Advanced Control Theory
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- Hugh Albert Ray
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1 Ava Corol Thory Rviw of Corol Sym
2 Irouio o orol Chibum L -Soulh Ava Corol Thory
3 Corol Sym Corol ym: A iroio of omo formig a ym ofiguraio ha will rovi a ir ym ro Targ Tmraur Corollr Tmraur Sor Har Roo Rlay/SCR Chibum L -Soulh Ava Corol Thory
4 Examl of Corol Auomobil rivig Chibum L -Soulh Ava Corol Thory
5 Ty of orollr Tyial orollr Emb Corollr : Sially ig orollr for ifi ym MPU, MCU, DSP, FPA Co for vlom uful for ma rouio PC Corollr Eay vlom, oly, ay a PLC Programmabl Logi Corollr Wily u i gral auomaio ym, qu orol U Lar iagram, Moular omoiio Chibum L -Soulh Ava Corol Thory
6 Corol Sym Trmiology Sym A hyial of omo ha ak a igal, a rou a igal Sigal A fuio rrig om variabl ha oai om iformaio abou h bhavior of a ym. igal A Sym igal B Chibum L -Soulh Ava Corol Thory
7 Corol Sym Trmiology Sym Pla: Th hyial obj o b oroll. Trmiology om from hmial ro la lik oil rfiri or owr la. Sor: Th vi ha allow you o maur variabl for la moiorig a for a variy of ohr uro I mhaial ym, i maur rur, for,, oiio,. Cf Auaor: Th vi ha au h ro o rovi h ouu. Th vi ha rovi h moiv owr o h ro Chibum L -Soulh Ava Corol Thory 7
8 Corol Sym Trmiology Sym Corollr: Th vi or oraio ha giv omma o h la. Th omma ar uually ba o h urr rforma of h la omar o h ir rforma of h la. Th orollr may or may o ilu h auaor, h vi ha for h orollr omma o h la. Corollr Pla Sor Chibum L -Soulh Ava Corol Thory 8
9 Corol Sym Trmiology Sigal Rfr: Wha you lik h maur ouu of your oroll la o b..g. ig for your rui orol. Fbak Maur ouu: Variabl ha ar omig ou of your la ha ar big omar o your rfr..g. maurm of your ar from whl or Chibum L -Soulh Ava Corol Thory
10 Corol Sym Trmiology Sigal Fbak Error: Diffr bw your rfr a your fbak igal..g. km/h iffr bw rfr a fbak. Pla Iu: Th igal ha i a from h orollr o h la o aff om aio..g. Throl variaio. Pla Ouu: all h igal omig ou of h la. No: o all ouu ar u i fbak Chibum L -Soulh Ava Corol Thory
11 Corol Sym Trmiology Sym a Sigal Pla, Corollr, a Sor all ym. uually hav om yami or iffrial quaio riio. rafr fuio rraio for h ym. Sigal arry iformaio bw h lm. rfr rror iu ouu Corollr Pla fbak Sor Chibum L -Soulh Ava Corol Thory
12 Corol Sym Trmiology Diurba Diurba: A uwa igal ha i o aibl o h orollr a o avrly aff h ym ouu Sym or igal??? iurba Corollr Pla Sor Chibum L -Soulh Ava Corol Thory
13 Corol Sym Trmiology Corol ym iagram wih iurba a oi Corol ym iagram wih ir a our loo Chibum L -Soulh Ava Corol Thory
14 Clo v. O loo orol Clo loo orol Th iffr of rfr a fbak ar u a a ma of orol. Clo-loo wih fbak O loo orol Th ouu ha o ff o orol aio uually imlr wih fwr omo Examl woul b imr;.g. oar. O-loo wihou fbak Chibum L -Soulh Ava Corol Thory
15 Clo v. O loo orol O-loo xaml Chibum L -Soulh Ava Corol Thory
16 Clo v. O loo orol Clo-loo xaml Chibum L -Soulh Ava Corol Thory
