Triangular Orthogonal Functions for the Analysis of Continuous Time Systems - Hardcover

Deb, Anish; Sarkar, Gautam; Sengupta, Anindita

Hardcover

ISBN 10: 0857289993 ISBN 13: 9780857289995

Verlag: Anthem Press, 2011

Inhaltsangabe

This book deals with a new set of triangular orthogonal functions, which evolved from the set of well-known block pulse functions (BPF), a major member of the piecewise constant orthogonal function (PCOF) family. Unlike PCOF, providing staircase solutions, this new set of triangular functions provides piecewise linear solution with less mean integral squared error (MISE).

After introducing the rich background of the PCOF family, which includes Walsh, block pulse and other related functions, fundamentals of the newly proposed set – such as basic properties, function approximation, integral operational metrics, etc. – are presented. This set has been used for integration of functions, analysis and synthesis of dynamic systems and solution of integral equations. The study ends with microprocessor based simulation of SISO control systems using sample-and-hold functions and Dirac delta functions.

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Anish Deb, Gautam Sarkar and Anindita Sengupta

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Triangular Orthogonal Functions for the Analysis of Continuous Time Systems

By Anish Deb, Gautam Sarkar, Anindita Sengupta

Wimbledon Publishing Company

Copyright © 2011 Anish Deb, Gautam Sarkar and Anindita Sengupta
All rights reserved.

ISBN: 978-0-85728-999-5

Contents

Preface, xi, 
Chapter 1: Walsh, Block Pulse, and Related Orthogonal Functions in Systems and Control, 1, 
Chapter 2: A Newly Proposed Triangular Function Set and Its Properties, 27, 
Chapter 3: Function Approximation via Triangular Function Sets and Operational Matrices in Triangular Function Domain, 59, 
Chapter 4: Analysis of Dynamic Systems via State Space Approach, 73, 
Chapter 5: Convolution Process in Triangular Function Domain and Its Use in SISO Control System Analysis, 83, 
Chapter 6: Identification of SISO Control Systems via State Space Approach, 101, 
Chapter 7: Solution of Integral Equations via Triangular Functions, 111, 
Chapter 8: Microprocessor Based Simulation of Control Systems Using Orthogonal Functions, 137, 
References, 152, 
Index, 153, 

CHAPTER 1

Walsh, Block Pulse, and Related Orthogonal Functions in Systems and Control

Orthogonal properties [1] of familiar sine–cosine functions have been known for over two centuries; but the use of such functions to solve complex analytical problems was initiated by the work of the famous mathematician Baron Jean-Baptiste-Joseph Fourier [2]. Fourier introduced the idea that an arbitrary function, even the one defined by different equations in adjacent segments of its range, could nevertheless be represented by a single analytic expression. Although this idea encountered resistance at the time, it proved to be central to many later developments in mathematics, science, and engineering.

In many areas of electrical engineering the basis for any analysis is a system of sine–cosine functions. This is mainly due to the desirable properties of frequency domain representation of a large class of functions encountered in engineering design. In the fields of circuit analysis, control theory, communication, and the analysis of stochastic problems, examples are found extensively where the completeness and orthogonal properties of such a system lead to attractive solutions. But with the application of digital techniques in these areas, awareness for other more general complete systems of orthogonal functions has developed. This "new" class of functions, though not possessing some of the desirable properties of sine–cosine functions, has other advantages to be useful in many applications in the context of digital technology. Many members of this class of orthogonal functions are piecewise constant binary valued, and therefore indicated their possible suitability in the analysis and synthesis of systems leading to piecewise constant solutions.

1.1 Orthogonal Functions and their Properties

Any time function can be synthesized completely to a tolerable degree of accuracy by using a set of orthogonal functions. For such accurate representation of a time function, the orthogonal set should be complete [1].

Let a time function f(t), defined over a time interval [0, T], be represented by an orthogonal function set Sn(t). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where, cn is the coefficient or weight connected to the nth member of the orthogonal set.

