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The Design and Implementation of Driving Mode Control for Vibrational Gyroscopes

The Design and Implementation of Driving Mode Control for Vibrational Gyroscopes

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    • 2008 American Control Conference FrB04.4 Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 2008 The Design and Implementation of Driving Mode Control for Vibrational Gyroscopes1 Lili Dong2, *, Member, IEEE, Qing Zheng2, Student Member, IEEE, David Avanesov2, Student Member, IEEE Abstract—The paper presents a novel oscillation controller for Since the early 1990s, more and more researchers have controlling the driving mode (or drive axis) of a vibrational been trying to improve the performance of the vibrational gyroscope to oscillate at a desired trajectory. The controller gyroscopes through control electronics. In [4, 5], a typical consists of a PD controller and an online Extended State Phase Locked Loop (PLL) is used to adjust the input Observer (ESO). The ESO is used to estimate the system states frequency till the output of drive axis is -900 out of phase and the discrepancy between the real system and the with the input indicating the resonance, and an Automatic mathematical model of the gyroscope. The PD controller drives Gain Control (AGC) loop is used to regulate the output the estimated discrepancy and the tracking error of the output amplitude. In this system, the input frequency is dependent of the driving mode to zero. Since the controller design does not on the mechanics of device and changing with environmental require exact information of system parameters, it is very robust against structural uncertainties of the gyroscope. The variations. As an alternative to the PLL control, an adaptive convergence of the estimation error of the ESO and the controller is developed in [6] to tune the closed-loop stability of the control system are theoretically proven. The frequency of drive axis to a fixed frequency chosen by the controller is validated by software and analog hardware designer. But in [6], the amplitude regulation of the drive axis implementations on a vibrational piezoelectric beam gyroscope. is disregarded. In [7], an adaptive oscillation controller is introduced without requiring an external driving input. Keywords: Vibrational gyroscopes, oscillation controller, However, the controller assumed an ideal model of the drive extended state observer, analog implementation. axis and did not consider the mechanical coupling terms between both axes. In [8], an adaptive controller is designed to drive the vibrations of two axes of the gyroscope operating in an adaptive mode where the movements of the mass along I. INTRODUCTION two axes are equal. However, most reported MEMS M EMS gyroscopes, also termed as vibrational gyroscopes are operating in a conventional mode [3] where gyroscopes, use vibrating elements to sense rotation the movement of the mass along the drive axis is relatively rates. Since the gyroscopes don’t have bearings, they can be large and the movement along the sense axis is very small. In easily miniaturized and batch fabricated on silicon or