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Book ChapterDOI

Triangular Orthogonal Functions for the Analysis of Continuous Time Systems: Convolution Process in Triangular Function Domain and Its Use in SISO Control System Analysis

About: The article was published on 2011-01-01. It has received 10 citations till now. The article focuses on the topics: Triangular function & Convolution.
Citations
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Journal ArticleDOI
TL;DR: An approach by using inverse fuzzy transforms based on the fuzzy partition with combination in collocation technique for the numerical solution of Fredholm integral equations of the second kind to reduce the problem to the linear system of equations.
Abstract: In this paper, we introduce an approach by using inverse fuzzy transforms based on the fuzzy partition with combination in collocation technique for the numerical solution of Fredholm integral equations of the second kind. The main advantage of this approach is to reduce the problem to the linear system of equations. We present the convergence theorem for this method. Finally, we give two examples to illustrate the efficiency of the proposed method.

16 citations


Cites background or methods from "Triangular Orthogonal Functions for..."

  • ...Ezzati and Najafalizadeh (2011) applied Chebyshev polynomials for nonlinear Volterra-Fredholm integral equations. In Maleknejad et al. (2003), Legendre wavelet was used to solve linear Fredholm and Volterra integral equations of the second kind....

    [...]

  • ...Ezzati and Najafalizadeh (2011) applied Chebyshev polynomials for nonlinear Volterra-Fredholm integral equations....

    [...]

  • ...…methods based on different wavelets, such as Legendre and Walsh were designed for analysis of control systems and various related applications (Maleknejad et al., 2003; Yousefi and Razzaghi, 2005), and approximating the solution of integral equations (Deb et al., 2007; Yousefi and Razzaghi, 2005)....

    [...]

  • ...Maleknejad and Kajani (2002) proposed piecewise constant basis functions to obtain numerical solution of integro-differential equations....

    [...]

  • ...Ezzati and Najafalizadeh (2011) applied Chebyshev polynomials for nonlinear Volterra-Fredholm integral equations. In Maleknejad et al. (2003), Legendre wavelet was used to solve linear Fredholm and Volterra integral equations of the second kind. Solving nonlinear Volterra integral equation of the second kind using Chebyshev polynomials was done in the work of Maleknejad et al. (2007). Several numerical methods based on different wavelets, such as Legendre and Walsh were designed for analysis of control systems and various related applications (Maleknejad et al....

    [...]

Journal ArticleDOI
TL;DR: In this article, the operational matrix of two-dimensional orthogonal triangular functions (2D-TFs) for 2D fractional integrals is derived and applied to a system of algebraic equations.
Abstract: In this paper, first, we derive the operational matrix of two-dimensional orthogonal triangular functions (2D-TFs) for two-dimensional fractional integrals. Then, we apply this operational matrix and properties of Two-dimensional orthogonal triangular functions to reduce two-dimensional fractional integral equations to a system of algebraic equations. Finally, in order to show the validity and efficiency, we present some numerical examples.

10 citations

01 Jan 2015
TL;DR: In this paper, the generalized triangular function operational matrices for approximating Riemann-Liouville fractional order integral in the triangular function (TF) domain are derived.
Abstract: This article introduces a new application of piecewise linear orthogonal triangular functions to solve fractional order differential-algebraic equations. The generalized triangular function operational matrices for approximating Riemann-Liouville fractional order integral in the triangular function (TF) domain are derived. Error analysis is carried out to estimate the upper bound of absolute error between the exact Riemann-Liouville fractional order integral and its TF approximation. Using the proposed generalized operational matrices, linear and nonlinear fractional order differential-algebraic equations are solved. The results show that the TF estimate of Riemann-Liouville fractional order integral is accurate and effective.

8 citations

29 Jan 2016
TL;DR: The knowledge is got, that the Laguerre-Volterra model can be relatively easy extended to higher in and outputs on SpiNNaker leading to a rapid prototyping and different scaling strategies could be easily implemented and tested, what is very helpful in the development phase of a neuro prosthesis.
Abstract: Illnesses as epilepsy or Alzheimers lead to strong life impairments for the affected persons and their caretakers. Damages in the brain are responsible for this behavior. One area of neuroscience addresses the healing processes of the affected areas. One approach, which is recently discussed for the future, contains the bridging of damaged regions with a neuro prosthesis. The brain activity functions of a healthy human should be replicated by such hardware in order to make a normal life possible again. This requires research, for instance, in the field of hardware platforms to estimate if they are appropriate for an artificial brain model. We examine, if the SpiNNaker system is a suitable platform to implement the Laguerre Volterra model. SpiNNaker is interesting, because it is a multiprocessor platform inspired by the human brain. We develop and implement strategies for parallelizations and distributions of the Laguerre-Volterra model on the single cores. As results, we got the knowledge, that the Laguerre-Volterra model can be relatively easy extended to higher inand outputs on SpiNNaker leading to a rapid prototyping. Furthermore different scaling strategies could be easily implemented and tested, what is very helpful in the development phase of a neuro prosthesis. This implementation can lay the foundation for a more portable spike-based VLSI hardware solution that can be implanted in the brain.

