Taylor Series Expansion Of Functions Research Articles

Modeling Nonlinear Physical Systems in a Real-Time Operating System Environment for Control and Cyber Threat Mitigation

Nonlinearity control and interconnection between physical systems are essential for many contemporary systems. Cyber Physical systems came about because of the growing number of security concerns with network control systems. Nonlinear physical systems are the primary subject of this study's investigation into control and cyber threat mitigation. Nonlinear physical systems are the focus of this study, which also delves into their control, modelling, and real-time implementation. Here, physical systems are defined as two kinds of benchmark nonlinear robotic systems: the Segway transporter and the cruise control system. Many see the Segway as an unstable and nonlinear physical system. By factoring in kinetic and potential energy, Lagrangian dynamics has allowed for the derivation of equations of motion. Next, a state-space model of the system is obtained by linearizing the nonlinear differential equations using Taylor series expansion of functions. This model is then transformed into two transfer functions, one for the tilt angle and one for the yaw angle. The Model Predictive Controller (MPC), GA Tuned PID, and PID are the building blocks of MATLAB's effective simulation study of nonlinear control.

Математическое моделирование нестационарности биотехнологического получения молочной кислоты: устойчивость

The article presents the calculated ratios of indicators determining the stationary states of the lactic acid production process. Three technologies that are most often mentioned in scientific publications are identified: the technology of using strains of microorganisms to produce biomass is a technology that is extremely rarely used; the fairly common technology of using strains of microorganisms to produce lactic acid with the consumption of the main substrate (most often glucose); the promising technology of obtaining lactic acid using, in addition to the main substrate, a component that reproduces the main substrate in the synthesis process. For each technology, the equations of material balance for stationary and non-stationary conditions, a generalized differential equation for non-stationary conditions, and a characteristic equation are given. The formulas for estimating the coefficients of differential equations and the coefficients of the characteristic equation are also given. The equations for non-stationary conditions according to the last two technologies are based on the use of the Taylor series expansion of functions with the preservation of only the first terms of the expansion, i. e. deviations from stationarity in small. The characteristic equation is formed using the eigenvalues . The methodology for all three technologies is given, which allows us to assess the stability of the considered stationary state – the Hurwitz method. For all three technologies, numerical results are obtained for estimating the coefficients of the characteristic equations Pi. Tabular values of the coefficients are given, according to which stability estimates for the dilution rate of 0.1 h–1, 0.2 h–1, 0.3 h–1 are obtained using determinants according to the Hurwitz matrix. The results of numerical estimates for the stability of stationary states for all three technologies are presented. The estimates were based on the indicators of constants published in scientific studies.

Efficient Linearized Model of Pressurizer System in Pressurized Water Reactors for Control Purposes

The pressurizer has an important role to ensure the protected operation of pressurized water reactor (PWR) by keeping the reactor coolant system pressure among allowed tolerances. In this paper, a non-equilibrium two-region nonlinear pressurizer (PZR) model was linearized using Taylor technique to introduce the linear models of pressurizer transients for the controller design purposes. Taylor series expansion of functions around an equilibrium point was calculated. The same assumptions of the reference model are considered in linearized model. The linearized model consists of five states; rate of changes for: mass of steam &water in the PZR, mass of water in the PZR, mass of water in the PC, steam temperature in PZR, and water temperature in PZR. The nonlinear and linear PZR models was implemented in MATLAB/Simulink. The linearized model of PZR is verified. The performance of linearized model of PZR and nonlinear model is consistent with small and acceptable errors. The closed-loop of the linearized model of PZR pressure and water level is implemented and investigated using standard Proportion-Integration-Differentiation (PID) controllers. Simulation results and the evaluation performance indicate that the proposed linear model of PZR is efficient, and it can be used for control purposes.

