View source for Lotka Shared OED

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The '''Lotka Shared Experimental Design problem''' looks for an optimal strategy to be performed on a fixed time horizon to control the biomasses multiple species as described in [[:LV Shared Resource]]. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. When measurements of the three state variables are performed becomes a degree of freedom.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The ODE from [[LV Shared Resource]] is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==

We are interested in estimating the parameters <math>\alpha_0, \alpha_1,</math> and <math>\alpha_2</math> of the Lotka-Volterra type predator-prey initial value problem
<p>
<math>
\begin{array}{rcl}
\dot{x}_0(t) & = &  x_0(t) - \alpha_0 x_0(t) x_1(t) - x_0(t) x_2(t), \\
\dot{x}_1(t) & = & - x_1(t) + \alpha_1 x_0(t) x_1(t) - c_1 x_1(t) u(t),  \\
\dot{x}_2(t) & = & -x_2(t) + \alpha_2 x_0(t) x_2(t) - c_2 x_2(t) u(t),
\end{array} 
</math>
</p>
where <math>u(\cdot)</math> is a control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 20</math> are fixed. We are interested in how to choose <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=0,1,2</math>. We use three different sampling functions, <math>w^i(\cdot), i=0,1,2,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (\alpha_0, \alpha_1, \alpha_2)</math>:
<p>
<math>
 \begin{array}{lll}
 \displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
 \text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & (1.5, 0.5, 1)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\ 
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
  \end{array}
</math>
</p>

The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 0,1,2,</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model: 

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\alpha_0</math> || align=right | 1.0  ||
|-
| align=center | <math>\alpha_1</math> || align=right | 1.0  ||
|-
| align=center | <math>\alpha_2</math> || align=right | 1.2  ||
|-
| align=center | <math>c_1</math> || align=right | 0.1 || 
|-
| align=center | <math>c_2</math> || align=right | 0.4 || 
|- 
| align=center | <math>t_\mathrm{f}</math> || align=right | 20 || Horizon of the control problem
|- 
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.1 || Regularization of Fisher matrix
|- 
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|- 
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|- 
| align=center | <math>M_0, M_1, M_2</math> || align=right | 4 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
 Image:Lotka_Shared_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:Casadi | CasADi]] at https://github.com/rlampel/PySHeRLOC

== Miscellaneous and Further Reading ==
The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.

== References ==
<biblist />

<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]
[[Category:Population dynamics]]