Mechanical Systems Part B: Digital Control Lecture BL4

Transcription

1 BL4: Mechancal Systems Part B: Dgtal Control Lecture BL4 Interretaton of Inverson of -transform tme resonse Soluton of fference equatons Desgn y emulaton Dscrete PID controllers

2 Interretaton of BL4: We saw that Z { } X Hence, a system wth a transfer functon of erforms a unt elay acwar shft oeraton Bloc agram: U Y U Smulaton agram: u q y u

3 Inverson of -transforms BL4: 3 By formula: X πj contour ntegraton not easy! By artal fracton eanson an tale of -transforms: teous for all ut elementary transforms By long vson: we have that X thus, gven X as a rato of olynomals n, we coul ean t as a ower seres y long vson to gve: X By smulaton n Matla:» mulsex

4 Inverson usng artal-fracton eanson Eamle: X a e a e BL4: 4 Most elementary -transforms have a ero at Hence reserve ths form y eanng X Hence, X a e a e a e a e hus, e a

5 Eamle of nverson usng long vson X BL4: 5.e., X Hence, {,.,.36,.448, }

6 Eamle, cont In fact, X n ths eamle s of the form X a e a e wth a.3 Carryng out the long vson n ths algerac form t s straghtforwar to show: BL4: Samle a a 3a 3 e e e X hus e a {,.,.36,.488,.59,.67,.738, }

7 Soluton of fference equatons y transform methos Just as we can solve lnear ornary fferental equatons y Lalace transform methos, so we can solve lnear fference equatons y - transform methos o hanle ntal contons, we ntrouce the Z one-se -transform: { } Shft theorem for one-se transform: Z { } m m m m BL4: 7

8 BL4: 8 Hence: { } X Z In general: { } X n n Z However, { } m m m m m m m m m Z Hence: { } X Z In general: { } n n n X n Z e.g. { } X Z

9 Eamle: Solve the fference equaton u.u.u e e where e for,,, ; u an u. -transform the fference equaton:.. U E u u.u e U E.... Inut: e E Resonse: U Hence: U e BL4:

10 Restore numerator : U.5.4 Recall from BL3:7 that { a } Z e a e - Hence Z he controller [ u ] outut s then:,,.,.,.6,.7,.79, 3 Comlete soluton { } 3 Comonents of soluton BL4: Resonse, u Samle,.857* * * NB: oscllaton at Nyqust frequency Samle,

11 Desgn y emulaton Frst, esgn a satsfactory contnuous comensator erhas allowng for the ZOH elay of / y nclung ths n the lant moel hen, gte the contnuous control law Several fferent aroaches to gtsaton are avalale. o llustrate, conser an ntegral control law: U s E s s Recall that any transfer functon can e realse as an nterconnecton of smle ntegrators In contnuous tme: u t t τ e τ τ Hence, at the -th samle nstant: At the net samle: u u u τ τ e τ τ e τ τ BL4:

12 We have: u u τ e τ τ he ntegral over one samle ero can e aromate n fferent ways; e.g. BL4: Euler's forwar rectangular rule eτ e e Euler's acwar rectangular rule eτ e e ustn's lnear or traeoal rule eτ e e τ τ u u e e e e U E E E/ U E τ

13 Dscretsaton of contnuous controller Hence, to scretse contnuous transfer functon usng: Euler s metho forwar rectangular rule relace s wth / Bacwar rectangular rule relace s wth / ustn s metho traeoal rule or lnear transformaton relace s wth Another alternatve s the Matche Pole-Zero metho see F, P & E-N match oles an eros usng e s match.c. gan BL4: 3

14 Matla functon c BL4: 4 SYSD CDSYSC,S,MEHODconverts the contnuous system SYSC to a screte-tme system SYSD wth samle tme S. he strng MEHOD selects the scretaton metho among the followng: 'oh' Zero-orer hol on the nuts. 'foh' Lnear nterolaton of nuts trangle a. 'tustn' Blnear ustn aromaton. 'rewar' ustn aromaton wth frequency rewarng. he crtcal frequency Wc s secfe last as n csysc,s,'rewar',wc 'matche' Matche ole-ero metho for SISO systems only. he efault s 'oh' when MEHOD s omtte. N.B. the 'oh' an 'foh' methos are not arorate for emulaton esgn -- they are for gettng a screte moel of a contnuous lant, who's nut comes from a ZOH or FOH DAC outut s samle see net lecture

15 Emulaton esgn of a PID controller he tetoo verson of a contnuous PID controller oerates on the error sgnal only: BL4: 5 U s E s K s s PID Plant R E U B s Y G c s A s A more ractcal esgn: ales Dervatve acton to the outut Y only, thus avong fferentaton of ra set-ont changes flters the Dervatve acton, so as not to amlfy hghfrequency measurement nose ales the Proortonal acton to only a fracton of the comman sgnal R, allowng control over the oston of the ero n the controller transfer functon

16 Practcal PID controller Wth these mofcatons, the controller actons are: R P K R Y K [E R] I K / s E D K s/ s/n Y E s I P D K U BL4: 6 s s / s N Y

17 ransfer functon of ractcal PID controller Y s R s K sa s K s B s [ s s ] B s BL4: 7 R E P I s D K U Re-arrange loc agram shows feeforwar effect of set-ont weghtng factor s s / N Zero at s / Y

18 Dscretaton of PID Varous methos are use. A oular one s: Proortonal acton P t K r t y t [ ] [ r y ] P K BL4: 8 Integral acton, aromate y Euler s metho K t K I t e τ τ I I e Dervatve acton, aromate y acwar fferences D t y t D t K N t t K N D D y y N N Control sgnal u t P t I t D t u P I D [ ]

19 K P K [ r y ] I I e K N D D [ y y ] N N BL4: 9 Intalsaton I ; D y_ol a /N* K*N*a K*/ Imlementaton of gtal PID controller See Åström & Wttenmar, Comuter Controlle Systems, 3r en,. 38, for C-coe mlementaton Cloc nterrut? Y Rea ADC r,y P K**r-y D a*d - *y-y_ol u P I D Wrte u DAC I I *r-y y_ol y N Kee ths elay to a mnmum

20 BL4: Dscrete transfer functon of ths emulaton of PID It s straghtforwar to show that hs mlementaton has the avantages that: the P, I an D contrutons to the control can e comute searately the controller s always stale the samle ole a goes to ero as See Franln, Powell & Worman 3.3 for an alternatve aroach to the tetoo PID controller [ ] [ ] Y a a a K a K K R a K a a K K U K Na K N a /,, where