Nonlinear Discrete Systems Research Articles

A New Friedman’s Model for Evolutionary Game Problem

The term game in game theory means a problem, where some of the people or groups (players) share a set of rules and regulations that create the conditions and events that make up the beginning of the game. For instance, in the trade market, the buyers and sellers of some commodities interact in a random way. The reputation of sellers effects on increasing of selling. e.g., honest sellers are more attractive than cheaters for the buyers and the buyers can examine the products or unexamined. In this paper, a non-linear discrete dynamical system of Friedman model was considered. Also, we proposed a new model of interaction between these two populations (buyers and sellers). investigate its limit run behavior where we found the limit converges to a fixed point (0,0) i.e., the sellers will always cheat and the sellers will not examine according to Friedman’s model which is denoted by the fixed point (0.0).

Online learning based neural network adaptive controller for efficient power tracking of PWR type reactor with unknown internal dynamics

An online learning neural network (NN) based adaptive scheme has been proposed for tracking the power level of higher order point kinetic pressurized water reactor (PWR) with unknown dynamics under local and global load following conditions and emergency conditions. The PWR type of nuclear reactors are linear parameter varying (LPV) systems whose parameters vary with power, ageing effects and changes in nuclear core reactivity with fuel burn up. The drastic changes in plant parameters should be accommodated in a reactor control system through on-line identification to ensures safe operation for the power plant which demands for adaptive control system. To track the demand changes which happens during the load following and emergency conditions considered, the PID controller parameters needs to be varied. However, the proposed NN controller is based on feedback linearisation of nonlinear discrete time system with unknown internal dynamics by using a multilayer neural network acting as function approximator. Online weight tuning algorithm based on a modified delta rule and projection algorithm are used for the update of the weights of the proposed neural network controller along with conventional PID controller. Simulation studies with the proposed controller on a non linear PWR core shows that the proposed algorithm exhibits improved performance in terms of lesser ISE/ IAE/ ITAE over using PID controller under load following conditions. This new proposed power level controller has self learning capability and it needs the measurement of all the state variables and the desired trajectory.

Event-triggered secure control of discrete systems under cyber-attacks using an observer-based sliding mode strategy

This paper is concentrated on the event-triggered adaptive sliding mode control (SMC) issue for a class of discrete uncertain nonlinear systems subject to false data injection attack (FDIA). The considered control input undergoes the loss of effectiveness (LE) and FDIA in a unified framework. Furthermore, in order to cater the engineering reality, the event-triggered mechanism is introduced in the sensor to observer (S-O) channel and observer to controller (O-C) channel with hope to reduce the communication load. By introducing a novel scheme, sufficient conditions are provided to ensure the asymptotic stability of resultant sliding mode dynamics with prescribed H∞ performance. Subsequently, to reduce the communication burden, an optimal strategy is proposed which can guarantee the stability simultaneously through choosing suitable weighting matrices. Moreover, a projection-based adaptive SMC law is designed to compensate the effects caused by the LE and FDIA, which can force the system state to be attracted into the pre-defined sliding manifold. Finally, some comparison simulations are presented to illustrate the validity of the provided control method.

Construction of exotical soliton-like for a fractional nonlinear electrical circuit equation using differential-difference Jacobi elliptic functions sub-equation method

The deal of this paper is to use the differential-difference Jacobi elliptic functions sub-equation method for constructing exact solutions of nonlinear electrical circuit including intrinsic fractional-order in the sense of Riemann–Liouville derivatives. According to the algorithm of a unified symbolic computation, we attain several solitons solutions as solitary waves train, singular kink-type soliton, doubly periodic solitons, grey and anti-grey soliton-like. These findings are emerged and constructed by means of the three Jacobi elliptic functions. These types of functions provide hyperbolic, trigonometric, exotic and doubly periodic fractional exact solutions which have not yet been reported in the studied model. For this studied model, The new solutions obtained are exotic soliton-like that have not been observed yet. And they provide new propagative’s modes through the cn, dn, sn Jacobi elliptic functions for the fractional nonlinear electrical pass band circuit. Therefore, further investigations on differential-difference Jacobi elliptic functions sub-equation method should help researchers to discover more soliton solutions for other nonlinear discrete systems.

