Polynomial methods for nonlinear control systems

Nowadays, the control theory operates with many various approaches and methods to solve a full range of problems. Various methods and techniques have been created for the analysis and modeling of control systems. One of the most popular and widely used approaches is based on an algebraic point of view. The idea of the algebraic approach is based on the vector spaces of differential one-forms over suitable fields of nonlinear functions. Upon the latter a polynomial framework can be built. Together they are well suited for solving problems for both continuous- and discrete-time cases. Moreover, using the tools based on differential one-forms and the related methods based on the theory of the skew polynomial rings, one can work with algebraic equations rather than with differential counterparts, what inherently is simpler.

Polynomial approach has been used so far to study problems like reduction of nonlinear i/o equations, linear i/o and transfer equivalence, controllability and used also in introducing the concept of transfer function into the nonlinear domain. Thus it has been already proved itself as a practical and reliable mathematical tool.

The developed theory and polynomial formalism in particular were implemented in the Mathematica Software package NLControl. To make it accessible by the external users we made this package available via webMathematica service.