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

Deb, Anish; Sarkar, Gautam; Sengupta, Anindita

Hardcover

ISBN 10: 0857289993 ISBN 13: 9780857289995

Publisher: Anthem Press, 2011

Synopsis

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.

About this title

About the Author

Anish Deb is Professor at the Department of Applied Physics, University of Calcutta, India.

Gautam Sarkar is the Labonyamoyee Das Professor at the Department of Applied Physics, University of Calcutta, India.

Anindita Sengupta is Senior Lecturer at the Department of Electrical Engineering, Bengal Engineering and Science University, India.

Excerpt. © Reprinted by permission. All rights reserved.

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 zero."

The Rademacher functions were found to be a true subset of the Walsh function set. The Walsh function set possesses the following properties all of which are not shared by other orthogonal functions belonging to the same class. These are:

(i) Its members are all two-valued functions,

(ii) It is a complete orthonormal set.

(iii) It has striking similarity with the sine-cosine functions, primarily with respect to their zero-crossing patterns.

1.2.4 Block pulse functions (BPF)

During the 19th century, the most important function set used for communication was block pulses. Voltage and current pulses, such as Morse code signals, were generated by mechanical switches, amplified by relays and detected by a variety of magnetomechanical devices. However, until recently, the set of block pulses received less attention from the mathematicians as well as application engineers possibly due to their apparent incompleteness. But disjoint and orthogonal properties of such a function set were well known. A set of block pulse functions [11–13] in the semi-open interval t [member of] [0, T) is shown in Fig. 1.4.

An m-set block pulse function is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, i = 0,1, 2, ..., (m - 1).

The block pulse function set is a complete [14] orthogonal function set and can be normalized by defining the component functions in the interval [0, T) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1.2.5 Slant functions

A special orthogonal function set known as the slant function set was introduced by Enomoto and Shibata [15] for image transmission analysis. These functions are also applied successfully to image processing problems [16].

Slant functions have a finite but a large number of possible states as can be seen from Fig. 1.5. The superiority of the slant function set lies in its transform characteristics, which permit a compaction of the image energy to only a few transformed samples. Thus, the efficiency of image data transmission in this form is improved. This is because the slant transform is designed to posses the following important properties [17]:

(i) constant basis vector,

(ii) slant basis vector (monotonically decreasing in constant size steps from maximum to a minimum amplitude),

(iii) sequency property,

(iv) fast computational algorithm,

(v) high-energy compaction.

1.2.6 Delayed unit step functions (DUSF)

Delayed unit step functions, shown in Fig. 1.6, were suggested by Hwang [18] in 1983. Though not of much use due to its dependency on BPFs, shown by Deb et al. [13], it certainly deserves to be included in the record of piecewise constant basis functions as a new variant.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

1.2.7 General hybrid orthogonal functions (GHOF)

So far the discussion centered on different types of orthogonal functions having a piecewise constant nature. The major departure from this class was the formulation of general hybrid orthogonal functions (GHOF) introduced by Patra and Rao [19–21]. While sine-cosine functions or orthogonal polynomials can represent a continuous function quite nicely, these functions/polynomials become unsatisfactory for approximating functions with discontinuities, jumps or dead time. For representation of such functions, undoubtedly piecewise constant orthogonal functions such as Walsh or block pulse functions, can be used more advantageously. But with functions having both continuous nature as well as a number of discontinuities in the time interval of interest, it is quite clear that none of the orthogonal functions/polynomials of continuous nature, or, for that matter, piecewise constant orthogonal functions are suitable if a reasonable degree of accuracy is to be achieved. Hence, to meet the combined features of continuity and discontinuity in such situations, the framework of GHOF proposed by Patra and Rao seemed to be more appropriate. Thus, the system of GHOF forms a hybrid basis which is both flexible and general.

Applications of GHOFs were sucessfully made by Patra and Rao in analyzing linear time invariant systems, converter fed and chopper fed dc motor drives, and in the area of self-tuning control. The main disadvantage of GHOF seems to be the required a priori knowledge about the discontinuities in the function, which are to be coincided with the segment boundaries of the system of GHOF to be chosen. This also requires a complex algorithm for better results.

1.2.8 Variants of block pulse functions

In 1995, a pulse-width modulated version of the block pulse function set was presented by Deb et al. [22,23] where, the pulse-width of the component functions of the BPF set was gradually increased (or, decreased) depending upon the nature of the square integrable function to be handled.

In 1998, a further variant of the BPF set was proposed by Deb et al. [24] where, the set was called sample and hold function (SHF) set and the same was utilized for the analysis of sampled data systems with zero-order hold.

1.3 Walsh Functions in Systems and Control

Among all the orthogonal functions outlined earlier in the chapter, Walsh function based analysis first became more attractive to the researchers from 1962 onwards [7, 25–27]. The reason for such success was mainly due to its binary nature. One immediate advantage is the task of analog multiplication. To multiply any signal by a Walsh function, the problem reduces to an appropriate sequence of sign changes, which makes this usually difficult operation both simple and potentially accurate [25]. However, in system analysis, Walsh functions were employed during early 1970s. As a consequence, the advantages of Walsh analysis were unraveled to the workers in the field compared to the use of conventional sine-cosine functions. Ultimately, the mathematical groundwork of the Walsh analysis became strong to lure interested researchers to try every new application based upon this function.

In 1973, it was Corrington [28] who proposed a new technique for solving linear as well as nonlinear differential and integral equations with the help of Walsh functions. In 1975, important technical papers relating Walsh functions to the field of systems and control were published. New ideas were proposed by Rao [29–35,37] and Chen [38–43]. Other notable workers were Le Van et al. [44], Tzafestas [45], Chen [46–49], Mahapatra [50], Paraskevopoulos [51], Moulden [52], Deb and Datta [53–55], Lewis [56], Marszalek [57], Dai and Sinha [58], Deb et al. [59–62], and others.

The first positive step for the development of the Walsh domain analysis was the formulation of the operational matrix for integration. This was done independently by Rao [29], Chen [38], and Le Van et al. [44]. Le Van sensed that since the integral operator matrix had an inverse, the inverse must be the differential operator in the Walsh domain. However, he could not represent the general form of the operator matrix that was done by Chen [38,39]. Interestingly, the operational matrix for integration was first presented by Corrington [28] in the form of a table. But he failed to recognize the potentiality of the table as a matrix. This was first pointed out by Chen and he presented Walsh domain analysis with the operational matrices for integration as well as differentiation:

(i) to solve the problems of linear systems by state space model [38];

(ii) to design piecewise constant gains for optimal control [39];

(iii) to solve optimal control problem [40];

(iv) in variational problems [41];

(v) for time domain synthesis [42];

(vi) for fractional calculus as applied to distributed systems [43].

Rao used Walsh functions for:

(i) system identification [29];

(ii) optimal control of time-delay systems [31];

(iii) identification of time-lag systems [32];

(iv) transfer function matrix identification [34] and piecewise linear system identification [35];

(v) parameter estimation [36];

(vi) solving functional differential equations and related problems [37].

(Continues...)

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Edition details

• Publisher: Anthem Press
• Publication date: 2011
• Language: English
• ISBN 10: 0857289993
• ISBN 13: 9780857289995
• Binding: Hardcover
• Edition number: 1
• Number of pages: 168