If a second-order system is overdamped, it is more diffi
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If a second-order system is overdamped, it is more difficult to determine the parameters z and t experimentally. One met
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If a second-order system is overdamped, it is more difficult to determine the parameters z and t experimentally. One method for determining the parameters from a step response has been suggested by R. C. Oldenbourg and H. Sartorius ( The Dynamics of Automatic Controls. ASME, p. 78, 1948), as described below. ( a ) Show that the unit-step response for the overdamped case may be written in the form S t r e r e r r r t r t ( ) 1 1 2 1 2 2 1 where r1 and r2 are the (real and negative) roots of t zt 2 2 s s 2 1 0 ( b ) Show that S ( t ) has an inflection point at t r r r r i ln 2 1 1 2 / / ( ) ( c ) Show that the slope of the step response at the inflection point dS t dt S t t ti i ( ) ( ) has the value S t r e r e r r r i r ti r ti r r ( ) ??? ??? 1 2 1 2 1 1 2 1 1 /( r2) ( d ) Show that the value of the step response at the inflection point is S t r r r r A B i i 1 1 2 S t 1 2 ( ) and that hence 1 1 1 1 2 S t S t r r i ( i ) ( ) ( e ) On a typical sketch of a unit-step response, show distances equal to 1 1 S t S t i S t i i ( ) ( ) and ( ) and hence present two simultaneous equations resulting from a graphical method for determination of r1 and r2 . ( f ) Relate z and t to r1 and r2 .
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