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Autonomous Oscillation Control Loop Design for Amplitude Controlled, Frequency Read-Out-Type Resonant Sensors Sangjun Park, Jin Woo Song, Byungjin Lee, Hyungjoo Yoon, Young Jae Lee, and Sangkyung Sung, Member, IEEE

Abstract—This paper suggests an autonomous oscillation control loop for frequency read-out-type resonant sensors that produces outputs of variable frequency depending on the input of an external physical quantity. The design goal of the oscillation loop is simultaneously to stabilize the resonance characteristics of the sensor and to automatically track the resonant frequency in order to guarantee highly reliable sensor performance. To this end, the concept of automatic gain control (AGC) is applied so that the loop is designed to maintain the oscillation amplitude as one control objective. The second control objective is to achieve resonance condition tracking even when external influences such as disturbance and noise exist. For the verification of the proposed control loop design, an example resonant sensor system is modeled, and a control loop and controller that accompany the system are also designed. Finally, the proposed loop performance was demonstrated via simulations which consider practical noise elements. The theoretical results were further verified via sensor’s transient responses and noise analysis. Index Terms—Automatic gain control (AGC), oscillation control, resonance condition, resonant sensor, tracking.

I. INTRODUCTION ENSORS that measure input amounts indirectly through resonant frequency variation of a dynamic system, which is driven by the physical input, are collectively referred to as frequency readout-type resonant sensors. Implementation of a resonant sensor essentially necessitates the use of a self-sustained oscillation loop that can either maintain or track oscillation amplitude and resonant frequency, which inherently serves as inertial energy conversion coefficients to characterize vibration systems [1]–[3], [6]–[8], [13], [14]. A suitable candidate to im-

S

Manuscript received August 7, 2010; revised January 19, 2011 and April 5, 2011; accepted April 23, 2011. Date of publication June 16, 2011; date of current version August 24, 2012. Recommended by Technical Editor J. M. Berg. This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded under Grant MEST (20110005527) and the Korea Foundation for International Cooperation of Science and Technology (KICOS) provided by the MEST under Grant K20601000001. S. Park, B. Lee, and S. Sung are with Konkuk University, Seoul 143-701, Korea (e-mail: [email protected]; [email protected]; [email protected]). J. W. Song is with Microinfinity Corporation Ltd., Suwon 443-270, Korea (e-mail: [email protected]). H. Yoon is with the Satellite Control System Department, Korea Aerospace Research Institute, Daejeon 305-333, Korea (e-mail: [email protected]). Y. J. Lee is with the Department of Aerospace Engineering, Konkuk University, Seoul 143-701, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2011.2157352

plement this kind of oscillation system is a loop construction based on the basic operation of a high-precision oscillator circuit that is often used in conventional electric oscillator circuits or quartz vibrators [9], [10]. The oscillator circuit based on conventional electric oscillators typically takes the configuration of LC tuned resonance system. As such, they usually require an external reference oscillator such as quartz. While these referencebased oscillator circuits have relatively high-frequency reliability, it is quite difficult to realize additional functions that can actively control a dynamically tuned resonance status over an input range required by the electromechanical sensors of interest. Implementing resonance sensors using a single feedback loop is another method that can be used [2], [3], [15]. In this case, the feedback loop is implemented by combining hard nonlinearity components, i.e., a limiter cascaded by a variable phase shifter. A self-sustained loop structure that has the addressed single feedback connection can be applied to a resonant sensor system employing a relatively high quality factor, and results in an easily achieved oscillation status. However, due to the resonator’s nonlinear dynamics and underlying phase and amplitude noise, the single-loop approach cannot guarantee an oscillation equilibrium with a fixed amplitude. This in turn may induce the nonlinearity and phase noise which will degrade overall oscillator system performance. This turns out to be a critical disadvantage when performing repeatability tests under varied initial conditions or between different samples manufactured via the same process, since the repeatability performance is subject to mechanical parameter variations. A number of studies into the micromechanical resonator design have reported on the adaptation of automatic amplitude control (AAC) or level control (ALC) for stabilized oscillation performance [13], [16]–[20], [23]. Generally, it is known that large oscillations induce the resonator nonlinearity and the 1/f3 phase noise, which is caused by a direct aliasing of 1/f noise with respect to the capacitive transducer and a beam stiffening Duffing effect. As oscillator limiting via resonator nonlinearity is dominant in introducing the 1/f3 noise term, the ALC scheme in previous works proposed resonator amplitude control that adjusts either the gain of the MOS trans-impedance amplifier [17]–[19] or the dc-bias voltage applied directly to the tuning electrode of the resonator [18]. On the other hand, the oscillator structure in [20] presented the control loop concept of AAC with a fixed gain at the preamplifier stage, which is similar to the automatic gain control (AGC) approach taken in [13] and [23], but is more focused on a CMOS level circuit design. With the implementation of a chopper stabilized peak detector,

