Control ENG HW_Part_7

Control ENG HW_Part_7 - = Y ( s) = = s +1 s + 2s + 5 2 s +1...

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Unformatted text preview: = Y ( s) = = s +1 s + 2s + 5 2 s +1 ( s + 1) 2 + 4 s +1 = L−1 (Y ( s )) = L−1 2 ( s + 1) + 4 using the table,we get Y (t ) = e − t Cos2t b) Y ( s ) = Y ( s) = s 2 + 2s s4 1 2 +3 2 s s Y(t)= L−1 (Y ( s )) = t + t 2 c) Y ( s ) = = = 2s ( s − 1) 3 2s − 2 + 2 ( s − 1) 3 2 2 + 2 ( s − 1) ( s − 1) 3 2 2 Y (t ) = L−1 + L−1 2 ( s − 1) 3 ( s − 1) t2 t 2(te + e = e t (t 2 + 2t ) 2 t = 3.7a) Y ( s) = As + B Cs + D 1 =2 + ( s 2 + 1) ( s + 1) ( s 2 + 1) thus ( As + B ) + (Cs + D )( s 2 + 1) = 1 = Cs 3 + Ds 2 + ( A + C ) s + ( B + D ) = 1 C=0,D=0 Also A=0;B=1 1 1 A B C D Y ( s) = 2 = = + + + 2 2 2 2 ( s + 1) (s + i) (s − i) (s + i) (s + i) (s − i) (s − i)2 A( s + i )( s − i ) 2 + B( s − i ) 2 + C ( s − i )( s + i ) 2 + D( s + i ) 2 = 1 ( A + C ) s 3 + ( − Ai + B + Ci + D ) s 2 + ( A − 2 Bi + C + 2 Di ) + ( − Ai − B + Ci − D ) = 1 Thus,A+C=0 -Ai+B+Ci+2Di=0 ; B=D A-2Bi+C+2Di=0 -Ai-B+Ci-D=1 Also D=-Ci;B=-Ci, A=-C,C=-i/4 A=i/4 ; B=-1/4; D=-1/4 Y ( s) = i/4 − 1/ 4 −i/4 − 1/ 4 + + + 2 (s + i) (s + i) (s − i) (s − i)2 Y (t ) = Y (t ) = − 1/ 4 −i/4 −1/ 4 i/4 + + + 2 (s + i) (s + i) ( s − i) ( s − i) 2 −1/ 4 −i/4 −1/ 4 i/4 + + + 2 (s + i) (s + i) (s − i) (s − i)2 ...
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