Elementary Operation Approach Research Articles

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

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.

Order evaluation to new elementary operation approach for MIMO multidimensional systems

ABSTRACTA new elementary operation approach (NEOA) for Roesser model realisation of multi-input and multi-output (MIMO) multidimensional (n-D) systems has been recently proposed, which can generate a lowest possible realisation but requires trying different processing manners for various possible realisations with different structures and orders. This paper develops two kinds of novel order evaluation methods and the corresponding algorithms for the realisation procedures A and B of the NEOA, respectively, such that the best processing manner for the lowest possible realisation can be determined with much less computational complexity. These new results enable us, for the first time, to know exactly the order of a realisation before actually conducting the realisation, and thus provide us a novel and high-efficiency way to explore the lowest possible realisation problem for MIMO n-D systems. Examples are given throughout this paper to illustrate the main idea as well as effectiveness of the proposed methods.

A novel elementary operation approach with Jordan transformation to order reduction for Roesser state-space model

This paper proposes a novel elementary operation approach to order reduction for the Roesser state-space model of multidimensional (n-D) systems by introducing a new kind of transformation, i.e., the Jordan transformation, which guarantees the establishment of an objective matrix with more general structure than the existing one. Then two basic order reduction techniques are developed which can overcome the difficulty encountered by the existing methods and reveal, for the first time, the fact that the order reduction is still possible even when the column (or row) blocks in the related n-D polynomial matrix are full rank. Furthermore, based on the Jordan transformation, an equivalence relationship between two Roesser models after using the elementary operations among the different blocks will be clarified. Although these operations do not directly lower the total order of the model, the partial orders can be changed so that it may nevertheless yield a possibility for further order reduction. It turns out that this new approach includes our previous elementary operation order reduction approach just as a special case. Examples are given to illustrate the details as well as the effectiveness of the proposed approach.

Elementary Operation Approach to Order Reduction for Roesser State-Space Model of Multidimensional Systems

This paper proposes a new order reduction approach for Roesser state-space model of multidimensional ( n-D) systems based on elementary operations, by inversely applying the basic idea adopted in the new elementary operation approach to the Roesser model realization of n-D systems. It will be shown first that the n-D order reduction problem can be formulated into an elementary operation problem of an n-D polynomial matrix obtained from the coefficient matrices of the given Roesser model. Based on this problem formulation, a basic order reduction procedure and three techniques are presented, by which the intrinsic relationship among the blocks with respect to different variables can be investigated to achieve a further possible order reduction. It turns out that the new proposed approach is applicable to a wider class of Roesser models than the existing reduction approaches and provides a possible way to explore the equivalence between two systems. Examples are given to illustrate the main idea as well as the effectiveness of the proposed approach.

A New Elementary Operation Approach to Multidimensional Realization and LFR Uncertainty Modeling: The MIMO Case

This paper proposes a new elementary operation approach (EOA) to multidimensional (-D) realization and linear fractional representation (LFR) modeling for multi-input and multi-output (MIMO) -D systems, as an extension of the new EOA proposed for the single-input and single-output (SISO) case by the authors recently. It is shown that, due to the substantial differences between the SISO and MIMO systems, the extension is not straightforward and further significant development is necessary. A matrix relation property among the associated matrices under the augmenting and admissible elementary operations is first revealed. Based on this matrix relation property, the realization problem for the MIMO -D case is formulated as an elementary operation problem of a certain -D polynomial matrix, which makes the extension possible. General constructive procedures are then established for the regular realizations based on the right and left matrix fraction descriptions (MFDs) of a given transfer matrix, respectively, such that one can easily implement this approach by a computer program in, e.g., MATLAB or Maple. Numerical and symbolic examples are provided to illustrate the main ideas and the effectiveness of the proposed approach.

A new elementary operation approach to multidimensional realization and LFR uncertainty modeling: the SISO case

In this paper, a new elementary operation approach is proposed to the problem of multidimensional (n-D) Roesser state-space model realization or linear fractional representation (LFR) uncertainty modeling. The new approach can overcome the main difficulties encountered by Galkowski's approach and always generates a standard or regular realization for a given causal n-D transfer function. In particular, the n-D realization problem is formulated as an elementary operation problem of a certain n-D polynomial matrix in a way totally different from Galkowski's approach so that the singularity problem can be completely avoided. A general constructive realization procedure and an algorithm for evaluating the realization order associated with this procedure will be proposed, which can be easily implemented by symbolic software in, e.g., MATLAB or Maple. Some further techniques for a realization with lower order will also be shown. Symbolic and numerical examples will be presented throughout the paper to illustrate the basic ideas as well as the effectiveness of the proposed approach.

State-space realizations of MIMO 2D discrete linear systems Elementary operation and variable inversion approach

Elementary operations are an alternative to the polynomial system matrix for studying basic systems theoretic properties of standard (1D) linear systems, such as state-space realizations. In the general area of nD systems, there is a basic need to analyse both rational and polynomial matrices in n indeterminates. This paper develops key results on using elementary operations and variable inversions to produce equivalent state-space realizations of MIMO 2D systems.

Elementary operation approach to state-space realizations of 2-D systems

In this paper, the method for obtaining various state-space realizations of the Roesser type for an assigned two-variable transfer function is developed. Two-variable elementary polynomial operations are employed. A new version of the general, state-space model is constructed. One also throws some new light on the problem of the mutual connections among the different 2-D state-space models such as the Roesser, the Fornasini-Marchesini, and the general one. Singular as well as standard (nonsingular) models are considered.