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PracticeProblemsMidterm_solu

# K s a in series with gs in a unity s b 1 ignoring

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Unformatted text preview: r the unit step input. 34. Consider the second-order system G( s ) = 1 s 2 + 2⇣ s + 1 We would like to add a transfer function of the form D(s) = feedback structure. K ( s + a) in series with G(s) in a unity( s + b) (1) Ignoring stability for the moment, what are the constraints on K , a, and b so that the system is Type 1? (2) What are the constraints placed on K , a, and b so that the system is both stable and Type 1? 9 Solution: (1) In a unity feedback structure, E (s) 1 = R (s) 1 + GD To be Type 1, there needs to be a pole at s = 0 in the product GD. Since there is no such pole in G, it must be supplied by D, thus, the answer is b = 0. (2) To assure stability, all poles of the closed loop must be in the left half plane, for which the criterion is by Routh. Thus the characteristic equation is s(s2 + 2⇣ s + 1) + K (s + a) = 0 and the Routh array is 1 2⇣ 2⇣ (1 + K ) 2⇣ aK 1+K aK aK Thus the requirements are ⇣>0 2⇣ (1 + K ) aK > 0 aK > 0 35. Consider the transfer function 1 s2 + s + 2 H (s) = Which of the following is the corresponding unit step response? Explain your choice and why you rejected the other three possibilities. (a) (b) 0.7 (c) 0.8 0.6 (d) 1.4 1 1.2 0.6 0.5 0.8 1 0.4 0.4 0.8 0.6 0.3 0.2 0.6 0.4 0.2 0.4 0 0.1 0 0 5 10 −0.2 0.2 0.2 0 5 10 0 0 5 10 0 0 5 10 Solution: The DC gain of H (s) equals 0.5. so (c) and (d) are immediately rejected because it corresponds to DC gain of 1. The plot (b) shows the correct DC gain, but it has undershoot which indicates the presence of a RHP zero, while H(s) has no zeros. The plot (a) is the correct one. 10...
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