天津大学2010届本科生毕业设计（论文）
CONCLUSION
Pendulous servo silicon accelerometers with Charge Controlled rebalance electronics have been developed with good overall clnamcteristics. The bias stability achieved is more than an oirder of magnitude better than a voltage rebalanced loop of the same g range. Higher performance all silicon accelerometers are being developed using
Silicon Direct Wafer Bonding and ME. These technologies can also be applied toward Coriolis angular rate sensors, and low cost, highly integrated IMUs.
天津大学2010届本科生毕业设计（论文）
Vibratory gyroscope controller design via modified automatic gain
control configuration （provenance：Sangkyung Sung 1,Woon-Tahk Sung 2 Jang Gyu Lee 2,and Taesam Kang
Department of Aerospace Information Engineering, Konkuk University, Seoul, Korea School of Electrical Engineering and Computer Science, Seoul National University Seoul, Korea）
Abstract: In this paper, a new design approach of force rebalance loop is
investigated for the vibratory rate sensor application. The proposed rebalance loop design takes advantages of a modified AGC loop configuration to simplify the system dynamics of oscillating characteristics. The proposed modification of AGC and rebalance strategy, which maintains controlled oscillation in the loop, acquires several advantages. First, it is possible to analyze and design the transient dynamics using a classical linear control theory. Also the control system to achieve the design objective is implemented using a relatively simple feedback loop. The practical application to vibratory gyroscope has shown that the force rebalance control with the proposed AGC configuration meets the design objective well from simulation results. Experiment is expected to verify the feasibility and performance of the proposed control scheme.
Keywords: force rebalance, rate sensor, AGC, oscillation, feedback, gyroscope, performance.
1. INTRODUCTION
A gyroscope is one of the key inertial sensors comprising the inertial measurement unit(IMU)in aerospace, industrial, and commercial filed. For its wide application and adoption into many products, high reliability and sensor performance is required. Usually the operation of gyroscope incorporating mechanical dynamics is to detect the variation of displacement induced by the Coriolis acceleration. In this, the signal detection in an open loop scheme is a simple and easy way to implement, but may generate many undesirable performance characteristics such as degraded scale factor linearity, limited dynamic range and bandwidth, and other mechanical restrictions. In this sense, the force rebalance of the moving mass has been a good breakthrough to overcome the performance limitation of the open loop sensor. The force rebalance loop is a feedback control system that maintains the proof mass at a designed state with reciprocally balanced forces from mechanical and electrical sources. In the past decade, there has been a variety of reports of adapting the rebalance scheme into MEMS vibratory gyroscopes[1-9].The control algorithm of[1,2]performs in an adaptive fashion. It attempts to not only allow the estimation of input angular rates, but also the compensation of gyroscope quadrature error as well as the mode tuning for enhanced sensitivity. However, the main defect of such an adaptive control approach is the difficulty in realizing an adaptive algorithm with electrical circuits.
天津大学2010届本科生毕业设计（论文）
Other researchers have found that the complications with electronic circuits are found to mainly arise from their controller implementation[4-6,8,9].In short, the literature survey presents a strong case for applying the force rebalance method to control a MEMS vibratory gyroscope. In this paper, a modified automatic gain control (AGC)loop design incorporating a double folded nonlinear feedback connection is proposed to obtain a simple force rebalance realization of the oscillating sensor and applied practically to the vibratory gyroscope to obtain an enhanced sensor performance. In section 2,we illustrate fundamentals of vibratory rate sensor and force rebalance scheme by introducing the AGC loop. In section 3,we investigate a loop transformation and controller design. Then we include simulation results in section 4 and conclusion and further research in section 5.
2. VIBRATORY GYROSCOPE AND FORCE REBALANCE
In this section, the operational principle and system dynamics for the mechanically vibrating gyroscope is described and the loop modeling is considered.Fig.1 shows the simple block diagram of the vibratory gyroscope, which illustrates the basic operational principle. In the figure, the simplified mechanical component of the vibratory gyroscope and rotating coordinate frame are depicted. Using the definition of driving mode axis and sensing mode axis, the proof mass connected with the fixed outer frame through spring undergoes oscillation in x-axis and introduces a perpendicular force in y-axis by the Coriolis acceleration.
