Editorial Manager(tm) for Proceedings of the Institution of Mechanical Engineers, Part I, Journal of Systems and Control Engineering Manuscript Draft

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1 Editorial Manager(tm) for Proceedings of the Institution of Mechanical Engineers, Part I, Journal of Sstems and Control Engineering Manuscript Draft Manuscript Number: JSCE48R1 Title: A Disturbance Rejection Based Control Sstem Design for Z-Axis Vibrator Rate Groscopes Article Tpe: Paper Kewords: MEMS groscopes, active disturbance rejection control, extended state observer, rate estimation, performance. Corresponding Author: Assistant Professor Lili Dong, Corresponding Author's Institution: First Author: Qing Zheng Order of Authors: Qing Zheng; Lili Dong Abstract: This paper presents a new control strateg known as active disturbance rejection control (ADRC) for controlling the sense axis of a vibrational MEMS groscope and detecting time-varing rotation rates. In the ADRC framework, the generalized disturbance, which is referred to as the combination of the internal dnamics and external disturbances, is estimated and compensated for in real time. It is assumed that the drive axis outputs an ideal sinusoidal signal. A force-to-rebalance mode of operation is applied to the sense axis of the groscope. Under this mode, the output of the sense axis is continuousl monitored and driven to zero b ADRC and the rotation rate is estimated b emploing the demodulation technique. Simulation results and performance analsis demonstrate the effectiveness and robustness of the ADRC against parameter variations and noises.

2 Manuscript A Disturbance Rejection Based Control Sstem Design for Z- Axis Vibrator Rate Groscopes 1 Qing Zheng and Lili Dong, * Abstract This paper presents a new control strateg known as active disturbance rejection control (ADRC) for controlling the sense axis of a vibrational MEMS groscope and detecting timevaring rotation rates. In the ADRC framework, the generalized disturbance, which is referred to as the combination of the internal dnamics and external disturbances, is estimated and compensated for in real time. It is assumed that the drive axis outputs an ideal sinusoidal signal. A force-to-rebalance mode of operation is applied to the sense axis of the groscope. Under this mode, the output of the sense axis is continuousl monitored and driven to zero b ADRC and the rotation rate is estimated b emploing the demodulation technique. Simulation results and performance analsis demonstrate the effectiveness and robustness of the ADRC against parameter variations and noises. Kewords: MEMS groscopes, active disturbance rejection control, extended state observer, rate estimation, performance. I. INTRODUCTION MEMS groscopes are inertial rate sensors. Most MEMS groscopes are vibrator. The use vibrating elements to sense 1 The work is supported b Ohio ICE (Instrumental, Control, and Electronic) consortium under grant The authors are with the Department of Electrical and Computer Engineering, Cleveland State Universit, Cleveland, OH 44115, USA. * The corresponding author. L.Dong34@csuohio.edu, Tel: , Fax: rotation rates and can be easil batch fabricated on crstal silicon or polsilicon [1]. A Z-axis MEMS groscope is sensitive to the rotation rate about the axis normal to the plane of silicon chip. The operating principle of the MEMS groscope is based on the energ transfer from driving mode (or drive axis) to sensing mode (or sense axis) of the MEMS groscopes caused b Coriolis acceleration, which is generated b vibrating a proof mass in a rotating reference frame. With the advancement of micromachining technolog, MEMS groscopes have been applied to automobiles, consumer electronics, GPS assisted inertial navigation, and so on []. However, fabrication imperfections and environmental variations produce undesirable coupling terms and frequenc mismatch between two axes, input and measurement noises, and parameter variations, which degrade the performance of the groscopes. Consequentl, a control sstem is essential to improve the performance and stabilit of MEMS groscopes. Since the 199 s, there has been substantial research on the control designs of MEMS groscopes. Both open-loop and feedback control methods are applied to the sense axis. Since feedback control greatl increases the bandwidth of the groscope and is able to compensate for fabrication imperfections, it has replaced open loop control in most MEMS groscopes [3]. However, almost all of the reported closed-loop control approaches [3-7] are based on precise 1

