Discrete Nonlinear Model Research Articles

Existence and uniqueness of almost periodic solutions to discrete mixed monotone hematopoiesis model

This paper is devoted to studying a generalized nonlinear discrete hematopoiesis model with delays. By applying mixed monotone operator fixed point theorem in an appropriate cone, we discuss the existence and uniqueness of positive almost periodic solution to this model. Four cases are classified based on parameters of the concerned model. The main results provided in the paper are more general than that of some previously known results, which are extended and complemented. Two particular examples are provided to illustrate that the obtained results, which are reasonable.

Optimization of Pressurized Tree-Type Water Distribution Network Using the Improved Decomposition–Dynamic Programming Aggregation Algorithm

Pressurized tree-type water distribution network (WDN) is widely used in rural water supply projects. Optimization of this network has direct practical significance to reduce the capital cost. This paper developed a discrete nonlinear model to obtain the minimum equivalent annual cost (EAC) of pressurized tree-type WDN. The pump head and pipe diameter were taken into account as the double decision variables, while the pipe head loss and flow velocity were the constraint conditions. The model was solved by using the improved decomposition–dynamic programming aggregation (DDPA) algorithm and applied to a real case. The optimization results showed that the annual investment, depreciation and maintenance cost (W1) were reduced by 22.5%; however, the pumps’ operational cost (p) increased by 17.9% compared to the actual layout. Overall, the optimal EAC was reduced by 15.2% with the optimized pump head and optimal diameter distribution of the network. This method demonstrated an intrinsic trade-off between investment and operational cost, and provided an efficient decision support tool for least-cost design of pressurized tree-type WDN.

Oba Filter for Online-Adaptive Nonlinear State Estimation With Interference

Interference, such as jamming, is widely encountered in practical applications, such as target tracking. Kalman-based nonlinear filters may not be feasible for nonlinear state estimation with nonlinearly transformed interference. This paper presents an online-based adaptive (oba) filter, which we have named the oba filter (OF), for state filtering and prediction of nonlinear discrete dynamic models with estimate constraints, nonlinearly transformed interference and noise. In the OF, noises and interference are approximated by discrete noises and interference; states are quantized; some undesired state estimates, which are not in near vicinities of the actual state values, are detected by a confidence level, and then state quantization levels are updated at the times when undesired estimates occur by using the measurement model; and states are recursively estimated by a suboptimal implementation of multiple composite hypothesis testing. The OF can prevent some undesired state estimates and outperform the unscented Kalman, regularized, sampling importance resampling (SIR), and auxiliary SIR particle filters for some nonlinear models with noise and interference. It is also suitable for state estimation with either missing observations or estimate constraints.

Multistability driven by inhibitory kinetics in a discrete-time size-structured chemostat model.

In this paper, we consider a discrete size-structured nonlinear chemostat model with inhibitory kinetics and multiple competing species. We show that, same as in models with monotone increasing nutrient uptake response functionals studied in the literature, one essential growth-limiting nutrient can support at most one species. More interestingly, we show that, differing from monotone increasing kinetics, inhibitory kinetics can induce several types of multistability and the competitive outcome is initial condition dependent. The species with the smallest lower break-even value is not always the winning species, and the winning species can exhibit sustained periodic oscillations and even chaotic irregular oscillations.

Predictive control for visual servoing control of cyber physical systems with packet loss

Networked robotic visual servoing system is a representative cyber-physical system (CPS). This paper studies the robust predictive control problem of networked robotic visual servoing system with packet losses and uncertainty. Firstly, according to eye-to-hand framework, the system is modeled as a nonlinear discrete model with packet loss based on image-based visual servoing (IBVS) approach, where the packet losses obey a Bernoulli distribution. Since the discretization error of the system is a norm-bounded parameter, a nonlinear system with bounded uncertainty is derived. With regard to the principle of the MPC and the stochastic system method, the upper bound of the MPC performance index and the min-max optimization problem with uncertainty are presented. Based on robust least-square approach, the parameter-dependent predictive controller is obtained and an iterative algorithm is presented to solve the horizon optimization problem. Finally, numerical simulations and experiments are proposed to verify the effectiveness of the proposed algorithm.