Optimal distributed filtering for nonlinear saturated systems with random access protocol and missing measurements: The uncertain probabilities case

The saturated-constrained distributed filtering problem is discussed for a class of discrete nonlinear systems with random access protocol (RAP) and missing measurements (MMs) under uncertain missing probabilities (UMPs). In order to prevent data congestions, the RAP is adopted to regulate the information transmission, where only one sensor is scheduled to have the communication right and transmit the information at each time step through a shared network. In addition, the phenomenon of MMs is considered, in which the UMPs are characterized by means of the nominal means and error bounds. The main purpose of this paper is to design a distributed filter under saturation constraint such that, for both RAP and MMs under UMPs, there exists an upper bound (UB) matrix of filtering error covariance and the corresponding distributed filter gain is constructed to minimize the trace of the received UB matrix by resorting to the matrix simplifying technique. Besides, the filtering algorithm performances are discussed regarding the boundedness and monotonicity of the presented distributed filtering scheme from the mathematical perspective. Finally, some comparisons are provided to illustrate the validity of the proposed time-varying distributed filtering algorithm.

Distributed resilient state estimation over sensor networks with random nonlinearities, fading measurements, and stochastic gain variations

AbstractThe distributed resilient state estimation problem of nonlinear discrete systems in sensor networks is investigated in this article. The system model under consideration involves three phenomena of incomplete information: randomly occurring nonlinearities, fading measurements, and random gain variations. The probabilistic characteristics of the above phenomena are depicted by three sets of independent random variables subject to more general random distribution. Based on the above model, by applying Lyapunov functional approach and random distribution solution method, the asymptotic stability in the mean square sense of the estimation error system with a given attenuation level is proved. Further, the estimator parameters are solved by introducing a novel linearization method. Finally, a numerical simulation is given to illustrate the validity of the theoretical results.

Adaptive multiple model control for a class of nonlinear discrete time systems: second-level adaption design approach

In this study, a multiple model control scheme based on a second-level adaption design approach is established for a class of nonlinear discrete time systems. The nonlinear function of systems is relaxed to zero-order bounded. In the first-level adaption, N identification models are employed to estimate unknown parameters, which can weaken the effect of unknown parameters. The convex hull property based on all identification models is constructed, which ensures that the convex hull identified parameters converge to the real value rapidly. Compared with the existing multiple model controller, the actual controller is developed by a convex hull for the identification of unknown parameters and it can avoid some potential drawbacks of the switch mechanism. In the constructed scheme, the discrete time system is bounded input bounded output (BIBO) and the tracking error is bounded. Finally, a simulation example is presented to validate the proposed scheme.

Higher order weak Galerkin methods for the Navier–Stokes equations with large Reynolds number

AbstractIn this article, we provide a class of numerical schemes to solve the steady state and non‐steady state Navier–Stokes equations with large Reynolds number. The high order weak Galerkin methods are numerically proposed and tested. We numerically find that the stabilizing term has a great impact on the robustness of the Jacobian matrices associated with the Newton's method in the procedure of solving the discrete nonlinear systems generated by weak Galerkin discretizations, especially for the case with large Reynolds number. The numerical investigations of several benchmarks are then carried out to demonstrate the efficiency and robustness of our proposed numerical methods. The errors for the Taylor–Green vortex flow are computed and compared with different weak Galerkin techniques. The lid driven flow and the flow around a cylinder are further numerically investigated and compared with existing references. The Navier–Stokes equations with large Reynolds number can be efficiently solved by the high order weak Galerkin methods with proper stabilizers.

Wave propagation in a continuum nonlinear phononic material with soft nonlinear elastic layers

This presentation discusses modeling the propagation of nonlinear waves in a continuum phononic material. This system is shown to exhibit amplitude-dependent response and energy transfer between frequencies, which are well studied phenomena of nonlinear discrete systems, but are relatively unexplored in fully elastic continuum models. The phononic material is composed of repeated layers of stiff and soft materials, and the nonlinearity is introduced through a nonlinear hyperelastic Gent model in the soft material. We analyze the dispersion of the linearized system using Bloch-wave analysis to identify band gaps in the system, and we use full-scale time-domainfinite element simulations to analyze the dynamic behavior of the system. Generation of zero frequency and second harmonic frequency amplitudes as well as their accumulation with distance are observed, and the influence of phononic band gaps on nonlinear wave propagation are described. We explore the influence of material nonlinearity in soft materials in continuum phononic materials as a mechanism for control of nonlinear wave propagation.

