Realization of nonlinear MIMO system on homogeneous time scales

Realization of nonlinear input-output equations in controller canonical form

“…From now on we make the assumption that H ∞ = {0}, which means that the system (1) is generically accessible in the sense that (1) does not admit autonomous elements in K, see [2]. Note that the latter is necessary for transforming a system (1) or (3) into the form (4), (5). Also, note that directly from the definition (7) of the sequence H i , one may conclude the following.…”

“…, n i − 1. Since the realization problem has a solution, the subspace H s+2 has to be integrable and one nth order state-space realization of (1) is (4), (5). Thus, by Theorem 3.3, the differentials of the states of system (4) form a basis for H s+2 .…”

“…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.…”

Krause's model of opinion dynamics on isolated time scales

“…However, with time, the new theory has found important applications in several fields that require modeling of discrete and continuous data simultaneously. We can mention here the calculus of variations, control theory, economics, and biology (e.g., [13][14][15][16][17][18][19][20][21][22][23][24]). For a general introduction to calculus on time scales and its applications, we refer the reader to the books [25,26].…”