ControlTheory - Revision history

Esden at 00:27, 10 March 2011

2011-03-10T00:27:13Z

← Older revision		Revision as of 18:27, 9 March 2011
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	--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)		--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)

			[[Category:Developer_Documentation]]

Poine at 17:12, 13 November 2009

2009-11-13T17:12:30Z

← Older revision		Revision as of 11:12, 13 November 2009
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	<math>X=\begin{pmatrix}z\\\dot{z}\end{pmatrix}</math>		<math>X=\begin{pmatrix}z\\\dot{z}\end{pmatrix}</math>

			<math>\dot{X}(t)=\begin{pmatrix}0&1\\0&0\end{pmatrix}X(t) + \begin{pmatrix}0\\1\end{pmatrix}a(t)</math>

			We don't really have an idea of the acceleration of our system, so let's asume it's 0 for now.
			When using this hypothesis and switching to discrete time (more on this later), we obtain the following equation:

			<math>X_{t+dt}=\begin{pmatrix}1&dt\\0&1\end{pmatrix}X_{t}</math>







	--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)		--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)

Poine at 16:23, 13 November 2009

2009-11-13T16:23:21Z

← Older revision		Revision as of 10:23, 13 November 2009
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	[[Image:se_ex3.png\|1024px]]		[[Image:se_ex3.png\|1024px]]

	And lag is bad for control as we'll show later. What would have happened if we had filtered the measurement before differentiating ? same punishment... so, is there a better way ?		And lag is bad for control as we'll show later. What would have happened if we had filtered the measurement before differentiating ? same punishment, as shown in the next plot produced by the following [[media:se_ex4.sce\|simulation]]...

			[[Image:se_ex4.png\|1024px]]

			so, is there a better way ?

	* Let's introduce Kinematics		* Let's introduce Kinematics

Poine at 16:08, 13 November 2009

2009-11-13T16:08:17Z

← Older revision		Revision as of 10:08, 13 November 2009
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	== Dynamic Systems ==		== Dynamic Systems ==

			* Notion of State
			The state of a dynamic system represents the minimal quantity of information required in order to be able to predict its future,
			knowing its equation of dynamic and its inputs.

			You can think of the state of a system as the frontier between past and future : two systems of identical dynamic and identical state will have the same trajectory in the future when applied the same inputs, whatever their past can be.

			* State Space Representation

	== State Estimation ==		== State Estimation ==
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	The noise gets amplified by the division by the sampling interval ( and we did not even introduce inaccuracy in that sampling interval ).		The noise gets amplified by the division by the sampling interval ( and we did not even introduce inaccuracy in that sampling interval ).
	In the above simulation, starting with a very good sensor having 1cm noise standard deviation and sampling at 100Hz, we obtain an estimate for the vertical speed having 1.4m of standard deviation. Pretty unusable...		In the above simulation, starting with a very good sensor having 1cm noise standard deviation and sampling at 100Hz, we obtain an estimate for the vertical speed having 1.4m/s of standard deviation. Pretty unusable...

	* Let's filter that noise out		* Let's filter that noise out
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	[[Image:se_ex2.png\|1024px]]		[[Image:se_ex2.png\|1024px]]

	~~Yeah~~ ! We reduced noise on our vertical speed estimate to 6cm/s. Did we achieve something interesting when doing so ? sadly no. As shown on the next plot where we have run the same algorithm on a moving vehicle, we have introduced lag in our estimate.		Yay ! We reduced noise on our vertical speed estimate to 6cm/s. Did we achieve something interesting when doing so ? sadly no. As shown on the next plot where we have run the same algorithm on a moving vehicle, we have introduced lag in our estimate.

	[[Image:se_ex3.png\|1024px]]		[[Image:se_ex3.png\|1024px]]

			And lag is bad for control as we'll show later. What would have happened if we had filtered the measurement before differentiating ? same punishment... so, is there a better way ?

			* Let's introduce Kinematics

			<math>X=\begin{pmatrix}z\\\dot{z}\end{pmatrix}</math>


	--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)		--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)

Poine at 14:57, 13 November 2009

2009-11-13T14:57:31Z

← Older revision		Revision as of 08:57, 13 November 2009
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	* Let's filter that noise out		* Let's filter that noise out

	A second idea would be to attempt and filter that noise. Let's run for example our estimate through a first order low pass filter. Here is the result of the [[media:se_ex2.sce\|simulation]		A second idea would be to attempt and filter that noise. Let's run for example our estimate through a first order low pass filter. Here is the result of the [[media:se_ex2.sce\|simulation]]

	[[Image:se_ex2.png\|1024px]]		[[Image:se_ex2.png\|1024px]]

	Yeah ! We reduced noise on our vertical speed estimate to 6cm/s. Did we achieve something interesting when doing so ? sadly no. As shown on the next plot		Yeah ! We reduced noise on our vertical speed estimate to 6cm/s. Did we achieve something interesting when doing so ? sadly no. As shown on the next plot where we have run the same algorithm on a moving vehicle, we have introduced lag in our estimate.

