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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

We present two oscillation control algorithms for resonant sensors such as vibratory gyroscopes. One control algorithm tracks the resonant frequency of the resonator and the other algorithm tunes it to the specified resonant frequency by altering the resonator dynamics. Both algorithms maintain the specified amplitude of oscillations. The stability of each of the control systems is analyzed using the averaging method, and quantitative guidelines are given for selecting the control gains needed to achieve stability. The effects of displacement measurement noise on the accuracy of tracking and estimation of the resonant frequency are also analyzed. The proposed control algorithms are applied to two important problems in a vibratory gyroscope. The first is the leading-following resonator problem in the drive axis of MEMS dual-mass vibratory gyroscope where there is no mechanical linkage between the two proof-masses and the second is the on-line modal frequency matching problem in a general vibratory gyroscope. Simulation results demonstrate that the proposed control algorithms are effective. They ensure the proof-masses to oscillate in an anti-phase manner with the same resonant frequency and oscillation amplitude in a dual-mass gyroscope, and two modal frequencies to match in a general vibratory gyroscope.

Robust operation of a resonant sensor, such as a vibratory gyroscope, requires the resonator to be driven at resonance with constant amplitude. However, both the amplitude and resonant frequency can vary due to environmental factors, such as changes in temperature or stiffness aging. Therefore, some form of oscillation control is needed to track the constantly changing resonant frequency or to tune it to a specified frequency while keeping the amplitude constant.

Automatic gain control (AGC) has generally been used to excite the resonator to track the reference amplitude. The application of AGC to the drive axis of a vibratory gyroscope is reported in [

There have also been a few studies on controllers which, instead of tracking the resonant frequency, tune it to a specified frequency chosen by the designer. The advantages of this method are that it can maintain consistent performance, since the sensor can retain the dynamic characteristics regardless of environmental factors such as temperature changes, and simplify the signal processing loop that uses the resonant frequency as its carrier frequency. In the literature, only Lyapunov based adaptive control schemes for vibratory gyroscopes are reported to place the resonant frequency at a specified frequency [

This paper presents two algorithms for controlling the frequency and amplitude of oscillation. One control algorithm tracks the resonant frequency and the other algorithm tunes it to the specified resonant frequency by altering the resonator dynamics. Both algorithms maintain the specified amplitude of oscillations. In the first algorithm, AGC and PLL structures are used to control the amplitude and track the resonant frequency. The displacement measurement is used to avoid using noisy velocity measurement. The second algorithm is similar to [

The averaging method is used to analyze the stability of the entire feedback system and the effects of the displacement measurement noise on tracking and estimation of the resonant frequency. The proposed control algorithms are applied to two important problems in a vibratory gyroscope to evaluate their performance. The first one is the leading-following resonator problem in the drive axis of a MEMS dual-mass vibratory gyroscope without a mechanical linkage between the two proof-masses. Adopting the proposed control algorithms enables the implementation of the following resonator which precisely traces the oscillation pattern of the leading resonator. The second one is the on-line modal frequency matching problem in a general vibratory gyroscope. Applying the first algorithm on the drive axis and the second algorithm on the sense axis enables the resonant frequencies of the drive and sense axes to be matched precisely, which significantly improves the gyroscope performance.

The equation of motion of a resonator is modeled as a spring-mass-damper system and described as a second-order differential equation as shown in (1):
_{n}

The control input used to excite the resonator at constant amplitude, _{0}, while tracking the resonant frequency of the resonator is given by:
_{0}, equal to zero. Therefore,

The AGC is composed of a rectifier, a low-pass filter, and a comparator [_{0}, we employ a proportional-integral control in the AGC. The proposed AGC in this paper is as follows:
_{p}_{I}_{a}_{p}

A PLL is composed of a phase detector, a PLL controller, and a voltage-controlled oscillator (VCO). Each can be written as follows [_{0}. The VCO control voltage, _{v}_{z}_{p}

This section follows the formulation in [_{p}_{I}_{v}_{z}_{a}_{p}

Transforming of the displacement and velocity,

In (7), _{p}_{p}_{p}_{0}, is calculated as:

If _{p}_{p}_{p}_{p}_{p}

A first-order approximation of the Taylor series expansion of the nonlinear function _{1}(x_{1})_{1}

Since

An equilibrium point of (16) is:

The Jacobian matrix of the averaged system (16) at the equilibrium point is:

The conditions for the above Jacobian matrix to be stable are:

_{n}_{n}_{0} = _{0} − _{0}π/2, the amplitude converges to the spcified value, _{0}, with a small deviation due to the displacement measurement noise.

As seen in the previous section, the displacement measurement noise not only causes errors in the amplitude control, but also affects the resolution of the resonant frequency tracking. Applying the covariance propagation equation, this section describes the effects of displacement measurement noise on the resolution of the resonant frequency tracking controller. The covariance propagation equation of (13) is defined as:

Expanding above equation using the Taylor series at the mean, _{1}

Applying the averaging method to (20) yields the covariance equations for (13) at steady state as:
_{1} is defined in (17) and:

The standard deviation, which is the averaged resolution of excitation frequency in the frequency tracking control, is derived as:
_{1}(5,5) denote (5,5) terms of the numerical solution of (22).

The frequency tuning control differs from the frequency tracking control in that it adjusts the dynamic characteristics of the resonator to match to a resonant frequency specified by the designer, instead of its own resonant frequency. This requires controlling the resonant frequency through the displacement feedback. Adopting the specified resonant frequency, _{s}_{s}_{p}_{ω}_{q}

Similarly to the previous section, transforming of the displacement and velocity,

Similarly to the previous section, an approximated time update of expectation of (31) is obtained as follows:

Applying the same method yields the nonlinear averaged dynamics of (33) as:

An equilibrium point of (34) is:

The Jacobian matrix of the averaged system (34) at the equilibrium point is:

The conditions for the above Jacobian matrix to be stable are:

If the feedback system is stable, actual resonant frequency is tuned to the specified resonant frequency since

Similarly to the previous section, applying the averaging method yields the following steady state covariance equation for (31):
_{2} is defined in (35) and:

The standard deviation, which is the averaged resolution of the frequency compensation in the frequency tuning control, is derived as:
_{2}(5,5) denote (5,5) terms of the numerical solution of (37).

