Chyi Hwang

Parameter identification via Laguerre polynomials

An operational matrix for the integration of the Laguerre vector whose elements are the Laguerre polynomials is introduced and applied to parameter identification of time invariant linear systems. Due to the unique property of the operational matrix, the algorithms to formulate the algebraic equations to estimate unknown parameters are recursive and suitable for computer programming. Examples with satisfactory results are given.

Laguerre operational matrices for fractional calculus and applications

The Laguerre operational matrices for the integration and differentiation of a Laguerre vector whose elements are Laguerre polynomials are generalized to fractional calculus for investigating distributed systems. The generalized operational matrices corresponding to 8, 1/8, 8/√(82+1) and exp (−8/(8+1)) are derived as examples. Comparison of the Laguerre series approximate inversions of irrational Laplace transforms with exact solutions is very satisfactory.

Laguerre series solution of a functional differential equation

This paper presents a Laguerre series method for the solution of a functional differential equation of the type (d/dt)y(t) = Ay(λt) + By(t), with given initial conditions. The method consists of the following steps : (1) represent y(t) and y(λt), respectively, by series of the Laguerre polynomials gi(t) and gi(λt) ; (2) expand gi(λt) into Laguerre series of gi(t) ; (3) integrate the Laguerre series approximation of the functional differential equation by an operational matrix approach. Two numerical examples are given with satisfactory results.

Analysis and optimal control of time-varying linear systems via shifted Legendre polynomials

This paper extends the application of shifted Legendre polynomial expansion to time-varying systems. The extension is achieved through representing the product of two shifted Legendre series in a new shifted Legendre series. With this treatment of the product of two time functions, the operational properties of the shifted Legendre polynomials are fully applied to the analysis and optimal control of time-varying linear systems with quadratic performance index.

A wavelet-Galerkin method for solving population balance equations

A new wavelet-Galerkin method is developed to solve the population balance equations which arise in the description of particle-size distribution of a continuous, mixed-suspension, mixed-product removal crystallizer with taking account of the effect of particle breakage. The class of Daubechies wavelets, which is both compactly supported and orthonormal, is adopted as the Galerkin bases. Some elegant results concerned with the exact evaluation of functions on wavelets and their derivatives and integrals are derived. These results along with the 2-scale relation which defines the wavelet bases make the Galerkin method feasible for the solution of population balance equations containing a scaled argument.

Solution of integral equations via Laguerre polynomials

Abstract A unique property of the convolution integral of Laguerre polynomials is applied to solve convolution integrla and three important types of integral equations: the first order integral equation of the first kind, the second order integral equation of the first kind, and the integral equation of the second kind. Recursive algorithms for solving these integral equations are developed and are particularly attractive to digital computation. Examples are given for illustration.

Transfer-function matrix identification in MIMO systems via shifted Legendre polynomials

A general algorithm is presented for the identification of the parameters in the transfer-function matrix of a multi-input—multi-output (MIMO) system. The approach adopted is that of expanding the system input and output variables in shifted Legendre series. The feasibility of the method lies in the generation of linear algebraic equations in the unknown parameters and initial conditions by means of an elegant operational matrix which relates shifted Legendre polynomials to their integrals. An example is included to illustrate the applicability of the proposed method.

A multipoint continued-fraction expansion for linear system reduction

This note presents a multipoint continued-fraction expansion (MCFE) to obtain reduced models for linear time-invariant systems. Algorithms for the expansion and inversion of the MCFE are outlined. A realization for the MCFE of a transfer function is suggested, and the corresponding MCFE canonical state-space model is established. A new similarity transformation matrix is constructed to transform a state-space model in the phase-variable canonical form to one in the MCFE canonical form. The attractive feature of the present MCFE is that it is general in form and the resulting reduced models approximate well the original system in both the frequency and time responses.

Optimal Routh approximations for continuous-time systems

A new combination of Routh stability criterion and integral squared error (ISE) criterion approach is proposed for the linear model reduction of high-order dynamic systems. The method consists of a two-step computational scheme. In the first step, the Routh approximation method is used to reduce the order of the denominator polynomial of the system transfer function. In the second step, the Routh table is used to derive a set of optimal coefficients of the numerator polynomial of reduced model such that the ISE between the unit step responses of the original and simplified system is reduced to a minimum. The advantages of the proposed method are that it does not actually evaluate the system time response in the step of minimizing the ISE, and the reduced model is stable if the original system is stable.

Solution of a functional differential equation via delayed unit step functions

Abstract A new set of delayed unit step functions (DUSFs) has been defined. Based on the approximations of the delay operators exp (−hs) and exp ( −αhs) o α  1) respectively by two new operational matrices of DUSFs ore derived. One is the integration matrix which relates DUSFs to their integrals, and the other is the stretch matrix which relates DUSFs to their stretched forms. By the use of these two operational matrices the solutions of a differential equation of the type [ydot] (t)= ay( are obtained in a series of DUSFs. The results obtained may be piecewise-constant or pointwise. Compared with Walsh or block pulse function approaches, the proposed method is simpler in the construction of its operational matrices, and is more amenable to computer programming.