Explanation Given : θ θ r e f = 53.176 4.6 s 2 + 31.281 s + 53.176 θ r e f ( t ) = 3 u ( t ) Concept Used: Use partial fraction expansion to obtain the amplitude of the sinusoids and residue value, which will tell us about the pole zero cancellation can be achieved or not. Use time domain response formulae for settling time, peak time to calculate values. Calculation: θ θ r e f = 53.176 4.6 s 2 + 31.281 s + 53.176 The input to the system is θ r e f ( t ) = 3 u ( t ) Thus, = > θ r e f ( s ) = 3 s θ ( s ) θ r e f ( s ) = 53.176 4.6 s 2 + 31.281 s + 53.176 And we have, = > θ r e f ( s ) = 3 s θ ( s ) = ( 53.176 4.6 s 2 + 31.281 s + 53.176 ) θ r e f ( s ) Substitute θ r e f ( s ) θ ( s ) = ( 53.176 4.6 s 2 + 31.281 s + 53.176 ) ( 3 s ) θ ( s ) = 1 s ( 159.528 4.6 s 2 + 31.281 s + 53.176 ) θ ( s ) = 159.528 s ( 4.6 s 2 + 31.281 s + 53.176 ) = A s + B s + C ( 4.6 s 2 + 31.281 s + 53.176 ) = A ( 4.6 s 2 + 31.281 s + 53.176 ) + s ( B s + C ) = 4.6 A s 2 + 31.281 A s + 53.176 A + B s 2 + C s = ( 4.6 A + B ) s 2 + ( 31.281 A + C ) s + 53.176 A On comparing both sides, we have = 53.176 A = 159.528 = > A = 3 = ( 4.6 A + B ) = 0 = > 4.6 × 3 + B = 0 = > B = − 13.8 = ( 31.281 A + C ) = 0 = > 31.281 × 3 + C = 0 = > C = − 93.843 θ ( s ) = 159.528 s ( 4.6 s 2 + 31.281 s + 53.176 ) = 3 s − 13.8 s + 93.843 ( 4.6 s 2 + 31.281 s + 53.176 ) θ ( s ) = 159.528 s ( 4.6 s 2 + 31.281 s + 53.176 ) = 3 s − 13.8 s + 93.843 ( 4.6 s 2 + 31.281 s + 53.176 ) θ ( s ) = 3 s − 13.8 s + 6.8 s 2 + 6.8 s + 11.56 Taking inverse Laplace transform on both sides, we get θ ( s ) = 3 s − 13.8 [ s s 2 + 6.8 s + 11.56 + 6.8 s 2 + 6.8 s + 11.56 ] θ ( s ) = 3 s − 13.8 [ s − 3.4 ( s − 3.4 ) 2 + ( 11.56 ) 2 + 6

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ISBN: 9781119185666

Author: NISE

Publisher: WILEY

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Chapter 4, Problem 28P

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A sketch and indicate Cfinal,Cmax,TP,Ts in the plot for θref(t)=3u(t) .

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Chapter 4, Problem 26P

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Chapter 4 Solutions

CONTROL SYSTEMS ENGINEERING

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A sketch and indicate C f i n a l , C max , T P , T s in the plot for θ r e f ( t ) = 3 u ( t ) .

A sketch and indicate Cfinal,Cmax,TP,Ts in the plot for θref(t)=3u(t) .

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Chapter 4 Solutions