Fornasini-Marchesini State-space Model Research Articles

Finite frequency model reduction for 2-D fuzzy systems in FM model

This paper deals with the problem of H∞ model reduction for two-dimensional (2D) discrete Takagi-Sugeno (T-S) fuzzy systems described by Fornasini-Marchesini local state-space (FM LSS) models, over finite frequency (FF) domain. New design conditions guaranteeing the FF H∞ model reduction are established in terms of Linear Matrix Inequalities (LMIs). To highlight the effectiveness of the proposed H∞ model reduction design, a numerical example is given.

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.

A novel constructive procedure to low-order Fornasini–Marchesini model realization

In this paper, a novel constructive procedure is proposed for the realization of a low-order Fornasini–Marchesini state-space model for a given multidimensional system. In particular, essential properties for the constructions of a backward shift space and a resolvent invariant space associated with the Gleason’s problem from a given transfer function matrix are investigated and some new sufficient conditions for such constructions are developed. Then, based on these conditions, a systematic constructive procedure is proposed for the multidimensional Fornasini–Marchesini model realization. It turns out that the new constructive procedure can generate Fornasini–Marchesini models with much lower orders and even the minimal realizations for a much larger class of multidimensional systems than the existing methods. Nontrivial examples are also provided to illustrate the basic idea as well as the effectiveness of the proposed procedure.

Dynamic output feedback [formula omitted] control for 2D fuzzy systems

In this paper, the dynamic output feedback H∞ control problem is considered for two-dimensional discrete fuzzy systems described by the second Fornasini–Marchesini state-space model. By introducing novel slacked matrices, sufficient criteria are established to guarantee an H∞ noise attenuation of the resulting closed-loop system. Then the proposed output feedback H∞ controller design method is applied to one-dimensional discrete-time fuzzy systems. It is worth mentioning that there is no rank constraints to system matrices. Moreover, the sufficient criteria are expressed as strict linear matrix inequalities, which can be effectively solved via MATLAB LMI toolbox. Finally, the effectiveness and advantage of the proposed approach are shown through two simulation examples.

Common eigenvector approach to exact order reduction for multidimensional Fornasini–Marchesini state-space models

ABSTRACTThis paper proposes an exact order reduction approach for the multidimensional (n-D) Fornasini–Marchesini (F–M) model by making use of the common eigenvector. Specifically, by introducing the concept of common eigenvectors, sufficient conditions of exact order reductions are developed for an n-D F–M model, which are able to simultaneously deal with n eigenvalues of the system matrices of the n-D F–M model. The obtained results reveal, for the first time, the internal connection between the multiple eigenvalues of the system matrices and the reducibility of the considered n-D F–M model. Then, a corresponding algorithm is proposed to exactly reduce the order of an n-D F–M model as much as possible. Examples are given to illustrate the details as well as the effectiveness of the proposed approach.

Control systems analysis for the Fornasini-Marchesini 2D systems model – progress after four decades

ABSTRACTThe development of a control systems theory for multidimensional linear systems, i.e. systems with more than one independent variable is an area where a very significant volume of research is based on representations for the underlying dynamics introduced in the 1970s. One of these is the Fornasini-Marchesini state-space model published in 1978. This paper reviews the significant developments in the succeeding four decades in both theory and applications, including examples where this model description is an enabler for the analysis of other classes of systems.

Stabilization of Nonlinear Fornasini–Marchesini Systems

This paper considers the 2D systems described by the Fornasini–Marchesini statespace model. Direct and converse theorems on the exponential stability of such systems are proved in terms of vector Lyapunov functions. The concepts of exponential passivity and a vector storage function are introduced for solving exponential stabilization problems. An example is given to illustrate the efficiency of the new results.

Control law design for discrete linear repetitive processes with non-local updating structures

Repetitive processes are a class of 2D systems where information propagation in one direction is of finite duration. These processes make a series of sweeps, termed passes, through a set of dynamics and on completion of each pass resetting to the starting position occurs ready for the start of the next pass. The control problem is that the previous pass output, termed the pass profile, acts as a forcing function on the current pass and can result in oscillations that increase in amplitude from pass-to-pass. In the case of discrete dynamics, these processes have structural links with 2D systems described by the well known Roesser and Fornasini–Marchesini state-space models but some applications require updating structures that cannot be represented by these models. This requirement arises either in adequately modeling the dynamics or as a result of the control law structure and requires the development of a systems theory for eventual use in applications. In this paper such a theory is advanced through the development of new control law design algorithms.

Stability of Distributed 3-D Systems Implemented on Grid Sensor Networks

Using the Givone-Roesser and the Fornasini-Marchesini state space models for m-D systems, a method for distributed information processing in rectangular grid based wireless sensor networks is presented. It can be used for distributed implementation of any general linear system on a grid sensor network. Nonlinearities caused by finite precision number representation schemes used for in node computations and inter node communication are modeled. Stability of the system is analyzed with special consideration given to the influence of inter-node communication on system dynamics. A necessary and sufficient condition for global asymptotic stability under finite precision arithmetic is established. Simulation results are presented to illustrate the theoretical results.

