Feedback. Joel Voldman* Massachusetts Institute of Technology *(with thanks to SDS)

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1 Feedback Joel Voldman* Massachusetts Institute of Technology *(with thanks to SDS) Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7 -

2 Outline > Motivation for using feedback > The uses of (linear) feedback > Feedback on Nonlinear Systems Quasi-static systems Oscillators Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-2

3 Why use feedback? > For actuators, how do you know when you have actuated? You can calibrate/calculate/etc., but what about drifts? > Adding a sensor can tell you where you are > Combining the sensor + actuator with feedback can keep you where you are Image removed due to copyright restrictions. An optical attenuator that uses MEMS actuator Senses optical output Uses feedback to control attenuation Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-3

4 Why use feedback? > For sensors, feedback can be used to enhance sensor response E.g., keep sensitivity constant > Must add an actuator to do this An accelerometer that uses MEMS tunneling sensor Electrostatic actuation Uses feedback to control tunneling current (and thus gap) Figure on p. 426 in: Liu, C.-H., and T. W. Kenny. "A High-precision, Wide-bandwidth, Micromachined Tunneling Accelerometer." Journal of Microelectromechanical Systems, no. 3 (September 2): IEEE. Figure 3a) on p. 426 in: Liu, C.-H., and T. W. Kenny. "A High-precision, Wide-bandwidth, Micromachined Tunneling Accelerometer." Journal of Microelectromechanical Systems, no. 3 (September 2): IEEE. Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-4

5 Feedback in MEMS > Since MEMS is often concerned with making sensors for measurements or actuators to do something, feedback is integral to the subject > Here we will examine some of the basic uses of feedback Limit sensitivity to variations Speed up system Stabilize unstable systems > At the end, we will look at feedback in nonlinear systems, which is useful for making oscillators Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-5

6 Outline > Motivation for using feedback > The uses of (linear) feedback > Feedback on Nonlinear Systems Quasi-static systems Oscillators Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-6

7 Example: A MEMS hotplate > Used for gas sensor > Heat up active material, which reacts with gas and changes resistance > Thermal MEMS is used because Low power Fast 33 µm Arrayable Wet SiO 2 Membrane low-stress SiN x, 2x 5 nm.33 mm mm Heater coil, bond pad TiN/Ti, 2/ nm Si wafer Image by MIT OpenCourseWare. Image removed due to copyright restrictions. Image by MIT OpenCourseWare. Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-7

8 The Canonical Feedback System > In controls, terminology refers to the plant, the controller, the state sensor, and the comparison point + _ Set point Sum Error K(s) Controller Control H(s) Plant Actual Output Output Sensed output M(s) Sensor Image by MIT OpenCourseWare. Adapted from Figure 5. in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2, p ISBN: Example: micro hotplate External amp Heater resistor + _ Set point temp Sum Error K(s) Controller Voltage H(s) Plant Temperature Output Measured temperature Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-8 M(s) Sense resistor Image by MIT OpenCourseWare. Adapted from Figure 5. in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2, p ISBN:

9 Adding Noise and Disturbances > Noise corrupts the sensor output > Disturbances modify the control input to the plant > In some cases, what we want to measure is the disturbance (a feedback-controlled accelerometer) Set point temp Convection Disturbance _ Error K(s) + Voltage Sensed output Controller (noisy) + M(s) _ Disturbed control 3 Hotplate resistor H(s) Plant Actual output Temperature Output Sense resistor Johnson noise Image by MIT OpenCourseWare. Adapted from Figure 5.2 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2, p ISBN: Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-9

10 Linear Feedback: Black s Formula > For a LTI system, we can use Laplace transforms to create an algebraic closed-loop transfer function Assume sensor has (perfect) unity transfer function X in ( s) X out ( s) K( s) [ X ( s) X ( s) ] in out X in (s) + _ Error K(s) Control Set point X out Feedback Path H(s) [ X ( s) X ( )] ( s) = H ( s) K( s) s in out Output X out (s) X out H ( s) K( s) ( s) = X in( s) + H ( s) K( s) Black s formula Image by MIT OpenCourseWare. Adapted from Figure 5.3 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2, p ISBN: X X out in ( s) ( s ) forward gain = loop gain Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7 -

