Final_07_Sol

# Final_07_Sol - MAE 143B LINEAR CONTROL ‘ Prof M Krstic...

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Unformatted text preview: MAE 143B LINEAR CONTROL ‘ Prof. M. Krstic FINAL . June 13, 2007 NAME: Sol U’ho "3 PID: 0 One page (front and back) of your own handwritten notes. 0 No graphing calculators. 0 Present your reasoning and calculations clearly. Random or inconsistent etchings will not be graded. 0 Write only on the paper provided. If you run out of space for a given problem, continue on the pages at the end of the set and indicate “Continued on page X.” o The problems are not ordered by difﬁculty. 0 Total points: 60 0 Time: 3 hours. adammmmw w. .1.K1_,:A::w1...;u:.-:_ "men-1.4.x 14:,:,':,,1:,; '_; J '_.:/.>:,,u/, H . e. , 4 . ./ , ‘ r, . . 'w . _ a Problem 1: Stability (4 points) Is the following polynomial stable? If not, how many eigenvalues are in the right—half plane? p(s)=s5+254+353+4s2+55+6 Problem 2: Root Locus (12 points) Sketch the root locus with respect to K for the equation 1 + K 0(5) = 0 for the following: (a) (4 points) (3 + 4)(s2 + 16) G(8) I _ 5(52 + 9) Describe the nature of the stability of the system for different values of gain K. (b) (4 points) 1 0(8) : (s + 2)(s2 + 25 + 2) Be sure to mark Where the Root Locus intersects the imaginary axis. When the Root Locus intersects the imaginary axis, What is the gain K? (c) (4 points) (5 + 4)(s + 2? 0(3) = 5(5 +1)2(5 +6) 3d K“- 7 5(51+0\)+(\$+4/)(g Z+/6)(I) : O ; 252+ qg1+&53 tea—4) lo” 54qlo‘lll‘lﬁj 97/546!” 'I} Marginal” giggle, ‘Br H1?) SVSlEM") Uo;~qule ~Q>F ©<K<oo ‘ £04, V (SM (S 1+2; +13 4’” :O 53+L/527Lés HM“ : O 3 l C . A ' 4&5 ~ 903+er +‘L/7‘h30 64w1+#+k\ + (wﬁewx : O Ser rm {mt 67W] 79; gm, WHA H290; ‘ 3.3 .V y ,4}. ﬁnnrifx iﬁgﬂﬁyuﬁﬁMEp Problem 3: Bode Plots (12 points) Sketch the Bode plots for the following open-loop transfer functions: (a) (41 points) on; — 10)(s + 100) C(s) : (s + l)2 (b) (71 points) ‘ _ s(s+100) G05) ~ 32 + 13+ 1 (C) ('1 paints) 0(8):7i0ﬂ__) - ‘ t " , 1 3232+105+100 r? ,V’ V, ‘x-_\ ’y—vi‘g :_ ,_ A‘ - .»/|<7;V‘J)H‘OM I 3)? M H (L '0 ~ )0 _ p 203g <UC)O§: 480 - 480 :71: -T‘\ \GUN , 048W" ’ (“520 ~ -90 {H 300 JO! “Oak 1:); \w """i‘f: /SJmU(L \$0099“ VOW, o :5: \ Nashufef / 54W" 3"” §50~ -, _, w cow I .ml - J40 cm, Problem 4: Nyquist Plots (12 points) Sketch the Nyquist plot for each of the transfer functions in problem 4. What does Nyquist’s stability criterion tell you about each of the systems? (4 points each part) 6) GCS):,O\ w (3+bl ‘ . GM: {oﬂ(~)o)()oo): «mo ) «may 480 GCM: tOl j <GCQ'ao): 06’ Fkné CbeS'1qgs: : ,ojﬂow "O\():)Uf100) (Wm 4/) : O‘ug / r track? 005}; im‘f‘fmdioh“ «C(5CJ' w): 0 i; ufHM4ﬁqlqooo W2: *‘H ¥°If Qmw 910000) N :0 '<'GQ\$O>: 010° G60”): ' / < 9mm“ LIMA CTO)5)A35: GQW) : *wl +IOOw3 _, x , x ,1 . WH +JWQ «(pH—'de : w4+7w1+<4q7ﬁw3w00wb M Cb) +‘l3 ‘ IM (Guam) :0 w(qq.qw‘vsoo):o blpox): 9913 w: 3.00)} 096’) (IQ—)EFSQCBOV‘ «a (42)) I C) WWW} : O (U : O -wf‘HmmJ' WNWL 56%; ' 961M: 3% K 10 A Cro§5mﬁ§t W”"“"*+"’W“ W‘ﬂ-Ioowluoaﬁo' wH-rooafwowd § 6 , ~~Iow +l2joWLI__IOOOW1>TL_’2Ow§T2/Oow3)d' Li 1 . Undth ‘V K 11 Problem 5: Bringing it all together (20'points) We can learn a lot about the behavior of a system from its Bode plot. Suppose we perform an experiment on a dynamic system, and we obtain the following Bode plot: Bode Diagram Magnitude (dB) Phase (deg) Frequency (rad/sec) (a) (4 points) Sketch the Nyquist plot. (b) (2 points) Determine the gain and phase margin(s). (c) (5 points) Determine the Ziegler—Nichols PID tuning parameters using the ultimate sensitivity method. (Hint: Find KL and Pu from the Bode plot.) (Use a straight edge to approximate the values the best you can) (d) (5 points) Determine the transfer function of the open—loop system. (Hint: Use Ku and the cooresponding w to ﬁnd C (Use powers of 10 for your breakpoints) (e) (4 points) Sketch the Root Locus to conﬁrm that high gain leads to instability. 12 LkJUV/X/ (“F/0V1 ﬂ ,5 m _, Eu" W706?!“ :: [.39] m: 3m :. 1%; Km; Eip'w: “6:134/ , k1 fV-H/iztz/g, kgqyzg w 13 7x\/ ‘ C5 - Ml {kg at 25+ mSJriOo), r ‘ W ‘ZW U56 kw CtmL w: \$0 Agkrm‘me éo' (72I'39r+g-’5*33.)-(100mg) “ 224 w ‘ ﬂy?g»2;' -> ' ‘, “ ' gv WWW(~40):sz J ' 3)+(‘Ll-O31+3€O7}5 (Q29-(,7r33—67t72§9+§g¢;)J’2739 \ m6”?! MIR ' . 7 _ :i *0/436X'L/02253CO2)+3€ (22 4+6. (/57) :0. 3457 :‘31/600 I 15 16 ...
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Final_07_Sol - MAE 143B LINEAR CONTROL ‘ Prof M Krstic...

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