Two-dimensional Discrete Systems Research Articles

Event-Triggered Ultimately Bounded Filtering for Two-Dimensional Discrete-Time Systems under Hybrid Cyber Attacks

This paper investigates the design problem of event-triggered ultimately bounded filter for two-dimensional discrete systems with time-varying delays subject to hybrid cyber-attacks. A bi-directional time-sequence event-triggered transmission scheme is developed under hybrid cyber-attacks to lighten the load of network bandwidth while preserving a satisfactory filtering performance. The impact of hybrid cyber-attacks, which occur in random patterns, on the filtering performance is also taken into account. In such a framework, an augmented system model accounting for the simultaneous presence of hybrid cyber-attacks and event-triggered transmission mechanism is established with respect to two-dimensional shift-varying discrete systems with time-varying delays. On the basis of the augmented model, sufficient conditions ensuring dynamic filtering error systems exponentially ultimately bounded in the mean-square sense are obtained in virtue of Lyapunov stability theory and the linear matrix inequality technique. Then, the asymptotic upper bound of the dynamic filtering error can be specifically quantified in terms of noise variance and cyber-attacks intensity. Furthermore, criteria for simultaneously designing the weighting matrix of event-triggered scheme and the filter gain matrix are derived by minimizing the asymptotic upper bound of filtering error. Finally, the validity of proposed ultimately bounded filtering algorithm is demonstrated by utilizing an example of the industrial heating exchange process.

Image encryption algorithms based on two-dimensional discrete hyperchaotic systems and parallel compressive sensing

Image encryption algorithms based on two-dimensional discrete hyperchaotic systems and parallel compressive sensing

Reduced-order state-space models for two-dimensional discrete systems via bivariate discrete orthogonal polynomials

This paper investigates the new model order reduction (MOR) methods via bivariate discrete orthogonal polynomials for two-dimensional (2-D) discrete systems. The 2-D discrete system is described by the Kurek model. First, we deduce algebraically the shift-transformation matrix of the classical discrete orthogonal polynomials of one variable. By means of the shift-transformation matrices, 2-D discrete systems are expanded in the spaces spanned by bivariate discrete orthogonal polynomials. The coefficient matrices are calculated from matrix equations. Then the reduced-order systems are produced by the orthogonal projection matrices defined by the coefficient matrices. Theoretical analysis shows that the reduced-order systems can match a certain number of coefficient vectors of the original outputs. Finally, one numerical example is simulated to demonstrate the feasibility and effectiveness of the proposed methods.

Sufficient criterion for instability of 2-D linear discrete systems

In this paper, a theorem for the instability checking of linear shift-invariant two-dimensional discrete systems is proposed. This is based on the results for the location of the roots of polynomials presented by Mori and the criterion for the stability of 2-D discrete systems presented by Murray and Delsarte et al. The theorem provides a new sufficient criterion which can be applied easily for the 2-D instability checking. This is demonstrated by detailed examples.

Robust $$H_{\infty }$$ filtering for uncertain two-dimensional discrete systems with state-varying delays

Robust $$H_{\infty }$$ filtering for uncertain two-dimensional discrete systems with state-varying delays

Reduced-order modeling methods via bivariate discrete orthogonal polynomials for two-dimensional discrete state-delayed systems

Reduced-order modeling methods via bivariate discrete orthogonal polynomials for two-dimensional discrete state-delayed systems

Frequency-Domain Based Iterative Learning Control for 2-D Discrete Systems

In the existing of traditional iterative learning control (ILC) results for two-dimensional (2-D) discrete systems with time-domain based analysis approach, fixed boundary states do not affect the complete convergence of P-type ILC law. However, it does affect the ILC convergence properties in the frequency domain. This paper first investigates the frequency-domain ILC tracking problem for 2-D discrete systems with different boundary states. An extended P-type ILC law is designed and a sufficient convergence condition of which can be derived through a rigorous mathematical proof. A simulation example is given to verify the effectiveness and validation of the proposed extended P-type ILC law. Finally, some comparison results on traditional P-type ILC law and D-type ILC law are presented.

Propagation of Non-Adiabatic Heterogeneous Solid Combustion Waves with Effect to Internal Microstructure and Heat Release

ABSTRACT Propagation of combustion front with effect to the external heat loss is studied by the two-dimensional discrete periodic and disorder systems. Discrete periodic systems are modeled by the regular arrangement of burnt and unburnt point heat sources characterized by the identical rate of heat release and different heat loss parameter. Discrete disorder systems, namely, S1, S2 and S3 are considered with the concatenation of regular burnt and unburnt point heat sources distributed by different shaping parameter of beta distribution function. The unburnt point heat sources in a discrete disorder system are characterized by the different heat loss parameter and incorporation of two categories of distributions for the rate of heat release namely (i) gamma (a = 20) and (ii) gamma (a = 2). The oscillations in ignition time delay, with the manifestation of bifurcation pattern, have been detected for discrete periodic systems. In a discrete disorder system, the external heat loss plays a crucial role in developing the fingering instabilities and determining the early onset of unstable combustion, quenching and combustion limit. Degree of randomness in the distribution of heat release for the disorder system affects the duration of combustion, burn rate without affecting the combustion limit. Computed values of burn rate and combustion limit from the discrete combustion model have been compared with the available data from experiment.