poly- [9], an adaptive control system is designed for both axes of silicon. The absence of bearing friction in MEMS gyroscopes the conventional MEMS gyroscopes where the mechanical also eliminates the principle cause of power consumption [1]. thermal noise and parameter variations are neglected. With the advantages of small size, low cost and low power In this paper, we aim to design an oscillation controller consumption, the MEMS gyroscopes have found broad for the drive axis of the conventional MEMS gyroscope applications in consumer and automotive markets. Some of where the coupling terms, the noise, and the parameter the well known automotive applications include vehicle variations are considered. We will apply a recently reported stability control, navigation assist, and rollover detection. Active Disturbance Rejection Controller (ADRC) [10] to the Examples of consumer applications are 3D input devices, drive axis. The ADRC is a combination of a linear observer robots, and camcorder stabilization. However, due to the high and a PD controller. The disturbance represents any production cost and low performance and reliability [1], most discrepancies between the mathematical model and a real of the applications of the MEMS gyroscopes have not system, hence is also taken as uncertainties of the system. reached any significant volume. The effects of imprecise The controller is built on the observed disturbance and micro-fabrication and environmental variations degrade the compensates for it in real time. As reported in [11, 12], performance of the MEMS gyroscope and consequently ADRC has been successfully applied to macro-systems such cause measurement errors of rotation rates [2]. Traditional as motion control, aircraft flight control, and jet engine mechanical balancing could reduce the effects of a limited control etc because of its robustness against parameter amount of the imperfection. But it is time consuming, variations and its few numbers of tuning parameters during expensive and difficult to perform on a small-size MEMS implementation. It is the first time we modify the controller gyroscope [3]. and extend its use to the MEMS gyroscopes. This paper is organized as follows. The dynamics of 1 The work is supported by Ohio ICE (Instrumental, Control, and Electronic) MEMS gyroscope is explained in Section II. The oscillation consortium under grant 0260-0652-10. controller is presented in Section III. Simulation results are 2 The authors are with the Department of Electrical and Computer shown in Section IV. It is followed by the analog Engineering at Cleveland State University, Cleveland, OH 44115. implementation and experimental results in Section V. *Corresponding author. Email: L.Dong34@csuohio.edu. Finally some concluding remarks are given in Section VI. 978-1-4244-2079-7/08/$25.00 ©2008 AACC. 4419
