modeling, identification, and control of micro-sensor prototypes · 2010-07-14 · mems...
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Modeling, Identification, and Control of Micro-Sensor Prototypes
Robert M’Closkey*Associate Professor
Mech. & Aero. EngineeringUniversity of California, LA
A. Dorian ChallonerScientist
Boeing Satellite SystemsEl Segundo, CA
Abstract— Micoelectromechanical systems (MEMS) are stim-ulating the move of modern digitial control “on-chip” fora growing number of sophisticated applications. One suchapplication is microinertial sensors of ever increasing capabilitybringing inertial awareness to virtually anything that moves.Modeling, identification and control of this particular genre ofMEMS developed through a Boeing and UCLA collaborationis presented as an illustration on the rising edge of thisrevolutionary transition in the realization and theory of moderncontrol systems.
I. I NTRODUCTION
The growing sophistication of microelectromechanical sys-tems (MEMS) is creating new opportunities for the develop-ment and application of modern digital control systems. Inan interesting application area to be discussed, sequentiallyproduced precision mechanical plants, with expensive 3D,point-by-point material removal are being replaced withmassively parallel, wafer-level, 2-D micro-machined siliconplants with integrally-machined sensors and actuators andintegral low power digital control. Chips with increasinglysophisticated designs will replace their much larger, conven-tional 3D counterparts, and with astonishing performance.The convergence of a precision mechanical plant along withsensors, actuators and the capability for highly complexand effective control computation on a single chip willrevolutionize control system engineering as digital wirelesschips have already done in communication engineering. For agrowing number of sophisticated control systems applicationsnothing will be “off-chip.” With such a revolution in controlsystem realization can theory remain invariant?
The MEMS application area discussed in this paper is in-ertial sensing leading to autonomous navigation for virtuallyanything that moves, analogous to wireless communicationapplications driving digital System on Chip or SOC. Theplant in this case is a precision inertial proof mass. Everyelement and phase of modern digital controls: plant, sensors,actuators, and computation, including plant model identifi-cation, self-test and adaptation will be distilled into a photomask set. This transformation of modern digital control willconverge advanced silicon micro machining as it evolvesfrom simple analog servos with sophisticated on-chip low-power digital signal processing driven by wireless communi-cations. These new digital control system realizations with
*Corresponding author. This work is supported by NSF grant ECS-9985046 and the Boeing Aerospace Company via UC-MICRO 03-060a.
every element in the loop in trade, including mechanicalplant, will give rise to new control paradigms and theories tomaximize performance per square mm of silicon real estate– the new control system performance index.
One application area of MEMS inertial sensors as perfor-mance improves will be for satellite attitude control. Sincethe beginning of the space age, zero-g attitude stability hasbeen a major concern leading the early Syncom satellitedesigners, in the absence of on-board digital computers, tospin stabilize the entire satellite. It was so well balancedagainst radiation pressure that remarkable drift rates< 0.001deg/h were achieved in the early 60’s. This quality of passivestability enabled ground based readout of attitude sensortelemetry and spin-synchronous thruster pulse precession foradjustment of attitude during orbit injection and on-orbitfrom time to time for attitude maintenance. The emerg-ing satellite communications era soon placed demands fornarrow, high power earth pointed beams and hence greatercontrol system sophistication. However, it was not until thelate 80’s that simple analog servos finally gave way toon-board three-axis digital control implemented in centralmicroprocessors linking a number of attitude sensors andactuators. As the availability of capable digital hardwareincreases and the performance demands increase the tran-sition to modern digital control is natural for optimizedperformance, improved robustness, increased versatility anddesign re-use, etc. This transition pertains increasingly to allsystem levels and can soon be anticipated for demandingMEMS applications, including inertial navigation as basicsensor performance improvement warrants. Even completesingle-chip satellites have been contemplated by JPL and TheAerospace Corporation [7]. Might initial realizations also bespin-stabilized like Syncom?
Modern digital control theory and practice was born in the60’s from a potent mixture of applied mathematics and thenew ability to do centralized digital computations in real timeand the ever present need to improve the performance of highvalue plants ($10M to $100M’s) in manufacturing, militaryand aerospace domains. The lathe and milling machinewere then the enabling factors of production for the precisemechanical plant, sensor and actuator units warranted bymodern digital control. Serial production of the requiredequipment with near “atom-at-a-time” material removal wasnot a limitation for such high value systems. Sensing andactuator units were distributed about the usually massive
plant and interconnected with the central processor (e.g.IBM 7090 or 360) with wire harnesses. The exorbitantexpense of custom control system development would beginwith a system block diagram and end in the source codefor the centralized processor being the prized domain ofthe control system engineer. Increasingly, with the adventof semiconductor technology the cost of signal processingwithin the sensors and actuators and the speed and size ofcentral processors improved by orders of magnitude enablingmodern control systems to be embedded in more and smallerplants. Still the high cost of a custom digital control systemdevelopment in the last century seemed an invariant, e.g.$5M for the three-axis attitude control of the Communica-tions Technology Satellite, perhaps a $60M plant c. 1976.Curiously, because of absence of space qualified computersa 500-IC digital differential analyzer paradigm was usedfor control system realization (this counterpoint betweenthe centralized algorithms of digital control implementedon “mainframes” and the distributed algorithms found inconstrained realizations continues in the on-chip era). Thehigh cost of these conventional control systems was largelybecause of the significant engineering teams needed forcustom selection, procurement and integration of the severaldiscrete sensor and actuator units into the physical plantas well as the centralized coding developed initially withsophisticated analytic models of the plant and finally onsome sort of end-to-end or closed loop system test-bed forvalidation. These plant models would begin with conceptualmodels of the sensor, actuator and plant dynamics derivedfrom physical principles or extensive system identificationinvolving some version of the open loop plant. For aerospacevehicles this was not possible so sophisticated real-timedigital mixed simulation systems evolved with sophisticateddynamics models to develop and then validate the finalcentral control code.