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linear VGA, and loop filter, the work in [20] reported that it was possible to prevent the resonator from entering the Duffing bifurcation point and greatly minimize the 1/f3 phase noise caused by the amplitude stiffening (A–S) effect. Despite demonstrating good performance, the previous works were primarily focused on the resonator’s equivalent modeling, nonlinearity analysis under specific resonator structures, some quantitative noise spectrum formulae which follow Leeson’s model, and IC level circuit design [16]–[20]. Consequently, the presented AAC or ALC schemes on oscillator control yield only fragmentary loop analysis methods and lack a systematic overview of the development of the oscillation loop analysis and design, specifically, from a control engineering viewpoint. For a synthetic design of a controller and other filters, rigorous mathematical development and a stability analysis of the control loop equations are required. From another point of view, since the application of the oscillator is to be a resonant sensor, in contrast to a static oscillator, the resonance characteristics are subject to a physically applied input that will yield a variation in the oscillation amplitude and phase. This may eventually affect the stability of the resonance owing to disturbance or environmental errors. As these limitations on tolerance result in a limited sensor input range, active amplitude control and a phase-matched loop configuration within the full input range and under loop noise are indispensable. In this respect, frequency domain tracking and time domain amplitude regulation are two primary design factors of the proposed oscillator system. To address these issues, this paper proposes a new design approach to the oscillator system which is based on a modified AGC configuration and includes mathematical developments for control system analysis. The presented AGC loop configuration relies only on the nonlinear loop analysis method for a synthetic control system design which contains dual feedback branches [5], [6]. This enables a quantitative design of the oscillator system and provides amplitude prediction and stability condition analysis in terms of the loop component parameters, i.e., controller, filter, and other electromechanical gains. The verification of the proposed approach is first done through simulations which test the designed parameter for each loop component under practical conditions. Moreover, the simulation demonstrates the reliability over the dynamic input range of the sensor and the loop’s internal errors. Finally, the presented oscillator system is implemented electronically with discrete devices for the experimental testing of the resonator. This testing provides fundamental results that show the input-dependent frequency variation with the desired oscillation amplitude which is predetermined via the reference value. The paper is organized as follows. Section II provides a brief explanation of an objective plant to which the proposed oscillation control system is applied and includes a description of its basic operation. Section III illustrates how the oscillator system and controller were analyzed and designed. Section IV presents the results from the simulations and experiments. Finally, Section V provides a conclusion.

Fig. 1. Block diagram of the variable stiffness mechanical model of a resonant sensor. The large proof mass on the left varies the mechanical stiffness of the double ended tuning fork resonator on the right with high sensitivity.

II. BASIC CONCEPTS AND PRINCIPLE This section illustrates the basic operational principles and explains the structure of an electromechanical resonant sensor for use in the oscillation control. Fig. 1 shows the sensor structure. The sensor has a double-ended tuning fork resonator connected to a large, acceleration sensitive, proof mass. The adapted drive and sensing mechanism takes a differential type electrostatic actuation and uses a capacitive transducer method. As is already known, the tuning fork mode enhances sensitivity whereas the differential sensing effectively cancels the parasitic effect caused by the stray capacitance. In order to explain the basic operation of the resonant system, let us consider a dynamic equation for a beam mass vibrating in the direction of the resonance axis, i.e., the Z-axis in Fig. 1. When there is little influence from damping in the second-order dynamic equation representing the mass–spring–damper, the resonant frequency ω z is determined mainly by the modulus of elasticity of the beam. If the beam’s modulus of elasticity on the resonance axis (Z-axis) changes with the axial force (from the Y-axis) due to an external proof mass, the system’s resonant frequency also changes in the Z-axis [17]. Therefore, the external input can be estimated by measuring the resonance frequency variations in the Z-axis. Note that in the aforementioned configuration, the input axis of the resonant sensor is the Y-axis and the oscillation axis is orthogonal to this input axis. Moreover, the vertical vibration of the beam mass is induced through electrostatic forces which are generated via the comb-shaped drive electrodes in the resonance axis; thus, the oscillation control mechanism is independent from the mechanical input axis. In the aforementioned design, the control and tracking of the resonance point can be implemented electrically by the control input signal with a drive voltage v AC . Now, assuming a variable stiffness due to a strain input in the second-order model, the natural frequency of the plant dynamics from Fig. 1 is given by k/m. Hence, the natural frequency deviation, given a stiffness variation Δk, is described as follows: Δωz =

(k + Δk) − m

k . m

(1)

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Fig. 2. Sensor frequency variation Δω z /ω z characteristics when spring stiffness varies due to strain change.

Consequently, the change in frequency ratio with respect to the change in stiffness ratio is given by Δk Δωz − 1. (2) = 1+ ωz k From (2), the relationship between change in stiffness ratio and the change in frequency ratio can be drawn. Fig. 2 shows a comprehensive plot of frequency variation when a varying strain is applied to the resonant sensor. Assuming a resonance model with negligible damping effects, the results are independent of the structure’s dimensions and geometry. As observed, when the input range of stiffness change is set to about 15% from the nominal value, the output as the ratio of frequency change is computed to be −7.6% to 7.2%. Note that, owing to its slight nonlinear property, the frequency change ratio gradually decreases as the stiffness change ratio increases. The maximum difference divided by the dynamic range using the first-order regression model is 0.65%, which would limit the sensor performance of linearity. Also the sensitivity and frequency resolution can be preliminarily calculated by defining the sensor’s nominal natural frequency. This analysis further establishes a reference data set for loop simulation results in the following sections. In conclusion, by referring to the analytic results, the proposed feedback control loop performance is investigated as to whether it sustains the oscillation at the analytically predicted frequency while simultaneously maintaining a constant amplitude for a wide dynamic range and robustness against disturbances. III. DESIGN OF THE OSCILLATION CONTROL LOOP This section includes the structural details of the proposed oscillation control loop and develops a mathematical model to describe it through a state-space approach. A. Control Loop Structure A conceptual block diagram of the presented oscillation control system, with simultaneous oscillation amplitude control and

Fig. 3.

