Microsyst Technol DOI 10.1007/s00542-015-2582-8
TECHNICAL PAPER
Drive‑mode control for an underactuated MEMS vibratory rate gyroscope Guangping He1 · Chenghao Zhang1 · Wei Sun1 · Quanliang Zhao1
Received: 6 February 2015 / Accepted: 20 May 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract It has been shown that the underactuated rate gyroscopes (URGs) hold appealing robust operational performance against the fluctuations of temperature, pressure and structure parameters. In order to further improve the performance of the URGs, the dynamics and the closedloop control issues of the URGs are investigated in this paper. For a three-DOFs gyroscope with two drive-modes, single sense mode, and single actuator, it is shown that the dynamics of the drive subsystem of the URG can be described as a second-order nonholonomic constraint system, and the dynamics of the drive subsystem of the URG can be transformed into a fourth-order linear system with the Brunovsky’s canonical forms by properly selecting the controlled output. Then a robust controller is presented to stabilize the underactuated system by solving a set of linear matrix inequalities. Additionally, the stability of the presented controller is analyzed and demonstrated by some simulations.
1 Introduction Conventional z-axis rate gyroscopes with single-oscillator show a narrow operational bandwidth (Yazdi et al. 1998; Acar and Shkel 2005). The mode matching between the drive-mode and the sense-mode is usually necessary to attain the maximum sensitivity. However, a conventional single-oscillator gyroscope based on exactly or closely
* Guangping He
[email protected] 1
Department of Mechanical and Electrical Engineering, North China University of Technology, Beijing 100041, People’s Republic of China
matching the drive and sense modes is extremely sensitive to variations in system parameters that shift the inherent frequencies and introduce quadrature errors (Painter and Shkel 2003). Batch-fabricated MEMS rate gyroscopes with single oscillator commonly deviate from the design target and also from device to device due to the inevitable fabrication imperfections. To achieve the mode matching condition and improve the robustness of the vibratory rate gyroscopes (VRGs), closed-loop controllers are generally adopted (Sung et al. 2009; Antonello et al. 2009; Leland 2006; Park et al. 2007; Dong and Avanesian 2009). However, the closed-loop controllers are commonly designed outside of the MEMS devices due to the difficulties in designing the on-chips controllers, which turns to enlarging gyroscopes size and reducing their potential using possibilities in some occasions such as rockets guidance system. To overcome the limitations of the VRGs with single oscillator and simplify the controllers, Acar and Shkel (Acar 2004; Acar and Shkel 2003, 2006) have designed several kinds of VRGs with multi-oscillators and obtained some measurement results based on the open-loop controllers. According to them, open-loop controlled VRGs with multi-oscillators show a wide bandwidth in frequency response and then dramatically improved the operational robustness. However, the accuracy of the VRGs presented in (Acar 2004) is not higher than 1 % for a three-DOFs MEMS gyroscope with double drive-modes controlled by an open-loop controller. A rate gyroscope with this accuracy grade could not be applied in navigation or guidance systems, where performances of better VRGs are wanted. It is worth noting that the inertial sensors are generally designed on the basis of precession effects, which is caused by the coupling motions of two different vector
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fields. The differential equations representing the coupling motions are generally not integrable for a dynamic system which the amounts of independent actuators is less than DOFs of system. Generally, the non-integrable constraints are the so-called nonholonomic constraints (Bloch 2003). A nonholonomic constraint system controlled by open-loop periodical inputs reveals the inherent “geometric phase” motions (Shapere and Wilczek 1989) that cause the drifts in the states of the overall system. The primary characteristic of the geometric phase motions only depends on the path of the periodical inputs rather than the styles (Shapere and Wilczek 1989). This indicates that the path of the periodical inputs determines the repeatability of the geometric phase motions of a nonholonomic system. Obviously, the precessions of the inertial sensors happen to be the geometric phase motions. Therefore, the closed-loop controllers are still required from the point of view of the real-world applications since the accuracy and the stability of the periodical inputs can significantly influence the performances of the inertial sensors. In recent years, some researches (Sung et al. 2009; Antonello et al. 2009; Leland 2006; Park et al. 2007; Dong and Avanesian 2009) involved the design of the closed-loop controllers for the VRGs. However, few of them are related to the underactuated rate gyroscopes (URGs) with multi-oscillators. Motivated by the relevant researches of Acar and Shkel (Acar 2004; Acar and Shkel 2003, 2006), the dynamics and the closed-loop control problems for the URGs are investigated in this paper. For a three-DOFs MEMS rate gyroscope with double drive modes actuated by single input, the dynamics of drive subsystem of the URG can be transformed into the fourth-order Brunovsky’s canonical form by a set of globally reversible transformations associated with the states and input. The main contributions of the paper can be mentioned here that, the drive subsystem of an URG is shown to be a second-order nonholonomic constraint system in Sect. 3. This result is crucial in understanding the necessity of the applications of the closed-loop controllers for all inertial sensors due to the nonholonomic constraints; We further show in Sect. 4 that the dynamics of the drive subsystem of the URG under consideration can be transformed into a higher-order linear system by properly selecting the controlled outputs. This result is important in understanding the stability of the zero dynamics for the underactuated systems controlled by output feedbacks; To stabilize the periodical excitation for the sense mode of the URG, a new robust controller is presented in Sect. 5 by solving a set of linear matrix inequalities (LMIs).