			[[Image:se_ex3.png\|1024px]]



	--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)		--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)

Poine at 14:32, 13 November 2009

2009-11-13T14:32:05Z

← Older revision		Revision as of 08:32, 13 November 2009
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	[[Image:se_ex2.png\|1024px]]		[[Image:se_ex2.png\|1024px]]

	Yeah ! We reduced noise on our vertical speed estimate to ~~0.15m~~/s		Yeah ! We reduced noise on our vertical speed estimate to 6cm/s. Did we achieve something interesting when doing so ? sadly no. As shown on the next plot



	--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)		--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)

Poine at 14:23, 13 November 2009

2009-11-13T14:23:04Z

← Older revision		Revision as of 08:23, 13 November 2009
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	* Direct Differentiation		* Direct Differentiation

	A first idea would be to take the difference between two consecutive altitude measurements and divide that quantity by the ~~time~~ between the two measurements. This is indeed a bad idea. Let's see one of the reasons why.		A first idea would be to take the difference between two consecutive altitude measurements and divide that quantity by the duration between the two measurements. This is indeed a bad idea. Let's see one of the reasons why.

	Our barometer being a real world instrument, it will ~~give us~~ noisy measurements ( think electrical noise, etc...). Let's use a little [[media:se_ex1.sce\|simulation]] to see what happens when you differentiate noisy measurements:		Our barometer being a real world instrument, it will output noisy measurements ( think electrical noise, etc...). Let's use a little [[media:se_ex1.sce\|simulation]] to see what happens when you differentiate noisy measurements:

	[[Image:se_ex1.png\|1024px]]		[[Image:se_ex1.png\|1024px]]

Poine at 14:18, 13 November 2009

2009-11-13T14:18:40Z

← Older revision		Revision as of 08:18, 13 November 2009
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	The noise gets amplified by the division by the sampling interval ( and we did not even introduce inaccuracy in that sampling interval ).		The noise gets amplified by the division by the sampling interval ( and we did not even introduce inaccuracy in that sampling interval ).
	In the above simulation, starting with a very good sensor having 1cm noise standard deviation, we obtain an estimate for the vertical speed having 1.4m of standard deviation. Pretty unusable...		In the above simulation, starting with a very good sensor having 1cm noise standard deviation and sampling at 100Hz, we obtain an estimate for the vertical speed having 1.4m of standard deviation. Pretty unusable...

			* Let's filter that noise out

			A second idea would be to attempt and filter that noise. Let's run for example our estimate through a first order low pass filter. Here is the result of the [[media:se_ex2.sce\|simulation]

			[[Image:se_ex2.png\|1024px]]

			Yeah ! We reduced noise on our vertical speed estimate to 0.15m/s



	--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)		--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)

Poine at 13:43, 13 November 2009

2009-11-13T13:43:20Z

← Older revision		Revision as of 07:43, 13 November 2009
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	In order to be able to control a vehicle, ie to steer it along a trajectory, a first prerequisite is to know its state.		In order to be able to control a vehicle, ie to steer it along a trajectory, a first prerequisite is to know its state.
	Typically, sensors are used to obtain this information. Sadly, sensors won't give you a straightforward knowledge of the state of the vehicle. The information is likely to be partial ( we have an altitude sensor, but nothing to measure vertical speed) and will also likely contain "errors" (bias, noise, latency, etc... ). You can always get more sensors or better sensors, but there's no such thing as a perfect sensor. The process of reconstructing the state of a dynamic system using a number of incomplete and erroneous measurements is called ''state estimation'' or ''state observation''.		Typically, sensors are used to obtain this information. Sadly, sensors won't give you a straightforward knowledge of the state of the vehicle. The information is likely to be partial ( we have an altitude sensor, but nothing to measure vertical speed) and will also likely contain "errors" (bias, noise, latency, etc... ). You can always get more sensors and/or better sensors, but there's no such thing as a perfect sensor. The process of reconstructing the state of a dynamic system using a number of incomplete and erroneous measurements is called ''state estimation'' or ''state observation''.

	Let's start with an example :		Let's start with an example :
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	We are interested in controlling the altitude of a vehicle. We measure this altitude using for example a barometer. As we will see later, the vertical speed of the vehicle is a mandatory information in order to be able to achieve a satisfactory control. So, how do we obtain the vertical speed information ?		We are interested in controlling the altitude of a vehicle. We measure this altitude using for example a barometer. As we will see later, the vertical speed of the vehicle is a mandatory information in order to be able to achieve a satisfactory control. So, how do we obtain the vertical speed information ?

	~~===~~ Direct Differentiation ~~===~~		* Direct Differentiation

	A first idea would be to take the difference between two consecutive altitude measurements and divide that quantity by the time between the two measurements. This is indeed a bad idea. Let's see one of the reasons why.		A first idea would be to take the difference between two consecutive altitude measurements and divide that quantity by the time between the two measurements. This is indeed a bad idea. Let's see one of the reasons why.

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	The noise gets amplified by the division by the sampling interval ( and we did not even introduce inaccuracy in that sampling interval ).		The noise gets amplified by the division by the sampling interval ( and we did not even introduce inaccuracy in that sampling interval ).
	In the above simulation, starting with a very good sensor having 1cm noise standard deviation, we obtain an estimate for the vertical speed having 1.4m of standard deviation. Pretty unusable...		In the above simulation, starting with a very good sensor having 1cm noise standard deviation, we obtain an estimate for the vertical speed having 1.4m of standard deviation. Pretty unusable...



			--[[User:Poine\|Poine]] 14:43, 13 November 2009 (CET)

Poine at 13:38, 13 November 2009

2009-11-13T13:38:58Z

← Older revision		Revision as of 07:38, 13 November 2009
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	The noise gets amplified by the division by the sampling interval ( and we did not even introduce inaccuracy in that sampling interval ).		The noise gets amplified by the division by the sampling interval ( and we did not even introduce inaccuracy in that sampling interval ).
			In the above simulation, starting with a very good sensor having 1cm noise standard deviation, we obtain an estimate for the vertical speed having 1.4m of standard deviation. Pretty unusable...