Block diagram of the proposed oscillation control loop.

resonant frequency tracking, is shown in Fig. 3. This electromechanical system contains second-order plant dynamics, dualbranched signal processing components and an analog multiplier for signal association. The structure takes advantage of the AGC scheme, that is, the feedback increment is generated in a negative way such that the loop signal continues to keep an equilibrium state [23]. The mass vibrates with an electrostatic force determined by the driving electrode, while the vibration characteristics of the mass are measured by displacement signals through the detection electrode. These signals then enter the amplitude maintenance control loop and the resonance-point detector. Next, by multiplying the amplitude maintenance control signals and the resonance-point detector signal, the drive voltage is generated and then authorized at the sensor drive input unit. This process allows the mass to maintain constant levels of amplitude at system’s resonance frequency. In Fig. 3, we can see that the resonance-point detector is implemented in the form of an analog differentiator through an electric circuit. Therefore, it is possible to autonomously adjust the feedback signal with the same phase as that applied to the plant input. Thus, the system automatically satisfies the Barkhausen criteria for a steady state of oscillation. Meanwhile, it is known that during the start-up phase of oscillation, the criteria cannot provide a resonance condition, as such, a more general excitation condition is needed. The essential requirement is to have at least one unstable pole during the initial transients, which can be further tested, e.g., via the extended Nyquist criteria [21], [22]. Nevertheless, in practice, it turned out that the feedback signal due to electrical noise fluctuation and large loop gain in a high Q-factor system easily yields an initial pulling-up status, which soon progresses to the resonance point. The resonance-point detector is implemented by an analog differentiator, which is composed of cascaded circuits with a first-order low pass filter (LPF) and high pass filter (HPF) such

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Fig. 4.

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Block diagram of amplitude maintenance control loop.

that the resonant frequency lies in the middle of a gain-increasing band of HPF. This configuration has the advantage of mitigating noise lying within the frequency band that is higher than the nominal resonance frequency, and thereby prevents highfrequency noise from being amplified by the differentiator. In another form, the differentiator function can be implemented via a second-order band-pass filter, whose transfer function can be simply represented as (1/Ri Cf )s . Z(s) = − 2 s + (1/Ri Ci + 1/Rf Cf )s + 1/Ri Ci Rf Cf (3) Here, Ri , and Ci are an input resistor and capacitor connected in series and Rf and Cf are a feedback resistor and capacitor connected in parallel. In (3), a combined set of electric components determines the pass-through and block-off band. In our application, the resonant inertial sensor’s undamped natural frequency is approximately 22 kHz. Therefore, considering the performance of the adopted op-amps, the differentiation blocking frequency determined via Rf and Cf is set to 1100 kHz, which is about 50 times higher than the system’s natural frequency. It is thought that an induced phase error of several degrees during circuit implementation provides noise constraints that we need to consider during the simulation work. Fig. 4 illustrates the composition of the amplitude maintenance control loop. The amplitude maintenance control loop transforms the displacement signal inputs into low-frequency signals through an envelope detector, compares them with reference value inputs, and then authorizes them to the controller. The objective function of the controller is to keep the tracking error between the reference value and controller input to effectively zero so that the oscillation amplitude is maintained to a constant value. In our loop design, a proportional-integralderivative controller is used for this. In the proposed oscillation loop, the envelope detector functions to extract the displacement signal of the mass in a lowfrequency band. The displacement signals, in the form of a sine wave, are passed through a full-wave rectifier and then converted into a positive low-frequency envelope signals via a low-pass filter. The transfer function of the low-pass filter can be described by the following simple transfer function: γ . (4) L(s) = s+λ

Fig. 5. Equivalent block diagram of the proposed loop and transformed block diagram of single-loop nonlinear model, respectively.

Here, γ and λ are parameters determining the cutoff frequency of the LPF such that it sufficiently removes higher order harmonics. In practice, the LPF is designed to have about a 500 to 1000 Hz cut-off frequency, which is arrived at by considering the resonant frequency range of the sensor structure and the target bandwidth of a typical mechanical sensor, i.e., 100 Hz. B. Loop Analysis and Control System Design In this section, loop analysis for the controller design is investigated. For this, it adopted a harmonic balance and averaging method, which leads to an efficient system modeling [11], [13]. Fig. 5(a) and (b) present an equivalent system diagram of the proposed oscillation control loop and its transformed single loop structure, respectively. Specifically in Fig. 5(a), the plant model is rearranged such that the output yields the velocity signal of the mass for the simplicity of the loop analysis. From a functional viewpoint, this structure is virtually equivalent to that presented in Fig. 3. In this structure, the inner loop simply meets the Barkhausen criteria for resonance at a steady state, while the outer loop comprises of a displacement control loop, where an integral operation is incorporated. In this figure, g, vz , vf , u, uv , vd , G, ka , ω z , and va represent the external input quantity, the velocity signals multiplied by gain, the detector output by displacement, the controller output, plant input, displacement, loop gain, plant gain, natural frequency, and envelope signal, respectively. The plant of the resonant sensor is modeled with second-order dynamics, where the system’s coefficients of elasticity vary depending on the inertial input signals (e.g., force, pressure, etc.). It is noted that this simple plant model originates from a differential drive/sense scheme. In this scheme, the higher order perturbations are neglected via frequency selective actuation while the system has a high q-factor, and the amplitude-controlled resonance mode prevents nonlinear bifurcation.

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For loop analysis, a dominant sinusoidal signal in the loop is assumed, since loop signals have harmonic balance characteristics originating from the LPF property of the plant and loop electronics. Given this property, when only the envelope of the high-frequency signal is considered, Fig. 5(a) can be converted into a single branched feedback control loop as shown in Fig. 5(b), with an introduction to a nonlinear statespace model. For the derivation of the loop conversion, we first assumed a pertained steady-state signal in the loop as η(t) = a(t) sin (ωr t + θ(t)), where a, θ denote the amplitude and phase, and ωr := ωz 1 − ζz2 is the damped natural frequency, respectively. Applying η(t) to the nominal plant dynamics in Fig. 5(a) and rearranging into sine and cosine term yields the following equation: ˙ 2 + 2ζz ωz a˙ + ω 2 a − Gka u˙ − Gka ua} {¨ a − a(ωr + θ) ˙ z

˙ u = −(kp σ˙ + ki σ + kd y)

(9)

where σ := (y − vr ) dτ . By applying the control law in (9) to the nonlinear state equation in (8), the resulting closed-loop system is formulated as x˙ = f (x, u) u = −kp σ˙ − ki σ − kd y˙ σ˙ = y − vr .

