Practical polynomial formulas in MIMO nonlinear realization problem

Abstract

The explicit and simple polynomial formulas are given for computation of the (differentials) of state coordinates from the set of nonlinear input-output (i/o) equations under assumption that the set of i/o equations is realizable in the statespace form. The paper addresses a very wide class of nonlinear control systems, that accommodates continuous-as well as two discrete-time cases, based either on the shift or difference operator. Two types of polynomial formulas are suggested, based either on polynomial left division or on the application of the concept of adjoint polynomials and implemented in Mathematica-based software.

Realization of nonlinear input-output equations in controller canonical form

“…The solution to the problem statement above is given in terms of the sequence of nonincreasing vector spaces H i , i ≥ 1, which are the subspaces of E and defined [3] recursively by…”

“…, ρ i , the subspace H s+2 is in a form (8). Based on the structure of H s+2 , as in the sufficiency proof of Theorem 3.4, the state equations can be written in the form (4). It remains to show that dy i ∈ H s+2 for i = 1, .…”

“…The general realization problem deals with possibilities to transform the equations (1) into the form (3) such that starting from the corresponding initial conditions, equal inputs yield equal output behaviors for systems (1) and (3). Precise definition of realization is given as follows.…”

“…The realization problem, i. e., transforming a set of higher order differential input-output (i/o) equations into a set of first order differential equations, called state equations, has been studied extensively for nonlinear control systems, both in continuous-and discretetime [4,5,8,9,13,18,19,20]. The reason is simple -most of the control theory is developed for systems described by the state equations, although system identification mostly yields the set of i/o equations.…”

“…For both cases we give necessary and DOI: 10.14736/kyb-2018-4-0736 sufficient conditions under which such special realization exists. The obtained conditions are restrictions of general solution, given in [3,5] and if the conditions are not satisfied, one can, without much additional computations, find a general state space realization. Additionally, we describe the structure of the i/o equations for which realization in the controller canonical form with linear output function is possible.…”