    • II. THE DYNAMICS OF MEMS GYROSCOPES presences of parameter variations, mechanical couplings, and mechanical-thermal noise. A MEMS gyroscope can be understood as a silicon proof mass attached to a rigid frame through springs and dampers. III. OSCILLATION CONTROLLER The mechanical structure of a MEMS gyroscope is shown in Fig.1. The proof mass is driven into resonance along the Both drive and sense axes of MEMS gyroscopes can be drive axis (X axis) in order to obtain the largest response and taken as lightly damped second-order systems. They are phase synchronization. As the rigid frame rotates around governed by the Newtonian law of motion. Then we could rotation axis (perpendicular to the page) with the rate of Ω, a rewrite the drive axis model as Coriolis acceleration is produced along sense axis (Y axis), && = f ( x, x, d ) + bud x & (2) which is perpendicular to both drive and rotation axes. The Coriolis acceleration provides the information of rotation where b is the coefficient of the controller (b=K/m), d is an rate. So we can determine the rotation rate through sensing external disturbance [12], and f ( x, x, d ) (or simply denoted & the vibration of sense axis. as f) accounts for all the other forces excluding the control effort ud [10], which is 2 f = −2ςωn x − ωn x − ω xy y + 2Ωy. & & (3) We assume b is known. If an observer is designed to estimate the f, we can take ud as Y 1 ˆ ud = (− f ( x, x, d ) + u0 ), & (4) X b ˆ where f is the estimated f, u0 is a controller to be determined. Then (3) becomes x & ˆ & && = f ( x, x, d ) − f ( x, x, d ) + u0 ≈ u0 . (5) Fig. 1: Mass-spring-damper structure of vibrational We suppose a desired signal r has the resonant frequency ω MEMS gyroscopes and the maximum amplitude A that the drive axis could Assuming the natural frequencies of both axes are same output. And r is represented by and the sense axis is operating in open loop, the vibrational r = A sin(ωt ). (6) MEMS gyroscope is modeled as Then our control goal is to drive the output signal x to the 2 K signal r. We have tracking error e=r-x. We can employ a && + 2ςωn x + ωn x + ω xy y − 2Ωy = x & & ud m common Proportional Derivative (PD) controller for u0 to (1) 2 1 drive tracking error e to zero. The controller is && + 2ςωn y + ωn y + ω xy x + 2Ωx = N (t ) y & & m u0 = k p e + k d e. & (7) where x and y are the displacement outputs of drive and sense If we take the partially unknown f as a generalized axes respectively, 2Ωx and 2Ω y are Coriolis accelerations, & & disturbance or the discrepancy between the real system and Ω is the rotation rate, ωn is the natural frequency of drive its nominal model, the controller will estimate it and and sense axes, ω xy y and ωxy x are Quadrature errors caused compensate for it actively, hence the name ADRC. by spring coupling terms between two axes, ς is damping A. Extended State Observer coefficient typically ranging from 10-3 through 10-1, m is the mass of the MEMS gyroscope, K is the controller gain The effectiveness of ADRC is dependent on the accurate including forward gain and actuator and sensor scale factors, estimation of the f. Consequently an Extended State Observer and ud is the control input for drive axis. The Quadrature (ESO) is developed to estimate the disturbance f in real time. errors are constant unknown signals, which need to be This can be achieved by using the linear state space canceled out in control effort ud. The rotation rate Ω is representation of drive axis model and augmenting the state assumed to be a constant unknown