Solution of discrete ARMA-representations via MAPLE

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Abstract

In Karampetakis et al. [Circuits Systems and Signal Processing 20 (2001) 89] closed formulae for the forward, backward and symmetric solutions of an ARMA-representation have been presented. Here these formulae are implemented in the symbolic computational language MAPLE and corresponding MAPLE code is provided.

Introduction

Consider the regular, discrete-time, autoregressive moving average (ARMA)-representation described by the matrix equationA(σ)y(k)=B(σ)u(k)whereA(σ)=A0+A1σ+⋯+AqσqR[σ]r×r,rankA(σ)=rB(σ)=B0+B1σ+⋯+BqσqR[σ]r×mwhere at least one of Aq, Bq is non-zero, σ denotes the advance operator (i.e. σiy(k)=y(k+i)), y(k):Z+Rr defines the output and u(k):Z+Rm defines the input of the system , .

In the case where A(σ)=σE−A∈R[σ]r×r and B(σ)=B∈Rr×m the ARMA-representation , reduces to the generalized state space (GSS)-representationEy(k+1)=Ay(k)+Bu(k)Further if E is non-singular (i.e. |E|≠0) then (1.2) reduces to the state space (SS)-representation.

In [3] closed formulae for the forward, backward and symmetric solutions of the ARMA-representation (1.1a) are presented. These formulae depend explicitly on the forward fundamental matrix Hk and the backward fundamental matrix Vk of A(z) both of which can be seen to be of fundamental importance in the analysis of discrete ARMA-representations. The closed formulae obtained can be seen to be natural extensions to the ones proposed by Lewis and Mertzios [4] for the GSS-representation (1.2) which correspondingly depend explicitly on the forward fundamental matrix φk and the backward fundamental matrix τk of (zEA).

In this paper we implement the closed formulae as given in [3] for the forward, backward and symmetric solutions of the ARMA-representation (1.1a) in the symbolic computational language MAPLE [1]. In Section 2 the definition and construction of Hk and Vk will be considered. In Section 3 the closed formulae for the solution of the ARMA-representation (1.1a) will be shown and these will be implemented via MAPLE in Section 4 where the corresponding MAPLE procedures will be presented. Finally in Section 5 these procedures will be illustrated via a numerical example.

Section snippets

Preliminaries

Consider the ARMA-representation (1.1a) and write it in the expanded formAqy(k+q)+⋯+A0y(k)=Bqu(k+q)+⋯+B0u(k)where we assume A(z) is regular (i.e. det[A(z)]≠0), k∈[0,N] and u(k) is non-zero for k=0,1,…,Nq. For such as case we can define the following

Definition 2.1

Forward fundamental matrix sequence of A(z)

Let the Laurent series expansion of A(z)−1 about z=∞ be given byA(z)−1=Hq̂rzq̂r+Hq̂r−1zq̂r−1+⋯+H1z+H0+H−1z−1+⋯where q̂r is the greatest order zero of A(z) at s=∞. Then the coefficient matrices HjRr×r, j⩽q̂r constitute the forward fundamental matrix

Solution of ARMA-representations

In [3] closed formulae for the forward, backward and symmetric solutions of the ARMA-representation (1.1a) were obtained. In this section each of these three types of solution are considered and the corresponding closed formulae provided.

Implementation via MAPLE

In this section we implement the formulae for the solution of the ARMA-representation (1.1a) as presented in Section 3, in the symbolic computational language MAPLE [1]. One obvious advantage of using MAPLE is that it enables the user to implement any one of the “built in” procedures inherent in it predominantly, in this case, those in the linear algebra package linalg which contains a collection of procedures for matrix manipulation. This results in further simplification of any program code

Examples

In this section we will consider the solution of the ARMA-representationσ2+5σ+6σ+102σ−53σ+210−10A(σ)y1(k)y2(k)y3(k)y(k)=001B(σ)u(k)using the MAPLE procedures presented in Section 4. Clearly from (5.1) we have q=2, r=3 and m=1.

The machine used is a SUN SPARC station10 (75 MHz SuperSPARC II). The last line of the output indicates the CPU time used in the computation. This is divided into three parts

  • (i)

    bytes used (integer) number of bytes of memory that have been requested up to that point in the

Conclusions

In this paper we have implemented the results presented in [3], concerned with the forward, backward and symmetric solution of an ARMA-representation, in the symbolic computational language MAPLE. We have also implemented the corresponding compatibility conditions for each such solution which define an admissible set of boundary conditions, in conjunction with the input sequence, for such a solution to exist. All the corresponding MAPLE code has been included and a concise illustrative example

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