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Microsyst Technol
2 The dynamics of the MEMS uneractuated vibratory rate gyroscopes Consider an n-DOFs mechanical system. Suppose that the generalized coordinate of the system in configuration space is q ∈ Rn, the kinematic energy of the system is denoted by ˙ ∈ R, and the potential energy of the system is T = T (q, q) U = U(q) ∈ R. Define L = T − U to be the Lagrangian, then the dynamics of the system can be written as ∂L d ∂T =τ − (1) dt ∂ q˙ ∂q
where τ ∈ Rm is the generalized forces. If m 0.
Then using the Lemma 1, we can prove the following theorem for designing the observer-based feedbacks (31), such that the closed-loop system (32) is globally exponentially stable (GES).
Microsyst Technol
Theorem 1 If there exist a set of constants γi > 0, yyi = 1, …, 4, such that the following bilinear-matrix-inequities (BMIs) are solvable
T S11 − S12 S−1 22 S12 < 0 P >0 1 P2 > 0
(35)
where
S11 =
¯ 11 A 0
S12 =
BKP−1 1 0
S22
0 , A22 0 I
P1−1 N1T 0
0 P2T M1
0 , P2T LM2
− γ11 P1−1 0 0 0 0 0 −γ1 P1−1 0 0 0 = 0 0 −γ2 I 0 0 , 0 0 0 −γ3 I 0 0 0 0 0 −γ4 I
¯ 11 = P−1 (A − BK)T + (A − BK)P−1 A 1 1 + γ2 M1 M1T + γ3 N1 N1T + γ4 N2 N2T
A22 = (A − LC)T P2 + P2 (A − LC)
Then the observer-based uncertain linear system (32) is GES. Proof See Appendix.
Remark 4 The main skill used in the proof of the theorem 1 is the dexterous utilizations of the Young’s inequality (34). To get the gains of the controller and the observer by solving the BMIs (35), one should use the substitutions
¯ = P−1 KT , L¯ = LT P2 K 1
(36)
T to transform the bilinear matrix terms (BMTs) P−1 1 (BK) , −1 T BKP1 , (LC) P2, and P2LC into the linear matrix terms (LMTs), such that the matrices S11, S12 and S22 of (35) can be rewritten as the linear matrices. Then by utilizing the LMI-solver in the Robust Control Toolbox of the Matlab software, the BMIs (35) can be resolved without any obstacles, and the gains of the controller and the observer can be given by
¯ T P1 , L = P−1 L¯ T K=K 2
(37)
Remark 5 In the relevant literatures, for instance (Lien 2004; Heemels et al. 2010), a priori choices of P1 = I and
P2 = I are commonly employed to change the BMTs in (35) to the LMTs, such that the obtained methods are rather conservative due to the unnecessary additional constraints P1 = I and P2 = I for the BMIs (35). However, by applying the substitutions (36) and the Young’s inequality (34), the BMIs (35) has not any additional constraints. Thus the approach presented by the Theorem 1 shows less conservation. These techniques are useful for searching for the feasible solutions of BMIs (35) in a larger region so that the uncertain linear system (32) could be stabilized in a larger domain of the uncertainties.