(10)

0 = f (xe , ue )

(5)

In (5), two associated equations that govern the amplitude and phase of the sinusoidal loop signal can be obtained. Focusing on the envelope signal only, the equation on the cosine term approximates the envelope dynamics of the loop into the firstorder equation as follows: a˙ + ζz ωz a = 0.5 · Gka ua.

At equilibrium, x˙ = 0 and σ˙ = 0 hold; therefore, the equilibrium point (xe ,σ e ) satisfies the following equations:

˙ + (2ζz ωz a − Gka ua) × sin(ωr t + θ) + {2a(ω ˙ r + θ) ˙ + aθ} ¨ cos(ωr t + θ) = 0. × (ωr + θ)

y(t) → vr as t → ∞ by a feedback control scheme, where a steady-state oscillation reaches an equilibrium amplitude. Following the integral control via the linearization method in [12], let us consider a linear feedback control law that contains a proportional-integral-derivative action such as

(6)

¨ ˙ In deriving (6), the scaled phase acceleration term θ/2(ω r + θ) is neglected since it is very small compared to ζ z ω z in the normal resonance mode. It can be observed that formulating the solution form of the differential equation yields a time-varying amplitude represented in terms of control input u. Thus, by defining the output of the plant dynamics with respect to u as (7) h[u(t)] := G · exp [0.5 · Gka u(τ ) − ζz ωz ]dτ. t

The scaled velocity signal can be obtained as va = h(u), which is the envelope of vz . It is observed that G is the voltage gain t and a(t) is given by exp{ 0 [0.5 · Gka u(τ ) − ζz ωz ]dτ }. For a state-space representation, let us define new state variables x1 := ln(a(t)) and x2 := vf (t). Then the state space equation of the nonlinear model containing h(u), loop gain G, detector, and LPF can be derived as ⎡ ⎤ 0.5 · ka Gu(t) − ζz ωz x˙ 1 ⎦. f (x, u) := = ⎣ 2γG x (t) (8) x˙ 2 e 1 − λx2 (t) πωr In deriving (8), the integrator and envelope detector are equivalently converted into 2 · (πωr )−1 , which results from the sinusoid averaging process [23]. Using the previous state variable definition, the system’s output y = vf (t) is represented by x2 . Finally, by extracting only envelope dynamics the transformed single-loop configuration is shown in Fig. 5(b). Hence, by introducing h(u) and separating a(t) from the coupled dynamic equation, the upper branch within the dotted line in Fig. 5(a) is simply converted into a single loop which solely governs the envelope dynamics. Now in Fig. 5(b), the design goal is

y e = vr (11)

T where xe := x1e x2e . Since the nonlinear equation (8) is continuous in time and locally Lipschitz, there exists an equilibrium point as the unique solution in a compact domain. Consequently, choosing control gains kp , kd , and ki at the equilibrium via an eigenstructure assignment completes the feedback loop design of the presented AGC control. The stability criterion of the control is given in the following section.

C. Control Loop Stability Analysis Assuming a steady-state oscillation, in this section, it is proved that the stability condition of the presented feedback loop can be restated in the form of the Kalman–Yakubovich– Popov (KYP) lemma [12] with an indirect Lyapunov candidate function. First, consider the change of variables given by ⎡ ⎤ ⎡ ⎤ x1 − x1e ξ1 ⎣ ξ2 ⎦ := ⎣ x2 − x2e ⎦ . (12) ξ3 σ − σe Then, the equation in (10) is represented by Gka kp Gka ki ξ˙1 = − (ξ2 + x2e − vr ) − (ξ3 + σe ) 2 2 Gka kd 2Gγ ξ 1 +x 1 e e − λ (ξ2 + x2e ) − ζz ωz − 2 πωr 2Gγ ξ 1 +x 1 e e − λ (ξ2 + x2e ) ξ˙2 = πωr ξ˙3 = (ξ2 + x2e ) − vr .

(13)

From (11), the steady-state values can be calculated as λπvr ωr x1e = ln 2Gγ x2e = vr σe = −

2ζz ωz . Gki ka

(14)

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Using (14), (13) is rearranged as Gka kp Gka ki Gka kd ξ˙1 = − ξ2 − ξ3 − [λvr eξ 1 − 1 − λξ2 ] 2 2 2 ξ˙2 = λvr eξ 1 − 1 − λξ2 ξ˙3 = ξ2 .

