Discrete Systems Research Articles

Higher-Order Iterative Learning Control Algorithms for Linear Systems

Iterative learning control (ILC) algorithms appeared in connection with the problems of increasing the accuracy of performing repetitive operations by robots. They use information from previous repetitions to adjust the control signal on the current repetition. Most often, information from the previous repetition only is used. ILC algorithms that use information from several previous iterations are called higher-order algorithms. Recently, interest in these algorithms has increased in the literature in connection with robotic additive manufacturing problems. However, in addition to the fact that these algorithms have been little studied, there are conflicting estimates regarding their properties. This paper proposes new higher-order ILC algorithms for linear discrete and differential systems. The idea of these algorithms is based on an analogy with multi-step methods in optimization theory, in particular, with the heavy ball method. An example is given that confirms the possibility to accelerate convergence of the learning error when using such algorithms.

Observation of Thouless pumping of light in quasiperiodic photonic crystals

Topological transport is determined by global properties of physical media where it occurs and is characterized by quantized amounts of adiabatically transported quantities. Discovered for periodic potential, it was also explored in disordered and discrete quasiperiodic systems. Here, we report on experimental observation of pumping of a light beam in a genuinely continuous incommensurate photorefractive quasicrystal emulated by its periodic approximants. We observe a universal character of the transport which is determined by the ratio between periods of the constitutive sublattices, by the sliding angle between them, and by Chern numbers of the excited bands (in the time-coordinate space) of the approximant, for which pumping is adiabatic. This reveals that the properties of quasiperiodic systems determining the topological transport are tightly related to those of their periodic approximants and can be observed and studied in a large variety of physical systems. Our results suggest that the links between quasiperiodic systems and their periodic approximants go beyond the pure mathematical relations: They manifest themselves in physical phenomena which can be explored experimentally.

Construction of inherent geometric pattern for equal-area hexagonal discrete global grid systems

ABSTRACT Discrete Global Grid Systems (DGGSs) are hierarchical frameworks that provide seamless global coverage and enable the efficient processing and analysis of heterogeneous geospatial data. As cell resolution becomes finer, similarities in cell attributes, such as cell size and shape, increase. Understanding these shared characteristics enhances our knowledge of DGGSs structures, improving their reliability and accuracy across a range of applications. However, existing methods for quantifying these characteristics are limited, often emphasizing overall analysis and visualization rather than detailed assessment. This paper introduces a geometric pattern quantification method for equal-area hexagonal DGGSs by calculating inter-level attribute similarities to identify a reference level, segmenting the spatial distribution image at this reference level, and fitting the boundaries of these sub-images to create iso-feature lines. The results show that this method improves the accuracy and efficiency of cell attribute calculations, with enhancements of at least 5.6 times for study regions and 13.8 times for sample points at lower levels. This pattern provides a robust basis for grid selection and optimization in regional applications, promoting the broader adoption of DGGSs across various fields.

Symmetric solutions to symmetric partial difference equations

This paper studies discrete systems defined by linear difference equations on the lattice Z n that are invariant under a finite group of symmetries, and shows that there always exist solutions to such systems that are also invariant under this group of symmetries.

Feedback projection synchronization of discrete chaotic systems and its application to speech encryption

Abstract Chaotic systems are widely used in secure communication due to their sensitivity to initial values, unpredictability, and complex motion trajectories. In this paper, we study the encryption method of chaotic synchronization and introduce a scaling factor based on traditional feedback control synchronization to achieve more accurate projection synchronization. The effectiveness and robustness of the method in chaotic systems are verified through theoretical proofs and numerical simulations. A chaotic masked speech encryption system utilizing bit similarity is designed; the structural similarity index (SSIM) of the decrypted signal with the original signal is as high as 0.992866, while the SSIM value of the encrypted signal with the original signal is only 0.000030, proving the efficiency and security of the encryption process. Additionally, we analyzed the data transmission process of the encryption system. The fusion of the control signal and the encryption sequence into one transmission sequence in the channel not only saves hardware and software design resources but also reduces inter-channel interference and conflict, improving the reliability and stability of the transmission. Experimental results show that the system performs well in terms of data transmission security and anti-interference capability.