17 Clo v. O loo orol Comario Co Sabiliy Prforma O-loo ym o or iml a l xiv a abl la ao go uabl Diurba or molig mimah a au rror ralibraio ar Clo-loo ym a abl la a go uabl, bu a uabl la a go abl mor robu o iurba a uraiy Thy uually hav br rforma Chibum L -Soulh Ava Corol Thory
18 Clo v. O loo orol O loo Clo loo + - wih mol uraiy + - % rror.% rror Chibum L -Soulh Ava Corol Thory
19 Corol ig rour. Suy h la a h orol objiv.. Mol h la if ary, imlify h mol. 3. Di h variabl o b oroll oroll ouu, h maurmor a maiula variabl auaor 4. Sl h orol ofiguraio a h y of orollr. 5. Di o rforma ifiaio 6. Dig a orollr. 7. Aalyz h rulig oroll ym 8. Simula h rulig oroll ym 9. Choo harwar a ofwar a imlm h orollr. Sofwar lab. T a valia h orol ym Harwar lab Chibum L -Soulh Ava Corol Thory
20 Lala raform Chibum L -Soulh Ava Corol Thory
21 Lala Traform Why Lala raform? Powrful for olvig liar ODE wih iiial valu Eay o al a Dira la fuio Comforabl for ioiu or omlx rioi fuio Lala Traform Chibum L -Soulh S-omai Prob. algbrai q. Origial Prob. iffrial q. AE Solvr Eay S-omai T-omai DE Solvr Diffiul Soluio o S-omai Prob. Soluio o origial Prob. Ivr Lala Traform Ava Corol Thory
22 Dfiiio Lala raform of a fuio f F f f Igral raform Ivr Lala raform of F F f πj j j F Chibum L -Soulh Ava Corol Thory
23 Ava Corol Thory Chibum L -Soulh Lala raform Th Lala raform i a liar oraio. Oby h riil of uroiio. ] [ ] [ x L a x a L x a a x x a a x x a L a x
24 Prori of Lala raform Shifig -omai Shifig -omai Diffriaio Igraio Fial valu horm L[ f ] F a L[ f T ] f L[ ] F f f f L[ ] L[ ] f F L[ f ] f lim F Iiial valu horm f lim F a T F Chibum L -Soulh Ava Corol Thory
25 Lala Traform uag i orol ym rimary u for Lala Traform. Coma rraio of: a Sigal iurba Corollr Pla b Sym Sor Chibum L -Soulh Ava Corol Thory
26 Ava Corol Thory Chibum L -Soulh Sigal S Ram a u a a a u a F
27 Ava Corol Thory Chibum L -Soulh Sigal Siuoi Cf i j j j Eulr formula i ] [i j j j j L F j j i f o f ] [o j j L F
28 Ava Corol Thory Chibum L -Soulh Sigal Imul fuio,, f } { lim lim lim lim lim { lim } { lim lim ] [ L F
29 Lala raform Sigal Chibum L -Soulh Ava Corol Thory
30 Sym Sym ar abra rraio of yamial homa. orr Coir a fr ro woul b oirig ju h ym. x ax ym F igal affig ym x ax ; x Chibum L -Soulh Ava Corol Thory
31 Sigal a Sym How igal aff ym a how ym gra igal Covoluio Covoluio Commuaiv Law: Diribuiv Law: Aoiaiv Law: f f Uuual Prori of Covoluio: Covoluio Thorm f g f g f g g f f g g f g f g f g v f g v f f f g f g Chibum L -Soulh Ava Corol Thory
32 Sigal a Sym How o igal f aff ym g? iu f ouu ovoluio g f F igal ym igal ym Chibum L -Soulh Ava Corol Thory
33 Covoluio orr xaml agai x ax ym F igal affig ym X F a ym igal ro yami of ym igal affig ym I h im omai x a f Covoluio Chibum L -Soulh Ava Corol Thory
34 Sigal a Sym Ex. Solv y '' y, y, y ' LHS: Y y y' Y TF: y, y', RHS: f F Ouu: y L F IC ro f Ram iu ih ouu y, y', Sym wih IC Chibum L -Soulh Ava Corol Thory
35 Sigal a Sym Viual xlaaio Examl igal, F ym Examl Chibum L -Soulh Ava Corol Thory
36 Molig of Corol ym Chibum L -Soulh Ava Corol Thory
