SURF 2019: Geometry of Control-Invariant Sets - Revision history

Murray at 03:40, 28 November 2018

2018-11-28T03:40:16Z

← Older revision		Revision as of 03:40, 28 November 2018
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	[[File:N_integrator.png\|200px]]		[[File:N_integrator.png\|200px]]

	For <math>n</math> = 2 a closed-form expression of the maximal control-invariant set contained inside the unit hypercube is known [3. eq. (53)], but for higher orders of <math>n</math> only numerical approaches are currently available. The figures above show numerical approximations for <math>n</math> equal to 2 and <math>n</math> equal to 3. ~~The objective of this project is to investigate geometrical properties of control-invariant sets via a combination of analytical and numerical techniques. In particular,~~		For <math>n</math> = 2 a closed-form expression of the maximal control-invariant set contained inside the unit hypercube is known [3. eq. (53)], but for higher orders of <math>n</math> only numerical approaches are currently available. The figures above show numerical approximations for <math>n</math> equal to 2 and <math>n</math> equal to 3.

			The objective of this project is to investigate geometrical properties of control-invariant sets via a combination of analytical and numerical techniques. In particular,
	* Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets for n > 2.		* Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets for n > 2.
	* Investigate incremental algorithms, i.e., if the set for n is known, can we characterize the set for n+1?		* Investigate incremental algorithms, i.e., if the set for n is known, can we characterize the set for n+1?
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	Familiarity with the following topics is desirable:		Familiarity with the following topics is desirable:

	* Advanced knowledge of linear ordinary differential equations.		* Advanced knowledge of linear ordinary differential equations.
	* Optimization.		* Optimization.

Murray at 03:38, 28 November 2018

2018-11-28T03:38:26Z

← Older revision		Revision as of 03:38, 28 November 2018
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			'''[[SURF 2019\|2019 SURF]]: project description'''
			* Mentor: Richard M. Murray
			* Co-mentor: Petter Nilsson

	The concept of a ''control-invariant'' set---a set <math>X</math> with the property that a controller can make the state <math>x(t)</math> of a system remain inside of <math>X</math> for all positive <math>t</math> ---is closely connected with safety and reliability of engineered systems [1]. Invariant sets can be used to construct controllers that make systems such as quadrotors avoid crashes [2]. Unfortunately, analytical expressions for control-invariant sets are not known for many important systems, and are also hard to compute numerically in high dimensions.		The concept of a ''control-invariant'' set---a set <math>X</math> with the property that a controller can make the state <math>x(t)</math> of a system remain inside of <math>X</math> for all positive <math>t</math> ---is closely connected with safety and reliability of engineered systems [1]. Invariant sets can be used to construct controllers that make systems such as quadrotors avoid crashes [2]. Unfortunately, analytical expressions for control-invariant sets are not known for many important systems, and are also hard to compute numerically in high dimensions.

Lnilsson at 00:21, 28 November 2018

2018-11-28T00:21:54Z

← Older revision		Revision as of 00:21, 28 November 2018
Line 7:		Line 7:
	[[File:N_integrator.png\|200px]]		[[File:N_integrator.png\|200px]]

	For <math>n</math> = 2 a closed-form expression of the maximal control-invariant set contained inside the unit hypercube is known [3. eq. (53)], but for higher orders of <math>n</math> only numerical approaches are available. The figures above show numerical approximations for <math>n</math> equal to 2 and <math>n</math> equal to 3. The objective of this project is to investigate geometrical properties of control-invariant sets via a combination of analytical and numerical techniques. In particular,		For <math>n</math> = 2 a closed-form expression of the maximal control-invariant set contained inside the unit hypercube is known [3. eq. (53)], but for higher orders of <math>n</math> only numerical approaches are currently available. The figures above show numerical approximations for <math>n</math> equal to 2 and <math>n</math> equal to 3. The objective of this project is to investigate geometrical properties of control-invariant sets via a combination of analytical and numerical techniques. In particular,

	* Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets for n > 2.		* Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets for n > 2.

