Singular Roesser Models Research Articles

Admissible transformation approach to Roesser state‐space model realization of singular multidimensional systems

AbstractThe problem of the singular Roesser (state‐space) model realization of non‐causal multivariate transfer (function) matrices is investigated. Specifically, the notion of so‐called admissible transformation is introduced, which allows to introduce and investigate a novel invariant matrix relationship on polynomial matrices. The singular multidimensional Roesser model (SNDRM) realization is transformed to the problem of how to reduce an initial polynomial matrix associated with the given transfer matrix into the objective polynomial matrix with a specific structure. Construction procedures are developed to obtain SNDRMs by applying admissible transformations on the initial polynomial matrix. Moreover, a separated standard form is introduced for an SNDRM so that the proposed approach can effectively treat both the left and right matrix fractional descriptions of a non‐causal multivariate transfer matrix in a unified manner. Non‐trivial examples are provided to illustrate the details and effectiveness of the proposed approach.

On the reduction of repetitive processes into singular and non-singular Roesser models

In this paper we present new methods for the reduction of a polynomial system matrix describing a discrete linear repetitive process, to equivalent singular and non-singular 2-D state space representations. Particularly, a zero coprime system equivalence transformation resulting in a singular Roesser state space model, preserving the core algebraic structure of the original system matrix, is proposed. As a second step utilizing the singular Roesser model introduced, we further reduce the system to a non-singular, zero coprime system equivalent Roesser model. Both models are constructed by inspection or by applying elementary matrix manipulations and have significantly smaller dimensions compared to similar reductions found in the literature.

Stability analysis and stabilization of 2-D singular Roesser models

This paper addresses the problem of controller design for 2-D singular Roesser models. Sufficient stability conditions are formulated in terms of strict linear matrix inequalities (LMIs) for 2-D singular systems described by the Roesser models. Then a set of sufficient conditions for existence of desired a state feedback controller are obtained. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

The state observer and compensator of a large class of 2-D acceptable singular systems

Under the assumption of nonexistence of the jump-mode, this paper gives a simple form of a large class of 2-D singular systems. This class of systems includes all the systems resulted from the linear state transformations of 2-D singular Roesser models and hence we called it the extended singular Roesser models. The criteria are presented for testing that via some suitable invertible linear state transformations, the given 2-D singular systems in the extended Roesser models can be converted equivalently to the classical Roesser models. Some results on 2-D classical singular Roesser model such as the reduced order regular observer and compensator design are generalized to the extended models.

Duality of 2-D singular systems of Roesser models

In this paper, the properties and concepts of dual systems of the two-dimensional singular Roesser models (2-D SRM) are studied. Two different concepts of the dual systems are proposed for the 2-D SRM. One is derived from the duality defined for two-dimensional singular general models (2-D SGM)-called the S-dual systems; the other one is defined based on 2-D SRM in a traditional sense-called the T-dual systems. It is shown that if a 2-D SRM is jump-mode free or jump-mode reachable, then it can be equivalently transformed into a canonical form of a 2-D SRM, for which the T-duality and the S-duality are equivalent. This will be of some perspective applications in the robust control of 2-D SRM.

Analysis and control of the jump modes behavior of 2-D singular systems—Part I: Structural stability

This paper considers the problem of structural stability of 2-D singular systems. Firstly, some properties of structural stability of 2-D general singular systems are presented. Sufficient and necessary conditions for the structural stability of the 2-D singular systems are given. Then, by extending the Lyapunov approach for the structural stability of 1-D continuous singular systems to the discrete case, a generalized Lyapunov equation approach to the analysis of the structural stability of 2-D singular Roesser models (2-D SRM) is proposed. The existence of a solution to the generalized Lyapunov equation gives a sufficient condition for the structural stability of the 2-D SRM.

H∞ Model Reduction of 2-D Singular Roesser Models

This paper discusses the problem of H? model reduction for linear discrete time 2-D singular Roesser models (2-D SRM). A condition for bounded realness is established for 2-D SRM in terms of linear matrix inequalities (LMIs). Based on this, a sufficient condition for the solvability of the H? model reduction problem is obtained via a group of LMIs and a set of coupling non-convex rank constraints. An explicit parameterization of the desired reduced-order models is presented. Particularly, a simple LMI condition without rank constraints is proposed for the zeroth-order H? approximation problem. Finally, a numerical example is given to illustrate the applicability of the proposed approach.

Elimination of Anticipation of a Class of Singular 2D Roesser Models by State Feedbacks

Necessary and sufficient conditions for the anticipation of a class of singular 2D Roesser models are established. Two methods of elimination of the anticipation of the model by suitable choice of feedback gain matrices are proposed. The methods are illustrated by numerical examples.

Bounded real lemma and robust [formula omitted] control of 2-D singular Roesser models

This paper discusses the problem of robust H ∞ control for linear discrete time two-dimensional (2-D) singular Roesser models (2-D SRM) with time-invariant norm-bounded parameter uncertainties. The purpose is the design of static output feedback controllers such that the resulting closed-loop system is acceptable, jump modes free, stable and satisfies a prescribed H ∞ performance level for all admissible uncertainties. A version of bounded realness of 2-D SRM is established in terms of linear matrix inequalities. Based on this, a sufficient condition for the solvability of the robust H ∞ control problem is solved, and a desired output feedback controller can be constructed by solving a set of matrix inequalities. A numerical example is provided to demonstrate the applicability of the proposed approach.

Geometric theory for the singular Roesser model

(A,E,B)-invariant and (E,A,B)-invariant subspaces for the two-dimensional singular Roesser model are investigated. These subspaces are related to the existence of the solutions when the boundary conditions are in these subspaces. Also, the existence of a solution sequence in certain subspaces derived from the invariant subspaces is shown. The boundary conditions that appear in the solution when some semistates in the solution are restricted to zero are also investigated. >

Transformation of the transfer function variables of the singular n‐dimensional roesser model

AbstractA singular Roesser model is presented for a non‐causal n‐dimensional (n‐D) system described by an n‐D transfer function with non‐monic denominator. Transformations of the transfer function variables (inversion, multivariable bilinear transformation) have been used to transform the given polynomial to monic form. These transformations are also discussed in state‐space description terms. Additionally, the matrix Q which relates the coefficients vector of the transformed polynomial to that of the original one is obtained using n‐way matrix methods.