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1 4 1 Vol. 4, No ACTA AUTOMATICA SINICA October, 214 MEMS Micro-electro-mechanical system (MEMS),, MEMS. MEMS Hurwitz, Popov,.,,. MEMS,,,,,,. MEMS., 214, 4(1): DOI /SP.J Zero Adjustment of MEMS Vibratory Gyroscope with Nonlinear Input WANG Wei 1 ZHAO Qing 1 LV Xiao-Yong 1 LI Xin 1 Abstract Considering the Micro-electro-mechanical system (MEMS) gyroscope nonlinearity, control input nonlinearity and external disturbance, a nonlinear control strategy based on hyperstability theory is proposed for MEMS gyroscope drifting caused by the non-superposition between the center of the geometry structure and the barycenter of the proof mass. System states can be regulated to zero level asymptotically by introducing a Hurwitz matrix to the MEMS state space model for the demand of system being strictly positive real, and choosing an appropriate control law with vector norm to satisfy the Popov integral inequality. Simulation results show the validity and robustness of the proposed nonlinear control strategy. Key words MEMS gyroscope, zero adjustment, hyperstability theory, nonlinear control Citation Wang Wei, Zhao Qing, Lv Xiao-Yong, Li Xin. Zero adjustment of MEMS vibratory gyroscope with nonlinear input. Acta Automatica Sinica, 214, 4(1): Micro-electromechanical system (MEMS),. MEMS, : ( ), Manuscript received July 3, 213; accepted March 5, 214 (HEUCFX4138), (LBH-TZ41), (213RFXXJ16), (NCET ) Supported by Fundamental Research Funds for the Central Universities (HEUCFX4138), Heilongjiang Postdoctoral Special Foundation (LBH-TZ41), Innovation of Science and Technology Talents in Harbin (213RFXXJ16), and New Century Excellent Talents Support Program (NCET ) Recommended by Associate Editor FANG Hai-Tao College of Automation, Harbin Engineering University, Harbin 151, ; ( ); ( ), MEMS, MEMS. MEMS,,.,,, : [1] H,,. [2 3],. [4 5] (Phase locked loop, PLL)

2 2172 4, [6] FPGA PLL. MEMS, [7]., Park [8] MEMS,,,, MEMS ( Park ). [9] Lyapunov,. [1 13], MEMS,. [14] MEMS,. [15 16], /. Park :, ( ) ( ).,, ( [17] [18] [19] ),, MEMS,,,.,,, Park. [2] (Popov).,,,, ( ), Popov,.,, [21 22],,. 1 Z- MEMS MEMS,,. 1 : x, y, x, y, Ω z, y 2mΩ z ẋ, Ω z. 1 MEMS. 1 MEMS Fig. 1 Simplified structural drawing of MEMS vibratory gyroscope, 1 - -,, MEMS (1) : mẍ + F d x + F r x = U x + 2mΩ z ẏ + mxω 2 z + my Ω z mÿ + F d y + F r y = U y 2mΩ z ẋ + myω 2 z mx Ω z (1), Fx d = c x ẋ + c xy ẏ, Fy d = c y ẏ + c xy ẋ, Fx r = k x x + k x3 x 3 + k xy y, Fy r = k y y + k y3 y 3 + k xy x. F d F r, U x U y,, 2mΩ z ẏ 2mΩ z ẋ, mxω 2 z myω2 z, my Ω z mx Ω z, c x k x c y k y,, c xy = c yx, k xy = k yx, Ω z. MEMS, (1) [23 24] :

