http://2009.igem.org/wiki/index.php?title=Special:Contributions/Bvanpary&feed=atom&limit=50&target=Bvanpary&year=&month=2009.igem.org - User contributions [en]2024-03-28T15:41:51ZFrom 2009.igem.orgMediaWiki 1.16.5http://2009.igem.org/Team:KULeuven/Design/Integrated_ModelTeam:KULeuven/Design/Integrated Model2009-10-13T19:06:22Z<p>Bvanpary: /* Proportional design (P controller) */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Intergated_Model}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Controller design=<br />
<br />
Because we want to optimize the design of the feedback loop in our system, we developed a more abstract block scheme of the bacterium. It shows each component as a block performing a specific task. The diagram is used to develop some theories about the performance of the feedback loop.<br />
<br />
==Proportional design (P controller)==<br />
<br />
Because the controller has to be implemented in 'biological technology', we choose the simplest possible design of <br />
controller, the proportional controller. The output signal in this type of controller is directly proportional to the error signal. The error signal is the substraction of the input and the control signal. however this type of controller has one importent flaw. When the input is a step function, as in most cases, there will be a steady state error. To make this steady state error as small as possible, the gain in the feedback loop must be as large as possible.<br />
The gain in the feedback loop can be adjusted by using low/high copy plasmids for the genes in the loop.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
==Proportional and Integral design (PI controller)==<br />
<br />
Because the steady state error on a step input signal will always be nonzero without an integrator, we considered to add some integral action, in the form of an proportional and integral controller. The input of the controlled system is, in this type of controller, a weighted sum of the error and the integral of that error signal. This way the staedy state error is avoided. However because of the limited amount of time and resources we stayed with the P controller design, which is more straightforward to implement in a biological systems.<br><br />
As an extention the integrator could be implemented in the cell by producing a species with a rate proportional to the amount of mRNA key.<br />
[[Image:PI.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-10-11T17:34:49Z<p>Bvanpary: /* Simulation */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Intergated_Model}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
In this section we consider the problem of choosing the amount of proportional action in the feedback loop, in the<br />
following sections terminology and concepts of linear control theory are used. Of course It's obvious that the system will not behave in a linear way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
==Stability==<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become larger as can be seen on the graph below. Eventually the oscillations become dominant and will destabilize the controlled system.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
==Tracking problem==<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
==Disturbance rejection==<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Vanillin_ReceptorTeam:KULeuven/Modeling/Vanillin Receptor2009-10-01T08:35:10Z<p>Bvanpary: /* Simulation */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Vanillin_Receptor}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
<br />
=Vanillin Sensor=<br />
<br />
<br />
<br />
== Biological Model ==<br />
<center><br />
[[image:Biologie_vanillin_receptor.png|center]]<br />
{{kulpart|BBa_K238003}}<br />
[[image:Biologie_antikey.png|center]]<br />
{{kulpart|BBa_K238001}}<br />
</center><br />
<br />
==Mathematical Model==<br />
[[image:Vanillin_receptor.png|650px|center]]<br />
<br />
<center><br />
{| width=80% style="border: 1px solid #003E81; background-color: #EEFFFF;"<br />
|+ ''Parameter values (Vanillin Sensor)''<br />
! width=15% | Name<br />
! width=15% | Value<br />
! width=40% | Comments<br />
! width=10% | Reference<br />
|-<br />
! colspan="4" style="border-bottom: 1px solid #003E81;" | Degradation Rates<br />
|-<br />
| d<sub>mRNA</sub><br />
| 2.3105E-3 s<sup>-1</sup><br />
|<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [4<html>]</html>]<br />
|-<br />
| d<sub>Proteins</sub><br />
| 1.9254E-5 s<sup>-1</sup><br />
| <br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [5<html>]</html>]<br />
|-<br />
! colspan="4" style="border-bottom: 1px solid #003E81;" | Transcription Rates<br />
|-<br />
| k<sub>transcription</sub><br />
| 0.00848 s<sup>-1</sup><br />
| estimate<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [6<html>]</html>]<br />
|-<br />
| k<sub>translation</sub><br />
| 0.167 s<sup>-1</sup><br />
| estimate<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [6<html>]</html>]<br />
|-<br />
! colspan="4" style="border-bottom: 1px solid #003E81;" | Phosphorylation Parameters<br />
|-<br />
| k<sub>autophosphorylation</sub><br />
| 0.00237 (s molecule)<sup>-1</sup><br />
