• Title/Summary/Keyword: singular perturbation method

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A FIFTH ORDER NUMERICAL METHOD FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS WITH NEGATIVE SHIFT

  • Chakravarthy, P. Pramod;Phaneendra, K.;Reddy, Y.N.
    • 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.

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Wavelet-based Analysis for Singularly Perturbed Linear Systems Via Decomposition Method (웨이블릿 및 시스템 분할을 이용한 특이섭동 선형 시스템 해석)

  • Kim, Beom-Soo;Shim, Il-Joo
    • Journal of Institute of Control, Robotics and Systems
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    • v.14 no.12
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    • pp.1270-1277
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    • 2008
  • A Haar wavelet based numerical method for solving singularly perturbed linear time invariant system is presented in this paper. The reduced pure slow and pure fast subsystems are obtained by decoupling the singularly perturbed system and differential matrix equations are converted into algebraic Sylvester matrix equations via Haar wavelet technique. The operational matrix of integration and its inverse matrix are utilized to reduce the computational time to the solution of algebraic matrix equations. Finally a numerical example is given to demonstrate the validity and applicability of the proposed method.

PARAMETER-UNIFORM NUMERICAL METHOD FOR A SYSTEM OF COUPLED SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATIONS WITH MIXED TYPE BOUNDARY CONDITIONS

  • Tamilselvan, A.;Ramanujam, N.;Priyadharshini, R. Mythili;Valanarasu, T.
    • Journal of applied mathematics & informatics
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    • v.28 no.1_2
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    • pp.109-130
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    • 2010
  • In this paper, a numerical method for a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with the mixed type boundary conditions is presented. Parameter-uniform error bounds for the numerical solution and also to numerical derivative are established. Numerical results are provided to illustrate the theoretical results.

Adaptive and Digital Autopilot Design for Nonlinear Ship-to-Ship Missiles (비선형 함대함 미사일의 적응 디지털 제어기 설계)

  • Im, Ki-Hong;Choi, Jin-Young
    • 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.

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Frequency Weighted Controller Reduction of Closed-Loop System Using Lyapunov Inequalities (Lyapunov 부등식을 이용한 페루프시스템의 주파수하중 제어기 차수축소)

  • Oh, Do-Chang;Jeung, Eun-Tae;Lee, Kap-Rai;Kim, Jong-Hae;Lee, Sang-Kyung
    • Journal of Institute of Control, Robotics and Systems
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    • v.7 no.6
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    • pp.465-470
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    • 2001
  • This paper considers a new weighed model reduction method using block diagonal solutions of Lyapunov inequalities. With the input and/or output weighting function, the stability of the reduced order system is guaranteed and an a priori error bound is proposed. to achieve this after finding the solutions of two Lyapunov inequalities and balancing the full order system, we find the reduced order systems using the direct truncation and the singular perturbation approximation. The proposed method is compared with other existing methods using numerical examples.

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An H Output Feedback Control for Uncertain Singularly Perturbed T-S Fuzzy Systems

  • Yoo, Seog-Hwan;Wu, Xue-Dong
    • Journal of the Korean Institute of Intelligent Systems
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    • v.19 no.6
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    • pp.840-847
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    • 2009
  • This paper deals with an $H_{\infty}$ output feedback controller design for uncertain singularly perturbed T-S fuzzy systems. Integral quadratic constraints are used to describe various kinds of uncertainties of the plant. It is shown that the $H_{\infty}$ norm of the uncertain singularly perturbed fuzzy system is less than $\gamma$ for a sufficiently small $\varepsilon$ > 0 if the $H_{\infty}$ norms of both the slow and fast subsystem are less than $\gamma$. Using this fact, we develop a linear matrix inequality based design method which is independent of the singular perturbation parameter $\varepsilon$. A numerical example is provided to demonstrate the efficacy of the proposed design method.

A Study on Singularly Perturbed Open-Loop Systems by Delta Operator Approach

  • Shim, Kyu-Hong;M. Edwin Sawan
    • Transactions on Control, Automation and Systems Engineering
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    • v.3 no.4
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    • pp.242-249
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    • 2001
  • In this paper, the open-loop state response of the two-time-scale systems by unified approach using the $\delta$-operator is presented with an example of the aircraft longitudinal dynamics. First, the $\delta$-operator system unifies both the continuous system and the discrete system simultaneously, and the $\delta$-operator approach improves the finite word-length characteristics. This saves more computing time than that of the discrete system. Second, the singular perturbation method by block diagonalization reduces the sizes and orders of the systems. This also reduces the floating-point operations (flops). The advantage of those two approaches is shown by comparing our results with the earlier ones in the illustrative example of the longitudinal motion of F-8 aircraft.

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SUBOPTIMAL VIBRATION CONTROL OF FLEXIBLE ROBOT BEARING SYSTEM BY USING A MAGNETIC BEARING

  • Lee, Chong-Won;Kim, Jong-Sun
    • 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.

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New method for LQG control of singularly perturbed discrete stochastic systems

  • Lim, Myo-Taeg;Kwon, Sung-Ha
    • 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.

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HYBRID DIFFERENCE SCHEMES FOR A SYSTEM OF SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATIONS

  • Priyadharshini, R.Mythili;Ramanujam, N.;Tamilselvan, A.
    • Journal of applied mathematics & informatics
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    • v.27 no.5_6
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    • pp.1001-1015
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    • 2009
  • In this paper, two hybrid difference schemes on the Shishkin mesh are constructed for solving a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with a small parameter multiplying the highest derivative. We prove that the schemes are almost second order convergence in the supremum norm independent of the diffusion parameter. Error bounds for the numerical solution and its derivative are established. Numerical results are provided to illustrate the theoretical results.

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