For today’s terrestrial platforms or plants such as computerdisk drives or automobile engines the shear production vol-ume can now warrant the expense of modern digital controlsystem development and integration for $100-1000 consumerproducts. MEMS are now opening up the possibility ofsimple control systems on a chip, e.g. accelerometers orgyroscopes, in the $1-10 range and some are already onthe market for safety systems and stabilization applicationsusing simple analog servos for proof mass control. TheADXL accelerometer and ADXRS gyroscope from AnalogDevices, in Figs. 1 and 2, respectively, perhaps best illustratethe state of the art of on-chip control in MEMS. Most ofthe sensor processing is open loop and only the gyroscopehas analog closed loop control on-chip to regulate the driveamplitude. The polysilicon BiMOS fabrication process, withthin films for the mechanical proof mass grown in-situ alongwith the electronics, is adequate for the application intended.Bulk micro-machined gyroscopes, with proof masses etchedfrom standard crystal silicon wafers and separately fabricated
electronics, such as the Draper tuning fork gyro [8] in Fig. 3,and Boeing/JPL post/cloverleaf gyro, shown in Figs. 4 and 7and discussed in the sequel, have demonstrated one to twoorders of magnitude better drift performance enabling tacticalnavigation performance with GPS aiding. The tuning forkdrive is closed loop but output axis is open loop while themechanically tuned post/cloverleaf has both drive and outputloops closed, originally with discrete analog electronics. Bothloops have now been realized in an off-chip low-powerdigital CMOS ASIC [5], [6], [3]. The notable BAE siliconring gyroscope [4] is also a mechanically tuned design withclosed-loop drive and output control. It employs a crystalsilicon bulk-micromachined structure and was initially elec-tromagnetically sensed and controlled. The latest generationis electrostatically forced and both the sensor excitationand output remain closed-loop. The Systron-Donner quartzrate sensor, Fig. 5, is a double tuning fork design quitesimilar in principle to the Draper single tuning fork. Thecurrent generation has an∼ 11 mm long fork and is etchedfrom quartz wafers and employs piezoelectric sensing andcontrol [16]. The in-plane drive tines are controlled similarto a crystal oscillator with an automatic gain control loop (notshown) while the pickup tines sense the out-of-plane Coriolisforce arising from in-plane inertial rate along an axis normalto the velocity of the tip masses of the driven tines. Onesuitable entry for modern digital control on chip is thus toadd value to the increasingly more precise and sophisticatedMEMS emerging for autonomous navigation in the 3D world.This paper then reviews the modeling, identification andcontrol performed at UCLA for the Boeing/JPL MEMS gyrosfrom the viewpoint of this emerging transition to higherperformance designs and modern digital control systems ona chip.
II. V IBRATORY RATE SENSORS
A. Theory of operation
A generic linear model for vibratory rate sensors consistsof a two degree-of-freedom (DOF) system with a skew-symmetric matrix that is modulated by the angular rate ofrotation of the sensor:
M~x+D~x+ΩS~x+K~x = ~f . (1)
In this modelM, D, andK are the real, positive definite 2×2mass, damping, and stiffness matrices, respectively, andΩand~f = [ f1, f2]T represent the sensor angular rate of rotationand applied actuation forces, respectively. These equationsare written in the sensor-fixed coordinates denoted by~x =[x1,x2]T and the forces~f are co-located with the sensingpickoffs. The reader is referred to [11] for a detailed introduc-tion to vibratory rate sensors. A more complete description ofthe sensor dynamics would include nonlinearities. A commonnonlinearity in these sensors arises from elastic stiffeningfor sufficiently large harmonic forcing in a neighborhood ofthe modes. Other nonlinear contributions can depend on the
Fig. 1. In the Analog Devices ADXL accelerometer, the electronics occupythe majority of the 3 mm2 chip area. The 2-axis device is not only extremelysmall, but also modest in power requirements and can measure both staticand dynamic acceleration. Operation is open loop.
Fig. 2. The iMEMS ADXRS angular rate-sensing gyro from AnalogDevices integrates an angular rate sensor and signal processing electronicsonto a single piece of surface-micromachined silicon. Drive axis velocitycontrol for the Coriolis sensor is implemented on-chip but the output axis,however, is open loop. Some on-board trim and test and perhaps bias thermalcompensation are also noted.
Fig. 3. The Draper tuning fork gyroscope micromachined from bulk orcrystal silicon uses off-chip electronics for closed loop control of in-planedrive amplitude of the two electrostatic comb-driven proof masses. Outputaxis operation is open loop and comprises capacitive sensing of out-of-planeproof mass motion due to Coriolis force arising from in-plane inertial ratealong an axis normal to the drive axis.
Fig. 4. Boeing/JPL Post Resonator Gyroscope (PRG) is a mechanicallytuned design employing closed loop control on both the drive axis to regulateamplitude and on the output axis to sense inertial rate. The post proof-massis rocked about one of the in-plane axes, the drive axis, using electrostaticsense and actuation and the Coriolis force arising from an out of planeinertial rate is electrostatically sensed and rebalanced about the second in-plane axis, or output axis.
manner in which forces are applied: electrostatic forces, forexample, produce a softening spring effect. Linear models,however, are quite adequate for describing the dynamics ofthe Boeing/JPL sensor so we will not consider any further thenonlinear elements associated with the sensor’s dynamics.
Control forces excitex1 into a sinusoidal response –this isusually accomplished by adrive control loop– then rotationat angular rateΩ about the sense axis transfers momentumfrom one DOF into the other DOF resulting in a change inamplitude and phase ofx2 from whichΩ may be inferred. Inpractice, a second control loop is used to null thex2 pickoffsignal and in this case the control effort is related toΩ. The
Fig. 5. Systron-Donner quartz rate sensor (QRS) is a double-tuningfork etched from quartz with piezoelectric sensing and control. Currentgeneration∼ 11 mm long fork is depicted on the left. Closed-loop automaticgain control maintains oscillation of the drive tines while the output axiscomprises open-loop sensing of motion of the pickup times.
second loop has been termed theforce-to-rebalanceloop orsimply, rebalance loop, in the literature. These two controlloops are common in vibratory rate sensors although theirimplementation may differ from one technology to another.
Analysis of the closed-loop signals illuminates certainlimitations imposed on the detection ofΩ so consider theclosed-loop system in Fig. 6. All signals have dimension twoin this diagram, and for steady state analysis we takeΩ to beconstant. Furthermore, we assume a sinusoidal steady-state isachieved after injection of an appropriate referencer so it isconvenient to represent the sensor model with its frequencyresponse function for this analysis,
P = (I − jωPSΩ)−1P,
whereP =
(−ω
2M + jωD+K)−1
.
The control force generated by feedback relative to thesensor output is~f = P−1~x = P−1 + jωSΩ. In order to detectΩ, the second component of the control force, denotedf2, isobserved. This is merely the(2,1) component ofP−1~x whichis obtained from the figure or the original equations (1),
f2 = (−ω2M21+ jωD21+K21+ jωS21Ω)x1+
(−ω2M22+ jωD22+K22)x2.