| Rate of autophosphorylation of VirA protein<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [2<html>]</html>]<br />
|-<br />
| k<sub>phosphorylation</sub><br />
| 0.00416 s<sup>-1</sup><br />
| Rate of phosphorylation of VirG by phosphorylated VirA.<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [3<html>]</html>]<br />
|-<br />
|}<br />
</center><br />
<br />
==Simulation==<br />
<br />
A scan of the influence of vanillin on the production of Anti Key is shown is the figure below.<br />
The steady state level of Anti Key is quite linearly dependent on the amount vanillin, a half rise time<br />
of 3 hours is estimated by this simulation.<br />
<br />
[[image:Vanillin_sensor.png|750px|center|thumb|The concentration of vanillin [µM] regulates the production mRNA AntiKey]]<br />
<br />
==Vanillin diffusion==<br />
<br />
Because VirA senses the vanillin concentration in the cytoplasm, it's important to estimate the diffusion of vanillin over the cell membrane. Also because vanillin is not actively removed from the extracellular medium, the rate of evaporation of vanillin is an important figure if we want to regulate the concentration of vanillin in the extracellular medium. A detailed analysis can be found in following document. [https://2009.igem.org/Image:Vanillin_sensing.pdf vanillin_sensing]<br />
<br />
The most important figures are that the time to reach steady state between the concentration of the inter and extracellular concentration of vanillin in the order of 10 ms, the half-life through evaporation of vanillin in water <br />
is 20 hours. <br />
<br />
An equivalent model (single cell) of vanillin in the aqueous medium and evaporation was conducted.<br />
<br />
[[Image:Diffusion_matlabmodel.jpg|center|thum|500px|Equivalent model of vanillin]]<br />
<br />
Following simulation shows the time-scale of evaporation of vanillin out of the lb(aqueous) medium, the slow<br />
evaporation rate of vanillin is not surprising considering its use in the perfume industries as an aroma in the ground note.<br />
<br />
[[Image:Evaporation.png|center|thumb|500px|Evaporation rate of vanillin out of lb medium]]<br />
<br />
If this evaporation rate would show to slow, several active techniques exists to speed the process of removing the vanillin from the extracellular medium.<br />
<br />
==Reference==<br />
<br />
[1] A. Vian et al., "Structure of the b-Galactosidase Gene from Thermus sp. Strain T2: Expression in Escherichia coli and Purification in a Single Step of an Active Fusion Protein," Department of Microbiology, University of Washington, Applied and Environmental Microbiology, Jun. 1998, p. 2187–2191<br />
<br />
[2] W.T. Peng et al., "The Phenolic Recognition Profiles of the Agrobacterium tumefaciens VirA Protein Are Broadened by a High Level of the Sugar Binding Protein ChvE," Department of Microbiology, University of Washington, JOURNAL OF BACTERIOLOGY, Nov. 1998, p. 5632–5638<br />
<br />
[3] S. Jin et al., "Phosphorylation of the VirG Protein of Agrobacterium tumefaciens by the Autophosphorylated VirA Protein: Essential Role in Biological Activity of VirG," Departments of Microbiology and Pharmacology University of Washington, Journal of Bacteriology, Sept. 1990, p. 4945-4950<br />
<br />
[4] J.A. Bernstein et al., “Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two-color fluorescent DNA microarrays,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, Jul. 2002, pp. 9697–9702<br />
<br />
[5] K. Nath et al., “Protein degradation in Escherichia Coli,” The Journal of Biological Chemistry, vol. 246, Nov. 1971, pp. 6956-6967<br />
<br />
[6] S.L. Gotta et al., “rRNA Transcription Rate in Escherichia Coli,” Journal of Bacteriology, vol. 173, Oct. 1991, pp. 6647-6649<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Vanillin_ReceptorTeam:KULeuven/Modeling/Vanillin Receptor2009-10-01T08:34:31Z<p>Bvanpary: /* Simulation */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Vanillin_Receptor}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
<br />
=Vanillin Sensor=<br />
<br />
<br />
<br />
== Biological Model ==<br />
<center><br />
[[image:Biologie_vanillin_receptor.png|center]]<br />
{{kulpart|BBa_K238003}}<br />
[[image:Biologie_antikey.png|center]]<br />
{{kulpart|BBa_K238001}}<br />
</center><br />
<br />
==Mathematical Model==<br />
[[image:Vanillin_receptor.png|650px|center]]<br />
<br />
<center><br />
{| width=80% style="border: 1px solid #003E81; background-color: #EEFFFF;"<br />
|+ ''Parameter values (Vanillin Sensor)''<br />
! width=15% | Name<br />
! width=15% | Value<br />
! width=40% | Comments<br />
! width=10% | Reference<br />
|-<br />
! colspan="4" style="border-bottom: 1px solid #003E81;" | Degradation Rates<br />
|-<br />
| d<sub>mRNA</sub><br />
| 2.3105E-3 s<sup>-1</sup><br />
|<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [4<html>]</html>]<br />
|-<br />
| d<sub>Proteins</sub><br />
| 1.9254E-5 s<sup>-1</sup><br />
| <br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [5<html>]</html>]<br />
|-<br />
! colspan="4" style="border-bottom: 1px solid #003E81;" | Transcription Rates<br />
|-<br />
| k<sub>transcription</sub><br />
| 0.00848 s<sup>-1</sup><br />
| estimate<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [6<html>]</html>]<br />
