• Journal of Mechanical Science and Technology
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• v.19 no.11
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• pp.1975-1987
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• 2005

An input time delay always exists in practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computers. In this paper a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption is proposed. The mathematical structure of the new discretization method is analyzed. On the basis of this structure the sampled-data representation of nonlinear systems with time-delayed multi-input is presented. The delayed multi-input general equation has been derived. In particular, the effect of the time-discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. Additionally, hybrid discretization schemes that result from a combination of the scaling and squaring technique (SST) with the Taylor series expansion are also proposed, especially under conditions of very low sampling rates. Practical issues associated with the selection of the method's parameters to meet CPU time and accuracy requirements, are examined as well. A performance of the proposed method is evaluated using a nonlinear system with time delay maneuvering an automobile.

• Journal of the Institute of Electronics Engineers of Korea SC
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• v.49 no.2
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• pp.17-25
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• 2012

A general discretization method for input-driven nonlinear continuous time-delay systems is proposed, which can be applied to general order sampling hold assumptions. It is based on a combination of Taylor series expansion and the theories of sampling and hold. The mathematical structure of the new discretization scheme is introduced in detail. The performance of the proposed discretization procedure is evaluated by two degrees of systems. The results show that the proposed scheme is applicable to control systems.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.583-593
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• 2022

Lienard's equations are important nonlinear differential equations with application in many areas of applied mathematics. In the present article, a new approach known as the modified fractional Taylor series method (MFTSM) is proposed to solve the nonlinear fractional Lienard equations with Caputo fractional derivatives, and the convergence of this method is established. Numerical examples are given to verify our theoretical results and to illustrate the accuracy and effectiveness of the method. The results obtained show the reliability and efficiency of the MFTSM, suggesting that it can be used to solve other types of nonlinear fractional differential equations that arise in modeling different physical problems.

• Journal of Power System Engineering
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• v.6 no.1
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• pp.96-101
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• 2002

In this paper, we address the problem of controlling an end-effector to track and grab a moving target using the visual servoing technique. A visual servo mechanism based on the image-based servoing principle, is proposed by using visual feedback to control an end-effector without calibrated robot and camera models. Firstly, we consider the control problem as a nonlinear least squares optimization and update the joint angles through the Taylor Series Expansion. And to track a moving target in real time, the Jacobian estimation scheme(Dynamic Broyden's Method) is used to estimate the combined robot and image Jacobian. Using this algorithm, we can drive the objective function value to a neighborhood of zero. To show the effectiveness of the proposed algorithm, simulation results for a six degree of freedom robot are presented.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.215-228
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• 2007

By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange's characteristics method (a new approach) for solving non linear fractional partial differential equations. The key of this results is the fractional Taylor's series $f(x+h)=E_{\alpha}(h^{\alpha}D^{\alpha})f(x)$ where $E_{\alpha}(.)$ is the Mittag-Leffler function.

• Journal for History of Mathematics
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• v.27 no.2
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• pp.127-138
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• 2014

In this paper we investigate how Newton discovered the generalized binomial theorem. Newton's binomial theorem, or binomial series can be found in Calculus text books as a special case of Taylor series. It can also be understood as a formal power series which was first conceived by Euler if convergence does not matter much. Discovered before Taylor or Euler, Newton's binomial theorem must have a good explanation of its birth and validity. Newton learned the interpolation method from Wallis' famous book ${\ll}$Arithmetica Infinitorum${\gg}$ and employed it to get the theorem. The interpolation method, which Wallis devised to find the areas under a family of curves, was by nature arithmetrical but not geometrical. Newton himself used the method as a way of finding areas under curves. He noticed certain patterns hidden in the integer binomial sequence appeared in relation with curves and then applied them to rationals, finally obtained the generalized binomial sequence and the generalized binomial theorem.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2005.06a
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• pp.840-843
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• 2005

For an analysis of nonlinear dynamic behavior in carbon nanotube(CNT) an electrostatic force of CNT was investigated. The boundary condition in the CNT was assumed to clamped-clamped case at both ends. This type of CNT is widely used as micro and nano-sensors. For larger gaps in between sensor and electrode the van der Waals force can be ignored. The electrostatic force can be expressed as linear form using Taylor series. However, the first term of the series expansion was investigated here. The electrostatic force From this study we can conclude that for larger gaps the electrostatic force play an important role in determining the deflections as well as the pull-in voltage of simply supported switches.

• Journal of Ocean Engineering and Technology
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• v.9 no.2
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• pp.53-60
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• 1995

A multi-leg spread mooring system for floating offshore structures is important, but the multi-leg static analysis is complicated due to the nonlinear behavior of each line and the effect of current which affects each line differently. The pretensioned position of the multi-leg mooring system obtained from the static equilibrium condition changes into a different position due to external loads and current. In this paper, the new position and the static tension at each line are caculated. The relation between the initial static equilibrium position and the new position due to the external loads is expressed in terms of the Taylor's series expansion. The Runge-Kutta $4^{th}$ method is employed in analyzing the 3-dimensional static cable nonlinear equations.

• Journal of the Korean Institute of Telematics and Electronics
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• v.20 no.5
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• pp.1-7
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• 1983

Generally, multiple-valued logic algebra is based on the number system of modulo-M. In this paper, characters a, b, c‥… each of them represents the independent state, are regarded as the elements of the symbolic multiple-valued logic. By using the set theory, the symbolic multiple - valued logic and their functions are defined. And Varation for the symbolic logic function due to the variation of a variable and their properties are suggested and analized. With these variations, the MacLaurin's and Taylor's Series expansions of the symbolic logic functions are proposed and proved.

• Proceedings of the KIPE Conference
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• 1998.11a
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• pp.73-77
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• 1998

The conventional sliding mode controller (SMCr) approach is often impractical or difficult when applied to high order process because the number of tuning parameters in the SMCr increases with the order of the plant. Camacho(1996) proposed the design of a fixed structure sliding mode controller based on a first order plus dead time approximation to the higher-order process. But, there are such problems as overshoot, settling time and command following. They are mainly due to the approximation errors of the time delay term by Taylor series. In this paper, in order to improve Camcho's method, a new Taylor approximation technique considering a weight is proposed.