The matrix subscripts denote the component of interest. Inan ideal sensor the off-diagonal terms in the mass, stiffnessand damping matrices are zero and if we further assumethat the controller gain is sufficiently large atω renderingx2≈ 0 then as a first order approximation the feedback forceis proportional toΩ and may be obtained by demodulatingf2 with respect to ˙x1. In a non-ideal sensor, the off-diagonaldamping produces a component inf2 that is in-phase withthe Ω-induced component. This represents a spurious ratethat cannot be distinguished from the actual rate and thisis the primary reason that damping is minimized to thegreatest degree possible in high performance vibratory ratesensors. The off-diagonal mass and stiffness matrix produce
Fig. 6. Angular rotation rate is depicted as a feedback loop around themechanical dynamics of the gyro.
terms which are 90 out of phase (quadrature) with theΩ-induced terms and hence may be rejected by phase-sensitivedemodulation with respect to ˙x1. In a loop with finite gainat ω, however, the quadrature and in-phase errors are furtherexacerbated by the fact thatx2 6= 0. The non-idealities in thesensor dynamics that produce rate-sensing errors are itemizedin [18], [19] for some simple models and can be identifiedaccording to their influence on the system matrices.
This analysis shows that even with feedback, the error inthe rate estimate can be sensitive to plant perturbations so itis fair to question the of role feedback in this application. Oneshould understand that the art of inertial instrument designresides in the ability to make plant perturbations zero, or atleast fixed, as a function of time, temperature, vibration inthe operating environment, etc. Feedback, however, is usefulfor a number of reasons even though it does not mitigateplant uncertainty for this particular problem. Here are a fewpoints justifying its use:
1) Feedback assists in generating an approximate plantinverse for the detection ofΩ. The complexity of thefeedback filters is often less than the complexity ofa “deconvolution” filter that would be required in anopen-loop setting.
2) It is sometimes possible to replace “mechanical” stiff-ness with “servo” stiffness with the benefit of the servostiffness being less susceptible to temperature effectsand possessing less hysteresis.
3) A high gain rebalance loop keeps thex2 signal nearzero, thereby improving the linearity of the responseand simplifying calibration algorithms.
4) In a high gain loop wherex2 ≈ 0, the quadratureterms in f2 are (−ω2M21+K21)x1 and hence may bedetermined by demodulatingf2 with respect tox1. Thisis essentially provides an on-line identification schemeof the quadrature terms.
The last point deserves more elaboration. The quadraturecomponent can be eliminated by orienting the generalizedeigenvector ofM and K associated with the driven modesuch that it’s projection onto thex2 pick-off is zero, andalso by reducing the frequency split between the modes.
Both methods are pursued in practice by a variety of tuningschemes that perturb the mass and stiffness matrices. Acomplementary method is to change the sensing pickoffframe by creating weighted sums of the pickoff signals.This approach leaves the plant dynamics unchanged butmodifies the sensor and actuator location on the vibratingstructure. Whatever method is employed, it is often necessaryto keep the quadrature signal small to avoid saturation ofthe rebalance loop amplifiers and to mitigate the effect ofdemodulation phase error, and so most high performancesensors employ aquad-nulling loop.
B. Boeing/JPL micromachined gyro overview
The foregoing discussion is general and applies to anyvibratory gyroscope. We discuss in this survey, however,many interesting system identification and control issues fora particular class of sensors: the Boeing/JPL micromachinedgyroscopes. This section provides some details on the phys-ical aspects of these sensors. The Boeing/JPL microgyros(Fig. 7) consist of a silicon micromachined plate suspendedabove a set of electrodes. The two modes that are exploitedin angular rate detection correspond to a two degree-of-freedom rocking motion of the plate that is parameterizedby the θ1 and θ2 angular coordinates in Fig 7. The centralpost strongly couples these rocking degrees of freedom viaa Coriolis term (the axis which is sensitive to angular ratesis the Ω-axis along the post). Various electrodes are usedto apply electrostatic forces at points on the plate or, as theplate deflects relative to the electrodes, capacitive sensing isused to measure the deflection. More details on the designand fabrication of these sensor may be found in [1], [21],[22], [23].
Relatively weak electrostatic forces are used to excite thesensor dynamics so an appreciable response amplitude canonly be achieved when the excitation power is concentratedin a neighborhood of the lightly damped sensor modes. Thus,it is necessary that the drive loop selectively excite oneof these modes, preferably at the frequency of maximumresponse because this make use of the sensors intrinsicgain. As mentioned in Section II-A there are advantages indesigning the sensor so thatω1 ≈ω2. The Boeing/JPL fabri-cation process aims to produce sensors with these degeneratedynamics and in fact can consistently produce devices withfrequency splits on the order of 0.2%.
There are multiple electrodes on the baseplate that canbe configured as sense pickoff or actuators. Two of theseelectrodes are selected for sensing, another two electrodesare designated as actuators, and the remaining electrodes areused for applying bias voltages to further reduce the detuningbetween the modes. The actuation electrodes, also commonlyreferred to as thedrive electrodes, can be driven directlyfrom a function generator or digital-to-analog converter(DAC). The sensing electrodes use a trans-impedance op-amp configuration to provide a buffered output voltage that
is proportional to the velocity averaged over the distributedelectrode. Maximum displacements of the sensor’s elasticstructure, on the order of 1 to 2µm, correspond to senseelectrode potentials of several hundred millivolts in thefrequency range of the important modes discussed below.When defining the equations of motion of this structure wefocus on the two closely spaced rocking modes that can bemodeled by (1) although we do not use angular coordinateparameterization in Fig. 7. Instead, the following perspectiveis adopted: the generalized coordinates are chosen to bethe coordinates established by the sensing electrodes; if wedenote the sensing electrode measurements asS1 andS2 then[x1 x2] := [S1 S2]; the electrostatic forces created by thepotentials applied to the pair of drive electrodes, denotedD1 andD2, are different from the generalized forces[ f1, f2]of (1) since the drive electrodes and the sensing electrodesare not colocated. Thus, we modify (1) to
M~S+C~S+K∫
~S= B~D, (2)
where~S := [S1, S2]T , ~D := [D1, D2]T , and B is a real,non-singular 2×2 matrix that specifies how the force appliedby each drive electrode couples into the coordinate framespecified by the sensing electrodes. The Coriolis term hasbeen omitted from this equation.
III. T HE M ICRO-SENSOR RESEARCH PARTNERSHIP
BETWEEN UCLA, BOEING AND JPL
As discussed in the Introduction, there has been an ever-present demand for high quality inertial stabilization for long-life satellite applications involving narrow-beam payloads.Gyroscopes, of the spinning mass variety, were initiallyqualified for intermittent operation during thrusting maneu-vers. In the early 1990’s Delco Systems, a division of thenHughes Aircraft Company completed the development andqualification of the first continuous duty solid state gyroscopefor satellites, the quartz Hemispherical Resonator Gyroscope(HRG). This was a conventionally-machined, precision tunedCoriolis vibratory sensor with capacitive sensing and actua-tion and closed loop drive and output axis control. Northrop-Grumman now owns and manufactures the HRG.
