Nonlinear Discrete Systems Research Articles

Measuring nonlinearity by means of static parameters in Bernoulli binary sequences distribution: A brief approach

This paper analyzes Bernoulli’s binary sequences in the representation of empirical nonlinear events, analyzing the distribution of natural resources, population sizes and other variables that influence the possible outcomes of resource’s usage. Consider the event as a nonlinear system and the metrics of analysis consisting of two dependent random variables 0 and 1, with memory and probabilities in maximum finite or infinite lengths, constant and equal to 1/2 for both variables (stationary process). The expressions of the possible trajectories of metric space represented by each binary parameter remain constant in sequences that are repeated alternating the presence or absence of one of the binary variables at each iteration (symmetric or asymmetric). It was observed that the binary variables [Formula: see text] and [Formula: see text] assume on time [Formula: see text] specific behaviors (geometric variable) that can be used as management tools in discrete and continuous nonlinear systems aiming at the optimization of resource’s usage, nonlinearity analysis and probabilistic distribution of trajectories occurring about random events. In this way, the paper presents a model of detecting fixed-point attractions and its probabilistic distributions at a given population-resource dynamic. This means that coupling oscillations in the event occur when the binary variables [Formula: see text] and [Formula: see text] are limited as a function of time [Formula: see text].

Finite-time memory fault detection filter design for nonlinear discrete systems with deception attacks

ABSTRACTIn this paper, the finite-time memory fault detection filter (MFDF) is designed for nonlinear discrete systems with randomly occurring deception attacks, where the phenomenon of the randomly occurring deception attacks is characterised by a Bernoulli distributed random variable with known probability. To be specific, the finite-horizon data are employed to construct the MFDF. The main purpose of this paper is to design the MFDF such that, for all nonlinearities, external disturbances and randomly occurring deception attacks, the resultant augmented system is finite-time stable and attains the performance. Accordingly, some sufficient conditions are developed to guarantee the existence of the desired fault detection filter parameters, where the solvability of the addressed problem is verified by the feasibility of certain matrix inequalities. Moreover, in order to show that the MFDF has better detection performance, a non-memory fault detection filter (NMFDF) is constructed to compare with the MFDF. Finally, two numerical simulations are utilised to illustrate the effectiveness of the proposed fault detection strategies and explain the superiority of the proposed MFDF.

Robust H2/H∞ control for a class of time-varying nonlinear stochastic systems with state- and control-dependent noises

ABSTRACTThis paper deals with the problem of robust control for a class of discrete time-varying nonlinear stochastic systems with state- and control-dependent noises (also called -dependent noises). In the addressed system, all system parameters are allowed to be time-varying, parameter uncertainties are supposed to be norm-bounded, and nonlinearities are assumed to satisfy the sector-bounded condition. A robust stochastic bounded real lemma is established for the uncontrolled system by means of the Riccati difference equation (RDE). Moreover, a necessary and sufficient condition is presented for the existence of finite-horizon robust control by virtue of the solvability of coupled RDEs, and an iterative algorithm is designed to solve these RDEs. Finally, a numerical example is utilised to illustrate the effectiveness of the derived results.

Control of continuous dynamical systems modeling physiological states

• The predictive control to remove periodic orbits and higher order orbits when the dynamic evolves in a continuous time. • The generalization to continuous-time systems of the results obtained on the suppression of periodic dynamics in multidimensional nonlinear discrete-time systems. • The control of some undesirable physiological states in the brain as well as in the heart. • The simulation results are based on the Rossler and the FitzHugh-Nagumo models. The aim of this work is a generalization to continuous dynamical systems of the results obtained for nonlinear discrete dynamical systems, i.e. the suppression of stable periodic dynamics, especially those corresponding to certain undesirable physiological states in the brain as well as in the heart. In most cases, complex pathologies are modeled by nonlinear dynamical systems with chaotic behavior and some pathological states correspond to periodic behaviors. We propose a new strategy using predictive control to suppress periodic and higher-order orbits when the dynamics evolves in continuous time. The feasibility and efficiency of our approach is illustrated using two models, the Rössler and FitzHugh-Nagumo models.

