EE_302_2010_HW4_soln - EE302 HW4 Solution Q1 Sketch root loci(by obtaining all relevant features as K 0 → ∞ for the unity feedback closed-loop

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Unformatted text preview: EE302 HW4 Solution 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 Sol’n. (a) We can write the characteristic equation (of the closed loop) as 1 + K ( s + 3) s ( s + 1 + j )( s + 1- j ) = 0 which has three roots. Therefore the root locus diagram (RLD) will have three branches. Open loop has m = 1 zero and n = 3 poles. Three branches will start from open-loop poles and two of them will sail to infinity along the asymptotes and the remaining one will approach the zero at s =- 3 as K → ∞ . Asymptotes will have angles ϕ =- 180 ◦ (2 ℓ + 1) n- m = ± 90 ◦ . Asymptotes will meet at the centroid σ ◦ which we calculate as σ ◦ = ∑ n i =1 p i- ∑ m i =1 z i n- m = (0- 1- j- 1 + j )- (- 3) 3- 1 = 1 2 where p i denotes open-loop pole and z i open-loop zero. Angle condition reveals that RLD will reside on the real axis only at the interval s ∈ [- 3 , 0]. To find where (if) the RLD crosses jω axis we exploit the characteristic equation. 0 = s 3 + 2 s 2 + ( K + 2) s + 3 K s = jω =- jω 3- 2 ω 2 + j ( K + 2) ω + 3 K = (- 2 ω 2 + 3 K )- j ( ω 3- ( K + 2) ω ) which implies- 2 ω 2 + 3 K = 0 ω 3- ( K + 2) ω = 0 whence either ω = ± √ 6 with K = 4 or ω = 0 with K = 0. Finally, we compute the departure angles. From the open loop pole p =- 1 + j RLD will depart by ψ which we obtain from ± 180 ◦ (2 ℓ + 1) = ∠ ( p + 3)- ( ∠ ( p + 0) + ∠ ( p + 1 + j ) + ψ ) = arctan(1 / 2)- (135 ◦ + 90 ◦ + ψ ) .-3.5-3-2.5-2-1.5-1-0.5 0.5-15-10-5 5 10 15 Root Locus Real Axis Imaginary Axis Figure 1: Root locus diagram for Q1(a). We compute ψ ≈ - 18 ◦ . Due to symmetry with respect to real axis, the departure angle from the conjugate pole p =- 1- j will be- ψ . This is all we have to have figured out to plot RLD for this example. The range of K for stability is 0 < K < 4....
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This note was uploaded on 01/30/2011 for the course EE 302 taught by Professor Erkmen during the Spring '10 term at Middle East Technical University.

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EE_302_2010_HW4_soln - EE302 HW4 Solution Q1 Sketch root loci(by obtaining all relevant features as K 0 → ∞ for the unity feedback closed-loop

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