The problems of identification, analysis and optimal control have been recently studied via orthogonal functions. The particular orthogonal functions used up to now are the Walsh, the block-pulse and the Laguerre functions. In this paper, the Chebyshev functions are introduced and solutions for the aforementioned problems are established. The algorithms proposed are analogous to those already derived for the Walsh, block-pulse and Laguerre functions. The Chebyshev series approach presented here appears to have certain advantages over other orthogonal series, and they may therefore be more suitable for the study of the problems of identification, analysis and optimal control.

Chebyshev series approach to system identification, analysis and optimal control

The Haar wavelets operational matrix of integration P is derived, which is similar to those previously derived for other types of orthogonal functions such as Walsh, block-pulse, Laguerre, Legendre and Chebyshev. A general procedure of forming this matrix P is summarized. This matrix P can be used to solve problems such as identification, analysis and optimal control, like that of the other orthogonal functions. However, in the process of solving practical problems, the different resolution bases can be selected on the local time domain but on the global time domain, in order to understand more clearly system behaviour on the local time domain. This advantage results from the property of Haar wavelet localization, and the other orthogonal functions do not have this property. A numerical example will be used to demonstrate this point.

The Haar wavelets operational matrix of integration

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.

Parameter identification via Laguerre polynomials

State analysis of linear time delayed systems via Haar wavelets

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 operational matrices for fractional calculus and applications

This paper is concerned with the determination of suboptimal feedback laws for the linear systems with quadratic performance criteria. The time-varying gains are approximated by the piecewise constant gains which are naturally determined by using Walsh functions. An increase of the number of intervals of Walsh functions enables us to approximate the true optimal control more closely ; and a decrease of the number of intervals makes the implementation easier. Therefore the proposed method is simple in theory and flexible in practice. The beginning part of the paper, being tutorial in nature, is on Walsh functions, the middle part developes an operational method for solving state equations and the final part is concentrated on the Walsh functions approach to the solution of piecewise constant gains of optimal control.

Walsh series analysis in optimal control

The original Nyquist criterion is based on the comparison of the encirclement of the frequency plot of the return ratio function with the number of poles and the number of zeros of the same function to determine the closed-loop stability of a feedback system. The extensions of the return ratio idea to the stability study of multi-variable feedback systems have used the same terminology and followed a similar course. For the multi-input—output case, the use of the Nyquist criterion or its extension is by no means a simple matter. This paper establishes a new frequency stability criterion which converts the Nyquist criterion from a return ratio oriented approach to a return difference oriented one. Instead of examining the encirclement of the return ratio function to a critical point, we examine the phase change of the positive frequency of the return difference function, and the number of zeros of the positive frequency of the return difference function. This result simplifies the stability study of multi-...

A general frequency stability criterion for multi-input-output, lumped and distributed-parameter feedback systems

The Routh algorithm is used to generate the parameters of the Hermite criterion, The new formulation is simpler than the original. The relationship among the Hermite, the symmetrical Hurwitz, the Kalman-Bertram, and the Routh criteria is naturally established.

A new formulation of the Hermite criterion

A real exponential describing function as an analysis tool for studying the transient response of a class of non-linear feedback systems was developed by Bickart (1966). In this paper describing functions are developed for similar analysis more suitable to higher-order non-linear feedback systems. The two classes of describing functions developed are identified as real exponential or complex exponential. Here, signals in ℒ2 ( − ∞, t] a space of the space of square integrable signals defined on (−∞, t ], are approximated by the sum of n signals in ℒ2 1, m (−∞, t)one-dimensional sub-spaccs of ℒ2 ( − ∞, t] having the mth function from the set of reversed time orthogonalized real or eponential functions as a basis. A system mapping ℒ2 1, m(−∞, t] into itself is associated with a system mapping ℒ2 1, m(−∞, t] into itself; the latter system is characterized by a gain—real or exponential describing function. These multiple one-dimensional system mappings give rise to approximation components of the response whos...

Real and complex-exponential describing functions for transient analysis of non-linear control systems†

A state-space procedure for the formulation and solution of mixed boundary value problems is established. This procedure is a natural extension of the method used in initial value problems; however, certain special theorems and rules must be developed. The scope of the applications of the approach includes beam, arch, and axisymmetric shell problems in structural analysis, boundary layer problems in fluid mechanics, and eigenvalue problems for deformable bodies. Many classical methods in these fields developed by Holzer, Prohl, Myklestad, Thomson, Love-Meissner, and others can be either simplified or unified under new light shed by the state-variable approach. A beam problem is included as an illustration.

C. F. Chen

Papers

Walsh series analysis in optimal control

A general frequency stability criterion for multi-input-output, lumped and distributed-parameter feedback systems

A new formulation of the Hermite criterion

Real and complex-exponential describing functions for transient analysis of non-linear control systems†

State space approach to mixed boundary value problems.