signal. In (1), mechanical & variables to include f [10]. We suppose f is unknown and thermal noise on the sense axis is represented by the random bounded. Let x1=x, x2= x , x3=f, and X=[x1, x2, x3] T, we have & force N(t). The drive axis displacement x(t) is usually so & X = AX + Bud + Eh large that the effects of thermal noise on the drive axis are (8) negligible and are disregarded [14]. For the conventional z = CX mode of operation, the displacement of the sense axis is very where small. It tends to be contaminated by the noise. Hence the noise on the sense axis can not be ignored. ⎡0 1 0 ⎤ ⎡0 ⎤ ⎡0 ⎤ A = ⎢0 0 1 ⎥ , B = ⎢b ⎥ , C = [1 0 0], E = ⎢0⎥ , h = f . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ & Our control objective is to force the drive axis to oscillate at specified amplitude and resonant frequency in the ⎢0 0 0 ⎥ ⎣ ⎦ ⎢0 ⎥ ⎣ ⎦ ⎢1 ⎥ ⎣ ⎦ 4420
    • Based on (8), a state observer is given by Now let us scale the observer estimation error ~i (t ) by x & ˆ ˆ ~ (t ) xi X = AX + Bu d + L( z − z ) ˆ (9) ωo−1 , i.e., let ε i (t ) = i , i = 1,2,3. Then (16) can be ˆ z = CX ˆ ω0−1 i rewritten as ˆ where the estimated state vector is X = [ x1 , x2 , x3 ]T , and the ˆ ˆ ˆ h( X , d ) , (17) vector of observer gain is L= [l1, l2, l3]T. We need to notice ε = ω0 Aε ε + Bε & ω02 ˆ that the key part of (9) is the third state of observer x3 , which ⎡ε1 ⎤ ⎡ −3 1 0 ⎤ ⎡0⎤ is used to approximate f. The characteristic polynomial of Where ⎢ε ⎥ , Aε = ⎢ − 3 0 1 ⎥ , Bε = ⎢ 0 ⎥ . ε = ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ observer is represented by ⎢ −1 0 0 ⎥ ⎢1 ⎥ ⎢ε 3 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ p ( s ) = s 3 + α1s 2 + α 2 s + α 3 . (10) Theorem 1: Assuming h(X,d) is bounded, there exists a constant σ > 0 and a finite time T1 > 0 such that If the observer gains are selected as l1 = 3ωo, l2 = 3 ωo2, l3= ωo3, and ωo>0, the characteristic polynomial becomes ~ (t ) ≤ σ , i = 1,2,3, ∀t ≥ T > 0 and ω > 0. Furthermore, xi 1 o p( s ) = ( s + ωo )3 . (11) ⎛ 1 ⎞ σ = Ο⎜ k ⎟ , for some positive integer k . Therefore we can change the observer gains through tuning ⎝ ωo ⎠ the unique parameter ω0, which is the bandwidth of the Proof: Solving (17), we can obtain observer. h ( X (τ ) , d ) ε ( t ) = e ω o Aε t ε ( 0 ) + ∫ e ω o Aε ( t −τ ) Bε t dτ . (18) 2 B. Control Algorithm 0 ωo After the observer is designed, the oscillation control of drive Let axis given by (4) becomes t ω A ( t −τ ) h ( X (τ ) , d ) p (t ) = ∫0 eo ε B ε 2 dτ . (19) 1 ωo u d = ( − x3 + u 0 ). ˆ (12) b Since h ( X (τ ) , d ) is bounded, that is, h ( X (τ ) , d ) ≤ δ , In order to minimize the number of tuning parameters of the where δ is a positive constant, we have controler u0, the controller parameters are chosen as Kp=ωc2 and Kd=2ωc, where ωc>0. Bε ⎤ h ( X (τ ) , d ) dτ t ω A ( t −τ ) Then (7) becomes ∫0 ⎡e o ε ⎣ ⎦i pi ( t ) ≤ 2 ωo (20) 2 u0 = ωc e + 2ωc e. & (13) δ ⎡ −1 Since the velocity of the movement along drive axis is not ≤ ωo ⎢ 3⎣ Aε Bε( ) + ( Aε eω i −1 o Aε t B ε ) ⎤, i⎥ ⎦ measurable, the controller u0 built on the observed velocity is ⎡0 0 − 1⎤ shown as where i = 1, 2,3. Since A −1 = ⎢ 1 0 − 3 ⎥ , one has ε ⎢ ⎥ u 0 = ω c2 (r − x1 ) + 2ω c (r − x 