Two-dimensional Discrete-time 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.

Development of time-limited model reduction technique for one-dimensional and two-dimensional systems with error bound via balanced structure

In this article, the two-dimensional model is decomposed into two one-dimensional models using the minimal rank-decomposition condition, and the model reduction is conducted on these two one-dimensional models using time-limited Gramians. The proposed framework works for both one-dimensional and two-dimensional systems. The suggested approach addresses the primary flaw in Gawronski & Juang’s problem of reduced-order model instability. Researchers revealed some stability preservation solutions to address this key flaw, which ensure the stability of one-dimensional reduced-order systems; nevertheless, these strategies result in large approximation errors. However, to the best of the authors’ knowledge, there is no literature available for the stability preserving time-limited-intervals Gramians based model reduction framework for the two-dimensional discrete-time systems. In comparison to other stability-preserving strategies, the proposed framework provides an a priori error-bound formulation that is easily computable. The simulation results show that the proposed framework performs well compared to other existing stability-preserving methods, demonstrating its usefulness.

Dynamics and chaos control for a novel model incorporating plant quality index and larch budmoth interaction

In this paper, we study some dynamics concerning the interaction of budmoth and plant quality index of larch situated in the Swiss Alps. Taking into account this interaction, a two-dimensional discrete-time system is formulated and discussed. The new model is formulated with an application of Holling type III functional response for the plant quality index. Furthermore, the proposed functional response is supported by actual observed data related to larch budmoth interaction. In addition to proving that solutions are uniformly bounded, the existence of biologically feasible fixed points and the local dynamics of the proposed model regarding its fixed points are also studied. It is shown that the proposed model undergoes period-doubling bifurcation around its coexistence with the implementation of the center manifold and normal form theories. Furthermore, the existence and direction of Neimark–Sacker bifurcation about positive fixed point are discussed. Taking into account the biological relevance of chaos control strategies, different methods of controlling chaos are implemented with their biological relevance. Numerical simulation is demonstrated to illustrate the theoretical discussion. Finally, theoretical discussion is validated with experimental and field data.

Degenerate Chenciner Bifurcation Revisited

Generic results for degenerate Chenciner (generalized Neimark–Sacker) bifurcation are obtained in the present work. The bifurcation arises from two-dimensional discrete-time systems with two independent parameters. We define in this work a new transformation of parameters, which enables the study of the bifurcation when degeneracy occurs. By the four bifurcation diagrams we obtained, new behaviors hidden by the degeneracy are brought to light.

Dynamics and Chaos Control for a Discrete-Time Lotka-Volterra Model

Bifurcation theory (center manifold and Ljapunov–Schmidt reduction, normal form theory, universal unfolding, calculation of bifurcation diagrams) has become an important and very useful means in the solution of nonlinear stability problems in many branches of engineering. The present study deals with qualitative behavior of a two-dimensional discrete-time system for interaction between prey and predator. The discrete-time model has more chaotic and rich dynamical behavior as compare to its continuous counterpart. We investigate the qualitative behavior of a discrete-time Lotka-Volterra model with linear functional response for prey. The local asymptotic behavior of equilibria is discussed for discrete-time Lotka-Volterra model. Furthermore, with the help of bifurcation theory and center manifold theorem, explicit parametric conditions for directions and existence of flip and Hopf bifurcations are investigated. Moreover, two chaos control methods, that is, OGY feedback control and hybrid control strategy, are implemented. Numerical simulations are provided to illustrate theoretical discussion and their effectiveness.

Fault‐tolerant control of two‐dimensional discrete‐time systems

A robust fault-tolerant controller design for a class of two-dimensional discrete-time systems with mixed actuator faults, stochastic uncertainties and non-linear perturbation in delayed states is proposed in this study. The authors' aim is to investigate the stochastic stabilisation issue with an adequate level of an extended dissipative performance index where the considered system contains failures occurring among the pre-specified subset of actuators with admissible uncertainties. More precisely, the considered stochastic system uncertainties are assumed to follow mutually uncorrelated Bernoulli distributed sequences along with their probability rules. Furthermore, a practical actuator fault model consisting of both linear and non-linear fault terms is proposed. By utilising some inequalities and extended dissipativity theory, the authors develop a robust stochastic stability criterion of the addressed system in the linear matrix inequality framework. Lastly, a simulation example is provided to verify the correctness and significance of the designed fault-tolerant control strategy.

Stability and constrained control for positive two-dimensional systems with delays in the second FM model

Abstract This paper addresses the stability and control problem of linear positive two-dimensional discrete-time systems with multiple delays in the second Fornasini–Marchesini model. The contribution lies in three aspects. First, a novel proof is provided to establish necessary and sufficient conditions of asymptotic stability for positive two-dimensional delayed systems. Then, a state-feedback controller is designed to ensure the non-negativity and stability of the closed-loop systems. Finally, a sufficient condition for the existence of constrained controllers is developed under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Two examples are given to validate the proposed methods.