In general, a dual-mass gyroscope has two proof-masses linked to each other by a mechanical beam and is designed to oscillate in anti-phase with the same resonant frequency and amplitude [

To evaluate the control performance, the proposed control algorithms are applied to the drive axis of a dual-mass gyroscope without a mechanical linkage between the proof-masses. The equation of motion for the drive axis of a dual-mass gyroscope is described by two second-order differential equations as follows:
_{1}, _{2} are the displacements of each proof-mass, _{1}, _{2} are normalized damping coefficients, _{1}, _{2} are natural frequencies, and _{1}, _{2} are control inputs.

The first proof-mass is set as the leading resonator, and the second one is set as the following resonator, which precisely traces the oscillation pattern of the leading resonator. The proposed frequency and amplitude controls are applied to each mass. The dual-mass gyroscope parameters are taken from a prototype fabricated at Sejong University, and the parameters are:

Considering likely manufacturing errors, 10% error is added to _{1} for _{2} calculation, and 10% error is added to _{1} for _{2} calculation. The PSD of the displacement measurement noise is assumed to be
_{1} and _{1} are calculated from (3) and (4), respectively. The control law for the second proof-mass is frequency tuning and amplitude control. In anti-phase, the second proof-mass should oscillate with the same resonant frequency as the first proof-mass, so the control law (25) is modified as:
_{1} is identical to (41) and _{2} and Δ

The control parameters are selected to meet the stability conditions in (18) and (36). _{1} (sec).

_{0} = 0.9_{1}, reaches the resonant frequency of the proof-mass in 0.4 second and then maintain its value. The averaged resolution of the excitation frequency from (23) is σ_{1} ≈ 0.004 Hz.

As soon as the estimation error becomes zero, the dynamic characteristic of the second proof-mass begins to have the same resonant frequency as the first proof-mass, as observed in _{2} ≈ 0.01% ×Δ_{0} = 5 μm, after some time.

Most vibratory gyroscopes rely on matching the resonant frequencies of drive and sense axes for high performance. However, manufacturing imperfections result in deviations of the resonant frequencies from their design values. Therefore, various tuning methods have been developed [

An alternative mode matching method is presented in this section using the proposed oscillation control algorithms. This method is adaptive to changes in the environment such that if the resonant frequency of sense axis is deviated from that of drive axis, the controller will continually compensate the frequency deviation. Therefore it can be used for on-line implementation.

In this method, the frequency tracking and amplitude control is used in the drive axis, and the frequency tuning control in the sense axis. The amplitude control is not used in the sense axis because the amplitude should be allowed to change according to the input angular rate. Since the drive axis control is same as that in previous section, only the sense axis is considered here. The sense axis of a vibratory gyroscope is modeled as:
_{y}_{y}_{y}

Because the frequency tuning control relies on the displacement measurement, it is required to excite the sense axis continually regardless of the presence of an angular rate input. Therefore we propose to insert fictitious angular rate into the sense axis, and the control law (25) is modified as follows:
_{x}_{x}_{0} is a fictitious angular rate, _{0} is the amplitude of drive axis oscillation, and Δ_{y}_{0}_{0} is the magnitude of sense axis oscillation, which is proportional to input angular rate.

Simulations are conducted with the same gyroscope data given by (40), where subscript 1 and 2 are replaced by

In this paper, two frequency and amplitude control algorithms are presented. One control algorithm excites the resonator at its own resonant frequency, and the other alters the resonator dynamics to place the resonant frequency at a specified frequency which is chosen by the designer. These control algorithms maintain specified amplitude of oscillations. The stability of the entire feedback system was analyzed using the averaging method, and the stability criteria were proposed so that it can be used as guidelines for selecting the control parameters. In addition, the effects of displacement measurement noise on the tracking and estimation of the resonant frequency were analyzed.

In order to evaluate the performance of the proposed control algorithms, we apply them to two important applications. The first one was application to the drive axis control loop for a dual-mass gyroscope without a mechanical linkage between two proof-masses. The second one was application to modal frequency matching in a vibratory gyroscope for high performance operation.

Simulation results agreed well with analytical analysis, and demonstrated the effectiveness of the proposed control algorithms. In addition, both the analytical and simulation studies showed that it is possible to make the two proof-masses of a dual-mass gyroscope without mechanical linkage to oscillate in anti-phase at the same resonant frequency and with the same amplitude. The proposed controller also enables the resonant frequencies of the drive and sense axes to be matched precisely by continual compensation the frequency deviations, which greatly improves the gyroscope performance.

Block diagram of frequency tracking and amplitude control.

Block diagram of frequency tuning and amplitude control.

(a) Excitation frequency, (b) Estimation error of the frequency difference.

Time response of drive axes.

(a) Frequency tuning estimation error, (b) Time response of sense axis.

Time response of angular rate estimates to (a) step input, (b) sinusoid input.

Non-dimensional values of the control parameters.

_{0}, _{p}_{I}_{a} |
5, 0.018, 1.5 × 10^{−5}, 0.05 |

_{0}, _{v}, K_{z}, λ_{p} |
0.9, 0.5, 1.5 × 10^{−4}, 0.5 |

_{ω}, λ_{q} |
1.5 × 10^{−4}, 0.5 |