Nonlinear Form Research Articles

Indirect Data-Driven Observer Design Using Neural Canonical Observer Structures

An indirect data-driven observer design approach for nonlinear discrete-time systems based on an input-output injection with neural canonical observer structures is proposed. An artificial neural network auto-encoder structure, trained with recorded state, input, and output data, is used for the identification of a system in a nonlinear Brunovsky observer form with output transformation. The neural approximations of the transformations and the input-output injection can be used to construct an observer with linear error dynamics using methods from linear control theory. The approach is demonstrated on two academic examples and on an industrially-motivated problem with a sampled continuous-time model.

Stable periodic solutions in scalar periodic differential delay equations

A class of nonlinear simple form differential delay equations with a $T$-periodic coefficient and a constant delay $\tau >0$ is considered. It is shown that for an arbitrary value of the period $T>4\tau -d_0$, for some $d_0>0$, there is an equation in the class such that it possesses an asymptotically stable $T$-period solution. The periodic solutions are constructed explicitly for the piecewise constant nonlinearities and the periodic coefficients involved, by reduction of the problem to one-dimensional maps. The periodic solutions and their stability properties are shown to persist when the nonlinearities are “smoothed” at the discontinuity points.

Nonlinear Optimal Control for a Gas Compressor Actuated by a Five-Phase Induction Motor

Abstract The article proposes a nonlinear optimal control method for the dynamic model of a gas centrifugal compressor being actuated by a five-phase induction motor (5-phase IM). To achieve high torque and high power in the functioning of gas compressors, 5-phase IM appear to be advantageous in comparison to three-phase synchronous or asynchronous electric machines. The dynamic model of the integrated compression system, which comprises the gas compressor and the 5-phase IM, is first written in a nonlinear and multivariable state-space form. It is proven that the electrically driven gas-compression system is differentially flat. Next, this system is approximately linearised around a temporary operating point that is recomputed at each sampling interval. The linearisation is based on first-order Taylor series expansion and uses the computation of the Jacobian matrices of the state-space model of the integrated system. For the linearised state-space description of the compressor and 5-phase IM, a stabilising optimal (H-infinity) feedback controller is designed. This controller achieves a solution to the nonlinear optimal control problem of the compressor and 5-phase IM system under model uncertainty and external perturbations. The feedback gains of the controller are computed by solving an algebraic Riccati equation at each iteration of the control method. Lyapunov analysis is used to demonstrate global stability for the control loop. Additionally, the H-infinity Kalman filter is used as a robust state estimator, which allows for implementing sensorless control for the gas compression system.

Mathematical modeling for anaerobic digestion under the influence of leachate recirculation

<abstract><p>In this paper, we proposed and studied a simple five-dimensional mathematical model that describes the second and third stages of the anaerobic degradation process under the influence of leachate recirculation. The state variables are the concentration of insoluble substrate, soluble substrate, produced hydrogen, acetogenic bacteria and hydrogenotrophic-methanogenic bacteria. The growth rates of used bacteria will be of general nonlinear form. The stability of the steady states will be studied by reducing the model to a 3D system. According to the operating parameters of the bioreactor described by the added insoluble substrate, soluble substrate and hydrogen input concentrations and the dilution rate, we proved that the model can admit multiple equilibrium points and we gave the necessary and sufficient assumptions for their existence, their uniqueness and their stability. In particular, the uniform persistence of the system was satisfied under some natural assumptions on the growth rates. Then, a question was answered related to the management of renewable resources where the goal of was to propose an optimal strategy of leachate recirculation to reduce the organic matter (either soluble or insoluble) and keep a limitation of the costs of the recirculation operation during the process. The findings of this work were validated by an intensive numerical investigation.</p></abstract>

Rheology of a Viscous-Plastic Liquid in a Porous Medium

The hydrodynamics of the capillary flow of a viscous-plastic liquid in cylindrical rectilinear pores is considered, as a result of which the structural velocity distribution over the pore cross section is obtained. Analytical solutions are proposed for the equations of hydraulic diffusion and nonlinear filtration for a non-Newtonian fluid in a cylindrical porous medium. It is noted that when a non-Newtonian fluid flows in a porous medium, the filtration equations take a nonlinear form due to the effective viscosity, shear, and yield stresses taken into account in its structure. The proposed solutions make it possible to evaluate the state of the porous medium and its main parameters (permeability, hydraulic diffusion, and effective viscosity coefficients). The obtained solutions are compared with existing experimental data for non-Newtonian oils.

