Koopman operator approach for computing structure of solutions and Observability of nonlinear finite state system
Abstract
Given a discrete dynamical system defined by a map in a vector space over a finite field called Finite State Systems (FSS), a dual linear system over the space of functions on the state space is constructed using the dual map. This system constitutes the well known Koopman linear system framework of dynamical systems, hence called the Koopman linear system (KLS). It is first shown that several structural properties of solutions of the FSS can be inferred from the solutions of the KLS. The problems of computation of structural parameters of solutions of nonlinear FSS are computationally hard and hence become infeasible as the number of variables increases. In contrast, it has been well known that these problems can be solved by linear algebra for linear FSS in terms of elementary divisors of matrices and their orders. In the next step, the KLS is reduced to the smallest order (called ROKLS) while still retaining all the information of the parameters of structure of solutions of the FSS. Hence when the order of the ROKLS is sufficiently small, the above computational problems of nonlinear FSS are practically feasible. Next, it is shown that the observability of the nonlinear FSS with an output function is equivalent to that of the ROKLS with an appropriate linear output map. Hence, the problem of nonlinear observability is solved by an observer design for the equivalent ROKLS. Such a construction should have striking applications to realistic FSS arising in Cryptology and Biological networks.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.10752
 arXiv:
 arXiv:2010.10752
 Bibcode:
 2020arXiv201010752A
 Keywords:

 Electrical Engineering and Systems Science  Systems and Control;
 Mathematics  Dynamical Systems
 EPrint:
 20 pages