General Orthogonal Polynomials Research Articles

Parameter identification of time-varying system via generalized orthogonal polynomials

An effective method is presented of using generalized orthogonal polynomials (GOPs) for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time-varying coefficients in the form of finite-order polynomials. It is based on the repeated integration of a differential equation and the integration operational matrix of the GOP which can represent all kinds of individual orthogonal polynomials. By expanding the state function and control function into a series of GOPs, the differential input-output equation is converted into a set of overdetermined linear algebraic equations. The unknown parameters are evaluated by using the least-squares method in conjunction with an orthogonal polynomial expansion. An example is given to illustrate the usefulness of the method and very satisfactory results are obtained.

Double generalized orthogonal polynomial series for the solution of integral equations

Double generalized orthogonal polynomial (GOP) series are developed to approximate the solutions of the Volterra integral equation and the Fredholm integral equation. This method simplifies computations of integral equations to the solutions of a linear algebraic equation in a matrix form. In addition, the computational complexity can be reduced remarkably; two examples are given. Very accurate results are obtained by using the proposed generalized orthogonal polynomial series expansion. The present approach is straightforward, convenient, and converges faster in finding approximations than other existing methods.

Solutions of convolution integral and integral equations via double general orthogonal polynomials

Double general orthogonal polynomials are developed in this work to approximate the solutions of convolution integrals, Volterra integral equations, and Fredholm integral equations. The proposed method reduces the computations of integral equations to the successive solution of a set of linear algebraic equations in matrix form; thus, the computational complexity is considerably simplified. Furthermore, the solutions obtained by the general orthogonal polynomials include as special cases solutions by Chebyshev polynomials, Legendre polynomials, Laguerre polynomials, or Jacobi polynomials. A comparison of the results obtained via several different classical orthogonal polynomial approximations is also presented.

Digital PID controller design via general discrete orthogonal polynomials

Based on the discrete orthogonal polynomials, a new computer-oriented algebraic method is proposed for designing a digital PID controller. The PID controller can approximately achieve an arbitrarily specified closed-loop response, and is determined by solving a linear equation using the least squares method. An example is given to demonstrate the availability of the proposed technique.

Analysis of discrete dynamic robot models

The discrete shift-transformation matrix of general orthogonal polynomials is introduced. The discrete shift-transformation matrix is employed to transform the difference equations, which describe the discrete dynamic robot model, into algebraic equations. Several lemmas are introduced which, together with the discrete shift-transformation matrix, solve for the joint positions and velocities of discrete dynamic robot models via discrete orthogonal polynomials approximations. The initial numerical experiment with a cylindrical coordinate robot shows the feasibility and applicability of discrete orthogonal polynomials approximations.

Identifiability and identification of linear distributed systems via double general orthogonal polynomials

A method is presented for identifying the parameters of linear parabolic distributed systems. The approach starts by reducing the linear distributed systems to lumped systems using double general orthogonal polynomials. Then a study of the identifiability is carried out for the lumped systems and, relying on the identifiability results obtained, we solve the identification problem. An illustrative example is given and good results are obtained using the proposed method.

A survey of general orthogonal polynomials for weights on finite and infinite intervals

We survey the analytic aspects of general o.p.'s (orthogonal polynomials) associated with weights on finite and infinite intervals, concentrating on their asymptotics. We consider also the associated o.p.'s on the circle, and briefly discuss o.p.'s for complex weights.

Analysis and parameter estimation of bilinear systems via generalized orthogonal polynomials

Abstract An effective method of using generalized orthogonal polynomials (GOP) for analysing and identifying the parameters of a process whose behaviour can be modelled by a bilinear equation is presented. The integration operational matrix and the operational matrix for the product of ti with the GOP vector are derived. These two kinds of operational matrices of the GOP are related to any type of individual orthogonal polynomial. By expanding the state and control functions into a series of GOP, the bilinear equation can be converted into a set of linear algebraic equations. The expansion coefficients of state variables are solved from these linear algebraic equations. The unknown parameters are evaluated by using the least squares method in conjunction with the individual orthogonal polynomial expansion. Two examples are given to illustrate the validity of the method. Very satisfactory results are obtained.

Analysis of time-varying scaled systems via general orthogonal polynomials

General orthogonal polynomials are introduced to analyze and approximate the solution of a class of scaled systems. Using the operational matrix of integration, together with the operational matrix of linear transformation, the dynamical equation of a scaled system is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the general orthogonal polynomials can be determined recursively by the derived algorithm. An illustrative example is given to demonstrate the validity and applicability of the orthogonal polynomial approximations.

MODELING OF CRYSTALLIZATION SYSTEMS VIA GENERALIZED ORTHOGONAL POLYNOMIALS METHOD

Abstract The population balance equation which is used to describe the dynamic of crystallization processes is simulated by the proposed generalized orthogonal polynomials (GOP). Examples of modeling include the ordinary differential equation for MSMPR crystallization processes, the partial differential equation for transient batch crystallization processes and the functional differential equation for breakage of crystals within the crystallizer. The key idea is to express the population density function by. a generalized orthogonal polynomials series. The ordinary (or partial, or functional) differential equation is then transformed into a set of algebraic (or ordinary differential, or algebraic) equations of expansion coefficients through the integration operation matrix. The advantage of using GOP approximation is that almost any kind of orthogonal polynomials can be used to approximate the system with very accurate results. Furthermore, it is very easy to calculate the expansion coefficients by using ...