|-<br />
| k<sub>translation</sub><br />
| 0.167 s<sup>-1</sup><br />
| estimate<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [6<html>]</html>]<br />
|-<br />
! colspan="4" style="border-bottom: 1px solid #003E81;" | Phosphorylation Parameters<br />
|-<br />
| k<sub>autophosphorylation</sub><br />
| 0.00237 (s molecule)<sup>-1</sup><br />
| Rate of autophosphorylation of VirA protein<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [2<html>]</html>]<br />
|-<br />
| k<sub>phosphorylation</sub><br />
| 0.00416 s<sup>-1</sup><br />
| Rate of phosphorylation of VirG by phosphorylated VirA.<br />
| [https://2009.igem.org/Team:KULeuven/Modelling/Vanillin_Production#References [3<html>]</html>]<br />
|-<br />
|}<br />
</center><br />
<br />
==Simulation==<br />
<br />
A scan of the influence of vanillin on the production of Anti Key is shown is the figure below.<br />
The steady state level of Anti Key is quite linearly dependent on the amount vanillin, a half rise time<br />
of 3 hours is estimated by this simulation.<br />
<br />
[[image:Vanillin_sensor.png|750px|center|The concentration of vanillin regulates the production mRNA AntiKey]]<br />
<br />
==Vanillin diffusion==<br />
<br />
Because VirA senses the vanillin concentration in the cytoplasm, it's important to estimate the diffusion of vanillin over the cell membrane. Also because vanillin is not actively removed from the extracellular medium, the rate of evaporation of vanillin is an important figure if we want to regulate the concentration of vanillin in the extracellular medium. A detailed analysis can be found in following document. [https://2009.igem.org/Image:Vanillin_sensing.pdf vanillin_sensing]<br />
<br />
The most important figures are that the time to reach steady state between the concentration of the inter and extracellular concentration of vanillin in the order of 10 ms, the half-life through evaporation of vanillin in water <br />
is 20 hours. <br />
<br />
An equivalent model (single cell) of vanillin in the aqueous medium and evaporation was conducted.<br />
<br />
[[Image:Diffusion_matlabmodel.jpg|center|thum|500px|Equivalent model of vanillin]]<br />
<br />
Following simulation shows the time-scale of evaporation of vanillin out of the lb(aqueous) medium, the slow<br />
evaporation rate of vanillin is not surprising considering its use in the perfume industries as an aroma in the ground note.<br />
<br />
[[Image:Evaporation.png|center|thumb|500px|Evaporation rate of vanillin out of lb medium]]<br />
<br />
If this evaporation rate would show to slow, several active techniques exists to speed the process of removing the vanillin from the extracellular medium.<br />
<br />
==Reference==<br />
<br />
[1] A. Vian et al., "Structure of the b-Galactosidase Gene from Thermus sp. Strain T2: Expression in Escherichia coli and Purification in a Single Step of an Active Fusion Protein," Department of Microbiology, University of Washington, Applied and Environmental Microbiology, Jun. 1998, p. 2187–2191<br />
<br />
[2] W.T. Peng et al., "The Phenolic Recognition Profiles of the Agrobacterium tumefaciens VirA Protein Are Broadened by a High Level of the Sugar Binding Protein ChvE," Department of Microbiology, University of Washington, JOURNAL OF BACTERIOLOGY, Nov. 1998, p. 5632–5638<br />
<br />
[3] S. Jin et al., "Phosphorylation of the VirG Protein of Agrobacterium tumefaciens by the Autophosphorylated VirA Protein: Essential Role in Biological Activity of VirG," Departments of Microbiology and Pharmacology University of Washington, Journal of Bacteriology, Sept. 1990, p. 4945-4950<br />
<br />
[4] J.A. Bernstein et al., “Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two-color fluorescent DNA microarrays,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, Jul. 2002, pp. 9697–9702<br />
<br />
[5] K. Nath et al., “Protein degradation in Escherichia Coli,” The Journal of Biological Chemistry, vol. 246, Nov. 1971, pp. 6956-6967<br />
<br />
[6] S.L. Gotta et al., “rRNA Transcription Rate in Escherichia Coli,” Journal of Bacteriology, vol. 173, Oct. 1991, pp. 6647-6649<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:Vanillin_sensor.pngFile:Vanillin sensor.png2009-10-01T08:33:21Z<p>Bvanpary: uploaded a new version of "Image:Vanillin sensor.png"</p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-10-01T08:14:48Z<p>Bvanpary: /* Tuning the controller */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious the system will not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
==Stability==<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant and ultimately destabilize the controlled system.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
==Tracking problem==<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
==Disturbance rejection==<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
=Simulation=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-10-01T08:14:21Z<p>Bvanpary: </p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious the system will not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
==Stability==<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant and ultimately destabilize the controlled system.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
