Maris Tõnso

Transfer Equivalence and Realization of Nonlinear Input-Output Delta-Differential Equations on Homogeneous Time Scales

Nonlinear control systems on homogeneous time scales are studied. First the concepts of reduction and irreducibility are extended to higher order delta-differential input-output equations. Subsequently, a definition of system equivalence is introduced which generalizes the notion of transfer equivalence in the linear case. Finally, the necessary and sufficient conditions are given for the existence of a state-space realization of a nonlinear input-output delta-differential equation.

Irreducibility conditions for discrete-time nonlinear multi-input multi-output systems

Abstract The purpose of this paper is to present necessary and sufficient condition for irreducibility of discrete-time nonlinear multi-input multi-output system which extends directly the corresponding condition for the linear case. The condition is presented in terms of the greatest common left divisor of two polynomial matrices related to the input-output equations of the system. The basic difference is that unlike the linear case the elements of the polynomial matrices belong to a non-commutative polynomial ring. This condition provides a bases for finding the equivalent minimal irreducible representation of the i/o equations which is a suitable starting point for constructing an observable and accessible state space realization.

Feedback linearization and lattice theory

Abstract The tools of lattice theory are applied to readdress the static state feedback linearization problem for discrete-time nonlinear control systems. Unlike the earlier results that are based on differential geometry, the new tools are also applicable for nonsmooth systems. In case of analytic systems, close connections are established between the new results and those based on differential one-forms. The Mathematica functions have been developed that implement the algorithms/methods from this paper.

Realization of discrete-time nonlinear input-output equations: polynomial approach

The algebraic approach of differential one-forms has been applied to study the realization problem of nonlinear input-output equations in the classical state space form, both in continuous- and discrete-time cases. Slightly different point of view in the studies of nonlinear control systems is provided by the polynomial approach in which the system is described by two polynomials from the non-commutative ring of skew polynomials that act on input and output differentials. Polynomial approach has been used so far to study the problems of reduction, input-output and transfer equivalence. The aim of the present paper is to apply the polynomial approach also to the realization problem. This allows to simplify the step-by-step algorithm given in terms of the sequence of subspaces of differential one-forms to check realizability and calculate the state coordinates in case the system is realizable. A new formula is presented which allows to compute the subspaces of one-forms directly from the polynomial description of the nonlinear system. The above method is noticeable less time-consuming, more direct and therefore better suited for implementation in computer algebra packages like Mathematica or Maple.

Realization of nonlinear MIMO system on homogeneous time scales

Abstract 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.

Irreducibility Conditions for Continuous-time Multi-input Multi-output Nonlinear Systems

The purpose of this paper is to present necessary and sufficient condition for irreducibility of continuous-time nonlinear multi-input multi-output system. The condition is presented in terms of the greatest common left divisor of two polynomial matrices related to the input-output equations of the system. The basic difference is that unlike the linear case the elements of the polynomial matrices belong to a non-commutative polynomial ring. This condition provides a basis for finding the equivalent minimal irreducible representation of the I/O equations which is a suitable starting point for constructing an observable and accessible state space realization

TRANSFER EQUIVALENCE AND REALIZATION OF NONLINEAR HIGHER ORDER INPUT/OUTPUT DIFFERENCE EQUATIONS USING MATHEMATICA

This paper presents a contribution to the development of symbolic computation tools for discrete-time nonlinear control systems. A set of functions is developed in Mathematica 3.0 that test if the higher order input/output difference equation is realizable in the classical state-space form, and for simple examples, also find such state equations. The approach relies on a new notion of equivalence of higher order difference equations which yields a minimal (i.e. accessible and observable) realization and generalizes the notion of transfer equivalence to the nonlinear case. The application of the developed functions is demonstrated on three examples obtained via identification.

Adjoint Polynomial Formulas for Nonlinear State-Space Realization

This paper focuses on computational aspects of the realization of nonlinear multi-input multi-output systems. Instead of the algorithmic solutions, provided in earlier works, the explicit formulas are presented, which enable to compute the differentials of the state coordinates directly from the polynomial description of the nonlinear system. The solution is based on the concept of adjoint polynomials and requires a minimal amount of computations. The formulas are implemented in computer algebra system Mathematica and made available online via web Mathematica tools.

Comparison of LPV and nonlinear system theory: A realization problem

Abstract The paper explores the utilization of the nonlinear realization theory to address the problem of transforming linear parameter-varying input–output (LPV-IO) equations into a state-space form with static dependence on the so-called scheduling parameter. The necessary and sufficient solvability conditions are given, and three additional subclasses of LPV-IO equations are suggested that are guaranteed to have a realization of the considered type.

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

The realization of nonlinear input-output equations in the classical state-space form can be studied by the polynomial approach in which the system is described by two polynomials from the non-commutative ring of skew polynomials. The aim of the present paper is to apply the polynomial methods to the realization problem. This allows to simplify the step-by-step algorithm based on certain sequences of subspaces of differential one-forms. The presented new formula allows to compute the differentials of the state coordinates directly from the polynomial description of the nonlinear system. This method is more clear, straight-forward and therefore better suited for implementation in different computer packages such as Mathematica or Maple. The developed theory and algorithm are illustrated by means of several examples.