The members of the function set Sn(t) are said to be orthogonal in the interval 0 ≤ t ≤ if for any positive integral values of m and n, we have

∫T0 Sm(t)Sn(t)dt = δmn (a constant) (1.2)

where, δmn is the Kronecker delta and δmn = 0 for m ≠ n. When m = n and δmn = 1, then the set is said to be an orthonormal set.

An orthonormal set is said to be complete or closed if for the defined set no function can be found which is normal to each member of the set satisfying equation (1.2).

Since only a finite number of terms of the series Sn(t) can be considered for practical realization of any time function f(t), right-hand side (RHS) of equation (1.1) has to be truncated and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

When N is large, the accuracy of representation is good enough for all practical purposes. Also, it is necessary to choose the coefficients cn in such a manner that the mean integral squared error (MISE) is minimized. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

This is realized by making

cn = 1/T ∫T0 f(t)Sn(t)dt (1.5)

For a complete orthogonal function set, the MISE in equation (1.4) decrease monotonically to zero as N tends to infinity.

1.2 Different Types of Nonsinusoidal Orthogonal Functions

1.2.1 Haar functions

In 1910, Hungarian mathematician Alfred Haar proposed a complete set of piecewise constant binary-valued orthogonal functions that are shown in Fig. 1.1 [3,4]. In fact, Haar functions have three possible states 0 and [+ or -] A where A is a function of [square root of (2)]. Thus, the amplitude of the component functions varies with their place in the series.

An m-set of Haar functions may be defined mathematically in the semi-open interval t [member of] [0,1) as given below:

The first member of the set is

har(0, 0, t) = 1, t [member of] [0,1)

while the general term for other members is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, j, n, and m are integers governed by the relations 0 ≤ j ≤ log2(m), 1 ≤ n ≤ 2j. The number of members in the set is of the form m = 2k, k being a positive integer. Following the above equation, the members of the set of Haar functions can be obtained in a sequential manner. In Fig. 1.1, k is taken to be 3, thus giving m = 8.

Haar's set is such that the formal expansion of given continuous function in terms of these new functions converges uniformly to the given function.

1.2.2 Rademacher functions

In 1992, inspired by Haar, German mathematician H. Rademacher presented another set of two-valued orthonormal functions [5] that are shown in Fig. 1.2. The set of Rademacher functions is orthonormal but incomplete. As seen from Fig. 1.2, the function rad(n, t) of the set is given by a square wave of unit amplitude and 2n-1 cycles in the semi-open interval [0,1). The first member of the set rad(0, t) has a constant value of unity throughout the interval.

1.2.3 Walsh functions

After the Rademacher functions were introduced in 1922, around the same time, American mathematician J.L. Walsh independently proposed yet another binary-valued complete set of normal orthogonal function Φ, later named Walsh functions [6,7], that are shown in Fig. 1.3.

As indicated by Walsh, there are many possible orthogonal function sets of this kind and several researchers, in later years, have suggested orthogonal sets [8–10] formed with the help of combinations of the well-known piecewise constant orthogonal functions.

In his original paper Walsh pointed out that, "... Harr's set is, however, merely one of an infinity of sets which can be constructed of functions of this same character." While proposing his new set of orthonormal functions Φ, Walsh wrote "... each function Φ takes only the values +1 and -1, except at a finite number of points of discontinuity, where it takes the value...

Details zur Ausgabe

• Verlag: Anthem Press
• Erscheinungsdatum: 2011
• Sprache: Englisch
• ISBN 10: 0857289993
• ISBN 13: 9780857289995
• Einband: Gebundene Ausgabe
• Anzahl der Seiten: 170
• Kontakt zum Hersteller: preigu GmbH & Co. KG; mail@preigu.de; ; Lengericher Landstr. 19; Osnabrück; Niedersachsen; 49078; Deutschland