Stability and l∞ Performance Analysis for Nabla Discrete Fractional Linear Positive System with Time-Varying Delays

In this paper, the stability and l∞-gain problem are investigated for the Nabla discrete fractional linear positive systems with bounded time-varying delays. First, a sufficient condition and a necessary condition are presented to ensure the system’s positivity. Then, based on the system’s positivity property, an asymptotically stable condition is established. Furthermore, it is demonstrated that the l∞-gain of such systems is determined by the system matrices and is independent of the magnitude of delays. Finally, numerical examples are provided to demonstrate the validity of the obtained results.

The Generative Generic-Field Design Method Based on Design Cognition and Knowledge Reasoning

Large language model (LLM) and Crowd Intelligent Innovation (CII) are reshaping the field of engineering design and becoming a new design context. Generative generic-field design can solve more general design problems innovatively by integrating multi-domain design knowledge. However, there is a lack of knowledge representation and design process model in line with the design cognition of the new context. It is urgent to develop generative generic-field design methods to improve the feasibility, innovation, and empathy of design results. This study proposes a method based on design cognition and knowledge reasoning. Firstly, through the problem formulation, a generative universal domain design framework and knowledge base are constructed. Secondly, the knowledge-based discrete physical structure set generation method and system architecture generation method are proposed. Finally, the application tool Intelligent Design Assistant (IDA) is developed, verified, and discussed through an engineering design case. According to the design results and discussion, the design scheme is feasible and reflects empathy for the fuzzy original design requirements. Therefore, the method proposed in this paper is an effective technical scheme of generative generic-field engineering design in line with the design cognition in the new context.

Smooth and Time-Optimal Trajectory Planning for Robots Using Improved Carnivorous Plant Algorithm

To improve the safety and reliability of robotic manipulators during high-speed precision movements, this paper proposes a method for smooth and time-optimal trajectory planning incorporating kinodynamic constraints. The primary objective is to use an evolutionary algorithm to determine a trajectory by considering time and jerk within the feasible path-pseudo-velocity phase plane region. Firstly, the path parameterization theory extracted the maximum pseudo-velocity projection curve from the kinodynamic constraints. Subsequently, the feasible region in the phase plane was defined through reachability analysis of discrete linear systems. Thereafter, we constructed the trajectory function using a cubic B-spline curve, optimizing its control points with an improved carnivorous plant optimization algorithm. Finally, the effectiveness and practicality of this method were verified through simulations on a 6-DOF manipulator.

Score filtering for contextualized noise suppression of Poisson‐distributed geophysical signals

AbstractGeophysical and remote sensing products that rely on Poisson‐distributed measurement signals, such as cosmic‐ray neutron sensing (CRNS) and gamma spectrometry, often face challenges due to inherent Poisson noise. Common techniques to enhance signal stability include data aggregation or smoothing (e.g., moving averages and interpolation). However, these methods typically reduce the ability to resolve detailed temporal (stationary data) and spatial (mobile data) features. In this study, we introduced a method for contextual noise suppression tailored to Poisson‐distributed data, utilizing a discrete score attribution system. This score filter evaluates each observation against eight different criteria to assess its consistency with surrounding values, assigning a score between 0 (very unlikely) and 8 (very likely) to indicate whether the observation is likely to act as noise. These scores can then be used to flag or remove data points based on user‐defined thresholds. We tested the score filter's effectiveness on both stationary and mobile CRNS data, as well as on gamma‐ray spectrometry and electromagnetic induction (EMI) recordings. In our examples, the score filter consistently outperformed established filters, for example Savitzky–Golay and Kalman, in direct competition when applied to CRNS time series data. Additionally, the score filter substantially reduced Poisson noise in mobile CRNS, gamma‐ray spectrometry and EMI data. The scoring system also provides a context‐sensitive evaluation of individual observations or aggregates, assessing their conformity within the dataset. Given its general applicability, customizable criteria and very low computational demands, the proposed filter is easy to implement and holds promise as a valuable tool for denoising geophysical data and applications in other fields.

