Multidimensional Discrete Systems Research Articles

On the analysis of positivity and stabilization of fractional multidimensional discrete linear systems

On the analysis of positivity and stabilization of fractional multidimensional discrete linear systems

Algebraic approach to chaos induced by snapback repeller

In this poster we present the results of [1]. We consider the problem of detecting chaotic behaviors in discrete dynamical systems. We propose an algebraic criterion for determining whether all the zeros of a given polynomial are outside the unit circle in the complex plane. This criterion is used to deduce critical algebraic conditions for the occurrence of chaos in multi-dimensional discrete systems based on Marotto's theorem. Using these algebraic conditions we reduce the problem of analyzing chaos induced by snapback repeller to an algebraic problem, and introduce an algorithmic approach to solve this problem by means of symbolic computation. The proposed approach is effective as shown by several examples and can be used to determine the possibility that all the fixed points are snapback repellers.

Analysis of Snapback Repellers Using Methods of Symbolic Computation

This paper presents an algebraic criterion for determining whether all the zeros of a given polynomial are outside the unit circle in the complex plane. This criterion is used to deduce critical algebraic conditions for the occurrence of chaos in multidimensional discrete dynamical systems based on a modified Marotto’s theorem developed by Li and Chen (called “Marotto–Li–Chen theorem”). Using these algebraic conditions we reduce the problem of analyzing chaos induced by snapback repeller to an algebraic problem, and propose an algorithmic approach to solve this algebraic problem by means of symbolic computation. The proposed approach is effective as shown by several examples and can be used to determine the possibility that all the fixed points are snapback repellers.

A multidimensional discrete digital chaotic encryption system

In this article, a new multidimensional discrete chaotic system is proposed, and the characteristics and advantages of the multidimensional discrete chaotic system are analyzed. The multidimensional discrete chaotic system is designed using digital technology, and the system has the complexity of hyper-chaotic system, and it can avoid the choice of step size, at the same time, the design of the circuit is simple, the resource occupancy rate is low, it is suitable for the design of digital system, and so on. It can be used in constructing chaotic sequence generator, chaotic encryption, and visual sensor networks.

Certified non-conservative tests for the structural stability of discrete multidimensional systems

In this paper, we present new computer algebra based methods for testing the structural stability of n-D discrete linear systems (with n >= 2). More precisely, we show that the standard characterization of the structural stability of a multivariate rational transfer function (namely, the denominator of the transfer function does not have solutions in the unit polydisc of \C^n) is equivalent to the fact that a certain system of polynomials does not have real solutions. We then use state-of-the-art computer algebra algorithms to check this last condition, and thus the structural stability of multidimensional systems.

Multidimensional Discrete Dynamical Systems with Slow Behavior

In this work we state sufficient conditions for convergence for nonhyperbolic fixed points of multidimensional discrete dynamical systems and analyze their speed of convergence. We introduce a new concept of slow convergence based on one dimensional discrete systems and we establish a general classification for slow discrete dynamical systems which is of paramount importance in numerical analysis and in the development of algorithms. A variety of two dimensional examples are presented to illustrate the diverse possibilities of convergence existing in multidimensional systems.

Feedback Linearization of Continuous and Discrete Multidimensional Systems

The problem of feedback linearization (FL) of continuous and discrete nonlinear MIMO systems is considered. The idea of FL method consists in converting the original nonlinear system into a linear one by means of feedback. Then, the methods of control theory for linear systems are used for system design. A widespread approach to FL design is based on the method of normal form, that uses a nonsingular transformation of system state variables z = T(x). In order to obtain a normal form of the nonlinear system in the neighbor of a some point, it is necessary to determine a special function - the system virtual output, for which a relative degree (in the case of single input) or a vector relative degree (in the case of multiple input) is determined. Applicability of the normal form method for FL is provided by the conditions of controllability and involutivity for the considered nonlinear system, which are not always true. Moreover, when developing a linearizing control law, the main difficulty lies in the transition from transformed variables z to state variables x of the original system. In this paper, we propose another approach, based on representing the original nonlinear system into a state-dependent coefficient form and applying the canonical similarity transformation z = T(x)x, that allow getting the system to canonical form, that considerably simplifies the FL problem. Such similarity transformation allow accomplishing linearization of system without determining of the virtual system output. Another advantage of the proposed method is that the technique of the transition from the transformed variables z to the state variables x of the original system is simpler. The results are illustrated by examples for continuous and discrete nonlinear systems.

