A new set of orthogonal functions and its application to the analysis of dynamic systems
01 Jan 2006-Journal of The Franklin Institute-engineering and Applied Mathematics (JOURNAL OF THE FRANKLIN INSTITUTE)-Vol. 343, Iss: 1, pp 1-26
TL;DR: It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution.
Abstract: The present work proposes a complementary pair of orthogonal triangular function (TF) sets derived from the well-known block pulse function (BPF) set. The operational matrices for integration in TF domain have been computed and their relation with the BPF domain integral operational matrix is shown. It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution. As a further study, the newly proposed sets have been applied to the analysis of dynamic systems to prove the fact that it introduces less mean integral squared error (MISE) than the staircase solution obtained from BPF domain analysis, without any extra computational burden. Finally, a detailed study of the representational error has been made to estimate the upper bound of the MISE for the TF approximation of a function f ( t ) of Lebesgue measure.
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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
TL;DR: In this paper, a numerical method to approximate the solution of linear stochastic Ito-Volterra integral equations driven by fractional Brownian motion with Hurst parameter was presented.
Abstract: In this paper, we present a numerical method to approximate the solution of linear stochastic Ito-Volterra integral equations driven by fractional Brownian motion with Hurst parameter $ H \in (0,1)$ based on a stochastic operational matrix of integration for generalized hat basis functions. We obtain a linear system of algebraic equations with a lower triangular coefficients matrix from the linear stochastic integral equation, and by solving it we get an approximation solution with accuracy of order $ \emph{O}(h^2)$. This numerical method shows that results are more accurate than the block pulse functions method where the rate of convergence is $ \emph{O}(h)$. Finally, we investigate error analysis and with some examples indicate the efficiency of the method.
8 citations
Cites background from "A new set of orthogonal functions a..."
...Hat functions and their properties [1,2,7,15] The family of the first (n+1) hat functions on [0, T] is defined as follows:...
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TL;DR: It is found that HF based approximation is a strong contender of approximations based upon orthogonal polynomials like Legendre polynomsials for solving third order non-homogeneous differential equations.
7 citations
TL;DR: In this paper, the authors studied the global dynamics of a polynomial differential system for a one-dimensional parametric subfamily and showed that there is an equilibrium point which is a global attractor.
Abstract: Recently several works have studied the following model of finance
x ˙ = z + ( y − a ) x , y ˙ = 1 − b y − x 2 , z ˙ = − x − c z , where a, b and c are positive real parameters. We study the global dynamics of this polynomial differential system, and in particular for a one–dimensional parametric subfamily we show that there is an equilibrium point which is a global attractor.
7 citations
Additional excerpts
...We consider the following polynomial differential system in R3 (see [1, 2, 3, 5, 6, 7]) ẋ = z + (y − a)x, ẏ = 1− by − x(2), ż = −x− cz, (1)...
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TL;DR: The sheet current model considering the end effects is applied to design a six-degree-of-freedom (6-DOF) active vibration isolation system (AVIS) driving by the voice coil motors and shows quicker response and smaller overshoot, i.e., the CNF controller achieves the dynamic damping ratio during the control.
Abstract: The sheet current model considering the end effects is applied to design a six-degree-of-freedom (6-DOF) active vibration isolation system (AVIS) driving by the voice coil motors (VCMs). Compared with the charge model, the sheet current method represents higher calculation efficiency. Then, the hybrid functions (HFs) are employed to identify the multi-input and multi-output (MIMO) AVIS, and the identification parameters are used to design the controller. To promote the control performance, the composite nonlinear feedback (CNF) controller is designed based on the identification parameters. Consequently, the simulations and experiments are carried out, compared with PID controller, the CNF controller shows quicker response and smaller overshoot, i.e., the CNF controller achieves the dynamic damping ratio during the control. Thus, the experimental results demonstrate the better performance of the CNF controller.
7 citations
References
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Book•
01 Jan 1970
TL;DR: This comprehensive treatment of the analysis and design of continuous-time control systems provides a gradual development of control theory and shows how to solve all computational problems with MATLAB.
Abstract: From the Publisher:
This comprehensive treatment of the analysis and design of continuous-time control systems provides a gradual development of control theoryand shows how to solve all computational problems with MATLAB. It avoids highly mathematical arguments, and features an abundance of examples and worked problems throughout the book. Chapter topics include the Laplace transform; mathematical modeling of mechanical systems, electrical systems, fluid systems, and thermal systems; transient and steady-state-response analyses, root-locus analysis and control systems design by the root-locus method; frequency-response analysis and control systems design by the frequency-response; two-degrees-of-freedom control; state space analysis of control systems and design of control systems in state space.
6,634 citations
TL;DR: In der Theorie der Reihenentwicklung der reellen Funktionen spielen die sog. orthogonalen Funktionensysteme eine fuhrende Rolle.
Abstract: In der Theorie der Reihenentwicklung der reellen Funktionen spielen die sog. orthogonalen Funktionensysteme eine fuhrende Rolle. Man versteht darunter ein System von unendlichvielen Funktionen $\phi_1 (s), \phi_2 (s),\ldots$, die in bezug auf die beliebige mesbare Punktmenge $M$ die Orthogonalitatseigenschaft $\int_{(M)}\phi_p(s)\phi_q(s)ds=0$ ($p
eq q, p, q=1,2,\ldots$), $\int_{(M)}(\phi_p(s))^2ds=1$ ($p=1,2,\ldots$) besitzen, wobei die Integrale im Lebesgueschen Sinne genommen sind. acces pdf
1,877 citations
01 Jan 1991
TL;DR: Continuous-time model-based system identification as mentioned in this paper is a well-established field in the field of control systems and is concerned with the determination of particular models for systems that are intended for a certain purpose such as control.
Abstract: System identification is a well-established field. It is concerned with the determination of particular models for systems that are intended for a certain purpose such as control. Although dynamical systems encountered in the physical world are native to the continuous-time domain, system identification has been based largely on discrete-time models for a long time in the past, ignoring certain merits of the native continuous-time models. Continuous-time-model-based system identification techniques were initiated in the middle of the last century, but were overshadowed by the overwhelming developments in discrete-time methods for some time. This was due mainly to the 'go completely digital' trend that was spurred by parallel developments in digital computers. The field of identification has now matured and several of the methods are now incorporated in the continuous time system identification (CONTSID) toolbox for use with Matlab. The paper presents a perspective of these techniques in a unified framework.
373 citations
TL;DR: In this paper, the Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems and a new set of orthogonal functions is derived from Walsh functions.
Abstract: The Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems. A new set of orthogonal functions is derived from Walsh functions. By using the new functions, the generalized Walsh operational matrices corresponding to √s, √(s2 + 1), e-s and e-√s etc. are established. Several distributed parameter problems are solved by the new approach.
207 citations