Jackson OED

Jackson OED
State dimension:	1
Differential states:	13
Discrete control functions:	3

The Jackson OED problem is a variation of the Jackson problem. It looks for optimal time intervals to measure the three states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale ODE model. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

Mathematical formulation

We are interested in estimating the parameters $k_1$ and $k_2$ of the initial value problem

$\begin{array}{rclll}\dot{x_1}(t)& =& -u(t)(k_1{x}_1(t)-k_2{x}_2(t)),& & t\in [0,t_f],x_1(0)=1,\\ \dot{x_2}(t)& =& u(t)(k_1{x}_1(t)-k_2{x}_2(t))-(1-u(t))k_3{x}_2(t),& & t\in [0,t_f],x_2(0)=0,\\ \dot{x_3}(t)& =& (1-u(t))k_3{x}_2(t),& & t\in [0,t_f],x_3(0)=0.\end{array}$

The initial values and $t_f$ are fixed. We are interested in how to choose the control $u$ and when to measure, with an upper bound $M$ on the measuring time. We can measure the states $x_1$ and $x_2$ directly, $h^1(x(t))=x_1(t)$ and $h^2(x(t))=x_2(t)$. We use two different sampling functions, $w_1(\cdot )$ and $w_2(\cdot )$ in the same experimental setting. This can be seen either as a two-dimensional measurement function $h(x(t))$, or as a special case of a multiple experiment, in which $u(\cdot ),x(\cdot )$, and $G(\cdot )$ are identical.

Now we formulate the OED problem with $\theta :=(k_1,k_2)$:

$\begin{array}{lll}\underset{x,G,F,z,w,u}{\min }& & \text{trace}(F^{-1}(t_f))\\ \text{subject to}\\ \dot{x}(t)& =& f(x(t),u(t),\theta )\\ \dot{G}(t)& =& f_x(x(t),u(t),\theta )G(t)+f_{\theta }(x(t),u(t),\theta )\\ \dot{F}(t)& =& \sum _{i=1}^{n_o}{w}_i(t)(h_x^i(x(t))G(t){)}^T(h_x^i(x(t))G(t))\\ \dot{z}(t)& =& w(t),\\ x(0)& =& x_0\\ G(0)& =& \frac{\partial x(0)}{\partial \theta }\\ F(0)& =& I\cdot {\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{g}},\\ z(0)& =& 0\\ u(t)& \in & {𝒰}\\ w(t)& \in & {𝒲}\\ z_i(t_f)& \le & M_i\end{array}$

The evolution of the symmetric matrix $F:[0,t_f]\to {\mathbb{ℝ}}^{2\times 2}$ is given by the weighted sum of observability Gramians $h_x^i(x(t))G(t),i=1,2,$ for each observed function of states.

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$k_1$	1	Interaction between $x_1$ and $x_2$
$k_2$	10	Interaction between $x_1$ and $x_2$
$k_3$	1	Growth of $x_3$ under complementary control
$t_{\mathrm{f}}$	1	Horizon of the control problem
${\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{g}}$	0.01	Regularization of Fisher matrix
${𝒰}$	[0,1]	Bounds of control function
${𝒲}$	[0,1]	Bounds of measurement function
$M_1,M_2$	0.2	Maximum measurement time

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  States, control, and sampling functions for a local optimum. Both measurement functions overlap.

References

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