Lnilsson at 00:21, 28 November 2018

2018-11-28T00:21:21Z

← Older revision		Revision as of 00:21, 28 November 2018
Line 1:		Line 1:
	The concept of a ''control-invariant'' set---a set <math>X</math> with the property that a controller can make the state <math>x(t)</math> of a system remain inside of <math>X</math> for all positive <math>t</math> ---is closely connected with safety and reliability of engineered systems [1]. Invariant sets can be used to construct controllers that make systems such as quadrotors avoid crashes [2]. Unfortunately, analytical expressions for control-invariant sets are not known for many important systems, and are also hard to compute numerically in high dimensions.		The concept of a ''control-invariant'' set---a set <math>X</math> with the property that a controller can make the state <math>x(t)</math> of a system remain inside of <math>X</math> for all positive <math>t</math> ---is closely connected with safety and reliability of engineered systems [1]. Invariant sets can be used to construct controllers that make systems such as quadrotors avoid crashes [2]. Unfortunately, analytical expressions for control-invariant sets are not known for many important systems, and are also hard to compute numerically in high dimensions.

	[[File:Double_int.png]] [[File:Triple_int.png]]		[[File:Double_int.png\|350px]] [[File:Triple_int.png\|350px]]

	Some of the most fundamental dynamical systems are linear <math>n</math>-order integrators. For <math>n</math> = 2 a closed-form expression of the maximal control-invariant set contained inside the unit hypercube is known [3. eq. (53)], but for higher orders of <math>n</math> only numerical approaches are available. The figures above show numerical approximations for <math>n</math> equal to 2 and <math>n</math> equal to 3. The objective of this project is to investigate geometrical properties of control-invariant sets via a combination of analytical and numerical techniques. In particular,		Some of the most fundamental dynamical systems are linear <math>n</math>-order integrators:

			[[File:N_integrator.png\|200px]]

			For <math>n</math> = 2 a closed-form expression of the maximal control-invariant set contained inside the unit hypercube is known [3. eq. (53)], but for higher orders of <math>n</math> only numerical approaches are available. The figures above show numerical approximations for <math>n</math> equal to 2 and <math>n</math> equal to 3. The objective of this project is to investigate geometrical properties of control-invariant sets via a combination of analytical and numerical techniques. In particular,

	* Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets for n > 2.		* Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets for n > 2.

Lnilsson at 00:18, 28 November 2018

2018-11-28T00:18:14Z

← Older revision		Revision as of 00:18, 28 November 2018
Line 1:		Line 1:
	The concept of a ''control-invariant'' set---a set <math>X</math> with the property that a controller can make the state <math>x(t)</math> of a system remain inside of <math>X</math> for all positive <math>t</math> ---is closely connected with safety and reliability of engineered systems [1]. Unfortunately, analytical expressions for control-invariant sets are not known for many important systems, and are also hard to compute numerically in high dimensions.		The concept of a ''control-invariant'' set---a set <math>X</math> with the property that a controller can make the state <math>x(t)</math> of a system remain inside of <math>X</math> for all positive <math>t</math> ---is closely connected with safety and reliability of engineered systems [1]. Invariant sets can be used to construct controllers that make systems such as quadrotors avoid crashes [2]. Unfortunately, analytical expressions for control-invariant sets are not known for many important systems, and are also hard to compute numerically in high dimensions.

	[[File:~~Example~~.~~jpg~~]]		[[File:Double_int.png]] [[File:Triple_int.png]]

	Some of the most fundamental dynamical systems are linear <math>n</math>-order integrators. For <math>n</math> = 2 a closed-form expression of the maximal control-invariant set contained inside the ~~unite~~ hypercube is known [2. eq. (53)], but for higher orders of <math>n</math> only numerical approaches are available. The ~~Figure~~ above ~~shows~~ numerical approximations for <math>n</math> equal to 2 and <math>n</math> equal to 3. The objective of this project is to investigate geometrical properties of ~~these~~ sets via a combination of analytical and numerical techniques. In particular,		Some of the most fundamental dynamical systems are linear <math>n</math>-order integrators. For <math>n</math> = 2 a closed-form expression of the maximal control-invariant set contained inside the unit hypercube is known [3. eq. (53)], but for higher orders of <math>n</math> only numerical approaches are available. The figures above show numerical approximations for <math>n</math> equal to 2 and <math>n</math> equal to 3. The objective of this project is to investigate geometrical properties of control-invariant sets via a combination of analytical and numerical techniques. In particular,

	* Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets of for n ~~larger than 3~~.		* Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets for n > 2.
	* Investigate incremental algorithms, i.e., if the set for $n$ is known, can we characterize the set for $n+1$?		* Investigate incremental algorithms, i.e., if the set for n is known, can we characterize the set for n+1?
	* Stretch goal: generalize the incremental ideas to differential extensions of general systems such as the quadrotor dynamics on $SE(3)$.		* Stretch goal: generalize the incremental ideas to differential extensions of general systems such as the quadrotor dynamics on SE(3).

