Sevomechanism Design Sevomechanism (sevo-sysem) is a conol sysem in which he efeence () (age, Se poin) changes as ime passes. Design mehods PID Conol u () Ke P () + K I ed () + KDe () Sae Feedback u() Kx() + S K x ( ) () Gain K is obained based on Pole Assignmen Opimal egulao design Sevo Seve / Slave Refeence sysem () Ax (), x () x y () Cx()
Refeence Sysem x () A x () x () x () c () Linea sysem which geneaes he efeence signal y () 3 Ex. Sep signal (Consan) () x( ) 3 () x (), x() 3 y( ) [ ] 3 y [ ] () x() { x2, x() 2, x2() Ex.2 Ramp signal (Polynomial of ) () x (), x() x() d + + 2( ) x y [ ] + () x() y() [ ] + Ex.3 igonomeic funcions x2, 2 x () (), () x2 x2, x2() x x x 2() cos, x() sin sin y [ ] () x() y () [ ] cos cos y x,
Sevo Poblem Obain a conol inpu u() so ha he sysem oupu y() coincides wih he efeence y (). () () () A x, x x, x R y () () () () Ax + bu, x x, x R y c () x() c () x () Deemine a conol inpu u*() so ha he sysem oupu y() y (),. y* () y () u*(): ideal inpu, x*(): ideal sae, y*(): ideal oupu *() Ax* ()+ bu* (), x* () x () A x(), x() x x* () c x () c n n
Goal y*( ) () Inenal Model Pinciple x () x() x* () x () y c * c cx( ) c x( ) x * x () () A * A () x + * () bu () A () + bu* () bu *( ) A A x () * ( ) ( ) c bs x () A A x () Inenal Model Pinciple A c A c b s x c Linea elaion u () s x () s Linea elaion b A A and s obained fom Inenal Model Pinciple give he ideal conol inpu u* ha achieves he Goal y*()y ()
Sevomechanism design based on Sae Feedback Conol x() x () : impossible in geneal x () x () x() x () x() * * ( ) u () * ε u () u() : diffeence beween ideal conol and eal conol x () * ε x () x () : diffeence beween ideal sae and eal sae Eo sysem () Ax () + bu (), x () x x ε ε ε ε Deemine uε so ha xε and uε conveges o zeo as fas as possible. Design Sae Feedback uεk xε fo Pole Assignmen fo A+b k Possible if (A,b) is conollable
Sevomechanism design based on Sae Feedback Conol u () k x () : Sae feedback law fo eo sysem ε ε uε u u * () () (), xε x x ( ) * * u u k x x () () () () * u s () x (), x * * () () () () x () coninued ( ) u () k x() + s k x () Feed fowad of efeence sysem sae Sae feedback fo oiginal sysem Obseve design
Block Diagam of Sae-feedback-based Sevo Mechanism ( s ) x u () k x() + k () Esimaed by Sae Obseve
Design Example Design a sevomechanism o follow he efeence wih convegence speed specified wih poles {-2,-3} fo he linea sysem: Seps fo soluion () () (), () 3 2 x + u x y () x() [ ] y () cos. Make a efeence sysem o geneae he efeence signal y () 2. Calculae and s fo he esulan sevo-sysem o saisfy he Inenal Model Pinciple A A bs, c c 3. Calculae he sae feedback gain k o ealize he specified convegence speed by pole assignmen 4. Simulae he behavio of he esulan sevomechanism (Make an augmened sysem fo () x() x () ) [ ]
Design example - sep () x (), x() y () x () [ ] x( ) sin x () cos 2 (Alenaive) x () y () cos, x () () sin 2 x () sin () () x, x 2() cos x2 () x2() () x (), x() y () x () [ ] x x 2 ( ) () cos sin
Design example - sep 2/3 xy, v zu w s A A, Sep 2 Pu s b c c ha is, And solve he pinciple: x y x y x y z u 3 2 z u z u 2, s 2 Sep 3 Oiginal chaaceisic polynomial is s 3 2 3 s + 2 2 s s s I + 2 + 3 [ v w], [ ] [ ] Fom he desiable poles {-2,-3}, he ideal chaaceisic polynomial is 2 ( s+ 2)( s+ 3) s + 5s+ 6 Hence he sae feedback gain k is obained as k [ k k ] [ 3 2 ] [ 3 3] 6 5
