Zbigniew Bartosiewicz

Noether's Theorem on Time Scales ?

Abstract We show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective Euler–Lagrange extremals.

Realizations of Nonlinear Control Systems on Time Scales

Nonlinear partially defined systems on an arbitrary unbounded time scale are studied. They include continuous-time and discrete-time systems. The main problem is to find necessary and sufficient conditions for an abstract input/output map to have a realization as a nonlinear system of a specific class on the time scale. The obtained results extend criteria of realizability of continuous-time polynomial systems. A simple construction of a realization is provided.

Irreducibility, reduction and transfer equivalence of nonlinear input-output equations on homogeneous time scales

The purpose of this paper is to present a necessary and sufficient condition for irreducibility of nonlinear input–output delta differential equations. The condition is presented in terms of the common left divisor of two differential polynomials describing the behaviour of the system defined on a homogenous time scale. The concept of reduction is explained. Subsequently, the definition of transfer equivalence based upon the notion of an irreducible differential form of the system is introduced, inspired by the analogous definition for continuous-time systems.

Control systems on regular time scales and their differential rings

The paper describes an algebraic construction of the inversive differential ring, associated with a nonlinear control system, defined on a nonhomogeneous but regular time scale. The ring of meromorphic functions in system variables is constructed under the assumption that the system is submersive, and equipped with three operators (delta- and nabla-derivatives, and the forward shift operator) whose properties are studied. The formalism developed unifies the existing theories for continuous- and discrete-time nonlinear systems, and accommodates also the case of non-uniformly sampled systems. Compared with the homogeneous case the main difficulties are noncommutativity of delta (nabla) derivative and shift operators and the fact that the additional time variable t appears in the definition of the differential ring. The latter yields that the new variables of the inversive closure, depending on t, have to be chosen to be smooth at each dense point t of the time scale.

On stabilisability of nonlinear systems on time scales

In this article, stabilisability of nonlinear finite-dimensional control systems on arbitrary time scales is studied. The classical results on stabilisation of nonlinear continuous-time and discrete-time systems are extended to systems on arbitrary time scales with bounded graininess function. It is shown that uniform exponential stability of the linear approximation of a nonlinear system implies uniform exponential stability of the nonlinear system. Then this result is used to show a similar implication for uniform exponential stabilisability.

Linear Input-Output Equivalence and Row Reducedness of Discrete-Time Nonlinear Systems

The problem of linear input-output (i/o) equivalence of mero morphic nonlinear control systems, described by implicit higher order difference equations, is studied. It is proved that any system is linearly i/o equivalent to a row-reduced form. The constructive algorithm is given for finding the required transformation. The latter amounts to 1) multiply the set of i/o equations ψ = 0 from left by a unimodular matrix A(δ), whose entries are non-commutative polynomials in the forward-shift operator δ, and 2) define certain multiplicative subset of the difference ring of analytic functions which introduces some inequations that should be satisfied.

Stability, stabilization and observers of linear control systems on time scales

Linear control systems defined on arbitrary time scales are studied. It is shown that the classical results on stabilization and detectability for linear continuous-time and discrete-time systems can be extended to systems on arbitrary time scales. These results depend on the exponential stability criteria, which are different for different time scales. The set of exponential stability, which appears in these criteria, is studied. It is shown that it may be empty, which leads to some pathologies in the system behavior.

Observability of linear positive systems on time scales

Two observability concepts for linear positive systems on time scales are studied. Necessary and sufficient conditions for both properties are established. Positive observability is characterized with the aid of a generalized observability Gram matrix. Particular cases of the continuous time scale and the discrete homogeneous time scale are studied in detail.

DYNAMIC FEEDBACK EQUIVALENCE OF NONLINEAR SYSTEMS ON TIME SCALES

Abstract The problem of dynamic feedback equivalence of nonlinear control systems on time scales is studied. Time scale is a model of time. Two most important cases are the real line (continuous time) and the set of integers (discrete time). Control systems on time scales include continuous-time and discrete-time systems. The delta algebra of a nonlinear control system is introduced. The main result says that two systems defined on a time scale are dynamically feedback equivalent if and only if their delta algebras are isomorphic. This is an extension of continuous-time and discrete-time versions.

Lyapunov functions in stability of nonlinear systems on time scales

Uniform stability and uniform asymptotic stability of nonlinear finite-dimensional systems on arbitrary time scales are studied. Sufficient as well as necessary conditions of stability are given with the aid of Lyapunov functions. They extend known stability criteria for continuous-time and discrete-time systems. The Massera lemma, needed to construct a Lyapunov function for uniformly asymptotically stable systems, is extended to time scales. The role played by the graininess function associated to the time scale is discussed. Some peculiarities that appear for time scales with an unbounded graininess function are exhibited.