Stability Analysis of Multi-Dimensional Linear Discrete System and Root Distribution Using Sign Criterion with Real Coefficients

A new idea was proposed to find out the stability and root location of multi-dimensional linear time invariant discrete system (LTIDS) for real coefficient polynomials. For determining stability the sign criterion is synthesized from the Jury’s method for stability which is derived from the characteristic polynomial coefficients of the discrete system. The number of roots lying inside or outside the unit circle and hence on the unit circle is directly determined from the proposed single modified Jury tabulation and the sign criterion. The proposed scheme is simple and the examples are given to bring out the merits of the proposed scheme which is also applicable for the singular and non-singular cases.

Recent progress in an algebraic analysis approach to linear systems

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized.

Stability and stabilisation of linear multidimensional discrete systems in the frequency domain

This paper gives a reasonably detailed review of advances in stability and stabilisation of linear multidimensional (N-D) discrete systems in the frequency domain. The emphasis is on the recent progress, especially in the past decade. The discussion will focus on two topics: (i) stability test.Determination of whether a given N-D (N ≥ 2) system is stable; (ii) stabilisation. Parameterisation of all stabilising compensators for a stabilisable N-D system. After reviewing the progress and several state of the art methods in these two topics with illustrative examples, some related issues are also briefly mentioned at the end.

Многомерная дискретная фазовая система с кусочно-линейной характеристикой

Многомерная дискретная фазовая система с кусочно-линейной характеристикой

Approximate Realization of Fractional-Order 2-D IIR Frequency-Planar Filters

Starting with the established concepts of multi-dimensional network resonance, practical-bounded-input-bounded-output (BIBO) stability of multi-dimensional discrete systems, and the integer order approximation of fractional order filters, a new method for the realization of integer-order approximations of 2-D fractional-order systems having frequency-planar beam shaped passbands is proposed. The method results in 2-D infinite impulse response (IIR) filters having nonseparable denominators in their transfer-functions while having guaranteed practical-BIBO stability for zero initial conditions in the corresponding discrete system. The resulting integer-order filters are shown to approximate 2-D fractional order frequency-planar beam shapes in both magnitude and phase leading to a new theoretical tool for use in the emerging area of multi-dimensional fractional-order circuits and systems. Although examples are provided for frequency-planar filters in two dimensions, the proposed methods are equally applicable to other types of transfer-functions having prototypes based on resistively-terminated passive networks defined over any integer number of dimensions.

Multidimensional dynamic processes studied by symbolic analysis in velocity-curvature space

A new computer-aided method for the symbolic analysis of discrete mappings and sequences is proposed that is based on a finite discretization of the velocity-curvature space. A minimum alphabet is introduced in a natural way. A number of initial analytical measures are defined that make it possible to study the dynamics structure of multidimensional discrete mappings, continuous systems, and dynamic processes. The proposed analytical method is tested by applying it to a logistic oscillator in the domain to the right of the period-doubling limit point, and the method is shown to be informative. It is revealed that the oscillation structure of the logistic map is highly asymmetric. The critical parameter values are found at which the geometric structure of the map trajectories changes qualitatively.

