Common Nonlinear Model Research Articles

Real-time control of a soft manipulator based on reduced order extended position-based dynamics

Soft robots are high-dimensional nonlinear systems coupled with both geometric and material nonlinearity. Control of such a system is complex and time-consuming. In this study, a real-time trajectory tracking control framework is established based on the reduced order extended position-based dynamics. In contrast to the common nonlinear model order reduction methods that require to collect a large number of data to create the motion subspace, this article's motion subspace is constructed based on the model configuration and material properties. The linear modes of the model and the related modal derivatives provide the reduced order matrix, which streamlines and increases the efficiency of model construction. Then, coupled with the instantaneous optimal control, a real-time reduced order model-based control framework of soft robots can be constructed. Experiments on trajectory tracking of a soft manipulator are conducted to verify the accuracy and efficiency of the proposed controller. The average error of all experiments is within 1 cm; and the single-step calculation time of the controller is about 0.057 s, which is less than the sampling period 0.1 s.

New non-augmented mixed finite element methods for the Navier–Stokes–Brinkman equations using Banach spaces

Abstract In this paper we consider the Navier–Stokes–Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixed-point equation whose operator involved maps the velocity space into itself. The well-posedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the Babuška–Brezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentum-conservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixed-point strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity, and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods.

A Nonlinear Autoregressive Model with Exogenous Variables for Traffic Flow Forecasting in Smaller Urban Regions

Data-driven forecasting methods have the problems of complex calculations, poor portability and need a large amount of training data, which limits the application of data-driven methods in small cities. This paper proposes a traffic flow forecasting method using a Nonlinear AutoRegressive model with eXogenous variables (NARX model), which uses a dynamic neural network Focused Time-Delay Neural Network (FTDNN) with a Tapped Delay Line (TDL) structure as a nonlinear function. The TDL structure enables the FTDNN to have short-term memory capabilities. At the same time, before the data is input into the FTDNN, the use of trend decomposition or differential calculation on the traffic data sequence can make the NARX model maintain long-term predictive capabilities. Compared with common nonlinear models, the FTDNN has structural advantages. It uses a simple TDL structure without the memory mechanism and the gated structure, which can reduce the parameters of the model and reduce the scale of data. Through the four-day data of Guilin City, the traffic volume forecast for five minutes is verified, and the performance of the NARX model is better than that of the SARIMA model and the Holt-Winters model.

Mathematical Model Simulation of Non-Linear Equations using MATLAB: Specific Volume of Gas with Van der Waals Equation

Computational physics is concerned with the application of numerical methods in solving physical problems. The van der Waals gas model is one of the most common non-linear models. This study simulated a mathematical model of a non-linear equation using MatLab for the case of the specific volume of gas in the equation of the state of van der Waals. This study aimed to determine the molar volume and compressibility factor, as well as describe the relationship between the compressibility factor and the reduced pressure. The method of the study is experimental. The independent variables are the reduced pressure and temperature values. The dependent variable is the determination of the value of the molar volume (V) and the compressibility factor (Z). The control variable, in the form of a function used in solving this case, is based on van der Waals equation with the gas used is ammonia. The fzero command can be used to solve f(x)=0 with a single variable. This program that has been running successfully can show various predictions in the form of reduction pressure, thus obtaining the values of the molar volume and compressibility factor using the ideal gas equation. There is a deviation in ammonia gas, the Z1 at high reduction pressures and Z1 at medium pressures. This study can provide contributions and benefits in the form of material enrichment of thermodynamics to understand how real gases behave. The ideal gas equation can be modified into the van der Waals equation.