37 Mahmaial Mol Mol ar ky lm i h ig a aalyi of orol ym qualiaiv mahmaial mol??? u y W mu mak a omromi b/w h imliiy of h mol v. h auray of h rul of aalyi Chibum L -Soulh Ava Corol Thory
38 Liar v. oliar ym Liar ym: h riil of uroiio hol Liariy i mahmai L V a W b vor a ovr h am fil K. A fuio f: V W i ai o b a liar ma if for ay vor x a y i V a ay alar α i K, h followig oiio ar aifi: Liariy i ym aiiviy homogiy A gral ym a b rib by oraor H, ha ma a iu x a a fuio of o a ouu y a y of blak box riio. Liar ym aify h rori of uroiio a homogiy. Chibum L -Soulh Ava Corol Thory
39 Liar Tim Ivaria Sym Liar Tim Ivaria = Liar & im ivaria A im-ivaria TIV ym i o who ouu o o xliily o im. If h iu igal x rou a ouu y, h ay im hif iu, x+, rul i a im-hif ouu y+ Tim ivaria ma ha h offii i h iffrial quaio ar oa a o hag wih r o im. W a aly imul ro & Lala raform i LTI ym Chibum L -Soulh Ava Corol Thory
40 Tim-Varyig Mol A im-varyig ym i a ym ha i o im ivaria i ouu xliily uo im Eg. a araf orol ym. Th ma of ful oumio hag u o ful oumio Dyami ym Liar Noliar Chibum L -Soulh Liar Tim Ivaria Our fou Liar Tim Varyig Ava Corol Thory
41 Trafr Fuio Aumig zro iiial oiio, ak h Lala Traform of boh i m Y by ky F m b k Y F ouu Y F m b k Trafr Fuio iu iu ouu Chibum L -Soulh Ava Corol Thory
42 Trafr Fuio Trafr Fuio: h raio of h Lala raform of h iu a ouu of a liar im-ivaria ym wih zro iiial oiio a zro-oi quilibrium. Raioal fuio i h omlx variabl L x : iu, y : ouu a y a y b m m a x b y m a x y m b x b x > m Chibum L -Soulh Ava Corol Thory
43 Ava Corol Thory Chibum L -Soulh Trafr Fuio -h orr ym i h high owr i h omiaor i. No: limi o im-ivaria, iffrial quaio i of h iu magiu. homogiy o iformaio o hyially ruur. MKS a RLC IC zro ] [ ] [ TF: a a a a b b b b X Y iu L ouu L m m m m
44 Ava Corol Thory Chibum L -Soulh Sym Pol a Zro Roo of N= : h ym zro z, z,, z m Roo of D= : h ym ol,,, No Sym Pol a zro: ral or ihr omlx ojuga air m m m m m m z z z z K D N a a a a b b b b umraor omiaor
45 Fbak Sym Chibum L -Soulh Ava Corol Thory
46 Fbak Sym Comar o o loo ym, fbak orol ha h followig avaag: Dra iiviy of h ym o variaio i h aramr of h ro Imrov rjio of h iurba Imrov maurm oi auaio Imrov ruio of h a-a rror of h ym Eay orol a ajum of h rai ro of h ym Chibum L -Soulh Ava Corol Thory
47 Fbak Sym Clo-loo ym ubj o a iurba a a maurm oi Aum LTI ym U V Y m Moly H Chibum L -Soulh Ava Corol Thory
48 Ava Corol Thory Chibum L -Soulh Sym Trafr Fuio Dfi Trakig rror: If w oir iu igal araly a u riil of uroiio, h ouu i giv by Y R E N H H D H R H Y
49 Ava Corol Thory Chibum L -Soulh Sym Trafr Fuio Aumig H=, iral igal ar giv by Ouu Maur ouu Corol iu N D R Y N D R Y m N D R U
50 Ava Corol Thory Chibum L -Soulh Sym Trafr Fuio ag of 4 Why iiviy rafr fuio? / / S T T : Comlimary iiviy rafr fuio : Siiviy rafr fuio S T h raio of h hag i h ym TF o h hag of a la TF
51 Ava Corol Thory Chibum L -Soulh Prforma Sifiaio oo orol rakig rror mall oo rfr rakig oo iurba rjio oo oi auaio N D R E T S S S R E S D E T N E