Approximation of Radiative Transfer for Surface Spectral Features

Remote sensing hyperspectral and more generally spectral instruments are common tools to decipher surface features in Earth and Planetary science. While linear mixture is the most common approximation for compounds detection (mineral, water, ice, etc...), the transfer of light in surface and atmospheric medium are highly non-linear. The exact simulation of non-linearities can be estimated at very high numerical cost. Here I propose a very simple non-linear form (that includes the regular linear area mixture) of radiative transfer to approximate surface spectral feature. I demonstrate that this analytical form is able to approximate the grain size and intimate mixture dependence of surface features. In addition, the same analytical form can approximate the effect of Martian mineral aerosols. Unfortunately, Earth aerosols are more complex (water droplet, water ice, soot,...) and are not expected to follow the same trend.

Impact of Political Risk on Foreign Direct Investment with Fourier Approach: The Case of Turkiye

Foreign direct investment (FDI) is a desirable form of capital inflows to emerging and developing countries. Such investments are less susceptible to crises and sudden breakdowns. Therefore, foreign direct investment between the political risks of using annual data for the years 1984-2019 which might influence Turkey has been tested. For this purpose, Fourier-based unit root and cointegration analysis are used. As a result of the Fourier unit root analysis, it is concluded that the FDI variable is difference stationary and the political risk variable is level stationary. For this reason, Yılancı et al. (2020), a fractional-frequency Fourier bootstrap autoregressive distributed lag (ARDL) boundary test was applied to the literature. This test, which is in nonlinear form, captures smooth transitional changes in long-term relationships. As a result of the analysis, it is seen that there is a cointegration relationship between foreign direct investments and political risk and the variables come to equilibrium in the long run. In addition, it is seen that the calculated long-term coefficient is positive and significant and the error mechanism works correctly in the short term. Therefore, as political stability is achieved foreign direct investment inflows will increase and the economies of the country will gain an advantage in reaching high levels of financial development. As a result, it is concluded that the Weak Partisan model is valid in the Turkish economy.

A study on modeling growth model of Adana pigeons

The study aims to determine a mathematical model that can be used to describe the growth of the Adana pigeon. Since pigeons have only one breeding season, just one or two pairs of baby pigeons are raised per year. Hatchlings sometimes die before reaching adulthood. For this reason, measurements can be taken for 10, 15 and 60 days periods. Related with this issue, only 43-days measurements of 68 pigeons are used over a 6-year period. The study is modelled by taking the day-to-day average of the data (43 days) of 68 pigeons. The study was conducted on 68 Adana pigeons in the interval between the age of 1 and 43 days. The growth of pigeon cub was measured by daily live weight until 1 to 43 days. The estimation is carried out by writing the specific Matlab codes. Classical growth functions used in animals are in nonlinear form. Various numerical methods have been developed to estimate parameters in nonlinear functions. Special program routines have been developed to implement these methods. In these nonlinear models, there are more than one parameter to be estimated. Therefore, the number of mathematical operations in estimating the parameters is large. The most used models in the literature are Brody, Bertalanffy, Logistic, Generalized Logistic, Gompertz, Richards, Negative Exponential, Stevens, and Tanaka. However, as far as is known, there is no published article for Adana pigeons that uses all of these models and compares which one is better. These models are Brody, Bertalanffy, Logistic, Generalized Logistic, Gompertz, Richards, Negative Exponential, Stevens, and Tanaka. The best analysis was done by the Richards model in terms of both the Mean Squared Error (MSE), mean absolute percentage error (MAPE) and (Coefficient of Determination) R2 .

Ridge Regression for Functional Form Identification of Continuous Predictors of Clinical Outcomes in Glomerular Disease