3 mathematical models of the groscopes and assume constant rotation rates. In realit, however, the rotation rate is time varing. The adaptive controller reported in [8, 9] is designed to approximate a time-varing rotation rate, but it is also a model-based controller, which is sensitive to all fabrication imperfections and parameter variations. In addition, the multiple tuning parameters in [8, 9] make it difficult to implement in real world situation. Dealing with such timevaring uncertain dnamics of MEMS groscopes makes the control problem challenging and criticall important. Since the sstem dnamics are onl partiall known, a solution that is insensitive to the uncertainties in sstem dnamics and is able to accuratel determine the rotation rate is needed. In this paper, a new control solution known as active disturbance rejection control (ADRC) is applied to the sense axis of MEMS groscopes [1, 11]. ADRC as a practical design method has been successfull applied to motion control [1, 13], aircraft flight control [14], voltage regulation in DC- DC power converter [15], and other mechanical sstems. The basic idea of this control strateg is to use an extended state observer (ESO) to estimate the generalized disturbance, which is the combination of the internal dnamics and external disturbances, in real time and dnamicall compensate for it. The use of ADRC is extended to MEMS. The force-torebalance control scheme on the sense axis is chosen because of the general success of the nulling-the-output approach in a broad range of precise sensing applications [3]. Under this control scheme, the output of the sense axis is driven to zero b ADRC. In addition, the ESO provides an estimate of the generalized disturbance which includes the unknown time varing rotation rate and unknown Quadrature error terms arising from mechanical imperfections. With the accurate estimation of ESO, the time-varing rotation rate is obtained through demodulation. This paper is organized as follows. The dnamics of MEMS groscope is described in Section II. The ADRC approach is presented in Section III and the rotation rate estimation law is developed in Section IV. The simulation results are shown in Section V and the robustness performance is analzed in Section VI. Finall, some concluding remarks are given in Section VII. II. THE DYNAMICS OF MEMS GYROSCOPES The mechanical structure of a Z-axis MEMS groscope is shown in Fig. 1. It can be understood as a proof mass attached to the rigid frame through dampers and springs. As the rigid frame is subject to a rotation rate (Ω) about the rotation axis (Z axis), a Coriolis acceleration is produced along the sense axis (Y axis) while the proof mass is driven to resonance along the drive axis (X axis), which is perpendicular to both the sense and rotation axes. The Coriolis acceleration is proportional to both the applied rotation rate and the amplitude of the velocit of the moving mass along the drive axis. Therefore the rotation rate can be determined through sensing the vibration of the sense axis. It is assumed there are no damping couplings and allow for the frequenc mismatch between both axes. The governing equations of the Z-Axis MEMS groscope [4] are

4 K && x + ςωnx& + ωnx + ωx Ω & = ud( t) m K && + ςω& + ω + ωxx + Ω x& = us( t) m where x() t and t ( ) are the drive axis and sense axis outputs respectivel, ω n and ω are natural frequencies of drive and sense axes, ς and ς are damping coefficients, (1) u d and u s are control inputs for the drive and sense axes, m is the proof mass, Ω x& and Ω & are Coriolis accelerations, Ω is an unknown time-varing rotation rate, ω x and ω x x are constant unknown Quadrature error terms caused b stiffness couplings between two axes, and K is a constant that accounts for sensor, actuator, and amplifier gains. There are also mechanicalthermal noises added to the input of the sense axis produced b the dampers. The power spectrum densit (PSD) of the noise is S n (f)=4k B TR N s () where K B is a Boltzman constant (K B = JK -1 ), T is the absolute temperature of the resistor, and R is a term associated with damping coefficient (R=ςω n m). Since the PSD of the noise is constant at all frequencies, the mechanical-thermal noise can be modeled as a zero-mean white noise [3]. Because the PSD S n (f)=δ =4K B TR according to [16], the mechanical thermal noise is represented b d ~ (, δ )=(,S n ). Rotation sensing is achieved b forcing the drive axis into a fixed amplitude vibration at the resonant frequenc, and measuring the displacement (t) of the sense axis. The operation of a conventional Z-axis MEMS groscope, where the movement along the drive axis is much larger than the one along sense axis, is based on the following equations: x = Acos( ωt) k & + & ζ ω + ω = us ωx x Ωx& m where A is the amplitude of X-axis vibration, and ω is the resonant frequenc of the X-axis. The basic idea of force-torebalance control strateg is that, if the output amplitude of the sense axis is driven to zero b feedback control, and since && & at stead state, then (3) becomes (3) 1 us ωxx+ Ω& x (4) m at stead state. This implies that the control signal is a part of the measurement of the rotation rate Ω. Based on the fact that Coriolis acceleration and Quadrature error are 9 degree outof-phase with respect to each other, the useful Coriolis acceleration signal can be separated from the undesirable Quadrature error term and the rotation rate can be determined. Therefore our control tasks are to force the output of the sense axis to zero and to estimate the rotation rate in the presence of noises. III. ACTIVE DISTURBANCE REJECTION CONTROL Most existing control approaches for MEMS groscope require the accurate model information of the plant. However, in practice, the factors such as the mechanical-thermal noise, the measurement noise, the unknown time varing rotation rate, and the unknown Quadrature error terms, introduce modeling errors and structural uncertainties in the MEMS groscope. The mechanical imperfection and environmental variations also introduce the parameter variations to the groscope model. ADRC is a natural fit for the MEMS groscope control due to its inherent disturbance rejection 3

