Search

The paper addresses a state space realization problem of a set of higher order delta-differential input–output equations, defined on a homogeneous time scale. The algebraic framework of differential one-forms is applied to formulate necessary and sufficient solvability conditions. This approach applies the total differential operator to analytic system equations to obtain the infinitesimal system description in terms of one-forms. This representation can be converted into polynomial system description by interpreting the polynomial indeterminate as the delta derivative acting on one-forms. The system description in terms of two matrices over skew polynomial ring is then used to derive explicit formulas for the differentials of state coordinates that significantly simplify the calculations. The formulas are found from the left quotients computed by the left Euclidean division algorithm. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to solve reduction and realization problems for discrete-time multi-input multi-output nonlinear control systems by applying the theory of non-commutative polynomials. First, the necessary and sufficient reducibility condition is presented in terms of the greatest common left divisor of two polynomial matrices associated with the set of the higher order input–output (i/o) difference equations of the system. The condition also provides a method for system reduction, i.e. for finding the irreducible representation of the set of the i/o equations, being transfer equivalent to the original system representation. Second, to solve the realization problem, a formula is presented for computing the differentials of the state coordinates directly from the polynomial description of the nonlinear system. The polynomial approach addressed in this paper is more direct and requires noticeably less computations than earlier methods represented in terms of subspaces of differential one-forms. [Copyright &y& Elsevier]

The nonlinear realization theory is recasted for time-varying single-input single-output nonlinear systems. The concept of realization has been extended to cover also the realizations with order greater than the order of input–output equation. The minimal realization problem is studied. The state realization is said to be minimal if it is either accessible and observable or its state dimension is minimal. In the linear case the two definitions are equivalent, but not for nonlinear time-invariant systems. It is shown that the two definitions remain equivalent for nonlinear systems under certain technical assumptions. Two alternative methods are presented for finding the minimal realization. [ABSTRACT FROM AUTHOR]

This paper addresses two related topics. First, a nonrealizable set of input–output equations can be made realizable in the classical state space form by adding a postcompensator whereas this is impossible in the continuous-time case. Construction of such postcompensator of minimal order is presented. Second, a general subclass of discrete-time systems is suggested, for which the observable subspace is non-integrable. The construction is based on the nonrealizable system, which is realized by adding a postcompensator, and then combined with the output of the original system. [ABSTRACT FROM AUTHOR]