INTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS

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Abraham Phelps
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1 adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC CONTROLS lag compnation...7 Laplac tranform... lad compnation...7 Maon' gain rul...4 opn loop ytm... 5, 0 P-D Controllr...8 pha lag compnation...7 pha lad compnation...7 P-I Controllr...8 PID... 0 PID Controllr...8 pol of a function..., 3 proportional drivativ...8 W u Laplac tranform bcau w ar daling with linar dynamic ytm and it i air than olving diffrntial quation. W don't u Fourir tranform bcau w ar daling with th tranint rpon and bcau a Fourir tranform won't handl a ytm that "blow up". LAPLACE TRANSFORM Th Laplac tranform i ud to convrt a function f(t) in th tim domain to a function F() in th domain, whr i a complx numbr: t F f t dt 0 f(t) i 0 for t<0. f(t) can "blow up" or b picwi. W ar fr to pick th valu of to mak th intgral convrg; howvr, onc th calculation i mad you can u th rult 0t f t, thn mut b 0 or vrywhr. For xampl if gratr to do th intgration. But th rult i F / 0, in which can b l than 0. Mic: jx σ+ jω, INDEX proportional intgral...8 proportional intgral drivativ...8 R()...5 r(t)...5 ramp...5 ridu..., 3 rpatd root...3 tability... tat vctor modl... 5, 6 tady tat tracking rror...6 tady-tat rpon... 0 tp...5 LAPLACE TRANSFORMS tracking rror... 5, 6 LaPlac tranform...6 tranint rpon... 0 trig idntiti...9 typ 0 ytm...5 typ ytm...5 typ ytm...5 unit ramp...5 unit tp...5 unity fdback...4 zro of a function..., 3 INVERSE LAPLACE TRANSFORM Th invr Laplac tranform i ud to convrt a function F() in th domain to a function f(t) in th tim domain, whr i a complx numbr: π j C+ j f t In th concptual viw, c i a ral numbr dfining a lin in th -plan a hown at right. All pol of F() mut li to th lft of thi lin. Pol ar alway ymmtric about th ral axi. C j t F d Imaginary axi Ral axi c+j c c-j Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag of 0

2 INVERSE LAPLACE TRANSFORM Scond Ordr Conjugat Pair Exampl 00 ( + + j0)( + j0) t co0 t f t t in0 t 0 F A cond ordr conjugat pol pair in th lft-hand id of th -plan rult in a dampd inuoid in th tim domain. SYSTEM STABILITY Stabl: A ytm i tabl i thr ar no root in th righthand plan and no rpatd root on th jω axi. Untabl: A ytm i untabl if thr ar any root in th right-hand plan or rpatd root on th jω axi. Aymptotically tabl: A ytm i aymptotically (vry) tabl if all root ar in th lft-hand plan. SOLUTION USING RESIDUES f t πj F d F C j C+ j t ridu of Th invr Laplac tranform can b found by taking th um of th ridu of F(). Th function F() ha a ridu at ach pol of th function. Thi mthod rquir that th function F() hav mor pol than zro: For xampl, thi function ha a zro Exampl: at -5 and pol at 0 and. Zro ar valu for that cau th 0( + 5) F( ) numrator to b zro; pol ar ( ) valu for that cau th dnominator to b zro. Th ridu of F() at a impl pol i found by taking th limit a follow: pol ( ) ridu lim pol F So for pol0 in th xampl abov, w hav: 0( + 5) t 0( 0+ 5) 0t 50 lim( 0) 0 ( ) ( 0 ) and for pol w hav: 0( + 5) lim( ) ( ) t t t t So w olv th invr Laplac tranform by f ( t) ridu of F( ) f t t ( 35 t + 5) Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag of 0