2 ) ˆ & ˆ (14) ⎢0 ⎣ 1 −3⎥ ⎦ ⎡-1⎤ ⎧1 Assuming accurate estimation of the states, the ideal closed- loop transfer function of controller is ( Aε−1 B ) i ⎢ ⎥ ⎪ i =1 = ⎢-3⎥ = ⎨ 3 . (21) ⎢-3⎥ i ⎪ i = 2,3 ⎣ ⎦ ⎩ 2 x ω c + 2ω c s (15) Since matrix Aε is Hurwitz, there exists a finite time T1 > 0 G (s) = = . r 2 2 s + 2ω c s + ω c such that From (15), we can see that this is a typically second-order ⎡eωo Aε t ⎤ ≤ 1 (22) critical-damping control system, where ωc is the only one ⎣ ⎦ ij ω 3 o tuning parameter for the control input ud. The details about how to tune the parameters of ADRC are introduced in [11]. for all t ≥ T1 , i, j = 1, 2,3 . Hence C. Stability Analysis ⎡ e ω o Aε t B ⎤ ≤ 1 (23) ⎣ ⎦i ωo3 1) Convergence of the ESO for all t ≥ T1 , i = 1, 2,3 . Note that T1 depends on ωo Aε . Let ~i t ) = xi (t ) − xi (t ), i=1, 2, 3. From (8) and (9), the x ˆ ⎡ s11 s12 s13 ⎤ ⎡ d11 d12 d13 ⎤ observer estimation error dynamics can be shown as Let Aε−1 = ⎢ s21 s22 s23 ⎥ and eωo Aε t = ⎢ d 21 d 22 d 23 ⎥ . ⎢ ⎥ ⎢ ⎥ ~ = ~ −l ~ & x1 x2 1 x1 ⎢ s31 s32 s33 ⎥ ⎢ d31 d32 d33 ⎥ ⎣ ⎦ ⎣ ⎦ ~ = ~ −l ~ & x x x (16) One has 2 3 2 1 ~ = h−l ~. & x3 3 x1 4421
    • ( Aε eω −1 o Aε t B ε ) i = si1d13 + si 2 d 23 + si 3 d33 Let e ( t ) = ⎡ e1 ( t ) , e2 ( t ) ⎤ , X ( t ) = ⎡ x1 ( t ) , x2 ( t ) , x3 ( t ) ⎤ , ⎣ ⎦ T % ⎣% % % ⎦ T ⎧ 1 (24) then ⎪ 3 ⎪ ωo ⎪ i =1 e ( t ) = Ae e ( t ) + AX X ( t ) & % % (31) ≤⎨ ⎪ 4 ⎡ 0 1 ⎤ ⎡ 0 0 0⎤ where Ae = ⎢ ⎥ and AX = ⎢ −k . − 1⎥ ⎪ω 3 % ⎪ o ⎩ i = 2,3 ⎣ −k1 −k2 ⎦ ⎣ 1 −k2 ⎦ for all t ≥ T1 . From (20), (21) and (24), we obtain Solving (32), we have e ( t ) = e Aet e ( 0 ) + e e ( ) A % X (τ ) dτ . t A t −τ pi ( t ) ≤ δ ⎡ 4 ⎤ 3δ 4δ 3+ 3 ⎥= 3 + 6 (25) ∫0 % X (32) 3 ⎢ ωo ⎢ ωo ⎥ ωo ωo ⎣ ⎦ According to (32) and Theorem 1, one has for all t ≥ T1 , i = 1, 2,3. ⎡ AX X (τ ) ⎤ = 0 ⎣ % % ⎦ i =1 Let ε sum ( 0 ) = ε1 ( 0 ) + ε 2 ( 0 ) + ε 3 ( 0 ) . It follows that ⎡ AX X (τ ) ⎤ = −k1 x1 (τ ) − k2 x2 (τ ) − x3 (τ ) ⎣ % % ⎦2 % % % (33) ⎡eωo Aε t ε ( 0 ) ⎤ = di1ε1 ( 0 ) + di 2ε 2 ( 0 ) + di 3ε 3 ( 0 ) ⎣ ⎦i ≤ k sumσ = γ for all t ≥ T1 t ε (0) (26) where ksum = 1 + k1 + k2 . Let ϕ ( t ) = ∫ e Ae ( t −τ ) AX X (τ ) dτ . % % 0 ≤ sum 3 ωo Define Ψ = [ 0 γ ] . It follows that T for all t ≥ T1 , i = 1, 2,3 . From (18), one has ϕi ( t ) = ∫ ⎡ e Ae ( t −τ ) AX X (τ ) ⎤ dτ t % 0⎣ ⎦i % ε i ( t ) ≤ ⎡eω Aε t ε ( 0 ) ⎤ + pi ( t ) . ⎣ o ⎦i (27) ≤∫ t ⎡ e Ae ( t −τ ) Ψ ⎤ dτ (34) 0⎣ ⎦i Let X sum ( 0 ) = X 1 ( 0 ) + X 2 ( 0 ) + X 3 ( 0 ) . % % % % According to X i (t ) % ( − ≤ Ae 1Ψ ) + (A −1 Aet e e Ψ ) . εi (t ) = i i and (25)-(27), we have ωo−1 i ⎡ k2 1⎤ ⎡ 2 1 ⎤ − − − − ε (0) i Since − Ae 1 = ⎢ k1 k1 ⎥ = ⎢ ωc ωc ⎥ , we have 2 X i ( t ) ≤ sum 3 ωo−1 + pi ( t ) ωo−1 % i ⎢ ⎥ ⎢ ⎥ ωo 0 ⎦ ⎢ 1 ⎢ 1 ⎣ ⎥ ⎣ 0 ⎥ ⎦ X sum ( 0 ) % 3δ 4δ (28) γ ≤ 3 ωo + ωo − i 4 + ωo − i 7 ( A Ψ) −1 e 1 = ωc2 (35) =σ for all t ≥ T1 , i = 1, 2,3. Q.E.D. ( − Ae 1Ψ ) 2 = 0. 