Acknowledgements. Control Design for a MEMS Accelerometer with Tunneling Sensing. Outline. Introduction. Force Feedback. Outline

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1 Acknoledgements Control Design for a MEMS Accelerometer ith Tunneling Sensing K. J. Åström Department of Automatic Control LTH, Lund University Kimberly Turner Group at UCSB (ME) Laura Oropeza-Ramos, Chris Burgner, Zi Yie Barry Demartini, Kari (Lukes) Moran Paul Hansma Group at UCSB (Physics) Johannes Kindt, Georg Schitter Georg Schitter UCSB and Vienna Forrest Breer Group at UCSB (ECE) Nitin Kataria Thank you for introducing me to a fascinating field for control applications Introduction Interesting and useful devices in dynamic development AFM, Accelerometers, Gyroscopes, Hard disks, Optical memories... Small scale Scaling of surface l 2 vs volume l 3 : surface effects important Oscillatory (nonlinear) dynamics ith lo damping Noise: Bronian motion, Johnson-Nyquist, tunneling, pink,... Parameter uncertainty and parameter variations Fast sampling MHz, challenging implementation Control is often mission critical, noise, robustness, dynamics, nonlinearities all have to be balanced Rich area for applying control BUT not standard control problems Force Feedback Classic idea ith tremendous impact Game changer in instrument design Open loop, all components matter Bandidth ω b = k/m Sensitivity = k a /k Invariant ω 2 b S= k a/m Closed loop, actuator only critical element Bandidth and sensitivity given by feedback Error signal also useful! 4. Performance Limits Design a Sensor not a Controller Key idea: Exploit error signal and not just the feedback signal Model of sensor system dt = Ax+ B + Bu x sensor state, signal to be measured u actuation signal. Design instrument to have and u co-located (B = kb). Model for signal to be measured d =, (constant but unknon) dt dt = A z, = C z (general) Characterized by A. Tune sensor to spectrum of acceleration to be measured, for example in automotive applications. System model Control Structure dt = A z, = C z y= Cx Standard controller structure based on Kalman filter and state feedback d ˆx dt = A ˆx+ B C ẑ+ Bu+ L x (y C ˆx) dẑ dt = A ẑ+ L (y C ˆx)= A ẑ+ L (y ŷ) u= K x ˆx K z ẑ. Design instrument to make B C close to B Determine filter gains L and L to give good estimates Determine feedback gains K and K to give small errors 1

2 Transfer function from to ŵ Gŵ = F(s) I+F(s) Instrument Transfer Function F(s)= C (si A ) 1 L (si A L x C) 1 B For A = (constant but unknon or sloly varying acceleration) the expression simplifies to Gŵ = L zc(si A+ L x C) 1 B s+ L z C(sI A+ L x C) 1 B, Gŵ ()=1 Gŵ does not depend on feedback gains K x and K z! Gŵ does not depend on B Gŵ depends on filter gains L x, L z Many design options: Optimize ith respect to disturbances and uncertainty Closed loop dynamics Choosing Feedback Gains dt = Ax+ B B K x ˆx B K z C ŵ =(A B K x )x+(b C B K z )z+ B K x x+ B K z z Physical interpretation Make effect of external signal small by matching B K z to B C (instrument design). The term(b C B K z )z vanishes if B K z = B C Make terms proportional to x and z small by good estimator design Choose K x to balance decay rate (eigenvalues of A B K x ) to disturbance amplification (B K x ) Design gains for robustness Shape the frequency response Gŵ Sensor Resolution Constant Acceleration, Fixed Estimator Gains dt = A z, = C z Augmented state x=(x; z) small abuse of notation A B C A a = B ], B A a =, C a = C, C a = [ C Kalman filter = A a xdt+ B a udt+dv dy= C a xdt+ de R x = Edv dv T, R e = Ede de T A a P+ PA a + R x PC T a R 1 y C ap=, L= Variance of estimate σ 2 ŵ = C apc T a Lx L = PC T a R 1 y dt = A z, = C z Augmented state ith small abuse of notation x=(x; z) A B C A a = B ], B A a =, C a = C, C a = [ C =(A a LC a )xdt+ B a udt+dv dy= C a xdt+ de R x = Edv dv T R e = Ede de T Variances of estimation error given by the Lyapunov equation A a P+ PA a + R x = Variance of estimate σ 2 ŵ = C apc T a The Tunneling Accelerometer 4. Performance Limits Courtesy of Laura Oropeza-Ramon Tunneling Tip To Control Problems Initialization - Move tip to tunneling position Tip starts at initial postion about 1 µm from the surface. No signal is available until tunneling starts about 1 nm= 1 Å from the surface. Controller is blind until tunneling occurs! Tip destroyed if it hits the mass! Steady state operation - Maintain tunneling Measure acceleration Tradeoff beteen bandidth and noise Courtesy of Laura Oropeza-Ramon 2