5 characteristics. The idea of ADRC is briefl introduced as follows. The sense axis of MEMS groscope can be taken as a slightl damped second-order sstem. The practical sstem of the sense axis can be rewritten as ( ς ω ω ω ) && = & + + x+ Ω x& + b u (5) K where b =. Define m x s ( ςω ω ω ) f = & + + x+ Ωx& (6) x ( 1 ) & ˆ ˆ ˆ ξ = Aξ + Bu + L ξ ξ ˆ C ˆ = ξ s 1 T where L = l1, l, l 3 is the observer gain. The observer gains are chosen such that the characteristic polnomial s + l s + l s+ l is Hurwitz. For tuning simplicit, all the observer poles are placed at ω o, characteristic polnomial of (9) to be λ 3 ( s) s l s l s l ( s ω ) 3 o 1 3 o (9) which results in the = = + (1) where f is referred to as the generalized disturbance. Then where ω o is the observer bandwidth of the sense axis and the sstem (5) becomes && = f + b u (7) s The basic idea of the ADRC is to obtain the estimated f, i.e., f ˆ, and to compensate for it in the control law in real time without an explicit mathematical expression of it. The ESO for the estimation purpose is presented as follows. Let ξ 1 =, ξ =, ξ3 = f Assuming & and ξ [ 1,, 3 ] T = ξ ξ ξ. f is differentiable and the derivative of f ( h = f & ) is bounded, the state space form of (5) is where & ξ = Aξ + Bu + Eh = Cξ s 1 A= 1, B b, C [1 ], E = = =. 1 An ESO for (8) is designed as (8) 3 l 1 = 3 ωo, l = 3 ωo, l3 = ωo. This makes ω o the onl tuning parameter for the observer. Hence the implementation process of the observer is much simplified. Once the observer is designed and well tuned, its outputs will track, &, f respectivel. B canceling the effect of f using ˆ ξ 3, ADRC activel compensates for f in real time. The control law is designed as follows. First, the control law u ξˆ 3 us = (11) b approximatel reduces the original plant (5) to && u (1) which is a simple control problem to deal with. A simple PD controller u = k ( r - ˆ ξ ) k ˆ ξ (13) 1 1 is usuall sufficient, where r is the desired trajector of the sense axis. The controller gains are selected so that the closed- 4

