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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 1
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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 2
Why use feedback?
> For actuators, how do you
know when you have
actuated?
• You can calibrate/calculate/etc.,
Image removed due to copyright restrictions.
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
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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 3
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 1 on p. 426 in: Liu, C.-H., and T. W. Kenny. "A High-precision,
Wide-bandwidth, Micromachined Tunneling Accelerometer." Journal of
Microelectromechanical Systems 10, no. 3 (September 2001): 425-433.
© 2001 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 10, no. 3 (September 2001): 425-433.
© 2001 IEEE.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 4
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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 5
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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 6
Example: A MEMS hotplate
> Used for gas sensor
Membrane
low-stress SiNx, 2x 500 nm
> Heat up active material, which
reacts with gas and changes
resistance
Heater coil, bond pad
TiN/Ti, 200/10 nm
Wet SiO2
1 mm
Si wafer
> Thermal MEMS is used because
• Low power
• Fast
• Arrayable
0.33 mm
330 µm
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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 7
The Canonical Feedback System
> In controls, terminology refers to the plant, the controller, the
state sensor, and the comparison point
1
Set point
+
_
Error
Sum
K(s)
Control
Actual Output
H(s)
Plant
Controller
Sensed output
1
Output
M(s)
Sensor
Image by MIT OpenCourseWare.
Adapted from Figure 15.1 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2001, p. 397. ISBN: 9780792372462.
Example: micro hotplate
1
Set point
temp
+
_
Sum
Heater resistor
External amp
Error
K(s)
Voltage
Controller
Measured temperature
Temperature
H(s)
1
Output
Plant
M(s)
Sense resistor
Image by MIT OpenCourseWare.
Adapted from Figure 15.1 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2001, p. 397. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 8
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)
Convection
Disturbance 2
1
+
_
Error
Set point
Sensed output
temp
(noisy)
K(s)
Controller
Voltage
M(s)
Sense resistor
+
+
Hotplate
Disturbed resistor
control
H(s)
Actual
output
Plant
+
_
1
Output
Temperature
3
Johnson noise
Image by MIT OpenCourseWare.
Adapted from Figure 15.2 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2001, p. 398. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 9
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 )
X in (s ) 1
Set point
+
_
Error
K ( s ) ⋅[ X in ( s ) − X out ( s )]
K(s)
1
H(s)
1
Control
X out ( s ) = H ( s ) ⋅Feedback
K ( s ) ⋅ [XPath
in ( s ) − X out ( s ) ]
⇓
X out ( s ) =
H ( s) ⋅ K ( s)
X in ( s )
1 + H (s) ⋅ K (s)
Black’s formula
1
Output
X out (s )
Image by MIT OpenCourseWare.
Adapted from Figure 15.3 in Senturia, Stephen D.
Microsystem Design. Boston, MA: Kluwer Academic
Publishers, 2001, p. 399. ISBN: 9780792372462.
X out ( s ) forward gain
=
X in ( s ) 1 − loop gain
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 10
Open-Loop Operation
> Control hot plate via calibration
Remember…
• Assume hotplate has 1st-order response
+
with s0~400 rad/s (f0~65 Hz)
IQ
• Assume controller has no dynamics
As
H ( s) = 0 0
K (s) = K0
s + s0
or drifts in system
Any deviations cause steady-state error
Troom
Tset
Tcal
RT
ΔT
=
=
I Q 1 + RT CT s 1
RT CT
RT CT
+s
+
+
-
RT
-
RT 1
> Works great if there are no disturbances
>
ΔT
CT
K(s)
H(s)
Thotplate
+
Thotplate = Troom + H ( s ) K ( s )(Tset − Tcal )
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 11
Open-Loop Operation