==Tracking problem==<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
==Disturbance rejection==<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Design/Integrated_ModelTeam:KULeuven/Design/Integrated Model2009-09-30T17:16:21Z<p>Bvanpary: /* Proportional and Integral design (PI controller) */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Controller design=<br />
<br />
Because we want to optimize the design of the feedback loop in our system, we developed a more abstract block scheme of the bacterium. It shows each component as a block performing a specific task. The diagram is used to develop some theories about the performance of the feedback loop.<br />
<br />
==Proportional design (P controller)==<br />
<br />
Because the controller has to be implemented in 'biological technology', we choose the simplest possible design of <br />
controller, the proportional controller. The output signal in this type of controller is directly proportional to the error signal. The error signal is the substraction of the input and the control signal. however this type of controller has one importent flaw. When the input is a step function, as in most cases, there will be a steady state error. To make this steady state error as small as possible, the gain in the feedback loop must be as large as possible.<br />
The gain in the feedback loop can be adjusted by using low/high copy plasmids for the genes involved in the production and sensing of vanillin.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
==Proportional and Integral design (PI controller)==<br />
<br />
Because the steady state error on a step input signal will always be nonzero without an integrator, we considered to add some integral action, in the form of an proportional and integral controller. The input of the controlled system is, in this type of controller, a weighted sum of the error and the integral of that error signal. This way the staedy state error is avoided. However because of the limited amount of time and resources we stayed with the P controller design, which is more straightforward to implement in a biological systems.<br><br />
As an extention the integrator could be implemented in the cell by producing a species with a rate proportional to the amount of mRNA key.<br />
[[Image:PI.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-29T13:04:42Z<p>Bvanpary: /* Stability */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious the system will not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant and ultimately destabilize the controlled system.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-29T13:03:56Z<p>Bvanpary: /* Tuning the controller */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious the system will not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
The transfer function from the blue light receptor to the output is, increasing the open loop gain will decrease<br />
the steady state level of vanillin.<br />
<br />
[[Image:Eqn1402.png]]<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant and ultimately destabilize the controlled system.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-29T13:03:35Z<p>Bvanpary: /* Tuning the controller */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious the system will not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional2.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
The transfer function from the blue light receptor to the output is, increasing the open loop gain will decrease<br />
the steady state level of vanillin.<br />
<br />
[[Image:Eqn1402.png]]<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant and ultimately destabilize the controlled system.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-29T13:00:29Z<p>Bvanpary: /* Tuning the controller */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious the system will not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
The transfer function from the blue light receptor to the output is, increasing the open loop gain will decrease<br />
the steady state level of vanillin.<br />
<br />
[[Image:Eqn1402.png]]<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant and ultimately destabilize the controlled system.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Design/Integrated_ModelTeam:KULeuven/Design/Integrated Model2009-09-29T12:59:28Z<p>Bvanpary: /* Proportional and Integral design (PI controller) */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Controller design=<br />
<br />
Because we want to optimize the design of the feedback loop in our system, we developed a more abstract block scheme of the bacteria. It shows each component as a block performing a specific task. The diagram is used to develop some theories about the performance of the feedback loop.<br />
<br />
==Proportional design (P controller)==<br />
<br />
Because the controller has to be implemented in 'biological technology', we choose the simplest possible design of <br />
controller, the proportional controller.<br />
The gain in the feedback loop can be adjusted by the use of low/high copy plasmids for the genes involved in production of vanillin.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