3 1 : MEMS ),, Ω z = ; 2), MEMS 3 2 khz, 5 /s,, mxω 2 z, myω 2 z ; 3), ( ),, U v, Φ( ), U i = Φ i (v i ), i = x, y., (1) : mẍ + F d x + F r x = Φ x (v x ) + 2mΩ z ẏ mÿ + F d y + F r y = Φ y (v y ) 2mΩ z ẋ (2) (2),, MEMS,, : ẊX = AX + BΦ(v) + f(x) + d(t) (3) X, Φ(v), f(x), d(t), : m A = 1 k x c x m k xy (c xy + 2mΩ z ) k xy (c xy 2mΩ z ) m k y c y [ ] B = 1, Φ(v) = Φ 1 x (v x ) m Φ y (v y ) 1 x X = ẋ y, f(x) = 1 k x3 x 3 m ẏ k y3 y 3 2, (3) : 1. 2, Φ( ) (Φ() = ), s il, s iu s il Φ i (v i )/v i s iu, i = x, y; 2. d(t), d(t) k(t), k(t) k(t). Fig. 2 2 Nonlinear function with sector bounded : 2 Φ i (v i ), f(x) d(t), X,. 1 (Popov [2] ). : ẊX = AX BΘ Y = CX (4) A B C, Θ,,. 1) (5) ( ). G(s) = C(sI A) 1 B (5) 2) (6), r. Y T Θdt r 2 (6) 1, (3) Hurwitz A H ẊX = A H X + (A A H )X + f(x) + d(t) + BΦ(v) (7) Θ = [(A A H )X + f(x) + d(t) + BΦ(v)] (8) : ẊX = A H X IΘ (9)

4 ),, A H = λi, λ, G(s) = (si A H ) 1. 2), : 1. 1) 2), (3) (1), (11) (12),. v(t) = BT X η(x, t) (1) X η(x, t) = [ (A A H)X + f(x) + ˆk(t)] s l λ min (BB T ) (11) ˆk(t) = α X (12) s l = min{s il i = x, y}, λ min ( ), ˆk(t) k(t), k(t) = ˆk(t) k(t), α >.., 1 s il v 2 i v i Φ i (v i ) s iu v 2 i, i = x, y (13) s l v T v v T Φ(v) s u v T v (14) s l = min{s il i = x, y}, s u = max{s iu i = x, y}. (8), X T Θ = X T [(A A H )X + f(x) + d(t)+ (1) (14) BΦ(v)] X [ (A A H )X + f(x) + k(t)] + X T BΦ(v) (15) v T Φ(v) = XT B η(x, t)φ(v) X s l v T X T B v = s l X η(x, X t)bt η(x, t) X s l λ min (BB T )η 2 (X, t) (16) X T BΦ(v) s l λ min (BB T )η(x, t) X (17) (17) (15) (11), X T Θ X [ (A A H )X + f(x) + k(t)]+ X T BΦ(v) X [ (A A H )X + f(x) + k(t)] s l λ min (BB T ) [ (A A H )X + f(x) + ˆk(t)] s l λ min (BB T ) X X [k(t) ˆk(t)] = X k (18) 2) k(t),, k(t) = ˆk(t). (12) (18) X T Θ X k(t) X α X T Θdt X (α X dτ X dτ)dt r 2, (6). 3 (19) (2),, Matlab/Simulink MEMS Park 3. : m = kg, k x = 8.98 N/m, k x3 =.16 N/m k y = N/m, k y3 =.14 N/m, k xy = 5 N/m c x = N s/m, c y = N s/m c xy =.429 [ 1 6 N s/m, Ω z = 3 rad/s ] ( cos v x )v x Φ(v) = ( sin v y )v y ( ), v x =.25 sin 416t V v y =.5 sin 511t V,. x() = m, y() = m ,, (.1 s),, Park ;, ; : ( ), 2mΩ z ẏ, 2mΩ z ẋ,,.,

1 : MEMS 2175, MEMS,., 6., d(t) (21) : d(t) = 1.5 1 6 sin 2 1 4 t (21) 2 1 6 sin 1 1 4 t : Hurwitz λ = 2 1 7, ˆk() =, α =.5, x() = 5 1 7 m, y() = 5 1 8 m. A.3 s (A 1.1A).7 s (1.1A.9A), Φ(v).5 s (.