6 loop characteristic polnomial s + ks+ k 1 is Hurwitz. For tuning simplicit, all the controller poles are placed at ωc. Then the approximate closed-loop characteristic polnomial is where k λ ( s) s k s k ( s ω ) = + + = + (14) c 1 c = ω, k = ω. This makes ω c, the controller 1 c c bandwidth, the onl tuning parameter for the controller of the sense axis. This process is called bandwidth parameterization [14], which greatl simplifies the ADRC tuning. The ESO (9) and the control law (11) and (13) constitute the force-torebalance control of the sense axis of the MEMS groscope. IV. ROTATION RATE ESTIMATION On the sense axis of the MEMS groscope, both Coriolis acceleration and Quadrature error terms are amplitude modulated signals centered at the resonant frequenc of the drive axis. The onl distinguishing characteristic between the two signals is that the have a relative phase shift of 9. Therefore the advantage of this characteristic can be taken to separate the undesired Quadrature errors from the useful Coriolis acceleration through the demodulation technique. The block diagram of ADRC for the sense axis control and rate estimation is shown in Fig., where a demodulation ( ) ω x + Ω x& = f + ς ω & + ω. (15) x Let q = ω x+ Ω& x. It is assumed that the rotation rate is a x constant or low frequenc signal. Then one has ( ω ) ( ω & ) ( ω ) q sin t = x+ Ωx sin t x ( ) ( ) ( ) 1 cos( ωt ) ( ) = ω Acos ωt sin ωt ΩAωsin ωt x 1 (16) = ωx Asin ωt ΩAω 1 = ωx Asin ( ωt) +ΩAωcos ( ωt) ΩAω where ω is much larger than the frequenc of the rotation 1 ω A ωt and x rate. In (16), the high frequenc signals sin ( ) ΩAω cos ( ωt) will be filtered out through a low pass filter (LPF). Therefore the rotation rate Ω can be demodulated from the signal q b multipling sin(ωt), dividing b a gain introduced from modulation/demodulation, and filtering the resultant signal with a LPF, that is q Ω= FLPF Aω sin ( ωt) (17) where F LPF (. ) represents the function of the LPF. With the information of the ESO, according to (15), the signal q can be estimated b ( ˆ ˆ ) ˆ 1 qˆ = f + ςωξ + ωξ. (18) block is used for estimating the rotation rate. In Fig., N Therefore the rotation rate can be determined b represents the mechanical-thermal noise input to the sense axis and N m represents the measurement noise (position noise) at the output of the sense axis [7]. &. In (5), one gets x = Acos( ωt) and x = Aωsin ( ωt) From (6), it follows that qˆ sin ( ωt) ˆ Ω= FLPF. Aω The transfer function of the low-pass filter is chosen as GLPF ( s) = 1 ( τ s + 1) (19) () 5

7 where τ is the time constant of the filter. V. SIMULATION RESULTS A control sstem based on ADRC is designed and simulated on a model of the Berkele Z-axis vibrator groscope [17. The ke parameters are K=.8338, ω = rad/sec, ς = , ω x = 6 rad /sec, m= 1 9 kg = where ω = rad/sec. Tpicall and x Acos( ωt) A 6 = 1 m. 5 A = in simulation units is used to represent this [3]. The design parameter b K m 8 = = The reference signal of the sense axis is r =. In the simulation, the mechanical-thermal noise and the measurement noise are applied to the sense axis. The PSD of mechanical-thermal 7 noise for the sense axis is N sec. The PSD of 7 measurement noise for the sense axis is N sec as reported in [7]. The controller and observer parameters for the sense axis are: ω c rad / sec, ωo 1 rad / sec = =. 5 The time constant of LPF is τ = sec. The output of the sense axis under the control of the ADRC is shown in Fig. 3. The stabilized output is around.1% of the uncontrolled amplitude of, which shows that the sense axis is driven to almost zero. For rotation rate estimation, the following cases are considered: 1) Ω =.1; ) Ω =.1sin ( π f t), and f rate =5Hz; 3) Ω ( π t) rate =.1sin f, and f rate =Hz. Without changing an tuning parameters, the rotation rate estimations for different cases are shown in Fig. 4 rate Fig. 6, which show the fast and accurate rotation rate estimation. To demonstrate the robustness of ADRC against parameter variations, the sstem parameters are changed as follows: the natural frequenc of the sense axis ω is increased b % and the magnitude of the Quadrature error term is increased b %. With the changed plant parameters, the output of the sense axis and the rotation rate estimations for different cases are shown in Fig. 7 - Fig. 9 respectivel. It can be observed that the sense axis output and rate estimations are almost the same as those without parameter variations. VI. PERFORMANCE ANALYSIS In this section, the stabilit robustness and the consistent input disturbance rejection of the ADRC against the sstem parameter uncertainties are analzed through frequenc response. Taking ( ω x x x ) + Ω& as a part of the input disturbance Di ( s ), the block diagram of the sense axis control sstem can be derived in the form of a two-degree-of freedom (DOF) closed-loop sstem [18], as shown in Fig. 1, where Gp() s represents the transfer function of the plant; R(), s Us() s, and Ys () represent the reference signal, control signal, and output respectivel. The transfer function from the control input u s to the output is given b Ys () K 1 Gp() s = =. U () s ms s s + ς ω + ω (1) 6