As the HRG was being developed and qualified for spaceapplications, Kaiser and Tang [21] at JPL were experimentingwith novel bulk silicon micromachined inertial sensors basedon tunneling and capacitive sensing techniques. Kubenaat then Hughes Research Laboratories, experimented withmethods of fabricating tunneling sensors using surface micro-machined nickel beams on a silicon substrate. This led to tun-neling rate sensor and accelerometer designs with closed loopoutput control being developed for the then Hughes Spaceand Communications unit, now Boeing Satellite Systems(BSS), see [9], [10]. With the advantages of conventionalvibratory gyroscopes for space established by the HRG, thepromise of MEMS was then to significantly reduce cost,
2-axis
-axis
1-axis
10 m
Capacitor
Gap
Fig. 7. A photo of the Boeing/JPL vibratory gyro prototype (left) and drawing indicating baseplate electrode layout (right). The cloverleaf structure isclearly visible in the photo and has three prominent modes that can be parameterized as two angular deflectionsθ1 and θ2, and an out-of-plane lineartranslation in the direction of theΩ-axis. The post along theΩ-axis strongly couples the two angular degrees of freedom with a Coriolis term and it isthis coupling that is exploited for detection of the sensor’s angular velocity resolved along theΩ-axis. The shaded area on the baseplate drawing revealsthe electrode layout.
JPL
UCLA Boeing Satellite Systems
Manufacture of MEMS gyros
and core experimentation
Modal identification testing tools
Digital ASIC prototype
Physical gyros to test
Experimentation data and analysis
Control algorithms
Mechanical design changes
General design collaboration
Fig. 8. Boeing MEMS technical cooperation and partnership with JPL andUCLA.
weight and power. By 1997, BSS had begun a partnershipwith JPL on the development of advanced silicon MEMSgyroscopes for space and with UCLA on complementarymodeling, identification and control of MEMS. The triadshown in Fig. 8 sketches out the contribution of each group.UCLA work under UC MICRO grant funding from BSS, theNational Science Foundation and the Jet Propulsion Lab, isdiscussed in the remainder of this paper.
A. Micro-sensor system identification
System identification plays an important role in not onlyproviding feedback compensation design models but alsogives critical information during post-manufacturing stepsduring which the detuned natural frequencies of the sensor
modes can be reduced by application of bias voltages tospecial electrodes on the baseplate. The bias voltages actuallyperturb the sensor dynamics by creating a force field that pro-duces an equivalent negative spring rate. Proper adjustmentof the bias voltages pulls the rocking mode frequencies closertogether.
A typical wide-band experimental frequency response isshown in Fig. 9. The sensor is tested as a 2-input/2-outputsystem and in this caseΩ = 0. The sensor/actuator pairs areindicated in the title of each subplot. The resonance thatcorresponds to a linear translation of the sensor’s elasticstructure along the post direction occurs near 2700 Hz. Thismode is not only easily excited by the control electrodes,but also by linear acceleration in the appropriate direction.The next lightly damped resonance near 4420 Hz actuallycomprises the two “rocking” modes of the cloverleaf. Thefrequency split between these modes is less than 5 Hz(0.11%) and so each mode cannot be individually resolved onthis scale. It is these modes that are strongly coupled via theCoriolis acceleration when the sensor is rotated. Althoughthe sensor was designed to exploit these two modes forangular rate sensing, it is quite evident that the wide-bandsensor dynamics are far richer than (2). Nevertheless (2) isan appropriate description if one is interested in a narrowfrequency window around the modes. The remaining lightlydamped resonances above 5 kHz are other flexural modes inthe sensor’s elastic structure. Parasitic capacitance between
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Fig. 9. Wide-band frequency response of the rate sensor. The modes thatare exploited for detection ofΩ are clustered near 4.4kHz. There is anothersignificant mode near 2.7 kHz that also requires active control.
the drive electrodes and sense electrodes causes the positivetrend in the slope up to 35 kHz at which point the signalconditioning amplifiers roll-off. Lastly, the sharp notches inclose proximity to the modes near 2700 Hz and 4420 Hz arecreated by a charge cancellation on the sensing electrodes dueto the opposite phase of parasitic capacitance-induced chargeand the charge created by motion of the cloverleaf. Theseplots were produced by concatenating results from multipletests with a dynamic signal analyzer and require a significanttime investment.
Although Fig. 9 represents the generic features for thisclass of rate sensor, it is not necessary to test the devicesover such a large frequency range since it is the Coriolis-coupled modes that must be quantified. Furthermore, eventhough all of the channels in Fig. 9 look similar, there doexist significant differences in the frequency responses in aneighborhood of the two rocking modes. This region mustbe examined more closely. Our solution to use an adaptivelattice filter for fitting a high-order two-input/two-outputARX model was motivated by the fact that accurate, rapidestimates of the two rocking modes be available for post-fabrication tuning. The object was to supplant the arduousand time consuming single-channel testing that was donewith a commercial signal analyzer. The main objection tousing the signal analyzer was the excessive testing timesrequired to produce results of adequate frequency resolution.The test data sets for the ARX modeling are generated byexciting the drive electrodes with periodic, but independent,chirp sequences. The chirp power is concentrated from 2kHz to 5 kHz in order to excite the translational mode near3 kHz in addition to the rocking modes at 4.4 kHz. Thebuffered sense electrode signals are synchronously sampledat 50 kHz. Relatively high ARX model orders (≈ 40) are
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Fig. 10. Frequency response of an ARX model identified from 5 secondsof I/O data (solid: magnitude; dash: phase). The sensor dynamics are excitedwith periodic, independent chirp sequences with power focused in the 2.5kHz to 5 kHz band. This span includes the modes of interest and leadsto a high-quality model that can be further analyzed. There is a singlemode near 2.7 kHz that corresponds to cloverleaf structure vibrating in adirection corresponding to theΩ-axis in Figure 7. The two rocking modesare clustered near 4.4 kHz.
necessary for capturing not only the “rigid body” dynamicsof the cloverleaf but also the dynamics of the pre-amps,anti-aliasing filters (8-poles/channel, two channels) and othersensor modes outside of the input power band that arenevertheless excited by vibration of the sensor case. Thehigh model orders ensure that the modes we want to identifyare estimated with little bias. Some of this work is reportedin [15]. Another fact that should not be overlooked is themultichannelnature of the experiments and models: as thefrequency split between the two rocking modes is reduced,the single channel identification problem becomes very ill-conditioned due to the fact that in the degenerate case, thesensor dynamics are unobservable and uncontrollable fromany SISO pairing of sensors and actuators. A frequencyresponse of the ARX model is shown in Fig. 10. The transla-tional mode and two rocking modes are captured. Figure 11shows the frequency response detail in a neighborhood of therocking modes. The phase includes lags from the anti-aliasfilters.