TABLE I PHYSICAL PARAMETERS OF THE EXAMPLE SENSOR

(15)

Hence, the system in (15) can be represented as a linear system with a feedback nonlinearity such that ξ˙ = Aa ξ + Ba uξ yξ = Ca ξ uξ = −ψ(yξ )

(16)

where the matrices are given by ⎡ Gka ki ⎤ Gka (kp − λkd ) − 0 − ⎢ 2 2 ⎥ Aa = ⎣ ⎦ 0 −λ 0 0 1 0 ⎡ Gk k ⎤

aenv = v r ·

a d

⎢ Ba = λvr ⎣

2 −1 0

⎥ ⎦

Ca = [ 1

0 0]

Finally, assuming a steady-state oscillation with a stabilizing controller, it is shown that the oscillation amplitude can be predicted in terms of the loop component values. Since the equivalent gain through detection electronics and the envelope detector is given as γ2G(λπ)−1 , and combined with the fact there is practically zero tracking error at the controller, it is obtained vr = vf , and the envelope amplitude satisfies vr = aenv · γ2G(λπ)−1 . This yields the following relation:

(17)

and ψ(x) := ex − 1. Now considering the system in (16), a Lyapunov candidate function is taken as V(ξ) = (1/2)ξ T Pξ, where P = PT > 0. Then, the derivative of V is depicted by 1 1 V˙ = ξ˙T P ξ + ξ T P ξ˙ 2 2 1 1 T T = ξ Aa + uTξ BaT P ξ + ξ T P ATa ξ + BaT uξ 2 2 1 = ξ T ATa P + P Aa ξ + ξ T P Ba uξ . (18) 2 Using the formulas in the KYP lemma, (18) yields √ 1 V˙ = − ξ T q T qξ + ξ T CaT ± q 2d uξ (19) 2 if Aa is Hurwitz, (Aa , Ba ) is controllable, (Aa , Ca ) is observable, and the transfer function in system (16) satisfies the positive real condition. Noting that d = 0 and yξ = Ca ξ in (16), (19) can be rewritten as 1 V˙ = − ξ T q T qξ + yξT uξ 2 1 = − ξ T q T qξ − yξT ψ(yξ ). (20) 2 Since yξT ψ(yξ ) ≥ 0, it implies that V˙ ≤ 0. Therefore, this concludes that the origin is asymptotically stable. In conclusion, the stability of the AGC loop incorporating the feedback control law in (9) is rephrased as the absolute stability problem of the nonlinear feedback system in (16) and can be proved in the form of the KYP lemma conditions, i.e., the positive realness of the transfer function and the matrix property of Aa , Ba , and Ca . In the following section, the KYP lemma is used for the loop stability test by taking advantage of the developed matrix representation in (17) and control gains.

λπ 2Gγ

(21)

where aenv represents the magnitude of the envelope when the steady-state oscillation is reached. Therefore, by using the steady-state equation in (21), the oscillation amplitude can be effectively determined through the reference voltage vr . It is noted that the sensor’s operational range and microresonator’s physical constraints are major factors in selecting vr . IV. CONTROL SYSTEM VERIFICATION AND DISCUSSION A. Numerical Simulation After incorporating the previously designed loop structure and controller, the feasibility of the resonant sensor’s control system is investigated via simulation. For simulation, the loop is constructed through the MATLAB SIMULINK program by assembling the functional components of the designed loop. The sensor to which the oscillation loop is applied was assumed to be an electromechanical sensor whose coefficient of stiffness varies in accordance with the applied mechanical stress. The sensor’s parameters used for the simulation are given in Table I. Parameters of the LPF in the loop are designed such that γ = λ = 300, which are arrived by considering the resonant sensor’s bandwidth. Also, the gain value at the detector, inclusive of the detection preamplifier, is set to G = 1.394 × 107 . Using the sensor parameters in Table I, the amplitude of the displacement signals within the oscillation loop is calculated to be 0.225 μm that is in accordance with (21). Meanwhile, the fluctuation of the coefficient of stiffness based on sensor inputs is assumed to be about 15% of the nominal value. As such, the corresponding range of the resonant frequency is from 23.550 to 20.332 kHz. Finally, the envelope-based controller, which is designed to maintain the oscillation amplitude, is as follows: 0.05s2 + s + 25 . (22) s With the controller in (22), kp = 2.5, ki = 62.5, and kd = 0.125 are obtained; thus, the resulting transformed system maK(s) = 2.5 ×

PARK et al.: AUTONOMOUS OSCILLATION CONTROL LOOP DESIGN

Fig. 6. Output characteristics of loop signal (v d , displacement converges fast to the regulated amplitude of 0.225 μm even when −0.8 N/m stiffness change is applied at 0.5 s).

Fig. 7. Output characteristics of the loop signal (resonant frequency is computed in each epoch by taking the inverse of each time interval of sinusoidal signal). From the figure, it is observed that the resonant frequency quickly converges to 20.33 kHz as predicted from the numerical simulation in Section II.

trices Aa , Ba , and Ca in (17) satisfy the stability condition in the KYP lemma, i.e., positive realness and Aa is Hurwitz, (Aa , Ba ) is controllable, and (Aa , Ca ) is observable. Fig. 6 illustrates the mass’ displacement signals obtained through the simulation when the sensor’s maximum input signals (input of −0.8 N/m) are applied at t = 0.5 s at the oscillation control loop. The reference input value within the loop is vr = 2 V; here, the magnitude of the displacement signal obtained from the simulation is 2.25 × 10−7 m, which matches the theoretical result in (21). In this figure, the vibration width, after the input signal is applied, has an overshoot of 9% and falling time of about 7 ms, yet it quickly converges to the steady-state value of 0.225 μm. Fig. 7 illustrates the output characteristics of Fig. 6 in the frequency domain. It can be observed that, following the applied input signal, the resonant frequency in a normal state enters a steady vibration of 20.33 kHz in accordance with the conver-

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Fig. 8. Sensor output characteristics of loop signal (v f , the envelope signal of displacement is extracted after passing through detector and shows complete matching with reference voltage value).

