain realization of SVD is available in Hungarian for a long time in the excellent book by Pál Rózsa [R48]. In our days SVD is a standard service (function) of software designed for the use in research, as e.g. INRIA’s SCILAB.
More recently, SVD, and its novel variant, the so called Higher Order Singular Value Decomposition (HOSVD) (e.g. [R49], [R50]) started to play an important role in several scientific fields as signal processing (e.g. [R51], [R52], [R53]), control applications in dealing with system models of Tensor Product (TP) form (e.g., the very interesting PhD Thesis by Zoltán Petres [R54] can be referred to in this context). The real variant of SVD was extensively used in the present Thesis, too.
My aim with providing this brief historical survey was to show that geometric way of thinking is a very useful and fruitful mode of problem-tackling in various fields. The use of the inventions by Hamilton, Grassmann, Hilbert, Banach, and Clifford in Physics and technical fields makes it possible
• To apply a “geometric way of thinking” with which we became familiar in our childhood in our playing house. Then we daily experienced the Euclidean Geometry of the reality around us.
Selection and use of adequate associations with simple pictures as vectors or directed quantities, linear combinations, basis vectors, orthogonality, orthogonal subspaces, tangents and tangent space of a surface in a given point, the notion of surfaces or hypersurfaces embedded in higher dimensional spaces became instinctive, hidden practice of our early years;
• To strengthen the above, almost “instinctive” associations with the aid of lucid, simple, aesthetic equations of algebraic relationships.
In the sequel its advantages will be shown in the field of nonlinear control.
For this purpose I try to give a brief survey on the prevailing, from certain point of view “classic” approaches.