3 Tunneling Model Block Diagram The tunneling current can be modeled by I=k t V ve α x φ n th n t n R Notice exponential dependence on x! Linearize di = α φ k t V ve α x φ = α φ I=k t I, δ I= k t I e δ x Notice that gain is proportional to current and that it is constant if e regulate the current ell. Typical parameters Tunneling current: I e = 2 na Effective distance beteen tip and sample, x e = 1 Å Tunneling gain: k t I e =.5 na/å=5a/m for I e = 2 na Amplifier gain: k va = V/A, V e =.4 V Sensor gain: k vx = 1 8 V/m=1 mv/å u Actuator F Mass z Tip I Amplifier Actuator: F= Nǫ h d (V + u) 2, δ F=k a δ u, k a = 2 Nǫ hv d Mass: m d2 z dt2+ c dt + kz= F+ m+ n th Tunneling tip: I=k t V ve α x φ, δ I=k t I e δ x+n t, k t = α φ Amplifier: V= k v (RI+ n R ) (simplified) y Preamplifier Noise Sources Notice tunneling current of the order of na! Additional capacitors needed to stabilize the circuit. Opamps also have dynamics. Thermal noise hite noise gives a force on the mass ith spectral density 4ck B T (dissipation fluctuation theorem), c damping coefficient, k B = [J/Kelvin] Boltzmann s constant and T temperature Tunneling noise modeled as shot noise hich is hite noise ith spectral density q 2I, here q = C is the charge of the electron and I is the current. Model resistors by an ideal resistor ith a voltage source in series representing the Johnson-Nyquist noise hich is hite noise ith spectral density 4k B T R Amplifier noise Pink noise (1/ f noise) Sensor Model Constant but unknon acceleration, simplified preamp model = A a xdt+ B a udt+dv, dy= C a xdt+ de, 1 A a = k/m c/m 1, B a= k a /m C y = k s, C = 1 R x = Edv dv T = diag(, 2ck B T/m 2, r ) R y = E(de) 2 = k 2 v(2k B T R+ R 2 q I ). Sensor transfer function l 3 k s Gŵ (s)= s 3 +(k s l 1 + c/m)s 2 +(k s (l 1 c/m+l 2 )+ k/m)+l 3 k s Pick l 1, l 2 and l 3 to shape the transfer function Gŵ (s) Trade-off beteen Bandidth and Variance Gain Phase Sensor Transfer Function ω B/ω α c = 1, ζ c =.5, ω B ω =.1,.1,.1, 1., and 1 Controller Architecture Choose filter gains to shape sensor transfer function Bandidth-variance compromise Design issues 1-2 σ 2 â [m2 /s 4 ] ω B/ω Physical interpretations! 3

4 First Attempt Initialize - Initiate tunneling, get safely from 1 µm to 1 nm Sitched integrating controller Regulate - maintain tunneling Hunt for Noise Sources Improved Experimental Set-up Originally very high noise levels Guide-lines from physical modeling very useful V t Redesign electronics: preamplifier, DAC ith better resolution Replace PC by National Instruments Compact Rio y [V] Improved Electronics Courtesy of Chris Burgner Control Signal has Long Term Drift 1/ f Model material irregularities as small RC systems in thermal equilibrium ith energy logarithmethically distributed a π ϕ(ω)= ω 2 d log a= + a2 2ω u [V] t [s] g / Hz Tunneling starts at t=2.4 Standard devviation σ= 3 mv.3 Å Frequency [Hz] Summary Interesting application area for control Systems ith lo damping Noise Truxal 1961: The design of feedback systems to effect satisfactorily the control of very lightly damped physical systems is perhaps the most basic of the difficult control problems. Thermal, Johnson-Nyquist, tunneling, 1/ f Noise model very important for improving electronics, tunneling current very small a fe na Integrated systems and control design A design frameork Insight and understanding Controller structure Design trade-offs State models are attractive numerically 4

5 References H. Rockstad, T. Kenny, J. Reynolds, W. Kaiser, and T. Gabrielson, A miniature high-sensitivity broad-band accelerometer based on electron tunneling transducer. Sensors and Actuators A, vol. 43, January P. Hartell, F. Bertsch, S. Miller, K. Turner, and N. MacDonald, Single mask lateral tunneling accelerometer. MEMS 98 pp , Nov 1997 T. Gabrielson Mechanical-thermal noise in micromachined acoustic andvibration sensors. IEEE Trans. Electron Devices. vol 4 13, Schitter, G. and Åström, K. J. DeMartini, B. E. and Thurner, P. J., Turner, K. L. and Hansma, P. K., Design and Modeling of a High-Speed AFM-Scanner. IEEE Transactions on Control Systems Technology", 15 (27) L.A. Oropeza-Ramos, N. Kataria, C.B. Burgner, K.J. Åström, F. Breer and K.L. Turner. Noise Analysis of a Tunneling Accelerometer base on state space stochastic theory. Hilton Head 28. Z. Yie, N. Kataria, C. Burgner, K. J. Åström, F. Breer, K. Turner. Control Design for Force Balance Sensors. ACC 29 K. J. Åström. Control and Estimation in Force FeedbackSensors. In Eleftheriou, E., Reza Moheimani, S.O. (Eds.) Control Technologies for Emerging Micro and Nanoscale Systems. Springer 211. Parameters Boltzmann s constant k B J/K Charge of electron q C Tunneling constant α /Å ev Tunneling barrier φ.5 ev Temperature T 293 K Mass m µg Resonant frequency f 4.2 khz Q-value Q 1 Actuator gain k a N/V Tunneling gain k t 4 A/m Preamp resistance R 1.2 MΩ Voltage gain k v 2 Sensor gain k s = k t k v R 21.6 MV/m 5