The main challenge in using the linearizationmethod as described in Makukula et al. [23,24] is how to generalize the method so as to find solutions of partial differential equations of the form (8)-(10). It is certainly not clear how the method may be applied directly to the terms on the right hand side of equations (8)-(10). For this reason, equations (8)-(10) are first simplified and reduced to sets of ordinary differential equa- tions by assuming regular perturbation expansions for f, θ, and j in powers of ξ (which is assumed to be small) as follows
Now we o ﬀ er some applications of the linearizationmethod established in the previous section. These applications are only a very limited sample of possibilities to use the results of linear oscillation theory when investigating (1.1), and are of rather straightforward character. More sophisticated applications, including looking for additional conditions under which linear equation (3.5) and half-linear equation (1.1) have the same oscillatory nature, regardless whether 1 < p ≤ 2 or p ≥ 2, is a subject of the present investigation.
Our main method is a combination of two numerical methods, quasi-linearizationmethod (QLM) and the LSEM. The QLM which is Newton-Raphson based, was originally proposed by . It is used to linearize the non-linear differential equation into an iterative sequence of linear differential equations. The resulting system of equations is solved using the LSEM.
of the linearizationmethod for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets. The conditions depend on the behavior of the controllability Gramian of the linearized system — the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero. The indicated asymptotic behavior occurs for a reasonably wide class of second- order nonlinear control systems but can be violated for systems of higher dimension. The results of numerical simulation illustrate the theoretical conclusions of the paper.
Feedback linearization is a controller design method that has been used in recent years. In this method, after using exact transformations of state and feedback, we use an input to delete nonlinear terms of the model and to linearize the system. This method is different from the linearizationmethod because we do not use the approximation of equations. After using this method, we can design a linear controller to control the system. Feedback linearization has some problems. For example, it requires exact information of the system .
Reduction of the number of the system unknowns for three rounds was not successful and searching of the solution was executed with XL method. When applying eXtended Linearizationmethod the authors consider every substitution box individually. In this case D parameter of XL method is equal to 3, i.e. each of 21 equa- tions for substitution box is multiplied by the unknown to 1 extent. Then we obtain 189 equations for one subs- titution box, the majority of these equations are linearly independent. For three encryption rounds after resulting in additional equations, the system includes 4536 equations with 192 unknowns and 2208 various monomials.
Abstract—This paper presents a direct fuzzy adaptive control for standalone Wind Energy Conversion Systems (WECS) with Permanent Magnet Synchronous Generators (PMSG). The problem of maximizing power conversion from intermittent wind of time-varying, highly nonlinear WECS is dealt with by an adaptive control algorithm. The adaptation is designed based on the Lyapunov theory and carried out by the fuzzy logic technique. Comparison between the proposed method and the feedback linearizationmethod is shown by numerical simulations verifying the effectiveness of the suggested adaptive control scheme.
On basis of linearizationmethod, extreme value theory and parameter estimation method, the nonlinear discrete-time D-MFAC controller in the compact form is derived from the NARX model when a new weighted one-step-ahead control input cost function including one-order term of output is adopted in this paper. With some simulations it illustrated mainly with the same initial parameters that 1) the system can be stabilized with the new D-MFAC controller in slower speed and longer settling time with the D-MFAC controller than that of the conventional MFAC controller with the same initial parameters normally, 2) there are overshoots in the D-MFAC controller which can not be reduced significantly.
method produces a slightly greater uncertainty than the bootstrap method. The slight difference is due to the differ- ences in re-sampling residues. While the Monte Carlo simulation generates residues based on a theoretical nor- mal distribution, the bootstrap method randomly takes the residues with replacement and no assumption is made about the underlying distributions. They are also compar- able to those approximated by the linear model obtained from the SigmaPlot software. However, by comparing the results of these three methods, the lower limit and upper limit of α and n obtained by the Monte Carlo and boot- strap methods are slightly greater than those obtained based on a linear assumption except for upper limit of α by the bootstrap method. Because the linearizationmethod is based on the assumption of normal distribution of parame- ters and linearity at the vicinity of the estimated parameter value, and it is more complicated in terms of calculation, the Monte Carlo and bootstrap methods may be preferred to the linearizationmethod to calculate the parameter un- certainties in spreadsheets. Furthermore, the Monte Carlo method may be preferred to the bootstrap method considering the less number of simulations required for the Monte Carlo method. However, if the number of mea- surements is too small to determine the probability distri- bution for Monte Carlo method, the bootstrap method may be superior.
