IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM

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1 Abstract IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM Alca Esparza Pedro Dept. Sstemas y Automátca, Unversdad Poltécnca de Valenca, Span The dentfcaton and control of a flexble transmsson usng the routnes avalable on the book webste s presented n detal. Identfcaton n closed loop s used n a second step for mprovng the qualty of the dentfed model and the acheved performance of the real system. An appendx provdes the results obtaned wth the data avalable on the book webste for the dentfcaton of a flexble transmsson (wth dfferent dynamcs).. Introducton The flexble transmsson system s depcted n fgure a and a vew of the system s shown n fgure b. It conssts of three pulleys joned by two flexble bands. The movement of the whole system s carred out by a DC motor whch s allocated next to the frst pulley and produces ts axs movement. Ths movement s transmtted to the other pulleys by the flexble lnkng bands. The objectve of the whole system s to control the angular poston of the thrd pulley. The DC motor has a servo-controller whose dynamcs s very fast comparng to that of the pulleys. The control of the system s carred out by a computer, whch s connected to the flexble transmsson by a nput-output acquston board, beng ts nput connected to the poston transducer, and ts output to the servo of the DC motor. The software used ether to gather data or to apply a controller to the system s the Real Tme Toolbox of Matlab. The consdered samplng frequency s 2 Hz. Fgure a. Flexble transmsson system Ths flexble transmsson has dfferent dynamc characterstcs wth respect to the one consdered n the book Dgtal control systems desgn, dentfcaton and mplementaton (see chapters 7, 8 and 9)

2 Fgure b. Vew of the flexble transmsson system 2. Identfcaton n open loop In order to obtan a model of the flexble transmsson, a set of data has been gathered from the system operatng n open loop (.e. wthout the RST controller). The nput appled to the system s a pseudorandom bnary sequence (PRBS) generated wth a 7 cell shft regster, and a frequency dvsor of 2 (p = 2). Two of such PRBS sequences have been appled to the plant nput (.e. a 58 data length vector). The centered nput-output data gathered from the real plant s stored n the fle flextransol.mat. The frst decson to be made s about the order of the model. Based on the physcal characterstcs of the system one can thnk that there must be two resonant modes and thus the model order must be four. But there s another way to decde about the model order. The Matlab functons estorderv.m or estorderls.m can be used to estmate the best order of the dentfed system., Usng estorderv.m, the result shown n fgure 2 s obtaned, pontng n fact at a fourth order system Norm max(na,nb+d) Fgure 2. Model order estmaton wth estorderv.m

3 A frst dentfcaton usng Recursve Least Squares (rls.m) shows that the correct model order s na=4, nb=3, d=. The dentfcaton and valdaton results are shown n fgure 3. B =[ ] A =[ ] Valdaton Whteness test.8 Whteness test for the OL dentfcaton Theoretcal lmt: Practcal lmt: WRN() max( WRN )=.845 >.5 Model not valdated Fgure 3. Results usng Recursve Least Squares method As the obtaned model does not pass the valdaton test, a new one must be sought. Another dentfcaton method, the Extended Output Error algorthm (xoloe.m), s used ths tme, consderng nc=na=4, nb=3, d=. Fgure 4 shows the obtaned results. B=[ ] A=[ ] C=[ ] Valdaton Whteness test 5 Theoretcal lmt: Practcal lmt:.5 Whteness test for the OL dentfcaton WRN() max( WRN )=.54 <.5 Model valdated Fgure 4. Results usng Extended Output Error method

