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Simplified Lumped Parameter MEMS Tuning Fork Gyroscope Model

Gabe Spradlin — Tue, 16 Apr 2013 17:52:02 GMT

Gabe Spradlin: /* Modeling and Simulation of the Tuning Fork Gyroscope */

{{Header|:Category:Sensors|:Category:MEMS||Sensors|MEMS}}
[[Category:Sensors]]
[[Category:Modeling]]
[[Category:MEMS]]

==Introduction to Simplified Lumped Parameter MEMS Tuning Fork Gyroscope Model==
[[Image:Generic two axis mass-spring-damper MEMS gyro.png|'''Figure 1: Generic 2-axis mass-spring-damper model'''|thumb|right|250px]]
{{SeeAlso|saMsg=[[Vibratory MEMS Gyroscopes]]<br>[[MEMS Gyro Modeling]]}}
The basics of vibratory MEMS gyro are presented in [[Vibratory MEMS Gyroscopes]]. The standard modeling technique is presented in [[MEMS Gyro Modeling]].

Typical is described concisely in Suhas et all<ref>Mohite et all, Abstract</ref>, quote
Among various MEMS sensors, a rate gyroscope is one of the most complex
sensors from the design point of view. The gyro normally consists of a
proof mass suspended by an elaborate assembly of beams that allow the
system to vibrate in two transverse modes.

In most cases, the FEM analysis becomes prohibitive and one resorts to
equivalent electrical circuit simulations... Here, we present a
simplified lumped parameter model of the tuning fork gyro and show how
easily it can be implemented using a generic tool like Simulink. The
results obtained are compared with those obtained from more elaborate
and intense simulations... The comparison shows that lumped parameter
Simulink model gives equally good results...

The standard proof mass model is shown in Figure 1.

===Main Design Parameters for Vibratory Gyroscopes===
The main design parameters are stiffness of the structure, damping coefficients and system response to applied rate. The stiffness is key to mode matching between the resonant axes and the driving frequency. Achieving the desired performance is an iterative modeling and simulation process.

==Gyroscope Structures==
As described in [[Vibratory MEMS Gyroscopes]], mode matching of the resonance is key to maximizing the signal to noise ratio (SNR). The quality (Q) factor is a measure of how large the resonant amplitude will be; when the sense and drive resonances are equal, the output signal is amplified be the Q factor. The larger the Q factor the better the SNR will be. The gyro structure can have a large impact on the Q factor.

===Structure with in-plane drive and sense modes===
This gyro structure type is symmetric and decoupled having both the drive and sense modes in-plane. In this structure the sensing is done with a pair of comb fingers identical to the fingers used for driving the structure. For more details on this structure and design see Suhas et all<ref>Mohite et all, pg. 759</ref>.

==Modeling and Simulation of the Tuning Fork Gyroscope==
As seen in Figure 1, the suspended proof mass acts as a single degree of freedom (DOF) system for both the drive and sense motions. The lumped parameters of the structure, mass and stiffness, are obtained using [[Vibration frequency by energy method|energy methods]]. The relevant boundary conditions are of the suspended beam model are the anchored end and the free end connected to the proof mass.

The lumped parameters are
{{Eqn|
eqnValue=<math>M_{d}=M_{s}=m_{p}+m_{b}+m_{c}-m_{e}</math>|
number=''|}}
where:
:<math>M_{d} / M_{s}</math> is the mass of the structure,
:<math>m_{p}</math> is the plate mass,
:<math>m_{b}</math> is the beam mass,
:<math>m_{c}</math> is the comb mass, and
:<math>m_{e}</math> is the etch hole mass.
{{Eqn|
eqnValue=<math>K_{d}=\frac{4Et\omega_{b}^{3}}{l_{b}^{3}}</math>|
number=''|}}
where:
:<math>K_{d}</math> is the in plane stiffness.
{{Eqn|
eqnValue=<math>K_{s}=\frac{4Et^{3}\omega_{b}}{l_{b}^{3}}</math>|
number=''|}}
where:
:<math>K_{s}</math> is the out of plane stiffness.
{{Eqn|
eqnValue=<math>B_{d}=\frac{\mu A}{\left( 1+2K_{n} \right)g}</math>|
number=for rarefied flow|}}
where:
:<math>B_{d}</math> is the slide film damping.
{{Eqn|
eqnValue=<math>B_{s}=\frac{3\mu \left( L_{p} \right)^4 n_{h}}{8g^3} \left [ 4ln \left( \eta \right) - 3+\frac{4}{\eta^2}-\frac{1}{\eta^4} \right ] + \frac{8\pi \mu t n_{h} \left( L_{p}^{2} - L_{h}^{2} \right)^2}{L_{p}^{4}}</math>|
number=''|}}
where:
:<math>B_{s}</math> is the squeeze file damping,
:<math>L_{p}</math> is the pitch,
:<math>L_{h}</math> is the edge length of hole, and
:<math>\eta=\frac{L_{p}}{L_{h}}</math>.

The in plane motion and the push-pull driving circuit is governed by the equation
{{Eqn|
eqnValue=<math>M_{d}\ddot{Y}+B_{d}\dot{Y}+K_{d}Y=F_{e}</math>|
number=(1)|}}
where:
:<math>F_{e}</math> is teh electrostatic driving force defined by
{{Eqn|
eqnValue=<math>F_{e}=\frac{2.28n\epsilon t V_{dc} V_{ac} \mbox{ sin} \left( \omega_{d} t \right)}{g_{c}}</math>|
number=''|}}
When a rotation rate Ω<sub>r</sub> is applied about the rotational axis. The Coriolis force causes motion along the sense direction (normal to the plane of excitation). The motion caused by the Coriolis force is
{{Eqn|
eqnValue=<math>M_{s}\ddot{Z}+B_{s}\dot{Z}+K_{s}Z=F_{c}</math>|
number=(2)|}}
where the Coriolis force is
{{Eqn|
eqnValue=<math>F_{c}=2M_{s}\Omega_{r}\dot{Y}\mbox{ sin} \left( \omega_{d} t \right)</math>|
number=''|}}
The (linearized) change in capacitance is
{{Eqn|
eqnValue=<math>\Delta C_{b}=\frac{\epsilon A_{e}}{g^2}Z</math>|
number=(3)|}}
Reduced order models can be developed using the governing equations (Eqn. 1-3). The model can be developed to design and predict the performance in the time and frequency domain.
{{Eqn|
eqnValue=<math>\frac{\Delta C_{b}}{\Omega_{r}}=\frac{\left( G_{1} \bar{G_{2}} G_{3} \right)}{\left[ M_{d}M_{s}s^4 + \left( M_{d}B_{s} + M_{s}B_{d} \right)s^3 + \left( M_{d}K_{s} + M_{s}K_{d} + B_{d}B_{s} \right)s^2 + \left( B_{d}K_{s} + B_{s}K_{d} \right)s + K_{d}K_{s} \right]}</math>|
number=(4)|}}
where:
:<math>G_{1}=2M_{s}</math>,
:<math>\bar{G_{2}}=\frac{2.28n\epsilon t V_{dc} V_{ac}}{g_{c}}</math> and
:<math>G_{3}=\frac{\epsilon A_{e}}{g^2}</math>
{{Eqn|
eqnValue=<math>\frac{\Delta C_{b}}{\Omega_{r}}=\frac{2YQ_{s}}{\omega_{s}}G_{3}</math>|
number=(5)|}}
where:
:<math>Y</math> is the amplitude of the drive mode and
:<math>Q_{s}=\frac{\sqrt{K_{s}M_{s}}}{B_{s}}</math> is the quality factor of the sense mode.

{{mbox|type=notice|text=This is the derivation of the lumped parameter model. When I find example parameters I'll create an example article.}}

==Resources==
*[http://www.iop.org/EJ/article/1742-6596/34/1/125/jpconf6_34_125.pdf?request-id=1eda8adc-59e6-402a-ae26-a5f1b946b8d5 Mohite, S., Patil, N., Pratap, R., "Design, modeling and simulation of vibratory micromachined gyroscopes.", Journal of Physics: Conference Series 34 (2006), pp. 757-763.]
*[http://cds.comsol.com/access/dl/papers/1913/Duwel_pres.pdf Thermoelastic Damping and Engineering for High Q MEMS Resonators]
*Boa, M. 2004 Micro Mechanical Transducers. 2nd. Elsevier. ISBN 044450558X
*Gaura, E., and Newman, R 2006 Smart MEMS and Sensor Systems. 1st. Imperial College Press. ISBN 1860944930

<amazon>044450558X</amazon><amazon>1860944930</amazon>

===Notes===
<references/>

Vibratory MEMS Gyroscopes

Gabe Spradlin — Tue, 16 Apr 2013 17:51:23 GMT

Gabe Spradlin: Created page with "{{Header|:Category:Sensors|:Category:MEMS||Sensors|MEMS}} Category:Sensors Category:Modeling Category:MEMS ==Introduction to Vibratory MEMS Gyroscopes== [[Image:G..."

{{Header|:Category:Sensors|:Category:MEMS||Sensors|MEMS}}
[[Category:Sensors]]
[[Category:Modeling]]
[[Category:MEMS]]

==Introduction to Vibratory MEMS Gyroscopes==
[[Image:Generic two axis mass-spring-damper MEMS gyro.png|'''Figure 1: Generic 2-axis mass-spring-damper model'''|thumb|right|300px]]
{{SeeAlso|saMsg=[[MEMS Gyro Modeling]]<br>[[Simplified Lumped Parameter MEMS Tuning Fork Gyroscope Model]]}}
All MEMS gyroscopes currently use vibrating proof masses. Those masses typically vibrate at a high frequency. As the sensor housing rotates in inertial space a Coriolis force is induced on the proof mass. The Coriolis force causes a vibration in an orthogonal plane and the amplitude of the orthogonal motion can be measured.

==Operating Principle of MEMS vibratory gyros<ref>Boa, pp. 16-19</ref>==
A simplified model of vibratory gyros is shown in Figure 1. The system has 2 orthogonal vibration modes; one mode corresponds to the vibration of the mass in the x-direction, the other the y-direction. The vibration frequency of the x-axis is ω<sub>x</sub> and the equivalent for the y-axis is ω<sub>y</sub>. Typically ω<sub>x</sub> is almost equal to ω<sub>y</sub>.

When powered on, the mass is driven in the x-direction with a driving frequency ω<sub>d</sub> which is close to ω<sub>y</sub>. When the entire MEMS gyro is rotated about the z-axis (out of the plane of the screen), an alternating force in the y-direction is caused by the Coriolis force. The amplitude of this vibration in the y-direction is used as a measure of the angular rate.

