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K 0 0 Σ Curtin University of Technology Department of Electrical and Computer Engineering CS301/CS603 Tutorial 6 1. Find necessary and sufficient conditions for all the roots of the equation s4 + a1s3 + a2s2 + a3s + a4 = 0 to lie in the open left half plane. Answer: The entries in the first column of the Routh array are 1, a1, a1a2 − a3 , a1 a3(a1a2 − a3) a1 − a1a4, a4 By Routh’s criterion, all the roots of the equation lie in the open left half plane if and only if all the above entries are positive. 2. Consider the closed-loop system shown in Figure 1. Determine the range of K for stability. Assume that K > 0. + R s− 2 Y (s+1)(s2 +6s+25) − Figure 1 Answer: The closed-loop transfer function is Y (s) = K (s − 2) From the Routh array R(s) s3 + 7s2 + (31 + K )s + 25 − 2K 1 31 + K 0 7 25 − 2K 0 192+9K 7 25 − 2K 0 0 it is seen that the range of K for ensuring closed-loop stability is 0 < K < 12.5. 3. Consider the servo system with tachometer feedback shown in Figure 2. Determine the ranges of stability for K and Kh. Note that Kh must be positive. Σ + − Σ + − K 20 (s+1)(s+4) Kh 1 s R Y Figure 2. Servo system with tachometer feedback. Answer: The closed-loop transfer function is Y (s) R(s) = 20K s3 + 5s2 + (4 + 20KKh)s+ 20K From the Routh array 1 4 + 20KKh 0 5 20K 0 4 + 20KKh − 4K 0 0 20K 0 0 the stability conditions are found to be K > 0 and Kh > K − 1 5K As Kh > 0, we are led to K > 0 and Kh > max ( 0, K − 1 5K )
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