Free algebras and noncommutative power series in the analysis of nonlinear control systems: an application to approximation problems

Volume 504 / 2014

Grigory M. Sklyar, Svetlana Yu. Ignatovich Dissertationes Mathematicae 504 (2014), 1-88 MSC: Primary 93B25; Secondary 93B11, 93B15, 93B17. DOI: 10.4064/dm504-0-1


The paper contains a consistent presentation of the approach developed by the authors to analysis of nonlinear control systems, which exploits ideas and techniques of formal power series of independent noncommuting variables and the corresponding free algebras. The main part of the paper was conceived with a view of comparing our results with the results obtained by use of the differential-geometric approach. We consider control-linear systems with $m$ controls. In a free associative algebra with $m$ generators (which can be thought of as a free algebra of iterated integrals), a control system uniquely defines two special objects: the core Lie subalgebra and the graded left ideal. It turns out that each of these two objects completely defines a homogeneous approximation of the system. Our approach allows us to propose an algebraic (coordinate-independent) definition of the homogeneous approximation. This definition provides the uniqueness of the homogeneous approximation (up to a change of coordinates) and gives a way to find it directly, without preliminary finding privileged coordinates. The presented technique yields an effective description of all privileged coordinates and an explicit way of constructing an approximating system. In addition, we discuss the connection between the homogeneous approximation and an approximation in the sense of time optimality.


  • Grigory M. SklyarInstitute of Mathematics
    University of Szczecin
    Wielkopolska 15
    70-451 Szczecin, Poland
  • Svetlana Yu. IgnatovichDepartment of Differential Equations and Control
    Kharkiv National University
    Svoboda Sq. 4
    61-022 Kharkiv, Ukraine

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