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Chapter 4 examples - EEL4657 Dr Haniph Latchman Chapter 4...

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EEL4657 - Dr. Haniph Latchman Chapter 4: Example Problems 1. The transfer function for a system with unity negative feedback is given by: G ( s ) = 1 ( s + 1)( s + 3) a) Find the closed-loop transfer function T ( s ) = Y ( s ) R ( s ) . b) Find the time response y ( t ) for a step input of r ( t ) = A. c) Use the final value theorem to find the steady-state value of y(t). d) Find the steady-state error. Solution a) We can use the formula for the closed-loop transfer function given in sec- tion 4.4.1 in the book. T ( s ) = G ( s ) 1 + G ( s ) = 1 ( s +1)( s +3) 1 + 1 ( s +1)( s +3) = 1 ( s + 1)( s + 3) + 1 = 1 s 2 + 4 s + 3 + 1 = 1 s 2 + 4 s + 4 T ( s ) = 1 ( s + 2) 2 1
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b) The output Y ( s ) is given by Y ( s ) = R ( s ) T ( s ) and we use the fact that the Laplace transform of the step input r ( t ) = A is R ( s ) = A s . Y ( s ) = R ( s ) T ( s ) = A · 1 s · 1 ( s + 2) 2 Y ( s ) = A 1 s ( s + 2) 2 Use partial fractions method to transform Y ( s ), we can remove the term A for now, realizing that it is a constant and we can include it later in the final equation Y ( s ) = 1 s ( s + 2) 2 = a s + b s + 2 + c ( s + 2) 2 by multiplying both sides by s ( s + 2) 2 and combining the like terms, we can solve for a , b , and c
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