Force rebalance control for a MEMS gyroscope using ascending frequency drive and generalized PI control

Transcription

1 MATEC Web of Conference 63, 53 (16) DOI: 1.151/ matecconf/ MMME 16 Force rebalance control for a MEMS gyrocope uing acending frequency drive and generalized PI control Chun-Hua HE 1,, Qin-Wen HUANG, Qian-Cheng ZHAO 1, Zhen-Chuan YANG 1, Da-Cheng ZHANG 1 and Gui-Zhen YAN 1 1 National Key Laboratory of Science and Technology on Micro/Nano Fabrication, Intitute of Microelectronic, Peking Univerity, Beijing 1871, P. R. China Science and Technology on Reliability Phyic and Application of Electronic Component Laboratory, No.5 Electronic Reearch Intitute of the Minitry of Indutry and Information Technology, Guangzhou, 5161, Guangdong, P. R. China Abtract. Thi paper ha propoed a novel force rebalance control method for a MEMS gyrocope uing acending frequency drive and generalized PI control. Theoretical analye of acending frequency drive and force rebalance control method are illutrated in detail. Experimental reult demontrate that the electrical anti-reonant peak are located at the frequency repone in the RFD ytem, which eriouly deteriorate the original repone characteritic. However, they are eliminated in the AFD ytem, and the electrical coupling ignal i alo uppreed. Beide, a for the force rebalance control ytem, the phae margin approximate to 6deg, gain margin are larger than 13dB, and enitivity margin are maller than 3.dB, which validate the control ytem i table and robut. The bandwidth of the force rebalance control ytem i meaured to be about 13.Hz, which accord with the imulation reult. The bia intability and angle random walk are evaluated to be 1.65deg/h and.6deg/h, repectively, which achieve the tactical level. Keyword: MEMS gyrocope, mode-matching, force rebalance control, acending frequency drive, electrical coupling uppreing. Introduction Electrical coupling i one of the major error ource for a MEMS gyrocope, which i induced by the direct coupling from the drive comb to the ene comb through the paraitic capacitance [1-]. It will bring about the non-ideal anti-reonance at the amplitude-frequency repone of drive mode and ene mode, which deteriorate the performance and affect the reliability of the cloed loop ytem. Therefore, it i ignificant to uppre the electrical coupling ignal by modifying the control and readout circuit. Electromechanical amplitude modulation (EAM) [3] method can modulate the actual vibration ignal by exerting a high-frequency carrier (uually everal MHz) to the public ma. Thu, coupling ignal can be eparated and eliminated from the ueful ene ignal eaily. However, a demodulation proce hould be added to retore the ueful information, which will complicate the control ytem. Furthermore, the circuit ha to handle the demodulation in high frequency domain, which increae the hardware cot, power conumption and high frequency noie. By introducing a drive modulation proce, acending frequency drive (AFD) method [4-5] i proved effective to uppre the electrical coupling for the drive mode of a MEMS gyrocope. Compared with EAM, it can implify the readout circuit ince it need no extra demodulation proce. In addition, compared to reonant frequency drive (RFD) method, the noie in AFD ytem can be further uppreed. Hence, in thi paper, AFD method will be applied to the force rebalance control ytem for ene mode of a MEMS gyrocope. Nowaday, force rebalance control for the ene mode i widely adopted in high-performance gyrocope, ince it can adjut the bandwidth, improve the nonlinearity, extend the meaure range [6-9]. A for EAM The Author, publihed by EDP Science. Thi i an open acce article ditributed under the term of the Creative Common Attribution Licene 4. (

