# Characterization of a class of spatially interconnected systems (ladder circuits) using two-dimensional systems theory

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## Abstract

This paper considers a class of spatially interconnected systems formed by ladder circuits using two-dimensional systems theory. The individual circuits in this class are described by hybrid (continuous/discrete) linear differential/difference equations in time (continuous) and spatial (discrete) variables and therefore have a two-dimensional systems structure. This paper shows that a ladder circuit model and models for 2-D dynamics have a well defined equivalence property and hence analysis tools can be transferred between them. Also the mechanism for transforming one to the other is established.

## Keywords

Spatially interconnected systems Ladder circuit networks Two-dimensional systems Zero coprime system equivalence## 1 Introduction

Spatially interconnected systems arise in a number of areas (D’Andrea and Dullerud 2003) (as one possible starting point for the literature) including electrical ladder networks/circuits, mechanical systems, composed, e.g, of masses and springs and heat transfer problems. For these systems, time and spatial dynamics can arise. Hence they can be treated as two dimensional, or 2-D, systems, which is the setting used in this paper.

In many cases, these systems are composed of a finite number of blocks or cells. Hence, for analysis, there is the option to embed the spatial dynamics into the system variables using a form of lifting based on the discrete (finite) spatial variable and obtain a system where time is the independent variable, referred to as a 1D system in some of the 2-D systems literature, see, e.g. Sulikowski et al. (2015). One drawback of this approach, for systems composed of a large number of cells, is the need to compute with matrices that have very large dimensions and hence possible computational issues. The 2-D systems setting does not involve increasing the dimensions of the matrices used to represent the dynamics.

The dynamics of 2-D systems are characterized by information propagation in two independent directions and the independent variables can both be discrete, both continuous or mixed, i.e. one continuous and one discrete. Spatially interconnected systems, including ladder circuits, can be considered as a class of 2-D systems where the independent variables are the time, which can be continuous or discrete and the node/cell number, which has to be discrete. However, left–right and right–left dependence between neighboring cells are often needed. Such systems are not causal along the spatial axis, i.e. along the node/cell direction. Hence, they cannot be modeled by the standard and extensively investigated 2-D (Roesser 1975a) and (Fornasini and Marchesini 1976) state-space models. Also the well known stability analysis methods for these systems are not applicable.

Repetitive processes are another class of 2-D dynamics that arise in the modeling of physical systems. These processes operate over a finite time duration and the 2-D structure arises as repeated passes are made through the dynamics and the output on any pass explicitly contributes to the dynamics produced on the next one. Background on these processes and the control problems they pose can be found in Rogers et al. (2007).

The links between linear repetitive processes and the Roesser and Fornasini–Marchesini state-space models has been investigated in previous research. For example, in Galkowski et al. (1998) the conditions for local controllability of linear repetitive processes have been derived via their transformation into the singular 2-D Roesser or Fornasini–Marchesini form and in Galkowski et al. (1999), building on previous work cited in this reference, the equivalence of their stability properties was investigated. This previous research established stability tests can, in some but not all, cases be interchanged. This is true for one class of repetitive processes but other classes can exhibit dynamics that have no Roesser or Fornasini–Marchesini state-space model representation. More recent work in this general area includes (Boudellioua et al. 2016, 2017; Galkowski et al. 2017) where new results on system equivalence are reported.

It is to be expected that at least some of these results would apply to spatially interconnected systems whose dynamics are written as 2-D linear systems. This is the motivation for the current paper where system equivalence between the state-space models of ladder circuits and those for the singular 2-D Roesser or Fornasini–Marchesini systems. The problem of reducing by equivalence a general 2-D polynomial matrix to the singular Roesser form has been considered previously, e.g. Pugh et al. (2005) developed a method for reducing a general 2-D polynomial matrix to such a form. Their method uses a two-step algorithm which is then adapted to the case of a general polynomial system matrix. In this paper, a direct method is developed to reduce a 2-D polynomial system matrix arising from a class of ladder circuit systems to an equivalent singular 2-D Fornasini–Marchesini or Roesser state-space model. The transformation used is shown to be zero coprime system equivalence of the two system matrices. This type of equivalence has previously been considered in the literature, e.g. Levy (1981), Johnson (1993), Boudellioua (2012), Pugh et al. (1996) and Pugh et al. (1998).

