Chapter 4. Simulations. 4.1 Introduction

Transcription

1 Chapter 4 Simulation 4.1 Introdution In the previou hapter, a methodology ha been developed that will be ued to perform the ontrol needed for atuator haraterization. A tudy uing thi methodology allowed the invetigation of the limitation that may be enountered while applying the ontroller experimentally. Thee limitation not only onern the operational range of the atuator ued a a ontrol atuator but alo problem of reonane of the tet bed itelf. Therefore, before going any further into the proe of developing the real time ative ontroller, it may be worth pending ome time reating different model that hould give a better inight into problem that may arie in experiment. Firt, a model of a tak atuator will be developed and applied to the theory of ontrol to help draw ome onluion on ontrollability limitation. Then, a imulation method uing experimental time hitory meaurement from a preliminary tet et-up will be onidered. Thi imulation will alo help to determine the ontrol limitation. 32

2 4.2 Simulation Uing a Model of the Atuator Model of a tak atuator A a preliminary tudy, a imple model of a tak atuator working againt a load ha been hoen to invetigate the limitation of the ontrol methodology developed in Chapter 3. A tak piezoeletri atuator i a train devie that trethe or ontrat when a voltage (of appropriate polarity) i applied to it [27]. The tak atuator an be modeled a a fore (alled F ) whih at in parallel to a pring and a damper. The fore, F, i alo alled the bloked fore. It repreent the amount of fore the atuator produe when uffiiently lamped on both ide uh that there i no diplaement. The tiffne (alled K ) of the pring model the internal tiffne of the atuator and the damper (whoe damping oeffiient i alled C ) model the trutural damping of the atuator. A the atuator i allowed to train, the fore due to the pring and the damper at againt the bloked fore to redue the overall fore output (alled F m ) of the atuator (Cf: Figure 4.1). Load v F m F K C Sample atuator Support Figure 4.1: Model of tak atuator 33

3 Uing a frequeny domain formulation, the total fore output of the atuator ating at a frequeny, ω, i given by: F Kv = F + jω m + C v (4.1) Where v i the veloity of the load on top of the atuator (Cf: figure 4.1). The fore and the veloity of the load are related to it own mehanial impedane. Uing bai mehani of material relationhip [26] to expre the internal tiffne in term of parameter uually given for an exiting tak atuator, a new expreion for K i: EA K = (4.2) L where E i Young modulu for the atuator, A i it ro-etional area and L i it length. For preliminary analyi, the trutural damping oeffiient, C, will be aumed to be proportional to the pring ontant K with the following relationhip: C=Kη/ω (4.3) where η, alo alled trutural damping fator, i a oeffiient whih i uually maller than unity.(i.e. typially η=0.1 for a moderately damped ytem). Finally, uing equation developed for piezoeletri material [27], an expreion for the bloked fore produed by the tak piezoeletri atuator ued a the ample atuator an be obtained. A the longitudinal fore and diplaement of the atuator are the only parameter of interet, the bloked fore will be equal to: F =K d 33 (4.4) 34

4 where d 33 i a piezoeletri train ontant for the ample atuator and i the eletrial input (voltage) to the atuator. Figure 4.2 how a model of the omplete ytem inluding both the ample and the ontrol atuator (Cf: Figure 3.1). C F K Control atuator M v C F K Sample atuator Figure 4.2: Dynami model of the tet apparatu Applying Newton eond law to thi ytem and auming harmoni exitation at frequeny ω, the equation of motion for the ma M i given by: 2 [ M ( K + K ) jω( C C )]w K d 33 K d33 = ω + (4.5) d where w i the vertial diplaement. Uing omplex notation we aume a olution: w(t)=ae jωt (4.6) 35

5 where A i the omplex amplitude of the diplaement. Therefore, with a imple derivation, the expreion for the orreponding veloity beome: dw( t) v = = jωw. (4.7) dt A een in Chapter 3, ubtituting equation (4.7) into equation (4.5), the veloity oeffiient (veloity of the ma normalized to voltage input) due to the two different atuator ating eparately (i.e.: v = v +v ) an be expreed a: v = ω( C + C ) + K d 33 2 j[ ω ω M ( K + K )] (4.8) and v = ω( C + C ) + K d 33 2 j[ ω ω M ( K. (4.9) + K )] From equation (4.1), the ame approah an be ued to expre the fore oeffiient meaured on the ma, M (i.e.: f = f +f ): K v f = K d + C jω v 33 + (4.10) and K v 0 Cv (4.11) jω f = + + With thoe four oeffiient (v, v, f and f ) the error ontribution E and E an be alulated for different atuator harateriti and for any deired impedane ondition Z d. Thee variou parameter may be varied independently to determine their effet on ontrollability. 36