Remark 6 It is worth mentioning that the given positive constants γi > 0, i = 1,…,4 in BMIs (35) are not exclusive. However, the optimizations of the given positive constants γi are still an open problem. Remark 7 Noteworthily and commonsensibly, the nonholonomic systems cannot be stabilized by any smooth, time-invariant, and pure state feedbacks, due to the Brockett’s theorem (Brockett 1983). The feasibility of stabilizing the nonholonomic system (20) by the observer-based linear controller (31) is guaranteed by the proposition 2, which shows that the nonholonomic system (20) can be transformed into a higher-order linear system by the output feedbacks. Under this case, the controllers of the higherorder linear systems have to use the dynamic feedbacks. In other words, the closed-loop controllers should use the feedbacks associated with both the states of the system and the time-derivates of them.
6 Numerical simulations For the linear system (28), the observer-based controller (31) can be easily designed by solving the BMIs (35). The Figs. 4 and 5 illustrate the responses of the closedloop system (28) controlled by the feedback (31) in order to stabilize the system to the origin. Where the initial states of the system are given by σ(0) = (1, 2, 3,−1) and e(0) = σ (0) − σˆ (0) = (−1, −1, −1, 1), the uncertain matrices are given by 0 0 0 −δ 0 0 −δ � � 0 , ∆C = 0 0 −δ 0 , ∆A = 0 δ 0 0 δ 0 0 0 for the purpose of clarity, where δ = 0.1 sin (2t). Without a doubt, other choices for the uncertain matrices ΔA and ΔCwill also be feasible. If the positive constants in BMIs
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Microsyst Technol
σ 1, σ 2 , σ 3 ,σ 4
10
σ1 σ2 σ3 σ4
5 0 -5
-10
0
5
10
15
time(s)
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Fig. 4 The state responses of the closed-loop system (32) for the third-order linear system (28)
Furthermore, the stability of the closed-loop URG system is simulated in the sequel, and then the sense output is obtained with considering the uncertainties of the structure parameters and the fluctuations of the spring’s stiffness. The nominal values of the physical parameters of the URG are listed in the Table 1, which are obtained from a specific design in (Acar 2004, p. 129) and are used in our simulations. Refer to the relations presented in Sect. 2, the nominal inherent frequencies of the URG can be calculated as: ω1x =
e1 , e2 , e3 , e4
2
e1
1
e2 e3
0 -1
e4 0
5
10
15
time(s)
20
Fig. 5 The errors of the observer’s outputs of the closed-loop system (32) for the third-order linear system (28)
Table 1 Physical parameters of the 3-dofs urg Symbol
Quantity
Unit
Nominal value
m1
Mass of the drive oscillator
kg
22.0 × 10−7
m2
Mass of the sense oscillator
kg
8.22 × 10−7
mf
Mass of the decoupling frame kg
1.7 × 10−7
k1x
Spring’s stiffness of the drive N/m oscillator
300
k2x
Spring’s stiffness of the decoupling frame
N/m
64
k2y
Spring’s stiffness of the sense N/m oscillator
90
c1x, c2x,c2y Damping coefficient of the URG
N/(m s−1) 5.0 × 10−4
(35) are selected to be γ1 = 0.03, γ2 = γ3 = γ4 = 500, then the parameters of the controller (31) are obtained as K = 13.7068 30.2477 25.8581 8.2985 , and
L = 6.4936
10.8471
8.6954
2.9087
T
by solving the BMIs (35) with considering the substitutions (36). The corresponding simulation results are plotted in Figs. 4 and 5.
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k1x m1 ≈ 11.677kHz, ω2x = k2x m2 ≈ 8.824 kHz.
The mass and frequency ratios are respectively given byµ = m2 /m1 ≈ 0.451, and η = ω2x /ω1x ≈ 0.756.By the relations (17) and (18), the two resonant frequencies of the drive-modes are calculated as ωd−n1 ≈ 1.175 kHz, and ωd−n2 ≈ 2.222 kHZ, while that of the sense mode is given by ωd − y ≈ 1.665 kHz, which satisfies the desired relation ωd−n1
