INPUT SHAPING FILTERS FOR THE CONTROL OF ELECTRICAL DRIVE WITH FLEXIBLE LOAD

Transcription

1 INPUT SHAPING FILTERS FOR THE CONTROL OF ELECTRICAL DRIVE WITH FLEXIBLE LOAD Goubej Martin, Skarda Radek and Schlegel Milo Deartment of Cybernetic, Unierity of Wet Bohemia in Pilen, Unierzitni 8, 36, Pilen, Czech Reublic fax : and cybernet@kky.zcu.cz Abtract: Thi aer deal with control of flexible mechanical ytem. The goal i to modify the inut ignal in order to minimize the reidual ibration excited during a motion of a ytem with flexible art. The filter i deigned in the time domain ia imule function analyi. Poible alication of the rooed olution i demontrated on two examle of flexible ytem - control of a with hanging and an electrical ero drie with attached flexible haft. The effect of nonlinearitie in the ignal ath caued by aturation of the ero loo controller i tudied. Variou oibilitie for the lacement of the filter are dicued. Keyword: Reidual ibration control, inut haing, with, flexible haft, motion control. INTRODUCTION The control of flexible tructure ytem uch a or robotic maniulator introduce eriou roblem with reidual ibration. Thee motioninduced ocillation are caued by the flexible art of the ytem and need to be attenuated in order to obtain recie behaior of the controlled ytem. Generally, there are three oible aroache to ure the unwanted ibration. Thee include mechanical daming, actie feedback control, oen-loo filtering method and their ariou combination. Mechanical comonent uch a ilent-block or ring-damer module can be introduced into machine deign in order to increae the tiffne of the contruction. Howeer, it i difficult to redict a dynamical behaior of the machine in the hae of deign; moreoer, mechanical damer are difficult to tune and mean additional exene. Cloed-loo actie daming method can achiee ery good reult becaue of the feedback, which uree nonlinearitie of the ytem and uncertainty in the mathematical model (Schlegel and Vecerek (3)), (Mertl et al. ()). The main diadantage of thi aroach i the neceity of feedback enor, comlicated controller deign and higher comutational cot. On the other hand, the oen-loo filtering method ue relatiely imle algorithm to modify the inut command in the feed forward ath in uch a way that the reulting inut ignal led to the ytem doe not excite the unwanted tranient and reidual ocillation. The adantage i imle deign and abence of feedback enor on the lant. The main drawback i the reduced robutne againt uncertainty in the ytem model reulting from oen-loo aroach. Thi aer deal with the lat mentioned aroach and ue o called Zero Vibration filter (ZV filter) (Singer and Seering (99)) for command haing of flexible tructure ytem (Huey et al. (7)), (Chen et al. (8)), (Sorenen and Singhoe (8)). The theory i alied to the roblem of the motion control of electrical ero drie with attached with or flexible haft.

2 . INPUT SHAPING FILTER DESIGN A general n-ule inut haing filter can be decribed in the form of imule function: n IS(t) = A i δ(t t i ), t < t i+, A i () i= Where A i mean amlitude of the i-th ule and δ i dirac function with t i time hift Reone of the haer in time domain can be determined by conolution with continuou inut ignal: (t) = = = h(τ)u(t τ)dτ ( n ) A i δ(τ t i ) u(t τ)dτ i= n A i w(t t i ) i= () It can be een, that the filter ha the form of um of time delayed alue of the inut weighted by coefficient A i. The original inut command i conoluted with the inut haer and the reulting ignal i then led to the controlled ytem. Thi ituation i illutrated in Fig.. The imule reone of erial connection of IS() and P () ha the form h(t) = A h (t t ) (t t )+A h (t t ) (t t ) (6) (t) i Heaid function. For time t > t, it hold y(t) = A h (t t ) + A h (t t ) ω = e ξωt C + S in(ω d t ψ) ξ C = S = A i e ξωti co(ω d t i ) i= A i e ξωti in(ω d t i ) i= ψ = arctan S C (7) (8) It can be een, that for minimizing the leel of reidual ibration after the econd ule, the following exreion ha to be fulfilled C + S = (9) By ubtituting (8) to (9), we obtain a nonlinear equation for A i and t i, i =,. With roer choice of alue t =, t = π ω d () the equation can be reduced to the condition ( A A e ξω π ω d ) = () Fig.. Reone of the ytem without and with ZV filter. The goal of the filter deign i to chooe the alue of amlitude A i and time delay t i uch that after the lat ule ha been led to the ytem, the amlitude of excited reidual ibration i equal to zero. The deign rocedure will be illutrated for - ule ZV filter and econd-order ytem. Conider linear ytem decribed by tranfer function ω P () = + ξω + ω (3) with imule function ω h (t) = ξ e tξω in(ω d t) () and two-ule haer with imule function IS(t) = A δ(t t ) + A δ(t t ) () Next, we get the econd condition becaue of a requirement of the unit tatic gain of the filter A + A = () By oling the algebraic equation () and () we get arameter of the filter: A = + K, A = K + K K = e ξπ ξ t =, t = π ω d, ω d = ω ξ (3) It can be hown that thoe alue are alid alo for the ytem with tranfer function in the form P () = τ + ω + ξω + ω () For the roer function of the ZV filter, the exact alue of natural frequency and daming coefficient of the ocillatory art of the ytem ha to be