3 RESIDUES: REPEATED ROOTS Whn thr i a rpatd root, th procdur for olution uing ridu chang. ( n ) ( n ) d ridu lim pol pol! d Exampl: F( ) ( + ) ( ) n ( ) n F t For xampl, thi function ha a zro at -5 and 3 pol at. ( + ) 3 ( ) ( 3 ) d t ridu lim ( 3) ( ) pol ( 3! ) d d t lim ( 0 50) pol d + d t t lim 0 50 pol d + d t t t lim 0t 0 50t pol d + + t t t t lim 0t + 0t + 0t + 50t pol 0 t 0 t 0 t 50 t t + t + t + t t t 70t + 0t ( 35t + 0t) So w olv th invr Laplac tranform by f ( t) ridu of F( ) and in thi ca thr i only on ridu o t f t 35t + 0t SOLUTION USING DIVISION Thi mthod mut b ud whn th numbr of zro i qual or gratr than th numbr of pol. Exampl: ( + ) 5 3 F( ) + 5 For xampl, thi function ha two zro at -3 and a pol at -5. W carry out th multiplication in th numrator and thn divid by th dnominator: f Th problm i now dividd into thr part: F ( ) 5, F ( ) 5, and 50 F3 ( ) + 5 Part and ar don by inpction and part 3 i by ridu a bfor: 5 d 5t f t δ ( t), f ( t) 5δ ( t), f3 ( t) 50 dt d f t 5 δ t + 5δ t + 50 dt Thi giv th rult: 5t not: δ(t) i th impul function, which i a ingl input pul having a larg amplitud, hort duration, and a plottd ara of on. FINDING THE DIFFERENTIAL EQUATION THAT DESCRIBES A TRANSFER FUNCTION Exampl: Givn th tranfr function: ( + ) ( + ) 0 5 G Prform th multiplication and, Y auming all initial condition ar 3 zro, writ: R + Thn cromultiply: 3 Y( ) + Y( ) 0R( ) + 50R 3 Tak th invr Laplac dy + dy 0 dr + 50Rt 3 dt dt dt tranform to gt: Thi diffrntial quation dcrib th original tranfr function abov. Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag 3 of 0

4 What if all initial condition ar not zro? Exampl: Givn th initial condition to th tranfr function abov: y( 0) d y ( 0) a b dt d y ( 0) c dt Working backward in th prviou xampl, tak th Laplac tranform of ach trm of th rult, incorporating th nw initial condition: 3 dy 3 d d L 3 Y( ) y( 0) y( 0) y ( 0) dt dt dt 3 Y a b c dy L Y a b dt dr L 0 0R( ) 0a dt L 50rt 50R { } So th Laplac tranform i: 3 Y a b c+ Y a b 0R 0a+ 50R Grouping trm w gt: 3 + Y R + a + a+ b+ 0a+ b+ c And dividing by ( 3 + ) giv u th rult: ( + ) R ( 0) ( ) 0 5 a b+ + c Y( ) Notic that th firt trm of th rult com from th original tranfr function and th cond trm i du to th initial condition. BLOCK DIAGRAMS Block diagram ar ud to rprnt tranfr function opration of a ytm. Som baic opration ar a follow: x G ( ) G G G( ) x R x G ( ) / G ( ) Σ x R - R - R MASON'S GAIN RULE G x Maon' gain rul i a mthod of finding th tranfr function of a block diagram. For an xampl of uing Maon' rul, MaonRul.pdf. M j M tranfr function or gain of th ytm M j gain of on forward path j an intgr rprnting a forward path in th ytm j th loop rmaining aftr rmoving path j. If non rmain, thn j. - Σ loop gain + Σ nontouching loop gain takn two at a tim - Σ nontouching loop gain takn thr at a tim + Σ nontouching loop gain takn four at a tim - M j j UNITY FEEDBACK SYSTEM R ( ) G( ) C ( ) Th tranfr function for thi ytm i C R G + G Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag 4 of 0