gence speed of the displacement signals. It is noted that the frequency change after the input is −1670 Hz, which yields a frequency change ratio of 7% from the nominal resonant frequency. This result matches with the analytic prediction in Fig. 2. Fig. 8 illustrates the displacement envelope output (vf ) in the loop with the previously discussed sensor input signal. It is demonstrated that the vibration envelope control signal is regulated at the resonant loop, while at the same time, it approaches and converges to the applied reference input value, i.e., vr = 2 V even under an external disturbance at 0.5 s. Observing its voltage variation is within 3–4 V. Due to this, we believe that there will be many practical circuit implementations which could use off-the-shelf operational amplifiers. As claimed, the resonant sensor performance is characterized by the frequency sensitivity with regard to input over the dynamic range. An input strain variation between −0.8 and 0.8 N/m is applied to the oscillation control loop. Fig. 9 shows the input–output curve where output is obtained by measuring the resonant frequency variation in the loop. In Fig. 9(a), the measured resonant frequency has a slight convex curve, since a slightly decreasing resonant frequency is detected as the change in input strain increases linearly within the dynamic range. Mathematically, the relationship can be modeled as the following second-order regression formula: 2 + 2010.3 × δstrain fr,m easured = −92.105 × δstrain

+22 000.0385

(23)

where fr,m easured and δstrain represent measured resonant frequency in the oscillation loop and input strain variation, respectively. Fig. 9(b) shows the frequency change ratio against the strain change, which plots together the analytic result in Section II. In is noted from the figure that the observed resonant frequency in the loop matches closely with the analytic prediction value by a frequency deviation less than 0.02%. Meanwhile, since the sensor resolution depends on the stability of the measured resonant frequency, performing sensitivity

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Fig. 10. Block diagram of noise and phase delay application for the resonant frequency stability analysis.

Fig. 9. Sensor output characteristics from loop simulation: (a) observed resonant frequency versus applied strain variation and (b) frequency change ratio, loop simulation result versus analytic result.

Fig. 11. Output characteristics of loop displacement signal, v d when phase delay is applied at 0.5 s. Upper right is an enlarged plot.

TABLE II LOOP ERROR TYPES CONSIDERED

mainly in the amplitude maintenance control loop, is assumed to be of the order of several tens of millivolts p-p. This quantity is typically measured during instrumentation. Fig. 10 shows the block diagram of the noise and phase uncertainty locating in the oscillation loop. Electrical white noise existing in the detection part is collectively considered through the voltage fluctuation at the plant input uv . Figs. 11 and 12 contain simulation results to illustrate the phase error effect on the displacement fluctuation and frequency fluctuation. For this, the oscillation control loop system is implemented and a phase delay was artificially applied as a step input at t = 0.5 s. The outcomes suggest that the amplitude and frequency converge to the steady-state rapidly after the phase perturbation is authorized. In terms of amplitude, it shows a magnitude deviation of below 0.05% from the nominal value, indicating that the control goal of the amplitude maintenance is hardly affected. As for the frequency fluctuation, it reached a steady-state even when a maximum phase difference of 12◦ was applied, showing a frequency deviation of about 22 mHz. In a similar way, Figs. 13 and 14 show outcomes of the simulation for the mass’ displacement fluctuation and frequency

analysis to electrical noise terms is essential. The major factors for error that could occur within the loop during implementation are phase uncertainty and white noise. Considering the noise-liable characteristics of the electromechanical systems, the phase perturbation and white noise problems are revealed to come mainly from the differentiator and the amplitude maintenance control loop. Supported by the PSPICE simulation and circuit experiment, the error constraints considered in the simulation are summarized in Table II. The phase error occurring in the loop is collectively approximated to a maximum of about 12◦ , which is arrived after considering the characteristics of the differentiator, preamplifier, and other electronics elements. The white noise, which occurs

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TABLE III ERROR EFFECT IN THE OSCILLATION LOOP

Fig. 12. Output characteristics of the measured resonant frequency when phase delay is applied at 0.5 s. Three different delay cases are plotted together.

Fig. 15. PCB board with electronics and onboard micro resonator structure for oscillation control experiment.

Fig. 13. Output characteristics of loop displacement signal, v d when loop noise is applied at 0.5 s. Upper right is an enlarged plot.

Because physical implementations of the control loop for electromechanical systems would usually be realized through analog circuits, validation of the loop performance with a realistic amount of phase uncertainty and white noise is essential. Table III shows the error factors and output deviation of the devised oscillation loop. It was confirmed through simulation that the displacement reliability due to phase and noise perturbation was within 0.05% from the nominal value while the stability of the resonance frequency within the loop was less than 20 mHz. Therefore, the outcomes of the simulation collectively verify that the proposed oscillation control loop is appropriate for a reliable resonant sensor whose input range is wide and resolution of frequency is highly aimed. B. Experiment Result

Fig. 14. Output characteristics of the measured resonant frequency when loop noise is applied at 0.5 s.

fluctuation when white noise was artificially applied at t = 0.5 s. After the loop was constructed, it shows a change of about 0.04% in terms of magnitude following a white noise application of about 30 mV, while maintaining the amplitude to within 1 mHz of the resonance frequency. This suggests that, in practice, a limited resonant frequency perturbation effect occurs in the oscillation loop under electrical noise.

The function block of the designed loop is fully implemented by analog electronics consisting of an op-amp, multiplier, diode, and RC elements. The envelope detector consists of a full wave bridge rectifier and LPF, the reference value input is implemented using a substractor and potentiometer, the PID controller with op-amp application circuits, and so on. Because the natural frequency of the developed resonant sensor lies in a high-frequency band, MC34184, which has a fast slew rate, is used. 1N4148T, which is robust against high frequency and sensitive to minute voltage changes, is used for the diode in the envelope detector. Meanwhile, MPY634, with its robustness to high-frequency noises, is used as the analog multiplier. Fig. 15 shows a printed circuit board (PCB) containing electronics for the designed resonant circuit with an onboard microresonator device. Each function block in the oscillator system is implemented via surface-mounted analog devices within the overall PCB size of 7 × 5 cm2 for easy installation on test apparatus. Fig. 16 presents the experimental results that show

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Fig. 16. Experimental result showing the dynamic response of the proposed oscillation control loop. Upper-left plot shows the initial transient when power is turned on, which consists of start-up phase and steady state with designed amplitude. Lower-right plot shows the step response when an external force is applied to the microresonator.