52 Ava Corol Thory Chibum L -Soulh Prforma Sifiaio Ahivig goo rfr rakig, iurba rjio, oi auaio imulaouly i o oibl Algbrai limiaio Choi bw oo rfr rakig a oi auaio Soluio? Sara h frquy omo i rfr, iurba, a oi T S
53 Tim Domai Aalyi Chibum L -Soulh Ava Corol Thory
54 Tim Domai Aalyi Afr h mahmaial mol of h ym i obai, aalyi of ym rforma i. Iu r Sym Ouu y Iu igal o a orol ym i ukow bu wha if o u kow iu igal A orrlaio b/w h ro hararii of a ym o a yial iu igal a h aabiliy of h ym o o wih aual iu igal Chibum L -Soulh Ava Corol Thory
55 Tim Domai Aalyi Iu Sym Ouu Tim ro y y y r Trai ro: go from IC o fial a Say a ro: h ym ouu bhav a aroah ifiiy Chibum L -Soulh Ava Corol Thory
56 Orr Sym Saar form Ty y r T Y R Y R T TF of orr ym Y Y Chibum L -Soulh Ava Corol Thory
57 Ava Corol Thory Chibum L -Soulh Orr Sym S ro Ui ro w/ zro iiial oiio T T T T Y R U r, / T y
58 Orr Sym S ro Chararii faur of a orr ym: A =T, y y T i. h ro y i 63.% of i oal hag A =, h lo of h ro y T / T T y T : im oa Chibum L -Soulh Ava Corol Thory
59 Ava Corol Thory Chibum L -Soulh Orr Sym Ram ro Ui ram ro w/ zro iiial oiio T T T T Y, / T T T y R U r
60 Orr Sym Ram ro Th rror bw h rfr & h ouu. r y y T T /T T /T = T lim T y Smallr T mallr Chibum L -Soulh Ava Corol Thory
61 Ava Corol Thory Chibum L -Soulh Orr Sym Imul ro Ui imul ro T Y, / T T y R r y y
62 Orr Sym Imul ro Ty y w/ y Comar Tim ro o a imul rfr igal i iial o a iiial oiio ro wih zro rfr. Ty y w/ y T TY Y Y T TY TY T T Y Y Y T T Y Chibum L -Soulh Ava Corol Thory
63 Ava Corol Thory Chibum L -Soulh Orr Sym Clo-loo J K J B J K K B J K R Y Y = Saar form of orr ym JK B J K,
64 Orr Sym Solv h followig iffrial quaio Ca. x x x, x, x Ca. x x x, x, x Chibum L -Soulh Ava Corol Thory
65 Orr Sym Dyami bhavior of orr ym a b rib i rm of amig raio a aural frquy Y R Chararii quaio omiaor of TF= Pol roo of hararii quaio j, Chibum L -Soulh Ava Corol Thory
66 Orr Sym Pol loaio If, If, If, If,,,,, j j Chibum L -Soulh Ava Corol Thory
67 Ava Corol Thory Chibum L -Soulh Orr Sym-S ro For uram a, Y a whr i i o y lim y
68 Ava Corol Thory Chibum L -Soulh Orr Sym-S ro For o am a, Y y o lim y y
69 Ava Corol Thory Chibum L -Soulh Orr Sym-S ro For riially am a, Y y lim y
70 Ava Corol Thory Chibum L -Soulh Orr Sym-S ro For ovram a, Y y lim y,,
71 Orr Sym-S ro Chibum L -Soulh Ava Corol Thory
72 Orr Sym-Imul ro Imul ro or Iiial oiio ro Y For y i For y lim y for Chibum L -Soulh Ava Corol Thory
73 Ava Corol Thory Chibum L -Soulh Highr Orr Sym Ui ro a a a a b b b b R Y m m m m or ro or ro r q b a a a a a a b b b b Y r k k k k k k k k j k q j j j m m m m r k k k k k k k q j j b a a y k j k j j i o
74 Highr Orr Sym Chibum L -Soulh Ava Corol Thory
75 Examl Domia ol Ex. Y R Y R Chibum L -Soulh Ava Corol Thory
76 Trai-Ro ifiaio or. y. Trai Ro Chararii o a ui iu Dlay im, : im rquir for rah 5% Ri im, r Pak im, : im rquir for ri from % o 9% or from % o % : im rquir for rah ak valu Maximum ovrhoo, M : Slig im, : im rquir for rah a ay % or 5% of fial valu y y y Chibum L -Soulh Ava Corol Thory
