Brief paperOrbital decompositions and integrable pseudosymmetries of control systems☆
Introduction
We study the problem of constructing local decompositions of control systems. In difference from well-known classical works (Fliess, 1985, Grizzle and Marcus, 1985, Hirshorn, 1982, Isidori, 1995, Isidori et al., 1981, Krener, 1977, Nijmeijer, 1983, Nijmeijer and van der Schaft, 1985, Respondek, 1982) we use orbital equivalences (Fliess et al., 1999) for transforming systems to decomposed forms. This is reason why the decompositions examined below are called orbital.
Since we use a wider group of transformations, we get a wider class of decomposable forms for a given system. Therefore, orbital decompositions will have wider application if one learns to construct them. At the moment, we can mention only two control problems that were investigated using decomposition: the construction of flat outputs (Schöberl & Schlacher, 2014) and the point-to-point steering problem (Belinskaya & Chetverikov, 2016).
In this paper, we present a method for constructing orbital decompositions. We use the geometry of infinite jets and characterize local orbital decompositions in terms of integrable pseudosymmetries. The notion of pseudosymmetry of evolution equations with one spatial variable was introduced in Sokolov (1988). We generalize this concept to control systems and introduce the notion of integrable pseudosymmetry. Note that orbital equivalences of systems can be interpreted as diffeomorphisms of infinite dimensional manifolds corresponding to the systems (see Krasil’shchik and Vinogradov (1999) or Fliess et al. (1999)). Any pseudosymmetry determines an involutive distribution on the corresponding infinite dimensional manifold. But involutive distributions on infinite-dimensional manifolds may be nonintegrable. Similarly to Chetverikov (1991), where integrable higher symmetries were described, we introduce an integrability condition for pseudosymmetries such that an integrable pseudosymmetry determines an integrable distribution. If an integrable pseudosymmetry depends on the state only, then its integrable distribution is a distribution invariant under dynamics (Isidori, 1995, Isidori et al., 1981). In this case the corresponding decomposition is classical. Like the invariant distribution in the classical case (see Isidori (1995)), the distribution of an integrable pseudosymmetry can be a fundamental tool in the analysis of the internal structure of nonlinear systems.
We prove that any orbital decomposition of a system determines and is determined by its integrable pseudosymmetry. Any pseudosymmetry is determined by two matrices of functions. The equations for these matrices are deduced. The method for constructing orbital decompositions is given and demonstrated by the example of the Kapitsa pendulum system.
Section snippets
Decompositions of control systems
Consider the system where is a smooth vector function, is the state, and is the control input. Here and throughout the following, smoothness is understood as infinite differentiability.
Suppose system (1) is orbitally equivalent (Fliess et al., 1999) to a system of the form where is the independent variable, . Then we say that system (2)–(3) is an orbital decomposition of system (1).
We
A geometric framework
A detailed exposition of geometry of infinitely prolonged differential equations can be found in Krasil’shchik and Vinogradov (1999).
Let us associate to system (1) an infinite dimensional manifold with coordinates where denotes the vector variable corresponding to the th order derivative of with respect to . Every solution of system (1) and a point determine a point of with the coordinates This point
Pseudosymmetries and decompositions
Let be the diffiety of system (1). We say that a set of vector fields on is an integrable pseudosymmetry of system (1) if the following conditions hold:
(i) the fields generate an involutive distribution of dimension on ;
(ii) the column satisfies the equality where is a square matrix of some functions on , is the product of the matrix and the column , and is a column of some functions on ;
(iii) there exists a ring of
Techniques of computing integrable pseudosymmetries
When and , Eq. (9) means that defines a higher symmetry of the system. The next theorem generalizes well-known theorems on higher symmetries (see Krasil’shchik and Vinogradov (1999)).
First consider the diffiety of system (1) and a -column of the form where , and , and are matrices of functions on of dimensions , and respectively, is the th degree of the -matrix differential operator acting on the
Proofs of theorems
First let us prove that the definition of integrable pseudosymmetry is invariant under -diffeomorphisms. Consider an integrable pseudosymmetry of a system and a -diffeomorphism . Since a diffeomorphism takes involutive distributions of dimension into involutive distributions of dimension , we see that the distribution generated by the fields is involutive and has dimension . Thus condition (i) is invariant under -diffeomorphisms.
Let and be the
Conclusion
The presented method allows one to construct both orbital and classical decompositions and consists of 2 stages:
(1) solving equation (16) using the splitting procedure, Theorem 8, and condition (iii);
(2) constructing decompositions for the found integrable pseudosymmetries.
In the case , the ring (see condition (iii)) coincides with , where the integer is determined by the matrices and (cf. with Example 11). In the case , we suggest first choosing and then solving equation
Vladimir Chetverikov graduated from the Moscow State University, Mechanical & Mathematical Faculty in 1983. He obtained the Ph.D. degree in Mathematics in 1988 and Doctor of Science degree in Mathematics in 2007. He is currently a full professor at the Bauman Moscow State Technical University. His research interests include theoretical aspects of nonlinear control, Lie–Backlund transformations of differential equations, invertible differential operators, Backlund transformations,
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Vladimir Chetverikov graduated from the Moscow State University, Mechanical & Mathematical Faculty in 1983. He obtained the Ph.D. degree in Mathematics in 1988 and Doctor of Science degree in Mathematics in 2007. He is currently a full professor at the Bauman Moscow State Technical University. His research interests include theoretical aspects of nonlinear control, Lie–Backlund transformations of differential equations, invertible differential operators, Backlund transformations, transformations of functional–differential equations.
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This work was partially supported by the Ministry of Science and Higher Education of the Russian Federation, project no. 0705-2020-0047 and the Russian Foundation for Basic Research , project no. 20-07-00279. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Warren E. Dixon under the direction of Editor Daniel Liberzon.