All MEMS gyroscopes use the Coriolis effect.
{{Eqn|
eqnValue=<math>F=2M\nu \times \Omega</math>|
number=''|}}
where:
:<math>F</math> is the force of the proof mass,
:<math>M</math> is the proof mass,
:<math>\nu</math> is the velocity of the mass, and
:<math>\Omega</math> is the angular velocity of the reference frame (or sensor housing).

===The role of mechanical resonance in vibratory gyros===
[[Image:Draper Tuning Fork Gyrosope.png|'''Figure 1: Draper Tuning Fork Gyroscopes'''|thumb|right|250px]]
The Coriolis force is typically weak. As a result mechanical resonance is used to amplify the motion and thus keep the signal to noise ratio high over the desired bandwidth. The driving frequency, ω<sub>d</sub>, and the 2 resonance frequencies, ω<sub>x</sub> and ω<sub>y</sub>, must be designed carefully.

==Types of Resonating MEMS Gyroscopes<ref>Sensormag</ref>==
All MEMS gyros require a resonating mass. The most common types of resonating MEMS gyros are
*Tuning Fork Gyros (TFG)
*Hemispherical Resonating Gryo (HRG) or Wine Glass Resonator Gyro
*Vibrating-Wheel Gyros
*Foucault Pendulum Gyros

==Resources==
*[http://www.sensorsmag.com/articles/0203/14/ SensorMag site on the basics of MEMS sensors]
*[http://www.iop.org/EJ/article/1742-6596/34/1/125/jpconf6_34_125.pdf?request-id=1eda8adc-59e6-402a-ae26-a5f1b946b8d5 Mohite, S., Patil, N., Pratap, R., "Design, modeling and simulation of vibratory micromachined gyroscopes.", Journal of Physics: Conference Series 34 (2006), pp. 757-763.]
*[http://cds.comsol.com/access/dl/papers/1913/Duwel_pres.pdf Thermoelastic Damping and Engineering for High Q MEMS Resonators]
*Boa, M. 2004 Micro Mechanical Transducers. 2nd. Elsevier. ISBN 044450558X
*Gaura, E., and Newman, R 2006 Smart MEMS and Sensor Systems. 1st. Imperial College Press. ISBN 1860944930

<amazon>044450558X</amazon><amazon>1860944930</amazon>

===Notes===
<references/>

MEMS Gyro Modeling

Gabe Spradlin — Tue, 16 Apr 2013 17:50:42 GMT

Gabe Spradlin: Created page with "{{Header|:Category:Sensors|:Category:MEMS||Sensors|MEMS}} Category:Sensors Category:Modeling Category:MEMS ==Introduction to MEMS Gyro Modeling== [[Image:Draper T..."

{{Header|:Category:Sensors|:Category:MEMS||Sensors|MEMS}}
[[Category:Sensors]]
[[Category:Modeling]]
[[Category:MEMS]]

==Introduction to MEMS Gyro Modeling==
[[Image:Draper Tuning Fork Gyrosope.png|'''Figure 1: Draper Tuning Fork Gyroscopes'''|thumb|right|250px]]
{{SeeAlso|saMsg=[[Vibratory MEMS Gyroscopes]]<br>[[Simplified Lumped Parameter MEMS Tuning Fork Gyroscope Model]]}}
MEMS gyroscopes rely on two principles. The first principle is the resonating vibration of the proof mass. The second principle is the Coriolis effect.

==Mechanical Resonator<ref>Thermoelastic Damping and Engineering for High Q MEMS Resonators, slides 5, 6</ref>==
The amplitude of a mechanical resonator can be described with the following equation
{{Eqn|
eqnValue=<math>x=x_{0}e^{\lambda t}</math>|
number='''Amplitude of Vibration'''|}}
The properties are as follows
:*<math>\mbox{Im} \left \{ \lambda \right \}</math> is the frequency of vibration,
:*<math>\mbox{Re} \left \{ \lambda \right \}</math> is the time scale over which the amplitude decays due to energy losses, and
:*<math>Q=\frac{W}{\Delta W}=\frac{\mbox{Im} \left \{ \lambda \right \}}{2 \mbox{Re} \left \{ \lambda \right \}}</math> is the quality factor.

There are reasons for designing a MEMS gyro with a high Q
:#higher gain
:#narrow frequency response
:#lower energy loss per cycle
Therefore high Q leads to higher performance MEMS devices.

Modeling of the MEMS gyro principles is key to designing
:*the geometry
:*the chosen materials
:*the resonant frequency
The Q is a result of these properties.

==Thermo-Elastic Damping<ref>Thermoelastic Damping and Engineering for High Q MEMS Resonators, slides 7</ref>==
Anyone who has dealt with MEMS gyros has come across the fact that they are temperature sensitive. One engineer suggested that what you really bought was a thermometer that happened to put out a rate too.

Mechanical engineers are familiar with the idea that a material stiffness changes with temperature. The hotter a metal gets the softer it gets. On a MEMS scale where the mechanism is extremely small. Coupling of stress, strain, and temperature become a means for energy loss. This is referred to as Thermo-elastic Damping (TED).

The reference material provides a detailed set of equations and examples.

==Resources==
*[http://www.iop.org/EJ/article/1742-6596/34/1/125/jpconf6_34_125.pdf?request-id=1eda8adc-59e6-402a-ae26-a5f1b946b8d5 Mohite, S., Patil, N., Pratap, R., "Design, modeling and simulation of vibratory micromachined gyroscopes.", Journal of Physics: Conference Series 34 (2006), pp. 757-763.]
*[http://cds.comsol.com/access/dl/papers/1913/Duwel_pres.pdf Thermoelastic Damping and Engineering for High Q MEMS Resonators]
*Boa, M. 2004 Micro Mechanical Transducers. 2nd. Elsevier. ISBN 044450558X
*Gaura, E., and Newman, R 2006 Smart MEMS and Sensor Systems. 1st. Imperial College Press. ISBN 1860944930

<amazon>044450558X</amazon><amazon>1860944930</amazon>

===Notes===
<references/>

IST MotionMaster -- EDR-6DOF-10-1200

Gabe Spradlin — Tue, 16 Apr 2013 17:47:56 GMT

Gabe Spradlin: Created page with "{{Header|:Category:Sensor Modeling|Sensor Fusion Example||Sensors}} Category:Sensors Category:Sensor Modeling Category:Modeling {{IST1}} {{MotionMaster2}} ==Mo..."

{{Header|:Category:Sensor Modeling|Sensor Fusion Example||Sensors}}
[[Category:Sensors]]
[[Category:Sensor Modeling]]
[[Category:Modeling]]

{{IST1}}

{{MotionMaster2}}

==MotionMaster -- EDR-6DOF-10-1200==

===Performance Data===

<div align="center">
{| style="background:transparent; color:black; font-size:120%; width:75%" align="center" valign="top" border="5"
|+ '''Table 1: MotionMaster -- EDR-6DOF-10-1200 Performance Specifications'''
! Rate BW (Hz) !! Acceleration BW (Hz) !! Rate Saturation (deg/sec) !! Acceleration Saturation
|- style="color: #C00000;" align="center" valign="top"
|
40
|
0 to 2000
|
1200
|
10
|
|}
</div>
[[Image:SensorTF_BW2000.png|'''Accelerometer Sensor with BW of 0 to 2000 Hz'''|thumb|right|375px]]
[[Image:SensorTF_BW40.png|'''Rate Sensor with BW of 40 Hz'''|thumb|left|375px]]

===Environmental Data===

<div align="center">
{| style="background:transparent; color:black; font-size:120%; width:75%" align="center" valign="top" border="5"
|+ '''Table 2: MotionMaster -- EDR-6DOF-10-1200 Environmental Specifications'''
! Rated Angular Rates (deg/sec) !! Rated Linear Acceleration (+/-g) !! Linear Acceleration Technology !! Operating Temperature (F)
|- style="color: #C00000;" align="center" valign="top"
|
1200.00
|
10.00
|
Piezoelectric, Piezoresistive
|
-40 to 158
|
|}
</div>

===Data for Sensors similar to the MotionMaster -- EDR-6DOF-10-1200===

<div align="center">
{| style="background:transparent; color:black; font-size:120%; width:75%" align="center" valign="top" border="5"
|+ '''Table 3: MotionMaster -- EDR-6DOF-10-1200 - Similar Sensors'''
! Product Name !! Rated Angular Rates (deg/sec) !! Rated Linear Acceleration (+/-g) !! Rate BW (Hz) !! Acceleration BW (Hz) !! Maximum Dimension (in) !! Weight (lb)
|- style="color: #C00000;" align="center" valign="top"
|
MotionMaster -- EDR-6DOF-10-1200
|
1200.00
|
10.00
|
40.00
|
0 to 2000
|
4.40
|
2.50
|
|- style="color: #C00000;" align="center" valign="top"
|
MotionMaster -- EDR-6DOF-10-300
|
300.00
|
10.00
|
40.00
|
0 to 2000
|
4.40
|
2.50
|
|- style="color: #C00000;" align="center" valign="top"
|
MotionMaster -- EDR-6DOF-100-1200
|
1200.00
|
100.00
|
40.00
|
0 to 2400
|
4.40
|
2.50
|
|- style="color: #C00000;" align="center" valign="top"
|
MotionMaster -- EDR-6DOF-100-300
|
300.00
|
100.00
|
40.00
|
0 to 2400
|
4.40
|
2.50
|
|- style="color: #C00000;" align="center" valign="top"
|
MotionMaster -- EDR-6DOF-10-1200
|
1200.00
|
10.00
|
40.00
|
0 to 2000
|
4.40
|
2.50
|
|- style="color: #C00000;" align="center" valign="top"
|
MotionMaster -- EDR-6DOF-100-150
|
150.00
|
100.00
|
40.00
|
0 to 2400
|
4.40
|
2.50
|
|- style="color: #C00000;" align="center" valign="top"
|
MotionMaster -- EDR-6DOF-100-600
|
600.00
|
100.00
|
40.00
|
0 to 2400
|
4.40
|
2.50
|
|- style="color: #C00000;" align="center" valign="top"
|
MotionMaster -- EDR-6DOF-10-150
|
150.00
|
10.00
|
40.00
|
0 to 2000
|
4.40
|
2.50
|
|- style="color: #C00000;" align="center" valign="top"
|
MotionMaster -- EDR-6DOF-10-600
|
600.00
|
10.00
|
40.00
|
0 to 2000
|
4.40
|
2.50
|
|}
</div>

{{Generic_Sensor_Model2}}

{{Sensor_Disclaimer1}}

Gain Margin Rule of Thumb

Gabe Spradlin — Tue, 16 Apr 2013 17:44:18 GMT

Gabe Spradlin: Created page with "{{Header|Gain Margin|Block Diagram}} Category:SISO Category:Classical Control Category:Rules of Thumb ==Gain Margin Rule of Thumb== The gain margin is one of the..."