5 1 : MEMS 2175, MEMS,., 6., d(t) (21) : d(t) = sin t (21) sin t : Hurwitz λ = 2 1 7, ˆk() =, α =.5, x() = m, y() = m. A.3 s (A 1.1A).7 s (1.1A.9A), Φ(v).5 s ( ),.9 s ( ) , 8 ( ),.4 s,,,. A Φ(v),,. Fig. 3 3 Open-loop schematic diagram of two modes Fig. 4 4 Open-loop displacement response of drive mode in each case Fig. 5 5 Open-loop displacement response of sense mode in each case

6 Fig. 6 6 Closed-loop control schematic diagrams of two modes Fig. 7 7 Position and velocity of two modes under the control actions Fig. 8 8 Control voltage and control force of two modes under the control actions 4 MEMS Hurwitz,,

7 1 : MEMS 2177,, MEMS.,,, MEMS Park. References 1 Li W L, Liu P X. Adaptive tracking control of an MEMS gyroscope with H-infinity performance. Journal of Control Theory and Applications, 211, 9(2): Fast B, Miklosovic R, Radke A. Active disturbance rejection control of a MEMS gyroscope. In: Proceedings of the 28 American Control Conference. Seattle, WA: IEEE, Zheng Q, Dong L L, Gao Z Q. Control and rotation rate estimation of vibrational MEMS gyroscopes. In: Proceedings of the 16th IEEE International Conference on Control Applications Part of IEEE Multi-conference on Systems and Control. Singapore: IEEE, Sung S, Sung W T, Kim C, Yun S, Lee Y J. On the modematched control of MEMS vibratory gyroscope via phasedomain analysis and design. IEEE/ASME Transactions on Mechatronics, 29, 14(4): Sung W T, Sung S, Lee J Y, Kang T, Lee Y J, Lee J G. Development of a lateral velocity-controlled MEMS vibratory gyroscope and its performance test. Journal of Micromechanics and Microengineering, 28, 18(5): ID5528, DOI: 1.188/ /18/5/ Luo Bing, Wang An-Cheng, Wu Mei-Ping. A drive control scheme based on phase-control for silicon micromechanical gyroscopes. Acta Automatica Sinica, 212, 38(2): (,,.., 212, 38(2): ) 7 Park S, Horowitz R. Adaptive control for the conventional mode of operation of MEMS gyroscopes. Journal of Microelectromechanical Systems, 23, 12(1): Park S, Horowitz R. New adaptive mode of operation for MEMS gyroscopes. Journal of Dynamic Systems Measurement and Control-Transactions of The ASME, 24, 126(4): Batur C, Sreeramreddy T, Khasawneh Q. Sliding mode control of a simulated MEMS gyroscope. ISA Transactions, 26, 45(1): Fei J T, Xin M Y. An adaptive fuzzy sliding mode controller for MEMS triaxial gyroscope with angular velocity estimation. Nonlinear Dynamics, 212, 7(1): Fei J T, Yang Y Z. Adaptive neural compensation scheme for robust tracking of MEMS gyroscope. In: Proceedings of the 212 IEEE International Conference on Systems, Man, and Cybernetics. Seoul, Korea: IEEE, Fei J T, Ding H F. Adaptive vibration control of microelectromechanical systems triaxial gyroscope using radial basis function sliding mode technique. Journal of Systems and Control Engineering, 213, 227(2): Mehran Z, Sahel S. A modified model reference adaptive control with application to MEMS gyroscope. Journal of Mechanical Science and Technology, 211, 25(8): Salah M H, Mclntyre M L, Dawson D M, Wagner J R, Tatlicioglu E. Sensing of the time-varying angular rate for MEMS Z-axis gyroscopes. Mechatronics, 21, 2(6): Mahyar F, Maysam Z P, Hassan S, Aria A. Parameter estimation and interval type-2 fuzzy sliding mode control of a z-axis MEMS gyroscope. ISA Transactions, 213, 52(6): Zhang Li-Jun, Yang Li-Xin, Guo Li-Dong, Sun Li-Ning. Adaptive output feedback control for piezoactuator-driven stage. Acta Automatica Sinica, 212, 38(9): (,,,.., 212, 38(9): ) 18 Zhu Sheng, Sun Ming-Xuan, Wang Xue-Jie, Li Yan-Jun, Fang Zhi-Gang. Robust repetitive control for a class of nonlinear systems with input deadzone. Acta Automatica Sinica, 213, 39(6): (,,,,.., 213, 39(6): ) 19 Li Jian-Xiong, Fang Yi-Ming, Shi Sheng-Li. Robust dynamic output-feedback control of hydraulic servo system with input saturation for rolling mill. Control and Decision, 213, 28(2): (,,.., 213, 28(2): ) 2 Popov V M. Hyperstability of Control Systems. New York: Springer-Verlag, Chang Kuo-Ming. Adaptive control for a class of chaotic systems with nonlinear inputs and disturbances. Chaos, Solitons and Fractals, 28, 36(2): Wu Xiao Jing. Research on Adaptive Control for Nonlinear Systems with Input and Output Constraint Characteristics [Ph. D. dissertation], Yanshan University, China, 212 (. [ ],,, 212) 11 Fei J T, Zhou J. Robust adaptive control of MEMS triaxial gyroscope using fuzzy compensator. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, 212, 42(6): Robert N D, Antonio L. Applications of microelectromechanical systems in industrial processes and services. IEEE Transactions on Industrial Electronics, 29, 56(4):