As an application of the ARX modeling consider Fig. 12.These pictures represent measurements of the vibratingcloverleaf measured by optical interferometry. The sensor isexcited at each of its modal frequencies and then the steady-state response of the cloverleaf surface is strobed at differentphases relative to the sinusoidal excitation –this builds mapof the surface displacement throughout its periodic response.The green bands indicate regions of zero displacement, soin fact the cloverleaf is “rocking” about these axes. Theseessentially reveal the generalized eigenvectors of the massand stiffness matrices since the transient time constants arelarger than the frequency split. This can be argued as follows.Consider (2) and letT be the matrix whose columns are
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Fig. 11. Zoomed version of Figure 10 showing the frequency responsedetail in in a neighborhood of the rocking modes. The model includes theanti-aliasing filter lag so the phase is not asymptotic to±90 as would beexpected from (2).
the generalized eigenvectors ofM and K, i.e. MTΛ = KTwhereΛ = diag(ω2
1 ,ω22) are the eigenvalues arranged on the
diagonal. Furthermore,TTMT andTTKT are both diagonalwith diagonal elements(m1,m2) and (k1,k2), respectively,and ω2
p = kp/mp, p = 1,2. The transformed damping ma-trix TTCT, however, is not diagonal in general and so it isrepresented by
TTCT =[c1 c3
c3 c2
].
The frequency response of (2) then becomes[−m1ω2 +k1 + jc1ω jc3ω
jc3ω −m2ω2 +k2 + jc2ω
]T−1S= jωTTBD,
(3)where ˆ denotes the frequency response representation. Whenω = ω1 (3) simplifies to
S=T
[c1 c3
c3 jm2∆ω +c2
]−1
TTBD,
=1δ
T
[jm2∆ω +c2 −c3
−c3 c1
]TTBD,
where ∆ω := (ω22 − ω2
1)/ω1 ≈ ω2 − ω1 and δ = c1c3 −c2
2 + jc2m2∆ω . The last expression is informative becauseit demonstrates that the steady state response of sensor isessentially a scaled version of thefirst column ofT when
m2∆ω >> cp, p = 1,2,3 (4)
regardless of the input direction. Thus, the cloverleaf platerocks about an axis orthogonal to the generalized eigenvectorcorresponding toω1. A similar argument can be made forsinusoidal excitation of the sensor with frequencyω2. Notethat (4) implies that the frequency split be larger than inverseof the open-loop sensor time constant (we can estimatethe open-loop sensor time constants to bem1/c1 ≈ m2/c2).When this condition is not satisfied the frequency response
is dominated by the damping when the forcing frequency isin a neighborhood of the modes.
Now turning our attention back to Fig. 12, the top twoplots represent the sensor’s response with zero potential onthe bias electrodes. The rocking mode frequency difference inthis case is approximately 3 Hz out of a nominal frequency of4.4 kHz. The bottom two plots represent the modal responseswith a non-zero potential on one bias electrode and in thiscase the frequency split has been reduced to less than 1 Hz.Not only have the modal frequencies changed, but also thegeneralized eigenvectors ofM andK as indicated by a changein orientation of the green bands. It is interesting to notethat the green bands, and hence the generalized eigenvectors,look roughly orthogonal for each case. This should comeas no surprise since in an ideal sensor with no fabricationirregularities, the mass and stiffness matrices are scalar-times-identity so small perturbations toM and K producea situation in which their generalized eigenvectors are nearlyorthogonal.
The ARX model yields estimates of the modal frequenciesas well as the generalized eigenvectors ofM and K, butusing only several seconds of I/O data and several secondsfor fitting and analyzing the ARX model (the same assump-tion applies when computing the generalized eigenvectorestimates, namely the frequency split is sufficiently largecompared to damping constant). Fig. 13 reveals the “tuningspace” for the gyro: the bias electrode potentials (denotedBT and B2 in the figure) are swept through a selectedrange, and I/O data are acquired at each point; an ARXmodel is fit to the data set and then subsequently analyzedfor the modal frequency split as well as the generalizedeigenvector orientation. The blue and red vectors in the figurerepresent the eigenvector estimates and the length of thevectors indicates the split. In this example, the split canbe reduced to less than 100 mHz, and the necessary biaspotentials can be obtained from the figure.
The ARX models are adequate for tuning the rockingmodes but it is difficult to obtain precise estimates of thedamping from short sequences of I/O data. This information,along with a more detailed picture of the mass and stiffnessmatrices associated with the two rocking modes, is usefulin the sensor development process where, for example, theeffect of a step taken during manufacturing can be analyzedfor its impact onM, K, and C. This is work in-progressbut we can report on an efficient method for generatingvery precise frequency response test data that can be usedin generating the system matrices by fitting the frequencyresponse data to (3). The idea is to use sine wave correlationtesting in a neighborhood of the rocking modes for itssuperior noise rejection capability. Sine wave testing ofthe open-loop sensor, however, requires marathon durationsbecause of the system time constants (Q can exceed 40,000for these sensors). Even if one could tolerate the lengthytests, the frequencydrift exhibited by these sensors precludes
Fig. 12. Top two figures: the vibrating cloverleaf imaged with a pulsed-
source interferometer when the bias potentials are zero. The green bandsreveal the orientation of the generalized eigenvectors ofM and K. Bottomtwo figures: same sensor with non-zero potentials on the bias electrodes.The modal frequencies and eigenvectors are perturbed. These experimentsrequire optical access to the sensor and several minutes of testing.
Fig. 13. Tuning the modes of the sensor to degeneracy with bias electrodes.
identification experiments lasting longer than a few minutes.For example, Fig. 25 shows frequency data obtained froma tracking control loop that selectively excites one of thesensor’s rocking modes to a stable amplitude independent ofits frequency. A real-time readout of the operating frequencyis provided by a frequency counter. This control loop will bediscussed in more detail in Section III-B. One can see fromthe figure that an identification experiment lasting more thena few minutes risks producing worthless results because thedynamics may have substantially changed during the courseof the experiment (this claim will be justified shortly).