{{Header|Gain Margin|Block Diagram}}
[[Category:SISO]]
[[Category:Classical Control]]
[[Category:Rules of Thumb]]

==Gain Margin Rule of Thumb==
The gain margin is one of the robustness measures of a closed loop system under control. A low gain margin can cause instability. Specifically, no measurement is perfect and therfore no model is perfect. When creating a [[LTI]] model from measured data there are always gain variations that are ignored as inconsequential. However, the lower the gain margin the more consequential those gain variations become.

The gain margin can be determined from a:
:*[[Bode Plot]] of the open loop transfer function - magnitude at the frequency where the phase crosses -180 deg
:*[[Nichols Plot]] of the open loop transfer function - magnitude at the points where the phase crosses -180 dB

===Minimum Recommended Gain Margin===
A good rule of thumb for the gain margin is

<div align="center">
{| style="background:transparent; color:black; font-size:120%; width:75%" align="center" valign="top" border="5"
|+ '''Table 1: Minimum Phase Margin Recommendations'''
! Application !! Minimum Phase Margin (20*log10 dB)
|- style="color: #C00000;" align="center" valign="top"
|
Tracking
|
6
|
|}
</div>

Please feel free to add the recommended minimum phase margin for the applications you are familiar with.

Gain Margin

Gabe Spradlin — Tue, 16 Apr 2013 17:43:43 GMT

Gabe Spradlin: Created page with "{{Header}} Category:Classical Control Category:SISO ==Introduction to Gain Margin<ref>Franklin, pg. 375</ref>== The gain margin (GM) is the factor by which the gain ..."

{{Header}}
[[Category:Classical Control]]
[[Category:SISO]]

==Introduction to Gain Margin<ref>Franklin, pg. 375</ref>==
The gain margin (GM) is the factor by which the gain is less than the neutral stability value. For the typical case it can be read directly from the open loop Bode plot.

The gain margin is is the factor by which the controller gain can be changed before the system goes unstable.

==Determining the Gain Margin==
The gain margin is determined from the open loop Bode plot by finding the frequency ω where
{{Eqn|
eqnValue=<math>\angle G\left(j\omega \right) = -180^\circ</math>|
number=''|}}
At ω the magnitude is the gain margin.

The system is unstable when
{{Eqn|
eqnValue=<math>\left | GM \right | < 1</math> - '''not''' in dB|
number=''|}}

==Gain Margin Guideline==
A good gain margin is > 9 dB.

==References==
*Franklin, G. F., Emami-Naeini, A., and Powell, J. D. 1993 Feedback Control of Dynamic Systems. 3rd. Addison-Wesley Longman Publishing Co., Inc. ISBN 0201527472

<amazon>0201527472</amazon>
===Notes===
<references/>

GNU Octave

Gabe Spradlin — Tue, 16 Apr 2013 17:42:08 GMT

Gabe Spradlin: Created page with "{{Header|:Category:Software|||Software}} Category:Software ==GNU Octave== ===Product Description=== According to the [http://www.gnu.org/software/octave/ GNU Octave] web..."

{{Header|:Category:Software|||Software}}
[[Category:Software]]

==GNU Octave==

===Product Description===
According to the [http://www.gnu.org/software/octave/ GNU Octave] website,
:GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with Matlab.

===License===
GNU Octave is free, open-source software, and is issued with a GNU General Public Licence ([http://www.gnu.org/copyleft/gpl.html GPL]).

===Platforms===
The GNU Octave software is available as source code or as binaries for the following operating systems:
*Microsoft Windows
*Linux (Packages are available for popular distributions such as Debian/Ubuntu, SuSE and Fedora/Red Hat)
*Mac OSX
*Sun Solaris
*OS/2

==Overview==

[[Image:GNU_Octave_CmdLine.png]]

GNU Octave can be extended using packages from the [http://octave.sourceforge.net/ Octave-Forge] repository.

[[Category:Needs Work]]

PSD

Gabe Spradlin — Tue, 16 Apr 2013 17:40:51 GMT

Gabe Spradlin: Created page with "{{Header|RMS|psdData Class Example|}} Category:Modeling Category:Stochastic Controls ==Introduction to PSDs in Controls== In reality many disturbances and noise sour..."

{{Header|RMS|psdData Class Example|}}
[[Category:Modeling]]
[[Category:Stochastic Controls]]

==Introduction to PSDs in Controls==
In reality many disturbances and noise sources are not well represented by Gaussian distributions. Instead the system will have random and periodic disturbances and noise. This type of behavior cannot be described with a probability distribution (such as Gaussian) but rather are better described by a Power Spectral Density (PSD).

For example, satellites often employ momentum wheels for attitude control. These momentum wheels spin at a constant speed for long periods of time and then they are sped up or slowed down to steer the spacecraft. This raises 2 issues. First, the disturbance is not truly random. Second, a measured PSD of the disturbance due to these wheels will show a spike at one wheel speed and then at another. Typically, the PSD is enveloped in these circumstances (this will be discussed below). There is, however, a random component to the disturbance created by the moementum wheels. Remember that they have their own controllers and sensors and random noise on the sensor will pass to the controller. The random noise will go through the controller (and plant) causing random fluctuations in the wheel speed. While these fluctuations are small they are not always inconsequential.

Note: While I have not taken a course on Stochastic Controls I believe that this falls into that category of control theory.

==PSD Definition (para-phrased from Wikipedia)==
The PSD is a positive real function of a frequency variable associated with a [[Stationary|stationary]] [[Stochastic Process|stochastic process]], or a deterministic function of time, which has dimensions of power per Hz. For example, many rate sensors will provide an output in Volts which can be converted to rad/sec. A PSD of the ambient noise from this rate sensor will have units of (rad/sec)^2 / Hz.

===Mathematical Definition of the PSD (heavily lifted from Wikipedia and needs rewrite and references)===
The PSD describes how the power of the signal or time series is distributed with frequency. Here power can be the actual physical power, or more often, for convenience with abstract signals, can be defined as the squared value of the signal (the actual power if the signal was a voltage applied to a 1-Ohm load). This instantaneous power (the mean or expected value of which is the average power) is then given by:
{{Eqn|
eqnValue=<math>P=s\left( t \right) ^{2}</math>|
number=''|}}
Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case. Fortunately, the Wiener-Khinchin therorem provides a simple alternative. The PSD is the Fourier transform of the autocorrelation function of the signal if the signal can be treated as a wide-sense stationary random process. The result is
{{Eqn|
eqnValue=<math>S\left( f \right)=\int_{-\infty}^{\infty} R\left( \tau \right) e^{-2 \pi i f \tau} d\tau</math>|
number=''|}}
The power of the signal in a given frequency band can be calculated by integrating over positive and negative frequencies,
{{Eqn|
eqnValue=<math>P=\int_{F_1}^{F_2} S \left( f \right) df + \int_{-F_1}^{-F_2} S \left( f \right) df</math>|
number=''|}}
The power spectral density of a signal exists if and only if the signal is a wide-sense stationary process. If the signal is not stationary, then hte autocorrelation function must be a function of two variables, so no PSD exists.
{{Eqn|
eqnValue=<math>G \left( f \right) = \int_{-\infty}^{f} S \left( f^{'} \right) df^{'}</math>|
number=Definition|}}

==Using PSDs in Controls==
PSDs are used in many places in controls. For time-domain simulation and performance analysis a PSD derived from measured sensor noise data can be turned into a time series of noise values at specifics times. When done correctly the PSD of these time series are close or identical to the original measured sensor noise PSD. Disturbances can be dealt with in a similar manner.

In frequency domain analyses and moedling the key equation is
{{Eqn|
eqnValue=<math>S_{YY}\left( \omega \right)=G^{*}\left( \omega \right)G\left( \omega \right)S_{XX}</math><ref>Sun, pp. ??-??</ref>|
number=''|}}
Various texts will make different claims about the input PSD. Some state the input PSD must be both [[Stationary|stationary]] and [[Ergodic|ergodic]] while others claim that it only needs to be weakly stationary. In practice the PSD is often used whenever it is consistent from measurement to measurement. In other words, if the magnitude of the PSD and [[RMS]] remain approximately constant from hour to hour or day to day then the PSD is good enough.

An example of how this is used, with MATLAB code, can be found in the article on [[Frequency Domain Modeling]].

===How PSDs are used in Controls===
PSDs are used in a variety of ways:
#Disturbance Modeling
#Noise Modeling
#Residual Error Characteristics

All of these applications are generically part of Frequency Domain Modeling. The basic purpose of frequency domain modeling is to determine the steady-state residual error due to disturbances and noise.

===PSDs and RMS===
Measured (and calculated) PSDs are basically a vector of magnitudes at specific frequencies. Obviously, dealing with this and discussing it are cumbersome. The [[RMS]] is a common metric. The RMS (or Root Mean Squared value) is essentially an integration under the PSD curve. This cumulative value allows a quicker, simpler discussion of total disturbance power and requirements. Like all metrics the RMS is not an all encompassing value in that a particular PSD may have a lot of RMS contribution from the lower frequencies where rejection is good. Therefore the disturbance RMS will be high and the residual (what's left over after rejection) is low but the system still may not meet performance specifications because the remaining RMS in the mid to high frequency ranges is unacceptable.

As an example consider a Hubble image. It may require that Hubble stares at the same object for minutes or even hours. Staring this long means that the camera is collecting photons for a long time but the final image will still likely be made up of several individual images taken over time. Low frequency drift can be subtracted out while mid to high frequency jitter will cause the image to blur. The control system's residual error RMS may be low due to very good low frequency rejection but the overall system performance (blurry images) is unacceptable.

