Oil Shale Pyrolysis

2015-12-08T17:17:19Z

ManuelTetschke: Created page with "{{Dimensions |nd = 1 |nx = 2 |nu = 1 |nw = 0 |nre = 2 }} The following problem is an example from the global optimal control literature and..."

{{Dimensions
|nd = 1
|nx = 2
|nu = 1
|nw = 0
|nre = 2
}}

The following problem is an example from the global optimal control literature and was introduced in <bibref>Wen1977</bibref>.

== Mathematical formulation ==

<math>
\begin{array}{ll}
\displaystyle \min_{u} & \displaystyle -x_1(t_N)^2 \\[1.5ex]
\mbox{s.t.} & \displaystyle \dot{x}_0(t) = -k_0x_0(t)-(k_2+k_3+k_4)x_0(t)x_1(t)\\
& \displaystyle \dot{x}_1(t) = k_0x_0(t)-k_1x_1(t) + k_2x_0(t)x_1(t)\\
& \displaystyle k_i = a_i e^{-u(t)\frac{b_i}{R}},\quad \forall i\in \{1,\dots,5\} \\ [1.5ex]
& \displaystyle t \in \left[t_0,t_N\right] \\
& \displaystyle u(t) \in \left[698.15/748.15,1\right]\\
& \displaystyle x(t_0) = (1,0)^T\\
\end{array}
</math>

where this is the normalized form with

<math> u(t)= \frac{1}{u_{temp}} </math>, with

<math> u_{temp} \in \left[698.15,748.15\right] </math>

== Parameters ==

{| class="wikitable"
|+State variables
|-
|Symbol
|Initial value (<math>t_0</math>)
|-
|<math>x_0(t)</math>
|<math>1</math>
|-
|<math>x_1(t)</math>
|<math>0 </math>
|}

{| class="wikitable"
|+Parameters
|-
|Symbol
|Value
|-
|<math>a_1</math>
|<math>8.86</math>
|-
|<math>a_2</math>
|<math>24.25</math>
|-
|<math>a_3</math>
|<math>23.67</math>
|-
|<math>a_4</math>
|<math>18.75</math>
|-
|<math>a_5</math>
|<math>20.7</math>
|-
|<math>b_1</math>
|<math>20.3</math>
|-
|<math>b_2</math>
|<math>37.4</math>
|-
|<math>b_3</math>
|<math>33.8</math>
|-
|<math>b_4</math>
|<math>28.2</math>
|-
|<math>b_5</math>
|<math>31.0</math>
|}

{| class="wikitable"
|+Control variable
|-
|Symbol
|Interval
|-
|<math>u(t)</math>
|[698.15/748.15,1]
|}

'''Measurement grid'''

== Reference solution ==

Coming soon.

== References ==
<bibreferences/>


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Oil Shale Pyrolysis