Optimal Model-Free Mean-Square Consensus for Multi-Agents with Markov Switching Topology

Due to the real applications, optimal consensus reinforcement learning with switching topology is still challenging due to the complexity of topological changes. This paper investigates the optimal consensus control problem for discrete multi-agent systems under Markov switching topologies. The goal is to design an appropriate algorithm to find the optimal control policies that minimize the performance index while achieving consensus among the agents. The concept of mean-square consensus is introduced, and the relationship between consensus error and tracking error to achieve mean-square consensus is studied. A performance function for each agent under switching topologies is established and a policy iteration algorithm using system data is proposed based on the Bellman optimality principle. The theoretical analysis shows that the consensus error realizes mean-square consensus and the performance function is optimized. The efficacy of the suggested approach is confirmed by numerical simulation using an actor–critic neural network. As a result, the value function is the optimum and the mean-square consensus can be reached using this technique.

Detection of Nonlinearity in Photonic Lattices

AbstractAlthough periodic photonic structures, especially associated with nonlinearity, play a prominent role in optics nowadays, effective detection of their nonlinearity still remains a critical challenge. Here, an approach is proposed to detect the nonlinearity of photonic lattices in a direct way. By properly launching structured beams, namely Airy beams, into the lattices, the nonlinear response function of the discrete system can be directly obtained in the nonlinearly‐shaped beam profiles. To be specific, a single Airy beam is utilized to map self‐defocusing nonlinearity, while self‐focusing nonlinearity, which is hard to visualize in the bulk case, is readily discerned by employing double Airy beams in photonic structures. The proposed method is validated numerically and experimentally by detecting different types of nonlinearities of photonic lattices fabricated in a nonlinear crystal. These findings introduce a promising route for characterizing the nonlinear response of optical structures, thereby broadening the scope of nonlinear measurement and is expected to be extended into other periodic photonic structures.

Huygens synchronization of three aligned clocks

AbstractThis study examines the synchronization of three identical oscillators arranged in an array and coupled by small impacts, wherein each oscillator interacts solely with its nearest neighbour. The synchronized state, which is asymptotically stable, is characterized by phase opposition among alternating oscillators. We analyse the system using a non-linear discrete dynamical system based on a difference equation derived from the iteration of a plane diffeomorphism. We illustrate these results with the application to a system of three aligned Andronov clocks, showcasing their applicability to a broad range of oscillator systems.

Assessing Stability in Discrete-Time Systems Impacted by Interference and State Delays: An Approach Using ISS

In this study, we introduce a criterion for addressing the input to state stability (ISS) problem inherent in nonlinear discrete systems featuring state delay, saturation overflow (SO), and external interference (EI). This paper proposes three novel theorems by employing the Lyapunov function to assure the stability of the targeted discrete time systems (DTS) under complex conditions such as state delay, SO, EI and the first theorem utilizes system information more effectively by considering EI, state delay and SO, the second theorem focuses on asymptotic stability (AS) analysis of the system when EIs are zero, the third theorem proposes the system is AS when state delays are zero. These theorems impose fewer constrains additionally these conditions are demonstrated to be more relaxed and computationally less intensive to existing criteria. The advantage of the suggested technique is proven through three numerical examples and comparison research.

Chaotic dynamics in a class of generalized memristive maps.