Approximate Laplace importance sampling for the estimation of expected Shannon information gain in high-dimensional Bayesian design for nonlinear models

One of the major challenges in Bayesian optimal design is to approximate the expected utility function in an accurate and computationally efficient manner. We focus on Shannon information gain, one of the most widely used utilities when the experimental goal is parameter inference. We compare the performance of various methods for approximating expected Shannon information gain in common nonlinear models from the statistics literature, with a particular emphasis on Laplace importance sampling (LIS) and approximate Laplace importance sampling (ALIS), a new method that aims to reduce the computational cost of LIS. Specifically, in order to centre the importance distributions LIS requires computation of the posterior mode for each of a large number of simulated possibilities for the response vector. ALIS substantially reduces the amount of numerical optimization that is required, in some cases eliminating all optimization, by centering the importance distributions on the data-generating parameter values wherever possible. Both methods are thoroughly compared with existing approximations including Double Loop Monte Carlo, nested importance sampling, and Laplace approximation. It is found that LIS and ALIS both give an efficient trade-off between mean squared error and computational cost for utility estimation, and ALIS can be up to 70% cheaper than LIS. Usually ALIS gives an approximation that is cheaper but less accurate than LIS, while still being efficient, giving a useful addition to the suite of efficient methods. However, we observed one case where ALIS is both cheaper and more accurate. In addition, for the first time we show that LIS and ALIS yield superior designs to existing methods in problems with large numbers of model parameters when combined with the approximate co-ordinate exchange algorithm for design optimization.

Gender and Age Patterns in NSGA Swim Competitions.

We estimate two common nonlinear models (quadratic and semilog) and one new model (exponential) of the time-age relationship in 500-yard freestyle swim times in the U.S. National Senior Games (ages 50 and up) in six biennial NSGA competitions (2009, 2011, 2013, 2015, 2017, and 2019) for 468 men and 587 women. We use OLS and quantile regression (25%, 50%, and 75%) separately for each gender. The semilog model predicts faster times than the quadratic or exponential models. Our hypothesis that women slow down faster than men after age 50 is supported by both models as well as by our unique within-gender comparisons. Our findings of a nonlinear performance decline agree with studies of elite swimmers (Olympic, FINA). Our first-time study of NSGA data provides new guidelines to inform senior competitors. Our findings will assist trainers and community organizations that support NSGA competitions to promote a healthy senior lifestyle.

Portfolio optimization using second order conic programming approach

In this paper, we examine the framework to estimate financial risk called conditional-value-at-risk (CVaR) and examine models to optimize portfolios by minimizing CVaR. We note that total risk can be a function of multiple risk factors combined in a linear or nonlinear forms. We demonstrate that, when using CVaR, several common nonlinear models can be expressed as second order cone programming problems and therefore efficiently solved using modern algorithms. This property is not shared with the more classical estimation of financial risk based on value-at-risk.

Simple Nonlinear Models for Glucose-Insulin Dynamics: Application to Intraperitoneal Insulin Infusion

The design of a model-based control method for an Artificial Pancreas requires a relatively simple and identifiable mathematical model to control glucose levels through hormone delivery. In this work we introduce new, simple nonlinear models to simulate data from experiments where insulin boluses are administrated in the peritoneal cavity. The models account for the delay between insulin administration and its nonlinear transport to other compartments. They were calibrated using experimental data from pigs. The results show that the suggested models are able to describe the data well, with average BIC value of 145. Moreover, the new models were compared with a common linear model which was not able to describe the data well, with BIC value of 920. They were also compared with a common nonlinear model which failed to represent insulin increases in the data and had BIC value of 637. Finally, profile likelihoods were applied for assessing the identifiability of one of the new models.

Adaptive optimal input design and parametric estimation of nonlinear dynamical systems: application to neuronal modeling

