Algebraic Methods for the Construction of Algebraic-Difference Equations With Desired Behavior

For a given system of algebraic and difference equations, written as an Auto-Regressive (AR) representation $A(\sigma)\beta(k)=0$, where $\sigma $ denotes the shift forward operator and $A\left( \sigma \right) $ a regular polynomial matrix, the forward-backward behavior of this system can be constructed by using the finite and infinite elementary divisor structure of $A\left( \sigma \right) $. This work studies the inverse problem: Given a specific forward-backward behavior, find a family of regular or non-regular polynomial matrices $A\left( \sigma \right) $, such that the constructed system $A\left( \sigma \right) \beta \left( k\right) =0$ has exactly the prescribed behavior. It is proved that this problem can be reduced either to a linear system of equations problem or to an interpolation problem and an algorithm is proposed for constructing a system satisfying a given forward and/or backward behavior.

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Construction of Algebraic and Difference Equations with a Prescribed Solution Space

International Journal of Applied Mathematics and Computer Science ◽

10.1515/amcs-2017-0002 ◽

2017 ◽

Vol 27 (1) ◽

pp. 19-32 ◽

Cited By ~ 1

Author(s):

Lazaros Moysis ◽

Nicholas P. Karampetakis

Keyword(s):

Difference Equations ◽

Time Horizon ◽

Polynomial Matrix ◽

Solution Space ◽

Elementary Divisor ◽

Polynomial Matrices ◽

Auto Regressive ◽

Finite Time Horizon ◽

Ar Representations ◽

Forward Operator

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.

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On the exact modelling of linear systems

IMA Journal of Mathematical Control and Information ◽

10.1093/imamci/dnz021 ◽

2019 ◽

Vol 37 (3) ◽

pp. 730-751

Author(s):

Georgia G Pechlivanidou ◽

Nicholas P Karampetakis

Keyword(s):

Inverse Problem ◽

Differential Operator ◽

Programming Language ◽

Computer Algebra ◽

Continuous Time ◽

Computer Algebra System ◽

Polynomial Matrix ◽

Main Theme ◽

Polynomial Matrices ◽

System A

Abstract It is well known that given the continuous-time AutoRegressive representation $A\left ( \rho \right ) \beta \left ( t\right ) =0,$ where $\rho $ denotes the differential operator and $A\left ( \rho \right ) $ a regular polynomial matrix, we can always construct the smooth behaviour of this system, by using the finite zero structure of $A\left ( \rho \right ) $. The main theme of this work is to study the following inverse problem: given a specific smooth behaviour, find a family of regular polynomial matrices $A\left ( \rho \right ) $, such that the system $A\left ( \rho \right ) \beta \left ( t\right ) =0$ has exactly the prescribed behaviour. Following an idea coming from Antoulas & Willems (1993) and Willems (1986, 1991) we present an algorithm which solve this problem and can be easily implemented either in a computer programming language like C++ or in a computer algebra system like Mathematica.

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