Jackson OED: Difference between revisions

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

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Latest revision as of 10:17, 26 March 2026

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

$\begin{array}{rcl} \dot{x_{1}} (t) & = & - u (t) (k_{1} x_{1} (t) - k_{2} x_{2} (t)), & t \in [0, t_{f}], x_{1} (0) = 1, \\ \dot{x_{2}} (t) & = & u (t) (k_{1} x_{1} (t) - k_{2} x_{2} (t)) - (1 - u (t)) k_{3} x_{2} (t), & t \in [0, t_{f}], x_{2} (0) = 0, \\ \dot{x_{3}} (t) & = & (1 - u (t)) k_{3} x_{2} (t), & t \in [0, t_{f}], x_{3} (0) = 0 . \end{array}$

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 $θ : = (k_{1}, k_{2})$ :

$\begin{array}{lll} \min_{x, G, F, z, w, u} & trace (F^{- 1} (t_{f})) \\ subject to \\ \dot{x} (t) & = & f (x (t), u (t), θ) \\ \dot{G} (t) & = & f_{x} (x (t), u (t), θ) G (t) + f_{θ} (x (t), u (t), θ) \\ \dot{F} (t) & = & \sum_{i = 1}^{n_{o}} w_{i} (t) (h_{x}^{i} (x (t)) G (t))^{T} (h_{x}^{i} (x (t)) G (t)) \\ \dot{z} (t) & = & w (t), \\ x (0) & = & x_{0} \\ G (0) & = & \frac{\partial x (0)}{\partial θ} \\ F (0) & = & I \cdot ε_{r e g}, \\ z (0) & = & 0 \\ u (t) & \in & 𝒰 \\ w (t) & \in & 𝒲 \\ z_{i} (t_{f}) & \leq & M_{i} \end{array}$

The evolution of the symmetric matrix $F : [0, t_{f}] \to ℝ^{2 \times 2}$ 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
$t_{f}$	1	Horizon of the control problem
$ε_{r e g}$	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

There were no citations found in the article.

@@ Line 22: / Line 22: @@
 </p>
-The initial values and <math>t_f</math> are fixed. We are interested in how to choose the control <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states <math>x_1</math> and <math>x_2</math> directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w^1(\cdot)</math> and <math>w^2(\cdot)</math> in the same experimental setting. This can be seen either as a two-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.
+The initial values and <math>t_f</math> are fixed. We are interested in how to choose the control <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states <math>x_1</math> and <math>x_2</math> directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w_1(\cdot)</math> and <math>w_2(\cdot)</math> in the same experimental setting. This can be seen either as a two-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 := (k_1, k_2)</math>:
@@ Line 30: / Line 30: @@
   \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),\theta) \\
+\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
-\quad \dot{G}(t) & = & f_x(x(t),\theta) G(t) + f_\theta(x(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=1}^{n_o} 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) & = & x_0 \\
 \quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
-\quad F(0) & = & 0, \\
+\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
 \quad z(0) & = & 0 \\
 \quad u(t) & \in & \mathcal{U} \\
@@ Line 46: / Line 46: @@
 The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{2 \times 2}</math> is given by the weighted sum of observability Gramians
-<math>h^i_x (x(t)) G(t), \ i = 1,2</math> for each observed function of states.
+<math>h^i_x (x(t)) G(t), \ i = 1,2,</math> for each observed function of states.
 == Parameters ==
@@ Line 63: / Line 63: @@
 |-
 | align=center | <math>t_\mathrm{f}</math> || align=right | 1 || Horizon of the control problem
+|-
+| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.01 || Regularization of Fisher matrix
 |-
 | align=center | <math>\mathcal{U}</math> || align=right | [0,1] || Bounds of control function
@@ Line 68: / Line 70: @@
 | align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
 |-
-| align=center | <math>\mathcal{M}</math> || align=right | 0.2 || Maximum measurement time
+| align=center | <math>M_1, M_2</math> || align=right | 0.2 || Maximum measurement time
 |}
@@ Line 76: / Line 78: @@
 <gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
-  Image:Dielectrophoretic_Particle_OED.png| States, control, and sampling functions for a local optimum.
+  Image:Jackson_OED.png| States, control, and sampling functions for a local optimum. Both measurement functions overlap.
 </gallery>