4 As can be seen, the Extended Output Error algorthm n open loop has succeeded n dentfyng a model for the plant. 3. Controller desgn Once the plant has been dentfed, a controller can be desgned n order to provde a good closed loop behavor. To do ths, the pole placement methodology has been chosen, and the Matlab program ppmaster has been used. The specfcatons are: Robustness: Modulus margn greater than.5, delay margn greater than a samplng perod. Regulaton dynamcs: - Domnant poles: two complex conjugate poles at the same frequency as the plant frst resonant mode, but wth a dampng factor of.8. - Two auxlary complex conjugate poles at the same frequency as the second resonant mode, wth a dampng factor of.5. - One ntegrator and one zero at -, the last openng the loop at hgh frequences. - Four addtonal auxlary poles at.3. Trackng: t s consdered T=P()/R(), and the same dynamcs as for regulaton (so Am=Bm=). The computed controller s dsplayed n fgure 5. R=[ ] S=[ ] T=.255 Fgure 5. RST controller obtaned by pole placement The system s closed loop step response has been smulated usng Smulnk. The block dagram s shown n fgure 6. All the correspondng transfer functon blocks are dscrete flters (whose coeffcents are ascendng powers of z - ). Fgure 6. Smulnk block dagram used for smulaton The smulated and the real closed loop behavor are shown n fgure 7, where a bg dfference between the desgned and the acheved closed loop s evdenced. Ths means that the plant model s not good enough n the frequency regons where the controller s operatng. So a new dentfcaton s needed. Then a closed loop dentfcaton wll be carred out for mprovng the model qualty.

5 .4 Step response.2 Real Ampltude Smulated Tme (sec) Fgure 7. Smulated and acheved closed loop behavour wth the RST controller 4. Identfcaton n closed loop To dentfy a new model whch explans the observed closed loop behavor, a new dentfcaton experment s carred out. The experment has been done n closed loop, usng the RST controller desgned above (fle RSTfrst.mat). The reference has been mposed to zero, and the same PRBS as that used for the open loop dentfcaton procedure has been appled to the plant nput (ru, as shown n fgure 8). The gathered data have been recorded n the fle datidcl.mat. Ths fle stores the exctaton sgnal (ru), the plant nput (u) and ts output (y). + T /S - ru + + u R Plant y ru S + - u Plant R/S y Fgure 8. Closed loop dentfcaton setup. Equvalent schemes To dentfy a new model, the Closed Loop Output Error wth Adaptve Fltered Observatons (afcloe.m) algorthm s used, wth the prevously desgned RST controller and the same orders as for the open loop dentfcaton. As the exctaton sgnal s appled to the controller output, n afcloe.m functon T must be set as S (see fgure 8). The frst dentfcaton gves the followng parameters: B=[ ] A=[ ]

6 One can see that the frst parameter of B s neglgble wth respect to the other ones. So, the dentfcaton s repeated wth na=4, nb=2, d=2. In order to valdate the plant models dentfed n closed loop, the whole closed loop s dentfed n open loop, usng the recorded sgnals y and ru. To carry out ths dentfcaton, an OLOE method has been used wth the orders na=9, nb=7, d=2 (the same orders as the computed closed loop system). As the resultng model passes the open loop correlaton test, ts poles have been consdered for comparson wth the followng closed loop dentfed plant models. B=[ ] A=[ ] Valdaton Uncorrelaton test 5.5 Uncorrelaton test of error and yhat Theoretcal lmt: Practcal lmt: o: dentfed poles; x: computed poles RN() max( RN )=48 >.5 Model not valdated B=[ ] A=[ ] Valdaton Uncorrelaton test Uncorrelaton test of error and yhat Theoretcal lmt: Practcal lmt:.5 Fgure 9. AFCLOE model o: dentfed poles; x: computed poles RN() max( RN )=. 492 >.5 Model not valdated Fgure. FCLOE model The result of the new plant dentfcaton usng the AFCLOE method s not successful, as shown n fgure 9, and a Closed Loop Output Error wth Fltered observatons (fcloe.m) algorthm s tested, usng the model prevously dentfed wth the AFCLOE method as the