Finite element approximation of a Keller–Segel model with additional self- and cross-diffusion terms and a logistic source

A fully finite element approximation of the Keller–Segel model, including self- and cross-diffusion terms and a logistic source, has been considered. The existence of a fully finite element approximation of the Keller–Segel model, some stability bounds of the solution and the convergence of the approximated solution have been shown. As a result, the existence of a solution to the nonlinear Keller–Segel model in Rd, d≤3 has been demonstrated. Moreover, an iterative approach to solving the resulted nonlinear discrete system has been introduced. Finally, some numerical results have been shown.

Converging outer approximations to global attractors using semidefinite programming

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

Periodic Solutions with Prescribed Minimal Period for 2 n th-Order Nonlinear Discrete Systems

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of 2 n th-order nonlinear discrete system are obtained. The main approach used in our paper is variational technique and the linking theorem. The problem is to solve the existence of periodic solutions with prescribed minimal period of 2 n th-order discrete systems.

Design of the modified fractional central difference Kalman filters under stochastic colored noises

For state estimation of discrete nonlinear fractional stochastic systems, this study presents two innovative modified fractional central difference Kalman filters. We consider a complicated scenario where the process noise or measurement noise in the system become colored noise. Firstly, the nonlinear function is linearized by utilizing the Stirling polynomial interpolation formula. Thus there is no need to calculate the Jacobi matrix for both algorithms, which means very few application limitations. Then, based on the augmented-state method, we develop an augmented state fractional central difference Kalman filter under the scenario of colored process noise. Afterwards, a state estimation algorithm for handling stochastic systems containing colored measurement noise is put forward by using the measurement expansion method. Finally, to perform the superiority of the developed algorithms, several simulations are carried out. As well, the algorithms derived in this paper are contrasted with the original fractional central difference Kalman filter and three other algorithms. Notably, a simulation with engineering significance for the state-of-charge estimation for lithium-ion batteries is also introduced, Aside from the commonly used numerical simulation. The results verify the superiority of the developed algorithms in sense of estimation accuracy and real-time performance.

A Novel Control Scheme for PWM Boost Converter based on Sliding Mode Control using Particle Swarm Optimization

Background: The PWM Boost converter is a strongly nonlinear discrete system, especially when the input voltage or load varies widely, therefore, tuning the control parameters of which is a challenge work. Objective: In order to overcome the issues, particle swarm optimization (PSO) is employed for tuning the parameters of a sliding mode controller of a boost converter. Methods: Based on the analysis of the Boost converter model and its non-linear characteristics, a mathematic model of a boost converter with a sliding mode controller is built firstly. Then, the parameters of the Boost controller are adjusted based on the integrated time and absolute error (ITAE), integral square error (ISE) and integrated absolute error (IAE) indexes by PSO. Results: Simulation verification was performed, and the results show that the controllers tuned by the three indexes all have excellent robust stability. Conclusion: The controllers tuned by ITAE and ISE indexes have excellent steady-state performance, but the overshoot is large during the startup. The controller tuned by IAE index has better startup performance and slightly worse steady-state performance.

A finite-element model for the Hasegawa–Mima wave equation

In a recent work Karakazian and Nassif (2021)[1], two of the authors have formulated the non-linear space-time Hasegawa–Mima plasma equation as a coupled system of two linear PDEs, a solution of which is a pair (u,w), with w=(I−Δ)u. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa–Mima problem with periodic boundary conditions were also derived. Using the Fourier basis in the space variables, existence of solutions were obtained. Implementation of algorithms based on Fourier series leads to systems of dense matrices. In this paper, we use a finite element space-domain approach to semi-discretize the coupled variational Hasegawa–Mima model, obtaining global existence of solutions in H2 on any time interval [0,T],∀T. In the sequel, full-discretization using an implicit time scheme on the semi-discretized system leads to a nonlinear full space-time discrete system. For the existence proof, a condition on the time step is imposed that is independent from the space mesh size h. For the uniqueness proof, a CFL-type of condition on the time step is imposed. Tests on a semi-linear version of the implicit nonlinear full-discrete system are conducted for several initial data, assessing the efficiency of our approach.

