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Elementary operation approach to Fornasini–Marchesini state-space model realization of multidimensional systems

Abstract

This paper proposes a novel realization approach to generate a low-order Fornasini–Marchesini (F-M) (second) model for multidimensional (n -D) systems. First, based on two developed matrix relation properties, the F-M model realization problem is converted to establishing a desired standard matrix by performing elementary operations on an associated matrix initialized from the given transfer matrix. Then, a general constructive procedure and two corresponding techniques to compute the F-M models are established for both the left and right matrix fractional descriptions (MFDs) of the given transfer matrix. It turns out that compared with the existing methods the proposed approach can produce an F-M model with lower order, and furthermore, this is the first method that can treat both the left and right MFDs for obtaining an F-M model realization in a unified manner. Non-trivial examples are given to illustrate the details and effectiveness of the proposed approach.

Introduction

Multidimensional (n-D) systems theory has been receiving considerable attention in the past few decades (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]), due to the potential value and extensive applications in image processing, wireless communication, complex dynamic systems, biological and medical sciences, etc. One of the fundamental problems in the field of n-D systems is the realization of a given transfer (function) matrix by a certain n-D local state-space model, typically by means of the Roesser model or the Fornasini–Marchesini (F-M) (second) model [19], [20], [21]. Unlike the one-dimensional (1-D) case, it is very difficult to obtain a minimal realization for n-D systems in general (see, e.g., [19], [20], [22], [23], [24]), since the realization order is quite sensitive to the specific structure of the given transfer matrix, e.g., the coefficients of the polynomials or the matrix fractional descriptions (MFDs) [25]. Thus, for a general n-D transfer matrix, it is desirable to obtain a state-space realization with lowest possible order.

The realization problem of the Roesser model in n-D systems has been investigated rather extensively (see, e.g., [10], [26], [27], [28]). One representative approach among them is the elementary operation approach, which has shown that the realization problem of a given n-D transfer matrix of a multi-input and multi-output (MIMO) system can be formulated as an elementary operation problem of an associated n-D polynomial matrix initialized from the given transfer matrix. For a certain given transfer matrix, the realization orders obtained from its left and right MFDs may differ greatly so that, in order to obtain a lower order realization, it is necessary to compute the models from the left and right MFDs, respectively, and then choose the one with a lower order [29].

It has been shown in [29] that the elementary operation approach is effective on both the left and right MFDs of the given transfer matrix due to the duality of the Roesser model. In the F-M model case, however, the duality between a transfer matrix and its transpose does not hold in general [12]. Thus, in order to obtain an F-M model with lowest possible order, it is necessary to fully explore the structural properties of the left and right MFDs and then some efforts [8], [12], [20], [30], [31] have been made based on the left MFD or the right MFD, separately. The first attempt for the n-D (n ≥ 3) F-M model realization has been made in [30] based on the so-called resolvent invariant space, which unfortunately usually generates an F-M model with rather high order as identified in [8]. To reduce the unnecessarily high order in [30], some further improvements have been suggested in [8], [20] by taking into account the row structural properties of the rational matrix associated with the left MFD of the given system. Considering that the approaches in [8], [20], [30] are, in fact, based on the left MFD, an innovative approach w.r.t. columns of transfer matrix has been obtained recently in [12] to deal with the right MFD of an n-D transfer matrix. However, all these approaches [8], [12], [20], [30] treat only the left MFD or only the right MFD, with great difficulty of dealing with the left and right MFDs unifiedly.

Moreover, due to the substantial different structures between the F-M model and the Roesser model, the elementary operation approach developed for the Roesser model cannot be straightforwardly applied to the F-M model case and further investigation and significant modification are necessary. Motivated by this fact, the purpose of this paper is to establish an elementary operation realization approach to the F-M model realization. The contributions of this paper are twofold. First, two matrix relation properties, which originate from [19] for the Roesser model, are successfully extended to the F-M model such that the F-M model realization problem of a given n-D transfer matrix can also be formulated as an elementary operation problem. Second, two different n-D polynomial matrices are established as the initially associated matrices based on the left and right MFDs of a given transfer matrix, respectively, which provide an effective way to treat these two kinds of MFDs in a unified manner, so that the difficulty encountered by all the existing F-M model realization methods [8], [12], [20], [30] can be significantly overcome for the first time. It turns out that the proposed approach always guarantees a lower order for the F-M model realization. Moreover, a general constructive procedure and two techniques for F-M model realization will be established, which can be implemented by a computer program, e.g., MATLAB or Maple, such that the realization can be generated automatically.