A fitted operator method for tumor cells dynamics in their micro-environment

In this paper, we consider a quasi non-linear reaction-diffusion model designed to mimic tumor cells’ proliferation and migration under the influence of their micro-environment in vitro. Since the model can be used to generate hypotheses regarding the development of drugs which confine tumor growth, then considering the composition of the model, we modify the model by incorporating realistic effects which we believe can shed more light into the original model. We do this by extending the quasi non-linear reaction-diffusion model to a system of discrete delay quasi non-linear reaction-diffusion model. Thus, we determine the steady states, provide the conditions for global stability of the steady states by using the method of upper and lower solutions and analyze the extended model for the existence of Hopf bifurcation and present the conditions for Hopf bifurcation to occur. Since it is not possible to solve the models analytically, we derive, analyze, implement a fitted operator method and present our results for the extended model. Our numerical method is analyzed for convergence and we find that is of second order accuracy. We present our numerical results for both of the models for comparison purposes.

Research on dynamic nonlinear input prediction of fault diagnosis based on fractional differential operator equation in high-speed train control system.

In order to control the nonlinear high-speed train with high robustness, the fractional order control of nonlinear switching systems is studied. The fractional order controller is designed for a class of nonlinear switching systems by the fractional order backstepping method. In this paper, a simple and effective online updating scheme of model coefficients is proposed by using the flexibility of the model predictive control algorithm and its wide range of model accommodation. A stochastic discrete nonlinear state space model describing the mechanical behavior of a single particle in a high-speed train is constructed, and the maximum likelihood estimation of the parameters of a high-speed train is transformed into an optimization problem with great expectations. Finally, numerical comparison experiments of motion characters of two high-speed trains are given. The results show the effectiveness of the proposed identification method.

Metro Traffic Modeling and Regulation in Loop Lines Using a Robust Model Predictive Controller to Improve Passenger Satisfaction

Automatic traffic regulation plays an important role in high-frequency metro lines. This paper introduces a discrete event nonlinear traffic model for metro loop lines based on the deviations from nominal departure times. This model considers the transfer of knock-on delay from one train to another one and improves on previous models. The model also includes uncertainty to accommodate variation in passenger demand over the course of a work day. The nonlinear uncertain traffic model is approximated by the linear uncertain model which facilitates the design of a robust predictive controller. The objective of control is increasing the passenger satisfaction by minimizing the deviation from the nominal time schedule and keeping headways close to the nominal headway. The linear robust model predictive controller (RMPC) is employed to compensate the disturbances and to regulate traffic in the presence of operational constraints. YALMIP toolbox was used to design the RMPC. Model verification and simulations were carried out using real data from Tehran metro line 4. The simulation results are presented to show the trueness of the model and effectiveness of the proposed control methodology.

Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

Author(s): Boeing, Geoff | Abstract: Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

Modeling of monetary and credit system indicators of the Russian Federation in multidirectional scenarios of oil market dynamics

The purpose of research – to build mathematical models that describe interrelationships between the key market indicators, significant for the Russian economy, and macroeconomic indicators of the monetary system. Materials and methods. In this study, we applied methods of model description, mostly used in control theory, meant for technical engineering, such as linear discrete transfer functions, space-state models and nonlinear Hammerstein-Wiener models. To identify these models, we used System Identification Toolbox from Matlab software package, mostly used for mechanical systems’ analysis. Based on the known input and output signals, a mathematical model was estimated. Time series of macroeconomic and market indicators for the period from January 10, 2008 to January 10, 2018 were used for identification. Results. Two prediction models were designed in this work. The first model describes a sequential transfer from the oil price and dollar- to-ruble exchange rate to the gross domestic product, then to M2 and then to loans. Dependencies between economic parameters are described by linear discrete transfer functions. There is only one difference in the second model’s general structure: the sequence of the last two transitions from the gross domestic product to loans, and then to M2. In addition, nonlinear Hammerstein-Wiener models describe last two transitions in the second model. As a result, predictions for macroeconomic indicators’ trends were given on different time horizons: three, seven and twelve years and with two differently directed scenarios of the oil market. The conclusion. Despite close values in the models accuracy estimation, they give similar results for matching scenarios, but different growth rates in general, in the forecast. Such a result in scenarios shows, that a sharp fall in oil prices has a stronger impact on given macroeconomic and market indicators, which, in its turn, shows the capability of the models to make correct trend predictions. In further studies, it is possible to move from macroeconomic indicators to their more particular components at meso- and micro levels.