The solution is to employ a damping controller to vastlyreduce the time constants in the closed-loop to facilitate rapididentification of closed-loop frequency responses from whichthe open-loop sensor frequency response can be extracted. Areduced-order ARX model is used to synthesize a controllervia the robust loop-shaping [17]. We won’t focus on thesynthesis procedure other than to note that it produces avery sensible controller from the point of view of increasingthe closed-loop damping. For example, Fig. 14 is a Nyquistplot of the ARX sensor model in a neighborhood of therocking modes (the mode near 2.7 kHz was excluded toavoid cluttering the figure –a similar argument can be madefor this mode too). The two loops in each plot are thedynamics associated with the two lightly damped rockingmodes and these plots reveal that both modes strongly coupleinto the off-diagonal channels, and to a lesser extent intothe diagonal channels. The ARX model of the loop gainPC where P is the full-order ARX sensor model andCis the 6-state synthesized controller, is shown in Fig. 15.The intriguing features of the loop gain are that it is nowdiagonally dominant, each diagonal channel has a different
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Fig. 14. Nyquist plot of the full-order ARX model from 4 kHz to 6 kHzis adapted from Figure 10. The translational mode is excluded with thisfrequency range.
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Fig. 15. A Nyquist plot of thePC from 4 kHz to 6 kHz demonstratesthat the controller has adjusted the loop phase to emulate velocity-to-forcefeedback (positive feedback convention for the implementation).
dominant mode, and the phase of the dominant modes isadjusted so that the drive electrode potential is 180 degreesout of phase sense electrode measurements thereby emulatingvelocity-to-force feedback.
The closed-loop system time constants are reduced by overa factor of 50 compared to the open-loop sensor. Since theopen-loop transients require approximately 4 seconds to ringdown to 10 percent of their initial amplitude, it is necessaryto wait on the order of 10 seconds to allow the transientscreated by switching the reference signal to decay to levelswhere the frequency response estimate is not affected. Incontrast, the closed-loop time for transient settling is reducedto 0.2 seconds. Thus, a detailed empirical frequency responseis identified in a neighborhood of the two rocking modes in
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Fig. 16. The sensor’s empirical frequency response in a neighborhood of therocking modes (solid: magnitude; dash: phase). The phase plot is associatedwith the heavy trace. The two other magnitude plots were obtained fromdata sets acquired 20 minutes apart and illustrate the slowly-varying natureof the gyro dynamics.
less than 2 minutes. The closed-loop transfer functions thatare identified are
H1 = PC(I −PC)−1, H2 = C(I −PC)−1,
and the open-loop sensor estimate is recovered from calcu-lating
P = H1H−12 .
Fig. 16 illustrates the frequency response estimates obtainedfrom this closed-loop testing procedure. The solid and dashedheavy traces represent a “nominal” case; the two additionallight solid traces represent the sensor’s frequency responsetaken 20 minutes and 40 minutes later. Note that the dy-namics slowly vary with time in such a way the frequencyvariable is scaled. This dependence can be observed in open-loop frequency response testing because the long durationsintroduce smearing of the frequency response estimate. Alsonote that the frequency response estimate clearly resolves thezeros –something that the ARX model has trouble capturing(compare Figure 16 to Figure 11).
This precise frequency response data can be used forfurther analysis. In fact, positive definite mass and stiffnessmatrices can be fit to the data –the damping matrix at thispoint is not constrained because the small parasitic capacitivecoupling from the drive to sense electrodes actually biasesthe damping estimate. More will be reported on these effortsat a later date. The identified mass and stiffness matrices,however, do permit computation ofdecouplingtransforma-tions applied at the sensor’s input and output that essentiallydiagonalizes its transfer matrix. Referring to (3), if a new setof outputs and inputs are defined as
S:=ToutS, whereTout = T−1
D :=T−1in D, whereTin = (TTB)−1
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Fig. 17. Empirical frequency response of the re-identified sensor includingthe I/O transformations. The frequency response is now diagonally dominantwith a different rocking mode in each diagonal channel.
then the transfer functionS/D = ToutHsensTin, where Hsens
denotes the sensor, has been diagonalized to the extentallowed by the damping matrix. Note thatS and D areconstructed from constant gain transformations of the nativesense electrode and drive electrode signals. This method ofdiagonalizing the sensor dynamics can be used in conjunctionwith the tuning methods discussed earlier to reduce thequadrature signals in the closed-loop sensor. The transfor-mations are realized with analog summing networks withprogrammable coefficients. The transformations are nearlyorthogonal matrices (maximum condition number is 1.2) andhence do not exacerbate any modeling uncertainty associatedwith the frequency response data. The re-identified sensor,including I/O these transformations, is shown in Figure 17.The off-diagonal channels are now nearly 40 dB smaller(in terms of peak magnitude comparison) than the diagonalchannels. This sensor is now in an ideal configuration for thedesign and implementation of the feedback compensation.This is the subject of the next section.
B. Micro-sensor control
Feedback compensation forms an integral part of manymicro-sensors and the vibratory rate gyros are no exception.As mentioned in Section II-A, feedback does not reduce theeffect of sensor uncertainties on the detection of the angularrotation rate because of the nature of the coupling of the rateinto the equations of motion. Feedback, however, is usefulproviding an approximate inverse of the sensor dynamicsand this simplifies subsequent signal processing. The primaryfeedback loops for vibratory gyros are:
1) Excitation loop.The harmonic excitation of a degreeof freedom, often a lightly damped mode, to a stableamplitude is required. There are numerous methodsusing feedback to achieve this, including self-excitingloops and phase-locked loops. A version of a self-
exciting loop called automatic gain control is discussedin some detail below.
2) Rebalance loop.The “sensing” pick-off signal is nulledby a high-gain feedback loop.
3) Quad-nulling loop.The quadrature signal nulling loopessentially provides real-time identification of the plantdynamics, albeit at the excitation frequency only. Thisloop typically has low-bandwidth and compensates forperturbations of the plant dynamics that have timeconstants on the order of minutes or hours (thermallyinduced drifts, for example –see Figure 16). The actua-tor associated with this loop has limited authority overmodifying the plant dynamics. The bias electrodes onthe Boeing/JPL microgyros are used for this purpose.
The reduction in sensing element size using microma-chining technology can mean reduced performance (largerrate noise density) due to relatively larger manufacturingtolerances so in order realize the advantages of MEMS sen-sors it is often necessary to incorporate on-chip electronicsfor a compact, but fully integrated and functional, sensor.Analog Devices has successfully commercialized MEMSaccelerometers and vibratory angular rate sensors with com-plete on-chip signal conditioning, control and self-test elec-tronics. It is understandable that the details of the algorithms,and especially their implementation, are not published bythe producers of commercially available micro-sensors. Theacademic environment, however, has exchanged ideas ondifferent approaches for error suppression and control. Morerecent efforts include [12], [20].