5 CLOSED LOOP SYSTEM R ( ) G( ) H( ) Th tranfr function for thi ytm i C R G + G H C( ) Th tranfr function for th opn loop ytm (th output i takn to b aftr H()) i + G( ) H( ) F Pol of th clod loop ytm ar zro of th opn loop ytm. Th clod loop ytm i untabl if F() ha zro in th right-hand plan. r(t), R() BASIC TYPES OF INPUTS Unit tp input, t > 0 r t R Unit ramp input r( t) t, t > 0 R Unit ramp input r( t) t, t > 0 3 R t 3 4 t 3 t BASIC TYPES OF SYSTEMS Typ 0 ytm no pol at th origin track a tp input with finit rror do not track a ramp input do not track a quar ramp input Typ ytm ha on pol at th origin track a tp input with zro rror track a ramp input with finit rror do not track a quar ramp input Typ ytm ha two pol at th origin track a tp input with zro rror track a ramp input with zro rror track a quar ramp input with finit rror STATE VECTOR MODEL Th tat vctor modl i anothr mthod of modling ytm. It i don in th tim domain and contain a t ordr diffrntial quation. Th olution i a vctor. X t AX t + bu t Stat Modl: for xampl whr A i a matrix w would hav: x t a a x t b u t x t a a + x t b and thi tranlat to: x t a x t a x t bu t x t a x t a x t bu t Th numbr of lmnt in th vctor ( in thi ca) corrpond to th ordr of th polynomial in th dnominator of th tranfr function. X(t) tat vctor, coniting of th output ignal and it drivativ X ( t) firt drivativ of th tat vctor A a quar matrix b a vctor u(t) ytm input ignal Output Equation: ct DX( t) c(t) ytm output ignal D a row vctor that alway ha a th firt lmnt and zro for th rmaining lmnt W pick a olution: x t ct x t ct Th olution i not uniqu, but it i what w u for thi typ of problm. For largr than a nd ordr polynomial w x t c t tc. would continu with 3 Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag 5 of 0

6 FINDING THE TRANSFER FUNCTION FROM A STATE MODEL Givn th tat vctor modl, th tranfr function may b found uing th formula: [ ] C DI A bu whr I i th idntity matrix. For xampl, givn x Ax+ bu, c Dx, w hav: 5 6 A 0, b, D [ 0] C U 0 0 [ 0] C U C U C U [ 0] adj [ 0 ] [ 0 ] ( + 5) + 6 For mor about finding th adjoint of a matrix, th fil Matric.pdf. 6 C( ) ( + 5) + 6 ( + 5) + 6 [ 0] U( ) + 5 ( + 5) + 6 ( + 5) + 6 C( ) 6 U( ) C( ) 6 and th tranfr function i U a m p l i t u d (t) TRACKING ERROR Th tracking rror i th diffrnc btwn th input and output of a ytm. ytm input ytm rpon (output) t r( t) c( t) tracking rror ( t) E() TRACKING ERROR, LAPLACE TRANSFORM E( ) R( ) C( ) Th Laplac tranform of th tracking rror of a ytm. For th ytm (no fdback) G( ) R( ) C For th ytm (unity fdback) C G + G R Th Laplac tranform of th tracking rror i E G R Th Laplac tranform of th tracking rror i E t + G STEADY STATE TRACKING ERROR R Th tracking rror of a ytm a t. Th tady tat tracking rror can b computd from E(), th LaPlac tranform of th tracking rror. Not that a t in th tim domain, 0 in th frquncy domain. 0 lim E o, for a unity fdback ytm, lim R 0 + G Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag 6 of 0