2 MATEC Web of Conference 63, 53 (16) DOI: 1.151/ matecconf/ MMME 16 method, a lot of reearche about the force rebalance control have been fulfilled and reported in our previou work [8-9]. In thi work, novel theoretical analyi of force rebalance control with AFD method will be conducted in detail. Theoretical analyi Electrical coupling uppreing The implified chematic of a Z-axi doubly decoupled MEMS tuning fork gyrocope i hown in Fig. 1 (a). It mainly include drive comb, drive-ening comb, ene comb, force feedback comb, and mode-matching comb. The key parameter of the teted gyrocope at room temperature are lited in Table 1. The reonant frequency of the ene mode will equal the reonant frequency of the drive mode d after mode-matching control. Fig. 1 (b) how that there are ome paraitic capacitance between the drive comb and ene comb ince they are cloe to each other in the layout. Table 1. Key parameter of the teted gyrocope at room temperature. Parameter Value Drive mode reonant frequency f d Hz Drive mode quality factor Q d Sene mode reonant frequency f Hz Sene mode quality factor Q 3 (a) The implified chematic of a Z-axi doubly decoupled MEMS tuning fork gyrocope (b) Electrical coupling analyi between the drive comb and ene comb of a MEMS gyrocope Fig. 1. Electrical coupling analyi of a MEMS gyrocope. In order to fulfill the differential detection, a DC carrier ignal V m i exerted to the proof ma. C d1 and C d are the differential drive capacitance, while C 1 and C are the differential ene capacitance. C p1 and C p tand for the differential paraitic capacitance between the drive comb and ene comb. V L and V R repreent the two differential drive ignal. Owing to the exitence of paraitic capacitance, drive ignal can be coupled to the ene comb, which will diturb the original detection ignal and deteriorate the cloed loop ytem tability and performance. Fig. and Fig. 3 depict the chematic of the readout circuit and AFD circuit for the ene mode of a MEMS gyrocope, repectively. In the readout circuit, the output of differential amplifier can be derived a: kg[ Vm( CC 1) ( VC R pvc L p1)] Vout 1 C Where, k g i the gain of the differential amplifier, and C f i the feedback capacitance of the charge amplifier. In the RFD circuit, the frequencie of V L and V R equal to thoe of C 1 and C, thu it i hard to ditinguih Vm( C C1) from ( VC R p VC L p1). Due to the influence of coupling ignal, there will be a non-ideal anti-reonance appeared at the amplitude-frequency repone [4-5]. However, in the AFD circuit, the frequency of V L and V R i far larger than that of C 1 and C, thu the coupling ignal ( VC R p VC L p1) can be eliminated by a uitable band-pa-filter (BPF). f

3 MATEC Web of Conference 63, 53 (16) DOI: 1.151/ matecconf/ MMME 16 Therefore, after filtered by BPF, y out can be decribed a: kv g m( CC 1) kv g mnh yout y C C d f f Where, y i the diplacement of the ene mode, i the permittivity, N i the number of ene capacitance, h i the thickne, and d i the gap. From equation (), the gain k cv of the readout circuit can be obtained a: kv g m N h kcv 3 Cd f Fig. 3. AFD circuit for the ene mode of a MEMS gyrocope. From Fig. 3, V L and V R can be yielded a: V ( V V in( t)) Pin( t) 4 R dc ac d c V ( V V in( t)) Pin( t) 5 L dc ac d c Where, V dc and V ac in( d t) are the DC and AC drive ignal. Pin( c t) i the high frequency carrier, and c? d. Take the noie n 1 and n into account, the drive ignal V 1 and V of the differential drive capacitance can be derived a: Fig.. The readout circuit for the ene mode of a MEMS gyrocope. V VR Vm n ( Vdc Vacin( dt)) Pin( ct) Vm n (6) V1 VL Vm n1 ( Vdc Vac in( dt)) Pin( ct) Vm n1 (7) From Fig. 1 (b), the energie of the NV1 h( l y) differential drive capacitance can be E1 9 d expreed a: E NV h( l y) 8 d Where, N i the number of drive capacitance, and l i the overlapped length. Then, the total drive force can be derived a: E E Nh( V V ) Nh F P V V t n n V n n (1) in( )co( ) ( ) in( ) ( ) in( )in( )] 1 1 tot [ dc ac in( d ) ( 1 m)( 1) y y d d PVacVdc dt ct n n1 VdcP ct n n1 Vm VacP dt ct Since the frequency repone of a gyrocope can be regarded a a BPF, the lower and higher frequency component will be uppreed, and the real ueful force can be yielded a: N hp Vdc ueful ac d d F V in( t) 11 Equation (11) indicate that the noie can be uppreed, and there i no need to demodulate the readout ignal y out ince the frequency of y i d, rather than c d. Hence, the gain k vf of the drive circuit can be expreed a: N hp Vdc kvf 1 d Force rebalance control AFD method i effective to uppre the coupling ignal, which i detailedly illutrated above. k cv and k vf of the AFD circuit are yielded in equation (3) and (1). Thu the force 3