====Lessons Learned====
Many times measured data of all types come with irregularities or oddities. Measured time series and PSD data is no different. Here are some lessons learned:
*Some analysis programs will take time series data and spit out a PSD. The minimum frequency of the PSD is often times automatically set to 0. Obviously, with measured data there is only a finite data length. With that finite data capture time frame comes a minimum measurable frequency (1 / time length of data capture). The result is that the low end frequencies of the PSD must be used with caution.
*Low frequency samples for PSD measurements take a long time to capture. As a result most PSDs have very few data points at lower frequencies. This leads to a coarseness in the measurements that can create errors in the [[RMS]] calculation. Specifically, when calculating the RMS (either forward or backward) the first point of the RMS is at the second frequency of the PSD. If the data is coarse at the low end then the method of calculation can lead to errors in the RMS calculation - especially when the lowest frequency point is at 0. However, interpolation can solve this problem.

====PSD Interpolation in MATLAB====
{{LogInterp}}

==References==
Sun, Jian-Qiao (2006). ''Stochastic Dynamics and Control, Volume 4''. Amsterdam: Elsevier Science. ISBN 0444522301.

<amazon>0444522301</amazon>
===Notes===
<references/>

PsdData Class Example

Gabe Spradlin — Tue, 16 Apr 2013 17:37:20 GMT

Gabe Spradlin: /* Download psdData Class */

{{Header|psdData Methods|psdData Properties}}

==Introduction to psdData Class Example==
{{Intro to psdData}}

==Example from PSD Data==
In controls for pointing and tracking, the customer's performance requirements often state that the system must "perform such that the residual error (RMS) is no greater than X assuming a disturbance spectrum as seen in Figure Y and Table Z." Let's assume that Table Z is a table of frequencies (in Hz) and magnitudes (in rad) representing an angular position disturbance envelope.

For the purposes of our example let's create some fake angular position PSD data.
<div align="center">
{| style="background:transparent; color:black; font-size:120%; width:75%" align="center" valign="top" border="5"
|+ ''Table 1: Strawman Angular Position Disturbance Requirement''
! Frequency (Hz) !! Magnitude (rad^2 / Hz)
|- style="color: #C00000;" align="left" valign="top"
|
0
|
1
|
|- style="color: #C00000;" align="left" valign="top"
|
1
|
1
|
|- style="color: #C00000;" align="left" valign="top"
|
10
|
1E-3
|
|- style="color: #C00000;" align="left" valign="top"
|
100
|
1E-9
|
|- style="color: #C00000;" align="left" valign="top"
|
1000
|
1E-9
|}
</div>
{{mbox-side|msg=Typically, when a PSD is referred to as a Strawman PSD it is a notional PSD which is meant to be representative of the system's final disturbance PSD envelope. The Strawman is used as a substitute for the actual disturbance PSD envelope in early stages of the system design because the whole system is immature and therefore creating an analysis intensive disturbance PSD is a waste of time - the PSD will change, substantially, several times by the end of the design process.}}
The resulting MATLAB Code is
>> freq = [0 ; 1 ; 10 ; 100 ; 1000];
>> mag = [1 ; 1 ; 1E-3 ; 1E-9 ; 1E-9];

Example Instantiation:
>> a = psdData('Strawman PSD Data', freq, mag, 'psd', [], 'interpolate', 5000);

Since the data is in ''rad'' and the default for the psdData Class is ''rad/sec'' we need to change the units
>> setUnits(a, 'Magnitude', 'rad', 'TimeUnit', 'Position')
{{mbox-side|msg=The Magnitude units can be set using 'magnitude', 'mag', 'base magnitude', or 'base mag'. Any combination uppercase and lowercase is fine.}}
{{mbox-side|msg=The Time units can be set using 'time', 'time unit', 'timeunit', 'timescale', or 'time scale'. Any combination uppercase and lowercase is fine.}}
The command above set the units for the magnitude to ''rad'' (radians).

===Plot===
[[Image:strawman_plot.png|'''Strawman PSD Data'''|thumb|right|200px]]
Plot Command:
>> plot([], a, 'rms legend', 'southwest')

For more detailed information on the plotting functions see the [[psdData Methods#psdData Methods: plot & plotTime|plot method]].

===Differentiate===
[[Image:strawman_diff_plot.png|'''Strawman PSD Differentiated'''|thumb|right|200px]]
Differentiate Command:
>> b = differentiate(a);

Plot Command:
>> plot([], b, 'rms legend', 'southwest')

Note that the Name property of the psdData class changes when integrate or differentiate are used to reflect the operation. The units are automatically updated as well. A more meaningful name might be
>> b.Name = 'Strawman Rate PSD Data';

==Example from Time Series Data==
[[Image:example_time_plotTime.png|'''Example Time Series Data'''|thumb|right|200px]]
[[Image:example_trim_plotTime.png|'''Time Series Data (Trimmed)'''|thumb|right|200px]]
[[Image:example_trim_plot.png|'''PSD of Time Series Data (Trimmed)'''|thumb|right|200px]]
Example Instantiation:
>> a = psdData('Example Time Series Data', time, data, 'time', window, 'interpolate', 5000);
where '''window''' determines the size of the window used in creating the PSD and can be any number of seconds up the maximum time of the vector '''time'''.

However, note that that the original data had a transient at the very begining. This is likely an artifact of the measurement process and a it would be more representative to create the PSD from the portion of the Time Series data that came after this transient had passed. It appears that the transient is over by second 2 so create a new psdData object like this
>> ind = find(time >= 2);
>> newTime = time(ind) - min(time(ind));
>> b = psdData('Trimmed Time Series Data', ...
newTime, ...
data(ind), ...
'time', ...
window, ...
'interpolate', 5000);
The variable ''ind'' captures all indices in the ''time'' vector that are greater than or equal to 2. However, for the PSD to be calculated correctly the ''time'' vector needs to start at 0. So the ''newTime'' variable was created to hold a shifted ''time'' vector. Now the psdData object will construct a PSD from the trimmed Time Series data set which is uncorrupted by the startup transient.

===Integrate===
Let's assume that this measured data is rate data in ''rad/sec'' - the psdData Class default. For our imaginary application we need a disturbance in angular position instead of rate. Therefore we need to ''integrate'' the PSD so that it represents a position disturbance. With the psdData class the code is simple
>> positionPSD = integrate(b);
>> positionPSD.Name = 'Angular Position Disturbance PSD';
The integrate method will automatically generate a name for the integrated PSD which, in this example, would be of the form ''Integrated Example Time Series Data''. However, I chose to rename the new PSD to give it a more meaningful name. This is important mainly in that the psdData Class name is used when the data is plotted.

Also note that the resulting psdData Class object of the integrate method (as well as the overloaded plus & minus and differentiate methods) will no longer have any time series data. There are several reasons for this but the main one is that direct manipulation of test data without any consideration of its source is problematic at best.

==Download psdData Class==
[[psdData Class|Download Here]]
{{TODO|Add file}}

==Notes==
{{psdData Notes}}

PSD enveloping

Gabe Spradlin — Tue, 16 Apr 2013 17:36:34 GMT

Gabe Spradlin: Created page with "{{Header|PSD|psdData Class Example}} Category:Stochastic Controls ==Introduction to PSD Enveloping== In stochastic controls disturbance..."

{{Header|PSD|psdData Class Example}}
[[Category:Stochastic Controls]]

==Introduction to PSD Enveloping==
In [[:Category:Stochastic Controls|stochastic controls]] disturbances and noises are described best with a PSD. At times the disturbances would best be described by a single spike at a single frequency. However, that single frequency may be any frequency within a given range of frequencies. This means that the control system will need to be capable of rejecting this spike disturbance at any of these frequencies. As a result the disturbance PSD is often enveloped. Specifically, this means that the PSD in this range of frequencies is defined as having the magnitude of the largest PSD spike but over the full range of frequencies instead of a single frequency.

===Example of PSD Enveloping===
In many instances, such as satellite attitude control, the disturbances are due to internal systems and the disturbance is deterministic. For disturbances caused by hardware such as momentum wheels the disturbance is consistent and at a relatively constant frequency.

A momentum wheel is a flywheel which stores angular momentum in a spinning mass. As the wheel is spun up or slowed down the angular momentum of the wheel changes. The net angular momentum must remain 0 so this change in the wheel's angular momentum is transferred to the spacecraft. This torques the spacecraft allowing for a change in attitude.

For long periods of time the momentum wheel can remain at the same speed leading to a disturbance at a specific level and specific frequency. Thus the control system needs to be able to reject this disturbance spike. However, when an attitude change is commanded the wheel speed will change and so will the magnitude and frequency of the disturbance.

Since the disturbance can be at any frequency (within a designed frequency range) the attitude control system must be capable of rejecting the largest disturbance the wheels can create at any speed (frequency) that the wheel's can operate at. Thus hte disturbance PSD is defined at the maximum magnitude over the entire operating range of the momentum wheels.

==Drawbacks to the PSD Enveloping Approach==
The wheels can induce a disturbance to any speed in its operating range so the control system must be capable of rejecting a disturbance at any of these frequencies. However, when the PSD is enveloped over the entire frequency range this often becomes a requirement that the control system is capable of rejecting disturbances at the maximum magnitude at all frequencies in the range all at once. This is, obviously, a much harder task than rejecting a disturbance at the maximum level at a single, narrow frequency.

This is known as an overly conservative assumption. The real problem that PSD enveloping can create is that it can force the design to be much more expensive in order to meet the requirement that the system be capable of rejecting simutaneous disturbances at all frequencies in the enveloped range. Any disturbance at a single frequency may be a simple matter to reject. However, being capable of rejecting disturbance at all frequencies at once may require a much larger actuator. That larger actuator can cause other issues which ripple throughout the system design.

Hysteretic Damping

Gabe Spradlin — Tue, 16 Apr 2013 17:19:17 GMT

Gabe Spradlin: /* Energy dissipated by Hysteretic DampingBeards, pp. 43-45 */

{{Header|Viscous Damping|:Category:Single DOF|||Single DOF}}
[[Category:Modeling]]
[[Category:Single DOF]]

==Introduction to Hysteretic Damping<ref>Beards, pp. 41-43</ref>==
Experiments on the damping that occurs in solid materials and structures which have been subjected to cyclic stressing have shown the damping force to be independent of frequency. This internal, or material, damping is referred to as hysteretic damping.

The viscous damping force <math>c\dot{x}</math> is dependent on the frequency of oscillation. Hysteretic damping is not dependent on frequency so <math>c\dot{x}</math> is not an adequate model. Hysteretic damping requires the damping force <math>c\dot{x}</math> to be divided by the frequency of oscillation <math>\omega_{n}</math>.

