Team:KULeuven/Modeling/Integrated Model

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__NOTOC__
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=Full model=
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=Tuning the controller=
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The complete model of our vanillin producing bacteria is shown in the next figure. The boxes around some species have now biological meaning
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In this section we consider the problem of choosing the amount of proportional action in the feedback loop, in the
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they merely serve to distinguish between the different subcomponents of our system.
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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.
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[[Image:Fullmodel.jpg|750px|center|thumb|Biological model of our system]]
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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.
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Because we want to optimize the design of the feedback loop in our system, we developed a block scheme of the
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[[Image:Proportional.JPG|750px|center|thumb|Block model of the system with proportional controller]]
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bacteria. Which can be used to develop some theories about its performance.
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[[Image:Blockmodel.jpg|750px|center|thumb|Block model of the system]]
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==Stability==
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=Control theory=
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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.
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Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The purpose is to design a controller who controls the system so that it behaves as wanted. There exists several criteria to measure the performance of the controller.
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[[Image:Stability.png|750px|center|thumb|Stability of system, overshoot and oscillations increases with increasing gain of controller.]]
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==Stability==
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==Tracking problem==
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Stability means that no matter what the input signal (the blue light) is, the output (vanillin concentration) will remain finite after an infinite amount of time.
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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:
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==Tracking problem==
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[[Image:Eqn7165.png]]
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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.
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We have simulated this behaviour in the non linear model of our bacteria.
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This criterium is an indication of how well the output well follow the wanted reference system, we want the difference between the output and the wanted reference signal as small as possible.
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[[Image:Tracking_error.png|750px|center|thumb|Tracking error on step input in function of amplitude and time]]
==Disturbance rejection==
==Disturbance rejection==
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Most controlled system are disturbed by other systems in their neighbour hood, in our case imagine someone adding a extra amount vanillin to the aqueous medium. We do not want to see these disturbances in our output of vanillin.
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Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by
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The disturbance rejection criteria indicates the ability of the system to reject those inputs.
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==Robustness==
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[[Image:Eqn3.png]]
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As with every model, our model of the bacteria is not perfect. Robustness is an property of a property, we say that the stabilizes the system in a robust way if also stabilizes all systems that are similar to the modelled system. It's then assumed that the controller will also stabilize the real system as it is also assumed to be similar to the modelled system.
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It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output.
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We simulated a constant addition of alien vanillin to the extracellular medium.
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=Biological implications=
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[[Image:Disturbance.png|750px|center|thumb|Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium]]
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Because the controller has to be implemented in 'biological technology', we optioned for the simplest possible design of
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On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.
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controller, the proportional controller.
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The gain in the feedback loop can be adjusted by the use of low/high copy plasmids for the genes involved in the the transduction of the signal in the feedback loop.
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Latest revision as of 17:34, 11 October 2009


Tuning the controller

In this section we consider the problem of choosing the amount of proportional action in the feedback loop, in the 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.

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.

Block model of the system with proportional controller

Stability

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.

Stability of system, overshoot and oscillations increases with increasing gain of controller.

Tracking problem

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:

Eqn7165.png

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. We have simulated this behaviour in the non linear model of our bacteria.

Tracking error on step input in function of amplitude and time

Disturbance rejection

Disturbance rejection is the ability to mask disturbances on the output, the transfer of disturbances d to the output y is given by

Eqn3.png

It now easily seen that we want to have a large enough enough open loop gain to reject disturbances on the output. We simulated a constant addition of alien vanillin to the extracellular medium.

Influence of disturbance on output, on t=6e5 we extract 2 molecules/s out of the extracellular medium

On the above figure it can be clearly seen that the influence of the perturbation on the output decreases with increasing open loop gain.


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