77 Ava Corol Thory Chibum L -Soulh Trai-Ro ifiaio or. y. Ri im: r y a i o i o r r r r r r y r y i o
78 Ava Corol Thory Chibum L -Soulh Trai-Ro ifiaio or. y. a a For a r r???
79 Ava Corol Thory Chibum L -Soulh Trai-Ro ifiaio or. y. From ym ol R Y, j a r agl a
80 Ava Corol Thory Chibum L -Soulh y Trai-Ro ifiaio or. y. Pak Tim,,3,, i i i o o i i o o i y y i o
81 Ava Corol Thory Chibum L -Soulh y y i o Trai-Ro ifiaio or. y. Maximum Ovrhoo M % %OvrhooPO i o y y y y y M
82 Trai-Ro ifiaio or. y. Slig Tim y y o i Aroximaio om from vlo fuio 4 for % ririo 5 for 5% ririo Chibum L -Soulh Ava Corol Thory
83 Ava Corol Thory Chibum L -Soulh Trai-Ro ifiaio or. y. y a whr i i o y
84 Ava Corol Thory Chibum L -Soulh Paramr Slio xaml Y Y J K KJ K K b K K K b J K R Y h h,
85 Ava Corol Thory Chibum L -Soulh Paramr Slio xaml.86 5% ririo 3,.48 % ririo 4.65 a.78 m,.5 N, r h h K B KJ K J K J K KJ K K b M
86 Ava Corol Thory Chibum L -Soulh Say-Sa Error i Fbak Sym Uiy fbak ym lim lim lim R E R Y R E R Y yumbr : N z z z z K l l N m m
87 Ava Corol Thory Chibum L -Soulh Ui iu Say-Sa Error i Fbak Sym or, For lim lim, For rror oa oiio : lim lim lim lim l m K N K z z K K N K K E
88 Ava Corol Thory Chibum L -Soulh Ui ram iu Say-Sa Error i Fbak Sym or lim, For lim, For or lim lim, For vloiy rror oa : lim lim lim lim lim l N m v v l m v l m v v v z z K K N K z z K K N z z K K N K K E
89 Say-Sa Error i Fbak Sym y y Ty ym ro o a ram iu Chibum L -Soulh Ava Corol Thory
90 Ava Corol Thory Chibum L -Soulh Ui arabolialraio iu Say-Sa Error i Fbak Sym or lim 3, For lim, For or lim lim, For alraio rror oa : lim lim lim lim lim 3 l N m a a l m a l N m a a a z z K K N K z z K K N z z K K N K K E
91 Say-Sa Error i Fbak Sym y y Ty ym ro o a araboli iu Chibum L -Soulh Ava Corol Thory
92 Say-Sa Error i Fbak Sym Chibum L -Soulh Ava Corol Thory
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Essential Skills Chapter f ( x + h) f ( x ). Simplifying the difference quotient Section. h f ( x + h) f ( x ) Example: For f ( x) = 4x 4 x, find and simplify completely. h Answer: 4 8x 4 h. Finding the
S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]
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S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:
Web-appendix 1: macro to calculate the range of ( ρ, for which R is positive definite
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International Journal of Pure and Applied Sciences and Technology
I. J. Pur Al. S. Thol.. 4-6 Iraoal Joural o Pur ad Ald S ad Tholog ISSN 9-67 Avalabl ol a www.joaaa. Rarh Par Blaral Lala-Mll Igral Traorm ad Alao S.M. Kharar * R.M. P ad J. N. Saluk 3 Darm o Egrg Mahma
Overview. Splay trees. Balanced binary search trees. Inge Li Gørtz. Self-adjusting BST (Sleator-Tarjan 1983).
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