==Hysteretic Damping: Equation of Motion==
The equation of motion is therefore
{{Eqn|
eqnValue=<math>m\ddot{x}+\left(\frac{c}{\omega_n}\right)\dot{x}+kx=0</math>|
number=''|}}

Structures under harmonic forcing experience stress that leads the strain by a constant angle, <math>\alpha</math>. For a harmonic strain, <math>\epsilon=\epsilon_{0}\mbox{ sin }\nu t</math>, where ν is the forcing frequency. The induced stress is
{{Eqn|
eqnValue=<math>\sigma=\sigma_{0}\mbox{ sin}\left(\nu t+\alpha \right)</math>|
number=''|}}
Hence
{{Eqn|
eqnValue=<math>\begin{alignat}{2}\sigma & =\sigma_{0} \mbox{ cos} \left(\alpha\right) \mbox{ sin}\left(\nu t\right)+\sigma_{0} \mbox{ sin} \left(\alpha\right) \mbox{ cos}\left(\nu t\right) \\ & =\sigma_{0} \mbox{ cos} \left(\alpha \right) \mbox{ sin}\left(\nu t\right)+\sigma_{0} \mbox{ sin} \left(\alpha\right) \mbox{ sin}\left(\nu t+\frac{\pi}{2}\right) \end{alignat}</math>|
number=''|}}
The first component of stress is in phase with the strain <math>\epsilon</math>; the second component is in quadrature with <math>\epsilon</math> and <math>\frac{\pi}{2}</math> ahead. Replacing <math>\frac{\pi}{2}</math> with <math>j=\sqrt{-1}</math> leads to,
{{Eqn|
eqnValue=<math>\sigma=\sigma_{0} \mbox{ cos}\left(\alpha\right) \mbox{ sin}\left(\nu t \right)+j\sigma_{0}\mbox{ sin}\left( \alpha \right) \mbox{ sin}\left(\nu t \right)</math>|
number=''|}}

===Hysteretic Damping: Loss Factor===
A complex modulus E<sup>*</sup> is formulated, where
{{Eqn|
eqnValue=<math>\begin{alignat}{2}E^{*} & =\frac{\sigma}{\epsilon}=\frac{\sigma_{0}}{e_{0}} \mbox{ cos} \left(\alpha \right)+j\frac{\sigma_{0}}{\epsilon_{0}} \mbox{ sin}\left(\alpha \right) \\ & = E^{'}+jE^{''} \end{alignat}</math>|
number=''|}}
where
:<math>E^{'}</math> is the in-phase or storage modulus, and
:<math>E^{''}</math> is the quadrature or loss modulus.

The loss factor <math>\eta</math>, which is a measure of the hysteretic damping in a structure, is equal to <math>\frac{E^{''}}{E^{'}}</math>, that is, <math>\mbox{tan}\alpha</math>.
Typically the stiffness of a structure cannot be separated from its hysteretic damping, so these quantities are considered as a single coefficient. The complex stiffness <math>k^{*}</math> is given by
{{Eqn|
eqnValue=<math>k^{*}=k\left(1+j\eta\right)</math>,|
number=''|}}
where
:<math>k</math> is the static stiffness and
:<math>\eta</math> the hysteretic damping loss factor.

===Hysteretic Damping: Equation of Free Motion for a single DOF system===
[[Image:Hysteretic Damping, Single DOF.png|'''Figure 1: Hysteretic Damping for single DOF system'''|thumb|center|400px]]
The equation of free motion for a single DOF system with hysteretic damping is therefore
{{Eqn|
eqnValue=<math>m\ddot{x}+k^{*}x=0</math>|
number=''|}}
Figure 1 shows a single DOF model with hysteretic damping of coefficient <math>c_{H}</math>.

The equation of motion is
{{Eqn|
eqnValue=<math>m\ddot{x}+\left(\frac{c_{H}}{\omega}\right)\dot{x}+kx=0</math>|
number=''|}}
Now if <math>x=Xe^{j\omega t}</math>,
{{Eqn|
eqnValue=<math>\dot{x}=j\omega x</math> and <math>\left(\frac{c_{H}}{\omega}\right)\dot{x}=jc_{H}x</math>|
number=''|}}
Reformulating the equation of motion it becomes
{{Eqn|
eqnValue=<math>m\ddot{x}+\left(k+jc_{H}\right)x=0</math>|
number=''|}}
Since
{{Eqn|
eqnValue=<math>k+jc_{H}=k\left(1+\frac{jc_{H}}{k}\right)=k\left(1+j\eta\right)=k^{*}</math>|
number=''|}}
we can write
{{Eqn|
eqnValue=<math>m\ddot{x}+k^{*}x=0</math>|
number=''|}}
That is, the combined effect of the elastic and hysteretic resistance to motion can be represented as a complex stiffness, <math>k^{*}</math>.

==Hysteretic Damping Loss Factors<ref>Lazan</ref>==
A range of values of η for some common engineering materials is given below. For more detailed information on meterial damping mechanisms and loss factors.

<div align="center">
{| style="background:transparent; color:black; font-size:120%; width:75%" align="center" valign="top" border="5"
|+ '''Table 1: Loss Factor for select materials'''
! Material !! Loss Factor (η)
|- style="color: #C00000;" align="left" valign="top"
|
Aluminum-pure
|
0.00002-0.002
|- style="color: #C00000;" align="left" valign="top"
|
Aluminum alloy-dural
|
0.0004-0.001
|- style="color: #C00000;" align="left" valign="top"
|
Steel
|
0.001-0.008
|- style="color: #C00000;" align="left" valign="top"
|
Lead
|
0.008-0.014
|- style="color: #C00000;" align="left" valign="top"
|
Cast iron
|
0.003-0.03
|- style="color: #C00000;" align="left" valign="top"
|
Manganese copper alloy
|
0.05-0.1
|- style="color: #C00000;" align="left" valign="top"
|
Rubber-natural
|
0.1-0.3
|- style="color: #C00000;" align="left" valign="top"
|
Rubber-hard
|
1.0
|- style="color: #C00000;" align="left" valign="top"
|
Glass
|
0.0006-0.002
|- style="color: #C00000;" align="left" valign="top"
|
Concrete
|
0.01-0.06
|}
</div>

==Energy dissipated by Hysteretic Damping<ref>Beards, pp. 43-45</ref>==
The energy dissipated per cycle by a force ''F'' acting on a system with hysteretic damping is <math>\int F dx</math>, where
{{Eqn|
eqnValue=<math>F=k^{*}x=k\left(1+j\eta\right)x</math>,|
number=''|}}
and ''x'' is the displacement.

For harmonic motion <math>x=X\mbox{ sin } \omega t</math>, so
{{Eqn|
eqnValue=<math>\begin{alignat}F & = kX\mbox{ sin}\left(\omega t\right)+j\eta kX\mbox{ sin}\left(\omega t\right) \\ & = kX\mbox{ sin}\left(\omega t\right)+\eta kX \mbox{ cos}\left(\omega t\right)\end{alignat}</math>.|
number=A|}}

Now
{{Eqn|
eqnValue=<math>\mbox{sin}\left(\omega t\right)=\frac{x}{X}</math>|
number=B.1|}}
{{Eqn|
eqnValue=<math>\mbox{cos}\left(\omega t\right)=\frac{\sqrt{\left(X^2-x^2\right)}}{X}</math>|
number=B.2|}}

Substituting the Eqn. (B.1) and Eqn. (B.2) into Eqn. (A)

{{Eqn|
eqnValue=<math>F=kx \pm \eta k \sqrt{\left(X^2-x^2\right)}</math>.|
number=C|}}

[[Image:Hysteretic Damping, Energy Dissipated.png|'''Figure 2: Energy Dissipated'''|thumb|center|400px]]
Eqn. (C) is an ellipse as shown in Figure 2. The energy dissipated is area of this ellipse.
{{TODO|Find image}}

Performing the integration of Eqn. (C) we get the total energy dissipated
{{Eqn|
eqnValue=<math>\int F dx & =\int_{0}^{x} \left(kx \pm \eta k \sqrt{\left(X^2-x^2\right)}\right) dx = \pi X^2 \eta k</math>.|
number=''|}}

==Notes==
*Beards, C. F. 1995 Engineering Vibration Analysis with Applications to Control Systems. ISBN 034063183X
*Lazan, B. J. 1968 Damping of Materials and Members in Structural Mechanics.

<amazon>034063183X</amazon>

===References===
<references/>

Forced Vibration in Second Order Systems

Gabe Spradlin — Tue, 16 Apr 2013 17:17:42 GMT

Gabe Spradlin: /* Forced Vibration is based on a Free Damped Vibration (Second Order Systems) Model */

{{Header|Second Order Systems|Fast Steering Mirror Example}}
[[Category:SISO]]
[[Category:Classical Control]]
[[Category:Modeling]]
[[Category:Single DOF]]

==Introduction to Forced Vibration in Second Order Systems<ref>Leigh, pg. 46</ref>==
Periodic excitation is a fact of life.
:*No rotating system is perfectly balanced.
:*Translational systems sit on vibratory platforms.

A free damped system under vibration cannot be controlled; it is a free system. Controlling a damped system under vibration requires the ability to inpart a force. This force is our control signal.

===Harmonic Forcing in Second Order (Damped) Systems===
Some periodic forces are harmonic. Some are not. Those forces which are not harmonic can be represented as a series of harmonic functions using Fourier analysis techniques. Since both harmonic and non-harmonic forcing functions can be represented with harmonic functions it makes sense to study the response of forced damped systems to a harmonic forcing function.

==Forced Vibration is based on a Free Damped Vibration (Second Order Systems) Model==
The equation of motion for a free damped vibration is (free second order system with viscous damping)
{{Eqn|
eqnValue=<math>m\ddot{x}+c\dot{x}+kx=0</math>|
number=Free Second Order System|}}
where
:''m'' is the mass,
:''c'' is the damping coefficient,
:''k'' is the spring stiffness, and
:''x'' is the displacement
and equilibrium is defined as
:''x=0'' at ''t=0''

For forced vibration the equation for viscous damping becomes
{{Eqn|
eqnValue=<math>m\ddot{x}+c\dot{x}+kx=F\mbox{ sin}\left( \nu t \right)</math>|
number=Forced Second Order System|}}
where
:''F'' is the amplitude of the forcing function,
:''ν'' is the frequency of the forcing function, and
:''t'' is time in seconds.

Below is more information for viscous damping, coulomb damping, and hysteretic damping.