Erratum to: $${{\mathcal{H}}_{\infty}}$$ control of linear multidimensional discrete systems

Erratum to: $${{\mathcal{H}}_{\infty}}$$ control of linear multidimensional discrete systems

$${{\mathcal H}_{\infty}}$$ control of linear multidimensional discrete systems

This paper presents a comprehensive investigation on the $${{\mathcal H}_{\infty}}$$ control problem of linear multidimensional (nD) discrete systems described by the nD Roesser (local) state-space model. A Bounded Real Lemma consisting of a series of conditions is first established for general nD systems. The proposed nD conditions directly reduce to their 1D counterparts when n = 1, and besides several sufficient conditions which include the existing 2D results as special cases, some necessary and sufficient conditions are also shown to explore further insights to the considered problem. By applying a linear matrix inequality (LMI) condition of the nD Bounded Real Lemma, the nD $${{\mathcal H}_{\infty}}$$ control problem is then considered for three kinds of control laws, namely, static state feedback (SSF) control, dynamic output feedback (DOF) control and static output feedback (SOF) control, respectively. The nD $${{\mathcal H}_{\infty}}$$ SSF and DOF control problems are formulated in terms of an LMI and LMIs, respectively, and thus tractable by using any available LMI solvers. In contrast, the solution condition of the nD $${{\mathcal H}_{\infty}}$$ SOF controller is not strictly in terms of LMIs, therefore an iterative algorithm is proposed to solve this nonconvex problem. Finally, numerical examples are presented to demonstrate the application of these different kinds of nD $${{\mathcal H}_{\infty}}$$ control solutions to practical nD processes as well as the effectiveness of the proposed methods.

Research of Method of Multidimensional Discrete Systems Iterative Identification

The method of iterative identification of multidimensional discrete systems is investigated. On the basis of this method the algorithms oriented on sequential recovering of separate clusters of modes called submodes are developed and proved. Therewith the dimension of approximating model is in accordance with the input data, that is important especially during identification with the bounded uncertainty.

A MODIFIED FREQUENCY-DOMAIN CRITERION FOR GRADIENT-LIKE BEHAVIOR OF DISCRETE PHASE SYSTEMS

Asymptotic behavior of a class of multidimensional discrete control systems with periodic nonlinearities and denumerable set of equilibria is investigated. By means of discrete version of Yakubovich-Kalman theorem and certain modification of Lur'e-Postnikov function a frequency-domain criterion which guarantees that every solution of a system tends to an equilibrium is obtained.

A Solvable Lie Algebra Condition for Stability of Linear Multidimensional Systems

This note analyzes exponential stability of a class of linear discrete multidimensional systems. Using a multidimensional comparison principle for estimating the system componentwise exponential convergence and a solvable Lie algebra condition, a sufficient condition for exponential stability of linear multidimensional systems is presented. The stability condition can be easily examined by computing the system matrices in finite steps. This is demonstrated by an example.

Effective algorithms for parametrizing linear control systems over Ore algebras

In this paper, we study linear control systems over Ore algebras. Within this mathematical framework, we can simultaneously deal with different classes of linear control systems such as time-varying systems of ordinary differential equations (ODEs), differential time-delay systems, underdetermined systems of partial differential equations (PDEs), multidimensional discrete systems, multidimensional convolutional codes, etc. We give effective algorithms which check whether or not a linear control system over some Ore algebra is controllable, parametrizable, flat or π-free.

Estimation of the Limit Accuracy of Discrete Systems for a Class of Dynamic Controllers Relative to the Output

A problem is considered for the estimation of the limit accuracy of multidimensional discrete systems for a definite class of dynamic controllers relative to the output, which differ in the fact that certain of the poles of a closed system prove to be zero ones, while the remaining poles are either the poles of a linear-quadratic controller or the poles of the Kalman filter that is dual with respect to this controller. Asymptotic properties of such controllers are investigated and for a minimum-phase object, a limit (minimum possible) control error is defined in the explicit form, for which the Euclidean norm of the vector of steady-state values of controllable variables is taken. An example of synthesis of a multidimensional discrete systems by the prescribed requirements for a static accuracy, which illustrates the obtained results, is given.