3 known. The error in the ytem model reult in non-zero reidual ocillation. If the model of the ytem cannot be determined exactly, more robut erion of the haer can be deigned by adding additional condition. [ C + S ] = () ω The reulting three-te haer i o called Zero Vibration Deriatie (ZVD) filter and achiee le enitiity with reect to arameter ariation at the cot of lower etoint reone of the ytem. _ Kfi _ Po.control Seed control Ki Gain K Gain Integrator i* i Current control Fig.. Schematic of the electrical ero drie in Simulink. u(t) te ZV haer A Gain 3 _a T.+ y(t) Kt Gain 3 Mz 3. CRANE WITH LOAD Tranort Delay t A Gain Fig.. Schematic of -te ZV filter in Simulink. Fig.. Schematic of the and ibration of it. * InOut ZV IS _a CRANE /M Ma a Friction coeff. Integrator Integrator Tau.+Om^ +*Ki*Om+Om^ LOAD Integrator Scoe modeled a the firt order ytem, the eed and oition control i realized uing tandard PI and P controller. The ero ytem contain alo the feed forward inut for lanned trajectory following. A human oerator et the deired oition of the. Without uing an inut haing filter, the motion of the induce the reidual ibration of the (Fig. 6). The winging can hit ome obtacle and an oerating eronnel ha to wait before the ocillation dam out and they can continue with a maniulation. Uing the ZV inut haer (Fig. ), the oerator command leading to the ero drie i filtered (Fig. 7) and the reulting moement doe not excite the ocillatory dynamic of the (Fig. 8). B Fig. 3. Schematic of the with electrical drie in Simulink. Moement of the without inut haer * Conider an electrical drien cargo with hanging (Fig. ). Such a ytem i deicted in block diagram in Fig. 3. It conit of electrical art of the drie with oition and elocity feedback and an ocillatory econd order ytem decribed by tranfer function P () = V l() V c () = τ + ω + ξω + ω (6) ω, ξ are natural frequency and daming of the ytem, V c i the eed of the, V l i the angular eed of the hanging The ytem (6) decribe the relationhi between motion of the and it. The force acting on the caued by the moement of the are omitted with reect to the ratio of it ma. The ubytem of the drie conit of three control loo in commonly ued cacade tructure (Fig. ). The dynamic of the current loo i & oition t[] Fig. 6. Moement of the without ZV-filter. The great benefit of the rooed technique i, that haer deign can be made for the econd order ytem of the winging and it filtering roertie tay unchanged een after aing through the dynamic of the ero drie. Howeer, thi reumtion i alid only in the cae, that all the control loo of the drie work in the linear mode. If any of the controller hit the aturation limit, the original filtered command ignal