7 startng one. The model dentfed usng the FCLOE method does not valdate ether (fgure ). Due to the unsatsfactory results t s necessary to try another algorthm: the Extended Closed Loop Output Error (XCLOE) s then used. Ths one fnally succeeds n provdng an acceptable model, as shown n fgure. From ths fgure and n the prevous fgures, t s possble to observe that the closed loop poles computed on the bass of the dentfed plant model are very smlar to those of the closed loop model dentfed n open loop. Wth the XCLOE one has the domnant poles (those on the rght part whose natural frequency s less than.s) very close to the domnant poles of the dentfed closed loop. B=[ ] A=[ ] H=[ ] Valdaton Uncorrelaton test.6.4 Uncorrelaton test of error and yhat Theoretcal lmt: Practcal lmt:.5.8 o: dentfed poles; x: computed poles RN() max( RN )=. 23 <.5 Model valdated -.8 Fgure. XCLOE model Fgure 2 shows the frequency response of both dentfed models: the one dentfed n open loop s drawn n dashed blue lne and the dentfed n closed loop s n sold red lne. 5. Controller desgn Agan, a controller s desgned usng ppmaster, and followng the same specfcatons as n secton 3. Ths tme, the consdered model s that dentfed n closed loop usng the XCLOE method (see secton 4). The controller parameters are stored n the fle RSTmodXCLOE.mat, and are: R=[ ] S=[ ] T=.7547 The expected and the acheved step responses are depcted n fgure 3, where just a slghtly dfference can be observed.

8 4 Bode Dagram 2 Magntude (db) Phase (deg) Frequency (rad/sec) Fgure 2. Comparson between the open loop (dashed) and the closed loop (sold) dentfed models.4 Smulated and real step response wth full controllers Tme (sec) Fgure 3. Smulated (red) and real (blue) step responses wth the RST controller desgned consderng the closed loop dentfed model

9 6. Appendx: Identfcaton of the flexble transmsson n closed loop usng the data from the webste In ths secton, the closed loop dentfcaton of a model for the flexble transmsson s revsted usng the data avalable on the book s webste. The same fle wth the centered data has been renamed (flexbook_bfdat.mat). The RST controller used to gather the data n closed loop operaton s stored n the fle RSTflexbook.mat. The whole closed loop s dentfed n open loop n order to compare ts poles wth those computed wth the new plant model. To obtan ths model a XOLOE has been appled, wth the orders na=8, nb=6, d=2. The frst dentfcaton s carred out usng the AFCLOE method, and the results are shown n fgure 4. B=[ ] A=[ ] Valdaton Uncorrelaton test 5 Uncorrelaton test of error and yhat Theoretcal lmt:.3563 Practcal lmt:.5.8 o: dentfed poles; x: computed poles RN() max( RN )= >.5 Model not valdated Fgure 4. Results for the model dentfed usng the AFCLOE algorthm Then a CLOE algorthm s tred out, gvng the results shown n fgure 5. It s worth to pont out that the closed loop poles are more far away from the dentfed ones than the prevous model. The maxmum value of the uncorrelaton test s also worse than the prevous one. The followng step s to try to dentfy the plant model usng the FCLOE method, consderng the prevously dentfed AFCLOE model as the startng one for ths new dentfcaton. The result s shown n fgure 6. Ths model s valdated, and t s possble to observe n the plot on the rght that the domnant closed loop poles (those wth a natural frequency less than s) are closer to the dentfed ones than the models prevously dentfed n ths secton.

10 B=[ ] A=[ ] Valdaton Uncorrelaton test.5 Uncorrelaton test of error and yhat Theoretcal lmt:.3563 Practcal lmt:.5.8 o: dentfed poles; x: computed poles RN() max( RN )= >.5 Model not valdated Fgure 5. Results for the plant model dentfed wth the CLOE algorthm B=[ ] A=[ ] Valdaton Uncorrelaton test.6.4 Uncorrelaton test of error and yhat Theoretcal lmt:.3563 Practcal lmt:.5.8 o: dentfed poles; x: computed poles RN() max( RN )= <.5 Model valdated Fgure 6. Results for the plant model dentfed usng the FLOE algorthm Acknowledgment Ths work was sponsored by Investgaton Incentve Programme of the Unversdad Poltécnca de Valenca (Span) and has been done at Laboratore d Automatque de Grenoble (CNRS/INPG/UJF).

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