Analysis and Identification of Linear Distributed Systems via Double General Orthogonal Polynomials

Double general orthogonal polynomials are used for the analysis, and the parameters estimation of linear distributed systems. The formulating of algorithms for analyzing and for estimating the parameters of linear distributed systems are straightforward and convenient for digital computation. Illustrative examples are given to demonstrate the availability of the proposed method

Application of Generalized Orthogonal Polynomials to Parameter Estimation of Time-Invariant and Bilinear Systems

Generalized orthogonal polynomials (GOP) which can represent all types of orthogonal polynomials and nonorthogonal Taylor series are first introduced to estimate the parameters of a dynamic state equation. The integration operation matrix (IOP) and the differentiation operation matrix (DOP) of the GOP are derived. The key idea of deriving IOP and DOP of these polynomials is that any orthogonal polynomial can be expressed by a power series and vice versa. By employing the IOP approach to the identification of time invariant systems, algorithms for computation which are effective, simple and straightforward compared to other orthogonal polynomial approximations can be obtained. The main advantage of using the differentiation operation matrix is that the parameter estimation can be carried out starting at an arbitrary time of interest. In addition, the computational algorithm is even simpler than that of the integral operation matrix. Illustrative examples for using IOP and DOP approaches are given. Very satisfactory results are obtained.

Analysis of stiff systems via the method of generalized orthogonal polynomials

The generalized orthogonal polynomials (GOP) which include all types of orthogonal polynomials are applied to the analysis of stiff systems. Using the idea of orthogonal polynomial functions which can be expressed by a power series and vice versa, the integration operational matrix of the GOP is derived. The state variables are expressed in a GOP series. An effective computational algorithm in which the whole time region is divided into several small time intervals is proposed. Only one or two terms of expansion coefficients are enough to get accurate results. The expansion coefficients are only required to be determined at the first time interval and can be used in the subsequent time intervals. A very simple recursive formula is developed to calculate the state functions. Four illlustrative examples are given and very satisfactory computational results are obtained.

New approach for parameter identification via generalized orthogonal polynomials

An effective method is presented for using generalized orthogonal polynomials (GOP) for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time-invariant coefficients. The method is based on the differentiation operational matrix of the GOP, which can represent all kinds of individual orthogonal polynomials. The main advantage of using the differentiation operational matrix is that parameter estimation can be made starting at any time of interest, without the restriction of starting at zero time. Using the concept of GOP expansion for a state function and a control function, the differential input-output equation is converted into a set of over-determined linear algebraic equations. The unknown parameters are evaluated by a weighted least-squares estimation method. Two examples are given to demonstrate the validity of the method and good results are obtained.

Solution of integral equations via generalized orthogonal polynomials

A new approximation method using a generalized orthogonal polynomial (GOP) is employed for solving integral equations. The integration operational matrix of the GOP, which can represent all kinds of individual orthogonal polynomial, is developed. The dependent variables in the integral equation are assumed to be expressed by a GOP series. A set of algebraic equations is obtained from the integral equation. The calculation of coefficients is straightforward and easy. Examples are given, and the results obtained from individual orthogonal polynomial approximations are compared with each other. It is found that nearly all individual orthogonal polynomials, except Hermite polynomials, offer excellent results.

Identification of a single-variable linear time-varying system via generalized orthogonal polynomials

The effective generalized orthogonal polynomials (GOPs) expansion is applied to estimating the parameters of a single-input-single-output linear time-varying system. It is based on the integration operational matrix for GOPs, which can represent all kinds of individual orthogonal polynomial. Using the concept of GOP expansion for a state function and a control function, the differential input-output equation is converted into a set of overdetermined linear algebraic equations. The unknown parameters are evaluated by a weighted least-squares method. An illustrative example is given. Very accurate results are obtained.

Generalization of generalized orthogonal polynomial operational matrices for fractional and operational calculus

A method is given for the approximation of generalized orthogonal polynomials (GOP) to solve the problems of fractional and operational calculus. A more rigorous derivation for the generalized orthogonal polynomial operational matrices is proposed. The Riemann-Liouville fractional integral for repeated fractional (and operational) integration is integrated exactly, then expanded in generalized orthogonal polynomials to yield the generalized orthogonal polynomial operational matrices. The generalized orthogonal polynomial operational matrices perform as sα(α ≥ αe R) in the Laplace domain and as fractional (and operational) integrators in the time domain. Using these results, inversions of the Laplace transforms of irrational and rational transfer functions are solved in a simple way. Very accurate results are obtained.

Analysis of time-varying delay systems via general orthogonal polynomials

General orthogonal polynomials are introduced in order to analyse and approximate the solution of a class of delay systems. Using the operational matrix of integration, together with the operational matrix of linear transformation, the dynamical equation of a delay system is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the general orthogonal polynomials can be determined recursively by use of the derived algorithm. Examples are given to demonstrate the validity and applicability of the orthogonal-polynomial approximations.

Analysis and parameter identification of time-delay linear systems via generalized orthogonal polynomials

A linear time-delay state equation is solved by the proposed generalized orthogonal polynomial (GOP) method. The parameter identification of such a system with time delay is also studied. The system is partitioned into several time intervals. Within a certain time interval, the state and control functions are assumed to be expressed by the GOP series. Time-delay differential equations are transformed into a series of algebraic equations of expansion coefficients. An effective algorithm is proposed to solve the time-delay system problem and to estimate the system parameters. By using such an effective computational algorithm, the calculation procedures are greatly simplified. Two illustrative examples are given to demonstrate the validity of the method. Very accurate results are obtained.

Analysis and optimal control of linear time-varying systems via general orthogonal polynomials

The operational matrix consisting of the product of two time functions, and the operational matrices for forward or backward integration consisting of general orthogonal polynomials are derived, respectively, for the analysis and optimal control of linear time-varying systems with a quadratic performance measure. The present results include results obtained via Chebyshev, Legendre, Laguerre, Jacobi, Hermite and ultraspherical polynomials as special cases.