8 citations


Additional excerpts

  • ...They build an orthogonal set [DA11] [IBW14]....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a new set of hybrid functions (HF) formed by the synthesis of sample-and-hold functions (SHF) and triangular functions (TF) is proposed.
Abstract: The present work uses a new set of hybrid functions (HF) formed by the synthesis of sample-and-hold functions (SHF) and triangular functions (TF). The SHF set is efficient for analyzing sample-and-hold control systems and the TF set have been employed for obtaining piecewise linear solution of control problems. After a brief review of the basic theory of HF, the operational matrices for integration in HF domain are also briefly discussed. Finally, this HF set is employed for the analysis and synthesis of homogeneous and non-homogeneous systems described via state space. Many examples are treated and the results are compared with the exact solutions and found to be attractively close. Since the HF set works with function samples, the computational burden in the presented method are much less than traditional ones.

4 citations


Cites background or methods from "Triangular Orthogonal Functions for..."

  • ...Following a similar procedure [4] for the components of the TF set comprised of four members, we integrate each member and express the result of integration, as before, in HF domain....

    [...]

  • ...Hybrid Function (HF) : A Combination of SHF and TF [7, 8] We can use a set of sample-and-hold functions and the RHTF set of triangular functions [4] to form a hybrid function set, which we name a...

    [...]

  • ...In 2003, orthogonal triangular functions (TF) [2-4] were introduced by Deb et al....

    [...]

References
More filters
Journal ArticleDOI
TL;DR: An approach by using inverse fuzzy transforms based on the fuzzy partition with combination in collocation technique for the numerical solution of Fredholm integral equations of the second kind to reduce the problem to the linear system of equations.
Abstract: In this paper, we introduce an approach by using inverse fuzzy transforms based on the fuzzy partition with combination in collocation technique for the numerical solution of Fredholm integral equations of the second kind. The main advantage of this approach is to reduce the problem to the linear system of equations. We present the convergence theorem for this method. Finally, we give two examples to illustrate the efficiency of the proposed method.

16 citations

Journal ArticleDOI
TL;DR: In this article, the operational matrix of two-dimensional orthogonal triangular functions (2D-TFs) for 2D fractional integrals is derived and applied to a system of algebraic equations.
Abstract: In this paper, first, we derive the operational matrix of two-dimensional orthogonal triangular functions (2D-TFs) for two-dimensional fractional integrals. Then, we apply this operational matrix and properties of Two-dimensional orthogonal triangular functions to reduce two-dimensional fractional integral equations to a system of algebraic equations. Finally, in order to show the validity and efficiency, we present some numerical examples.

10 citations

01 Jan 2015
TL;DR: In this paper, the generalized triangular function operational matrices for approximating Riemann-Liouville fractional order integral in the triangular function (TF) domain are derived.
Abstract: This article introduces a new application of piecewise linear orthogonal triangular functions to solve fractional order differential-algebraic equations. The generalized triangular function operational matrices for approximating Riemann-Liouville fractional order integral in the triangular function (TF) domain are derived. Error analysis is carried out to estimate the upper bound of absolute error between the exact Riemann-Liouville fractional order integral and its TF approximation. Using the proposed generalized operational matrices, linear and nonlinear fractional order differential-algebraic equations are solved. The results show that the TF estimate of Riemann-Liouville fractional order integral is accurate and effective.

8 citations

29 Jan 2016
TL;DR: The knowledge is got, that the Laguerre-Volterra model can be relatively easy extended to higher in and outputs on SpiNNaker leading to a rapid prototyping and different scaling strategies could be easily implemented and tested, what is very helpful in the development phase of a neuro prosthesis.
Abstract: Illnesses as epilepsy or Alzheimers lead to strong life impairments for the affected persons and their caretakers. Damages in the brain are responsible for this behavior. One area of neuroscience addresses the healing processes of the affected areas. One approach, which is recently discussed for the future, contains the bridging of damaged regions with a neuro prosthesis. The brain activity functions of a healthy human should be replicated by such hardware in order to make a normal life possible again. This requires research, for instance, in the field of hardware platforms to estimate if they are appropriate for an artificial brain model. We examine, if the SpiNNaker system is a suitable platform to implement the Laguerre Volterra model. SpiNNaker is interesting, because it is a multiprocessor platform inspired by the human brain. We develop and implement strategies for parallelizations and distributions of the Laguerre-Volterra model on the single cores. As results, we got the knowledge, that the Laguerre-Volterra model can be relatively easy extended to higher inand outputs on SpiNNaker leading to a rapid prototyping. Furthermore different scaling strategies could be easily implemented and tested, what is very helpful in the development phase of a neuro prosthesis. This implementation can lay the foundation for a more portable spike-based VLSI hardware solution that can be implanted in the brain.

8 citations

Journal ArticleDOI
TL;DR: In this paper, a new set of hybrid functions (HF) formed by the synthesis of sample-and-hold functions (SHF) and triangular functions (TF) is proposed.
Abstract: The present work uses a new set of hybrid functions (HF) formed by the synthesis of sample-and-hold functions (SHF) and triangular functions (TF). The SHF set is efficient for analyzing sample-and-hold control systems and the TF set have been employed for obtaining piecewise linear solution of control problems. After a brief review of the basic theory of HF, the operational matrices for integration in HF domain are also briefly discussed. Finally, this HF set is employed for the analysis and synthesis of homogeneous and non-homogeneous systems described via state space. Many examples are treated and the results are compared with the exact solutions and found to be attractively close. Since the HF set works with function samples, the computational burden in the presented method are much less than traditional ones.

4 citations