8 Converting the ADRC equations into the transfer function form, one obtains G YD i bs s + ( l1+ kd) s+ ( l + kp + kl d 1) () s = s D s D s D s D s D d4 + d3 + d + d1 + d (7) ( kl p 1+ kl d + l3) s + ( kl p + kl d 3) s+ kl p ( 1+ d) + ( + p + d 1) 1 Gc () s = b s l k s l k k l s and 3 kp( s + l1s + ls+ l3) p 1+ d p + d 3 + p 3 () Gf () s =. (3) ( k l k l l ) s ( k l k l ) s k l The loop gain transfer function G () s and the closed-loop sstem transfer function G () s are YR lp G () s = G () s G (); s (4) lp c p where d = p 3, d1 = ω( + p + d 1) + p + d 3, d = ω( 1+ d) + ςω( + p+ d 1) + p 1+ d + 3, d3 = ω + ςω 1+ d + + p + d 1 d4 = ςω + 1+ d D k l D l k k l k l k l D l k l k k l k l k l l D ( l k ) l k k l, D l k. Equation (7) shows that GYD i () s tends to zero when s approaches zero and infinit. To find out how it behaves in the middle range frequencies and how it is affected b parametric uncertainties, the Bode plots of (7) with different ω values are shown in Fig. 1, demonstrating excellent disturbance rejection properties that are unaffected b the plant parametric G YR Gf () s Gc() s Gp() s () s =. 1 + G ( s) G ( s) c p (5) uncertainties. Similar insensitivit to changes in performed. ς is Furthermore, the transfer function from the input disturbance to the output is given b G YD The loop gain Bode plots at different i Gp () s () s =. (6) 1 + G ( s) G ( s) c p ω are shown in Fig. 11. Fig. 11 demonstrates that the gain margin, phase margin, and crossover frequenc are almost immune to changes in ω. Similar insensitivit to changes in ς is performed, but the plots are not shown here due to the space limitation. These show the strong robustness of ADRC against the variations of sstem parameters. Now the input disturbance rejection performance is analzed as follows. The transfer function from the input disturbance to the output is VII. CONCLUDING REMARKS A novel control approach of active disturbance rejection is used to control the sense axes of a Z-axis MEMS groscope. Based on the accurate estimation of the generalized disturbance b ESO, a demodulation technique is used to estimate the rotation rate. Since ADRC does not require an accurate mathematical model of the sstem, it is a good fit for the control and rate estimation of the MEMS groscope. The simulation results demonstrate the high tracking performance of the sense axis as well as the fast and accurate estimation of the rotation rates in the presence of noises and parameter variations. The performance analsis shows a remarkable level of consistenc in stabilit margins against significant plant parameter variations. Similar consistenc is also found in 7