6 4.2.2 Effet on Controllability A model of the entire ytem (Cf: Figure 4.2) an be obtained by ubtituting value for the variable (K, C, M, d 33, ). In the following example, both the ontrol and ample atuator have been hoen to be idential. Chooing a juntion ma M=280g and given value for the other oeffiient, that give a new ytem whoe natural frequeny i 600Hz. Subtituting equation (4.8) through (4.11) into equation (3.8) and (3.9) allow to ee how both error E and E behave over a range of frequenie for different deired impedane ondition. It ha een previouly, in the limitation of ontrol in Chapter 3, that it i deirable that E i never ignifiantly maller than E. If thi happen, then the ontrol atuator ha to drive very hard in order to ahieve the deired impedane ondition (i.e.: H beome large). Figure 4.3 how the error ontribution due to the ample (olid line) and ontrol (dahed line) atuator in four different impedane ondition. In the four different ae, the deired impedane ondition i alway real (a for the ae of a damper) but with a magnitude varying from a low level (Z d =.1) to a high level (Z d =10e4). A one an ee with the plot, below the reonant frequeny of the ytem, for all of the different impedane ae, the error level of the ontrol atuator i higher than the error level of the ample atuator. Thi implie good ontrollability (the ontroller will not have to deliver a high ontrol ignal). Above the natural frequeny of the ytem, the oppoite happen. E beome maller than E and a large input voltage will be required by the ontrol atuator to ahieve good ontrol (bring the error, E, to zero). Furthermore, it an be notied that the differene between the two error term i more ignifiant for low impedane ondition. Under high impedane ondition, it i eay to ee from equation (3.8) and (3.9) that the term f and f beome almot inignifiant in their ontribution to the error. Then, ine 37

7 the veloity term, for idential atuator, are the ame in magnitude (equation (4.8) and (4.9)) both error are the ame for high impedane ondition (E E ): the ontrollability i good over the entire bandwidth of atuation. One onluion that an be made i that to provide good ontrol, it i neeary for the tet et-up to have a reonane frequeny well above the frequeny range of interet. Figure 4.3: Contribution to the error from the ample atuator and the ontrol atuator over variou impedane requirement 38

8 4.2.3 Effet of Control Atuator Stiffne To determine the effet of ontrol atuator tiffne on performane, imulation were onduted in whih the ontrol atuator tiffne, K, wa varied. Figure 4.4 how the importane of making the ontrol atuator tiffer than the ample atuator (K =2*K ) ine thi ondition alway give, under the reonant frequeny, a maller error E, ompared to E. Thi hould, therefore, provide better ontrol. On the ontrary, when K =0.5*K (Cf: Figure 4.5), it an be een that the error E i alway maller that the error E. Thi differene between the ontrol and the ample atuator will automatially lead to ontrol authority problem. Thi an be eaily undertood ine, for a piezoeletri atuator, tiffne and bloked fore are diretly proportional (Cf: equation (1.1)). Therefore, the atuator with higher tiffne will have the higher fore output. Beaue of thi relationhip between fore and tiffne, the atuator with greater tiffne will have a greater ontrol apability ompared to the other one. Thi explain why it i important for the ontrol atuator to be the one with the greater tiffne. 39

9 Figure 4.4: Contribution to the error from the ample atuator and the ontrol atuator when K =2*K 40

10 Figure 4.5: Contribution to the error from the ample atuator and the ontrol atuator when K =0.5*K 4.3 Simulation Uing Meaured Data Experimental Rig Having a better idea about the ontrol limitation, an experimental rig wa built. The rig hown in figure 4.6 ha been deigned to be uffiiently tiff ompared to the atuator ued, in order to prevent reonane of the tet rig from being within the 41

11 bandwidth of ontrol. The frame i ompoed of 2 teel rod (length: 8 inhe, diameter: 7/8 inh ) on the ide, 2 aluminum plate (Length: 5.5 inhe, height: 1 inh), one horter teel rod on the middle to enure a tati ompreive load (not neeary with the PCB atuator ued in the firt experiment). Figure 4.6: Experimental rig howing the two atuator onneted in erie via a fore gauge. The ample and ontrol atuator ued on thi rig are flextenional atuator model 710MO2 from PCB (thee devie were hoen mainly for onveniene and will be later ubtituted with the 1_3 tube array piezoeletri atuator provided by MSI). A tati load ell (model 31/ from Senote) i inluded in the tet et-up o that the tati preure ould be monitored. A the PCB atuator do not require any ompreive load, the preure applied with the middle rod will be kept to a minimum in thi ae. The dynami fore tranduer, model 208A03, and the aelerometer, model 309A, 42

12 whih were ued on the et-up to meaure the fore and veloity ignal, repetively, are hown on Figure 4.7. They were alo ening devie from PCB. Figure 4.7: Main devie ued on the tet rig To enure that the frame of the tet rig wa uffiiently rigid, the vibration level at five poition on the rig were monitored while the ontrol atuator wa driven with white noie. The ontrol atuator wa hoen over the ample atuator ine it generate the highet level of exitation on the rig. Thee poition are hown in Figure 4.8. Depite a reonant frequeny around 900Hz, the vibration of the enter ma wa more than 40dB larger than the vibration at all of the other poition for a frequeny range of interet from 0 to 1600Hz. Therefore it wa onluded that, for thee atuator, the tet rig wa uffiiently rigid and the bae had a large enough input impedane. It wa important to fulfill thi requirement 43