2178 4 24 Liu Yu, Lu Yong-Le, Li Lei-Lei, Fang Zhen, Ren Chun-Hua, Pan Ying-Jun. Solid Vibration Gyroscope and Navigation Technology. Beijing: China Aerospace Press, 21. 49 51 (,,,,,.

degree from Harbin Engineering University in 26.

MEMS. E-mail: majorzq@gmail.com (ZHAO Qing Master student at the College of Automation, Harbin Engineering University.

His research interest covers adaptive sliding mode control theory and its application in the MEMS gyroscope control field.). 29. MEMS. E-mail: lengfeng56@163.com (LV Xiao-Yong Ph. D.

8 Liu Yu, Lu Yong-Le, Li Lei-Lei, Fang Zhen, Ren Chun-Hua, Pan Ying-Jun. Solid Vibration Gyroscope and Navigation Technology. Beijing: China Aerospace Press, (,,,,,.. :, ). 26.,.. wangwei47@hrbeu.edu.cn (WANG Wei Professor at the College of Automation, Harbin Engineering University. He received his Ph. D. degree from Harbin Engineering University in 26. His research interest covers high-precision inertial device and strapdown inertial navigation, advanced control theory and its application. Corresponding author of this paper.) MEMS. majorzq@gmail.com (ZHAO Qing Master student at the College of Automation, Harbin Engineering University. He received his bachelor degree from Harbin Engineering University in 212. His research interest covers adaptive sliding mode control theory and its application in the MEMS gyroscope control field.). 29. MEMS. lengfeng56@163.com (LV Xiao-Yong Ph. D. candidate at the College of Automation, Harbin Engineering University. He received his bachelor degree from Harbin Engineering University in 29. His research interest is structure desigh of MEMS gyroscope.) xinxin-forever@sohu.com (LI Xin Lecturer at the College of Automation, Harbin Engineering University. She received her Ph. D. degree from Harbin Engineering University in 211. Her research interest covers modeling of high-precision inertial device and strapdown inertial navigation system.)

(L d L q ). i d = 0,, [8].,,,,, / (maximum torque per ampere, MTPA) [9],. MTPA [9] [10],,,. MTPA, MTPA.,,,. MTPA,,,.,. 2 (Model of PMSM) d-q, :

(L d L q ). i d = 0,, [8].,,,,, / (maximum torque per ampere, MTPA) [9],. MTPA [9] [10],,,. MTPA, MTPA.,,,. MTPA,,,.,. 2 (Model of PMSM) d-q, : 32 2 2015 2 DOI: 10.7641/CTA.2015.40361 Control Theory & Applications Vol. 32 No. 2 Feb. 2015 /,,, (, 266071) : (PMSM), / (MTPA). MTPA,.., MTPA.. : ; ; ; / : TP273 : A Adaptive sliding mode maximum torque