4 inut command 8 6 Inut command haing original command ZV filter haed command t[] Fig. 7. Comarion of the original and haed command ignal. & oition 8 6 Moement of the with ZV IS * t[] the ret-to-ret moement with reect to limitation for elocity, acceleration and deceleration (otionally alo with contraint for deriatie of acceleration and deceleration). The ignal of deired oition, elocity and acceleration led to the feed forward inut of the ero controller enure the correct tracking of the lanned trajectory and alo reent the aturation effect. Thi configuration i dilayed in Fig.. The block AVS comute the deired trajectory, which i led to the ero drie controller. The inut haer deign remain the ame, the filter ha to be laced before all of the ero control inut. Reone of the ytem can be een in Fig.. There are no reidual ibration een in the reence of nonlinearitie in the ero control loo. All the controller work in the linear mode, becaue of trajectory lanning block. The reulting trajectory of the moement i no more time-otimal, neerthele, it i ubotimal with reect to the demand for attenuation of winging ibration. The imortant notice i that the filtered trajectory ignal doe not iolate the default contraint for elocity, acceleration and deceleration due to the unity gain of the filter. Figure how the difference between t-otimal and real cure of oition, elocity and acceleration. The reented examle can be reduced eaily to the cae, only the elocity of the and hould be controlled. Fig. 8. Moement of the with ZV-filter. i corruted and the reult i non-zero amlitude of the excited reidual ibration. Thi ituation i deicted in Fig. 9, aturation ha been laced to the current control loo. It can be een, that the ibration ha not been canceled out comletely due to the nonlinearity in the drie control loo. ZV inut haer with aturated ero controller * * * AVS a TRAJECTORY GENERATOR ZV FILTERS In Out ZV IS In Out ZV IS In Out ZV IS3 _a CRANE /M Ma Friction coeff. a Integrator Integrator Tau.+Om^ +*Ki*Om+Om^ LOAD Integrator Fig.. Schematic of the with electrical drie and t-otimal trajectory generator (block AVS) in Simulink. B Scoe & oition 8 6 oition Filtered t otimal trajectory t otimal trajectory haed trajectory elocity t[] Fig. 9. Moement of the with ZV-filter and aturated ero controller. A oible olution of thi roblem i to add an uerior ytem for trajectory lanning. After the human oerator et a new deired oition, thi block comute the time otimal trajectory for acceleration t[] Fig.. T-otimal and ZV filtered trajectory for the.

5 & oition & oition T otimal moement with ZV haer T otimal moement without inut haer * t[] Fig.. Comarion of the time-otimal and timeotimal ZV filtered trajectory of. * * AVS a TRAJECTORY GENERATOR ZV FILTERS In Out ZV IS In Out ZV IS In Out ZV IS3 [] From [] From _a T /J Gain 3 DRIVE SHAFT /J Gain Gain B B Gain DRIVE LOAD Integrator FLEXIBLE JOINT Integrator Integrator Integrator 3 Fig.. Schematic of the flexible haft with the electrical drie and ZV filter in Simulink. be deried from the block cheme (Fig. ). The dynamic of the current loo can be omitted, becaue the electrical time contant of the drie i negligible with reect to mechanical time contant of the whole ytem. K [] Goto [] Goto θ () () = num() den() (8) Fig. 3. Schematic of the flexible haft.. FLEXIBLE SHAFT Conider a ytem coniting of electrical drie from the reiou examle and attached flexible haft (Fig. 3). Dynamic of the ytem can be decribed by the et of two equation J θ + B θ = T K(θ θ ) J θ + B θ = K(θ θ ) (7) J, J are moment of inertia of the motor and the haft B, B are coefficient of daming K i torion ring contant T i torque θ, θ are angular dilacement of motor and the haft The goal of the control ytem i to control the oition of the haft without exciting the reidual ibration on the flexible end. Here, the deign of the filter i not o traightforward a in the reiou cae. The back roagation of the torque from the end of the haft to the drie can not be omitted analogouly to the. The entire ytem including the drie and flexible feedback ha to be analyzed in order to find the frequencie, which hae to be attenuated. Tranfer function between etoint command and angle of the flexible end of the haft can and num() = KK θ K t K + KK θ K t K I den() = J J + (B J + J [K t K + B ]) + [KJ + B (K t K + B ) + J (K t K I + K θ K t K + K)] 3 + [K(K t K + B ) + J KθK t K I + B (K t K I + K θ K t K + K)] + [K(K t K I + K θ K t K + K) + B K θ K t K I K ] + KK θ K t K I K t, K P, K I, K θ are arameter of controller i oition etoint From the obtained tranfer function, the location of the ole can be analyzed. For the tyical configuration of the arameter, ero control loo are et to achiee table etoint reone without oerhoot and reumtion of J > J, the reulting ytem ha two air of ocillatory ole and one table real ole. In mot cae, only the lower ole air lying cloer to the imaginary axi need to be canceled out by the filter. The natural frequency and daming can be eaily comuted from the location of the ole. Next, the ZV filter can be deigned. If the econd air of ocillatory ole till caue unaccetable leel of ibration, a econd ZV filter for fater ole can be added and erial-connected to the firt one. The roblem with nonlinearitie in the control loo due to aturation of ero controller remain the ame and alo the olution i identical to the - roblem. For the roer function of the filter, trajectory lanning block hould be added

