By K B Datta, B M Mohan

ISBN-10: 9810218893

ISBN-13: 9789810218898

Normally, box conception is taught via canonical quantization with a heavy emphasis on excessive strength physics. notwithstanding, the recommendations of box thought are acceptable in addition and are generally utilized in a number of different parts of physics reminiscent of consdensed subject, nuclear physics and statistical mechanics. the trail imperative technique brings out this option so much basically. during this e-book, the trail fundamental process is constructed intimately thoroughly in the context of quantum mechanics. hence, it truly is utilized to varied parts of physics Ch. 1. Orthogonal services in platforms and regulate: A historic point of view -- Ch. 2. Least Squares Approximation of indications -- Ch. three. sign Processing in non-stop Time area -- Ch. four. research of Time-Delay structures -- Ch. five. id of Lumped Parameter structures -- Ch. 6. id of disbursed Parameter structures -- Ch. 7. id of Linear Time-Varying and Nonlinear allotted Parameter structures -- Ch. eight. optimum keep watch over of Linear structures

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Additional resources for Orthogonal Functions in Systems and Control

Example text

Tii (t) r + 2ru (l) = 0 r w h i c h w h e n differentiated r times results in the differential equation 2 l - (2i)rz4 +2) (1 - t )u ; (t) + 2 ( r - l)tu \t) r + 1 ) r) (i) - r(r - l)u (t) r + 2 r ( r - l ) u < ( i ) + 2ru \t) T+1 r ) r r T = 0 where we have used L e i b n i t z f o r m u l a for the r t h d e r i v a t i v e of a p r o d u c t , v i z . « _ = = (-l)"(- ) a 1 - 2 - m (a) = t w 1 -2--m a ( a + 1) • - . ( a + fc - l ) , ( a ) 0 = 1 and a is a real n u m b e r . 48) w h i c h is k n o w n as Legendre differential equation.

As 7 ' s are positive, we must have m = 0 , m = 0 , 1 , . . , r - 2. 28 follows f r o m E q . 2 . 3 1 . r-l(<) where g , - - ^ ) is a p o l y n o m i a l of degree at most r — I . 28 by w(t)(j> _ (t) r and t a k i n g E q . 2 . 32 into cosideration, we 1 integrate this p r o d u c t over [*o,*/] to get 0 = a f'tw{t) _ (t) {t)dt = a T T T I Io l 0r- w{t) flr7rA- + r c T T i - {t) + g _ , ( t ) 4> (t)dt + c T + c ', _ r r r 7 r _! 33 respecively is k n o w n as the three-term recurrence relation for a set of o r t h o g o n a l polynomials.

T h e i n t e r v a l [ i , t / ] over w h i c h each class is defined depends on the n a t u r e of the b o u n d a r y of a physical p r o b l e m and the mean integral square error is a g u i d i n g factor i n selecting the w e i g h t i n g f u n c t i o n w(t). We shall now t r y to get a solution t o the following p r o b l e m : P r o b l e m Given \to,tj) and w(t), determine the set of orthogonal polynomials 4> (t)'s of degree r,r = 0 , 1 , . . , n . w h i c h w i l l m i n i m i z e the i n t e g r a l square error i n representing an a r b i t r a r y square integrable f u n c t i o n .

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Orthogonal Functions in Systems and Control by K B Datta, B M Mohan


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