Nonlinear Autoregressive Moving Average-L2 Model Based Adaptive Control of Nonlinear Arm Nerve Simulator System

This paper considers the trouble of the usage of approximate strategies for realizing the neural controllers for nonlinear SISO systems. In this paper, we introduce the nonlinear autoregressive moving average (NARMA-L2) model which might be approximations to the NARMA model. The nonlinear autoregressive moving average (NARMA-L2) model is an precise illustration of the input–output behavior of finite-dimensional nonlinear discrete time dynamical systems in a neighborhood of the equilibrium state. However, it isn't always handy for purposes of neural networks due to its nonlinear dependence on the manipulate input. In this paper, nerves system based arm position sensor device is used to degree the precise arm function for nerve patients the use of the proposed systems. In this paper, neural network controller is designed with NARMA-L2 model, neural network controller is designed with NARMA-L2 model system identification based predictive controller and neural network controller is designed with NARMA-L2 model based model reference adaptive control system. Hence, quite regularly, approximate techniques are used for figuring out the neural controllers to conquer computational complexity. Comparison were made among the neural network controller with NARMA-L2 model, neural network controller with NARMA-L2 model system identification based predictive controller and neural network controller with NARMA-L2 model reference based adaptive control for the preferred input arm function (step, sine wave and random signals). The comparative simulation result shows the effectiveness of the system with a neural network controller with NARMA-L2 model based model reference adaptive control system. Index Terms--- Nonlinear autoregressive moving average, neural network, Model reference adaptive control, Predictive controller DOI: 10.7176/JIEA/10-3-03 Publication date: April 30 th 2020

Average Predictive Control for Nonlinear Discrete Dynamical Systems

We explore the problem of stabilization of unstable periodic orbits in discrete nonlinear dynamical systems. This work proposes the generalization of predictive control method for resolving the stabilization problem. Our method embodies the development of control method proposed by B.T. Polyak. The control we propose uses a linear (convex) combination of iterated functions. With the proposed method auxiliary, the problem of robust cycle stabilization for various cases of its multipliers localization is solved. An algorithm for finding a given length cycle when its multipliers are known is described as a particular case of our method application. Also, we present numerical simulation results for some well-known mappings and the possibility of further generalization of this method.

On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation

This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.

On stability of nonlinear nonautonomous discrete fractional Caputo systems

Conditions to establish Mittag-Leffler stability of solutions for nonlinear nonautonomous discrete Caputo-like fractional systems just from the linear associated system is shown. Mittag-Leffler stability for linear systems is tackled pointing out properties the matrix must satisfy. Additionally features on solutions for linear systems are included in order to establish the equivalence between uniform asymptotical stability and exponential stability. Converse-like-Lyapunov Theorem is developed and a particular result to show Mittag-Leffler stability from a condition on a linear part of the full system is worked.

On Problem of Partial Detectability for Nonlinear Discrete-Time Systems

Discrete (finite-difference) systems are widely used in modern nonlinear control theory. One of the main problems of a qualitative study of such systems is the problem of stability of the zero equilibrium position, which has great generality. In most works, such a stability problem is analyzed with respect to all variables that determine the state of the system. However, for many cases important in applications, it becomes necessary to analyze a more general problem of partial stability: the stability of the zero equilibrium position not for all, but only with respect to some given part of the variables. Such a problem is often also considered as auxiliary problem in the study of stability with respect to all variables. In this way, the corresponding concepts and problems of detectability of the studied system arise, which play an important role in the process of analysis of nonlinear controlled systems. Then, more general problems of partial detectability were posed, within the framework of which the situation was studied when stability from a part of variables implies stability not with respect to all, but with respect to more part of the variables. This article studies a nonlinear discrete (finite-difference) system of a general form that admits a zero equilibrium position. Easily interpreted conditions are found on the structural form of the system under consideration that determine its partial detectability, for which stability over a given part of the variables of the zero equilibrium position means its stability with respect to the other, more part of the variables. In this case, the stability with respect to the remaining part of the variables is uncertain and can be investigated additionally. In the process of analyzing this problem of partial detectability, the concept of partial null-dynamics of the system under study is introduced. An application of the obtained results to the stabilization problem with respect to part of the variables of nonlinear discrete controlled systems is given.

