Tubular Reactor - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T13:46:48Z

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:Tubular_Reactor.png\| States and discretized control for a local optimum.		Image:Tubular_Reactor.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* References */

2025-08-21T11:19:12Z

References

← Older revision		Revision as of 11:19, 21 August 2025
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	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:~~Path~~-~~constrained~~ arcs]]		[[Category:Sensitivity-seeking arcs]]

RobertLampel: /* Miscellaneous and Further Reading */

2025-08-20T14:59:06Z

Miscellaneous and Further Reading

← Older revision		Revision as of 14:59, 20 August 2025
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	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	This formulation detailed description can be found in [[#apmonitor\|[1]]].		This formulation and a detailed description can be found in [[#apmonitor\|[1]]].

	== References ==		== References ==

RobertLampel: /* References */

2025-08-20T09:49:58Z

References

← Older revision		Revision as of 09:49, 20 August 2025
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	== References ==		== References ==
	<span id="~~BH75~~">[1]</span> ~~Arthur E Bryson and Yu-Chi Ho. Applied Optimal Control: Optimization, Estimation and Control. CRC Press, 1975. <br>~~		<span id="apmonitor">[1]</span> https://apmonitor.com/do/index.php/Main/DynamicOptimizationBenchmarks <br>
	~~<span id="openmdao">[2]</span> Multidisciplinary Optimal Control Library:~~ https://~~openmdao~~.~~org~~/~~dymos~~/~~docs~~/~~latest~~/~~examples/bryson_denham/bryson_denham.html~~<br>


	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:Path-constrained arcs]]		[[Category:Path-constrained arcs]]

RobertLampel: /* Miscellaneous and Further Reading */

2025-08-20T09:49:38Z

Miscellaneous and Further Reading

← Older revision		Revision as of 09:49, 20 August 2025
Line 32:		Line 32:

	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	~~The Bryson-Denham problem is a variation of the double integrator problem [[#BH75\|[1]]].~~ This formulation detailed description can be found in [[#~~openmdao~~\|[2]]].		This formulation detailed description can be found in [[#apmonitor\|[1]]].

	== References ==		== References ==

RobertLampel: /* Reference Solutions */

2025-08-20T09:47:48Z

Reference Solutions

← Older revision		Revision as of 09:47, 20 August 2025
Line 28:		Line 28:

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

RobertLampel: /* Mathematical formulation */

2025-08-20T09:43:10Z

Mathematical formulation

← Older revision		Revision as of 09:43, 20 August 2025
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	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{w} && -x_2(1) dt \\		\displaystyle \min_{w} && -x_2(1) \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x_1}(t) & = & -(w(t) + \frac{1}{2} \cdot w(t)^2) \cdot x_1(t),\\		\quad \dot{x_1}(t) & = & -(w(t) + \frac{1}{2} \cdot w(t)^2) \cdot x_1(t),\\

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 1 }} The '''Tubular Reactor problem''' is a two-dimensional ODE model. It aims to maximize the value of the second differential state at the end of the time interval. The optimal integer control functions exhibits a singular arc. == Mathematical formulation ==
$\begin{array}{lll} \displaystyle \min_{w} && -x_2(1) dt \\ \text{subject to} \..."$

2025-08-20T09:42:59Z

Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 1 }} The '''Tubular Reactor problem''' is a two-dimensional ODE model. It aims to maximize the value of the second differential state at the end of the time interval. The optimal integer control functions exhibits a singular arc. == Mathematical formulation == <math> \begin{array}{lll} \displaystyle \min_{w} && -x_2(1) dt \\ \text{subject to} \..."

New page

{{Dimensions
|nd = 1
|nx = 2
|nw = 1
}}

The '''Tubular Reactor problem''' is a two-dimensional [[:Category:ODE model|ODE model]]. It aims to maximize the value of the second differential state at the end of the time interval.

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

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{w} && -x_2(1) dt \\
\text{subject to} \\
\quad \dot{x_1}(t) & = & -(w(t) + \frac{1}{2} \cdot w(t)^2) \cdot x_1(t),\\
\quad \dot{x_2}(t) & = & w(t) \cdot x_1(t), \\
\quad x(0) &=& (1, 0)^T, \\
\quad w(t) & \in & [0, 5]
\end{array}
</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:Bryson-Denham.png| States and discretized control for a local optimum.
</gallery>

== Miscellaneous and Further Reading ==
The Bryson-Denham problem is a variation of the double integrator problem [[#BH75|[1]]]. This formulation detailed description can be found in [[#openmdao|[2]]].

== References ==
[1] Arthur E Bryson and Yu-Chi Ho. Applied Optimal Control: Optimization, Estimation and Control. CRC Press, 1975. 
[2] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/bryson_denham/bryson_denham.html 

[[Category:MIOCP]]
[[Category:Path-constrained arcs]]