The Boeing/JPL post-resonator gyro (PRG) shown inFig. 4 has demonstrated 0.1 deg/hr bias stability with discreteanalog implementations of the control and signal processingcomponents. The excitation loop in this sensor is calledautomatic gain control (AGC), a block diagram of which isshown in Fig. 18. The idea is simple: a measurement of theoscillator velocity is fed back to the forcer and depending ofthe sign of the feedback gain, the oscillation is damped (whenoscillator amplitude is larger than the reference amplitudeR)or destabilized (when the oscillator amplitude is smaller thanthe reference amplitude). The feedback gain is determined byoutput of the the PI compensation which is preceded by anamplitude detection scheme. When the oscillator has reacheda steady-state amplitude the AGC adds the same amountof energy as is dissipated by the damping. The primaryadvantage to using the AGC is that it isadaptivedue to thefact that the oscillator is always excited atωn, even whenωn is slowly varying with time. The rebalance loop for thepost-resonator gyro is a traditional high-gain feedback loop.
The discrete analog electronics for the PRG are effective,but not very flexible or power-efficient. Our research groupat UCLA has developed, under funding from Boeing, anapplication specific integrated circuit (ASIC) that implementsa digital AGC, rebalance loop, and signal demodulation forin-phase and quadrature component detection, in a very low-
Fig. 18. Simplified AGC for self-excitation of the oscillator to a stableamplitude.
power and programmable platform. The next section providesa few details of the ASIC performance when integrated withthe Boeing/JPL microgyro. Efforts to integrate the ASIC withthe PRG are underway.
ASIC details
The ASIC architecture, shown in Fig. 19, employs anonlinear automatic gain control for the drive loop, a linearfilter for the rebalance loop and also provides filtering anddemodulation of the rebalance signal for detection of its in-phase and quadrature components. The linear filters are real-ized with an FIR implementation. The ASIC is a low-powersignal processor –its core consumes only(0.37µW/tap) fs,where fs is the servo rate in kHz. The efficient hardware im-plementation that produces these low power requirements isdescribed in [5]. We focus on designing the filters in Fig. 19and subsequently demonstrate the closed-loop performancewith this ASIC and a microgyro.
The AGC is nonlinear, however, the distinct time scalesin the closed-loop system –a fast oscillator frequency anda slower time scale for the modulated amplitude– can beexploited to analyze the closed-loop system response asdemonstrated in [14], [6]. These references analyze thecontinuous-time case shown in Fig. 18 and reveals howthe response of the oscillator depends upon the variouscontroller parameters. In contrast to the thoroughly analyzedcontinuous-time models, the ASIC filters are purely discrete-time and FIR. The discrete-time implementation is pursuedbecause of the flexibility it imparts to the ASIC for adaptingto a wide range of sensor dynamics. The choice of FIRfilters is due to the desire to avoid limit cycling becauseall computations use fixed-point arithmetic. Consequently,rigorous analysis of the closed-loop system consisting of asampled-data sensor model and a controller employing fixed-point implementations of FIR filters is quite complicated andone relies on simulation to identify issues associated withover/underflow and scaling.
The main features of the architecture in Fig. 19 are 1)the seven, 128-tap, fully programmable FIR filters and aproportional-integral section for the automatic gain control,2) fixed-point computation with 18-bit input data precision,18-bit output data precision, 18-bit coefficient precision, and
20-bit internal data precision (24-bits for the integrator), and3) programmable gainsK1 throughK8 that can implement theTin andTout transformations discussed in Section III-A. TheAGC consists of FIR1, the rectifier, FIR2, the proportional-integral (PI) compensation and programmable limiters on theintegrator and AGC gain. The rebalance loop compensationis implemented by FIR3. The ASIC is hosted on a brassboardthat also contains the signal converters and an interfaceto program the ASIC from a personal computer. A highperformance, 24-bit, audio codec provides the analog-to-digital and digital-to-analog conversion. The brassboard alsohosts the antialiasing and smoothing filters, and six 12-bitADCs for gyro biasing.
The decoupled sensor dynamics in a neighborhood of 4420Hz permits a greatly simplified design process because theoff-diagonal terms in Fig. 17 can be ignored. The wide-bandresponse in Fig. 9 demonstrates that large loop gain can beachieved in a neighborhood of 2705Hz and 4420Hz withrelatively modest controller gain. In the sequel, the sensoris designated byP, the control hardware with ASIC filtersis designated byC, and thei j th channel ofP is denotedPi j , with the same convention applying toC. Our designprocess ignores the off-diagonal coupling in bothC andP butthe analysis of the designs in Fig. 23 includes the couplingeffects at least from the point of view of nominal stability.The automatic gain controller is defined by specifying FIR1,FIR2, the reference amplitudeR, and the gainsKP and KI .The plant model for this loop is the(1,1) channel in Fig. 17.
A useful heuristic emerged from [14] that is somewhatindependent of the filter implementation details and that maybe applied here to the design of the AGC filters: the two-time scale behavior of the closed-loop system allows us todesign the AGC filters by assuming the compensator is afixed, linear filter. That is, from the sensor’s perspective, thecompensation can be treated as a fixed filter for any givensensor response because the resonances in the sensor aremuch higher in frequency than the bandwidth of the AGCgain. Here we refer to theAGC gainas the subsystem thatmodulates the feedback signal in Fig. 19 to distinguish itfrom thecompensatorthat is defined as the AGC gain timesFIR1. A typical AGC gain as a function of its input signalamplitude is shown in Fig. 20. This gain modulates the outputof FIR1 to determine the control effort. Using this as a guide,we design FIR1 to adjust the loop gain phase to be zero at thehigher frequency rocking mode so under positive feedbackthis mode is destabilized, but as the amplitude approachesthe reference value, the AGC gain is reduced as seen inFigure 20. Furthermore, the compensation is also designed toattenuate the translational mode near 2.7 kHz. The loop gainfor the AGC channel (P11C11 using our notation) is shownin Figure 21.
The rebalance loop is a linear, high-gain filter implementedby FIR3. One objective of this loop is to achieve disturbancerejection at the 2.7 kHz translational mode. The primary
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Fig. 19. The ASIC employs eight programmable 128-tap FIR filters in addition to the PI compensation for the AGC.
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Fig. 20. The steady state properties of the AGC are shown in two formats.The top plot shows the compensator output amplitude as a function of theinput amplitude when the input is a sinusoid corresponding to the higherfrequency rocking mode (approximately 4428 Hz). Very little excitationamplitude is required to maintain a constant amplitude response for thismode so the sensor’s final amplitude is very near where the graph crosseszero. The bottom plot is theAGC gainand is the top plot normalized bythe input magnitude.
control, however, is to reject the Coriolis-induced disturbanceat the 4423 Hz rocking mode caused by sensor rotation. Asin the AGC design, there are interpolation constraints on thephase of the control filters at these modes in order to achieveoptimal damping of the modes. Figure 22 shows the loop gainP22C22 associated with the rebalance channel.
