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EE_302_2010_HW4

# EE_302_2010_HW4 - C s 1 s 6 s 2-4 s 8(a Let us use...

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EE302 HW4 Due: 8 April 2010 Q1. Sketch root loci (by obtaining all relevant features) as K : 0 → ∞ for the unity feedback closed-loop systems with open-loop transfer functions given below. (a) G ( s ) = K s + 3 s ( s 2 + 2 s + 2) (b) G ( s ) = K s + 2 s 4 - 1 (c) G ( s ) = K ( s + 1) 2 s 3 + 8 Q2. Consider the figure given below where N ( s ) and D ( s ) are polynomials with real coeﬃcients and K > 0. K N ( s ) D ( s ) Let s 1 ( K ) denote a closed-loop pole for a given K . Let arg( z ) denote the argument of a complex number z . That is, if arg( z ) = θ then z = | z | (cos θ + j sin θ ). Let t s ( K ) denote the settling time (if exists) for the unit-step response of the closed loop. Let d N and d D denote the degree of polynomials N ( s ) and D ( s ), respectively. Figure out whether the below given scenarios are possible or not. When impossible, explain; otherwise provide an example that would realize the scenario. (a) lim K →∞ arg( s 1 ( K )) = 72 . (b) lim K →∞ arg( s 1 ( K )) = 72 . (c) d N = 73, d D = 83, and lim K →∞ t s ( K ) = 93sec. (d) Suppose d N = 0. For each K 1 > 0 there exists K 2 and K 3 satisfying K 3 > K 2 > K 1 such that t s ( K 2 ) exists but t s ( K 3 ) does not. Q3. Consider the system below.

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Unformatted text preview: C ( s ) 1 ( s + 6)( s 2-4 s + 8) (a) Let us use proportional control, i.e., C ( s ) = K . Sketch the root locus plot. (No need to ﬁnd the departure angles.) Can we stabilize this system with proportional control? (b) Let us now try proportional derivative control, i.e., C ( s ) = K (1 + K d s ). Find the range of K d such that for each K d in that range we can ﬁnd some K > 0 to stabilize the system. (c) Let C ( s ) be a PD controller with K d = 1. Sketch the root locus plot. Find the value of K for which system displays sustaining oscillations. Determine the frequency of oscillations. Q4. Consider the below system. Take m = 1kg and b = 1Ns/m. Sketch the root locus plot as the spring gets stiﬀer, i.e., k : 0 → ∞ N/m. What can we say about the relation between the settling time and the stiﬀness of the spring? k x f m f x b...
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EE_302_2010_HW4 - C s 1 s 6 s 2-4 s 8(a Let us use...

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