2) Convergence of the ADRC Since Ae is Hurwitz, there exists a finite time T2 > 0 such Let [ r1 , r2 , r3 ] = [ r , r , &&] T T & r and ei ( t ) = ri ( t ) − X i ( t ) , that i = 1, 2. ⎡e Aet ⎤ ≤ 1 (36) Theorem 2: Assuming that h is bounded, there exist a ⎣ ⎦ ij ω 3 c constant ρ > 0 and a finite time T3 > 0 such that for all t ≥ T2 , i, j = 1, 2. Note that T2 depends on Ae . Let ei ( t ) ≤ ρ , i = 1, 2, ∀t ≥ T3 > 0, ωo >0 and ωc >0. Furthermore, ⎡o o ⎤ e Aet = ⎢ 11 12 ⎥ and esum ( 0 ) = e1 ( 0 ) + e2 ( 0 ) . It follows ⎛ 1 ⎞ ⎛ 1 ⎞ ⎣ o21 o22 ⎦ ρ = Ο⎜ k ⎟ and ρ = Ο ⎜ q ⎟ for some positive integers k that ⎝ ωo ⎠ ⎝ ωc ⎠ ⎡e Aet e ( 0 ) ⎤ = oi1e1 ( 0 ) + oi 2 e2 ( 0 ) and q . ⎣ ⎦i (37) Proof: From (12) and (14), one has esum ( 0 ) ≤ k1 ( r - x1 ) + k2 (r - x2 ) − x3 + && ˆ & ˆ ˆ r 3 ωc ud = b for all t ≥ T2 , i = 1, 2. Let T3 = max {T1 , T2 } . We have k1 ⎡ r1 − ( x1 − x1 ) ⎤ + k2 ⎡ r2 − ( x2 − x2 ) ⎤ − ( x3 − x3 ) + r3 ⎣ % ⎦ ⎣ % ⎦ % (29) = γ b k1 ( e1 + x1 ) + k2 ( e2 + x2 ) − ( x3 − x3 ) + r3 % % % (e Aet Ψ ) i ≤ ωc3 (38) = . b for all t ≥ T3 , i = 1, 2 , and It follows that e1 = r1 − x1 = r2 − x2 = e2 & & & ⎧1 + 2ωc γ ⎪ 2 3 e2 = r2 − x2 = r3 − ( x3 + bud ) ⎪ ωc ωc (A ) ⎪ & & & −1 Aet i =1 (30) e e Ψ ≤⎨ (39) = r3 − x3 − ⎡ k1 ( e1 + x1 ) + k 2 ( e2 + x2 ) − ( x3 − x3 ) + r3 ⎤ ⎣ % % % ⎦ i ⎪γ = − k1 ( e1 + x1 ) − k2 ( e2 + x2 ) − x3 . ⎪ 3 % % % ⎪ ωc i = 2 ⎩ 4422
    • for all t ≥ T3 . From (34), (35), and (39), we obtain a controller gain ωc as 5×104 and an observer gain ωo as 25×104. Fig. 2 (a) shows the real output x of drive axis and ϕi ( t ) ≤ ( Ae−1Ψ )i + ( Ae−1e A t Ψ )i e the reference signal r in the first 1ms. Fig. 2 (b) shows the ⎧ γ 1 + 2ωc γ (40) output x and r in one period which is around 0.1ms as we ⎪ 2+ desired. We can see that after initial deviation, the output x ⎪ ωc ωc2 ωc3 i =1 ≤⎨ can track the reference signal r very well. Fig. 3 shows the ⎪γ tracking error between the reference signal r and the real ⎪ω 3 ⎩ c i=2 output of drive axis x in different time ranges. The stabilized for all t ≥ T3 . From (32), one has peak error is around 0.7% of the desired amplitude of the output x. Our simulation results demonstrate the robustness ei ( t ) ≤ ⎡e Aet e ( 0 ) ⎤ + ϕi ( t ) . ⎣ ⎦i (41) of the oscillation controller against parameter variations (within 40%) and mechanical-thermal noise. According to (34), (35), and (37), we have The Output of Drive Axis ⎧ esum ( 0 ) γ γ (1 + 2ωc ) 0.1 ⎪ 3 + 2+ 5 ⎪ ωc ⎪ ωc ωc 0.05 ei ( t ) ≤ ⎨ i =1 x and r r 0 ⎪ esum ( 0 ) + γ x -0.05 ⎪ 3 ⎪ ⎩ ωc i =2 -0.1 ( ) ( ) 0 0.2 0.4 0.6 0.8 1 ⎧ 1 + 2ωc + ωc σ 1 + 2ωc + ωc (1 + 2ωc ) σ 2 2 ⎪ esum ( 0 ) + Time (seconds) -3 x 10 + (a) ⎪ ω3 ωc 2 ωc 5 c ⎪ i =1 =⎨ 0.1 ⎪ ( ⎪ esum ( 0 ) + 1 + 2ωc + ωc σ ) 2 0.05 x and r 0 r 3 ⎪ ωc x ⎩ i =2 -0.05 ≤ρ -0.1 0.035 0.035 0.035 0.035 0.035 0.03510.03510.03510.0351 (42) Time (seconds) ⎧ esum ( 0 ) (1 + 2ωc + ωc ) σ 2 (b) ⎪ for t ≥ T3 , i = 1, 2, where ρ = max ⎨ + Fig. 2: The output of drive axis and the reference signal ⎪ ωc ωc2 3 ⎩ Tracking Error + (1 + 2ω c + ωc2 ) (1 + 2ωc ) σ esum ( 0) + (1 + 2ωc + ωc2 ) σ ⎫ ⎪ 0.06 , ⎬. Q.E.D. ωc5 ωc3 ⎪ 0.04 Error e ⎭ 0.02 IV. SIMULATION RESULTS 0 -0.02 We simulate the oscillation controller on the model of a 0 0.2 0.4 0.6 0.8 1 piezoelectrically driven vibrational beam gyroscope [6]. The Time (seconds) x 10 -3 (a) vibrational beam gyroscope consists of a 20mm-long steal -3 x 10 beam and four piezoelectric strips which are attached to each side of the cross-section of the beam. Two of the strips are 1 functioning as actuators, and the other two are acting