Denbigh Reaction

Denbigh Reaction
State dimension:	1
Differential states:	3
Discrete control functions:	1

The Mountain Car problem s based on the system of chemical reactions initially considered by Denbigh (1958), which was also studied by Aris (1960) and more recently by Luus (1994):

$\begin{array}{rl}A+B& \to X\\ X& \to Q\\ X& \to Y\\ A+X& \to P\end{array}$

where $X$ is an intermediate, $Y$ is the desired product, and $P$ and $Q$ are waste products. The optimal control problem is to find $T(t)$ (the temperature of the reactor as a function of time) so that the yield of $Y$ is maximized at the end of the given batch time $t_f$.

Its dynamics are given by a three-dimensional ODE model. The optimal integer control functions exhibits a bang bang structure.

Mathematical formulation

$\begin{array}{lll}\underset{u}{\max }& & x_3(t_f)\\ \text{subject to}\\ \dot{x_1}(t)& =& -k_1(t)\cdot x_1(t)-k_2(t)\cdot x_1(t),\\ \dot{x_2}(t)& =& k_1(t)\cdot x_1(t)-k_3(t)+k_4(t)\cdot x_2(t),\\ \dot{x_3}(t)& =& k_3(t)\cdot x_2(t),\\ k_i(t)& =& k_i^{*}\cdot \exp \left(\frac{-E_i}{T(t)}\right),i=1,\dots ,4,\\ T(t)& \in & [273,415]\mathrm{\forall }t\in [0,t_f]\\ x(0)& =& (1,0,0{)}^T\end{array}$

Parameters

Parameters

Symbol	Value
$E_1$	$10^3$
$E_2$	$10^7$
$E_3$	$10$
$E_4$	$10^{-3}$
$k_1^{*}$	$3\cdot 10^3$
$k_2^{*}$	$6\cdot 10^3$
$k_3^{*}$	$3\cdot 10^3$
$k_4^{*}$	$0$
$t_f$	$10^3$

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  States and discretized control for a local optimum.

Miscellaneous and Further Reading

This formulation and a detailed description can be found in [1].

References

[1] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/mountain_car/mountain_car.html
[2] Andrew William Moore. Efficient memory-based learning for robot control. Technical Report UCAM-CL-TR-209, University of Cambridge, Computer Laboratory, November 1990. URL: https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-209.pdf, doi:10.48456/tr-209.
[3] Alexey A Melnikov, Adi Makmal, and Hans J Briegel. Projective simulation applied to the grid-world and the mountain-car problem. arXiv preprint arXiv:1405.5459, 2014.