Objective. Many physical models of biological processes including neural systems are characterized by parametric nonlinear dynamical relations between driving inputs, internal states, and measured outputs of the process. Fitting such models using experimental data (data assimilation) is a challenging task since the physical process often operates in a noisy, possibly non-stationary environment; moreover, conducting multiple experiments under controlled and repeatable conditions can be impractical, time consuming or costly. The accuracy of model identification, therefore, is dictated principally by the quality and dynamic richness of collected data over single or few experimental sessions. Accordingly, it is highly desirable to design efficient experiments that, by exciting the physical process with smart inputs, yields fast convergence and increased accuracy of the model. Approach. We herein introduce an adaptive framework in which optimal input design is integrated with square root cubature Kalman filters (OID–SCKF) to develop an online estimation procedure that first, converges significantly quicker, thereby permitting model fitting over shorter time windows, and second, enhances model accuracy when only few process outputs are accessible. The methodology is demonstrated on common nonlinear models and on a four-area neural mass model with noisy and limited measurements. Estimation quality (speed and accuracy) is benchmarked against high-performance SCKF-based methods that commonly employ dynamically rich informed inputs for accurate model identification. Main results. For all the tested models, simulated single-trial and ensemble averages showed that OID–SCKF exhibited (i) faster convergence of parameter estimates and (ii) lower dependence on inter-trial noise variability with gains up to around 1000 ms in speed and 81% increase in variability for the neural mass models. In terms of accuracy, OID–SCKF estimation was superior, and exhibited considerably less variability across experiments, in identifying model parameters of (a) systems with challenging model inversion dynamics and (b) systems with fewer measurable outputs that directly relate to the underlying processes. Significance. Fast and accurate identification therefore carries particular promise for modeling of transient (short-lived) neuronal network dynamics using a spatially under-sampled set of noisy measurements, as is commonly encountered in neural engineering applications.

Generation of Dangerous Disturbances for Flight Systems

This paper addresses the generation of dangerous disturbances in problems of aircraft control. The method is based on the construction of repulsive disturbances in linear differential games using a dynamic programming approach. The computation involves solving a large number of linear programs. For this purpose, a fast algorithm efficiently working in low dimensions is proposed. Several examples of constructing dangerous disturbances are given. First, a simple linear differential game is considered to demonstrate the main features of the method. Second, a linearized model of the longitudinal motion of an aircraft is considered. The model includes flight and servomechanism dynamics, pilot and disturbance inputs, and a controller supporting the piloting process. This model is investigated in two setups: as a control law clearance problem and as a conflict control task. Third, a common nonlinear model of aircraft dynamics with reference to flight in a vertical plane is considered, and the problem of takeoff in wind shear conditions is addressed. For all above-listed problems, simulations showing the efficiency of the constructed disturbances are presented.

Shuffled Frog-Leaping Algorithm for Optimal Design of Open Channels

AbstractOpen channels are important water structures for irrigation, water supply, power generation, and drainage. The primary concern of designing a channel is to determine optimum dimensions while minimizing construction costs. In this paper, a memetic metaheuristic algorithms called “shuffled frog-leaping algorithm” (SFLA) is used to optimize sampling design variables of composite open channels. The results of optimization using SFLA and a generic language for interactive general optimization (LINGO) software are compared with other metaheuristic algorithms, such as genetic algorithm (GA), simulated annealing (SA), Lagrange multiplier method (LMM), ant colony optimization (ACO), improved ant colony optimization (IACO), and particle swarm optimization (PSO). Four common nonlinear models in open-channel designing are considered in this study: (1) model I (M1), which applies an equivalent roughness coefficient without any sectional division following the Manning equation; (2) model II (M2), which consider...

Vehicle rollover avoidance by application of gain-scheduled LQR controllers using state observers

Many researches have been conducted in the area of control applied to vehicle dynamics, aiming at reducing the possibility of the occurrence of the type of accident known asrollover. In this research, based on a common nonlinear model and its linearisation, a method for properly selecting matrices for solving the Riccati equation considering different speeds was proposed. The method showed in which ways speed really influences the choice of controller gains. By developing the dynamic equations for the yaw- and roll-coupled motions and modelling of controllers and state observers, it is possible to compare the efficacy of this control strategy using both linear and nonlinear simulations using Matlab. Significant results were obtained regarding the reduction of the rollover coefficient for a double-lane change manoeuvre at different speeds, thus indicating advantages of using this controller in practical cases.