Fig.2 illustrates the conceptual diagram of system dynamics, where the transfer function is derived from the following dynamic equations,
where m is mass, and
are damping coefficients, and spring constants,
is an angular rate applied in z-axis
respectively, corresponding to each axis. Here
and denotes a driving force. To derive the sensing mode transfer function in Fig.2, it is assumed that the sensing mode velocity is much smaller than the driving force input in(1),thus the cross coupling term -2mis neglected. By observing the system characteristics in Fig.2, it is easily identified that the forced oscillation in sensing mode results from the coupled dynamics of both driving mode oscillation and sensing
天津大学2010届本科生毕业设计（论文）
mode oscillation. Therefore the sensing mode output due to the angular rate input is always a velocity modulated signal. In this background, the design objective of force rebalance loop is to improve the performance indices such as dynamic range, output linearity and bandwidth, which is accomplished by way of suppressing the sensing mode oscillation induced by the Coriolis acceleration. Then by measuring the control input that suppresses the sensing mode oscillation, the applied angular rate is computed. A simple suppression of the sensing mode oscillation is achieved by introducing an AGC loop as given in Fig.3,in which the system purpose is to regulate the plant output with a given reference value. In the figure, u denotes the controller output, error,
denotes the modulated control signal, denotes a natural frequency,
denotes the input acceleration
denotes the
denotes a damping ratio and
conversion gain for the velocity signal. Using the scheme, by setting the reference r=0, the control objective is to cancel out the external Coriolis input, using the control gain. Then with a successful controller design, if the output y is well regulated, the displacement in sensing mode is properly controlled too.
3. AGC FEEDBACK LOOP DESIGN
3.1 Loop transformation and controller design
In Fig.3,the feedback loop uniquely contains a nonlinear function h(·),where the input-output characteristics is to normalize the amplitude of the oscillation signal with the unit gain and imposes any deviation on neither frequency nor phase. In the figure, it is assumed that the applied Coriolis acceleration is given by
（t）sin（
t+）.
This is because the resonant frequency of sensing and driving mode dynamics are same to each other or can be tuned to match, if different, in most vibratory gyroscopes. Also note that the plant dynamics is chosen to contain integral term in the numerator, thus acts like a band pass filter in which there is no deviation with regard to the phase. Since the input Coriolis acceleration has common frequency and phase with those of sensing mode dynamics, it is easily shown that the amplitude of input Coriolis acceleration,
can be factored out and then combined together with the controller
output, u. Physically, it implies that the angular rate signal, can be extracted from the sinusoidal signal and combined with the controller output signal, u. The resulting
天津大学2010届本科生毕业设计（论文）
compensation error is denoted by e, which is multiplied by the normalized velocity signal to generate a control input for a stabilizing AGC loop. After restructuring the loop, a more simplified loop transformation can be done with the help of the harmonic balance property and low pass filtering dynamics of the loop. Assuming a dominant sinusoidal signal as the loop solution, the system governing equation produces differential equations which are responsible for the amplitude and phase, respectively. Then by arranging the amplitude-related equation, the loop dynamics in Fig.3 is approximated by the equivalent diagram in Fig.4.In figure 4,note that the gain of 2/π is included due to the gain reduction while averaging the absolute value of a sinusoidal signal envelope. By observing the block diagram, it is identified that the transformed system describes only the envelope dynamics in low frequency ranges.
In the next, using the equivalent model in figure 4,a stabilizing AGC loop for a force rebalance is designed. In order to avoid impractical control gain, we take an approach of a biased oscillation where the design objective is to maintain a uniform oscillation amplitude under varying angular rate inputs. With this objective, the reference is set to a non-zero constant value and a proportional-integral controller is applied to achieve zero error value in the steady state[11]. Let?s consider a linear feedback control law that contains an integral action such as
u=K(S) , where σ denotes the error, y-r. The low pass filter is designed such that the cut-off frequency is small enough to filter out higher order harmonics and large enough to pass out transients of the envelope signal. For example, by incorporating the first order low pass filter and a proportional-integral control law in(3)into the loop equation, the overall transfer function from the envelope of Coriolis acceleration to oscillation amplitude can be obtained as
where
and
represent the proportional and integral gain, respectively
and represent the denominator and numerator coefficients of the low pass filter, respectively. Lastly by assigning the stabilizing controller gains, the envelope