The memory effects of the memristors in nonlinear systems make the systems generate complicated dynamics, which inspires the development of the applications of memristors. In this article, the model of the discrete memristive systems with the generalized Ohm's law is introduced, where the classical Ohm's law is a linear relationship between voltage and current, and a generalized Ohm's law is a nonlinear relationship. To illustrate the rich dynamics of this model, the complicated dynamical behavior of three types of maps with three types of discrete memristances is investigated, where a cubic function representing a kind of generalized Ohm's law is used, and this cubic function is a simplified characteristic of the famous tunnel diode. The existence of attractors with one or two positive Lyapunov exponents (corresponding to chaotic or hyperchaotic dynamics) is obtained, and the coexistence of (infinitely) many attractors is observable. A hardware device is constructed to implement these maps and the analog voltage signals are experimentally acquired.

Experimental performance verification of an intelligent detection and assessment scheme for disturbances and imbalances of three-phase synchronous machine output using coherence estimators

In this article, power quality disturbances, such as voltage and current unbalances, series and shunt faults, are monitored in three-phase synchronous machines. Because the imbalances affect machine efficiency and service life, it is essential to figure out whether the three-phase voltages/currents are balanced or unbalanced. An integrated detection algorithm is proposed to detect the unbalanced voltages/currents and various abnormal conditions precisely and quickly, which is based on the coherence estimator. Most existing detection methods require more than one cycle time to determine the contingency conditions, while the proposed technique detects these situations in a fast and discrete protection system. The suggested scheme acquires instantaneous measurements of the three-phase voltages and currents taken at the synchronous machine terminals in order to calculate fifteen coherence coefficients that are utilized to detect and assess any unbalance or trouble accurately and swiftly. A new proposal for imbalance and disturbance indicators based on the coherence estimators is developed in this study. Multiple test cases have been conducted to validate the proposed algorithm capabilities using a real power system, which includes a motor-generator set, a three-phase load and measurement transformers. The experimental results have been revealed that the relay reliability and accuracy percentages have been found to be 98.28% and 98.52%, respectively. The quantitative findings of the imbalance and disturbance assessment are recorded.

Dynamics of a delayed discrete-time predator prey model proposed from a nonstandard finite difference scheme

Existing literature established stability in a delayed logistic Lotka–Volterra predator- prey model in terms of equilibrium analysis. However, several researchers did not construct the time-series analysis in such models. It has also been observed that both the RK4 methods and the inbuilt ‘dde23’ MATLAB solver were unable to generate stable solutions. This motivated us to develop a nonstandard scheme to capture the numerical solutions which are well consistent with the analytical equilibrium analysis of continuous delayed predator–prey model. In this paper, we will propose a nonstandard finite difference (NSFD) scheme for a delayed predator–prey model. We shall prove that the developed scheme preserves the qualitative behavior of the system, including the local stability of the equilibrium, and stability switching for any step size h=1m,m∈Z+. It is observed that the discretized system shows the occurrence of a Neimark-Sacker bifurcation. Moreover, the convergence analysis of the numerical scheme establishes first-order convergence. The bifurcation diagram and comparison of delay τ−sequence generated by NSFD with the ones obtained by analytical means have been discussed graphically.

Disconnected Stationary Solutions in 3D Kolmogorov Flow and Their Relation to Chaotic Dynamics

This paper aims to investigate the nonlinear transition to turbulence in generalized 3D Kolmogorov flow. The difference between this and classical Kolmogorov flow is that the forcing term in the x direction sin(y) is replaced with sin(y)cos(z). This drastically complicates the problem. First, a stability analysis is performed by deriving the analog of the Orr–Sommerfeld equation. It is shown that for infinite stretching, the flow is stable, contrary to classical forcing. Next, a neutral curve is constructed, and the stability of the main solution is analyzed. It is shown that for the cubic domain, the main solution is linearly stable, at least for 0<R≤100. Next, we turn our attention to the numerical investigation of the solutions in the cubic domain. The main feature of this problem is that it is spatially periodic, allowing one to apply a relatively simple pseudo-spectral numerical method for its investigation. We apply the method of deflation to find distinct solutions in the discrete system and the method of arc length continuation to trace the bifurcation solution branches. Such solutions are called disconnected solutions if these are solutions not connected to the branch of the main solution. We investigate the influence of disconnected solutions on the dynamics of the system. It is demonstrated that when disconnected solutions are formed, the nonlinear transition to turbulence is possible, and dangerous initial conditions are these disconnected solutions.