Fig. 17. Experimental result showing the frequency change due to external input. Frequency computation rate is 20 Hz. The external forced is applied via gravity at around 1884 s. Gravity sensitive frequency variation with about 36 Hz/g is observed with the fixed oscillation amplitude.

the dynamic response of the proposed oscillation system. Each plot in the figure represents a scaled displacement signal in the loop. The upper-left plot shows the initial transient response when power is turned on. It shows both a start-up transient and steady-state oscillation. At a steady state through amplitude control, the scaled displacement is measured as about 1.6 V. This corresponds to the applied reference voltage of 2 V since envelope detector gain is set to 1.25. The lower-right plot shows the step response when an external disturbance is applied. It is observed that the displacement amplitude quickly converged, within 50 ms, to the regulated amplitude. Through observation of the oscillation response during the initial excitation and the amplitude recovery after disturbance, the basic performance of the loop has been demonstrated. Fig. 17 presents experimental results showing the frequency change due to an external acceleration. The frequency is computed using the Agilent 53132A with 20 Hz update rate. The external input is applied via grav-

Fig. 18. Allan deviation curve of the prototype resonant accelerometer is drawn. Line extension is done to compute noise components.

itational force at around 1884 s through the attitude change using the rate table, Acutronic BDS-357. While maintaining a constant oscillation amplitude, the gravity-sensitive frequency variation was found to be about 36 Hz/g. Finally, the noise characteristic of the prototype resonant accelerometer is analyzed. For this, the Allan deviation is computed by measuring and averaging static sensor data during cluster time period. Fig. 18 presents the Allan deviation curve with the cluster time from 0 to 550 s. It is observed that the velocity random walk is dominant in the short cluster time. By extending a straight line with a slope of –0.5, the velocity random walk (i.e., white noise) of the accelerometer is estimated as 0.15 mg/s1 / 2 . Also note that the bias instability can be estimated as 0.023 mg by measuring a flat region in the curve. Further observation concludes that the quantization noise term is negligible in the short cluster times while the rate ramp term explicitly appears in the long cluster times around several hundreds of seconds [24]. A further environmental test for sensor performance verification is planned for the future. V. CONCLUSION This paper has suggested an oscillation control loop appropriate for a frequency readout type, resonant sensors that employ resonant frequency variation as an output signal. By considering a typical electromechanical system, a second-order dynamic equation was derived for the plant model, while other electrical components constitute each loop block in the presented feedback control system. The design goal of the control loop was to realize automatic tracking of the resonance point and the maintenance of a constant amplitude. To this end, a feedback system with a double-branch structure, which is a variation of the AGC loop, was devised. The controller was designed to work through an envelope-based transformation and analysis. A simulation was performed using practical parameters of the sensor and other electronics. Basic loop performance was

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investigated through this simulation and factors such as the noise generated in practical circuits were taken into account. Finally, experimental verification was performed and the results were presented.

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[22] V. Singh, “Discussion on Barkhausen and Nyquist stability criteria,” Analog Integr. Circuits Signal Process., vol. 62, no. 3, pp. 327–332, 2010. [23] R. T. M’Closkey and A. Vakakis, “Analysis of a microsensor mutomatic gain control loop,” in Proc. Amer. Control Conf. San Diego, CA, 1999, pp. 3307–3311. [24] N. El-Sheimy, H. Hou, and X. Niu, “Analysis and modeling of inertial sensors using Allan variance,” IEEE Trans. Instrum. Meas., vol. 57, no. 1, pp. 140–149, Jan. 2008.