> Response is sensitive to variations in controller,
plant, and disturbances
Troom=23 ºC, K0=1.2
40
A0=1
Tcal=23 ºC
Troom=25 ºC, K0=1
38
Troom=23 ºC, K0=1
36
Temp (C)
34
32
30
28
Tset=37 ºC
26
24
22
0
0.2
0.4
0.6
0.8
1
tim e (s)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 12
Feedback use #1: limit sensitivity to variations
Troom
> Add in term
proportional T
set
to error
> This is called
proportional
control
+
+
K(s)
-
H(s)
Thotplate
+
HK
1
Tset +
Troom
1 + HK
1 + HK
A0 s0
⋅ K0
s + s0
1
=
Tset +
Troom
A0 s0
A0 s0
⋅ K0
⋅ K0
1+
1+
s + s0
s + s0
Thotplate =
Closed-loop TF Thotplate =
A0 K 0 s0
s + s0
Tset +
Troom
s + s0 (1 + A0 K 0 )
s + s0 (1 + A0 K 0 )
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 13
Close the loop
> Error Æ 0 as K0 increases,
despite
• Variations in device (A0)
• Variations in plant (K0)
• Disturbances (Troom)
Open-loop
40
Set-point
38
Temp (C)
36
> In limit of large K0, system
responds “perfectly”
• Though DC error never goes
exactly to zero
42
K0=10
34
32
K0=1
30
28
Heater A0=1.2
Troom=25 ºC
26
24
0
0.05
0.1
tim e (s)
0.15
Tset=37 ºC
Steady-state (DC) Error
ε ( s ) s =0 = Tset − Thotplate =
0.2
1
1
Tset −
Troom
1 + A0 K 0
1 + A0 K 0
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 14
Feedback use #2: increase system bandwidth
42
> Settling time goes down
Thotplate
Tset
A0 K 0 s0
= H cl ( s ) ≈
s + s0 (1 + A0 K 0 )
36
Temp (C)
A0 s0
s + s0
K0=10
34
32
K0=1
30
28
Heater A0=1.2
Troom=25
26
24
0
s0 → s0 (1 + A0 K 0 )
Set-point
38
> Bandwidth goes up
H ( s) =
Open-loop
40
0.05
0.1
tim e (s)
0.15
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 15
0.2
Controlling a 2nd-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 1999): 433-438.
© 1999 IEEE.
X out ( s ) ⎛ 1 ⎞
1
=⎜ ⎟
F (s)
⎝ k ⎠ sˆ 2 + 1 sˆ + 1
Q
mω0
where: sˆ = s , ω0 = k , Q =
m
b
ω0
H ( s) =
•Set Q=1/2 (critically damped)
•Set k=1 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 1999): 433-438.
© 1999 IEEE.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 16
Proportional control of 2nd-order system
> Use ideal controller K(s)=K0
X out ( s )
H ( s) K ( s)
=
=
> This gives us two overall poles:
• Two from SMD H(s)
• None from controller K(s)
X in ( s )
1 + H (s) K (s)
order system
• Q of closed-loop response
increases with increasing DC
gain
2
1.5
K0=100
1
> This means that our critically
damped system is now
underdamped
1
sˆ + ( K 0 + 1)
Q
Qcl = Q K 0 + 1
• Decreasing DC error as K0
> Some differences:
sˆ 2 +
ω0,cl = K 0 + 1
> Some results are same as 1stincreases
• System speeds up
K0
K0=10
0.5
• This can be bad or fatal for our
system
0
0
2
4
6
8
10
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 17
Control of complex systems
> Dynamics of closed-loop system are determined by
H(s)K(s)
> Thus, behavior seen with 2nd-order SMD system will
also occur with 1st-order thermal system coupled to
1st-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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 18
Single-Pole Controller (Real amp)
> Take SMD and control
with
1st-order
controller
> The system now has
three poles
> When going to large K0,
the system goes
unstable
• This happens if one of
the roots has real
positive part
> Routh test can be used
K0
K ( s) =
1 + sˆτˆ
⇓
τˆ = ω0τ
controller
time constant
X out ( s)
K0
= 3
X in ( s ) τˆsˆ + (1 + 2τˆ )sˆ 2 + (2 + τˆ )sˆ + K 0 + 1
Routh test for third - order system :
a3 s 3 + a2 s 2 + a1s + a0
All coefficients ( an ) must have same sign
AND a2 a1 > a0 a3
⇓
to find maximum gain
K0 <
1⎛1
1⎞
⎜⎜ + τˆ + ⎟⎟
Q⎝Q
τˆ ⎠
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 19
2nd-order vs. 3rd-order systems
> Stable system at all loop
> Unstable system at
gains
sufficiently high loop
gain
5
15
^τ = 1
5
Imaginary
Imaginary
10
0
-5
0
-10
-15
-3
-2
-1
Real
0
-5
-6
1
-5 -4 -3 -2 -1
Real
0
1
2
Image by MIT OpenCourseWare.
Adapted from Figure 15.6 in: Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 403. ISBN: 9780792372462.
Image by MIT OpenCourseWare.
