Denbigh Reaction: Difference between revisions

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

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Revision as of 09:25, 21 August 2025

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):

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

where X is an intermediate, Y is the desired product, and P and Q are waste products.

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

Mathematical formulation

$\begin{array}{lll} \max_{u} & x_{3} (t_{f}) \\ subject to \\ \dot{x_{1}} (t) & = & - k_{1} (t) \cdot x_{1} (t) - k_{2} (t) \cdot x_{1} (t), \\ \dot{x_{2}} (t) & = & k_{1} (t) \cdot x_{1} (t) - k_{3} (t) + k_{4} (t) \cdot x_{2} (t), \\ \dot{x_{3}} (t) & = & k_{3} (t) \cdot x_{2} (t), \\ k_{i} (t) & = & k_{i}^{*} \cdot \exp (\frac{- E_{i}}{T (t)}), i = 1, \dots, 4, \\ x (0) & = & (1, 0, 0)^{T}, \\ T (t) & \in & [273, 415] \forall t \in [0, t_{f}] \end{array}$

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.

@@ Line 24: / Line 24: @@
 <math>
   \begin{array}{lll}
-  \displaystyle \min_{u} && t_f \\
+  \displaystyle \max_{u} && x_3(t_f) \\
   \text{subject to} \\
-\quad \dot{x}(t) & = & v(t),\\
+\quad \dot{x_1}(t) & = & -k_1(t) \cdot x_1(t) - k_2(t) \cdot x_1(t),\\
-\quad \dot{v}(t) & = & 0.001 \cdot u(t) - 0.0025 \cdot \cos(3 \cdot x(t)), \\
+\quad \dot{x_2}(t) & = & k_1(t) \cdot x_1(t) - k_3(t) + k_4(t) \cdot x_2(t),\\
-\quad x(0) &=& -0.5, \\
+\quad \dot{x_3}(t) & = & k_3(t) \cdot x_2(t),\\
-\quad v(0) &=& 0, \\
+\quad k_i(t) & = & k_i^* \cdot \exp\left( \frac{-E_i}{T(t)} \right), \ i=1,\ldots,4, \\
-\quad x(t_f) &=& 0.5, \\
+\quad x(0) &=& (1, 0, 0)^T, \\
-\quad v(t_f) & \geq & 0, \\
+\quad T(t) & \in & [273, 415] \ \quad \forall t \in [0,t_f]
-\quad u(t) & \in & [-1, 1] \ \quad \forall t \in [0,t_f]
    \end{array}
 </math>