Minimal realization of nonlinear MIMO equations in state-space form: Polynomial approach

Explicit formulas for the state coordinates in nonlinear MIMO realization problem on homogeneous time scales

“…By condition (8) the subspace H 4 is not closed, and therefore, the i/o equations (18) do not admit the classical state-space realization, until the time scale is not specified. However, in [3] it was shown that the system is realizable in the continuous-time case, i.e. for T = R.…”

“…Again, these replacements are not applied automatically, since the polynomial object OreP has no information about them. Addition of polynomials may be performed by Mathematica standard "+" operator, but multiplication requires the special function OreMultiply[p,q,R], which computes the product of polynomials p and q using commutation rule (3). The left division of polynomials p and q may be performed by LeftQuotientRemainder[p,q,R].…”

“…Adapting these programs for systems defined on homogeneous time scales required a serious improvement of the low-level functions operating with the elements of K . For instance, the forward shift 3 and backward shift functions were supplemented so that they could perform jump operators and ordinary time derivative was replaced by delta-derivative function. However, the polynomial functions, like multiplication and left division, required very few modifications.…”

Practical polynomial formulas in MIMO nonlinear realization problem

“…Note that the results of this paper may be also understood as the generalization of the results in [13] and [4] where the results for shift-operator-based discrete-time and continuoustime cases are presented. However, the results for differenceoperator-based case are new.…”

“…If the input u applied to (3) (for any initial state x 0 ) generates an output y such that u and y satisfy equation (4), the set of i/o equations (4) is called realizable and (3) is called a realization of (4).…”