Adapted from Figure 15.5 in: Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 402. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 20
Effect of controller bandwidth
> Controller bandwidth < Plant
bandwidth causes controller to
dominate overall response
τ=1
Phase (deg)
Magnitude (dB)
Bode Diagram
τ=100
Bode Diagram
50
0
0
-50
-50
-100
-100
0
-45
-90
-135
-180
-225
-270
10-2
Q=1/2
ω0=1
K0=6
10-1
100
Frequency (rad/sec)
101
Controller
-150
0
-45
-90
-135
-180
-225
-270
2
10
10-4
Plant
10-2
100
Frequency (rad/sec)
102
Overall
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.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechan
ical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 21
Effect of controller bandwidth
> Controller bandwidth > Plant
bandwidth causes plant to
dominate overall response
τ=1
Q=1/2
ω0=1
K0=6
Phase (deg)
Magnitude (dB)
Bode Diagram
Bode Diagram
50
0
0
-50
-50
-100
-150
-100
0
-45
-90
-135
-180
-225
-270
10-2
τ=0.01
10-1
100
Frequency (rad/sec)
101
Controller
102
-200
0
-45
-90
-135
-180
-225
-270
10-2
Plant
100
102
Frequency (rad/sec)
104
Overall
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Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 22
PI control
Proportional - Integral (PI) Control :
> Add pole at s=0
⎛ β⎞
K ( s ) = K 0 ⎜1 + ⎟
s⎠
⎝
> This gives K(0) Æ ∞
• And thus no DC error
> Benefits
• As long as β≠0, will get
perfect DC tracking, but
it may take awhile
1.8
1.6
K0=100, β=0
1.4
1.2
K0=1, β=2
1
• Completely insensitive
0.8
to changes in plant at
DC
0.6
K0=1, β=0.1
K0=1, β=0
0.4
> Drawbacks
• Additional pole means
possibility of ringing
and instability
0.2
0
0
20
40
60
80
100
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
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JV: 2.372J/6.777J Spring 2007, Lecture 17 - 23
PID control
> Final generic term is to
Proportional - Integral - Differential (PID) Control :
⎛ β
⎞
K ( s ) = K 0 ⎜1 + + γs ⎟
s
⎝
⎠
add in differential
feedback
• Anticipate future
10
8
instability due to
integral and
proportional control
> Methods exist to tune
PID controllers
Imaginary Axis
> “Tame” ringing and
6
γ = 0.03
4
β = 2.2
γ=0
2
β = 2.2
0
-2
-4
-6
-8
-10
-1.5
-1
-0.5
Real Axis
0
0.5
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Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 24
Stabilization of unstable systems
Piyabongkarn (2005), IEEE Trans. Control Systems Tech.
> Use of feedback
#3: Stabilize an
unstable system
> Stabilize
Figure 2 on p. 139 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 13, no. 1
(January 2005): 138-145 © 2005 IEEE.
.
electrostatic
actuator beyond
pull-in
> Most approaches
use feedback to
approximate
charge control
Figure 6 on p. 140 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 13, no. 1 (January 2005): 138-145. © 2005 IEEE.
no feedback
Figure 11 on p. 144 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 13, no. 1 (January 2005): 138-145. © 2005 IEEE.
with feedback
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 25
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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 26
Control for MEMS
> Electrostatic traps for cells
Exploded Pixel
> The goal is to trap single cells at
IMD hole
C
Metal 2
Assembled Pixel
each site
PECVD SiO2
Metal 1
> System is currently run open loop
Thermal SiO2
Silicon
> Could we do better if we ran closedloop?
Image by MIT OpenCourseWare.
> Need to sense: optical or electrical
> Need to actuate
A
B
Bead
• This is hard…
Cell
Image by MIT OpenCourseWare.
100 µm
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Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 27
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 2007.
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JV: 2.372J/6.777J Spring 2007, Lecture 17 - 28
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 2007.
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JV: 2.372J/6.777J Spring 2007, Lecture 17 - 29
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
X/X0
⎡F⎤
X out = X 0 tan −1 ⎢ ⎥
⎣ F0 ⎦
π/2
0
−π/2
-20 -15 -10 -5
1
Error
Error
C ontrol
F=K0(Xin-Xout)
C ontrol
Image by MIT OpenCourseWare.
Adapted from Figure 15.11 in Senturia, Stephen
D. Microsystem Design . Boston, MA: Kluwer
Academic Publishers, 2001, p. 412.