11 Open-Loop Operation > Control hot plate via calibration Assume hotplate has st -order response with s ~4 rad/s (f ~65 Hz) Assume controller has no dynamics A s H ( s) = K() s = K s + s > Works great if there are no disturbances or drifts in system > Any deviations cause steady-state error Remember + I Q C T ΔT R Δ T R T T RC = = T T I Q + RTCTs + s RC T - T R T T room T set T cal + - K(s) H(s) + + T hotplate T = T + H() s K()( s T T ) hotplate room set cal Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7 -

12 Open-Loop Operation > Response is sensitive to variations in controller, plant, and disturbances A = T cal =23 ºC T room =23 ºC, K =.2 T room =25 ºC, K = T room =23 ºC, K = Temp (C) T set =37 ºC time (s) Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-2

13 Feedback use #: limit sensitivity to variations > Add in term proportional to error > This is called proportional control T set Closed-loop TF + - K(s) H(s) HK T = T + T + HK + HK As K hotplate set room T room s+ s = Tset + As As + K + K s+ s s+ s room AKs s+ s T = T + T hotplate set room s+ s( + AK) s+ s( + AK) + + T T hotplate Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-3

14 > Error as K increases, despite Variations in device (A ) Variations in plant (K ) Disturbances (T room ) > In limit of large K, system responds perfectly Though DC error never goes exactly to zero Close the loop Temp (C) Open-loop Set-point K = K = Heater A =.2 T room =25 ºC time (s) T set =37 ºC Steady-state (DC) Error ε () s = T T = T T = s set hotplate set room + AK + AK Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-4

15 Feedback use #2: increase system bandwidth > Settling time goes down > Bandwidth goes up Thotplate AKs = Hcl () s T s + s + A K set H ( s) = A s s + s s + ( A ) s K ( ) Temp (C) Open-loop Set-point K = K = Heater A =.2 T room = time (s) Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-5

16 Controlling a 2 nd -order system > Vibration sensor Really just an z-axis accelerometer > Use feedback to keep gap constant > In this case, control signal measures vibration > Mechanical plant is a SMD Figure 4 on p. 435 in: Bernste in, J., R. Miller, W. Kelley, and P. Ward. "Low-noise MEMS Vibration Sensor for Geophysical Applications." Journal of Microelectromechanical Systems 8, no. 4 (December 999): IEEE. Xout () s H() s = = Fs () k s + s + Q 2 ˆ ˆ where: ˆ s, k mω s = ω =, Q= ω m b Set Q=/2 (critically damped) Set k= for convenience Figure 6 on p. 435 in: Bernste in, J., R. Miller, W. Kelley, and P. Ward. "Low-noise MEMS Vibration Sensor for Geophysical Applications." Journal of Microelectromechanical Systems 8, no. 4 (December 999): IEEE. Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-6

17 Proportional control of 2 nd -order system > Use ideal controller K(s)=K > This gives us two overall poles: Two from SMD H(s) None from controller K(s) > Some results are same as st - order system Decreasing DC error as K increases System speeds up > Some differences: Q of closed-loop response increases with increasing DC gain > This means that our critically damped system is now underdamped This can be bad or fatal for our system Xout () s H() s K() s K = = X () () () in s + H s K s Q ω = K +, cl Q = Q K + cl K = ( ) 2 sˆ + sˆ+ K + K = Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-7

18 Control of complex systems > Dynamics of closed-loop system are determined by H(s)K(s) > Thus, behavior seen with 2 nd -order SMD system will also occur with st -order thermal system coupled to st -order controller > What happens when we add an additional pole? Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-8

19 Single-Pole Controller (Real amp) > Take SMD and control with st -order controller > The system now has three poles > When going to large K, the system goes unstable This happens if one of the roots has real positive part > Routh test can be used to find maximum gain X X out in ( s) ( s) = ˆˆ τs a K + Routh test s K( s) 2 ( + 2 ˆ τ ) sˆ + ( 2 + ˆ τ ) + a All coefficients ( a 3 3 AND 3 < Q n K = + sˆ ˆ τ s a + a a > K s + a a + τˆ + Q a τˆ sˆ + K for third - order system: ) must have same sign 2 3 ˆ τ = ωτ controller time constant Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-9