In this work, a delayed reaction-diffusion predator-prey model with stage structure and continuous harvesting for predator is constructed. The boundedness, existence and uniqueness of the model is investigated. The existence and uniqueness of the global solution of the system are proved. The local and global stability of the constant equilibria are discussed by the linearizationmethod and the method of upper and lower solutions, respectively. By using the linearizationmethod and the method of upper and lower solutions, the local and global stability of the constant equilibria of the system are obtained, respectively. By Theorem 3.1, one can see that the equilibria E 1 (0,0) and
The protonation constant values of different four triazole derivatives have been determined in ethanol %50 – water %50 mixtures by spectrophotometric methods. The electronic absorption spectra of different four triazole derivatives at various pH values at 190-400 nm intervals were recorded. The calibration of the electrode system was done potentiometrically by Gran’s method. Data were calculated using the linearizationmethod with Henderson-Hasselbach equation. The obtained results are in good accordance with potentiometric values.
for solving 2D problems. We compare three diﬀerent methods: two iterations of New- ton’s method (denoted by N in tables and ﬁgures), and the linearizationmethod, ﬁrst in its standard form (denoted by L), then combined with the operator splitting (de- noted by S), in terms of the computational eﬃciency (cost). We also comment on the accuracy of the schemes. Second, we study the ﬁnite-extinction phenomenon for the porous-medium equation with strong absorption employing our scheme. We present the experimental results of the numerical extinction time values for various spatial and temporal step sizes. All our programs, written in Matlab, are implemented on a PC with Pentium III processor at 933 MHz.
A second order oscillator with nonlinear restoring force and nonlinear damping is considered: it is subject to both external and internal (parametric) excitations of Gaussian white noise type. The nonlinearities are chosen in such a way that the associated Fokker-Planck-Kolmogorov equation is solva- ble in the steady state. Different choices of some system parameters give rise to different and interesting shapes of the joint probability density function of the response, which in some cases appears to be multimodal. The problem of the determination of the power spectral density of the response is also ad- dressed by using the true statistical linearizationmethod.
Oscillators with fractional order restoring forces and relativistic oscillators are investigated in this paper by using the Equivalent Linearizationmethod based on a weighted averaging. Averaging value is calculated in a new way called the weighted averaging value by introducing a weighted coefficient function. The amplitude-frequency relationships are presented by considering three nonlinear oscillators. The obtained solutions have been compared with approximate analytical solutions, exact solutions and numerical solutions. Comparisons show the reliability of this method.
Although equivalent linearization of base isolation systems has been addressed by many research works (Liu et al., 2014d; Zordan et al., 2014), most of them assume the combined system to be SDOF systems, as described in Chapter 4. In other words, the superstructure is assumed to be a rigid body. However, with increasing the height (or natural period) of the superstructure, higher mode effect should be also taken into account; otherwise large errors could be obtained due to the SDOF assumption. This Chapter evaluates the accuracy of equivalent linearization meth- ods to predict the maximum displacement of isolation interface in multi-storey base- isolated buildings. The superstructure is considered to be a shear-type frame build- ing. Three-, six-, nine- and twelve-storey versions of the superstructure frame are considered and modeled as multi-degree-of-freedom (MDOF) systems with lumped masses. Based on the specified parameters, a comprehensive parametric study is performed to examine the influence of the ratio between equivalent period and su- perstructure period on the estimation accuracy of the maximum displacement of iso- lation interface under seismic excitations. Results show the prediction accuracy is significantly affected by this ratio and the equivalent linearizationmethod proposed in Chapter 4 is further improved in order to get more accurate results.
Integral and integro-differential equations play an im- portant role in characterizing many social, biological, physical and engineering problems; for more details see [1-3] and references cited therein. Nonlinear integral and integro-differential equations are usually hard to solve analytically and exact solutions are rather difficult to be obtained. Many numerical methods have been studied such as the Legendre wavelets method , the Haar functions method [5,6], the linearizationmethod , the finite difference method , the Tau method [9,10], the hybrid Legendre polynomials and block-pulse functions , the Adomian decomposition method [12,13], the Taylor polynomial method [14-16] and the collocation approach (for linear case) .
water %50 mixtures. Plot of calculated log = A − A HA A − − A all triazole derivatives as a function of pH in studied media are shown in Figure 3-4. All the values presented are the average of at least 5 measurements and the standard deviations of each are listed. The corresponding pKa values for all compounds, obtained from the spectrophotometric methods using the linearizationmethod with Henderson-Hasselbach equation in ethanol %50 – water %50 mixtures are given in Table 1.
Abstract. The simplified analysis of the parallel composite isolation system (PCI) was carried out based on equivalent linearizationmethod. Through the derivation, the restoring force model expressed in the form of basal shear force and displacement was converted to the basal shear force coefficient and displacement model. According to the new code for seismic design of buildings, the application of the simplified analysis method was illustrated by an example. The results showed that the simplified analysis method could be conveniently and effectively used for preliminary design of the PCI system, which laid a foundation for the popularization and application of the PCI system.
On the other hand, the incremental part plays an important role in analysis convergence. Selecting suitable parameters in predictor steps, could make an excessive impression on the rate of convergence, specially in highly non-linear problems [12,13]. For example, a proper extrapolation in the incremental part can avoid divergence [14,15]. Incremental-iterative techniques, as a solution of non-linear problems, are also able to combine with Neural Networks, Boundary Element Method and Normal Flow Algorithm [16-18].