4 MATEC Web of Conference 63, 53 (16) DOI: 1.151/ matecconf/ MMME 16 rebalance control ytem for the ene mode can be introduced a Fig. 4. Where, -(t) and q(t) are input angular rate and quadrature coupling ignal, repectively. F c, F q and F b repreent Corioli force, quadrature force, and feedback force, repectively. m p A v, k qf and k vf are correponding force coefficient of them. m and m p repreent the ene ma and proof ma, repectively. A v tand for the amplitude of the drive velocity. k dc i the converion coefficient from diplacement to capacitance, and i the demodulation phae. The tranfer function of the ene mode G (), BPF B(), low-pa-filter (LPF) L(), and generalized-pi (G-PI) controller P() are defined a follow: k / m G () Q B () L () P dc / b b bbb l lll k ( ) p 16 ( )( ) 1 3 K dc / m Q Fig. 4. Force rebalance control ytem with AFD method for the ene mode of a MEMS gyrocope. Where, b and b are the damping ratio and centre frequency of BPF, l and l are the damping ratio and cut-off frequency of LPF, k p, 1, and 3 are the key parameter of G-PI controller. There are two control loop hown in Fig. 4, one i utilized to rebalance the Fb () - j F () Where, complex frequency i equal to j. Thu the tranfer function of the cloed loop ytem T() and the enitivity function S() can be yielded a: D () T ()= D ( ) Corioli force, while another i ued to rebalance the quadrature force. A for thi force control ytem, the tranfer function of the open loop ytem D() can be deducted eaily baed on Fig. 4: j D() jk k [ e Lj ( - j ) Pj ( - j )- e Lj ( + j ) Pj ( + j )] Bj ( ) G( j) (17) cv vf d d d d ()= 1 S 1+ D ( ) 19 4

5 MATEC Web of Conference 63, 53 (16) DOI: 1.151/ matecconf/ MMME 16 Experimental tet Swept frequency tet The implified chematic of a Z-axi doubly decoupled MEMS tuning fork gyrocope i hown in Fig. 5. It mainly include an analog circuit and a digital circuit. Swept frequency tet of the ene mode are conducted in the RFD ytem and AFD ytem, repectively. Experimental reult demontrate that the electrical anti-reonant peak are located at the amplitude-frequency and phae-frequency repone in the RFD ytem, which eriouly deteriorate the original repone characteritic, a hown in Fig. 6 (a). Due to the exitence of electrical anti-reonant peak, it i very hard to deign a uitable G-PI controller to guarantee the reliability and robutne of the force rebalance control ytem. Therefore, it i urgent to uppre the electrical coupling. On the other hand, in Fig. 6 (b), there are no anti-reonant peak appeared in the AFD ytem, and the electrical coupling ignal i eliminated, which indicate the theoretical analyi i accurate. Gyrocope FPGA Analog circuit Fig. 5. The implified chematic of a Z-axi doubly decoupled MEMS tuning fork gyrocope. Force rebalance control tet 1 After the theoretical deduction above, equation (3) and (1)-(19) can be utilized to deign the cloed loop ytem. Simultaneouly, imulation baed on thee equation are accomplihed with Matlab oftware, a hown in Fig. 7 and Fig. 8. In Fig. 7, owing to narrow-band force rebalance control, there are two croover frequencie appearing in the frequency-amplitude repone of D(). Thu, imilar to our previou analyi [9], there are two correponding phae margin (PM l and PM r ), gain margin (GM l and GM r ), cut-off frequencie (f tl and f tr ) and enitivity margin (M l and M r ), a marked in the figure. The key tability indice of the feedback control ytem obtained by Matlab imulation are lited in Table. According to claical linear control theory, the force rebalance control ytem i table and robut enough. Beide, the bandwidth (BW lr ) of T() i calculated to be 13.18Hz, which mean the force outide thi narrow-band will never be balanced. ADC / DAC Amplitude (db) Phae (deg) Digital circuit Anti-reonant peak Reonant frequency drive Acending frequency drive (a) Amplitude-frequency repone Anti-reonant peak Reonant frequency drive Acending frequency drive (b) Phae-frequency repone 5