On local stability of stochastic delay nonlinear discrete systems with state-dependent noise

We examine the local stability of solutions of a delay stochastic nonlinear difference equation with deterministic and state-dependent Gaussian perturbations. We apply the degenerate Lyapunov–Krasovskii functional technique and construct a sequence of events, each term of which is defined by a bound on a normally distributed random variable. Local stability holds on the intersection of these events, which has probability at least 1−γ,γ ∈ (0, 1). This probability can be made arbitrarily high by choosing the initial value sufficiently small. We also present a generalization to systems where a condition for stability is expressed in terms of the diagonal part of the unperturbed system, and computer simulations which illustrate our results.

PRECONDITIONED ITERATIVE METHOD FOR REACTIVE TRANSPORT WITH SORPTION IN POROUS MEDIA

This work deals with the numerical solution of a nonlinear degenerate parabolic equation arising in a model of reactive solute transport in porous media, including equilibrium sorption. The model is a simplified, yet representative, version of multicomponents reactive transport models. The numerical scheme is based on an operator splitting method, the advection and diffusion operators are solved separately using the upwind finite volume method and the mixed finite element method (MFEM) respectively. The discrete nonlinear system is solved by the Newton–Krylov method, where the linear system at each Newton step is itself solved by a Krylov type method, avoiding the storage of the full Jacobian matrix. A critical aspect of the method is an efficient matrix-free preconditioner. Our aim is, on the one hand to analyze the convergence of fixed-point algorithms. On the other hand we introduce preconditioning techniques for this system, respecting its block structure then we propose an alternative formulation based on the elimination of one of the unknowns. In both cases, we prove that the eigenvalues of the preconditioned Jacobian matrices are bounded independently of the mesh size, so that the number of outer Newton iterations, as well as the number of inner GMRES iterations, are independent of the mesh size. These results are illustrated by some numerical experiments comparing the performance of the methods.

On the synthesis of a polynomial controller for enlarging the stability domain for nonlinear discrete system

This paper aims to propose an algebraic method for estimating the stability region for nonlinear discrete systems. The main contribution is the improvement of an existing technique to enlarge the initial stability region in which the asymptotic stability is guaranteed. To deal with the maximization issue, a state feedback controller is proposed to extend the largest estimation of attraction domain of nonlinear polynomial discrete systems. The proposed approach is illustrated and validated on an isothermal stirred tank reactor.

On the combination of kernel principal component analysis and neural networks for process indirect control

ABSTRACT A new adaptive kernel principal component analysis (KPCA) for non-linear discrete system control is proposed. The proposed approach can be treated as a new proposition for data pre-processing techniques. Indeed, the input vector of neural network controller is pre-processed by the KPCA method. Then, the obtained reduced neural network controller is applied in the indirect adaptive control. The influence of the input data pre-processing on the accuracy of neural network controller results is discussed by using numerical examples of the cases of time-varying parameters of single-input single-output non-linear discrete system and multi-input multi-output system. It is concluded that, using the KPCA method, a significant reduction in the control error and the identification error is obtained. The lowest mean squared error and mean absolute error are shown that the KPCA neural network with the sigmoid kernel function is the best.

One-Step Prediction for Discrete Time-Varying Nonlinear Systems With Unknown Inputs and Correlated Noises

One of the most important problems in the estimation theory is the one-step prediction. The goal of this problem is to determine the predictions of states in the next time step. This paper focuses on the one-step prediction for nonlinear dynamic systems. The system under investigation involves unknown inputs and the system noises are correlated. In this approach, using the Taylor series expansion for nonlinear functions, a new augmented state nonlinear predictor is proposed for discrete time-varying nonlinear systems. This predictor is obtained by solving a deterministic min-max optimization based on the regularized least squares problem. Moreover, to reduce the computational complexity of the prediction solution, using a nonlinear transformation, we propose a two-stage predictor including lower order estimators for states and unknown inputs. Finally, a frequency modulated signal model is considered to illustrate the effectiveness and performance of the proposed approaches in comparison with the existing estimation methods.