Dynamic interaction between adjacent buildings through nonlinear soil during earthquakes

This paper evaluates the effect of Structure-Soil-Structure Interaction (SSSI) between two buildings given different parameters of the buildings, inter-building spacing, and soil type. A two-dimensional simple discrete nonlinear model is proposed that is described by a set of nonlinear differential equations of motion. A nonlinear phenomenological Bouc-Wen model, for the soil directly underneath the foundations, linear rotational interaction spring between buildings and linear behaviour of buildings are assumed. The seismic ground motion employed is spectrally matched with EC8 elastic spectra. The results showed that there are both unfavourable and beneficial configurations of the two buildings that produce important differences between nonlinear SSSI and nonlinear SSI (the uncoupled building case). Importantly it is demonstrated that the adverse effects of SSSI can be more pronounced when the nonlinear soil behaviour is assumed.

Reproducing the nonlinear dynamic behavior of a structured beam with a generalized continuum model

In this paper we study the coupled axial-transverse nonlinear vibrations of a kind of one dimensional structured solids by application of the so called Inertia Gradient Nonlinear continuum model. To show the accuracy of this axiomatic model, previously proposed by the authors, its predictions are compared with numeric results from a previously defined finite discrete chain of lumped masses and springs, for several number of particles. A continualization of the discrete model equations based on Taylor series allowed us to set equivalent values of the mechanical properties in both discrete and axiomatic continuum models. Contrary to the classical continuum model, the inertia gradient nonlinear continuum model used herein is able to capture scale effects, which arise for modes in which the wavelength is comparable to the characteristic distance of the structured solid. The main conclusion of the work is that the proposed generalized continuum model captures the scale effects in both linear and nonlinear regimes, reproducing the behavior of the 1D nonlinear discrete model adequately.

Research of Enterprise Storage Ecosystem Based on Storage Theory and Nonlinear Discrete Optimization

Warehousing and transferring strategies are an important part of business operations. The issue of optimal warehousing and transferring strategy is studied in this paper. Wal-Mart in Wuhan serves as an example to establish a (s, S) random storage strategy model, a Markov chain model, and a nonlinear discrete programming model, aiming at maximizing the profit per cycle of every branch and further maximizing the company’s total profit per cycle. Among them, the random storage strategy model establishes a security zone of inventory for every branch, that is, it can meet consumers’ demand without spending too much storage costs. The Markov chain model is used to get the probability of losing sales opportunities in every branch. The nonlinear discrete programming model takes into account the horizontal transferring among branches, which further maximizes the company’s overall profit expectations. The three models above can be used to formulate inventory strategies, assess risks, and provide advice for every branch in order to form a complete storage ecosystem and provide constructive suggestions for the company’s operations.

Complex dynamics analysis for a two-stage Cournot duopoly game of semi-collusion in production

In this paper, a two-stage oligopoly game of semi-collusion in production is analyzed and expounded, where at first stage all firms compete in R&D and at second stage all firms coordinate the production activities in order to make their joint profit maximized. Not only the local stability of equilibriums, but also the existence, stability and direction of flip bifurcation of the discrete nonlinear model are investigated by using the normal form method and the center manifold theory. Then, the validity of the theoretical analysis is justified through numerical simulation. We find that the model we built can exhibit very complex dynamical behaviors, but it cannot undergo the Neimark–Sacker bifurcation. Also the coexistence of attractors is found through numerical simulation, and their basins of attraction are simulated. At last of this paper, the chaotic motion of the proposed model is controlled by delayed feedback control method.

A Nonlinear Discrete Velocity Relaxation Model For Traffic Flow

We derive a nonlinear 2-equation discrete velocity model for traffic flow from a continuous kinetic model. The model converges to scalar Lighthill--Whitham--Richards-type equations in the relaxatio...