ISBN: 9780792372462.
K0
Xin
In1
0 5 10 15 20
F/F0
f(u)
Plant
1
Out1
Xout
Controller
Feedback Path
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 30
Feedback in Nonlinear Systems
> Controller “linearizes” in nonlinear system
> This also occurs in op-amps
1.0
⎡ K ( X − X out ) ⎤
X out = X 0 tan −1 ⎢ 0 in
⎥
F
0
⎣
⎦
0.04
0.03
0.02
0.5
X in − X out
0.0
0
Error
⎡X ⎤
F0
tan ⎢ out ⎥
K0
⎣ X0 ⎦
≈ 0 for K 0 >> F0
X in − X out =
Xout
0.01
-0.01
-0.02
-0.5
-0.03
-1.0
-1
-0.5
0
0.5
1
-0.04
Xin
Imag e by MIT OpenCourseWare
Adapted from Figure 15.12 in: Senturia, Stephen D. Microsystem Design. Boston,MA:
Kluwer Academic Publishers, 2001, p. 413. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 31
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 2007.
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JV: 2.372J/6.777J Spring 2007, Lecture 17 - 32
Example: Resonant RLC Circuit
> While a linear amplifier can theoretically produce an undamped
linear system, it cannot create an oscillator
+ vC -
vs
+
-
C
iL
vo
L
R
+ vC C
iL
+
L
R
-
dvC 1
= iL
dt
C
dvC 1
= iL
dt C
diL 1
= (vS − vC − iL R )
dt L
v
diL 1
= ( v0 − vC − v0 ) = − C
dt L
L
vo
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 33
Example: Resonant RLC Circuit
> No stable limit cycle
> Vary gain A of op-amp circuit
2000
Vc
20
A>1
1000
10
0
0
-1000
-10
-2000
-2000
20
0
-20
-20
2000
20
A=1, Vc0=11.5
10
10
0
0
-10
-10
-20
-20
0
-20
-20
20
IL
A<1
+
0
20
A=1, Vc0=5
0
20
Image by MIT OpenCourseWare.
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JV: 2.372J/6.777J Spring 2007, Lecture 17 - 34
Adding a limiter creates an oscillator
> Add in arctangent
state
eqns.
limiter
⎛v ⎞
vs = V1 tan ⎜ 0 ⎟
⎝ V2 ⎠
−1
dvC 1
= iL
dt C
⎤
⎛ Ai R ⎞
diL 1 ⎡
= ⎢V1 tan −1 ⎜ L ⎟ − vC − iL R ⎥
dt L ⎣
⎝ V2 ⎠
⎦
For
+ Vc+
iL
R
-
A
Amplifier
Limiter
V5
V0
Image by MIT OpenCourseWare.
Adapted from Figure 15.15 in Senturia, Stephen D. Microsystem Design. Boston,
MA: Kluwer Academic Publishers, 2001, p. 416. ISBN: 9780792372462.
AV1
>1
V2
iL R
⎛ Ai R ⎞
V1 tan −1 ⎜ L ⎟
⎝ V2 ⎠
Net gain
Net loss
iL R
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007.
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JV: 2.372J/6.777J Spring 2007, Lecture 17 - 35
Marginal Oscillator
15.0
> Gradual limiting leads to
nearly sinusoidal
waveforms for weakly
damped resonators
iL
5.0
-10.0
-15.0
90
4
10
Damped
92
94
t
96
98
100
Image by MIT OpenCourseWare.
Adapted from Figure 15.17 in: Senturia, Stephen D. Microsystem Design
.
Boston, MA: Kluwer Academic Publishers, 2001, p. 418. ISBN: 9780792372462.
103
102
15.0
101
10.0
100
10-1
10-2
0
0.5
1
1.5
2
Frequency (Hz)
Image by MIT OpenCourseWare.
2.5
3
Capacitor voltage
Power spectral density
0.0
-5.0
105
10-3
Damped
10.0
5.0
0.0
-5.0
-10.0
-15.0
-15
-10
Adapted from Figure 15.18 in: Senturia, Stephen D. Microsystem Design.
Boston,MA: Kluwer Academic Publishers, 2001, p. 419. ISBN: 9780792372462.
-5
0
5
10
15
Inductor current
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JV: 2.372J/6.777J Spring 2007, Lecture 17 - 36
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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 37
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 Æ 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 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 17 - 38