Jackson OED: Difference between revisions

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

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Revision as of 07:36, 27 January 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 $t_{f} = 1$ 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:

$\begin{array}{lll} \min_{y, G, F, z, w} & trace (F^{- 1} (t_{f})) \\ subject to \\ \dot{y} (t) & = & f (y (t), θ) \\ \dot{G} (t) & = & f_{y} (y (t), θ) G (t) + f_{θ} (y (t), θ) \\ \dot{F} (t) & = & \sum_{i = 1}^{n_{o}} w_{i} (t) (h_{y}^{i} (y (t)) G (t))^{T} (h_{y}^{i} (y (t)) G (t)) \\ \dot{z} (t) & = & w (t), \\ y (0) & = & y_{0} \\ G (0) & = & \frac{\partial y (0)}{\partial θ} \\ F (0) & = & 0, \\ z (0) & = & 0 \\ w (t) & \in & 𝒲 \\ z_{i} (t_{f}) & \leq & M_{i} \end{array}$

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

Parameters

We use $t_{f} = 8$ , $α = - 0.75$ , and $c = 1$ . The upper bound on the measurement time intervals is chosen as $M_{1} = M_{2} = 2$ .

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

There were no citations found in the article.

@@ Line 17: / Line 17: @@
 \dot{x_1}(t) &=& -u(t) (k_1 x_1(t) - k_2 x_2(t)), && t \in [0,t_f], \quad 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], \quad x_2(0) = 0, \\
-\dot{x_2}(t) &=& (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_2(0) = 0.
+\dot{x_3}(t) &=& (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_3(0) = 0.
 \end{array}
 </math>
 </p>
-The initial values and <math>t_f = 10</math> are fixed. We are interested in how to fish and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states 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 = 1</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: