Difference between revisions of "Lotka Experimental Design"

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</math>
 
</math>
 
</p>
 
</p>
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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>\Ge(\cdot)</math> are identical. The experimental design problem then reads
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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
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For <math>t \in [t_0, t_f]</math> almost everywhere the mixed-integer optimal control problem is given by
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<p>
 
<p>
 
<math>
 
<math>
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\begin{array}{llcl}
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\begin{array}{lll}
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  \displaystyle \min_{x, w} & x_2(t_f)   \\[1.5ex]
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  \displaystyle \min_{x,G,F,z^1,z^2,u,w^1,w^2} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
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  \mbox{s.t.} & \dot{x}_0(t) & = & x_0(t) - x_0(t) x_1(t) - \; c_0 x_0(t) \; w(t), \\
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  \text{subject to} \\
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& \dot{x}_1(t) & = & - x_1(t) + x_0(t) x_1(t) - \; c_1 x_1(t) \; w(t), \\
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\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),\\
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& \dot{x}_2(t) & = & (x_0(t) - 1)^2 + (x_1(t) - 1)^2, \\[1.5ex]
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\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),\\
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& x(0) &=& (0.5, 0.7, 0)^T, \\
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\quad \dot{G_{11}}(t) & = & f_{x11}(\cdot) \; G_{11}(t) + f_{x12}(\cdot) \; G_{21}(t) + f_{p12}(\cdot), \\
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& w(t) &\in& \{0, 1\}.
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\quad \dot{G_{12}}(t) & = & f_{x11}(\cdot) \; G_{12}(t) + f_{x12}(\cdot) \; G_{22}(t), \\
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\end{array}  
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\quad \dot{G_{21}}(t) & = & f_{x21}(\cdot) \; G_{11}(t) + f_{x22}(\cdot) \; G_{21}(t), \\
 +
\quad \dot{G_{22}}(t) & = & f_{x21}(\cdot) \; G_{12}(t) + f_{x22}(\cdot) \; G_{22}(t) + f_{p24}(\cdot), \\
 +
\quad \dot{F_{11}}(t) & = & w^1(t) G_{11}(t)^2 + w^2(t) G_{12}(t)^2, \\
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\quad \dot{F_{12}}(t) & = & w^1(t) G_{11}(t) G_{12}(t) + w^2(t) G_{12}(t) G_{22}(t), \\
 +
\quad \dot{F_{22}}(t) & = & w^1(t) G_{21}(t)^2 + w^2(t) G_{22}(t)^2, \\
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\quad \dot{z^1}(t) & = & w^1(t), \\
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\quad \dot{z^2}(t) & = & w^2(t), \\[1.5ex]
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\quad x(0) &=& (0.5, 0.7), \\
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\quad G(0) &=& F(0) = 0, \\
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\quad z^1(0) &=& z^2(0) = 0, \\ [1.5ex]
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\quad u(t) & \in & \mathcal{U}, \; w^1(t) \in \mathcal{W}, \; w^2(t) \in \mathcal{W}, \\
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\quad 0    & \le & M - z(t_f)
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  \end{array}
 
</math>
 
</math>
 
</p>
 
</p>
  
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Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. The decision, whether the fishing fleet is actually fishing at time <math>t</math> is denoted by <math>w(t)</math>.
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with
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<math>f_{x11}(\cdot) = \partial f_1(\cdot) / \partial x_1 = p_1 - p_2 x_2(t) - p_5 u(t)</math>,  
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<math>f_{x12}(\cdot) = - p_2 x_1(t)</math>,  
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<math>f_{x21}(\cdot) = p_4 x_2(t)</math>,  
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<math>f_{x22}(\cdot) = -p_3 + p_4 x_1(t) - p_6 u(t)</math>, and
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<math>f_{p12}(\cdot) = \partial f_1(\cdot) / \partial p_2 = -x_1(t) x_2(t)</math>,  
 +
<math>f_{p24}(\cdot) = \partial f_2(\cdot) / \partial p_4 = x_1(t) x_2(t)</math>. 
 +
 
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Note that the state <math>F_{21}(\cdot) = F_{12}(\cdot)</math> has been left out for reasons of symmetry.
  
 
== Parameters ==
 
== Parameters ==
  
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These fixed values are used within the model.
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We use <math>t_f=12</math>, <math>p_1 = p_2 = p_3 = p_4 = 1</math>, and <math>p_5 = 0.4</math>, <math>p_6 = 0.2</math>. The upper bound on the measurement time intervals is chosen as <math>M=4</math>.
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<math>
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\begin{array}{rcl}
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[t_0, t_f] &=& [0, 12],\\
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(c_0, c_1) &=& (0.4, 0.2).
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\end{array}
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</math>
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== Reference Solutions ==
 
