Toy OED: Difference between revisions

Toy OED
State dimension:	1
Differential states:	4
Discrete control functions:	1

Latest revision as of 13:57, 29 January 2026

The Toy OED problem looks for an optimal measurement strategy to determine a single parameter in a one-dimensional ODE model, where can directly measure the single state.

The optimal integer control functions shows bang bang behavior.

Mathematical formulation

For a single parameter $p$ the original initial value problem is given by $\dot{x} (t) = p \cdot x (t), t \in [0, t_{f}], x (0) = x_{0} .$

We assume both $x_{0}$ and $t_{f}$ to be fixed and are only interested in when to measure, with an upper bound $M$ on the measuring time. We can measure the state directly, i.e. $h (x (t)) = x (t)$ . Thus, the experimental design problem simplifies to:

$\begin{array}{lll} \min_{x, G, F, z, w} & 1 / F (t_{f}) \\ subject to \\ \dot{x} (t) & = & p \cdot x (t), \\ \dot{G} (t) & = & p \cdot G (t) + x (t), \\ \dot{F} (t) & = & w (t) \cdot G (t)^{2}, \\ \dot{z} (t) & = & w (t), \\ x (0) & = & x_{0}, \\ G (0) & = & F (0) = z (0) = 0, \\ w (t) & \in & 𝒲, \\ 0 & \leq & M - z (t_{f}) \end{array}$

Parameters

These fixed values are used within the model:

$x_{0} = 1; t_{f} = 1; 𝒲 = [0, 1]; M = 0.2; p \in {- 0.5, - 2}$

Reference Solutions

Here is one local solution to the above control problem.

Reference solution plots
States and measurement control for different choices of $p$ .

Miscellaneous and Further Reading

The Toy OED problem was introduced by Sebastian Sager in [Sager2013]Author: Sager, S.
Journal: SIAM Journal on Control and Optimization
Number: 4
Pages: 3181--3207
Title: Sampling Decisions in Optimum Experimental Design in the Light of Pontryagin's Maximum Principle
Url: http://mathopt.de/PUBLICATIONS/Sager2013.pdf
Volume: 51
Year: 2013
, which contains further details.

References

[Sager2013]

Sager, S. (2013): Sampling Decisions in Optimum Experimental Design in the Light of Pontryagin's Maximum Principle. SIAM Journal on Control and Optimization, 51, 3181--3207

