<abstract xmlns="http://www.w3.org/1999/xhtml"><p>This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations <italic>A</italic>(<italic>σ</italic>)<italic>β</italic>(<italic>k</italic>) = 0, where <italic>σ</italic> denotes the shift forward operator and <italic>A</italic>(<italic>σ</italic>) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of <italic>A</italic>(<italic>σ</italic>). This work deals with the inverse problem of constructing a family of polynomial matrices <italic>A</italic>(<italic>σ</italic>) such that the system <italic>A</italic>(<italic>σ</italic>)<italic>β</italic>(<italic>k</italic>) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg <italic>et al</italic>. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.</p></abstract>

This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ)β(k) = 0, where σ denotes the shift forward operator and A(σ) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ). This work deals with the inverse problem of constructing a family of polynomial matrices A(σ) such that the system A(σ)β(k) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.

Construction of Algebraic and Difference Equations with a Prescribed Solution Space

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ)β(k) = 0, where σ...

Construction of Algebraic and Difference Equations with a Prescribed Solution Space

Published Online: May 04, 2017

Page range: 19 - 32

Received: May 25, 2016

Accepted: Oct 12, 2016

DOI: https://doi.org/10.1515/amcs-2017-0002

Keywords
algebraic and difference equations, behavior, exact modeling, auto-regressive representation, discrete time system, higher order system

© by Lazaros Moysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Construction of Algebraic and Difference Equations with a Prescribed Solution Space

Published Online: May 04, 2017

Page range: 19 - 32

Received: May 25, 2016

Accepted: Oct 12, 2016

DOI: https://doi.org/10.1515/amcs-2017-0002

Keywordsalgebraic and difference equations, behavior, exact modeling, auto-regressive representation, discrete time system, higher order system

© by Lazaros Moysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
algebraic and difference equations, behavior, exact modeling, auto-regressive representation, discrete time system, higher order system