## 2 State-space models and associated transfer functions

In this paper, the general problem considered is the equivalence between ladder circuits and commonly used Roesser and Fornasini–Marchesini models for 2-D linear systems. In the remainder of this section, the relevant state-space models are introduced.

### 2.1 Ladder circuits

*x*(

*p*,

*t*) and

*u*(

*p*,

*t*), respectively, denote the state and input vectors. In this paper, the particular case of an active ladder circuit of the from of Fig. 1 is considered, which for onward analysis is considered in the form of Fig. 2, where the controlled sources \(i(p,t)=\gamma U_c(p-1,t)\) and \(E(p,t)=u(p,t)\) have been added to the nodes as possible control input variables but could also be an intrinsic part of a particular circuit.

*p*in Fig. 2 is defined as

### 2.2 2-D singular Fornasini–Marchesini model

*x*(

*i*,

*j*) is the state vector,

*u*(

*i*,

*j*) is the input vector,

*y*(

*i*,

*j*) is the output vector,

*E*, \(A_{0}\), \(A_{1}\), \(A_{2}\),

*B*,

*C*and

*D*are constant real matrices of compatible dimensions and

*E*is a singular matrix. In this paper, the model (1) is continuous in time

*t*and therefore the differential version of singular 2-D Fornasini–Marchesini state-space model, is required, i.e.

*p*denotes the node number and the matrices are as in (1). If

*E*is nonsingular in either model then the standard (or nonsingular) model is obtained.

### Note 1

In this paper differential dynamics are considered. If, however, the system is sampled then the analysis still applies with the shift operator \(z_{1}\) in (6) defined as a forward shift operator in *t*.

### 2.3 2-D singular Roesser model

*E*is square and singular. In this model, static in both directions

*i*and

*j*links between sub-vectors are allowed. If

*E*is nonsingular then (as in the Fornasini-Marchesini model) the standard model is obtained. The boundary conditions are \(x^h(0,j) = f(j),\, j \ge 0\) and \(x^v(i,0) = d(i),\, i \ge 0,\) where the \(n_1 \times 1\) vector

*f*(

*j*) and the \(n_2 \times 1\) vector

*d*(

*i*) have known constant entries.

## 3 Polynomial system matrix descriptions and system equivalence

*T*,

*U*,

*V*and

*W*are polynomial matrices with elements in \({\mathbb {R}}[z_1,z_2]\) of dimensions \(n\times n, n\times l, m\times n\) and \(m\times l,\) respectively. The meaning of the operators \(z_1\) and \(z_2\) depend on the case considered as detailed in the previous section.

*n*-D systems. One of these for 2-D system matrices is zero coprimeness, see, e.g. Levy (1981), and Johnson (1993). This equivalence may be viewed as an extension of Fuhrmann’s strict system equivalence (Fuhrmann 1977) from 1-D to 2-D systems and is defined as follows.

### Definition 1

The system matrix can be used to study critical systems properties such as controllability, observability and stability. In the case of 2-D linear systems, the zero structure of their associated system matrices, see, e.g. Zerz (2000), Johnson (1993), Levy (1981), Pugh et al. (1998), and Pugh et al. (1996), is of particular interest. One result is the following.

### Lemma 1

## 4 Transformation of the ladder circuit model to a differential Fornasini–Marchesini model

### Theorem 1

### Proof

## 5 Equivalence of the ladder circuit and the differential singular Roesser model

### Theorem 2

### Proof

## 6 Conclusions

In this paper, an equivalent representation is obtained in the form of 2-D Fornasini–Marchesini and Roesser singular state-space models for a given system matrix arising from a hybrid linear ladder circuit system. The exact connections between the original system matrix with its corresponding 2-D singular forms have been developed and shown to be zero coprime system equivalence. Also the zero structure of the original polynomial system matrix is preserved, making it possible to analyze the polynomial system matrix in terms of its associated 2-D singular form. Moreover, the transformation matrices include identity sub-matrices and this suggests that these transformations can be generated by finite sequences of elementary row/column operations together with trivial inflation/deflation of the polynomial system matrices. This area is the subject of ongoing research. Also the implications of these results in terms of the structure and design of control laws is also under investigation.

## Notes

### Acknowledgements

This work has been supported by Sultan Qaboos University (Oman) and also by the National Science Centre in Poland, Grant No. 2015/17/B/ST7/03703.

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