ABSTRACT
The main objective of the thesis entitled
"Some Aspects of Analysis and Identification of Continuous Dynamical Systems via Orthogonal Functions" is to explore the potentialities of orthogonal polynomials and Sine-Cosine functions in (a) analysis of linear time-invariant continuou systems containing time-delay(s) (b) identification of
linear time- invariant single-input single-output lumped parameter systems and (c) identification of linear time- invariant distributed parameter systems of order one and two and then to compare all the orthogonal function approaches applied to above three major problems to assess the relative merits and demerits of each approach in each problem. The
important contributions in each chapter of this thesis containing four chapters in all are as follows •
Chapter I s Orthogonal Functions/ Signal Representations and Time- Domain Ope r a t i o n s .
This chapter develops some mathematical tools necessary to investigate the problems considered in the subsequent chapters* The necessity of 'shift* for infinite range orthogonal polynomials such as Laguerre and Hermite polynomials is stressed and demonstrated its usefulness in representation of square-integrable functions by these
polynomials. Uncertainity of representation of signals characterized by square-integrable functions over a finite range by infinite range orthogonal polynomials is brought
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out* Considering the inherent filtering features of orthogonal functions, it is demonstrated that Chebyshev second kind and Legendre polynomials and Sine~Cosine
functions are better in this respect than Block-Pulse funct
ions while Chebyshev polynomials of first kind can be used only to <3terministic situations and not to stochastic
situations in view of nonavailability of appropriate numeri
cal integration technique to carry out the Chebyshev first kind spectrum in noisy environment* The error introduced by integration operational matrix which converts integral calculus to linear algebra approximately in the sense of least squares is analysed for each orthogonal system. The derivative operational matrix which can reduce differential calculus to linear algebra is introdticed for Chebyshev
second kind, Legendre, Laguerre and Hermite polynomial systems and for Sine~Cosine functions as it is not yet available for these orthogonal systems-
Chapter II • Analysis of Time-Delay Systems
This chapter deals with analysis of linear time- invariant continuous dynamical systems containing single
delay, multi-delay, or piecewise constant delay using delay operational matrix approach and time-partition technique.
A unified approach to obtain delay operational matrix of any class of orthogonal polynomial systems or of Sine~
Cosine functions from its derivative operational matrix is
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developed* For polynomial systems, the delay-integration operational matrix is obtained from the product of the
above delay operational matrix and conventional integration operational matrix- Since for Sine~Cosine functions this property does not exist, a new integration operational
matrix having a close relation with the conventional integra
tion operational matrix is developed for these basis functi
ons -
Using the above operational matrices, three new algorithms for the analysis of systems containing delay in both state and control are proposed. The first two algori
thms are to be used via Block-Pulse functions only while the third algorithm is to be used via orthogonal polynomials or Sine-Cosine functions- It is shown that infinite range orthogonal polynomials must be applied cautiously for the analysis of delay systems via the third algorithm.
In order to analyse single-delay, multi-delay or piecewise constant delay systems via Block-Pulse functions and orthogonal polynomials, a powerful technique called time-partition method which merely relies on integration operational matrix is introduced* This time-partition technique eliminates all the difficulties mentioned above for the analysis of delay systems- Finally, the analogy between time-partition method and single-term piecewise constant basis functions approach is discussed-
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C hapter I I I . : Identification of Lumped Parameter Systems
This chapter deals with identification of parameter and initial conditions of linear time- invariant single~input single-output continuous dynamical systems-
The concept of 050MRI is extended to all classes of orthogonal polynomials and Sine~Cosine functions- It is
shown that OSGMRI does not exist for infinite ranee ortho
gonal polynomials. Making use of OSCMRI concept a unified identification algorithm is developed which can be used in conjunction with any class of orthogonal systems to estimate parameters, and sometimes depending upon the situation,
initial conditions in noise-free and noisy environments- It is found Hermite polynomial approach for system identifica
tion is computationally not attractive and impractical in some sense while Fourier series approach is computationally la b o r i o u s .
Chapter IV
'•
Identification of Distributed Parameter SystemsThis chapter deals with identification of parameters as well as initial and boundary conditions of linear time- invariant continuous time distributed parameter systems of order one and two.
A unified approach for the identification of the most general second-order distributed parameter system is
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presented by using the concept of OSOMRI . Depending upon the order of differentiation, three different models are derived from the general model by setting certain parameter(
of it equal to zero and the corresponding sets of identi- fication equations are deduced directly from those of the general model.
The practical limitations of the existing and the proposed identification approaches are discussed. It is observed that the identification scheme based on orthogonal functions simply fails under certain conditions. To
detect these conditions and to avoid the same, a linear
independence test for the columns of linear algebraic system resulting from the original partial differential equation model is suggested. Further the conditions under which
the proposed unified identification approach via Slock-Pulse functions works successfully are established. I*t_is pointed out that identification via Haar or Walsh functions is not possible under the conditions when Block-Pulse approach fail
