• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.419-435
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• 2021

This paper deals with numerical treatment of singularly perturbed differential equations involving small delays. The highest order derivative in the equation is multiplied by a perturbation parameter 𝜀 taking arbitrary values in the interval (0, 1]. For small 𝜀, the problem involves a boundary layer of width O(𝜀), where the solution changes by a finite value, while its derivative grows unboundedly as 𝜀 tends to zero. The considered problem contains delay on the convection and reaction terms. The terms with the delays are approximated using Taylor series approximations resulting to asymptotically equivalent singularly perturbed BVPs. Inducing exponential fitting factor for the term containing the singular perturbation parameter and using central finite difference for the derivative terms, numerical scheme is developed. The stability and uniform convergence of difference schemes are studied. Using a priori estimates we show the convergence of the scheme in maximum norm. The scheme converges with second order of convergence for the case 𝜀 = O(N^-1) and for the case 𝜀 ≪ N^-1, the scheme converge uniformly with first order of convergence, where N is number of mesh intervals in the domain discretization. We compare the accuracy of the developed scheme with the results in the literature. It is found that the proposed scheme gives accurate result than the one in the literatures.

• Journal of Institute of Control, Robotics and Systems
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• v.22 no.9
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• pp.679-685
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• 2016

This paper addresses an observer-based output-feedback sampled-data controller design problem for nonlinear systems in Takagi-Sugeno (T-S) form including singular perturbations. The design condition is represented in terms of linear matrix inequalities. The separation principle is also investigated.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.679-702
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• 2014

Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used.

• Kyungpook Mathematical Journal
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• v.60 no.3
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• pp.629-645
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• 2020

A uniformly convergent numerical method is developed for solving singularly perturbed 1-D parabolic convection-diffusion problems. The developed method applies a non-standard finite difference method for the spatial derivative discretization and uses the implicit Runge-Kutta method for the semi-discrete scheme. The convergence of the method is analyzed, and it is shown to be first order convergent. To validate the applicability of the proposed method two model examples are considered and solved for different perturbation parameters and mesh sizes. The numerical and experimental results agree well with the theoretical findings.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.441-452
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• 2009

In this paper, a fifth order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been using for delay. Similar boundary value problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, first we use Taylor approximation to tackle terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system and is solved using the boundary conditions. Several numerical examples are solved and compared with exact solution. It is observed that present method approximates the exact solution very well.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1035-1054
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• 2010

We consider a mixed type singularly perturbed one dimensional elliptic problem with discontinuous source term. The domain under consideration is partitioned into two subdomains. A convection-diffusion and a reaction-diffusion type equations are posed on the first and second subdomains respectively. Two hybrid difference schemes on Shishkin mesh are constructed and we prove that the schemes are almost second order convergence in the maximum norm independent of the diffusion parameter. Error bounds for the numerical solution and its numerical derivative are established. Numerical results are presented which support the theoretical results.

• Proceedings of the KIEE Conference
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• 2005.10b
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• pp.619-621
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• 2005

This paper proposes apractical design method for ship-to-ship missiles' autopilot. When the pre-designed analogue autopilot is implemented in digital way, theygenerally suffer from severe performance degradation and instability problem even for a sufficiently small sampling time. Also, aerodynamic uncertainties can affect the overall stability and this happens more severely when the nonlinear autopilot is digitally implemented. In order to realize a practical autopilot, two main issues, digital implementation problem and compensation for the aerodynamic uncertainties, are considered in this paper. MIMO (multi-input multi-output) nonlinear autopilot is presented first and the input and output of the missile are discretized for implementation. In this step, the discretization effect is compensated by designing an additional control input. Finally, we design a parameter adaptation law to compensate the control performance. Stability analysis and 6-DOF (degree-of-freedom) simulations are presented to verify the proposed adaptive autopilot.

• 제어로봇시스템학회:학술대회논문집
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• 1995.10a
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• pp.432-435
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• 1995

In this paper a new approach to obtain the solution of the linear-quadratic Gaussian control problem for singularly perturbed discrete-time stochastic systems is proposed. The alogorithm proposed is based on exploring the previous results that the exact solution of the global discrete algebraic Riccati equations is found in terms of the reduced-order pure-slow and pure-fast nonsymmetric continuous-time algebraic Riccati equations and, in addition, the optimal global Kalman filter is decomposed into pure-slow and pure-fast local optimal filters both driven by the system measurements and the system optimal control input. It is shown that the optimal linear-quadratic Gaussian control problem for singularly perturbed linear discrete systems takes the complete decomposition and parallelism between pure-slow and pure-fast filters and controllers.

• 제어로봇시스템학회:학술대회논문집
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• 1989.10a
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• pp.255-259
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• 1989

A suboptimal output feedback controller is designed and applied to a flexible rotor bearing system in order to control the unstable or lilghtly damped vibrations. The reduced order model is the truncated modal equation of the distributed parameter system obtained through the singular perturbation. The instability problem arising from the spillover effects caused by the uncontrolled high frequency modes is prevented through the constrained optimization by incorporating the spillover term into the performance index. The efficiency of the proposed method is demonstrated experimentally with a flexible rotor by using a magnetic bearing.

• Proceedings of the KIEE Conference
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• 1994.11a
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• pp.6-8
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• 1994

In this paper an attempt is made to understand the voltage stability when the power system networks are represented by the differential-algebraic equations (DAEs) form. The problem is analyzed by interpreting the shape of constraint manifold, based on the singular perturbation model. The global picture or constraint manifold is given to show how the local shape or constraint manifold can be used to guess for the system behavior. The gradient analysis is used systematically to obtain a local shape or the constraint manifold.