Control of Heat Equation with Actuator Placement: Difference between revisions

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

← Older edit Newer edit →

Revision as of 15:44, 23 February 2016

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
). 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.

Mathematical formulation

$\begin{array}{llcl} \min_{u, v, w} & J (u, v) = | | u (\cdot, \cdot, t_{f}) | |_{2, Ω}^{2} + 2 | | u (\cdot, \cdot, \cdot) | |_{2, Ω \times T}^{2} + \frac{1}{500} \sum_{l = 1}^{L} | | v_{l} (\cdot) | |_{2, T}^{2} \\ s.t. & \frac{\partial u}{\partial t} (x, y, t) - κ Δ u (x, y, t) = \sum_{l = 1}^{9} v_{l} (t) f_{l} (x, y) & in & Ω \times T \\ u (x, y, t) = 0 & on & \partial Ω \times T \\ u (x, y, 0) = 100 \sin (π x) \sin (π y) & in & Ω \\ - M w_{l} (t) \leq v_{l} (t) \leq M w_{l} (t) for all l \in {1, \dots, L} & in & T \\ \sum_{l = 1}^{L} w_{l} (t) = 1 & in & T \\ w_{l} (t) \in {0, 1} for all l \in {1, \dots, L} & in & T . \end{array}$

Parameters

These fixed values are used within the model.

$\begin{array}{rcl} Ω & = & [0, 1] \times [0, 2], \\ L & = & 9, \\ κ & = & 0.01, \\ t_{f} & = & 10 . \end{array}$

The parameter $κ$ describes the thermal dissipativity of the material in the domain $Ω$ , it may vary in space. The parameter $L$ indicates the number of possible actuator locations. They are distributed as indecated in the picture.

Reference solution

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--323

@@ Line 12: / Line 12: @@
 This problem is governed by the heat equation and is adapted from Iftime and Demetriou (<bib id="Iftime2009"/>).
 Its goal is to choose a place to apply an actuator in a given area depending on time.
-We consider a rectangle <math>\Omega=[0,1]\times[0,2]</math> with the boundary <math>\partial\Omega</math> and the time horizon <math>T=[0,10]</math> as the domains.
 The objective function is quadratic, its first term captures the desired final state <math>\bar{u}\equiv 0</math>, 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.
@@ Line 25: / Line 24: @@
 \begin{array}{llcl}
-\min\limits_{u,v,w}~~ &J(u,v)=||u(\cdot,\cdot,10)||_{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]
+\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]
@@ Line 44: / Line 43: @@
 <math>
 \begin{array}{rcl}
+\Omega &=& [0,1] \times [0,2],\\
 L &=& 9, \\
 \kappa &=& 0.01,\\