Online optimization based model predictive control on two wheel Segway system

The current work explores the modelling, control and realistic execution of nonlinear segway system in both simulation andreal-time. The Segway vehiclehas been consideredas a physical system that is nonlinear and unstable. Equations of motions have been derivedusing Lagrangian dynamics by considering kinetic energy and potential energy. Then, the nonlinear differential equations have linearised using Taylor series expansion of functions to acquire a state space model of the system and convert it into two transfer functions that corresponds to tilt angle and yaw angle. These are the controlparametersto stabilize the Segway system. Effective simulation analysis of nonlinear controlis madeup with PID, GA Tuned PID and Model Predictive controller (MPC) byMATLAB. And also, an electro-mechanical model of Segway vehiclehas developedforreal-timecontrol analysis. It is autonomously sensed and actuated through embedded processor tomeetthe desired response. An online optimization techniquehas involvedsolving dynamic optimization problems in MPC for superior performance. This innovative realisation paves the approach for the young researchers to consider the essence of analysis in nonlinear applications.

A reference-modified density functional theory: An application to solvation free-energy calculations for a Lennard-Jones solution.

In the conventional classical density functional theory (DFT) for simple fluids, an ideal gas is usually chosen as the reference system because there is a one-to-one correspondence between the external field and the density distribution function, and the exact intrinsic free-energy functional is available for the ideal gas. In this case, the second-order density functional Taylor series expansion of the excess intrinsic free-energy functional provides the hypernetted-chain (HNC) approximation. Recently, it has been shown that the HNC approximation significantly overestimates the solvation free energy (SFE) for an infinitely dilute Lennard-Jones (LJ) solution, especially when the solute particles are several times larger than the solvent particles [T. Miyata and J. Thapa, Chem. Phys. Lett. 604, 122 (2014)]. In the present study, we propose a reference-modified density functional theory as a systematic approach to improve the SFE functional as well as the pair distribution functions. The second-order density functional Taylor series expansion for the excess part of the intrinsic free-energy functional in which a hard-sphere fluid is introduced as the reference system instead of an ideal gas is applied to the LJ pure and infinitely dilute solution systems and is proved to remarkably improve the drawbacks of the HNC approximation. Furthermore, the third-order density functional expansion approximation in which a factorization approximation is applied to the triplet direct correlation function is examined for the LJ systems. We also show that the third-order contribution can yield further refinements for both the pair distribution function and the excess chemical potential for the pure LJ liquids.

Analysis of Fabrication Errors on Micro-Spring's Elastic Coefficient

Because fabrication errors have a great influence on micro-springs elastic coefficient, and it is difficult to reduce the influence through fabrication process methods at this stage, the authors propose solutions through design process to reduce the sensitivity of Elastic coefficient of fabrication errors. The authors derived error propagation formula through multivariate function Taylor series expansion method, and made a quantitative analysis of the influence of fabrication errors, then summarized the influence rules of the size parameters on dispersibility of elastic coefficient K, finally proposed the method to degrade device performance dependent on the fabrication precision.

Taylor Series Expansion in Discrete Clifford Analysis

Discrete Clifford analysis is a discrete higher-dimensional function theory which corresponds simultaneously to a refinement of discrete harmonic analysis and to a discrete counterpart of Euclidean Clifford analysis. The discrete framework is based on a discrete Dirac operator that combines both forward and backward difference operators and on the splitting of the basis elements \(\mathbf{e}_j = \mathbf{e}_j^+ + \mathbf{e}_j^-\) into forward and backward basis elements \(\mathbf{e}_j^\pm \). For a systematic development of this function theory, an indispensable tool is the Taylor series expansion, which decomposes a discrete (monogenic) function in terms of discrete homogeneous (monogenic) building blocks. The latter are the so-called discrete Fueter polynomials. For a discrete function, the authors assumed a series expansion which is formally equivalent to the Taylor series expansion in Euclidean Clifford analysis; however, attention needed to be paid to the geometrical conditions on the domain of the function, the convergence and the equivalence to the given discrete function. We furthermore applied the theory to discrete delta functions and investigated the connection with Shannon sampling theorem (Bell Sys Tech J 27:379–423, 1948). We found that any discrete function admits a series expansion into discrete homogeneous polynomials and any discrete monogenic function admits a Taylor series expansion in terms of the discrete Fueter polynomials, i.e. discrete homogeneous monogenic polynomials. Although formally the discrete Taylor series expansion of a function resembles the continuous Taylor series expansion, the main difference is that there is no restriction on discrete functions to be represented as infinite series of discrete homogeneous polynomials. Finally, since the continuous expansion of the Taylor series expansion of discrete delta functions is a sinc function, the discrete Taylor series expansion lays a link with Shannon sampling.