==Proportional and Integral design (PI controller)==<br />
<br />
Because without integral action the steady state error on a step input signal will be nonzero, we considered to add some integral action. But because of the limited amount of time and resources we stuck with the P controller design, which is orders of magnitudes simpler to implement effectively biological systems.<br />
<br />
[[Image:PI.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
Integral action could be implemented in the cell by producing a species with a rate proportional to the amount of mRNA key.<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Design/Integrated_ModelTeam:KULeuven/Design/Integrated Model2009-09-29T12:56:41Z<p>Bvanpary: /* Controller design */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Controller design=<br />
<br />
Because we want to optimize the design of the feedback loop in our system, we developed a more abstract block scheme of the bacteria. It shows each component as a block performing a specific task. The diagram is used to develop some theories about the performance of the feedback loop.<br />
<br />
==Proportional design (P controller)==<br />
<br />
Because the controller has to be implemented in 'biological technology', we choose the simplest possible design of <br />
controller, the proportional controller.<br />
The gain in the feedback loop can be adjusted by the use of low/high copy plasmids for the genes involved in production of vanillin.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
==Proportional and Integral design (PI controller)==<br />
<br />
Because without integral action the steady state error on a step input signal will be nonzero, we considered to add some integral action. But because of the limited amount of time and resources we stuck with the P controller design, which is orders of magnitudes simpler to implement effectively biological systems.<br />
<br />
[[Image:PI.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-29T12:55:43Z<p>Bvanpary: /* Stability */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
The transfer function from the blue light receptor to the output is, increasing the open loop gain will decrease<br />
the steady state level of vanillin.<br />
<br />
[[Image:Eqn1402.png]]<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant and ultimately destabilize the controlled system.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-29T12:54:48Z<p>Bvanpary: /* Stability */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
The transfer function from the blue light receptor to the output is, increasing the open loop gain will decrease<br />
the steady state level of vanillin.<br />
<br />
[[Image:Eqn1402.png]]<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:Eqn1402.pngFile:Eqn1402.png2009-09-29T12:54:39Z<p>Bvanpary: </p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-29T12:53:42Z<p>Bvanpary: /* Stability */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
The transfer function from the blue light receptor to the output is, increasing the open loop gain will decrease<br />
the steady state level of vanillin.<br />
<br />
[[Image:Eqn1401.png]]<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:Eqn1401.pngFile:Eqn1401.png2009-09-29T12:52:30Z<p>Bvanpary: </p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T21:03:30Z<p>Bvanpary: /* Disturbance rejection */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T21:02:27Z<p>Bvanpary: /* Robustness */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T21:02:15Z<p>Bvanpary: /* Disturbance rejection */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T21:01:02Z<p>Bvanpary: /* Disturbance rejection */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
[[Image:Disturbance.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:Disturbance.pngFile:Disturbance.png2009-09-28T21:00:32Z<p>Bvanpary: </p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:54:46Z<p>Bvanpary: /* Disturbance rejection */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.<br />
We simulated a constant addition of alien vanillin to the extracellular medium.<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:52:24Z<p>Bvanpary: /* Disturbance rejection */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
[[Image:Eqn3.png]]<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:Eqn3.pngFile:Eqn3.png2009-09-28T20:52:11Z<p>Bvanpary: </p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:51:07Z<p>Bvanpary: /* Disturbance rejection */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:48:15Z<p>Bvanpary: /* Stability */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:47:16Z<p>Bvanpary: /* Stability */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
[[Image:Stability.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:Stability.pngFile:Stability.png2009-09-28T20:46:51Z<p>Bvanpary: </p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:45:37Z<p>Bvanpary: /* Stability */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