===Viscous Damping<ref>Leigh, pp. 46-52</ref>===
{{SeeAlso|saMsg=[[Vibration Isolation Example]]}}
For full derivation see [[Viscous Damping]].
[[Image:One DOF, Free Damped Vibration.png|'''Figure 1: Viscous Damping for single DOF system'''|thumb|right|300px]]

Figure 1 shows the model for a single degree of freedom system with forcing function and viscous damping. The equation of motion is
{{Eqn|
eqnValue=<math>m\ddot{x}+c\dot{x}+kx=F\mbox{ sin}\left(\nu t\right)</math>|
number=Second Order System with Viscous Damping}}
The solution to <math>m\ddot{x}+c\dot{x}+kx = 0</math> is presented in [[Viscous Damping]]. It also represents the complementary function. It demonstrates that the initial vibration will quickly dissipate. For sustained motion simple harmoinc motion at the frequency of excitation is assumed. That solution is <math>x=X\mbox{ sin}\left( \nu t - \phi \right)</math> and leads to
{{Eqn|
eqnValue=<math>\dot{x}=X\nu\mbox{ cos}\left( \nu t - \phi \right)=X \nu \mbox{ sin}\left( \nu t - \phi + \frac{1}{2}\pi \right)</math>|
number=''}}
and
{{Eqn|
eqnValue=<math>\ddot{x}=-X\nu^2 \mbox{ sin}\left(\nu t - \phi \right)=X\nu^2 \mbox{ sin}\left(\nu t - \phi + \pi \right)</math>|
number=''}}
Making the correct substitutions the equation of motion for a second order system with viscous damping becomes
{{Eqn|
eqnValue=<math>mX\nu^2 \mbox{ sin} \left(\nu t - \phi + \pi \right) + cX\nu \mbox{ sin}\left(\nu t - \phi + \frac{\pi}{2} \right) + kX \mbox{ sin} \left(\nu t - \phi \right) = F \mobx{ sin} \left( \nu t \right)</math>|
number=''}}
[[Image:Viscous Damping Vector Diagram.png|'''Figure 2: Force vector diagram'''|thumb|right|200px]]
The vector diagram in Figure 2 leads to
{{Eqn|
eqnValue=<math>F^2 = \left(kX-mX\nu^2 \right)^2 + \left( cX\nu \right)^2</math>|
number=''}}
or
{{Eqn|
eqnValue=<math>X = \frac{F}{\sqrt{\left( \left(k-m\nu^2 \right)^2 + \left(c\nu \right)^2 \right)}}</math>|
number=''}}
and
{{Eqn|
eqnValue=<math>\mbox{tan} \phi = \frac{cX \nu}{\left( kX - mX \nu^2 \right)}</math>|
number=''}}
At steady-state the equation for a second order system with viscous damping is
{{Eqn|
eqnValue=<math>x=\frac{F}{\sqrt{\left( \left( k-m\nu^2 \right)^2 + \left( c \nu \right)^2 \right)}} \mbox{ sin} \left( \nu t - \phi \right)</math>|
number=''}}
where
{{Eqn|
eqnValue=<math>\phi = \mbox{tan}^{-1} \left( \frac{c\nu}{k-m\nu^2} \right)</math>|
number=''}}
The equation of transient motion for a second order system with viscous damping is
{{Eqn|
eqnValue=<math>x = Ae^{-\zeta \omega t} \mbox{ sin} \left(\omega \sqrt{\left(1-\zeta^2 \right)^{2} t} + \alpha \right)</math>|
number=''}}
where
:<math>\zeta = \frac{c}{2 \sqrt{k m} }</math>
The following definitions will make the transient equation more convenient
{{Eqn|
eqnValue=<math>\omega=\sqrt{\frac{k}{m}}</math> '''rad/s''' and <math>X_{s}=\frac{F}{k}</math>|
number=''}}
Then the dynamic magnification factor, <math>X/X_{s}</math>, is
{{Eqn|
eqnValue=<math>\frac{X}{X_{s}}=\frac{1}{\sqrt{\left \{ \left [ 1 - \left( \frac{\nu}{\omega} \right)^2 \right ]^2 + \left [2\zeta \frac{\nu}{\omega} \right ]^2 \right \} }}</math> and <math>\phi = \mbox{tan}^{-1} \left( \frac{2\zeta \left(\frac{\nu}{\omega} \right)}{1-\left( \frac{\nu}{\omega} \right)^2} \right)</math>|
number=''}}
where
:<math>X_{s}</math> is the static deflection of the system under a steady force <math>F</math> and
:<math>X</math> is the dynamic amplitude
Mechanical vibration becomes more important as ''X/X''<sub>''s''</sub> gets larger and ν gets closer to ω. As ν gets closer to ω a small harmonic force can produce a large amplitude of vibration. This is known as '''resonance''' and occurs when the forcing frequency is equal to the natural frequency (i.e. ν/ω = 1). The maximum value of ''X/X''<sub>''s''</sub> can be determined by
{{Eqn|
eqnValue=<math>\left( \frac{\nu}{\omega} \right)_{\left( X/X_s \right)_{max}} = \sqrt{1-2\zeta^2} \approx 1</math> for smal ζ|
number=''}}
and
{{Eqn|
eqnValue=<math>\left( \frac{X}{X_s} \right)_{max} = \frac{1}{\left( 2\zeta \sqrt{\left( 1-\zeta^2 \right)} \right)}</math>|
number=''}}

From the other [[Second Order Systems]] article
{{Eqn|
eqnValue=<math>M_p = \left| H \left( j \omega_p \right) \right| = \begin{cases} 1, & \mbox{if }\zeta \ge \frac{1}{\sqrt{2}} \\ \frac{1}{2 \zeta \sqrt{1-\zeta^2}}, & \mbox{if }0 \le \zeta < \frac{1}{\sqrt{2}}\end{cases}</math>|
number=''|}}
Note the similarities to <math>\frac{X}{X_s}</math>.

====Q factor====
For small values of ζ, <math>X/X_{s} \approx 1/2\zeta</math>. <math>1/2\zeta</math> is a measure of the damping in a system; this is known as the ''Q'' factor.

====Alternative solution for Second Order Systems with Viscous Damping====
An alternative solution to the equation of motion for a second order system with Viscous Damping can be obtained by substituting <math>F\mbox{ sin} \left( \nu t \right)=\mbox{Im}\left( Fe^{j\nu t} \right)</math>. Then <math>m\ddot{x} + c\dot{x} + kx = Fe^{j\nu t}</math> and an assumed solution of <math>x=Xe^{j\nu t}</math> leads to
{{Eqn|
eqnValue=<math>\left( k-m\nu^2 \right) X + jc\nu X = F</math> or <math>X=\frac{F}{\left(k-m\nu^2 \right) +jc\nu}</math>|
number=''}}
where
:<math>j=\sqrt{-1}</math>
Therefore
{{Eqn|
eqnValue=<math>X=\frac{F}{\sqrt{\left( \left( k-m\nu^2 \right)^2 + \left( c\nu \right)^2 \right)}}</math>|
number=''}}

====Rotating Second Order Systems with Viscous Damping====
Rotating systems with an ''unbalance'' produce excitation force proportional to the square of the excitation frequency. The necessary variables are
:<math>m_r</math> is the unbalanced mass,
:<math>r</math> is the effective radious, and
:<math>\nu</math> is the rotating angular speed.
Therefore the excitation force is
:<math>m_{r}r\nu^2</math>
If applied to a system like Figure 1, the component of the force in the direction of motion is <math>m_{r}r\nu^2\mbox{ sin}\left( \nu t \right)</math> and the amplitude of vibration is
{{Eqn|
eqnValue=<math>X=\frac{\left( \frac{m_r}{m} \right) r \left( \frac{\nu}{\omega} \right)^2}{\sqrt{\left( \left( 1 - \left( \frac{\nu}{\omega} \right)^2 \right)^2 + \left( 2\zeta \nu \omega \right)^2 \right)}}</math>|
number=''}}
The value of ''ν/ω'' for maximum ''X'' can be found by differentiating the equation above to get
{{Eqn|
eqnValue=<math>\left( \frac{\nu}{\omega} \right)_{X_{max}} = \frac{1}{\sqrt{1-2 \zeta^2}}</math>|
number=''}}
The peak response occurs when ''ν > ω''. Also,
{{Eqn|
eqnValue=<math>X_{max}=\left( \frac{m_r}{m} \right) \frac{r}{2\zeta \sqrt{1 - \zeta^2}}</math>|
number=''}}

===Coulomb Damping<ref>Leigh, pp. 69-70</ref>===
[[Image:One DOF, Free Coulomb Damped Vibration.png|'''Figure 3: Coulomb Damping for single DOF system'''|thumb|right|300px]]
For full derivation see [[Coulomb Damping]].

Figure 3 shows the model for a single degree of freedom system with forcing function and coulomb damping. The equation of motion is
{{Eqn|
eqnValue=<math>m\ddot{x}+kx \pm F_{d}=F\mbox{ sin}\left(\nu t\right)</math>|
number=Second Order System with Coulomb Damping}}
Note that this equation is non-linear. This is due to the friction force always opposing the direction of motion.

The motion can be discontinuous if ''F''<sub>''d''</sub> is large compared to ''F''. However, in most systems ''F''<sub>''d''</sub> is small. This fact allows us to create a linear approximation.

Express ''F''<sub>''d''</sub> in terms of ''c''<sub>''d''</sub> - the viscous damping coefficient.
{{Eqn|
eqnValue=<math>c_{d}=\frac{4F_{d}}{\pi \nu X}</math>|
number=''|}}

{{Eqn|
eqnValue=<math>X=\frac{F}{\sqrt{\left [ \left( k-m\nu^2\right)^2 + \left(c_{d}\nu \right)^2 \right]}}</math>|
number=''|}}

{{Eqn|
eqnValue=<math>X=\frac{F}{\sqrt{\left [ \left( k-m\nu^2\right)^2 + \left( \frac{4F_{d}}{\pi X} \right)^2 \right]}}</math>|
number=''|}}

{{Eqn|
eqnValue=<math>\frac{X}{X_{s}}=\frac{\sqrt{1-\left( \frac{4F_{d}}{\pi F} \right)^2}}{1-\left( \frac{\nu}{\omega} \right)^2}</math>|
number=''|}}
where
:<math>\frac{4F_{d}}{\pi F}<1</math> or <math>F_{d}>\left( \frac{\pi}{4} \right) F</math>

At resonance the amplitude is not limited by Coulomb friction.