@@ Line 10: / Line 10: @@
 == Mathematical formulation ==
+For a single parameter <math>p</math> the original initial value problem is given by
-We are interested in estimating the parameters <math>p_2</math> and <math>p_4</math> of the Lotka-Volterra type predator-prey fish initial value problem
-<p>
 <math>
-\begin{array}{rcl}
+  \dot{x}(t) = p \cdot x(t), \quad t \in [0, t_f], \quad x(0) = x_0.
-\dot{x_1}(t) &=& p_1 \; x_1(t) - p_2 x_1(t) x_2(t) - p_5 u(t) x_1(t), \; t \in [0,t_f], \quad x_1(0) = 0.5, \\
-\dot{x_2}(t) &=& - p_3 \; x_2(t) + p_4 x_1(t) x_2(t) - p_6 u(t) x_2(t), \; t \in [0,t_f], \quad x_2(0) = 0.7,
-\end{array}
 </math>
-</p>
-where <math>u(\cdot)</math> is a fishing control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 12</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 experimental design problem then reads
+We assume both <math>x_0</math> and <math>t_f</math> to be fixed and are only interested in when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the state directly, i.e. <math>h(x(t)) = x(t)</math>. Thus, the experimental design problem simplifies to:
 <p>
 <math>
   \begin{array}{lll}
-  \displaystyle \min_{x,G,F,z^1,z^2,u,w^1,w^2} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
+  \displaystyle \min_{x,G,F,z,w} && 1 / F(t_f) \\
   \text{subject to} \\
-\quad \dot{x_1}(t) & = &  p_1 \; x_1(t) - p_2 x_1(t) x_2(t) - p_5 u(t) x_1(t),\\
+\quad \dot{x}(t) & = &  p \cdot x(t),\\
-\quad \dot{x_2}(t) & = &  - p_3 \; x_2(t) + p_4 x_1(t) x_2(t) - p_6 u(t) x_2(t),\\
+\quad \dot{G}(t) & = & p \cdot G(t) + x(t), \\
-\quad \dot{G_{11}}(t) & = & f_{x11}(\cdot) \; G_{11}(t) + f_{x12}(\cdot) \; G_{21}(t) + f_{p12}(\cdot), \\
+\quad \dot{F}(t) & = & w(t) \cdot G(t)^2, \\
-\quad \dot{G_{12}}(t) & = & f_{x11}(\cdot) \; G_{12}(t) + f_{x12}(\cdot) \; G_{22}(t), \\
+\quad \dot{z}(t) & = & w(t), \\
-\quad \dot{G_{21}}(t) & = & f_{x21}(\cdot) \; G_{11}(t) + f_{x22}(\cdot) \; G_{21}(t), \\
+\quad x(0) &=& x_0, \\
-\quad \dot{G_{22}}(t) & = & f_{x21}(\cdot) \; G_{12}(t) + f_{x22}(\cdot) \; G_{22}(t) + f_{p24}(\cdot), \\
+\quad G(0) &=& F(0) = z(0) = 0, \\
-\quad \dot{F_{11}}(t) & = & w^1(t) G_{11}(t)^2 + w^2(t) G_{21}(t)^2, \\
+\quad w(t) &\in& \mathcal{W}, \\
-\quad \dot{F_{12}}(t) & = & w^1(t) G_{11}(t) G_{12}(t) + w^2(t) G_{21}(t) G_{22}(t), \\
-\quad \dot{F_{22}}(t) & = & w^1(t) G_{12}(t)^2 + w^2(t) G_{22}(t)^2, \\
-\quad \dot{z^1}(t) & = & w^1(t), \\
-\quad \dot{z^2}(t) & = & w^2(t), \\[1.5ex]
-\quad x(0) &=& (0.5, 0.7), \\
-\quad G(0) &=& F(0) = 0, \\
-\quad z^1(0) &=& z^2(0) = 0, \\ [1.5ex]
-\quad u(t) & \in & \mathcal{U}, \; w^1(t) \in \mathcal{W}, \; w^2(t) \in \mathcal{W}, \\
 \quad 0    & \le & M - z(t_f)
    \end{array}
@@ Line 47: / Line 34: @@
 </p>
-with
+== Parameters ==
-<math>f_{x11}(\cdot) = \partial f_1(\cdot) / \partial x_1 = p_1 - p_2 x_2(t) - p_5 u(t)</math>,
+These fixed values are used within the model:
-<math>f_{x12}(\cdot) = - p_2 x_1(t)</math>,
+<p>
-<math>f_{x21}(\cdot) = p_4 x_2(t)</math>,
+ <math>
-<math>f_{x22}(\cdot) = -p_3 + p_4 x_1(t) - p_6 u(t)</math>, and
+  x_0 = 1; \quad t_f = 1; \quad \mathcal{W} = [0,1]; \quad M = 0.2; \quad p \in \{-0.5, -2\}
-<math>f_{p12}(\cdot) = \partial f_1(\cdot) / \partial p_2 = -x_1(t) x_2(t)</math>,
+ </math>
-<math>f_{p24}(\cdot) = \partial f_2(\cdot) / \partial p_4 = x_1(t) x_2(t)</math>.
+</p>
-Note that the state <math>F_{21}(\cdot) = F_{12}(\cdot)</math> has been left out for reasons of symmetry.
 == Reference Solutions ==
@@ Line 61: / Line 46: @@
 Here is one local solution to the above control problem.
-== Source Code ==
+<gallery caption="Reference solution plots" widths="500px" heights="250px" perrow="1">
+ Image:Toy OED.png| States and measurement control for different choices of <math>p</math>.
-Model descriptions are available in
+</gallery>
-* [[:Category:AMPL | AMPL]] at [[Lotka Experimental Design (AMPL)]]
-* [[:Category: VPLAN | VPLAN code]] at [[Lotka Experimental Design (VPLAN)]]
-== Variants ==
-There are several alternative formulations and variants of the above problem, in particular
-* a prescribed time grid for the control function <bib id="Sager2006" />, see also [[Lotka Experimental Design (AMPL)]],
-* no fishing, i.e., <math>u \equiv 0</math>,
-* different fishing control functions for the two species,
-* different parameters and start values.
 == 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" />.
+The Toy OED problem was introduced by Sebastian Sager in <bib id="Sager2013" />, which contains further details.
 == References ==
@@ Line 88: / Line 61: @@
 [[Category:ODE model]]
 [[Category:Bang bang]]
-[[Category:Population dynamics]]