Strategies of refueling, sailing speed and ship deployment of containerships in the low-carbon background

Abstract To solve the problems of refueling, ship speed determination and ship deployment of containerships, based on the nonlinear relationship between ship sailing speed, fuel consumption and carbon emissions, a corresponding nonlinear mixed integer programming model is developed by taking container transportation cost, fuel consumption cost and carbon emissions cost in multiple service routes as the total cost objective function, and ship refueling ports, refueling amounts, sailing speed and ship deployment quantities in the route as decision variables. The optimal solution can be obtained by transforming the nonlinear discrete mixed integer programming model into a nonlinear continuous programming model. The proposed model is applied to an Asia-Europe service route of China Shipping (Group) Company. The results show that the optimal refueling policy, speed selection and ship deployment can be determined using the proposed model and solution method. The impact of fuel price, container transportation quantity, ship tanker capacity and carbon tax rate on ship refueling strategy, speed and deployment is investigated. Some insights obtained from example analysis are provided.

Discrete Time Formulation of Quasi Infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability

The quasi infinite horizon NMPC formulation developed by Chen and Allgöwer [1998] provides a systematic method to characterize the terminal set needed to ensure nominal stability of a system controlled using finite horizon NMPC formulation at an unstable operating point. Their approach, however, has been developed for continuous time NMPC formulations. Real time implementation of an NMPC scheme using microprocessors, however, makes model discretization inevitable. In this work, a discrete time version of the quasi infinite NMPC formulation is developed, which is particularly suited for operating a system at an unstable operating point. Also, a novel approach is developed for characterization of the terminal region that makes use of the discrete time nonlinear model and that provides sufficient degrees of freedom to characterize the terminal region. The nominal closed loop system is then shown to be equi-asymptotically stable when controlled using the proposed discrete time QIH-NMPC. Efficacy of the proposed approach is demonstrated using a benchmark problem given by Chen and Allgöwer [1998]. The simulation study shows that the terminal regions for the discrete time case are relatively smaller than the terminal region for the continuous time case reported by Chen and Allgöwer [1998]. Also, the proposed discrete time QIH-NMPC is able to steer the state trajectories from specified feasible initial conditions to locations inside the terminal set within the specified prediction horizon and later converge to the unstable operating point.

Discrete Time Approximation of Continuous Time Nonlinear State Space Models

In this paper we discuss the problem of approximating a continuous time nonlinear state space model with an explicit discrete time state space model. Two steps are made to do so. The first one is to replace the classical zero-order-hold (ZOH) excitation signal assumption by a more general concept of low pass (LP) signals. It will be shown that opposed to the ZOH-property, the LP-property is maintained for a wide class of (non)linear systems, including cascaded and closed loop systems. The second step is based on a result from the signal processing theory that states that a BL-signal can be exactly predicted from its past samples by a fixed causal linear filter. This result is generalized to the discrete time integration of LP-signals by introducing an error bound that can be made arbitrarily small by increasing the sample frequency. Both ideas are then combined to bound the approximation errors of an explicit discrete time nonlinear state space approximation for a continuous time nonlinear state space model. The order of the decay of the approximation error as a function of the sample frequency is given. It is shown that eventually the aliasing error is the dominating error, at a cost of a (slightly) increased model complexity. These results are directly applicable to nonlinear system identification, and will be experimentally verified on the identification of a closed loop nonlinear system (the silverbox) using a discrete time nonlinear state space model.

Nonlinear Discrete Mechanical Model of Steel Rings

AbstractCircular rings are used in various engineering structures because of their nonlinear properties. The use of large assemblies of steel rings in flexible ring nets is widespread in recent tec...

Two-degree-of-freedom H-infinity control of combustion in diesel engine using a discrete dynamics model

This paper proposes an H-infinity combustion control method for diesel engines. The plant model is the discrete dynamics model developed by Yasuda et al., which is implementable on a real engine control unit. We introduce a two-degree-of-freedom control scheme with a feedback controller and a feedforward controller. This scheme achieves both good feedback properties, such as disturbance suppression and robust stability, and a good transient response. The feedforward controller is designed by taking the inverse of the static plant model, and the feedback controller is designed by the H-infinity control method, which reduces the effect of the trubocharger lag. The effectiveness of the proposed method is evaluated in simulations using the nonlinear discrete dynamics model.