as Error e 0 sensors. The beam is driven to resonance through one of the piezoelectric actuators. The deformation (or displacement) of -1 the beam is transformed to voltage output through the other 1 1.1 1.2 1.3 1.4 piezoelectric actuator. The voltage output of the actuator is Time (seconds) x 10 -3 (b) proportional to its displacement. As the output is 100mv, the corresponding displacement of the beam gyroscope is Fig. 3: The tracking error e approximately 1 micrometer. The natural frequency and V. ANALOG IMPLEMENTATION AND damping coefficient of the gyroscope are ωn=63881.1rad/s EXPERIMENTAL RESULTS and ς=0.0005 N ⋅ s / m . The desired frequency of drive axis is ω=65973.4 rad/s (f=10.17KHz), which is approximately We conduct the analog implementation of the ADRC onto equal to its natural frequency. Normally the displacement the drive axis of the vibrational gyroscope because of fast output amplitude of drive axis is A=10-6 m. So we choose response and low cost of the analog circuit. The setup 100mv as the desired amplitude of the reference signal. We diagram of the Analog Implementation of the ADRC Using use A=0.1 in simulation units to represent this. Then the Discrete Components (AIUDC) is shown in Fig.4. In Fig.4, a reference signal r = 0.1sin (65973.4t). The PSD of saturator is used to limit the amplitude of the control signal mechanical-thermal noise is 4.22×10-26 N2sec according to that prevents the fragile vibrational gyroscope from being [8]. We assume the magnitude of Quadrature error term is broken by possibly large magnitude of the input. We choose 0.1% of natural frequency according to [9]. We suppose the 100mv as the desired amplitude of the reference signal. The rotation rate is constant and Ω = 0.1rad / s. The coefficient other parameters of the beam gyroscope are subject to the b=K/m=271780942.56. For the ADRC and ESO, we choose variations of ±10% in the original values in practice. 4423
    • The output of drive axis Reference 0.1 signal 0.08 AIUDC Saturator 0.06 0.04 0.02 Xout ( v ) Actuator Vibrational 0 gyroscope -0.02 -0.04 Fig. 4: Setup diagram of analog implementation of ADRC -0.06 The reference signal and the real output of the drive axis -0.08 are given in Fig. 5. The tracking error between the real output -0.1 0 20 40 60 80 100 120 140 160 180 200 and the reference signal is shown in Fig.6. The peak error is Time (micro-seconds) about 10% of the original amplitude, which is bigger than the Fig.7: The output of the drive axis one in simulation. We believe this is mainly caused by the electronic noise coming from the circuit. However, the VI. CONCLUDING REMARKS analog implementation of the ADRC produces a very fast We explored a new application of the ADRC for response as shown in Fig. 7 where the output of the drive axis controlling the driving mode of vibrational gyroscopes. The reaches the desired output within 0.14ms only. In [6], the stability analysis shows that both the estimation error and the same beam gyroscope is used for testing an adaptive tracking error of the drive axis output are bounded and the controller which is also implemented by an analog circuit. It upper bounds of the errors monotonously decrease with