Dynamic Formation of Customers’ Quality Expectations

In this paper, based on well-established behavioral theories, we propose a new model for dynamic formation of customer’ quality expectations. The model is described by a one-dimensional non-linear discrete dynamical system which depends on several parameters with links to behavioral theories. Applying stability analysis, we investigate the dynamic properties of the non-linear system. We find those domains in the parameter space in which the equilibrium point of the system (the fixed point) and the periodic orbits are stable and in which they are unstable. These domains correspond to the cases customers will converge to a specific expectation of the offering, while in other cases their expectations will oscillate between a small number of points periodically. These results can help identify target segments for companies and pave the way for interventions that foster more accurate customer expectations.

Geometric shaping optimization of 64-APSK constellation in discrete nonlinear frequency division multiplexing systems.

We experimentally demonstrated a geometric shaped (GS) 64-ary amplitude phase shift keying (64-APSK) eigenvalue transmission. The signal is modulated on the scatter coefficient of a single eigenvalue and linear minimum mean square error (LMMSE) estimator is used to reduce the noise. The channel response is estimated by transmitting a normally distributed 64-APSK constellation through a communication link. Based on the polar coordinates distribution of the received constellation, the diameter distributions for each circle can be obtained so that circles with larger noise can obtain larger judgment width. After optimization, the experimental results show that the Q-factor gain is 1.13 dB under 22 dB received optical signal to noise ratio (OSNR) configuration and 0.88 dB after 900 km transmission compared with normally distributed APSK configuration.

Distributed filtering stochastic parameter learning algorithm for discrete-time random nonlinear systems with delay

Distributed filtering algorithm for a discrete time random nonlinear stochastic systems associated with state delay for the distributed wireless communication sensors are discussed in this paper. Stochastic Parameter Learning Algorithm (SPLA) is tend to aim at obtaining a collection of stochastic filter parameters in a finite limited time horizon, that minimize the traces of the upper limits which is permitted to reduce the error variance matrices of the concerned stochastic filter system’s states and delay measurements. Filter gain values of the filter derived by the determination of Riccati type difference equations, estimates systems states with delay. Two different filter rules are taken into account for the SPLA discrete time random nonlinear systems with steady state space equations model. Zero mean distinct covariance matrix along with the constructive state values and constant time delay are focused in compatible dimensions. The variance of the projected systems predicted noise and the actual estimated noise are validated through numerical examples.

Output Feedback Controller for a Class of Unknown Nonlinear Discrete Time Systems Using Fuzzy Rules Emulated Networks and Reinforcement Learning

A model-free adaptive control for non-affine discrete time systems is developed by utilising the output feedback and action-critic networks. Fuzzy rules emulated network (FREN) is employed as the action network and multi-input version (MiFREN) is implemented as the critic network. Both networks are constructed using human knowledge based on IF–THEN rules according to the controlled plant and the learning laws are established by reinforcement learning without any off-line learning phase. The theoretical derivation of the convergence of the tracking error and internal signal is demonstrated. The numerical simulation and the experimental system are given to validate the proposed scheme.

A framework for 3D tracking of frontal dynamic objects in autonomous cars

Both recognition and 3D tracking of frontal dynamic objects are crucial problems in an autonomous vehicle, while depth estimation as an essential issue becomes a challenging problem using a monocular camera. Since both camera and objects are moving, the issue can be formed as a structure from motion (SFM) problem. In this paper, to elicit features from an image, the YOLOv3 approach is utilized beside an OpenCV tracker. Subsequently, to obtain the lateral and longitudinal distances, a nonlinear SFM model is considered alongside a state-dependent Riccati equation (SDRE) filter and a newly developed observation model. Additionally, a switching method in the form of switching estimation error covariance is proposed to enhance the robust performance of the SDRE filter. The stability analysis of the presented filter is conducted on a class of discrete nonlinear systems. Furthermore, the ultimate bound of estimation error caused by model uncertainties is analytically obtained to investigate the switching significance. Simulations are reported to validate the performance of the switched SDRE filter. Finally, real-time experiments are performed through a multi-thread framework implemented on a Jetson TX2 board, while radar data is used for the evaluation.