9 disturbance rejection performance. Since most MEMS sensors have similar control problems to the MEMS groscope, i.e., amplitude regulation, disturbance rejection, and minimizing the effects of fabrication imperfections, ADRC provides a new effective solution to the problems. REFERENCES [1] S. Park and R. Horowitz, Adaptive Control for Z-axis MEMS Groscopes, In Proceedings of American Control Conference, pp.13-18, 1. [] Y. Yazdi, F. Aazi, and K. Najafi, Micromachined inertial sensors, in Proceedings of the IEEE, vol. 86, no.8, pp , [3] S. Park, Adaptive control strategies for MEMS groscopes, PhD Dissertation, The Universit of California Berkel,. [4] R. Leland, Lapunov based adaptive control of a MEMS groscope, in proceedings of American Control Conference, pp ,. [5] R. P. Leland, Y. Lipkin, and A. Highsmith, Adaptive oscillator control for a vibrational groscope, in Proceedings of American Control Conference, Denver, CO, pp , June 3. [6] A.M. Shkel, R. Horowitz, Ashwin A. Seshia, Sungsu Park, and Roger T. Howe, Dnamicsw and control of micromachined groscopes, in Proceedings of the American Control Conference, pp , [11] Z. Gao, Active disturbance rejection control: a paradigm shift in feedback control sstem design, In Proceedings of the American Control Conference, pp , 6. [1] Q. Zheng and Z. Gao, Motion control opmitimization: problem and solutions, Interntional Journal of Intelligent control and sstems, vol. 1, no. 4, pp , 6. [13] Y. X. Su, B. Y. Duan, C. H. Zheng, Y. F. Zhang, G. D. Chen, and J. W. Mi, Disturbance- rejection high-precision motion control of a Stewart Platform, IEEE Transactions on Control Sstem Technolog, vol. 1, no. 3, 4. [14] Y. Huang, K. Xu, and J. Han, Flight control design using extended state observer and non-smooth feedback, In Proceedings of the IEEE Conference on Decision and Control, pp. 3-8, 1. [15] B. Sun and Z. Gao, A DSP-based active disturbance rejection control design for a 1KW H-bridge DC-DC power converter, IEEE Trans. on Industrial Electronics, vol. 5, no. 5, pp , 5. [16] R. P. Leland, Mechanical-thermal noise in MEMS groscope, IEEE Sensor Journal, Vol.5, No.3, pp.493-5, 5. [17] W. A. Clark, Micromachined vibrator rate groscope, PhD Dissertation, The Universit of California Berkel, [18] G. Tian and Z. Gao, Frequenc response analsis of active disturbance rejection based control sstem, Submitted to the IEEE Multi- Conference on Sstems & Control. [7] S. Park and R. Horowitz, Adaptive control for the conventional mode of operation of MEMS groscopes, Journal of Microelectromechanical Sstems, vol. 1, no. 1, pp , 3. [8] L. Dong and R.P. Leland, The adaptive control sstem of a MEMS groscope with time-varing rotation rate, in Proceedings of the American Control Conference, pp , 5. [9] L. Dong, Adaptive control sstem for a vibrational MEMS groscope with time-varing rotation rates, PhD Dissertation, The Universit of Alabama, 5. [1] Z. Gao, Scaling and parameterization based controller tuning, In Proceedings of the American Control Conference, pp , 3. 8

10 * Revision Notes - list of changes and response to referees Dear Editor and Reviewers of the paper JSCE48: With man thanks for our time and consideration, I would like to address the corrections into the previous paper submission. In the new version of the paper, we have considered man of the comments as following: Response to Comments of Reviewer 1: Regarding the reviewer s comments on b, it is assumed in the paper that b, which is the ratio between the spring constant k and the mass m of the MEMS groscope, is a fixed known constant. Therefore, the robustness test on b is not included in the revised version. Responses to Comments of Reviewer : 1. Regarding the reviewer s comments on the mathematical representation of mechanical thermal noise, the terms used in the representation are explained at the end of the second paragraph of section II. Another reference paper is also added to the paper in this paragraph.. Regarding the reviewer s comments on the parameter k of equation (3), the value of k is not one. There is a tpo in the equation (3). The coefficient of u s has been changed from m 1 to m k. In addition to the corrections above, we also made the following changes. 1. Rewrite the paper with the stle in the third person;. Figures are separated from the main bod of the text. All the figures are included into the file entitled figures ; 3. A list of figure captions is provided and included into a separate file entitled List of figure captions ; 4. A list of notation is provided and included into a separate file entitle List of notation ; 5. The spacing of manuscript (including references) is changed to double line spacing across the page. 6. The issues of Editorial Checklist are addressed. Regards, Lili Dong Assistant Professor Department of Electrical and Computer Engineering Cleveland State Universit

