Difference between revisions of "Control of Heat Equation with Actuator Placement"

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     & u(x,y,0) = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]
 
     & u(x,y,0) = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]
 
     & -M w_l(t)\leq v_l(t)\leq M w_l(t) \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]
 
     & -M w_l(t)\leq v_l(t)\leq M w_l(t) \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]
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     & \sum\limits_{l=1}^L w_l(t) = W &\text{ in } & T\\[10pt]
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     & \sum\limits_{l=1}^L w_l(t) \leq W &\text{ in } & T\\[10pt]
 
     & w_l(t)\in \{0,1\} \text{ for all } l\in \{1,\dots,L\} &\text{ in } &T.
 
     & w_l(t)\in \{0,1\} \text{ for all } l\in \{1,\dots,L\} &\text{ in } &T.
  

Revision as of 18:17, 23 February 2016

Control of Heat Equation with Actuator Placement
State dimension: 1
Differential states: 1
Continuous control functions: 9
Discrete control functions: 9
Path constraints: 3
Interior point equalities: 2


This problem is governed by the heat equation and is adapted from Iftime and Demetriou ([Iftime2009]Author: Orest V. Iftime; Michael A. Demetriou
Journal: {A}utomatica
Number: 2
Pages: 312--323
Title: {O}ptimal control of switched distributed parameter systems with spatially scheduled actuators
Volume: 45
Year: 2009
Link to Google Scholar). Its goal is to choose a place to apply an actuator in a given area depending on time. The objective function is quadratic, its first term captures the desired final state \bar{u}\equiv 0, the second term regularize the state over time and the third term regularize the continuous controls. The constraints are a source budget, which limits the quantity of placed actuators, and the two-dimensional heat equation with some source function. Additionally, we assume Dirichlet boundary conditions and initial conditions. Originally, the problem formulation was non-convex. We overcome this issue by substitution of v(t)w_l(t) by v_l(t) and adding the Big M formulation.


Mathematical formulation

\begin{array}{llcl} \min\limits_{u,v,w}~~ &J(u,v)=||u(\cdot,\cdot,t_f)||_{2,\Omega}^2 +2||u(\cdot,\cdot,\cdot)||_{2,\Omega\times T}^2+\frac{1}{500}\sum\limits_{l=1}^L||v_l(\cdot)||^2_{2,T} & \\[10pt] \text{ s.t.} ~~~~ &\frac{\partial u}{\partial t}(x,y,t)- \kappa \Delta u(x,y,t)=\sum\limits_{l=1}^9 v_l(t) f_l(x,y) &\text{ in }&\Omega\times T\\[10pt] & u(x,y,t) =0 &\text{ on } &\partial\Omega\times T \\[10pt] & u(x,y,0) = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt] & -M w_l(t)\leq v_l(t)\leq M w_l(t) \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt] & \sum\limits_{l=1}^L w_l(t) \leq W &\text{ in } & T\\[10pt] & w_l(t)\in \{0,1\} \text{ for all } l\in \{1,\dots,L\} &\text{ in } &T. \end{array}

Parameters

We define the source term for all locations l\in \{1,\dots,L\} and a fix parameter \varepsilon\in \R_+: f_l(x,y) = \frac{1}{\sqrt{2\pi\varepsilon}}e^{\frac{-((x_l-x)^2+(y_l-y)^2)}{2\varepsilon}} where (x_l,y_l) is the coordinate of the mesh point of the lth possible actuator location.

The parameters used are:

\begin{array}{rcl} \Omega &=& [0,1] \times [0,2],\\ \kappa &=& 0.01,\\ L &=& 9, \\ W &=& 1,\\ t_f &=&10,\\ \varepsilon &=& 0.01 . \end{array}


The parameter \kappa describes the thermal dissipativity of the material in the domain \Omega , it may vary in space: \kappa(x,y) . The parameter L indicates the number of possible actuator locations. They are distributed as indicated in the picture. The source budget is limited by W and t_f denotes the final time.

Discretization

To solve the problem we apply a "first discretize, then optimize" approach an discretize the components of the problem. For the heat equation, we use a five-point-stencil in space and the implicit euler in time. For i=1,\dots, N-1, j=1,\dots, M-1, and k=0,\dots, T_n-1, this yields: 

   \frac{1}{h_t}(u_{i,j,k}-u_{i,j,k-1}) -  \frac{\kappa_{i,j}}{h_x h_y}(u_{i-1,j,k}+u_{i+1,j,k}+u_{i,j-1,k}+u_{i,j+1,k}-4u_{i,j,k}) =   \sum\limits_{l=1}^L(s_{k+1,l}+s_{k  ,l} )f_l(ih_x,jh_y),

with u_{i,j,k}=u(ih_x,jh_y,kh_t), the stepsizes h_x,h_y in space, and the stepsize in time h_t, respectively.

It holds for the source buget with the discretized binary controls w_{l,k} for all k\in \{0,\dots,T_n\}: \sum\limits_{l=1}^L w_{k,l}\leq W.

This condition is called SOS-W conditon.

We remark that the number of discretized binary variables does not depend on the space discretization but it depends on the time discretization. Thats why we taged this problem containing mesh-independend and as mesh-dependend integer variables.

Source Code

References

[Iftime2009]Orest V. Iftime; Michael A. Demetriou (2009): {O}ptimal control of switched distributed parameter systems with spatially scheduled actuators . {A}utomatica, 45, 312--323Link to Google Scholar
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