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Final EX 2005

# Final EX 2005 - EML4312 Spring 2005 Final Exam Name...

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EML4312: Spring 2005 Final Exam Name 1. (Stability Analysis - 15 points) Consider the following closed-loop block diagram where H FF = ( s 1) ( s 2) ( s 2 +6 s +9)( s +6) and H FB = K ( s 3) ( s 1) ( s 2) where K is a positive scalar constant. Determine the values of K that make the system stable. Figure 1: Solution: From the block diagram, the transfer function is H ( s )= ( s 1) ( s 2) ( s 2 s s +6)+ K ( s 3) Continued Fraction Expansion M = s 3 +12 s 2 +(45+ K ) s +54 3 K M 1 = s 3 K ) s M 2 =12 s 2 3 K M 1 M 2 = s 3 K ) s 12 s 2 3 K M 1 M 2 = 1 12 s + 1 μ 12 s 2 3 K (486 + 15 K ) s M 1 M 2 = 1 12 s + 1 μ 12 486 + 15 K s + μ 54 3 K (486 + 15 K ) s 1

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M 1 M 2 = 1 12 s + 1 μ 12 486 + 15 K s + 1 μ 486 + 15 K 54 3 K s If K> 0 then μ 12 486 + 15 K > 0 and 486 + 15 0 ,andi f 54 3 0 implies that K< 54 3 . Therefore, the system is stable for all K such that 0 54 3 =18 . 2
2. (Root Locus - 10 points) Determine the characteristic equation for a system with the following Root Locus: Solution: x x -1 Real-axis Imaginary-axis Figure 2: Root Locus From the root locus: M =1+ K 1 s ( s +1) = s ( s +1)+ K = s 2 + s + K 3

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3. (Lead Compensator Design - 20 points) For a system with open-loop poles at s =0 , 1
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Final EX 2005 - EML4312 Spring 2005 Final Exam Name...

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