This paper is organized as follows. Some preliminaries are presented in the next section. In Section 3, two matrix relation properties for the associated n-D polynomial matrices are stated for the realization of a given n-D transfer matrix in the F-M model, then an elementary operation approach to the F-M model realization is proposed by following these properties. In Section 4, a general constructive procedure and two techniques are then proposed for the lower order F-M realization based on the left and right MFDs of the given transfer matrix, respectively. In Section 5, two examples and further analyses are given to illustrate the effectiveness and properties of the new proposed approach. Finally, conclusions are given in Section 6.

Section snippets

The F-M state-space model

The F-M model for an n-D MIMO linear discrete system is described by [23], [32], [33] x ( i 1 + 1 , i 2 + 1 , … , i n + 1 ) = A 1 x ( i 1 , i 2 + 1 , … , i n + 1 ) + ⋯ + A n x ( i 1 + 1 , … , i n − 1 + 1 , i n ) + B 1 u ( i 1 , i 2 + 1 , … , i n + 1 ) + ⋯ + B n u ( i 1 + 1 , … , i n − 1 + 1 , i n ) , y ( i 1 , … , i n ) = C x ( i 1 , … , i n ) + D u ( i 1 , … , i n ) , where x ( i 1 , … , i n ) ∈ R r is the (local) state vector, u ( i 1 , … , i n ) ∈ R q , y ( i 1 , … , i n ) ∈ R p are the input vector and the output vector, respectively, with A 1 , … , A n ∈ R r × r , B 1 , … , B n ∈ R r × q , C ∈R p ×r , D ∈R p ×q . The size of the state vector x ( i 1 , … , i n ) , i.e., r, is called the order or

F-M model realization based on elementary operations

It is worth noting that comparing Eq. (3) with Eqs. (5) and (6) gives H ˜ ( z 1 , … , z n ) = R ( M ) = M 21 M 11 − 1 M 12 by just letting M 21 ≜ C , M 11 ≜ I r − ∑ i = 1 n A i z i , M 12 ≜ ∑ i = 1 n B i z i . This fact reveals that for a given n-D strictly causal transfer matrix H ˜ ( z 1 , … , z n ) , once a polynomial matrix M r = [ I r − ∑ i = 1 n A i z i ∑ i = 1 n B i z i C X ] can be established by following Lemmas 1–2, then a corresponding F-M model ( A, B , C) can be readily extracted from Mr in Eq. (12) for H ˜ ( z 1 , … , z n ) . More specifically, we have an F-M model ( A, B , C) with A = ( A 1 , … , A n ) ,

Realization techniques for the F-M model of n-D systems

In the previous section, it has been revealed by Theorems 1 and 2 that the problem of an F-M model realization can be converted to finding an objective matrix Mr in Eq. (12) by performing appropriate elementary operations on an initial matrix Mp in Eq. (15) or M p + q in Eq. (21) of a given transfer matrix. This section is devoted to developing a general procedure and two techniques for such an implementation transformation. First, a general realization procedure for the F-M model realization is

Examples and analyses

In this section, two examples will be provided first to show more details on the proposed techniques and then some analyses will be given to reveal further properties of the new method.

Example 1

Consider the following strictly causal 3-D transfer matrix given by [29] in Example 3.1: H ˜ ( z 1 , z 2 , z 3 ) = [ a 1 z 1 + a 2 z 2 z 3 + a 3 z 3 2 1 + b 1 z 2 z 3 + b 2 z 2 z 3 2 a 4 z 3 + a 5 z 2 z 3 1 + b 3 z 3 + b 4 z 1 z 2 z 3 a 6 z 2 z 3 + a 7 z 3 2 1 + b 1 z 2 z 3 + b 2 z 2 z 3 2 a 8 z 3 + a 9 z 1 z 2 z 3 1 + b 3 z 3 + b 4 z 1 z 2 z 3 ] , which can be expressed by the following right MFD: H ˜ ( z 1 , z 2 , z 3 ) = N R ( z 1 , z 2 , z 3 ) D R − 1 ( z 1 , z 2 , z 3 ) where D R

Conclusions

Based on the matrix relation properties extended from [19], an elementary operation approach to the F-M model realization has been proposed by finding a desired standard matrix Mr in Eq. (12) from the initial matrix Mp in Eq. (15) or M p + q in Eq. (21), which are based on the left and right MFDs, respectively. Moreover, a general constructive procedure and two techniques to generate low-order F-M models have been established such that one can easily implement this kind of approach by a computer

Acknowledgment

This work was supported by the Natural Science Foundation of Gansu Province of China (18JR3RA286), the Fundamental Research Funds for the Central Universities (No.lzujbky-2018-128) and the Japan Society for the Promotion of Science (JSPS.KAKENHI15K06072).

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