13 ine it met the original aumption of no diplaement of the bottom ide of the ample atuator. Knowing the tet truture modal harater to be appropriate for the bandwidth of ontrol, atuator haraterization tet ould then be onduted. Figure 4.8: Tranfer funtion of the diplaement on the rig when the ontrol atuator i driven with white noie Appliation of the Simulation To tet the methodology, a ontrol imulation for an eaily reproduible ae wa onduted. The deired impedane wa et to be the one of a 100g ma (i.e.: Z d =jω0.1, where j i uh that j 2 =-1 and ω i the frequeny in rad/). Uing meaured data, the goal wa then to imulate the behavior of an inertial atuator with a 100g inertial ma. The reult ould then be ompared with the performane of a real inertial atuator with a 100g inertial ma. 44

14 To ahieve thi, the four tranfer funtion of the fore and veloity in both ae where the ontrol and the ample atuator are driven eparately were meaured, F =f /, V =v /, F =f / and V =v / repetively. After uing thee four tranfer funtion to ompute the tranfer funtion of the different impedane, Z= f / * /v and Z= f / * /v, and etting the deired impedane a the one of that 100g ma, the ontrol filter, H (equation (3.10)), ould then be eaily alulated. Figure 4.9 explain the priniple of thi imulation tehnique. Figure 4.9: Priniple of the ontrol imulation uing meaured data Thi imulation wa ued to ompare predited performane with a meaured impedane from a real load. Thi omparion would then reveal ome limitation that ontroller might enounter later. In the experimental ae the fore produed by thi inertial atuator wa meaured under an atuator haking a 100g ma (Cf: Figure 4.10, drawing on the top left). For the imulation, a fore gauge on the bae of tet rig under the ample 45

15 atuator wa ued to monitor the atuator fore output (Cf: Figure 4.10, piture on the bottom left). Thi extra fore gauge (that wa not mount on the initial tet rig (Figure 4.6)) wa neeary to get the meaurement of a omparable fore that the one meaured in the experimental ae. Two additional tranfer funtion between the input to the atuator and the fore input to the bae of the tet rig were meaured (f b and f b ) and uing the priniple of uperpoition the total fore input to the bae of the tet rig ould then be alulated a: F t =(f b +H.f b ). (4.9) Thi method i hown in Figure 4.9. The total fore alulated uing thi method i an etimate of the fore that would be reated at the bae of the rig if the ontroller were imulating the behavior of a 100g load. Figure 4.10 ompare thi fore with the fore meaured uing an atual 100g load on the ample atuator. For the fore to be aurately modeled, ertain adjutment had to be made. Firt, part of the ma of the fore gauge (about half of it) i already inluded in the meaured fore output for the real inertial atuator. The meaured data for the tranfer funtion of the fore (f and f ) orrepond then to an inertial atuator with a little higher ma (approximately 115g whih i equal to the 100g ma plu around 15g due to the ma of the fore gauge itelf). Thi explain why uing thi data, the imulation predit a lower natural frequeny than the one meaured on the experimental et up (Figure 4.10). To adjut for thi, an 85g load wa hoen a the deired impedane, taking the pre-exiting 15g due to the fore gauge into aount in the imulation. With thi adjutment, both natural frequenie of the inertial atuator and the imulation mathed very well. A eond adjutment wa required beaue of a phae mathing problem between the fore gauge and the aelerometer. For a lightly damped reonane ondition only a few degree of phae mimath an tranform a reative impedane (i.e.: a ma or a pring) into a ignifiant real impedane (i.e.: damping). Thi aount for the damped reonane ondition predited uing the imulation (Figure 4.10). It appear then, that the enor devie meauring both fore and veloity have to be perfetly phae mathed to 46

16 enure a perfet meaure of the impedane. A a omplex value, the meaure of the impedane require a good preiion not only of it magnitude but alo of it phae. Therefore, to enure good ontrol thoe phae mimathe alo have to be ompenated. After thee effet (ma and phae) were aounted for, the fore predited uing meaured data from the rig and meaured fore from the real inertial atuator mathed very well (Cf: Figure 4.10, dahed plot). Figure 4.10: Comparion between the fore out of an inertial atuator and the one obtained with ontrol imulation for the orreponding deired impedane 47

17 4.4 Conluion At thi point of the reearh, all the methodology to reate a tet et-up, whih would allow the uer to imulate a wide range of load impedane and to ae the performane of the atuator under thee ondition, ha been developed and teted through variou imulation. Some pratial limitation due mainly to ontrol authority, reonant frequenie of the ytem and meaurement auray due to the enor devie themelve (problem of phae mimath) have been raied. However thee limitation do not look totally inurmountable and they hould eaily be overome making improvement to the rig. Therefore, the next tage i to develop the real time DSP (Digital Signal Proeing) oftware that would allow to perform the ative feedforward ontrol of impedane (Cf: Chapter 5). 48