Building credibility for human systems integration in model‐based systems engineering

AbstractRobust and trusted digital human representations are necessary to successfully account for human considerations in model‐based systems engineering (MBSE). Multiple domains and modeling frameworks leverage verification, validation, and accreditation (VV&A) processes to characterize when and under what conditions a model is valid to establish credibility. A literature review was completed on mathematical, physics‐based, software development, discrete event simulation, agent‐based, system dynamics, and MBSE models with the goal of proposing a process for performing VV&A on digital engineering (DE) and MBSE models for sociotechnical systems. However, this research also revealed the need for a broader framework to characterize the risk associated with using these models for making high‐consequence decisions. While accomplishing the literature review, another approach to building credibility was identified that is used heavily in the financial industry, namely model risk management (MRM). This process is extended by leveraging MRM approaches from within the financial community to propose a framework for sociotechnical model users to characterize the risk of using MBSE models to make programmatic decisions. The primary contribution of this work is to document a meta‐analysis of model VV&A while proposing an alternative approach to characterizing and communicating credibility that was discovered during this analysis. This approach could be a viable option for ensuring the credibility of human systems integration in MBSE models.

Development of a control algorithm for a boost converter with the possibility of dynamic correction of control system parameters

RELEVANCE of the study lies in the development of an algorithm for controlling a DC voltage converter capable of providing a high-quality transient process with a wide change in the input parameters of the control object, affecting its nonlinear properties. purpose. THE PURPOSE. To consider methods for the development of continuous and discrete control systems for a step-up DC voltage converter and to obtain an analytical dependence of accounting for the nonlinear properties of the converter that affect the quality of regulation of the output voltage of the voltage stabilizer METHODS. The method of the average geometric root was chosen as a method for calculating the coefficients of the DC voltage converter control system, and the current circuit is adjusted using the technical optimum method, and the voltage circuit using the symmetric optimum method. The obtained analytical solutions of mathematical models and simulation models were compared in the SimInTech software environment. results. RESULTS. The simulation results show that the analytical solution of the mathematical model of the voltage stabilization system has a shorter transition time than the simulation results. This is due to the fact that in the mathematical model according to which the continuous control algorithm was synthesized, the presence of a power input filter, as well as the degree of their precharge, is not taken into account. Because of this, the transition time on simulation models is slower, applied 4 times than in the mathematical model. conclusion. CONCLUSION. When comparing the algorithms with each other, it can be seen that the algorithm obtained by converting a continuous algorithm by the Tusten method has the shortest transition time than all other algorithms. The continuous algorithm and the algorithm obtained by the inverse Euler transformation method have a speed close to each other, and the algorithm obtained by the direct Euler transformation turns out to be the slowest.

Static output feedback control for positive 2D discrete systems in Roesser model

Abstract This paper investigates the output stabilization problem for multi-input multi-output positive 2D systems described by a linear discrete-time Roesser system. This kind of systems have the property that the horizontal and vertical states take non-negative values whenever the initial boundaries are non-negative. Conditions for the existence of desired static output feedback controllers guaranteeing the resultant closed-loop system to be positive and asymptotically stable are obtained. Also, the synthesis of static output feedback controllers, including the requirement of positivity of the controllers, is solved. These results are then extended to the case of uncertain plants. All of the proposed conditions are expressed in terms of Linear Programming, which can be easily verified by using standard numerical software. Finally, several numerical examples are included to illustrate the validity and the effectiveness of the developed results.