Simple charge transfer model for one electron oxidation and reduction processes: Describing reactive sites in benzocarbazolediones and gallates

By making use of a second order functional Taylor series expansion, we have derived a simple charge transfer model based on the idea that for the oxidation and reduction of chemical species through electrochemical methods, exactly one electron is removed or added. Thus, the global energy change for the oxidation and the reduction processes is given by the first ionization potential I and the electron affinity A, respectively, and they are expected to correlate with the half wave potential. For the local energy change one finds that it is given by the product of I times the Fukui function for charge donation, for the oxidation case, and by the product of (− A) times the Fukui function for charge acceptance, for the reduction case. These expressions are applied to study the substituent effects in specific sites of two families of compounds, gallates and benzocarbazolediones, as representatives of compounds presenting oxidizable and reducible groups.

STRUCTURE OF CONFINED HARD ELLIPSES USING MOLECULAR DENSITY FUNCTIONAL THEORY

Density functional (DF) theory is used to study confined two-dimensional molecular fluids. The approximate DF used here is based upon a functional Taylor series expansion of the excess grand potential about the density of a uniform molecular liquid. This formalism is applied to obtain the density profiles of the hard-ellipse fluid confined between two parallel hard walls. The required direct correlation function is obtained using Percus–Yevick and hypernetted chain approximations for low- and high-number density, respectively. Both the restricted orientation model and the extension of this model are used. Generally, we obtained that the number density of the hard ellipses with the major axes parallel to the wall is larger near the walls than the other directions. To check the results, we show that for isotropic hard-ellipse fluids the average number density in the middle of the wall is almost equal to the bulk density and as we expect the average number density of the ellipses at the wall is nearly equal to the amount of the reduced pressure, βP. We also perform Monte Carlo simulation and find reasonable agreement with our results.

Highly charged colloidal suspensions in planar slits

A simple density functional approximation, which is based both on the density functional Taylor series expansion of the one-particle direct correlation function (DCF) and on the exact contact value theorem for a structureless hard wall with infinity wall distance, has been developed to study the structural properties of a highly charged colloidal suspension and compared with the computer simulation. Two kinds of two-particle DCFs, which are the rescaled mean spherical approximation (RMSA) and the Rogers-Young (RY) closure relation, have been used to calculate the local concentration profiles of a highly charged colloidal suspension confined in charged silt walls as input. The calculated results show that for a structureless hard wall the present approximation is better than the hypernetted chain (HNC) functional approximation and describes well the structural properties of a confined colloidal suspension compared with the computer simulation. However, the agreement between the computer simulation and the theory for the density profiles deteriorates with increasing the wall-particle interaction.

Density functional approximations for confined classical fluids

A density functional approximation, which is based on both the density functional Taylor series expansion of the one-particle direct correlation function and the exact contact value theorem for a hard wall, has been proposed to study the structural properties of confined classical fluids. The approximation has been applied to calculate the density profiles of sticky hard-sphere fluids confined in structureless hard walls. The calculated density profiles have shown that the present approximation compares very well with the results from the computer simulation. Furthermore, a density functional perturbative approximation, which is based on both the weighted-density approximation for the repulsive part of potential and the present approximation for the attractive part of potential, has been developed to predict the density profiles of model fluids with the attractive part of potential and has been applied to calculate the density profiles of hard-sphere Yukawa fluids near a planar slit. The calculated results also show that the proposed perturbative approximation is a significant improvement upon those of the modified version of the Lovett-Mou-Buff-Wertheim, and compares very well with the computer simulation.