One can show that the proportional gain can not be infinitely large due to stability problems, since we lose phase margin if we increase the proportional gain. Oscillations will become dominant until breakdown of the controller.<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:41:13Z<p>Bvanpary: /* Tracking problem */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:41:04Z<p>Bvanpary: /* Tracking problem */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
<br />
[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:Tracking_error.pngFile:Tracking error.png2009-09-28T20:39:59Z<p>Bvanpary: </p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:39:24Z<p>Bvanpary: /* Tracking problem */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
We have simulated this behaviour in the non linear model of our bacteria.<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:34:33Z<p>Bvanpary: /* Tracking problem */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain. As we wish to eliminate the transfer of disturbances to the output of our system, we have to maximize the loop gain of the system.<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:33:30Z<p>Bvanpary: /* Tracking problem */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7165.png]]<br />
<br />
with T the total open loop gain.<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:Eqn7165.pngFile:Eqn7165.png2009-09-28T20:33:09Z<p>Bvanpary: </p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/File:Eqn7166.pngFile:Eqn7166.png2009-09-28T20:32:31Z<p>Bvanpary: uploaded a new version of "Image:Eqn7166.png"</p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:08:42Z<p>Bvanpary: /* Tuning the controller */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most useful ways of investigating the behaviour of closed loop system is the investigation of the open loop system. The open loop system is the system which one becomes if you 'remove' the differentiator.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7166.png]]<br />
<br />
with T the total open loop gain.<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:07:38Z<p>Bvanpary: /* Tracking problem */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most usefull ways of investigating the behaviour of closed loop system is the investigation of the open loop system.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7166.png]]<br />
<br />
with T the total open loop gain.<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T20:07:21Z<p>Bvanpary: /* Tracking problem */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most usefull ways of investigating the behaviour of closed loop system is the investigation of the open loop system.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
We are mostly interested in the ability of the controlled system to follow a step input signal, for a linear system the steady state tracking error is:<br />
<br />
[[Image:Eqn7166.png]]<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:Eqn7166.pngFile:Eqn7166.png2009-09-28T20:07:00Z<p>Bvanpary: </p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T19:45:22Z<p>Bvanpary: /* Tuning the controller */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most usefull ways of investigating the behaviour of closed loop system is the investigation of the open loop system.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Modeling/Integrated_ModelTeam:KULeuven/Modeling/Integrated Model2009-09-28T19:44:37Z<p>Bvanpary: /* Design of the controller */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Tuning the controller=<br />
<br />
Here we consider the problem of choosing the amount of proportional action in the feedback loop, in <br />
following sections terminology and concepts used in linear control theory are used. It's of course obvious will<br />
not behave in a linearly way. But the concepts and design strategies of linear control theory can be translated to non-linear control theory.<br />
<br />
One of the most usefull ways of investigating the behaviour of closed loop system is the investigation of the open loop system.<br />
<br />
[[Image:Proportional.JPG|750px|center|thumb|Block model of the system with proportional controller]]<br />
<br />
=Stability=<br />
<br />
=Tracking problem=<br />
<br />
=Disturbance rejection=<br />
<br />
=Robustness=<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Design/Integrated_ModelTeam:KULeuven/Design/Integrated Model2009-09-28T19:44:02Z<p>Bvanpary: /* Proportional and Integral design (PI controller) */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Controller design=<br />
<br />
==Proportional design (P controller)==<br />
<br />
Because we want to optimize the design of the feedback loop in our system, we developed a more abstract block scheme of the bacteria. It shows each component as a block performing a specific task. The diagram is used to develop some theories about the performance of the feedback loop.<br />
<br />
Because the controller has to be implemented in 'biological technology', we choose the simplest possible design of <br />
controller, the proportional controller.<br />
The gain in the feedback loop can be adjusted by the use of low/high copy plasmids for the genes involved in production of vanillin.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
==Proportional and Integral design (PI controller)==<br />
<br />
