Linear Quadratic Regulator - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T13:46:00Z

Reference Solutions

← Older revision		Revision as of 13:46, 28 November 2025
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	Here is one local solution to the above control problem.		Here is one local solution to the above control problem.

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:LQR.png\| States and discretized control for a local optimum.		Image:LQR.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel at 13:21, 21 August 2025

2025-08-21T13:21:22Z

← Older revision		Revision as of 13:21, 21 August 2025
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	}}		}}

	The '''Linear Quadratic Regulator problem''' is a one-dimensional toy [[:Category:ODE model\|ODE model]] which is especially suited for multiple shooting solvers. It aims to minimize a quadratic Lagrange term.		The '''Linear Quadratic Regulator problem''' is a popular one-dimensional toy [[:Category:ODE model\|ODE model]] which is especially suited for multiple shooting solvers. It aims to minimize a quadratic Lagrange term.

	The optimal control function exhibits a [[:Category:Sensitivity-seeking arcs\|singular arc]].		The optimal control function exhibits a [[:Category:Sensitivity-seeking arcs\|singular arc]].
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	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	The problem description and further references can be found ~~in the PhD thesis of Michael Ernst Geiger~~ [[#~~GeigerPhD~~\|[1]]].		The problem description and further references can be found on Wikipedia [[#Wikipedia\|[1]]].

	== References ==		== References ==
	<span id="~~GeigerPhD~~">[1]</span> ~~"Adaptive Multiple Shooting for Boundary Value Problems and Constrained Parabolic Optimization Problems" by M~~. E. ~~Geiger~~ <br>		<span id="Wikipedia">[1]</span> https://en.wikipedia.org/wiki/Linear%E2%80%93quadratic_regulator <br>


	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:Sensitivity-seeking arcs]]		[[Category:Sensitivity-seeking arcs]]

RobertLampel at 13:16, 21 August 2025

2025-08-21T13:16:06Z

← Older revision		Revision as of 13:16, 21 August 2025
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	The '''Linear Quadratic Regulator problem''' is a one-dimensional toy [[:Category:ODE model\|ODE model]] which is especially suited for multiple shooting solvers. It aims to minimize a quadratic Lagrange term.		The '''Linear Quadratic Regulator problem''' is a one-dimensional toy [[:Category:ODE model\|ODE model]] which is especially suited for multiple shooting solvers. It aims to minimize a quadratic Lagrange term.

	The optimal ~~integer~~ control ~~functions~~ exhibits a [[:Category:Sensitivity-seeking arcs\|singular arc]].		The optimal control function exhibits a [[:Category:Sensitivity-seeking arcs\|singular arc]].

	== Mathematical formulation ==		== Mathematical formulation ==

RobertLampel: /* Mathematical formulation */

2025-08-21T13:15:21Z

Mathematical formulation

← Older revision		Revision as of 13:15, 21 August 2025
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	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x}(t) & = & a \cdot x(t) + b \cdot u(t),\\		\quad \dot{x}(t) & = & a \cdot x(t) + b \cdot u(t),\\
	\quad x(0) &=& 1,		\quad x(0) &=& 1
	\end{array}		\end{array}
	</math>		</math>
	</p>		</p>


	== Parameters ==		== Parameters ==

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 1 |nw = 1 }} The '''Linear Quadratic Regulator problem''' is a one-dimensional toy ODE model which is especially suited for multiple shooting solvers. It aims to minimize a quadratic Lagrange term. The optimal integer control functions exhibits a singular arc. == Mathematical formulation ==
$\begin{array}{lll} \displaystyle \min_{x,w} && \int_0^{1..."$

2025-08-21T13:14:15Z

Created page with "{{Dimensions |nd = 1 |nx = 1 |nw = 1 }} The '''Linear Quadratic Regulator problem''' is a one-dimensional toy ODE model which is especially suited for multiple shooting solvers. It aims to minimize a quadratic Lagrange term. The optimal integer control functions exhibits a singular arc. == Mathematical formulation == <math> \begin{array}{lll} \displaystyle \min_{x,w} && \int_0^{1..."

New page

{{Dimensions
|nd = 1
|nx = 1
|nw = 1
}}

The '''Linear Quadratic Regulator problem''' is a one-dimensional toy [[:Category:ODE model|ODE model]] which is especially suited for multiple shooting solvers. It aims to minimize a quadratic Lagrange term.

The optimal integer control functions exhibits a [[:Category:Sensitivity-seeking arcs|singular arc]].

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{x,w} && \int_0^{10} 10 \cdot (x(t)-3)^2 + 0.1 \cdot u(t)^2 dt \\
\text{subject to} \\
\quad \dot{x}(t) & = & a \cdot x(t) + b \cdot u(t),\\
\quad x(0) &=& 1,
\end{array}
</math>


== Parameters ==
We choose <math>a = -1</math> and <math>b = 1</math>.

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
Image:LQR.png| States and discretized control for a local optimum.
</gallery>

== Miscellaneous and Further Reading ==
The problem description and further references can be found in the PhD thesis of Michael Ernst Geiger [[#GeigerPhD|[1]]].

== References ==
[1] "Adaptive Multiple Shooting for Boundary Value Problems and Constrained Parabolic Optimization Problems" by M. E. Geiger 

[[Category:MIOCP]]
[[Category:Sensitivity-seeking arcs]]