A report on the computer algebra of differentiation TAYLOR

The report describes the TAYLOR software for computing the coefficients in Taylor-series expansions of functions. The order of the expansion is unlimited, the functions may be specified in different forms, including implicitly and by a differential equation.

Structural optimization based on second-order approximations of functions and dual theory

Based on the dual theory of nonlinearly mathematical programming and the second-order Taylorseries expansions of functions, an efficient algorithm for structural optimum design has been developed. The main advantages of this method are the generality in use, the efficiency in computation, and the capability in identifying automatically the set of critical constraints. On the basis of the virtual work principle, formulas expressed through element stresses for the first- and second-order derivatives of nodal displacement and stress with respect to design variables are derived. By introducing Saint-Venant's principle, the computational effort involved in the Hessian matrix associated with the iterative expression can be significantly reduced. This method is especially suitable for optimum design of large-scale structures. Several typical examples have been optimized to test its power.

An efficient method for structural optimization

Based on the dual theory of nonlinear mathematical programming and the second order Taylor series expansions of functions, an efficient algorithm for structural optimum design has been developed. The main advantages of this method are the generality in use, the efficiency in computation and the capability in identifying automatically the set of active constraints. On the basis of the virtual work principle, formulas in terms of element stresses for the first and second order derivatives of nodal displacement and stress with respect to design variables are derived. By applying the Saint-Venant's principle, the computational efforts involved in the Hessian matrix associated with the iterative expression can be significantly reduced. This method is especially suitable for optimum design of large scale structures. Several typical examples have been optimized to test its uasefulness.

A UNIFIED OPTIMALITY CRITERIA METHOD IN STRUCTURAL DESIGN

On the basis of the Kuhn-Tucker condition, two iterative expressions for determining optimal values for design variables and Lagrangian multipliers are derived. The convergence of this method has been proved mathematically to be of order two. This method can be used to solve problems with various kind of behavioral constraints as long as the second order Taylor series expansions of functions involved in the problems can be generated, and therefore it has general significance. Based on the virtual work principle. formulas for computing the first and second order derivatives of nodal displacements and stresses with respect to design variables are given. The computational effort involved in the Hessian matrix associated with the iterative expression can be significantly reduced by using Saint-Venant's principle, and the efficiency of the optimization procedures can be further improved. Computational results for several typical examples show that the method presented in this paper performs satisfactorily in comparison with other competing techniques.

Mechanical response and the initial value problem

Solutions to canonical mechanical equations are considered as functionals of a source function σ, which parametrizes the source. The coefficients of the functional Taylor series expansions in σ for the solutions are constructed from an infinite sequence of time ordered Poisson brackets, which in turn can be expressed in terms of the derivatives of the solution to the homogeneous equation with respect to its initial canonical momenta. Thus the functional Taylor series expansion is formally defined once the initial value problem for the possibly nonlinear homogeneous equation has been solved. A time ordered exponential operator acting on the homogeneous solution, which may be viewed as a ’’response’’ extension of the Lie series operation, simply represents the series expansion.

Variational Principle for Eigenvalue Equations

A variational principle is developed to provide an estimate of an arbitrary functional of the eigen-functions of a set of eigenvalue equations. It is shown that the variational formalism is equivalent to a functional Taylor series expansion of the desired functional about the trial functions. The relationship of this work to perturbation theory is considered, and it is shown that the formalism can be used to construct higher-order variational principles, i.e., those for which first-order errors in the trial functions leads to an nth-order error in the desired functional. Finally, it is shown that the variational principle of Borowitz and Vassell for estimating off-diagonal matrix elements, as well as the usual Rayleigh quotient, are special cases of the principle presented here.