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

Abstract

The paper presents a computation-oriented necessary and sufficient accessibility condition for the set of nonlinear higherorder input-output differential equations. The condition is presented in terms of the greatest common left divisor of two polynomial matrices, associated with the system of input-output equations. The basic difference from the linear case is that the elements of the polynomial matrices belong to a non-commutative polynomial ring. 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]. 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. Note that the single-input single-output (SISO) case was studied in [27]. For discrete-time nonlinear systems the accessibility property was examined in [16], though the paper itself was focused on finding the minimal realization of the set of i/o equations and accessibility was studied only from the viewpoint of irreducibility of the i/o description of the system. The irreducibility problem for the SISO case has also been treated in [12], being a generalization of [27] to the systems defined on homogeneous time-scales.The paper is organized as follows. Section 2 describes the differential field and a polynomial matrix representation, associated to nonlinear systems. Using the polynomial matrix description, in Section 3 necessary and sufficient accessibility condition is given. Section 4 is devoted to the system reduction and concept of transfer equivalence. In Section 5 the obtained results are compared with the results of [4] and illustrated by two examples, one realizable in the state-space form and the other not. Section 6 draws the conclusions and drafts some future goals of study.