A#22_Solutions - Assignment #22 - Solutions — p.1...

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Unformatted text preview: Assignment #22 - Solutions — p.1 ECSE—2410 Signals & Systems — Spring 2009 Due Tue 04/21/09 1(20). Find the closed loop system Y (S) X (S) in terms of transfer functions A(s), B(s), C(s), D(s). transfer function, , expressed amalgam—g My; 1 a; a 52‘ ZAP’SICAgtADi-CBHB =7} 3, 1+AB+AI>+BL 2(45). In each case, sketch the root locus plots (K>0) for the following open loop transfer functions for unity feedback systems. Find the following, if they exist. Breakaway points and points of re—entry. Direction of travel of the roots as K varies from 0 to infinity. All asymptotes, asymptote angles and asymptote locations (centroids). Any j waxis crossings. <a>G<s>=S+1‘ P%Ezi-:7 i W»:@ 136°; A50 media Caz/QC, 4L 2 S 6m Cauwka BI? “Lg: ” (:23Z g2” 65:.) ~21»; {re-L 9» H Ca so Tag mafia? (110564th syslem rs avjlgflgyyfl Wag-2.. Assignment #22 — Solutions - p.12 ECSE—2410 Signals & Systems — Spring 2009 Due Tue 04/21/09 2. Continued. 1 k (b) G(S) = =- 5(52 + 5+1) SECS+iZ 1+ Assignment #22 - Solutions - p.3 ECSE~2410 Signals & Systems — Spring 2009 Due Tue 04/21/09 2 2. Continued. (c) G(S) : (5 +1) Wars-i =7 .1 gage? (90°, M5 024 +9 @0ka AL. .4]; 3i? . , ’3 FF, 1 :- “ “L 2 ~— S 2 W 69 @9 2 —- gig ; fl [[3150 332” 35 249-01: O #> 3(51-1) «2,5,4; 4'8 (PM! 6L 952m: W S} (“27) 2 l 9 __fl . ‘ fl 3 ‘- 3 cans x€1\@s-5lwf K“ GAD; Lt } Szm’g w~m0§s: Spa 1 Z}; r”? 3a+l==O -— 3:51} 37' A6 /?’<‘ % walk :1 Assignment #22 - Solutions - p.4 ECSE-2410 Signals & Systems - Spring 2009 Due Tue 04/21/09 2. Continued. Note. (a) G(s) : S :1 . This problem is simple enough that we can solve it analytically. s Root Locus 1 num==[l l]; O 8 den=[1 0 0]; ' sys=tf(num,den); 0.6 k=[0: .01: 50]; rlocus(sys,k) 0-4 axis([-2.5 0 —l 1]) E Real Axis 0'2 fl : O E E -2.5 -0.2 -0.4 —0.6 -O.8 -1 Note. We can find the exact equation of the locus for simple geometric figures such as this one. KQ+U 2 S Char. equation. 1+ KG(S)H1(S) = 0 21+ => sz + Ks + K = 0 In the complex plane let s = x + jy, and subs. back into char. equation. (x+ jy)2 +K(x+ jy)+ K =0 Now equate the real and imaginary parts separately. xZ—y?+K»+K=0 2w+mzo Solve the second equation for K and substitute into the first equation. x2 + 2x + y2 = 0 Next complete the square x2+2x+1—1+y2=0 OI‘ (x+l)2+y2 =1, which is a circle of radius 1, centered at (x = -l, y = 0). Assignment #22 — Solutions - p.5 ECSE—2410 Signals & Systems - Spring 2009 Due Tue 04/21/09 3(15). The proportional-plus—integtral (PI) controller shown is used to improve the steady-state error “PI controller” “plant” Y(s) K of the feedback system shown by integrating out any residual error signal. Suppose ~—’ 2 0.1. KP (a) Sketch the root locus for varying Kp. KGZQ: (an Qggpkgg .3. 54:58:34) Esrlome» ~l (b) Find the value of Kp needed to place a pole of the closed loop at s=-5. 6&9»??? tagger.” genesis-m) =0 $L$+z) e MSW ‘ sen) W= 80 "El '3 Hf “‘ ll” "” 0530M *3 ‘/ (p g I (c) What is the steady—state error for the value of K p in (b) when the input is x(t)=tu(t)? on i “L 2.8.2. QQWKW‘ Ms:— ‘l" “t M . z a shaman; 1;” ‘ , > SS Siva-0 sa—vo [+537 1) :ZLLMLM - 1W... 47‘? 2551 a, $0339“): 80f”, 2 2 = s g SQSmggé—T #2))(0 Assignment #22 ~ Solutions - p.6 ECSE-2410 Signals & Systems — Spring 2009 Due Tue 04/21/09 4(20). Text 11.56, modified as follows. [9» "‘1wa ((Ufiumlnauwl i 1 0th“) (deceit/watts“ 0% Can’t} @415” 1-§w-vf My 21:. (1.8 m/Seg? Luafl 01V Veal (Pendulmml =- iM: L a Bast: Egg: o1 when; Newtmé 2*! [Mr : Iwfi‘a'fivguiv 626691271521ch L CEQ z iSM<®ffi>~ 41+) C4959)+L M), Rf?» 3%” W33 Lsz :gaa Hat-9+wa d1“? 03546012515911 4249.) (60 Fax 6184743. Cm 3141!. Us leatmk. 2%) mM wwbww W w; $53 $1 gk 1.17915) HOT 01?" 0 (0+0 HP“) I‘g “9 -= i (to) Fm. fhofwzfi‘vhfl/é 69774137911» G59 ’91 9&1 g 0 CL S. Elem c; WEE. t 1 FE: K s: lad in dAfSWMCQ. Gigolo , {iii‘ “aw/$1)?“ 5m“ ($566053 Aims) M mm?“ “mm dew 13:21” (C) FVL FD Control; QwsléafirW—g; ; R‘hd 141; k1. $0 W EMMA,me MO‘HW Wvévziofig £31? Wang 39c .3: waging I E ~99 . Assignment #22 - Solutions - pg; 7 ECSE—2410 Signals & Systems - Spring 2009 Due Tue 04/21/09 4. Continued. t. a” 9—9- : L ski ._._. W: D) .1. 523+ J“ (“13 i + K( L. L. 9kg: WW ...
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This note was uploaded on 05/03/2009 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.

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A#22_Solutions - Assignment #22 - Solutions — p.1...

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