Study on the Curvature Reducing Method of Non-linear Regression Model

The method to reduce the non-linear strength (curvature)of non-linear regression model was studied in this paper. Firstly, the reference point of the non-linear strength was analyzed. Based on the definition of curvature cubic matrix, a computing method of curvature cubic matrix was proposed based on the Cholesky disassembling. Then the common ways to reduce the non-linear strength was also discussed. Pointed at some common non-linear models in real engineering applications, such as non-linear models used for multiple-measurement and mutual-calibration of different instruments, or non-linear models prior information, a new least square method with weight was given, which can evidently reduce the curvature of these multi-structure non-linear regression models, therefore evidently reduce the non-linear strength. Finally, the Numerical simulation results were given to validate the effectiveness and feasibility of this weighted least square method. The method to reduce the non-linear strength (curvature) of non-linear regression model was studied in this paper. Firstly, the reference point of the non-linear strength was analyzed. Based on the definition of curvature cubic matrix, a computing method of curvature cubic matrix was proposed based on the Cholesky disassembling. Then the common ways to reduce the non-linear strength was also discussed. Pointed at some common non-linear models in real engineering applications, such as non-linear models used for multiple-measurement and mutual-calibration of different instruments, or non-linear models with prior informations, a new least square method with weight was given, which can evidently reduce the curvature of these multi-structure non-linear regression models, therefore evidently reduce the non-linear strength. Finally, the Numerical simulation results ware given to validated the effectiveness and feasibility of this weighted least square method.

On Parameter Identifiability in Non-Linear Biophysical Models

The quantitative modeling of biological phenomena allows for a deeper understanding of underlying mechanisms as well as the determination of biophysical parameters. However, we note that in many systems, the unique identification of relevant parameters is not possible with common experimental methods. The parameters of a model are said to be identifiable if there a unique point in parameter space that leads to an optimal agreement with particular data. In common non-linear models, however, there is not a unique map from parameter space to data space, and the confusion resulting from this lack of parameter identifiability may be slowing progress in many fields of biophysics. We use a simple example model to show analytically that this problem often results from rank-difficient regression, i.e., there are an infinite number of ways for the model parameters to fit the data equally well. Further, we use the same model to show that the identifiability problem can be resolved if additional experimental data is included for model constraint. Unfortunately, most models of interest will not be amenable to analytical examination. We present a numerical method based on Markov chain Monte Carlo (MCMC) sampling which can be used for any data and any model and will allow the diagnosis of these issues. We provide example uses of MCMC to asses parameter identifiability in a variety of systems. This method is an important tool that provides the ability to assess parameter identifiability for a given model and various potential experimental manipulations. This new kind of power analysis will lead to more precise conclusion and more fruitful experimentation.

New linearized two-parameter infiltration equation for direct determination of conductivity and sorptivity

A new two-parameter (the sorptivity, S [ L / T 0.5 ], and saturated soil hydraulic conductivity, K s [ L / T ]) vertical infiltration equation is proposed. A three-parameter accurate approximate equation is derived. The proposed two-parameter equation is a specific solution that is approximately located at the middle of the domain of real soils defined by two “limiting” behavior soils. According to the proposed equation, plotting the square of cumulative infiltration, i [ L ], divided by time, t [T], as a function of i , a linear line is obtained and it is easy to determine S squared as the intercept and K s as the slope of the line and to test the adequacy of the proposed equation. The equation approaches Philip semi-analytical infinite-series solution at small and moderate times, whereas the infiltration rate approaches the constant K s at large times. It is expressed in a form explicit in i as function of t and vice versa. The equation is validated through comparison with other common nonlinear models (with two- and three-parameters) on a number of reference soils from the literature (both real and analytical) The parameter estimates of the equation are practically insensitive to the number of data points used in fitting, as well as to random measurement errors. Two illustrative infiltration tests for more realistic conditions (non-uniform soil and initial moisture profiles) were also performed. The tests demonstrate the advantage of the linearized equation over other common nonlinear models to be able visually detect and eventually eliminate abnormalities of infiltration process.

Asymptotic rejection of unknown sinusoidal disturbances in nonlinear systems

This paper deals with global disturbance rejection of nonlinear systems. The disturbance is assumed to be sinusoidal with completely unknown phases, amplitude, and frequencies, but the number of distinct frequencies or the order of the corresponding unknown linear exosystem is known. Different from the common structural assumptions of nonlinear systems needed in literature for disturbance rejection of nonlinear systems, the proposed method only requires the information of control design with a known Lyapunov function when the system is disturbance-free, and a mild assumption needed for internal model design. The proposed disturbance rejection algorithm extends complete global rejection of unknown sinusoidal disturbances for nonlinear dynamic systems beyond the common nonlinear models such as the strict feedback forms and the output feedback forms.