Design example - sep 4 ( ) Ax( ) + bu (), x() x, x R, y( ) x( ) A x x x x R () n () (), (),, y() c x ) k x( ) + x( ) u ( ( s k ) k : deemined so ha x ε x*-x quickly conveges o () Ax() + b k x() + k x () { ( s ) } ( ) ( ) ( A bk x + b s k ) x () + ( ) Ox() + A x ( ) ( s ) x () A+ bk b k x() x(), () () () O A x x x y () c x() y() () c x n () Ax (), (), x R 2 y () Cx (), y R c n+ n
Design example Design/Simulaion pogam /3 H Pocess L n 2; n 2; m ; A 88, <, 8 3, 2<<; b 88<, 8<<; c 88, <<; x 8, <; AJ 3 2 N bj N c^h L x_j N A_J N c_^h L x_j N H Refeence geneao L A 88, <, 8, <<; c 88, <<; x 8, <; Pin@"A", A êê MaixFom, " b", b êê MaixFom, " c^", c êê MaixFom, " x_", x êê MaixFomD Pin@"A_", A êê MaixFom, " c_^", c êê MaixFom, " x_", x êê MaixFomD
Design example Design/Simulaion pogam 2/3 H Inenal Model Pinciple L 88, 2<, 82, 22<<; s 88s, s2<<; Eq 8.A A. b.s, c. c<; Sol Flaen@Solve@Flaen@EqD, Flaen@8, s<ddd; ê. Sol; J s s ê. Sol; Pin@"", êê MaixFom, " s^", s êê MaixFomD N s^h 2 2L H Pole Assignmen fo Eo Sysem L ChCoef Flaen@ake@A, DD; H Coefficiens of chaceisic equaion L OpPole 8 2, 3<; OpChPoly ; H Opimal chaaceisic polynomial L Do@OpChPoly OpChPoly Hs OpPole@@iDDL, 8i,, n<d OpCoef 8<; H Opimal coefficiens L Do@OpCoef Append@OpCoef, Coefficien@OpChPoly, s, idd, 8i,, n <D Feedback gain fo σ^ 8 2, 3< is k^h 3 k 8ChCoef OpCoef<; Pin@"Feedback gain fo σ^ ", Ldes, " is k^", k êê MaixFomD 3 L
Design example Design/Simulaion pogam 3/3 H Make Simulaion Sysem HAugmened SysemL L M Join@A + b.k, able@, 8i, 2<, 8j, 2<DD; M2 Join@b.Hs k.l, AD; AA anspose@join@anspose@md, anspose@m2ddd; Pin@"AA", AA êê MaixFom, " CC", CC êê MaixFomD Pin@"Eigenvalues of AA", Eigenvalues@AADD M Join@c, able@, 8i, <, 8j, 2<DD; M2 Join@able@, 8i, <, 8j, 2<D, cd; CC anspose@join@anspose@md, anspose@m2ddd; H Rewie fom SaeEquaion L i n 4; vx able@x@id@d, 8i, n<d; AA vx Join@x, xd; j k f 3 Pi; Equ able@x@id @D HAA.vxLPi, 8i,, n<d; IC able@x@id@d vxpi, 8i,, n<d; DE Join@Equ, ICD; Sol NDSolve@DE, vx, 8,, f<d; y 6 5 5 5 z { CCJ N Plo@Evaluae@CC.vx ê. SolD, 8,, f<, PloRange All, Fame ue, FameLabel 8" @secd", "yhl & yhl"<, DefaulFon > 8"Helveica", 5<, GidLines Auomaic, PloSyle 88RGBColo@,, D, hickness@.3d<, 8RGBColo@,, D, hickness@.7d<<d
Refeence & Oupu.5 y HL &yhl -.5-2 4 6 8 @secd
Saes of efeence sysem & plan.5 x HL &xhl -.5-2 4 6 8 @secd
Response fo anohe iniial sae AJ 3 2 N bj N c^h L x_j N y HL &yhl.5 -.5 x HL &xhl.5 -.5-2 4 6 8 @secd - 2 4 6 8 @secd
Pole assignmen fo fase convegence Feedback gain fo σ^ 8 5, 6< is k^h 27 i y 3 29 A j z k { CCJ N 9 L y HL &yhl.5 -.5 x HL &xhl.5 -.5-2 4 6 8 @secd - 2 4 6 8 @secd
Pole assignmen fo slowe convegence Feedback gain fo σ^ 8.5,.6< is k^h 2.7.9 L i.... y.3..7. AA j z k { CCJ N y HL &yhl.5 -.5 x HL &xhl.5 -.5-2 4 6 8 @secd - 2 4 6 8 @secd
Response fo fase efeence A_J 2 2 N c_^h L x_j i 2 j N y z k s^i 2 { 2 M y HL &yhl.5 -.5 x HL &xhl.5 -.5-2 4 6 8 @secd - 2 4 6 8 @secd
Design Pacice Design a sevomechanism o follow he efeence wih convegence speed specified wih poles {-4±2j} fo he linea sysem: () () (), () 3 2 x + u x y () x() [ ] y () sin2