Bidimensional Jordan form and linear discrete dynamical systems with application in red blood cell production

Jordan canonical form (JCF) is one of the most important, and useful, concepts in linear algebra. Mathematics, physics, biology, science and engineering undergraduates often find the first application of real JCF in the discipline of differential equations (continuous models) to solving systems of differential equations. In this work, we apply real JCF to understand, through analytical and numerical methods, the two-dimensional linear discrete dynamical systems. After that, we present a schematic model of red blood cell production. An advantage of discrete models over continuous models is that they readily yield numerical exploration, whether by calculator or computer (in our case we will use Mathematica software). This fact provides us with a valuable pedagogical resource because we can find a simple model to show the links between mathematics and other areas of knowledge without the need for sophisticated mathematical concepts beyond linear algebra. Following Paul Halmos's suggestion, ‘The only way to learn mathematics is to do mathematics’, some classroom exercises are given to better understand the subject, and a project for further student investigations is suggested.

Iterative learning control of two-dimensional discrete systems in General model

In the existing iterative learning control (ILC) results for two-dimensional linear discrete systems (2-D LDS), they focus on 2-D LDS Fornasini–Marchesini model. To the best of our knowledge, the 2-D LDS Fornasini–Marchesini model is a particular case of 2-D LDS General model. This paper is first concerned with the ILC problem for 2-D LDS General model with nonidentical boundary states. A modified ILC law under three-dimensional (3-D) framework is proposed. By using 2-D LDS stability theory, perfect tracking on 2-D desired surface trajectory except the boundaries can be achieved. In addition, ILC convergence analysis for 2-D LDS General model with direct transmission from inputs to outputs is discussed. Two illustrative examples are given to show the effectiveness and feasibility of the presented ILC approach.

A New Sufficient Criterion for the Stability of 2-D Discrete Systems

During the past few decades, two and higher dimensional systems have been extensively applied in many areas of research. The representation of the 2-D systems in the frequency domain is usually given by its transfer function. The bounded-input bounded-output (BIBO) stability of the two dimensional discrete systems depends on the zeros of the characteristic polynomial which is the denominator of this transfer function. In this paper, a new sufficient criterion for the stability of two-dimensional linear shift-invariant discrete systems is presented. The new criterion is based on the sufficient condition for stable polynomials with complex coefficients and the stability criterion for 2-D discrete systems proposed by Murray and Delsarte et al. . The new criterion is non-conservative for the stability testing of 2-D discrete systems. It is shown that the proposed sufficient criterion is simple enough to be applied for the stability checking of the 2-D discrete systems. The utility of the proposed criterion is demonstrated by examples.

Adaptive ILC of Tracking Nonrepetitive Trajectory for Two-dimensional Nonlinear Discrete Time-varying Fornasini-Marchesini Systems with Iteration-varying Boundary States

Most of adaptive iterative learning control (AILC) algorithms focus on one-dimensional (1-D) systems, rather than two-dimensional (2-D) systems. This brief is first concerned with AILC for 2-D nonlinear discrete time-varying Fornasini-Marchesini system (NDTVFMS) with nonrepetitive reference trajectory under iteration-varying boundary states. By using Lyapunov analysis method, it can guarantee that the ultimate tracking error tends to zero asymptotically, and make all identified parameters and system signals to be bounded as iteration number goes to infinity. Two illustrative examples are used to validate the effectiveness of the designed AILC approach.

Convergence Analysis of Iterative Learning Control for Two Classes of 2-D Linear Discrete Fornasini–Marchesini Model

This paper first investigates convergent property of two iterative learning control (ILC) laws for two kinds of two-dimensional linear discrete systems described by the first Fornasini–Marchesini model (2-D LDFFM with a direct transmission from inputs to outputs and 2-D LDFFM with input delay). Different from existing ILC results for 2-D LDFFM, this paper provides convergence analysis in a three-dimensional (3-D) framework. By using row scanning approach (RSA) or column scanning approach (CSA), it is theoretically proved no matter which method is adopted, perfect tracking on the desired reference surface is accomplished. In addition, linear matrix inequality (LMI) technique is utilized to computer the learning gain of the ILC controller. The effectiveness and feasibility of the designed ILC law are illustrated through numerical simulation on a practical thermal process.