11 Figure Figures: Z Ω Spring Damper Mass X Rigid frame Fig. 1 The mechanical structure of a Z-axis MEMS groscope. r ( t) v ( t) v1 ( t) v3 ( t) 1/ b us ( t) N N m ( t) ˆ1 ξ ˆ ξ ξˆ 3 ˆq ˆΩ Fig. Block diagram of the ADRC and rate estimation. 4 The output of sense axis Output x 1-4 The stabilized sense axis output Output Time(s) Fig. 3 The output of the sense axis.

12 . The estimated Ω.1 Ω(rad/sec) Time(s) Fig. 4 The rotation rate estimation when =.1 Ω.. The estimated Ω Ω(rad/sec) The stabilized estimated Ω Ω(rad/sec) actual Ω estimated Ω Time(s) Fig. 5 The rotation rate estimation as f rate =5Hz.. The estimated Ω Ω(rad/sec) The stabilized estimated Ω Ω(rad/sec) actual Ω estimated Ω Time(s) Fig. 6 The rotation rate estimation as f rate =Hz.

13 4 The output of sense axis Output x 1-4 The stabilized sense axis output Output Time(s) Fig. 7 The output of the sense axis with parameter variations.. The estimated Ω.1 Ω(rad/sec) Time(s) Fig. 8 The estimation of Ω (=.1)with parameter variations.. The estimated Ω Ω(rad/sec) The stabilized estimated Ω Ω(rad/sec) actual Ω estimated Ω Time(s) Fig. 9 The rotation rate estimation as f rate =Hz with parameter variations.

14 Di () s R() s Gf () s Gc () s Us ( s) G () p s Y() s Fig. 1 Block diagram of the sense axis control sstem. The frequenc responses of the open-loop sstem with different ω Magnitude (db) 1-1 ω 1 =.1ω ω = ω 1 ω ω - 9 ω =1ω Phase (deg) ω ω ω Frequenc (rad/sec) Fig. 11 The loop gain Bode plots with different ω values. -8 Bode Diagram Magnitude (db) ω 1 =.1ω ω = ω =1ω Phase (deg) Frequenc (rad/sec) Fig. 1 The Bode plots of G () s with different ω values. YD i

15 Table List of figure captions: Fig. 1 The mechanical structure of a Z-axis MEMS groscope. Fig. Block diagram of the ADRC and rate estimation. Fig. 3 The output of the sense axis. Fig. 4 The rotation rate estimation when Ω =.1. Fig. 5 The rotation rate estimation as f rate =5Hz. Fig. 6 The rotation rate estimation as f rate =Hz. Fig. 7 The output of the sense axis with parameter variations. Fig. 8 The estimation of Ω (=.1)with parameter variations. Fig. 9 The rotation rate estimation as f rate =Hz with parameter variations. Fig. 1 Block diagram of the sense axis control sstem. Fig. 11 The loop gain Bode plots with different ω values. Fig. 1 The Bode plots of G () s with different ω values. List of notation: b : coefficient of the control law; f : generalized disturbance; h : derivative of generalized disturbance; m : proof mass; r : desired trajector of the sense axis; YD i u d : control input for the drive axis; u s : control input for the drive axis; x() t : drive axis output; t ( ) : sense axis output; K B : Boltzman constant; T L = l, l, l : observer gain; 1 3 N : the mechanical-thermal noise input to the sense axis; N m : the measurement noise at the output of the sense axis; R: a term associated with damping coefficient; S n: power spectrum densit (PSD) of the noise;

16 T: absolute temperature of the resistor; ω c : controller bandwidth; ω n : natural frequenc of the drive axis; : observer bandwidth of the sense axis; ω o ω : natural frequenc of the sense axis; ς: damping coefficient of the drive axis; ς : damping coefficient of the sense axis; Ω : rotation rate.