REFERENCES [1] D. Joachim and L. Lin, “Selective polysilicon deposition for frequency tuning of MEMS resonators,” in Proc. IEEE Micro. Electro. Mech. Syst., 2002, pp. 727–730. [2] B. L. Norling, “Superflex: A synergetic combination of vibrating beam and quartz flexure accelerometer,” J. Inst. Navigat., vol. 34, no. 4, pp. 337– 353, 1988. [3] B. L. Lee, C. H. Oh, Y. S. Oh, and K. Chun, “A novel resonant accelerometer; variable electrostatic stiffness type,” in Proc. Int. Conf. S. S. Sensors Actuators, 1999, pp. 1546–1549. [4] S. Sung, J. G. Lee, T. Kang, and J. Song, “Development of a tunable resonant accelerometer with self-sustained oscillation loop,” in Proc. IEEE Natl. Conf. Aerosp. Electron. Dayton, OH, Oct. 2000, pp. 354–361. [5] A. Lechner, A. Richardson, B. Hermes, and M. Ohletz, “A design for testability study on a high performance automatic gain control circuit,” in Proc. IEEE 16th VLSI Test Symp. Monterey, CA, Apr. 1998, pp. 376– 385. [6] D. Green, “Lock-in, tracking, and acquisition of AGC-aided phase-locked loops,” IEEE Trans. Circuits Syst. I, vol. 32, no. 6, pp. 559–568, Jun. 1985. [7] M. Aikele, K. Bauer, W. Fickerb, F. Neubauerc, U. Prechtelb, J. Schalkb, and H. Seidela, “Resonant accelerometer with self-test,” Sens. Actuators A, Phys., vol. 92, pp. 161–167, 2001. [8] V. Ferrari, A. Ghisla, D. Marioli, and A. Taroni, “Silicon resonant accelerometer with electronic compensation of input-output cross-talk,” Sens. Actuators A: Phys., vol. 124, pp. 258–266, 2005. [9] M. V. Andres, K. W. H. Foulds, and M. J. Tudor, “Non-linear vibrations and hysteresis of micromachined silicon resonators designed as frequency-out sensors,” Electron. Lett., vol. 23, pp. 952–954, 1987. [10] E. Vittoz, M. Degrauwe, and S. Bitz, “High-performance crystal oscillator circuits: Theory and application,” IEEE J. Solid-State Circuits, vol. 23, no. 3, pp. 774–783, Jun. 1988. [11] S. Yun, S. Sung, Y. J. Lee, and T. Kang, “Oscillation amplitude-controlled resonant accelerometer design using automatic gain control loop,” Korean Soc. Aeron. Space Sci. J., vol. 36, pp. 674–679, 2008. [12] H. K. Khalil, Nonlinear Systems, 2nd ed. Upper Saddle River, NJ: PrenticeHall, 1996. [13] W. Sung, S. Sung, J. Lee, T. Kang, Y. Lee, and J. G. Lee, “Development of a lateral velocity-controlled MEMS vibratory gyroscope and its performance test,” J. Micromech. Microeng., vol. 18, 055028, 2008. [14] S. Sung, W. Sung, Y. J. Lee, S. Yun, and C. Kim, “On the mode matched control of MEMS gyroscope via phase domain approach,” IEEE/ASME Trans. Mechatronics, vol. 14, no. 4, pp. 446–455, 2009. [15] C. Comi, A. Corigliano, G. Langfelder, A. Longoni, A. Tocchio, and B. Simoni, “A resonant microaccelerometer with high sensitivity operating in an oscillating circuit,” IEEE J. Microelectromech. Syst., vol. 19, no. 5, pp. 1140–1152, Oct. 2010. [16] C. T.-C. Nguyen and R. T. Howe, “An integrated CMOS micromechanical resonator high-Q oscillator,” IEEE J. Solid-State Circuits, vol. 34, no. 4, pp. 440–455, Apr. 1999. [17] T. A. Roessig, R. T. Howe, and A. P. Pisano, “Nonlinear mixing in surface micromachined tuning fork oscillators,” in Proc. IEEE Int. Freq. Control Symp., May, 1997, pp. 1270–1270. [18] S. Lee and C. T.-C. Nguyen, “Influence of automatic level control on micromechanical resonator oscillator phase noise,” in Proc. IEEE Int. Freq. Control Symp, May 2003, pp. 341–349. [19] Y.-W. Lin, S. Lee, S. Li, Y. Xie, Z. Ren, and C. T.-C. Nguyen, “Series-resonant VHF micromechanical resonator reference oscillators,” IEEE J. Solid-State Circuits, vol. 39, no. 12, pp. 2477–2491, Dec. 2004. [20] L. He, Y. Ping Xu, and M. Palaniapan, “A CMOS Readout Circuit √ for SOI Resonant Accelerometer With 4-μg Bias Stability and 20-μg/ H Z resolution,” IEEE J. Solid-State Circuits, vol. 43, no. 6, pp. 1480–1490, Jun. 2008. [21] F. He, R. Ribas, C. Lahuec, and M. Jezequel, “Discussion on the general oscillation startup condition and the Barkhausen criterion,” Analog Integr. Circuits Signal Process., vol. 59, no. 2, pp. 215–221, 2009.

Sangjun Park received the B.S. degree in aerospace information engineering from Konkuk University, Seoul, Korea, in 2010, where he is currently working toward the M.S. degree. His current research interests include microinertial sensors, inertial sensor resonance control loop design and analysis, GNSS, and navigation systems.

Jin Woo Song received the Ph.D. degree in electrical engineering from Seoul National University, Seoul, Korea, in 2002. He is the Chief Technical Officer of Microinfinity Corporation, Ltd., Seoul, Korea. His research interests include robust and optimal control, MEMS device control, MEMS inertial sensor design and control, inertial navigation systems, indoor navigation systems, and robot localization.

Byungjin Lee received the B.S. degree in aerospace engineering in 2010 from Konkuk University, Seoul, Korea, where he is currently working toward the M.S. degree in the Department of Aerospace Information Engineering. His research interests include flight dynamics and control system design of rotary UAVs, virtual instrumentation-based avionics design, integrated navigation systems, and instrumentation.

Hyungjoo Yoon received the Ph.D. degree in aerospace engineering from the Georgia Institute of Technology, Atlanta, in 2004. He is currently a Senior Researcher in the Satellite Control System Department, Korea Aerospace Research Institute, Daejeon, Korea. His research interests include satellite attitude dynamics and control, orbit mechanics and control, adaptive and robust control of MIMO systems, adaptive filters, and highprecision optical system control.

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Young Jae Lee received the Ph.D. degree in aerospace engineering from the University of Texas, Austin, in 1990. He is currently a Professor in the Department of Aerospace Engineering, Konkuk University, Seoul, Korea. His research interests include navigation filters and systems, integrity monitoring of global navigation satellite system (GNSS) signals, attitude determination, orbit determination, and GNSS-related engineering problems.

Sangkyung Sung (M’07) received the Ph.D. degree in electrical engineering from Seoul National University, Seoul, Korea, in 2003. He is currently an Associate Professor in the Department of Aerospace Engineering, Konkuk University, Seoul, Korea. His research interests include inertial sensors and MEMS mechatronics, vision-aided integrated navigation filters, GNSS applications, and flight control of unmanned aerial robot systems.