6 to atify the demand for the linear function of the ero loo controller. ability to ure the meaurement noie (Huey et al. (7)). drie&haft oition drie&haft oition Flexible haft control drie oition flexible haft oition command t[] & oition & oition ZV haer in the cloed loo * t[] Fig.. Comarion of the flexible haft moement without and with ZV filter. Figure how the reult of flexible haft control. Figure on the to illutrate the control without the inut haer, the lower one with the ZV filter. Uing the filter, the ibration ha been comletely canceled out.. SHAPING FILTER IN THE CLOSED LOOP _ Kfi Po.control _ Seed control Ki K Gain Gain 3 _a Integrator i* i inut filter Artifical aturation InOut ZV IS Real aturation T.+ Current control Fig. 6. Schematic of ero loo controller with ZV filter. The inut haing filter can be laced alo in the cloed loo of the ero drie. The mot uitable oition for the filter i the eed controller loo (Fig. 6). The adantage of thi aroach i, that the aturation effect of the oition and eed loo doe not affect the filtering roertie of the haer. For eery real electrical drie, the current loo contain aturation limit due to the limited uly oltage of the ower inerter. Therefore, the alue of the maximal current that the inerter i able to delier to the motor i limited. Thi maximal alue change with actual rotational eed of the drie becaue of the back electromotie-force acting againt the ower uly. For the roer function of the filter, an artificial aturation block hould be laced before the filter (Fig. 6). The aturation limit hould be choen in uch a way that the outut of the filter neer hit the limit of the real aturation of the current loo. Thi enure that the frequency band of the ignal remain unaffected and the reulting motion of the drie doe not excite the ibration of the flexible. Next, there exit ome tudie indicating the Kt Gain 3 Fig. 7. Sytem reone a) without ZV filter b) with ZV filter in cloed loo. The main drawback i that the inertion of the filter introduce an additional dynamic, which can affect the behaior of the cloed loo. For the large natural frequencie, which hae to be attenuated, the delay of the filter can be mall with reect to the dynamic of the drie. Smaller alue of the natural frequency mean lower filter, which ha to be taken into account while tuning the loo controller. The effect of the filter inide the cloed loo i illutrated in the (Fig. 7). The figure on the to rereent the moement of the with without the inut haer. The drie controller are tuned in order to achiee fat etoint reone without oition oerhoot. After introducing the filter to the cloed loo, the oerall dynamic of the drie ha changed and the reult i the ocillatory moement of the a dilayed in the lower figure. Een if the drie doe not excite the reidual ibration of the hanging, the controller need to be re-tuned in order to achiee deired etoint reone. The integration of the filter inide the cloed loo can caue een the untable dynamic. It can be aumed that by adding an additional delay to the ytem, the oerall dynamic of the cloed loo become wore. 6. CONCLUSION Thi aer reent the alication of the inut haing filter to the control of electrical ero drie with attached flexible. The goal i to minimize any tranient and reidual ibration induced by the moement of the drie. Firtly, the Zero Vibration filter i deried. The next art reent it utilization for the control of a with hanging and a drie with attached flexible haft. The reult how ignificant imroement of the cloed loo behaior and attenuation

7 of the unwanted ocillation. The ZV filter can be eaily imlemented in a real time control ytem and ome tudie how it better erformance comared to conentional notch filter (Huey et al. (7)). In the cae of model uncertainty, more robut filter erion can be deigned at the cot of increaing the delay of the filter. Next, the effect of aturation of the ero loo controller i dicued and the olution i rooed by adding a trajectory lanning block. The lat art deal with the oibility of lacing the filter inide the cloed loo of the drie. The adantage and drawback of thi aroach are dicued. Parameter of ero drie controller (8) K ϕ = K = 3 K t = K I =. Reference K-S Chen, T-S Yang, K-S Ou, and J-F Yin. Deign of command haer for reidual ibration ureion in duffing nonlinear ytem. Mechatronic, 8. doi:.6/j.mechatronic J.R. Huey, K.L. Sorenen, and W.E. Singhoe. Ueful alication of cloed-loo ignal haing controller. Control Engineering Practice, 6: , 7. J. Mertl, M. Schlegel, M. Cech, and P. Balda. Aktini tlumeni ibraci loatek aerodynamickem tunelu. M. Schlegel and O. Vecerek. Samonataujici e regulator ro elmi labe tlumené ytemy N.C. Singer and W.P. Seering. Prehaing command imut to reduce ytem ibration. Journal of Dynamic Sytem, Meaurment and Control, :76 8, 99. K.L. Sorenen and W.E. Singhoe. Commandinduced ibration analyi uing inut haing rincile. Automatica, :39 397, 8. APPENDIX Parameter of with model (6) ω n = rad ξ =. τ =. M = kg Parameter of flexible haft model (7) J = kg m J = kg m K =. B, B =.