the takes a much longer time (almost 3 seconds) for the adaptive increase of observer and controller bandwidths. The control system in [6] to reach the steady-state response. In simulation and analog results validated the oscillation addition, with the application of the ADRC, the vibrational controller with the chosen controller and observer gains. In beam gyroscope achieved resonance for a bigger range of the our future research, we plan to implement the ADRC on the input frequency (150Hz) than the range given in [6] (100Hz). sense axis of the vibrational gyroscope. We will also explore 0.15 The reference signal and the real output of the drive axis the possibility of using the ADRC to other MEMS sensors. REFERENCES 0.1 [1] N. Barbour, G. Schemidt, “Inertial sensor technology trends,” IEEE 0.05 Sensor Journal, Vol.1, No.4, pp. 332-339, Dec. 2001. [2] Y. Yazdi, F. Ayazi, and K. Najafi, “Micromachined inertial sensors,” Xout (v) 0 in Proceedings of IEEE, Vol. 86, No.8, pp.1640-1659, August 1998. [3] S. Park and R. Horowitz, “Adaptive control for the conventional mode -0.05 of operation of MEMS gyroscopes,” Journal of Micro- electromechanical Systems, Vol.12, No.1, pp. 101-108, Feb. 2003. -0.1 [4] C. Acar, S. Eler, and A. Shel, “Concept, implementaion, and control reference signal of Wide Bandwidth MEMS Gyroscopes,” in Proceedings of American real output Control Conference, Arlington, VA, pp.1229-1234, June 2001. 0 20 40 60 80 100 120 140 160 180 200 [5] R. T. M’Closkey, A. Vakakis, “Analysis of a micromsensor automatic Time (micro-seconds) gain control loop,” in Proceedings of American Control Conference, San Diego, CA, pp. 3307-3311, June 1999. Fig.5: The output of the drive axis and reference signal [6] R. P. Leland, “Adaptive tuning for vibrational gyroscopes,” in Tracking error Proceedings of 40th IEEE Conference on Decision and Control, 0.05 Orlando, FL, pp. 3447-3452, Dec. 2001. 0.04 [7] R. P. Leland, Y. Lipkin, and A. Highsmith, “Adaptive oscillator 0.03 control for a vibrational gyroscope,” in Proceedings of American Control Conference, Denver, CO, pp. 3347-3352, June 2003. 0.02 [8] S. Park, “Adaptive control strategies for MEMS gyroscopes,” Ph.D. 0.01 dissertation, University of California, Berkeley, 2000. Error e (v) 0 [9] L. Dong and R. P. Leland, “The adaptive control System of a MEMS gyroscope with time-varying rotation rate,” in Proceedings of -0.01 American Control Conference, Portland, OR, pp. 3592-3597, June -0.02 2005. -0.03 [10] Z. Gao, “Active disturbance rejection control: a paradigm shift in -0.04 feedback control system design,” in proceedings of American Control Conference, Minneapolis, MN, June 2006. -0.05 0 20 40 60 80 100 120 140 160 180 200 [11] Z. Gao, “Scaling and bandwidth-parameterization based controller Time (micro-seconds) tuning,” in Proceedings of American Control Conference, Denver, CO, Vol. 6, pp.4989-4996, June 2003. Fig. 6: The tracking error between the reference signal and [12] Z. Gao, Y. Huang, J. Han, “A novel motion control design approach the real output of the drive axis based on active disturbance rejection,” in IEEE Conference on Decision and Control, Orlando, FL, Vol.5, pp. 4578-4585, Dec. 2001. 4424