	Familiarity with the following topics is desirable:		Familiarity with the following topics is desirable:
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	[1] Blanchini, F. (1999). Set invariance in control. Automatica, 35(11), 1747–1767. https://doi.org/10.1016/S0005-1098(99)00113-2		[1] Blanchini, F. (1999). Set invariance in control. Automatica, 35(11), 1747–1767. https://doi.org/10.1016/S0005-1098(99)00113-2

	[2] Ames, A. D., Xu, X., Grizzle, J. W., & Tabuada, P. (2017). Control Barrier Function Based Quadratic Programs for Safety Critical Systems. IEEE Transactions on Automatic Control, 62(8), 3861–3876. https://doi.org/10.1109/TAC.2016.2638961		[2] https://www.youtube.com/watch?v=rK9oyqccMJw

			[3] Ames, A. D., Xu, X., Grizzle, J. W., & Tabuada, P. (2017). Control Barrier Function Based Quadratic Programs for Safety Critical Systems. IEEE Transactions on Automatic Control, 62(8), 3861–3876. https://doi.org/10.1109/TAC.2016.2638961

Lnilsson: Created page with "The concept of a ''control-invariant'' set---a set $X$ with the property that a controller can make the state $x(t)$ of a system remain inside of $X..."$

2018-11-28T00:05:42Z

Created page with "The concept of a ''control-invariant'' set---a set <math>X</math> with the property that a controller can make the state <math>x(t)</math> of a system remain inside of <math>X..."

New page

The concept of a ''control-invariant'' set---a set <math>X</math> with the property that a controller can make the state <math>x(t)</math> of a system remain inside of <math>X</math> for all positive <math>t</math> ---is closely connected with safety and reliability of engineered systems [1]. Unfortunately, analytical expressions for control-invariant sets are not known for many important systems, and are also hard to compute numerically in high dimensions.

[[File:Example.jpg]]

Some of the most fundamental dynamical systems are linear <math>n</math>-order integrators. For <math>n</math> = 2 a closed-form expression of the maximal control-invariant set contained inside the unite hypercube is known [2. eq. (53)], but for higher orders of <math>n</math> only numerical approaches are available. The Figure above shows numerical approximations for <math>n</math> equal to 2 and <math>n</math> equal to 3. The objective of this project is to investigate geometrical properties of these sets via a combination of analytical and numerical techniques. In particular,

* Search for closed-form expressions or cheap-to-evaluate algorithms that characterize control-invariant sets of for n larger than 3.
* Investigate incremental algorithms, i.e., if the set for $n$ is known, can we characterize the set for $n+1$?
* Stretch goal: generalize the incremental ideas to differential extensions of general systems such as the quadrotor dynamics on $SE(3)$.

Familiarity with the following topics is desirable:

* Advanced knowledge of linear ordinary differential equations.
* Optimization.
* Programming in Matlab and/or Python (the figures above were created with code from https://github.com/pettni/pcis).

----

'''References'''

[1] Blanchini, F. (1999). Set invariance in control. Automatica, 35(11), 1747–1767. https://doi.org/10.1016/S0005-1098(99)00113-2

[2] Ames, A. D., Xu, X., Grizzle, J. W., & Tabuada, P. (2017). Control Barrier Function Based Quadratic Programs for Safety Critical Systems. IEEE Transactions on Automatic Control, 62(8), 3861–3876. https://doi.org/10.1109/TAC.2016.2638961

SURF 2019: Geometry of Control-Invariant Sets - Revision history

Murray at 03:40, 28 November 2018

Murray at 03:38, 28 November 2018

Lnilsson at 00:21, 28 November 2018

Lnilsson at 00:21, 28 November 2018

Lnilsson at 00:18, 28 November 2018

Lnilsson: Created page with "The concept of a ''control-invariant'' set---a set X with the property that a controller can make the state x(t) of a system remain inside of X..."

Lnilsson: Created page with "The concept of a ''control-invariant'' set---a set $X$ with the property that a controller can make the state $x(t)$ of a system remain inside of $X..."$