20 2 nd -order vs. 3 rd -order systems > Stable system at all loop gains > Unstable system at sufficiently high loop gain 5 5 τ ^ = Imaginary 5-5 Imaginary Real Real Image by MIT OpenCourseWare. Adapted from Figure 5.6 in: Senturia, Stephen D. Microsystem Design. Image by MIT OpenCourseWare. Boston, MA: Kluwer Academic Publishers, 2, p. 43. ISBN: Adapted from Figure 5.5 in: Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2, p. 42. ISBN: Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-2

21 Effect of controller bandwidth > Controller bandwidth < Plant bandwidth causes controller to dominate overall response τ= Q=/2 ω = K =6 τ= Magnitude (db) Phase (deg) Bode Diagram Bode Diagram Frequency (rad/sec) Frequency (rad/sec) Controller Plant Overall Image by MIT OpenCourseWare. Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechan. ical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-2

22 Effect of controller bandwidth > Controller bandwidth > Plant bandwidth causes plant to dominate overall response Magnitude (db) Phase (deg) -5 τ= Bode Diagram Q=/2 ω = K =6 τ= Frequency (rad/sec) Frequency (rad/sec) Controller Plant Overall Bode Diagram Image by MIT OpenCourseWare. Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-22

23 PI control > Add pole at s= > This gives K() And thus no DC error > Benefits As long as β, will get perfect DC tracking, but it may take awhile Completely insensitive to changes in plant at DC > Drawbacks Additional pole means possibility of ringing and instability Proportional - Integral (PI) Control : K =, β= K =, β=2 K( s) β = K + s K =, β=. K =, β= Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-23

24 PID control > Final generic term is to add in differential feedback Anticipate future > Tame ringing and instability due to integral and proportional control > Methods exist to tune PID controllers Proportional - Integral- Differential (PID) Control : Imaginary Axis K( s) β = K + + γs s γ =.3 β = 2.2 γ = β = Real Axis Imag e by MIT OpenCourseWare. Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-24

25 Stabilization of unstable systems > Use of feedback #3: Stabilize an unstable system Piyabongkarn (25), IEEE Trans. Control Systems Tech. > Stabilize electrostatic actuator beyond pull-in Figure 2 on p. 39 in: Piyabongkarn, D., Y. Sun, R. Rajamani, A. Sezen, and B. J. Nelson. "Travel Range Extension of a MEMS Electrostatic Microactuator." IEEE Transactions on Control Systems Technology 3, no. (January 25): IEEE.. > Most approaches use feedback to approximate charge control Figure 6 on p. 4 in: Piyabongkarn, D., Y. Sun, R. Rajamani, A. Sezen, and B. J. Nelson. "Travel Range Extension of a MEMS Electrostatic Microactuator." IEEE Transactions on Control Systems Technology 3, no. (January 25): IEEE. no feedback Figure on p. 44 in: Piyabongkarn, D., Y. Sun, R. Rajamani, A. Sezen, and B. J. Nelson. "Travel Range Extension of a MEMS Electrostatic Microactuator." IEEE Transactions on Control Systems Technology 3, no. (January 25): IEEE. with feedback Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-25

26 Stabilization of unstable systems > However: All potentially unstable modes must be both observable and controllable Observable means that the sensor provides state information about the mode Controllable means that the control inputs can modify the mode If a mode has both attributes, it can be stabilized (at least in theory) with feedback > Adding sensors to a system improves observability of modes > Adding actuators improves controllability > This can be generalized from unstable to unwanted Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-26

27 Control for MEMS > Electrostatic traps for cells > The goal is to trap single cells at each site > System is currently run open loop > Could we do better if we ran closedloop? Exploded Pixel IMD hole Metal 2 C Assembled Pixel PECVD SiO 2 Metal Thermal SiO 2 Silicon Image by MIT OpenCourseWare. > Need to sense: optical or electrical > Need to actuate This is hard Bead A B Cell µm Image by MIT OpenCourseWare. Image by MIT OpenCourseWare. Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-27

