Irreducibility Conditions for Continuous-time Multi-input Multi-output Nonlinear Systems

Realization of discrete-time nonlinear input-output equations: polynomial approach

“…Slightly different point of view in the studies of nonlinear control systems is provided by the polynomial approach in which the system is described by two polynomials from the noncommutative ring of skew polynomials that act on input and output differentials. Polynomial approach has been used so far to study the reduction of nonlinear input-output equations [5], [6], [7], the input-output [8] and transfer equivalence [5], [6], [9], controllability [10] and used also in introducing the concept of transfer function into the nonlinear domain [11], [12], [13], [14]. The aim of the present paper is to apply the polynomial approach to the realization problem.…”

Polynomial accessibility condition for the multi-input multi-output nonlinear control system

“…Compared with [13], in the present paper irreducible system representation (accessible subsystem) is related to the recently introduced concept of transfer matrix of the nonlinear system [8,9], whereas that in [13] was based on the notion of an irreducible differential form, associated with the control system [6]. Moreover, the notion of the transfer equivalence of systems is now defined via the equality of transfer matrices, exactly like in the linear case, and the reduction problem is straightforwardly addressed.…”

“…Moreover, the notion of the transfer equivalence of systems is now defined via the equality of transfer matrices, exactly like in the linear case, and the reduction problem is straightforwardly addressed. Furthermore, the proof of the main theorem is improved (while only the sketch of the proof was given in [13]) and the role of non-uniqueness of the greatest common left divisor to the solution is explained. Finally, comparison with alternative algebraic accessibility criteria is added.…”

“…The condition found provides a basis for finding the accessible representation of the set of input-output equations, which is a suitable starting point for the construction of an observable and accessible state space realization. Moreover, the condition allows us to check the transfer equivalence of two nonlinear systems.The results of this paper are based on the conference paper [13]. Compared with [13], in the present paper irreducible system representation (accessible subsystem) is related to the recently introduced concept of transfer matrix of the nonlinear system [8,9], whereas that in [13] was based on the notion of an irreducible differential form, associated with the control system [6].…”

“…The results of this paper are based on the conference paper [13]. Compared with [13], in the present paper irreducible system representation (accessible subsystem) is related to the recently introduced concept of transfer matrix of the nonlinear system [8,9], whereas that in [13] was based on the notion of an irreducible differential form, associated with the control system [6].…”