A Novel Approach Based on Terminal Sliding Mode Control for Optimization of Nonlinear Systems

It is difficult for many discrete nonlinear systems to quickly meet control requirements and establish accurate dynamic models. This study presents a control strategy that combines model-free adaptive control and discrete terminal sliding mode control to solve these problems. A novel discrete terminal sliding surface is designed in this strategy. It not only avoids the establishment of system models but also achieves quick response. Firstly, a perturbation observer is established to estimate perturbation of the system and time - varying parameter pseudo - gradient is estimated from input and output data of the system. Secondly, the controller combining the partial form dynamic linearization model free adaptive control and discrete terminal sliding mode control is designed. The stability and convergence of the proposed strategy are proved in theory. Thirdly, the proposed strategy is compared with PID technique and the method of combining model free adaptive control(MFAC) with discrete sliding mode control in chemical system and water tank system respectively. Simulation examples demonstrate that the proposed strategy is superior to others in terms of fast response. This study provides a reference for solving the control problem of discrete nonlinear systems.

On the Stability and Stabilization of Nonlinear Non-Stationary Discrete Systems

The study of many biological, economic and other processes leads to its modeling based on discrete equations. Currently, the need to develop a mathematical apparatus for the qualitative analysis of discrete equations is caused by the creation of digital control systems, processors and microprocessors as well as discrete methods of signal transmission in automatic control systems and other theoretical and technical problems. One of the important areas of the qualitative analysis of discrete equations is the stability problem. The main method for studying the stability of nonlinear differential, discrete, and other types of equations is the direct Lyapunov method. The aim of this paper is to develop the direct Lyapunov method in the study of the limiting behavior and asymptotic stability of nonlinear nonstationary discrete equations using the comparison principle. New theorems are proved that are applied in the stability problem of the well-known epidemiological model as well as in solving the stabilization problem of a nonlinear discrete controlled system. An example is shown illustrating a qualitative difference in the conditions of stabilization of non-stationary and stationary discrete systems.

Search for Cycles of Nonlinear Periodic Discrete Systems Using the Method of Average Predictive Control

Search for Cycles of Nonlinear Periodic Discrete Systems Using the Method of Average Predictive Control

Exponential Stabilization of Discrete Nonlinear Time-Varying Systems

We consider a discrete nonlinear control time-varying system x(k + 1) = f(k, x(k), u(k)), k ∈ ℕ, x ∈ ℝn, u ∈ ℝr. A control process of this system is a pair (x(k), u(k))k ∈ N consisting of a control (u(k))k∈N and some solution (x(k))k∈N of the system with this control. We assume that the control process is defined for all k ∈ N. We have obtained sufficient conditions for uniform and non-uniform (with respect to the initial moment) exponential stabilization of the control process with any pregiven decay of rate. Exponential convergence to zero of the deviation of both the state vector and the control vector is guaranteed. The result is based on the property of uniform complete controllability (in the sense of Kalman) for a system of linear approximation.

Observer-based sliding mode control for discrete nonlinear systems with packet losses: an event-triggered method

In this paper, the observer-based output feedback sliding mode control (SMC) problem is investigated for discrete delayed nonlinear systems subject to packet losses under the event-triggered strategy. It is assumed that the packet losses may occur in the control channel from the sensor to the observer. A suitable compensation strategy via the Bernoulli distributed random variable is used to reduce the effects of packet losses. In order to avoid the phenomenon of network congestion during the networked transmission, an event-triggered mechanism is introduced to determine if the last released measurement needs to be updated. Based on the zero-order-hold (ZOH) measurement, an output feedback observer is designed to reconstruct the system state. This method can facilitate the design of the discrete-time sliding surface. A sufficient condition is proposed to guarantee the stochastic stability of sliding mode dynamics systems by using linear matrix inequality (LMI) method, and a novel observer-based sliding mode controller is synthesized to force the trajectories of the error systems onto the pre-designed sliding mode surface within finite time. Finally, an example is given to illustrate the validity of the proposed theoretical result.

Improved finite frequency <i>H</i><SUB align="right">∞ filtering for Takagi-Sugeno fuzzy systems

This paper investigates the design problem of H∞ filtering for discrete nonlinear systems in the Takagi-Sugeno (T-S) form. Our aim is to design a new filter guaranteeing an H∞ performance level in specific finite frequency (FF) ranges. Using the well-known generalised Kalman Yakubovich Popov lemma, Finsler's lemma, sufficient conditions for the existence of H∞ filters for different FF ranges are proposed and then unified in terms of solving a set of linear matrix inequalities (LMIs). Two examples are given to illustrate the effectiveness and the less conservatism of the proposed approach in comparison with the existing methods.