28 Control for MEMS > Many MEMS devices/systems are run open loop why? > Open loop Does not need additional sensors or actuators» These increase fab complexity, chip size, cost, etc. But is sensitive to perturbations > Closed loop Requires extra complexity More stable performance > If you don t need closed-loop control, don t use it Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-28

29 Outline > Motivation for using feedback > The uses of (linear) feedback > Feedback of Nonlinear Systems Quasi-static systems Oscillators Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-29

30 Feedback in Nonlinear Systems > Can no longer use nice algebraic forms > However, the same idea still holds: The controller pre-distorts the control signal so as to compensate for nonlinearities in the plant π/2 X out = X tan F F X/X π/ F/F 2 Image by MIT OpenCourseWare. X in K Error Control In Error Control Controller F=K (X in -X out ) f(u) Plant Out Adapted from Figure 5. in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2, p. 42. ISBN: X out Feedback Path Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-3

31 Feedback in Nonlinear Systems > Controller linearizes in nonlinear system > This also occurs in op-amps X out tan ( ) K X X in out = X F F X out Xin Xout = tan K X X X for K >> F in out X out Error X in Imag e by MIT OpenCourseWare Adapted from Figure 5.2 in: Senturia, Stephen D. Microsystem Design. Boston,MA: Kluwer Academic Publishers, 2, p. 43. ISBN: Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-3

32 Resonators, Oscillators and Limit Cycles > Resonator: a passive element that exhibits underdamped oscillatory behavior > Oscillator: a resonator plus an active gain element that compensates the resonator losses and results in steady oscillatory behavior > Limiting: a required nonlinearity in either the resonator or gain element > Limit Cycle: stable closed path in state space Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-32

33 Example: Resonant RLC Circuit > While a linear amplifier can theoretically produce an undamped linear system, it cannot create an oscillator v s v C - C i L L R v o + v C - C i L L R + - v o dv dt C = C i L dv dt C = C i L di dt L = L ( v v i R) S C L dil vc = ( v vc v) = dt L L Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-33

34 Example: Resonant RLC Circuit > No stable limit cycle > Vary gain A of op-amp circuit 2 A> 2 A< V c A=, V c = A=, V c = I L Image by MIT OpenCourseWare. Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-34

35 Adding a limiter creates an oscillator > Add in arctangent limiter v s v = V tan V2 state eqns. dv dt C = i C dil Ai R = dt L V L V tan vc ilr 2 L + Vci L R + A - Amplifier Limiter V V 5 AV > For V2 ir L Image by MIT OpenCourseWare. Adapted from Figure 5.5 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2, p. 46. ISBN: V Net gain ir L tan AiLR V 2 Net loss Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-35

36 > Gradual limiting leads to nearly sinusoidal waveforms for weakly damped resonators Power spectral density Frequency (Hz) Marginal Oscillator Image by MIT OpenCourseWare Inductor current Adapted from Figure 5.8 in: Senturia, Stephen D. Microsystem Design. Boston,MA: Kluwer Academic Publishers, 2, p. 49. ISBN: Image by MIT OpenCourseWare. Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-36 i L Capacitor voltage Damped Damped t Image by MIT OpenCourseWare. Adapted from Figure 5.7 in: Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2, p. 48. ISBN:

37 Oscillator Parting Comments > Do not confuse a resonator with an oscillator > The oscillator is the combined result of a resonator with a suitably designed circuit > The oscillator is intrinsically nonlinear > The limit cycle obeys its own dynamics, which can be discovered by analyzing the perturbation of a limit cycle and the time required to recover Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-37

38 Conclusions > When properly designed, feedback can Reduce sensitivity to variations Decrease response time of system Control output with zero DC error Stabilize unstable systems > But it may be too complicated or unnecessary for your MEMS part a systems issue > All elements in the feedback path have poles, and these can cause instabilities > Numerous methods exist to analyze control systems in frequency, time, root-locus, and state-space domains Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 27. JV: 2.372J/6.777J Spring 27, Lecture 7-38