The ASIC filters were designed without regard to cross-
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Fig. 21. P11C11 using empirical sensor and control hardware data. TheAGC excites the higher frequency rocking mode by proper shaping of theloop phase (not shown). The second rocking mode is not excited becauseof the decoupling achieved in the plant, cf. Figure 17, and the translationalmode at 2.7 kHz is attenuated and not regulated by the AGC.
coupling in the controller. Subsequent tests of the ASIC haveconfirmed that peak magnitudes of the off-diagonal terms areapproximately -80dB. In other words, maxω |Cpq| ≈ −80dB,p 6= q – these results are not shown for sake of brevity.The small off-diagonal magnitudes justifies neglecting theseterms in the design process. The off-diagonal terms in theplant, however, can be more problematic. These terms arethe (1,2) and (2,1) channels in Figure 17. Furthermore,the translation mode near 2.7 kHz couples strongly into allchannels in the sensor. Thus, a more rigorous argument for
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Fig. 22. The magnitude of the rebalance channel,P2C22, is shown. Thischannel regulates both the lower frequency rocking mode (∼ 4423 Hz) andthe translational mode (∼ 2.7 kHz) to zero using linear high gain feedback.
the stability condition of the closed-loop system for differentAGC gains is desired and can be obtained by application ofthe multivariable Nyquist criterion using all channels of plant(sensor) and controller (ASIC plus brassboard hardware)frequency response data. This criterion does not explorethe robustness of the stability condition (namely, destabilizethe higher frequency rocking mode, but dampen the otherrocking mode and translational mode) but answers moredefinitively the nominal stability condition. A Nichols chartof the return difference for various values of the AGC gainis the easiest way to view the results. This chart is shownin Fig. 23 and reveals that the AGC does indeed initiallydestabilize the mode at 4428 Hz when the loop is closedabout a quiescent sensor (AGC measurement signal is muchsmaller than the reference value upon loop closure andcorresponds toδ = 1 in the figure). As the amplitude ofthis mode grows, the AGC gain is reduced (seeδ = 0.1)until the desired reference value for the response amplitude isachieved, at which point the AGC causes the return differenceto pass through the origin at a frequency very near 4428 Hz.
When both AGC and rebalance loops are closed abouta quiescent sensor, the transients in Fig. 24 are recorded.Ideally, the S2 and D2 signals are completely decoupledfrom the AGC signals (S1 and D1). This is largely the case,although some cross coupling does occur in the sensor. TheS1 signal is essentially the isolated rocking mode at 4428 Hzresponding to the AGC feedback. Its initial growth rate iseasily predicted from simple sensor frequency response andAGC parameters. Steady state is reached after 0.05 secondswhere it is noted that the drive amplitude is quite small dueto the very light damping. The bandwidth of the AGC is26 Hz and is tested by applying a small perturbation to thereference value. The advantage of the AGC is revealed inFig. 25 where it is shown how the controller tracks the shifts
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Fig. 23. Yes, this really is a Nichols chart. It represents the loci ofdet(I−P∆C) for ∆ = [δ ,0;0,1], δ = 1,1/10,1/328, whereδ representsa scale of the maximum AGC gain (the rebalance gain is fixed),P isthe decoupled sensor dynamics (Figure 17), andC is the controller. Thefrequency span covers 1 kHz to 6 kHz and the empirical frequency responsedata of the sensor and controller are used to create these plots. The plotscorresponding to the AGC gains1,1/10 imply that the closed-loop systemhas two unstable closed-loop poles. When the AGC gain is reduced to1/328 the return difference determinant passes through the origin indicatinga sustained oscillation is achieved. The frequency markers reveal that thehigher frequency rocking mode is excited as intended (ωA1 = 4427.63 HzandωA2 = 4427.64 Hz.)
Fig. 24. AGC transients when loops are closed about a quiescent sensor.
in modal frequency over a period of 1 hour. Snapshots of thedynamics at different times were shown in Figure 16 yet theAGC is able to follow the trend. This is indeed confirmedby the fact that the AGC control effort magnitude changesby only 5% over the same interval.
The performance of the rebalance loop is quantified bymeasuring the closed-loop sensitivity function for the (2,2)channel. Due to the decoupled plant dynamics in a neighbor-hood ofω1 andω2, and because of the low gain of the AGCloop atω3, we can treat this channel is linear even when the
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AGC is operational. Furthermore the(2,2) channels of theinput and output sensitivity functions are equal (the controlleris diagonal) so only one measurement is made and shown inFig. 26. Since the loop gainP22C22 can be made greaterthan one only in a neighborhood of the rocking mode andtranslational mode, the reduction in sensitivity occurs nearthese frequencies. The primary objective of the rebalancecontrol is to regulate rocking mode at 4423 Hz to zero and thesensitivity function can be used to quantify the closed-looptime constant. In fact, the 50 dB of attenuation at 4423 Hzcorresponds to a closed-loop bandwidth of 20 Hz. It is alsoevident from the sensitivity plot that disturbance rejection of-45 dB is achieved at 2.7 kHz.
Acknowledgment: The authors wish to acknowledge thesupport of the JPL Microdevices Lab, and in particular DeanWiberg, Karl Yee, Kirill Shcheglov, Sam Bae, Brent Blaes,and Tuan Tran and alumni, Tony Tang and Roman Gutierrez.
IV. REFERENCES
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[2] Chen, Y.-C., Hui, J., and M’Closkey, R.T., “Closed-loop Identification of a Micro-sensor,” AcceptedIEEEConference on Decision and Control,Dec. 2003, Maui,HI.
[3] Chen, Y.-C., M’Closkey, R.T., Tran, T., and Blaes,B.,“A control and signal processing integrated circuitfor the JPL-Boeing micromachined gyroscopes,”Sub-mitted. Contact first author for draft.
[4] Fell, C., Hopkin, I., Townsend, K., “A Second Gener-ation Silicon Ring Gyroscope,”Proc. Symposium GyroTechnology, 1999, Stuttgart, Germany.
[5] Grayver, E. and Daneshrad, B., “Word-serial Architec-tures for Filtering and Variable Rate Decimation,”VLSIDesign, Vol. 14(4). pp. 363–72, 2002.
[6] Grayver, E., and M’Closkey, R.T., “Automatic GainControl ASIC for MEMS Gyro Applications,”Proc.2001 American Control Conference, Vol. 2, pp. 1219–22, 2001.
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