Stability analysis of linear 2-D systems

The present paper is concerned with stability analysis of linear two-dimensional systems described by Fornasini–Marchesini state-space model. Necessary and sufficient conditions for asymptotic stability of the systems are obtained first. Several simple stability criteria are derived via the nonnegative matrix theory, which are sharper than those in literature. When all the parameter matrices are nonnegative, the criteria are necessary and sufficient for stability of the system. Illustrative examples are provided.

ビヘイビアアプローチに基づく２‐Ｄ離散時間システムのＬｙａｐｕｎｏｖ安定解析

This paper is concerned with a Lyapunov stability analysis of a two-dimensional (2-D) discrete-time system described by a high-order partial difference equation in the behavioral setting. We first formulate the autonomy and asymptotic stability of 2-D discrete-time behavior in terms of the characteristic set over Z2. A sufficient condition for asymptotic stability is derived in terms of quadratic difference forms. This condition can be numerically checked by solving a certain four-variable polynomial matrix equation. It also turns out that the present result generalizes some existing stability conditions for the Fornasini-Marchesini state-space model to the behavioral setting.

Simple criteria for stability of two-dimensional linear systems

Stability criteria for two-dimensional (2-D) systems described by the Fornasini-Marchesini state-space model are presented. The criteria are sharper than the reported results in literature. Illustrative examples are provided.

Transfer Function Computation for Multidimensional Systems

A theoretically interesting technique is proposed for the determination of the coefficients of the determinantal polynomial and the coefficients of the adjoint polynomial matrix of a given n-D system, described by the Fornasini-Marchesini state space model. The proposed algorithms are based on the discrete Fourier transform (DFT), and easily can be implemented. An example is included to illustrate the application of the algorithm.

New 2D models and a transition matrix for discrete linear repetitive processes

This paper reports further results on the development of a control systems theory for discrete linear repetitive processes. In particular, Roesser and Fornasini-Marchesini state space model equivalent descriptions for the dynamics of these processes are constructed and then used to develop new stability tests which involve only computations on matrices with constant entries. Also, they are used to develop a transition matrix, or fundamental matrix sequence, for these processes which is then used to define and characterize so-called local reachability and controllability properties in the form of matrix rank-based tests.

Stability of two-dimensional discrete systems with periodic coefficients

Two-dimensional (2-D) discrete systems with periodic coefficients are considered for stability. These systems are called periodically shift variant (PSV) digital filters and have many applications in signal processing that include the filtering of 2-D signals with cyclostationary noise, scrambling of digital images, and implementation of multirate filter banks. In this paper, the filters are formulated in the form of the well known Fornasini-Marchesini state-space model with periodic coefficients. This PSV model is then studied for stability. Two sufficient conditions and one necessary condition are established for asymptotic stability. Some examples are given to illustrate the results.

New algorithms for the derivation of the transfer-function matrices of 2-D state-space discrete systems

New algorithms for the derivation of the transfer-function matrices of two-dimensional (2-D) discrete systems from the Roesser and Fornasini-Marchesini state-space models are presented. Two key steps in developing the algorithms are as follows. First, the transfer-function matrix is reformulated in terms of the characteristic polynomials of the matrices involved. Second, an efficient algorithm for the determination of 1-D polynomial coefficients is developed and is, in turn, used to determine the coefficient matrices of the 2-D transfer-function matrix. The proposed algorithms are computationally efficient and reliable. The efficiency of the algorithms is illustrated by comparing the proposed method with two existing methods through examples.

Stability of the 2-D state-space system with overflow and quantization

Two-dimensional (2-D) digital filters are investigated with and without the nonlinear effects of overflow and quantization when implemented in fixed-point arithmetic. A sufficient condition is established for the stability of the linear Fornasini-Marchesini state-space model. The same system is then studied for stability when implemented using two's complement arithmetic. The combined effects of overflow and quantization are considered, and sufficient conditions for stability are derived. >

New stability results for 2-D discrete systems based on the Fornasini-Marchesini state space model

New conditions for the internal stability of a 2-D discrete system described by the Fornasini-Marchesini (FM) local state-space model are presented. Further, the stability margin of a 2-D discrete system described by a (FM) state-space model is considered and methods to compute it or to obtain lower bounds are developed. These methods are based on the various forms of the Lyapunov equation for 2-D systems. Finally, a sufficient condition for the system to remain stable under parameter variations is presented.

Two's complement quantization in two-dimensional state-space digital filters

Two-dimensional (2-D) state-space digital filters are investigated for stability with and without quantization. A sufficient condition is obtained for the stability of the linear Fornasini-Marchesini (1976) state-space model. The same model is then studied for stability when implemented using two's complement truncation quantization, and a sufficient condition for asymptotic stability of the nonlinear filter is obtained. In the process, a theorem is proved which gives a sufficient condition for the stability of a one-dimensional (1-D) state-space digital filter under the same type of quantization.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>