Fault Diagnosability Analysis of Two-Dimensional Linear Discrete Systems

In this article, a systematic fault diagnosability evaluation, including fault detectability and isolability, is established in a quantitative manner for two-dimensional systems. With ingenious data formulation, a parity relation of two-dimensional systems is first established, then the Kullback-Leibler divergence is employed as the key measure for the diagnosability analysis based on the established parity relation. The basic idea is to quantify the distribution differences among each fault scenario-related system dynamic behavior. Explicit necessary and sufficient condition for fault diagnosability is further derived based on the appropriately introduced definitions corresponding to the two directions evolving system properties. Finally, the effectiveness of the proposed method is verified by two examples.

Robust Observer Design for Two-Dimensional Discrete Positive Switched Systems With Delays

The design problem of the robust positive observer for a class of two-dimensional (2-D) positive discrete switched systems in the presence of delays and uncertainties is considered in this brief. The delays and uncertainties in this brief are assumed to be time-varying and polytopic type, respectively. Firstly, the multiple Lyapunov functional (MLF) approach has been used to formulate some sufficient asymptotical stability conditions for 2-D switched systems with uncertainties, by choosing a suitable co-positive Lyapunov-Krasovskii functional (CPLKF). Secondly, under the assumption that the system states are not measurable, a robust observer design scheme is formulated that helps in the approximation of the corresponding non-negative states. It also ensures the asymptotic convergence of the associated error system. Finally, an appropriate example is discussed to show the usefulness of the proposed results.

Optimal state and fault estimation for two-dimensional discrete systems

An optimal state and fault estimation scheme is proposed for two-dimensional discrete systems subject to either deterministic disturbances or stochastic disturbances (noises). A direct solution to the deterministic estimation problem is obtained first, based on a well-designed regularized least squares problem with a dynamic constraint of a two-dimensional singular system, augmented from the original system state and unknown disturbance. After proving the solution equivalence between the deterministic scenario and the stochastic one in the sense of optimal state and fault estimation, a unified solution, based on a Riccati-like equation recursion, can be established by weighting parameterization for two-dimensional systems in deterministic and stochastic cases. The unified solution also works as the optimal state observer and generalized Kalman filter for two-dimensional singular systems. Generalization discussions concerning different system descriptions with respect to fault as well as the implementations of the proposed estimator are also presented. Simulation illustrates the effectiveness of the proposed method.

Parameter-dependent robust H∞ filtering for uncertain two-dimensional discrete systems in the FM second model

Abstract This paper deals with the problem of robust $H_{\infty }$ filtering for uncertain two-dimensional discrete systems in the Fornasini–Marchesini second model with polytopic parameter uncertainties. Firstly, a new $H_{\infty }$ performance criterion is derived by exploiting a new structure of the parameter-dependent Lyapunov function. Secondly, based on the criterion obtained, a new condition for the existence of a robust $H_{\infty }$ filter that ensures asymptotic stability, and a prescribed $H_{\infty }$ performance level of the filtering error system, for all admissible uncertainties is established in terms of linear matrix inequalities. Finally, two examples are given to illustrate the effectiveness and advantage of the proposed method.

Robust Iterative Learning Control of 2-D Linear Discrete FMMII Systems Subject to Iteration-Dependent Uncertainties

In this article, robust and convergent properties of iterative learning control (ILC) laws are investigated for two kinds of two-dimensional (2-D) linear discrete systems (LDS) in the second Fornasini-Marchesini's model (FMMII). The iteration-dependent uncertainties in reference trajectories, boundary states, and disturbances are considered in the ILC designs of 2-D FMMII. By virtue of the lifting matrix/vector technique, the ILC processes of two kinds of 2-D FMMII are transformed into 2-D linear inequalities. As a result, robust and convergent conditions are obtained for the proposed ILC laws. It is theoretically proved that by using the proposed ILC laws, under the discussed iteration-dependent uncertainties, the ILC tracking errors of 2-D FMMII can be driven into a residual range, the bound of which is relevant to the bound parameters of uncertainties. In particular, when the iteration-dependent uncertainties of 2-D FMMII are progressively convergent in iteration domain, an accurate tracking to the desired reference trajectory can be realized except at the boundaries. Numerical simulations are used to illustrate the validity and feasibility of the designed ILC laws.

Graded discrete integrable systems and Darboux transformations

We present the Darboux transformations for a novel class of two-dimensional discrete integrable systems named as graded discrete integrable systems, which were firstly proposed by Fordy and Xenitidis within the framework of graded discrete Lax pairs very recently. In this paper, the graded discrete equations in the coprime case and their corresponding Lax pairs are derived from the discrete Gel’fand–Dikii hierarchy by applying a transformation of the independent variables. The construction of the Darboux tranformations is realised by considering the associated linear problems in the bilinear formalism for the graded lattice equations. We show that all these graded equations share a unified solution structure in our scheme.

Discrete Double-Sided Quaternionic Fourier Transform and Application

In the present article, we first introduce the definition of the discrete double-sided quaternionic Fourier transform (DQFT) and obtain its inverse. We give examples how to compute the DQFT. We derive a discrete version of the duality property of the DQFT. We finally present an application of the DQFT to study the two-dimensional discrete linear time-varying systems.