Control Systems on Regular Time Scales and their Differential Rings

Control Systems on Regular Time Scales
and their Differential Rings
Zbigniew Bartosiewicz† , Ülle Kotta? , Ewa Pawluszewicz‡ ,
Malgorzata Wyrwas†
†
Faculty of Computer Science, Bialystok University of Technology
Wiejska 45a, 15-351 Bialystok, Poland
[email protected]; [email protected]
‡
?
Department of Mathematics, University of Aveiro
3810-193 Aveiro, Portugal
[email protected]
Institute of Cybernetics at Tallinn University of Technology
Akadeemia tee 21, 12618 Tallinn, Estonia
[email protected]
Control systems defined on regular time scales and operators associated to such systems are studied. Regular time scales allow to unify the continuous- and discretetime systems and accommodate also non-uniformly sampled systems. A mathematical formalism that allows to study nonlinear control systems on non-homogeneous
time scales is developed. The key applications of the formalism are a construction of
a differential ring associated with the nonlinear control system on regular but nonhomogeneous time-scale and a construction of its inversive closure. The constructed
rings are equipped with some operators whose properties are studied.
The presented theory is an extension of the algebraic framework given in [1] for
control systems defined on homogeneous time scales. Since the extension from homogeneous to non-homogeneous but regular case is far from being trivial, so the
difficulties are discussed. Compared with the homogeneous case, the main difficulties are non-commutativity of the delta derivative and shift operators and the fact
that the additional time variable t appears in the ring associated to the control
system.
[1] Bartosiewicz, Z., Kotta, Ü., Pawluszewicz, E., Wyrwas, M, Algebraic formalism
of differential one-forms for nonlinear control systems on time scales, Proc. Estonian
Acad. of Sci. Phys. Math., (2007), 264–282.