6 MATEC Web of Conference 63, 53 (16) DOI: 1.151/ matecconf/ MMME 16 Fig. 6. Swept frequency tet of the ene mode in RFD Amplitude (db) Phae (deg) D() GMl=13.4dB PMl=61.47deg PMr=6.54deg GMr=13.59dB Amplitude (db) Phae (deg) - -4 and AFD method. T() -6 ftl= hz ftr=9553.7hz BWlr = ( ftr - ftl )/ = 13.18Hz -3dB Fig. 7. The frequency repone of the open loop ytem D() and cloed loop ytem T(). Amplitude (db) S() Ml=3.15dB Mr=3.4dB Fig. 8. The frequency repone of the enitivity function S(). Table. Key tability indice of the feedback control ytem obtained by Matlab imulation. Type Key tability indice Phae margin PM (deg) PM l =61.47 PM r =6.54 Gain margin GM (db) GM l =13.4 GM r =13.59 Senitivity margin M (db) M l =3.15 M r =3.4 Cut-off frequency f t (Hz) f tl = f tr = Bandwidth BW (Hz) BW lr =( f tr - f tl )/=13.18Hz The bandwidth (BW) of the force rebalance control ytem i meaured to be about 13.Hz, which approximate to the imulation reult, a hown in Fig. 9. It further verifie the theoretical analyi of the cloed loop ytem i precie. The cale factor of the MEMS gyrocope with cloed loop controlled ene mode i 65.9mV/deg/ with nonlinearity of.3%. Fig. 1 illutrate that the bia intability and angle random walk are evaluated to be 1.65deg/h and.6deg/h, repectively, which reache the tactical level. 6

7 MATEC Web of Conference 63, 53 (16) DOI: 1.151/ matecconf/ MMME 16 Am plitude (db) Tr() BW=13.Hz -3dB Fig. 9. Bandwidth tet reult of the cloed loop control ytem. deg/h Bia intability: 1.65deg/h ARW:.6deg/qrt(h) Original data Fitting data Averaging time () Fig. 1. Allan variance graph of bia drift of the MEMS gyrocope with cloed loop controlled ene mode. Summary In thi paper, a novel force rebalance control method ha been preented for a MEMS gyrocope uing acending frequency drive and generalized PI control. Theoretical analye of acending frequency drive and force rebalance control method are illutrated in detail. Experimental reult demontrate that the electrical anti-reonant peak are located at the amplitude-frequency and phae-frequency repone in the RFD ytem, which eriouly deteriorate the original repone characteritic. However, they are eliminated in the AFD ytem, and the electrical coupling ignal i alo uppreed. On the other hand, the frequency repone of the force rebalance control ytem indicate that the phae margin approximate to 6deg, gain margin are larger than 13dB, and enitivity margin are maller than 3.dB, which validate the control ytem i table and robut. The bandwidth of the force rebalance control ytem i meaured to be about 13.Hz, which verifie the theoretical analyi i accurate. The bia intability and angle random walk are evaluated to be 1.65deg/h and.6deg/h, repectively, which reache the tactical level. Acknowledgement Thi work i partially upported by the National Natural Science Foundation of China under Grant No Reference [1] Lee, H. Ko, et al., Non-ideal behavior of a driving reonator loop in a vibratory capacitive microgyrocope, Microelectronic J., vol. 39, 8, pp [] M.S. Weinberg and A. Kourepeni, Error ource in in-plane ilicon tuning-fork MEMS gyrocope, J. Microelectromech. Syt., vol. 15, 6, pp [3] Acar, A.M. Shkel, Structurally decoupled micromachined gyrocope with pot-releae capacitance enhancement, J. Micromech. Microeng., 5, [4] J. Cui, Z. Y. Guo, Z.C. Yang, et al., Electrical coupling uppreing for a microgyrocope uing acending frequency drive with -DOF PID controller, Proc. Tranducer 11, pp. -5. [5] J. Cui, Z.Y. Guo, Z.C. Yang, et al., Electrical coupling uppreion and tranient repone improvement for a microgyrocope uing acending frequency drive with a -DOF PID controller, J. Micromech. Microeng., 11, [6] S. Sonmezoglu, S.E. Alper, T. Akin, An Automatically Mode-Matched MEMS Gyrocope With Wide and Tunable Bandwidth, J. Microelectromech. Syt., Vol. 3, No., 14, pp [7] R. Mirjalili, H. Wen, D. E. Serrano, and F. Ayazi, Subtrate-decoupled ilicon dik reonant having degenerate gyrocopic 7

8 MATEC Web of Conference 63, 53 (16) DOI: 1.151/ matecconf/ MMME 16 mode with Qin exce of 1-million, Proc. TRANSDUCERS 15, pp [8] C.H. He, Q.C. Zhao, Y.X. Liu, et al., Cloed loop control deign for the ene mode of micromachined vibratory gyrocope, Sci. China Tech. Sci. 13, Vol. 56, pp [9] C.H. He, D.C. Liu, Q.C. Zhao, et al., A novel narrow-band force rebalance control method for the ene mode of MEMS vibratory gyrocope, Meaurement, vol. 6, 15, pp