===Hysteretic Damping<ref>Leigh, pp. 70-71</ref>===
[[Image:Hysteretic Damping, Single DOF.png|'''Figure 4: Hysteretic Damping for single DOF system'''|thumb|right|300px]]
For full derivation see [[Hysteretic Damping]].

Figure 4 shows the model for a single degree of freedom system with forcing function and hysteretic damping. The equation of motion is
{{Eqn|
eqnValue=<math>m\ddot{x}+k^{*}x=F\mbox{ sin}\left(\nu t\right)</math>|
number=Second Order System with Hysteretic Damping}}
where
:<math>k^{*}=k\left(1+j\eta\right)</math>
:<math>x=\frac{F\mbox{ sin}\left(\nu t\right)}{\left(k-m\nu^2\right)+j\eta k}</math>
and
:<math>\frac{X}{X_{s}}=\frac{1}{\sqrt{\left(\left [ 1-\left(\frac{\nu}{\omega}\right)^2\right ] ^2 + \eta^2\right)}}</math>
Note this result requires the <math>c=\frac{\eta k}{\nu}</math> substitution.

====Q factor====
If <math>c=\frac{\eta k}{\nu}</math>, at resonance <math>c={\nu}\sqrt{km}</math>, then <math>\eta=2\zeta=\frac{1}{Q}</math> .

==References==
Leigh, J. R. 2004 Control Theory. ISBN 0863413390

<amazon>0863413390</amazon>

===Notes===
<references />

Initial Value Theorem

Gabe Spradlin — Tue, 16 Apr 2013 17:16:29 GMT

Gabe Spradlin: Created page with "{{Header|:Category:Classical Control|Final Value Theorem||Classical Control}} Category:Classical Control Category:Digital Control ==Introduction to Initial Value The..."

{{Header|:Category:Classical Control|Final Value Theorem||Classical Control}}
[[Category:Classical Control]]
[[Category:Digital Control]]

==Introduction to Initial Value Theorem==
In mathematics, the '''initial value theorem''' is a theorem used to relate [[frequency domain]] expressions to the [[time domain]] behavior as time approaches [[zero]].<ref>http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html, 4/3/09</ref>

The Laplace Transform of <math>x\left(t\right)</math> is
{{Eqn|
eqnValue=<math>\mathcal{L} \left[ x\left( t \right) \right]=X\left(s\right)</math>|
number=Definition}}
where
{{Eqn|
eqnValue=<math>X\left( s \right) = \int_{0}^\infty x \left( t \right) e^{-st}\,dt</math>|
number=Definition|}}
be the (one-sided) [[Laplace transform]] of <math>x\left(t\right)</math>. The initial value theorem then states<ref>Cannon, pg 567</ref>
{{Eqn|
eqnValue=<math>\lim_{t\to 0}x\left( t \right)=\lim_{s\to\infty}{sX \left( s \right)}</math>|
number=''|}}

==Continuous Time form of the Initial Value Theorem==
{{Eqn|
eqnValue=<math>\lim_{t\to 0}x\left( t \right)=\lim_{s\to\infty}{sX \left( s \right)}</math>|
number=Continuous Time|}}

===Franklin et all's version<ref>Franklin et all, pg 105</ref>===
The '''Initial Value Theorem''' states that it is always possible to determine the initial vlaue of the time function <math>f \left( t \right)</math> from its Laplace Transform. Mathematically this can be stated as:
{{Eqn|
eqnValue=<math>\lim_{s \to \infty} sF \left( s \right) = f \left( 0^{+} \right)</math>|
number=Eqn. 3.28|}}

====Proof====
{{Eqn|
eqnValue=<math>\mathcal{L} \left \{ \frac{df}{dt} \right \}=sF \left(s \right) - f \left( 0^{-} \right)=\int_{0^{-}}^\infty e^{-st} \frac{df}{dt} dt</math>|
number=Eqn. 3.29|}}
Consider when <math>s \to \infty</math> and rewrite as
{{Eqn|
eqnValue=<math>\int_{0^{-}}^\infty e^{-st} \frac{df \left( t \right)}{dt} dt=\int_{0^{+}}^\infty e^{-st} \frac{df \left( t \right)}{dt} dt + \int_{0^{-}}^{0^{+}} e^{-st} \frac{df \left( t \right)}{dt} dt</math>|
number=''|}}
Taking the limit of Eqn. 3.29 as <math>s \to \infty</math>, we get
{{Eqn|
eqnValue=<math>\lim_{s \to \infty} \left[ sF \left(s \right) - f \left( 0^{-} \right) = \lim_{s \to \infty} \left[ \int_{0^{-}}^{0^{+}} e^{0} \frac{df \left( t \right)}{dt} dt + \int_{0^{+}}^\infty e^{-st} \frac{df \left( t \right)}{dt} dt \right]</math>|
number=''|}}
The 2nd term on the right side approaches 0 because <math>e^{-st} \to 0</math>. So
{{Eqn|
eqnValue=<math>\lim_{s \to \infty} \left[ sF \left(s \right) - f \left( 0^{-} \right) = \lim_{s \to \infty} \left[ f \left(0^{+} \right) - f \left( 0^{-} \right) \right] = f \left(0^{+} \right) - f \left( 0^{-} \right)</math>|
number=''|}}
or
{{Eqn|
eqnValue=<math>\lim_{s \to \infty} sF \left( s \right) = f \left( 0^{+} \right)</math>|
number=''|}}

====Example====
Find the initial value of the signal
{{Eqn|
eqnValue=<math>Y \left( s \right) = \frac{3}{s \left( s-2 \right)}</math>|
number=''|}}
Answer:
{{Eqn|
eqnValue=<math>y \left( 0^{+} \right) = \lim_{s \to \infty}sY \left( s \right)= \lim_{s \to \infty} s\frac{3}{s \left( s-2 \right)} = s \frac{1}{s^{2}} = \frac{\infty}{\infty^{2}}= 0</math>|
number=''|}}

==Discrete Time form of the Initial Value Theorem==
{{Eqn|
eqnValue=<math>\lim_{t\to 0}x\left( t \right)=\lim_{z\to\infty}{X \left( z \right)}=\lim_{z\to\infty}\frac{z-1}{z}{X \left( z \right)}</math>|
number=Discrete Time|}}

==References==
*http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html
*Robert H. Cannon, ''Dynamics of Physical Systems'', Courier Dover Publications, 2003
*Franklin, G. F., Emami-Naeini, A., and Powell, J. D. 1993 Feedback Control of Dynamic Systems. 3rd. Addison-Wesley Longman Publishing Co., Inc. ISBN 0201527472

<amazon>0201527472</amazon>
===Notes===
<references/>

Final Value Theorem

Gabe Spradlin — Tue, 16 Apr 2013 17:16:00 GMT

Gabe Spradlin: Created page with "{{Header|:Category:Classical Control|Initial Value Theorem||Classical Control}} Category:Classical Control Category:Digital Control ==Introduction to Final Value The..."

{{Header|:Category:Classical Control|Initial Value Theorem||Classical Control}}
[[Category:Classical Control]]
[[Category:Digital Control]]

==Introduction to Final Value Theorem<ref>Franklin et all, pp. 102-105</ref>==
The Final Value Theorem allows the evaluation of the steady-state value of a time function from its Laplace transform. The final value theorem is only valid if <math>X \left( s \right)</math> is stable (all poles are in th left half plane). If all poles are in the left half plane then
{{Eqn|
eqnValue=<math>\lim_{t \to \infty} y \left( t \right) = \lim_{s \to 0} sY \left( s \right)</math>|
number=3.26|}}

==Continuous Time Final Value Theorem<ref>Franklin et all, pp. 102-105</ref>==
{{Eqn|
eqnValue=<math>\lim_{t \to \infty} y \left( t \right) = \lim_{s \to 0} sY \left( s \right)</math>|
number=3.26|}}

===Proof===
{{Eqn|
eqnValue=<math>\mathcal{L} \left \{ \frac{df}{dt} \right \}=sF \left(s \right) - f \left( 0^{-} \right)=\int_{0^{-}}^\infty e^{-st} \frac{df}{dt} dt</math>|
number=''|}}
Consider when <math>s \to 0</math> and rewrite as
{{Eqn|
eqnValue=<math>\lim_{s \to 0} \left[sF \left(s \right) - f \left( 0^{-} \right) \right] = lim_{s \to 0} \left(\int_{0}^\infty e^{-st} \frac{df}{dt} dt \right) = \lim_{t \to \infty} \left[ f \left( t \right) - f \left( 0 \right) \right]</math>|
number=''|}}
Then
{{Eqn|
eqnValue=<math>\lim_{t \to \infty} f \left( t \right) = \lim_{s \to 0} sF \left( s \right)</math>|
number=Result 1|}}
Partial fractions can be used to see this another way
{{Eqn|
eqnValue=<math>F \left( s \right) = \frac{C_{1}}{s-p_{1}}+\frac{C_{2}}{s-p_{2}}+ \cdots + \frac{C_{n}}{s-p_{n}}</math>|
number=''|}}
If <math>p_{1}=0</math> and all other <math>p_{i}<0</math> then <math>C_{1}</math> becomes the steady-state value of <math>f \left( t \right)</math> and
{{Eqn|
eqnValue=<math>C_{1}=\lim_{t \to \infty} f \left( t \right) = sF \left(s \right) |_{s=0}</math>|
number=Result 2|}}
As you can see Result 1 is the same as Result 2.

===Example===

====The Right Way====
Find the steady-state value of the system
{{Eqn|
eqnValue=<math>Y \left( s \right) = \frac{3 \left( s+2 \right)}{s \left(s^2+2s+10 \right)}</math>|
number=''|}}
Applying the final value theorem:
{{Eqn|
eqnValue=<math>y \left( \infty \right) = sY \left( s \right) |_{s=0}=\frac{3*2}{10}=0.6</math>|
number=''|}}

====The Wrong Way====
Find the final value of the signal
{{Eqn|
eqnValue=<math>Y \left( s \right) = \frac{3}{s \left( s-2 \right)}</math>|
number=''|}}
Answer:
{{Eqn|
eqnValue=<math>y \left( \infty \right) = sY\left( s \right) |_{s=0}=-\frac{3}{2}</math>|
number=''|}}
However,
{{Eqn|
eqnValue=<math>y \left( t \right) = \left( -\frac{3}{2} + \frac{3}{2}e^{2t} \right) 1 \left( t \right)</math>|
number=''|}}

Remember that the final value theorem is only valid when all the poles are in the left half plane (<math>p \le 0</math>). Clearly this system has a pole at +2 and therefore the final value theorem does not apply.

