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MEM355W07-08.HW2.soln_stud

# MEM355W07-08.HW2.soln_stud - MEM355 PERFORMANCE ENHANCEMENT...

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Unformatted text preview: MEM355: PERFORMANCE ENHANCEMENT OF DYNAMIC SYSTEMS Home work #2 solution Problem 1: Unity (negative) feedback system, with the loop transfer function 2 450( 8)( 12)( 15) ( ) ( 38)( 2 28) s s s G s s s s s + + + = + + + Assume the closed loop system is stable. (we need to verify this). This is a type-1 system (presence of one integrator). Hence, it will yield finite error for type-1 (t 1 ) input, zero errors for inputs of lower types, and infinite errors for higher type inputs. The (finite) errors could be evaluated using the error constants, K i . Input Error constant error 25 U(t) ) ( lim ( ) or p s K G s G = = ( ) 1 1 or p e K = = + 37 tU(t) 1 ( ) 450(8)(12)(15) lim ( ) (38)(28) s or v K sG s G = = ) 1 ( 1 1 (38)(28) 450(8) 37 (12)(15) or v e K = = 94 (t 2 /2) U(t) ) 2 ( 2 lim ( ) o s r a K s G s G = = ( ) 2 2 1 or a e K = = Alternately, one could compute the closed loop transfer function, and E(s), the laplace of the error, use the final value theorem to determine the error in time-domain. Such as…. 2 ( ) ( ) 1 1 ( ) 450( 8)( 12)( 15) ( ) 1 ( ) 1 1 ( 38)( 2 28) cl cl C s G E s G s s s s R s G R s G s s s s = = = = + + + + + + + + + 2 2 ( 38)( 2 28) ( ) ( ) ( 38)( 2 28) 450( 8)( 12)( 15) s s s s E s R s s s s s s s s + + + = + + + + + + + (check stability: It is stable, with roots at {-455, -13.34 ± 4.5i, -7.2} . Then by FVT (final value theorem), lim ( ) s e sE s G G = . Thus.....
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MEM355W07-08.HW2.soln_stud - MEM355 PERFORMANCE ENHANCEMENT...

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