Three Tank OED

Three Tank OED
State dimension:	1
Differential states:	21
Discrete control functions:	6

The Three Tank OED problem is a variation of the Three Tank multimode problem. It looks for optimal time intervals to measure the two 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 $c_1,c_2,$ and $c_3$ of the initial value problem

$\begin{array}{rclll}{\dot{x}}_1(t)& =& -\sqrt{x_1(t)}+c_1{u}_1(t)+c_2{u}_2(t)-u_3(t)\sqrt{c_3{x}_1(t)},& & t\in [0,t_f],x_1(0)=2,\\ {\dot{x}}_2(t)& =& \sqrt{x_1(t)}-\sqrt{x_2(t)},& & t\in [0,t_f],x_2(0)=2,\\ {\dot{x}}_3(t)& =& \sqrt{x_2(t)}-\sqrt{x_3(t)}+u_3(t)\sqrt{c_3{x}_1(t)},& & t\in [0,t_f],x_3(0)=2.\end{array}$

Additionally, the controls $u_1,u_2,$ and $u_3$ are constrained by:

$\sum _{i=1}^3{u}_i(t)=1\mathrm{\forall }t\in [0,t_f]$

The initial values and $t_f=12$ are fixed. We are interested in how to choose the controls $u_i,i=1,2,3,$ and when to measure, with an upper bound $M$ on the measuring time. We can measure the states directly, i.e., $h^i(x(t))=x_i(t),i=1,2,3$. We use three different sampling functions, $w^i(\cdot ),i=1,2,3,$ in the same experimental setting. This can be seen either as a three-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 $\theta :=(p_1,p_2)$:

$\begin{array}{lll}\underset{x,G,F,z,w,u}{\min }& & \text{trace}(F^{-1}(t_f))\\ \text{subject to}\\ \dot{x}(t)& =& f(x(t),u(t),\theta )\\ \dot{G}(t)& =& f_x(x(t),u(t),\theta )G(t)+f_{\theta }(x(t),u(t),\theta )\\ \dot{F}(t)& =& \sum _{i=1}^3{w}_i(t)(h_x^i(x(t))G(t){)}^T(h_x^i(x(t))G(t))\\ \dot{z}(t)& =& w(t),\\ x(0)& =& (2,2,2)\\ G(0)& =& \frac{\partial x(0)}{\partial \theta }\\ F(0)& =& I\cdot {\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{g}},\\ z(0)& =& 0\\ x_1(t)& \in & {𝒳}\\ u(t)& \in & {𝒰}\\ w(t)& \in & {𝒲}\\ z_i(t_f)& \le & M_i\end{array}$

The evolution of the symmetric matrix $F:[0,t_f]\to {\mathbb{ℝ}}^{3\times 3}$ is given by the weighted sum of observability Gramians $h_x^i(x(t))G(t),i=1,2,3,$ for each observed function of states.

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$p_1$	1	Unknown parameter
$p_2$	1	Unknown parameter
$t_{\mathrm{f}}$	10	Horizon of the control problem
${\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{g}}$	0.1	Regularization of Fisher matrix
${𝒳}$	[-0.5,$\mathrm{\infty }$]	Bounds of $x_1$
${𝒰}$	[-1,1]	Bounds of control function
${𝒲}$	[0,1]	Bounds of measurement function
$M_1,M_2$	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.

References

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