An Efficient Integer-Preserving Stability Test for Discrete-Time Systems

The paper presents an efficient integer-preserving version for the author’s stability test for discrete-time linear systems. A first naive solution that satisfies this constraint is shown to have an explosive (severely exponential) growth of the magnitude of the integers. Then a simple, but far from obvious, new recursion form is established that has a more restrained (linear) growth of coefficients. A qualitative evaluation of computing time shows that the new test form is most efficient. Its possible usefulness for determining stability constraints for filters and systems with designable parameters is illustrated by a numerical example. Its capacity to offer better numerical accuracy for high-degree polynomials is also illuminated. Additional applications may arise from its usability over other algebraic rings. The latter capacity was demonstrated recently by implementing it into an efficient stability test for two-dimensional discrete-time systems.

Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits

We show how to obtain information about the dynamics of a two-dimensional discrete-time system from its homoclinic and heteroclinic orbits. The results obtained are based on the theory of ‘trellises’, which comprise finite-length subsets of the stable and unstable manifolds of a collection of saddle periodic orbits. For any collection of homoclinic or heteroclinic orbits, we show how to associate a canonical ‘trellis type’ which describes the orbits. Given a trellis type, we show how to compute a ‘graph representative’ which gives a combinatorial invariant of the trellis type. The orbits of the graph give the dynamics forced by the homoclinic/heteroclinic orbits in the sense that every orbit of the graph representative is ‘globally shadowed’ by some orbit of the system, and periodic, homoclinic/heteroclinic orbits of the graph representative are shadowed by similar orbits.

Stability Testing of Two-Dimensional Discrete-Time Systems by a Scattering-Type Stability Table and Its Telepolation

Stability testing of two-dimensional (2-D) discrete-time systems requires decision on whether a 2-D (bivariate) polynomial does not vanish in the closed exterior of the unit bi-circle. The paper reformulates a tabular test advanced by Jury to solve this problem. The 2-D tabular test builds for a real 2-D polynomial of degree (n1, n2) a sequence of n2 matrices or 2-D polynomials (the ‘2-D table’). It then examines its last polynomial - a 1-D polynomial of degree 2n1n2 - for no zeros on the unit circle. A count of arithmetic operations for the tabular test is performed. It shows that the test has O(n6) complexity (assuming n1 = n2 = n)- a significant improvement compared to previous tabular tests that used to be of exponential complexity. The analysis also reveals that, even though the testing of the condition on the last polynomial requires O(n4) operations, the count of operations required for the table’s construction makes the overall complexity O(n6). Next it is shown that it is possible to telescope the last polynomial of the table by interpolation and circumvent the construction of the 2-D table. The telepolation of the tabular test replaces the table by n1n2 + 1 stability tests of 1-D polynomials of degree n1 or n2 of certain form. The resulting new 2-D stability testing procedure requires a very low O(n4) count of operations. The paper also brings extension for the tabular test and its simplification by telepolation to testing 2-D polynomials with complex valued coefficients.

Stability determination of two-dimensional discrete-time systems

In determining root distribution of univariate polynomials with real or complex-valued coefficients, the Bistritz tabular form offers a significant computational advantage. Stability studies of two-dimensional (2-D) discrete-time systems involve univariate polynomials possessing parameter-dependent coefficients, where the parameter takes values on the unit circle in the complex plane. This paper investigates the application of Bistritz tabular form in determining stability of 2-D discrete-time systems, and for this purpose we present two algorithms. Both algorithms utilize a recent result that has established the relationship between Schur-Cohn minors and the entries of the Bistritz tabular form corresponding to a given polynomial. A comparison between the use of the modified Jury table and the Bistritz table in stability checking of 2-D discrete-time systems is also presented.

A new stability test for two-dimensional discrete systems using bilinear continuous fractions

An algorithm for testing the stability of single-input single-output (SISO) two-dimensional discrete-time systems is presented, using a method in the z-domain exclusively. It is based on expanding the bilinear discrete reactance function into z-domain continuous fractions in a mixed fashion. It is faster than B.M. Karan and M.C. Srivastava's algorithm (1986) and compares favorably with the commonly used tabular form of testing stability of 2-D systems. Two examples are given to illustrate the method and its practical implementation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An algorithm for the computation of the transfer function matrix of generalized two-dimensional systems

A generalized Leverrier's algorithm is developed for the computation of the transfer function matrix of a singular two-dimensional discrete-time system. The algorithm is a recursion in terms of the original system matrices, and does not require the inversion of a polynomial matrix.