Introduction: Penalized regression models can be used to identify and rank risk factors for poor quality of life or other outcomes. They often assume linear covariate associations, but the true associations may be nonlinear. There is no standard, automated method for determining optimal functional forms (shapes of relationships) between predictors and the outcome in high-dimensional data settings. Methods: We propose a novel algorithm, ridge regression for functional form identification of continuous predictors (RIPR) that models each continuous covariate with linear, quadratic, quartile, and cubic spline basis components in a ridge regression model to capture potential nonlinear relationships between continuous predictors and outcomes. We used a simulation study to test the performance of RIPR compared to standard and spline ridge regression models. Then, we applied RIPR to identify top predictors of Patient-Reported Outcomes Measurement Information System (PROMIS) adult global mental and physical health scores using demographic and clinical characteristics among N = 107 glomerular disease patients enrolled in the Nephrotic Syndrome Study Network (NEPTUNE). Results: RIPR resulted in better predictive accuracy than the standard and spline ridge regression methods in 56–80% of simulation repetitions under a variety of data characteristics. When applied to PROMIS scores in NEPTUNE, RIPR resulted in the lowest error for predicting physical scores, and the second-lowest error for mental scores. Further, RIPR identified hemoglobin quartiles as an important predictor of physical health that was missed by the other models. Conclusion: The RIPR algorithm can capture nonlinear functional forms of predictors that are missed by standard ridge regression models. The top predictors of PROMIS scores vary greatly across methods. RIPR should be considered alongside other machine learning models in the prediction of patient-reported outcomes and other continuous outcomes.

Study of optical solitons for Kudryashov’s Quintuple power-law with dual form of nonlinearity using two modified techniques

The main purpose of this article is to explore the analytic soliton solutions of a nonlinear Schrodinger equation (NLSE), including Kudryashova’s quintuple power-law with dual form nonlinearity by presenting two efficient analytical techniques. The present equation is a significant model which deals with a variety of nonlinear phenomenons in the fields of optical fibers. Modified G′G2 -expansion technique and exp(−φ(ρ)) technique explore kink-singular, bright, dark and cupson-singular form of soliton solutions. The present techniques are simple and efficient because these techniques produced a rational, trigonometric solution of the nonlinear partial differential equation. The above two techniques are novel and have not been applied yet on present model.

A numerical performance of the novel fractional water pollution model through the Levenberg-Marquardt backpropagation method

The aim of the current work is to provide the importance and significance of the fractional order (FO) derivatives for solving the nonlinear water pollution model (FWP) model. The FO derivative to solve the water pollution model is provided to get more precise results. The investigations through the non-integer and nonlinear mathematical form to define the fractional water pollution model are also provided in this study. The composition of the fractional water pollution model is classified into three classes, execution cost of control, system competence of industrial elements and a new diagnostics technical exclusion cost. The mathematical FWP system is numerically studied by using the artificial neural networks (ANNs) along with the Levenberg-Marquardt backpropagation method (ANNs-LMBM). Three different cases using the FO derivative have been examined to present the numerical performances of the FWP model. The data is selected to solve the mathematical FWP system is 70% for training and 15% for both certification and testing. The exactness of solver is observed through the comparison of the results. To ratify the aptitude, validity, constancy, and exactness of the ANNs-LMBM, the replications using the regression/correlation, state transitions, and error histograms are also described.

Relaying Data With Joint Optimization of Energy and Delay in Cluster-Based UAV-Assisted VANETs

Vehicular networks are known for their dynamic topology, high mobility, and frequent disconnections. Unmanned aerial vehicles (UAVs) have been recently used as instant communication relays to bridge the communication gaps between terrestrial vehicles to improve connectivity in vehicular networks and overcome the aforementioned problems. Despite the existing work in the literature where each vehicle connects directly to UAVs, this work studies how clustering and different densities of vehicles affect delay and energy efficiency in a UAV-based vehicular network integrated with 5G technology. Consequently, this work addresses the problem of UAV enabling vehicular ad-hoc networks (VANETs) in a highway scenario, where UAVs serve as an effective complement to forward data packets between vehicles in the absence of sufficient fixed infrastructures in an emergency situation. The main objective of this work is to minimize the delay while maximizing the energy efficiency by minimizing the power consumption and maximizing the total data rate under realistic conditions, while nonorthogonal multiple access (NOMA) is also adopted as an alternative answer for the effective utilization of limited bandwidth. The free-flowing traffic follows a Poisson stochastic process where each vehicle is assigned a random speed selected from a truncated Gaussian distribution. To this end, a novel modification of fast global <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -means is adopted to partition vehicles, allowing communication between clusters by vehicle-to-vehicle links, while data packets between clusters are relayed through UAVs. By computing the convex approximation of the objective function and the constraints, the original problem with the mixed-integer, nonconvex, and nonlinear form is solved by the proposed iterative inner penalty function algorithm. Finally, extensive simulations are conducted to validate the superiority of the proposed method in terms of various metrics. The results indicate that the relaying task in the proposed UAV-assisted VANET based on 5G technology is perfectly suited to enhance the network connectivity.

