Control ENG HW_Part_61

# Control ENG HW_Part_61 - will be negative or zero is untrue...

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0 60 25 2 0 120 50 4 2 0 120 40 10 4 2 0 10 . 2 ). 3 ( 2 ) 10 4 2 ( 2 3 2 3 2 3 2 = + + + = + + + = + + + + = + + + + s s s s s s s s s s s s s s Routh Array 1 25 2 60 -10/2 The system is unstable at Kc = 2. 14.4 Prove that if one or more of the co-efficient (a 0 ,a 1 ,….a n ) of the characteristic equation are negative or zero, then there is necessarily an unstable root Characteristic equation : 0 .... .......... .......... .......... 1 1 0 = + + + - n n n a x a x a 0 ) / ... .......... .......... / ( 0 1 0 1 0 = + + - a a x a a x a n n n 0 ......... .......... .......... , 0 ) .( .......... )......... )( ( 2 1 2 1 0 < = - - - n n have We x x x a α As we know the second co-efficient a 1 /a 0 is sum of all the roots 2 / ) 1 ( 1 1 2 0 1 - = = = n j j i n i a a Therefore sum of all possible products of two roots will happen twice as 2 1 dividing the total by 2. And 0 0 / ) 0 0 ( 0 2 0 2 > > < < > a a a j i j i Similarly

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) ,....... 1 ( 0 0 / 0 0 ) 1 ( ) 1 ( 0 / 0 1 ) 1 ( ) ( ) 1 ( 0 0 0 0 n j for a so a a case both in again is a a so is sum the and is odd j if a a so is sum the and is even j if roots j of products possible aoll of sum a a j j j j j j j j = > > > < - - = > > - = - = 14.5 Prove that the converse statement of the problem 14.4 that an unstable root implies that one or more co-efficient
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Unformatted text preview: will be negative or zero is untrue for all co-efficient ,n&gt;2. Let the converse be true, always .Never if we give a counter example we can contradict. Routh array 2 3 2 3 1 3 1 2 1 3 2 s s s s s s s-+ + + System is unstable even when all the coefficient are greater than 0; hence a contradiction, 14.6 Deduce an expression for Routh criterion that will detect the Presence of roots with real parts greater than for any rectified &gt;0 Characteristic equation ... .......... .......... .......... 1 1 = + + +-n n n a x a x a Routh criteria determines if for any root, real part &gt; 0. Now if we replace x by X such that...
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## This note was uploaded on 11/13/2011 for the course COP 4355 taught by Professor Koslov during the Spring '10 term at University of Florida.

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Control ENG HW_Part_61 - will be negative or zero is untrue...

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