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en:research_ts [2014/10/02 18:55] Juri Belikov |
en:research_ts [2014/10/03 14:07] (current) Juri Belikov |
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<div classes #research_head_main>Control systems on time scales</div> | <div classes #research_head_main>Control systems on time scales</div> | ||
- | In the control theory some of the results obtained for continuous and discrete time are very similar so a natural question is whether these two cases may be unified in one more general theory. Calculus on time scales, originated in 1988 by Stefan Hilger, seems to deliver a perfect language for unification. A time scale is a model of time and it is defined as an arbitrary nonempty closed subset of the reals. Since there are infinitely many time scales, we can work with, the other feature of the time scales calculus is extension. Therefore the main goal is to unify and extend the theoretical and computational tools for analysis of nonlinear control systems defined on time scales. Up to now, the algebraic formalism of differential one-forms, associated to nonlinear control systems defined on homogeneous time scales, has been developed. Since homogeneous time scales are models of continuous or uniformly sampled time (discrete time), the developed formalism unifies the existing theories for continuous- and discrete-time systems. Recently, the realization problem for single-input single-output nonlinear delta differential equations defined on homogeneous time scales has been studied by using the language of differential forms. Necessary and sufficient conditions have been given for the existence of a state-space realization of the nonlinear input-output delta differential equations. Moreover, we have obtained necessary and sufficient conditions for irreducibility of nonlinear input-output delta differential equations defined on homogeneous time scales both in terms of subspaces of one-forms, classified according to their relative degrees, and in terms of the common left factor of two differential polynomials describing the behavior of the system. Recently, the framework has been extended for non-homogeneous but regular time scales. Our future goal is to extend the results described above for a general time scale. | + | In the control theory some of the results obtained for continuous and discrete-time are very similar. So, a natural question is whether these two cases may be unified in one more general theory. Calculus on time scales, originated in 1988 by Stefan Hilger, seems to deliver a perfect language for unification. A time scale is a model of time and it is defined as an arbitrary nonempty closed subset of the reals. Since there are infinitely many time scales, we can work with, the other feature of the time scales calculus is extension. Therefore, the main goal is to unify and extend the theoretical and computational tools for analysis of nonlinear control systems defined on time scales. |
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+ | Up to now, the algebraic formalism of differential one-forms, associated to nonlinear control systems defined on homogeneous time scales, has been developed. Since homogeneous time scales are models of continuous or uniformly sampled time (discrete-time), the developed formalism unifies the existing theories for continuous- and discrete-time systems. Recently, various nonlinear control theory problems were studied. In particular, the realization problem for single-input single-output nonlinear delta differential equations defined on homogeneous time scales has been studied by using the language of differential forms. Necessary and sufficient conditions have been given for the existence of a state-space realization of the nonlinear input-output delta differential equations. Moreover, we have obtained necessary and sufficient conditions for irreducibility of nonlinear input-output delta differential equations defined on homogeneous time scales both in terms of subspaces of one-forms, classified according to their relative degrees, and in terms of the common left factor of two differential polynomials describing the behavior of the system. Recently, the framework has been extended for non-homogeneous but regular time scales. Our future goal is to extend the results described above for a general time scale. | ||