==DiscreteTime Final Value Theorem==
{{Eqn|
eqnValue=<math>\lim_{t \to \infty} x \left( t \right) = \lim_{s \to 0} sX \left( s \right) = \lim_{z \to 1} \left( z-1 \right) X \left( z \right)</math>|
number=''|}}

==References==
*Franklin, G. F., Emami-Naeini, A., and Powell, J. D. 1993 Feedback Control of Dynamic Systems. 3rd. Addison-Wesley Longman Publishing Co., Inc. ISBN 0201527472

<amazon>0201527472</amazon>
===Notes===
<references/>

Electro-Optical Sensors

Gabe Spradlin — Tue, 16 Apr 2013 17:13:12 GMT

Gabe Spradlin: Created page with "{{Header|:Category:Sensors|:Category:Modeling||All Sensors|All Modeling Articles}} Category:Sensors {{TODO|Some sensors still need explaining. And there are new sensors c..."

{{Header|:Category:Sensors|:Category:Modeling||All Sensors|All Modeling Articles}}
[[Category:Sensors]]

{{TODO|Some sensors still need explaining. And there are new sensors coming out all the time that will need to be added.}}

==Introduction to Electro-Optical Sensors==
{{SeeAlso|saMsg=[[:Category:Sensors|All Sensors]]}}
Electro-Optical sensors are a category of sensors that turn light energy into electrical energy. The sensors can be common such as the light/dark sensors on kids' night lights - I suspect these are just simple solar cells that tell the light to run on when the voltage is appproximately 0. Electro-Optical sensors can also be much more complicated. A common but complex electro-optical sensor is the CCD at the heart of digital cameras. CCDs are also at the heart of many scientific imaging satellites.

In my laser communications work I come across a variety of electro-optical sensors. The most common are LECs, Quad Cells, FPAs and CCDs.

==A Quick Overview of Electro-Optical Sensors==

===LECs===
This section is adapted from [http://www.udt.com/application-notes/AN-Position-Sensing-Photodiodes.pdf UDT's Position Sensing Photodiode Application Note].

Lateral effect cells (LECs) are Position Sensor Devices (PSDs) which are continuous single element planar diffused photodiodes with no gaps or dead areas.

LECs provide direct readout of a light spot displacement across the entire active area. The analog output is directly proportional to both the position and intensity of a light spot present on the detector active area. A light spot present on the active area will generate a photocurrent, which flows from the point of incidence through the resistive layer to the contacts. This photocurrent is inversely proportional to the resistance between the incident light spot and the contact. When the input light spot is exactly at the device center, equal current signals are generated. By moving the light spot over the active area, the amount of current generated at the contacts will determine the exact light spot position at each instant of time. These electrical signals are proportionately related to the light spot position from the center.

The main advantage of LECs is their wide dynamic range. They can measure the light spot position all the way to the edge of the sensor. They are also independent of the light spot profile and intensity distribution that effects the position reading in the segmented diodes. The input light beam may be any size and shape, since the position of the centroid of the light spot is indicated and provides electrical output signals proportional to the displacement from the center. The devices can resolve positions better than 0.5 μm. The resolution is detector / circuit signal to noise ratio dependent.

===Quad Cells===
This section is adapted from [http://www.udt.com/application-notes/AN-Position-Sensing-Photodiodes.pdf UDT's Position Sensing Photodiode Application Note].

Segmented PSD’s, like Quad Cells, are common substrate photodiodes divided into either two or four segments (for one or two-dimensional measurements, respectively), separated by a gap or dead region. A symmetrical optical beam generates equal photocurrents in all segments, if positioned at the center. The relative position is obtained by simply measuring the output current of each segment. They offer position resolution better than 0.1 μm and accuracy higher than LECs due to superior responsivity match between the elements. Since the position resolution is not dependent on the SNR of the system, as it is in LECs, very low light level detection is possible. They exhibit excellent stability over time and temperature and fast response times necessary for pulsed applications. They are however, confined to certain limitations, such as the light spot has to overlap all segments at all times and it can not be smaller than the gap between the segments. It is important to have a uniform intensity distribution of the light spot for correct measurements. They are excellent devices for applications like nulling and beam centering.

===CCDs===
[[Image:CCD Picture.jpg|'''A specially developed CCD used for ultraviolet imaging in a wire bonded package.'''|thumb|right|250px]]
Charge Coupled Devices (CCDs) are an array of photodiodes which can be thought of as a bucket for photons. The fuller the bucket gets the more current that bucket outputs when the readout circuitry empties the bucket. Often the array in a CCD is very large meaning larges amounts of parallel circuitry for emptying those buckets. It also leads to a readout rate of the full array in the 10s of Hz.

Additionally, CCD readout circuits typically readout a column of array pixels in sequencial order. Figure 3 presents a simple graphic of how the CCD reads out the data. To be specfic, the CCD reads the pixel at the bottom of the column first then moves the current from the buckets above down 1 pixel. Then the circuit empties the next pixel and moves the rest down 1. This process is repeated until all the pixel buckets are emptied.
[[Image:CCD Readout Diagram.png|'''Figure 3: Simplified CCD Readout Diagram'''|thumb|left|300px]]
The way a CCD is readout creates a spatial noise across the imager. There is electrical noise in each pixel which is accumulated as the current from 1 pixel bucket down to the next. Also, light continues to shine on the CCD as the pixels are read out by the circuitry. This means that the pixel data from the pixel at the bottom of the column is read first and thus has less noise than the pixel at the top. This leads to even more spatial noise across the CCD.

<div align="center">
{| style="background:transparent; color:black; font-size:120%; width:25%" align="center" valign="top" border="5"
|+ '''CCD Applications'''
! Applications
|- style="color: #C00000;" align="left" valign="top"
|
Digital Cameras
|- style="color: #C00000;" align="left" valign="top"
|
Space Imaging
|- style="color: #C00000;" align="left" valign="top"
|
Star Trackers
|}
</div>

===CMOS===
Complementary Metal Oxide Semiconductor (CMOS) is an imager similar to CCDs. In CCDs each pixel gathers light and electrical potential is transferred to a number of pixels before becoming an analog voltage. This analog voltage then needs to converted to a digital number. In CMOS detectors each pixel has its own photon to voltage conversion. Many CMOS detectors have amplifiers, noise corrections and an analog to digital converter so that each pixel is output as a digital number.

In CCDs all of the pixel area can be devoted to gathering light making them more sensitive than a CMOS. Also, since all of the pixels are read out through a small number of charge to voltage and voltage to digital number converters the uniformity across a CCD is better than a CMOS detector.

===FPAs===
Focal Plane Arrays (FPAs) are similar to CMOS except that CMOS and CCDs are for visible light while FPAs are used for infrared. (Infrared light is invisible to CMOS and CCDs because they are made of silicon.) Like a CMOS, an FPA can have circuitry for each pixel. And like a CMOS uniformity across the array can be a problem.

To correct the non-uniformity of the FPA the FPA can be tested with a uniform light source. Each pixel can be read and compared to an ideal value. The non-uniformities come from a difference in electrical circuitry and responsivity to light from pixel to pixel. This is observed as a change in photon to voltage gain. Each pixel gain difference can be accumulated into a table. This table can then be applied as a correction factor.

==A Quick Overview of Electro-Optical Sensor Noise==
[[Image:Generic Random Noise Model.png|'''Figure 1: Generic Random Noise Model'''|thumb|left|350px]]
All sensors have noise. Electro-optical sensor noise is generally an accumulation of Shot Noise, Johnson (thermal) Noise, Dark Current Noise, and others specific to the sensor such as Beat Noise. All noises are random but Johnson and Dark Current noise have a "fixed" RMS magnitude. Shot noise is signal strength dependent.

{{SeeAlso|saMsg=[[Basic Noise Modeling]]}}
In most of my modeling the sim is too complex to let it run for more than a few seconds of sim time. These few seconds take 10s of minutes, sometimes hours, of real time to complete. In reality the RMS magnitude of the Dark Current and Johnson noise will be approximately constant. However, Shot noise is signal strength dependent and signal strength can change very rapidly. Firgure 1 show a generic random noise model with a fixed RMS magnitude.

For Dark Current and Johnson noise the model in Figure 1 is adequate when you set the Band Limited White Noise block to have an RMS of 1 and the constant block to the RMS value you desire. For Shot noise the constant needs to be replaced with a connection to a time varying source like a gain connected to a signal power measure.

PI Control

Gabe Spradlin — Tue, 16 Apr 2013 17:10:21 GMT

Gabe Spradlin: Created page with "{{Header|:Category:SISO|:Category:Controller Design||SISO|Controller Design}} Category:SISO Category:Controller Design Category:Classical Control ==Introduction ..."

{{Header|:Category:SISO|:Category:Controller Design||SISO|Controller Design}}
[[Category:SISO]]
[[Category:Controller Design]]
[[Category:Classical Control]]

==Introduction to Proportional-Integral Controller (PI) Design==
A PI-Lead controller is a proportional gain in parallel with an integrator; both in series with a lead controller. The proportional gain provides fast error response. The integrator drives the system to a 0 steady-state error.

==Mathematics of the Proportional-Integral Controller==
[[Image:PI Controller Examples.png|'''Figure 2: PI Controller Example'''|thumb|right|350px]]
The Proportional-Integral (PI) Controller is a proportional controller (simple gain <math>k_{p}</math>) and an integrator <math>\left(\frac{k_{i}}{s}\right)</math>. Examples of PI controllers with different zeros are on the right.
{{Eqn|
eqnValue=<math>K\left(s\right)=k_{p} + \frac{k_{i}}{s}</math>|
number=''|}}
{{Eqn|
eqnValue=<math>K\left(s\right)=\frac{k_{p}s}{s} + \frac{k_{i}}{s}</math>|
number=''|}}
{{Eqn|
eqnValue=<math>K\left(s\right)=\frac{k_{p}s + k_{i}}{s}</math>|
number=''|}}
{{Eqn|
eqnValue=<math>K\left(s\right)=k_{p}\frac{s + \frac{k_{i}}{k_{p}}}{s}</math>|
number=Eqn. 1|}}

The MATLAB command to create a PI controller is
PIcomp = @(z) tf([1 z],[1 0]);
where
:z is the zeros in a PI controller.

The MATLAB command presented ignores the proportional gain, <math>k_{p}</math>. The proportional gain is set so that the open loop system can achieve the desired open loop crossover frequency.

==See Also==
*[[Standard Controller Forms]]
*[[PI-Lead Control]]