Performance of a vibration isolator with sigmoidal force-deflection curve

This paper presents a theoretical insight on the performance of a vibration isolator consisting of a combination of linear mechanical springs arranged to achieve a specific form of geometric nonlinearity. In particular, positive and negative stiffness nonlinearities are combined to achieve a sigmoidal shape of the force-deflection curve, which is proven to be beneficial for vibration isolation purposes when the amplitude of vibration is relatively large. Such behaviour is fundamentally different from that of the classical quasi-zero-stiffness isolator, which presents a low dynamic stiffness at the equilibrium configuration and is thus effective for relatively low amplitude of vibration. The analytical findings are validated by numerical simulations, providing useful guidelines for the design of such isolators.

Projection-Based Containment Control of Multiple Nonlinear Systems With Switching Topologies

This article addresses the distributed containment control problem of nonlinear multiagent systems with directed jointly connected switching topologies. It is assumed that the dynamics of agents are unknown nonlinear pure-feedback forms. The novel adaptive containment-control laws are designed to solve the case without considering the formation among leaders first and then the case of leaders’ formation. Different from the existing results, a condition of average dwell time is given by using the smooth parameter projection. Uniformly ultimate boundedness of the closed-loop system is analyzed in detail. Simulations show the feasibility and effectiveness of the proposed scheme.

Benchmarking solutions of the Folgar–Tucker-Equation and its reduction to a linear problem for non-linear closure forms

The Folger-Tucker-Equation (FTE) arises from a modification of the Jeffery equation and describes the orientation of elongated particles in a flow by means of the orientation tensor A . Though the FTE represents the most widespread and commercially used relaxation equation for A in simulations of injection molding of short-fiber reinforced fluids, analytical solutions of the formulation applied in industry are hardly investigated. Previous work focused on the solution of the underlying (modified) Jeffery’s equation and its integration to the orientation tensor rather than a direct analytical solution of the FTE. The present paper firstly introduces a lemma that reduces nonlinear formulations of the FTE to linear problems. Its solution is numerically computational less intensive than the one of the original problems. Secondly, this paper presents analytical benchmarking solutions of the FTE to validate simulation algorithms. Thirdly, a closer look at two dimensional solutions enables a deeper understanding of the mathematical behaviour of this popular differential equation. The influence of rotational components in the velocity field on the orientation process is analysed. Finally an extension to the three dimensional case is discussed shortly. • We present analytical solutions to the Folgar-Tucker-Equation applied in industry. • The solutions provide an effective benchmark to validate numerical algorithms. • A lemma is proposed reducing nonlinear formulations of the FTE to linear problems. • The lemma reduces computational effort for its solution.

Efficient Removal of Phosphate from Wastewater by a Novel Phyto-Graphene Composite Derived from Palm Byproducts.

The increased demand for clean water especially in overpopulated countries is of great concern; thus, the development of eco-friendly and cost-effective techniques and materials that can remediate polluted water for possible reuse in agricultural purposes can offer a life-saving solution to improve human welfare, especially in view of climate change impacts. In the current study, the agricultural byproducts of palm trees have been used for the first time as a carbon source to produce graphene functionalized with ferrocene in a composite form to enhance its water treatment potential. Scanning electron microscopy (SEM), energy-dispersive X-ray spectroscopy, X-ray diffraction (XRD), ultraviolet-visible, Fourier transform infrared spectroscopy, zeta potential, thermogravimetric analysis, and Raman techniques have been used to characterize the produced materials. SEM investigations confirmed the formation of multiple sheets of the graphene composite. Data collected from the zeta potential revealed that graphene was supported with a negative surface charge that maintains its stability while XRD elucidated that graphene characteristic peaks were evident at 2θ = 22.4 and 22.08° using palm leaves and fibers, respectively. Batch adsorption experiments were conducted to find out the most suitable conditions to remove PO4 3- from wastewater by applying different parameters, including pH, adsorbent dose, initial concentration, and time. Their effect on the adsorption process was also investigated. Results demonstrated that the best adsorption capacity was 58.93 mg/g (removal percentage: 78.57%) using graphene derived from palm fibers at 15 mg L-1 initial concentration, pH = 3, dose = 10 mg, and 60 min contact time. Both linear and non-linear forms of kinetic and isotherm models were investigated. The adsorption process obeyed the pseudo-second-order kinetic model and was well fitted to the Langmuir isotherm.