== Reference Solutions ==
  
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If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of [http://en.wikipedia.org/wiki/Pontryagin%27s_minimum_principle Pontryagins maximum principle]. The optimal solution contains a singular arc, as can be seen in the plot of the optimal control. The two differential states and corresponding adjoint variables in the indirect approach are also displayed. A different approach to solving the relaxed problem is by using a direct method such as collocation or Bock's direct multiple shooting method. Optimal solutions for different control discretizations are also plotted in the leftmost figure.
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bla
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The optimal objective value of this relaxed problem is <math>x_2(t_f) = 1.34408</math>. As follows from MIOC theory<bibref>Sager2008</bibref> this is the best lower bound on the optimal value of the original problem with the integer restriction on the control function. In other words, this objective value can be approximated arbitrarily close, if the control only switches often enough between 0 and 1. As no optimal solution exists, two suboptimal ones are shown, one with only two switches and an objective function value of <math>x_2(t_f) = 1.38276</math>, and one with 56 switches and <math>x_2(t_f) = 1.34416</math>.
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<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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* a prescribed time grid for the control function <bibref>Sager2006</bibref>, see also [[Lotka Experimental Design (AMPL)]],
 
* a prescribed time grid for the control function <bibref>Sager2006</bibref>, see also [[Lotka Experimental Design (AMPL)]],
 +
* no fishing, i.e., <math>u \equiv 0</math>,
 
* different fishing control functions for the two species,
 
* different fishing control functions for the two species,
 
* different parameters and start values.
 
* different parameters and start values.

Revision as of 10:13, 31 August 2012

Lotka Experimental Design
State dimension: 1
Differential states: 12
Discrete control functions: 2
Interior point equalities: 12


The Lotka Experimental Design problem looks for an optimal fishing strategy to be performed on a fixed time horizon to control the biomasses of both predator as prey fish as described in Lotka Volterra fishing problem. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. In addition to the fishing decision the question when measurements of the two state variables are performed becomes a degree of freedom.

The mathematical equations form a small-scale ODE model. The ODE from Lotka Volterra fishing problem is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

The optimal integer control functions shows bang bang behavior.

Mathematical formulation

We are interested in estimating the parameters p_2 and p_4 of the Lotka-Volterra type predator-prey fish initial value problem

\begin{array}{rcl} \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}

where u(\cdot) is a fishing control that may or may not be fixed. The other parameters, the initial values and t_f = 12 are fixed. We are interested in how to fish and when to measure, with an upper bound M on the measuring time. We can measure the states 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. The experimental design problem then reads

\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) \\ \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_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_{11}}(t) & = & f_{x11}(\cdot) \; G_{11}(t) + f_{x12}(\cdot) \; G_{21}(t) + f_{p12}(\cdot), \\ \quad \dot{G_{12}}(t) & = & f_{x11}(\cdot) \; G_{12}(t) + f_{x12}(\cdot) \; G_{22}(t), \\ \quad \dot{G_{21}}(t) & = & f_{x21}(\cdot) \; G_{11}(t) + f_{x22}(\cdot) \; G_{21}(t), \\ \quad \dot{G_{22}}(t) & = & f_{x21}(\cdot) \; G_{12}(t) + f_{x22}(\cdot) \; G_{22}(t) + f_{p24}(\cdot), \\ \quad \dot{F_{11}}(t) & = & w^1(t) G_{11}(t)^2 + w^2(t) G_{12}(t)^2, \\ \quad \dot{F_{12}}(t) & = & w^1(t) G_{11}(t) G_{12}(t) + w^2(t) G_{12}(t) G_{22}(t), \\ \quad \dot{F_{22}}(t) & = & w^1(t) G_{21}(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}

with f_{x11}(\cdot) = \partial f_1(\cdot) / \partial x_1 = p_1 - p_2 x_2(t) - p_5 u(t), f_{x12}(\cdot) = - p_2 x_1(t), f_{x21}(\cdot) = p_4 x_2(t), f_{x22}(\cdot) = -p_3 + p_4 x_1(t) - p_6 u(t), and f_{p12}(\cdot) = \partial f_1(\cdot) / \partial p_2 = -x_1(t) x_2(t), f_{p24}(\cdot) = \partial f_2(\cdot) / \partial p_4 = x_1(t) x_2(t).

Note that the state F_{21}(\cdot) = F_{12}(\cdot) has been left out for reasons of symmetry.

Parameters

We use t_f=12, p_1 = p_2 = p_3 = p_4 = 1, and p_5 = 0.4, p_6 = 0.2. The upper bound on the measurement time intervals is chosen as M=4.

Reference Solutions

bla

  • Reference solution plots

  • Optimal relaxed control determined by an indirect approach and by a direct approach on different control discretization grids.

  • Differential states and corresponding adjoint variables in the indirect approach.

  • Control and differential states with only two switches.

  • Control and differential states with 56 switches.


Source Code

Model descriptions are available in

Variants

There are several alternative formulations and variants of the above problem, in particular

  • a prescribed time grid for the control function <bibref>Sager2006</bibref>, see also Lotka Experimental Design (AMPL),
  • no fishing, i.e., u \equiv 0,
  • 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 <bibref>Sager2006</bibref> and revisited in his PhD thesis <bibref>Sager2005</bibref>. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bibref>Sager2011d</bibref>.

References

<bibreferences/>

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