7 PHASE LAG COMPENSATION Pha lag compnation rduc th high frquncy gain to zro at th location of th dird pha margin. Th pha lag compnator hift th zro croing downward to th location of th dird pha margin by adding a pol and zro blow thi point. A ngativ pha hift occur, but not at th zro-croing point. Gain db Pha angl 0 T Amplitud Pha at nw zrocroing point ) Find th valu of K that atifi th valu pcifid for th tady-tat tracking rror. ( ramp) lim KG( ) 0 ) Draw th bod plot of KG() and find th frquncy at which th dird pha margin occur. Thi will b th compnatd zro-croing point ω 0. Dtrmin th amount of db gain hift rquird to adjut th plot to cro zro at thi point (a downward hift i ngativ). 3) Find th valu of a uing th db gain hift found abov. 4) Now find T. 0loga db gain hift 0 at ω 5) Th compnating factor for th ytm tranfr function i: G lag ( ) 0 + at + T 6) And th nw tranfr function i Glag ( ) KG( ) ω ω PHASE LEAD COMPENSATION Pha lad compnation hift th zro-croing point and rduc th pha angl at that point by adding a nw pol and zro to th tranfr function. Th pha lad compnator hift th zro croing lightly upward to a point midway btwn th addd pol and zro. Th pha plot i bowd upward, with th maximum ffct occurring at th nw zro-croing frquncy ω max. Gain db Pha angl 0 nw zrocroing point at T Amplitud ω Pha ) Find th valu of K that atifi th valu pcifid for th tady-tat tracking rror. ( ramp) lim KG( ) 0 ) Draw th bod plot of KG() and find th uncompnatd pha margin. 3) Find th valu of a uing th pcifid pha margin plu a 5 fudg factor and th uncompnatd pha margin. in in PM 5 PM a φ + max comp. uncomp. a + 4) Uing a, find th uncompnatd gain at th frquncy which will bcom th nw zro-croing point. Not that in thi xprion a factor of 0 i ud intad of 0 bcau thi gain i locatd midway up th 0 db/dcad lop a hown abov. Gain 0log a Find th nw zro-croing point ω max by locating th frquncy on th uncompnatd bod plot that ha th abov gain. Thi will alo b th point at which th compnator produc maximum pha hift. 5) Now find T. ω max T a 6) Th compnating factor for th ytm tranfr function i: G lad ( ) + at + T ω 7) And th nw tranfr function i Glad ( ) KG( ) Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag 7 of 0

8 PID CONTROLLERS PID tand for proportional intgral drivativ: intgral drivativ proportional kt t d p + K I ( t ) dt + K d 0 dt or K K I p + + K W won't covr thi controllr, but w will covr th P-D and th P-I controllr. d P-D CONTROLLERS Th P-D controllr add a zro at (K p /K d ). If l than 45 of pha hift i rquird thn th gain will not chang. E( ) "P" Controllr K p E( ) K p +K d "P-D" Controllr G( ) G( ) P-I CONTROLLERS Th P-I Controllr olution may b obtaind uing th P-D olution tchniqu. E() "P-I" Controllr ( ) E( ) K p + KI H( ) "P-I" Controllr, rdrawn + ( K K p ) I K p H( ) ) Givn th tranfr function H(), find th valu of K p and K d that would achiv P-D compnation for th tranfr function H()/. Th will b th valu for K p and K I rpctivly in th P-I controllr. ) Th compnatd tranfr function i K p KI + Kp H "P-D" Controllr, rdrawn E() K p + K d K p G( ) ) Find th valu of K p that atifi th valu pcifid for th tady-tat tracking rror. ( ramp) lim KG p ( ) 0 ) Draw th bod plot of K p G() and find th uncompnatd pha margin. 3) If w do not nd to incra th pha margin by mor than 45, thn ω 0 will not chang. U ω 0 from th plot and olv for K d. Kd tan PM + 5 PM ω comp. uncomp. 0 K p If w do nd to incra th pha margin by mor than 45, thn u th following xprion to find th uncompnatd gain at th nw ω 0. Rad th nw ω 0 from th plot and plug in to th abov xprion to find K d. Gain 0log + tan PMcomp. + 5 PM 4) Th compnatd tranfr function i K p K K d + p G uncomp. Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag 8 of 0

9 GENERAL TRIG IDENTITIES Hr ar om idntiti w u: ± θ j co θ ± j in θ Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag 9 of 0

10 clod loop ytm compnat for diturbanc by mauring th output rpon and rturning that through a fdback path to compar with th input at th umming junction. opn loop ytm an input or "rfrnc" i applid to a controllr that driv a proc. Thr i no fdback compnation. PID proportional + intgral + drivativ, or 3-mod controllr. impl man not rpatd or duplicatd tady-tat rpon th approximation to th dird or commandd rpon tranint rpon th chang from on tat to anothr GLOSSARY Tom Pnick tomzap@dn.com AutomaticControl.pdf 5/0/000 Pag 0 of 0

(1) Then we could wave our hands over this and it would become:

(1) Then we could wave our hands over this and it would become: MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and