Enhanced higher-order modulational instability in a parity–time-symmetric fiber Bragg grating system with modified saturable nonlinearity

With the use of cubic, quintic, and septic nonlinearities, we demonstrate the influence of modified nonlinear saturation on modulational instability (MI) in a nonlinear complex parity–time (PT)-symmetric fiber Bragg grating (FBG) structure. Using a modified coupled nonlinear Schrodinger equation and linear stability analysis, we derive a dispersion relation for instability gain spectra in a complicated PT-symmetric system. Our main aim is to examine the MI in non-Kerr nonlinearities with nonlinear saturation in three PT-symmetric regimes: below threshold point, at threshold point (breaking point), and above threshold point. The occurrence of MI is known to be problematic at the PT-symmetry threshold point in a standard FBG structure (A.K. Sharma, 2014). At the same time, MI can exist in the normal group velocity dispersion domain when the modified nonlinear saturation effect is used. With the help of a modified form of saturable nonlinearity, we discovered that MI could exist in all three regimes in a complex PT-symmetric FBG structure. In anomalous group velocity dispersion alone, we found bistability behavior in a PT-symmetric FBG structure with higher-order saturable nonlinearity. In the presence of a modified nonlinear saturation effect and higher-order non-Kerr nonlinearities, we found a novel type of dynamics in the PT-symmetric FBG structure. All alterations in the photonic device bandgap directly result from changes in the refractive index of the medium caused by the interaction of PT-symmetric potential with the cubic–quintic–septic and modified form of nonlinear saturation. As a result, we provide approaches for generating and managing the MI in a complex PT-symmetric FBG structure under the influence of the modified nonlinear saturation effect.

Defence expenditure and public debt in Greece: A non-linear relationship

The aim of this paper is to consider and evaluate the widely–used argument attributing the growth of the public debt of Greece in recent decades to the country’s defence expenditure. We employ a two-equation model that focuses on the determinants of the military debt and how this has affected the public debt. The source of our military debt data is the Hellenic General Accounting Office, an option taken to cope with confidentiality issues. Finally, we insist on focusing on the expenditure on defence equipment imports rather than the total defence spending that the literature mostly covers, for reasons extensively analysed. Our methodology firstly assumes a linear relationship between military debt and public debt. We solve the system using 3SLS to obtain consistent and efficient estimates to derive interesting policy implications. The employment of a non-linear quadratic form not previously adopted proves to be superior in terms of performance and reveals the complexity of the above relationship. These point to the conclusion that the defence equipment purchases made since the beginning of the eighties have not had a decisive impact on the public debt of Greece. The defence equipment purchases made since the beginning of the eighties represent a very low percentage of the total public debt of Greece. Moreover, these are considered to be an investment activity rather than consumer spending, which has been acknowledged in the literature as the primary cause of Greece’s high indebtedness and the ensuing economic crisis.

Analysis of oscillations in 1D nanostructures influenced by different concrete potential functions

A family of nonlinear second-order partial differential equations based on concrete potential functions were formulated and analyzed. The occurrence of linear and nonlinear wave solutions were dependent on the interaction potential introduced into the nanostructure and on the initial conditions introduced to the system. The family of nonlinear equations were numerically analyzed because at present, it is difficult (or not possible) to obtain their exact solutions. The law of linear growth was introduced into the nanostructure and space variables were transformed in terms of new variables. The accreting nanostructure was linearly growing for the interaction potentials introduced. Due to the nonlinear form of the formulated equations, the method of lines approach was used to obtain the numerical solutions of the governing equations. Copper atoms were used to in all simulations and calculations.

Tropical cyclone rapid intensification and the excitation of inertia-gravity waves on the edges of an evolving potential vorticity structure

The problem of tropical cyclone rapid intensification is reduced to a potential vorticity (PV) equation and a second order, inhomogeneous, partial differential equation for the azimuthal wind. The latter equation has the form of a Klein-Gordon equation, the right-hand side of which involves the radial derivative of the evolving PV field. When the PV field evolves rapidly, inertia-gravity waves are excited at the edges of the evolving PV structure. In contrast, when the PV field evolves slowly, the second order time derivative term in the Klein-Gordon equation is negligible, inertia-gravity waves are not excited, and the equation reduces to an invertibility principle for the PV. The above concepts are presented in the context of an axisymmetric shallow water model, in both its linear and nonlinear forms. The nonlinear results show a remarkable sensitivity of vortex intensification to the percentage of mass that is diabatically removed from the region inside a given absolute angular momentum surface.