Algebraic-difference equation, behavior, exact modeling, auto-regressive representation, discrete time system, higher order system, descriptor system
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.
Moysis, Lazaros and Karampetakis, Nicholas.
"Algebraic Methods for the Construction of Algebraic-Difference Equations With Desired Behavior",
Electronic Journal of Linear Algebra,
Volume 34, pp. 1-17.