Jackson

Jackson
State dimension:	1
Differential states:	6
Discrete control functions:	3

The Jackson problem is a classical benchmark in optimal control. This description is taken from [1].

It consists of controlling a three-dimensional system in which the first two states interact linearly under the effect of a single control input, while the third state accumulates based on the complementary control. The objective is to minimise the third state at the final time, while satisfying bounds on states and control, as well as initial and terminal conditions. The problem exhibits singular arcs, making it a useful benchmark for testing direct transcription and nonlinear programming methods.

Mathematical formulation

$\begin{array}{lll}\underset{u}{\min }& & x_3(t_{\mathrm{f}})\\ \text{subject to}\\ \dot{x_1}(t)& =& -u(t)(k_1{x}_1(t)-k_2{x}_2(t)),\\ \dot{x_2}(t)& =& u(t)(k_1{x}_1(t)-k_2{x}_2(t))-(1-u(t))k_3{x}_2(t),\\ \dot{x_3}(t)& =& (1-u(t))k_3{x}_2(t),\\ x(0)& =& (1,0,0{)}^T,\\ x_{(}t)& \in & [0,1.1]\mathrm{\forall }t\in [0,t_{\mathrm{f}}],i\in \{1,2,3\}\\ u(t)& \in & [0,1]\mathrm{\forall }t\in [0,t_{\mathrm{f}}]\end{array}$

Parameters

These fixed values are used within the model:

Symbol	Value
$m$	2.2 $\mathrm{k}\mathrm{g}$
$J$	0.05 $\mathrm{k}\mathrm{g}\mathrm{\cdot }{\mathrm{m}}^{\mathrm{2}}$
$r$	0.2 $\mathrm{m}$
$\mathrm{m}\mathrm{g}$	4 $\mathrm{N}$
$\mu$	1

The weight $\mu$ balances control effort and transition time.

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] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/ducted_fan.md