Because without integral action the steady state error on a step input signal will be nonzero, we considered to add some integral action. But because of the limited amount of time and resources we stuck with the P controller design, which is orders of magnitudes simpler to implement effectively biological systems.<br />
<br />
[[Image:PI.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/File:PI.JPGFile:PI.JPG2009-09-28T19:43:32Z<p>Bvanpary: </p>
<hr />
<div></div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Design/Integrated_ModelTeam:KULeuven/Design/Integrated Model2009-09-28T19:38:04Z<p>Bvanpary: /* Simulations */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Controller design=<br />
<br />
==Proportional design (P controller)==<br />
<br />
Because we want to optimize the design of the feedback loop in our system, we developed a more abstract block scheme of the bacteria. It shows each component as a block performing a specific task. The diagram is used to develop some theories about the performance of the feedback loop.<br />
<br />
Because the controller has to be implemented in 'biological technology', we choose the simplest possible design of <br />
controller, the proportional controller.<br />
The gain in the feedback loop can be adjusted by the use of low/high copy plasmids for the genes involved in production of vanillin.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
==Proportional and Integral design (PI controller)==<br />
<br />
Because without integral action the steady state error on a step input signal will be nonzero, we considered to add some integral action. But because of the limited amount of time and resources we stuck with the P controller design, which is orders of magnitudes simpler to implement effectively biological systems.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Design/Integrated_ModelTeam:KULeuven/Design/Integrated Model2009-09-28T19:37:49Z<p>Bvanpary: /* Proportional and integral design (PI controller) */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Controller design=<br />
<br />
==Proportional design (P controller)==<br />
<br />
Because we want to optimize the design of the feedback loop in our system, we developed a more abstract block scheme of the bacteria. It shows each component as a block performing a specific task. The diagram is used to develop some theories about the performance of the feedback loop.<br />
<br />
Because the controller has to be implemented in 'biological technology', we choose the simplest possible design of <br />
controller, the proportional controller.<br />
The gain in the feedback loop can be adjusted by the use of low/high copy plasmids for the genes involved in production of vanillin.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
==Proportional and Integral design (PI controller)==<br />
<br />
Because without integral action the steady state error on a step input signal will be nonzero, we considered to add some integral action. But because of the limited amount of time and resources we stuck with the P controller design, which is orders of magnitudes simpler to implement effectively biological systems.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
= Simulations =<br />
In order to check the set-up of the model and to estimate the behaviour of the real bacterium, we performed a number of simulations. <br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanparyhttp://2009.igem.org/Team:KULeuven/Design/Integrated_ModelTeam:KULeuven/Design/Integrated Model2009-09-28T19:37:28Z<p>Bvanpary: /* Controller design */</p>
<hr />
<div>{{Team:KULeuven/Common2/BeginHeader}}<br />
{{Team:KULeuven/Common/SubMenu_Project}}<br />
{{Team:KULeuven/Common2/EndHeader}}<br />
__NOTOC__<br />
<br />
=Controller design=<br />
<br />
==Proportional design (P controller)==<br />
<br />
Because we want to optimize the design of the feedback loop in our system, we developed a more abstract block scheme of the bacteria. It shows each component as a block performing a specific task. The diagram is used to develop some theories about the performance of the feedback loop.<br />
<br />
Because the controller has to be implemented in 'biological technology', we choose the simplest possible design of <br />
controller, the proportional controller.<br />
The gain in the feedback loop can be adjusted by the use of low/high copy plasmids for the genes involved in production of vanillin.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
==Proportional and integral design (PI controller)==<br />
<br />
Because without integral action the steady state error on a step input signal will be nonzero, we considered to add some integral action. But because of the limited amount of time and resources we stuck with the P controller design, which is orders of magnitudes simpler to implement effectively biological systems.<br />
<br />
[[Image:Proportional10.JPG|750px|center|thumb|Block model of the system with proportional controller (Simulink)]]<br />
<br />
= Simulations =<br />
In order to check the set-up of the model and to estimate the behaviour of the real bacterium, we performed a number of simulations. <br />
<br />
{{Team:KULeuven/Common2/PageFooter}}</div>Bvanpary