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Unformatted text preview: EE263 Summer 2010-11 Laurent Lessard EE263 homework 6 1. Optimal choice of initial temperature profile. We consider a thermal system described by an n-element finite-element model. The elements are arranged in a line, with the temperature of element i at time t denoted T i ( t ). Temperature is measured in degrees Celsius above ambient; negative T i ( t ) corresponds to a temperature below ambient. The dynamics of the system are described by c 1 T 1 = a 1 T 1 b 1 ( T 1 T 2 ) , c i T i = a i T i b i ( T i T i +1 ) b i- 1 ( T i T i- 1 ) , i = 2 , . . ., n 1 , and c n T n = a n T n b n- 1 ( T n T n- 1 ) . where c R n , a R n , and b R n- 1 are given and are all positive. We can interpret this model as follows. The parameter c i is the heat capacity of element i , so c i T i is the net heat flow into element i . The parameter a i gives the thermal conductance between element i and the environment, so a i T i is the heat flow from element i to the environment ( i.e. , the direct heat loss from element i .) The parameter b i gives the thermal conductance between element i and element i + 1, so b i ( T i T i +1 ) is the heat flow from element i to element i + 1. Finally, b i- 1 ( T i T i- 1 ) is the heat flow from element i to element i 1. The goal of this problem is to choose the initial temperature profile, T (0) R n , so that T ( t des ) T des . Here, t des R is a specific time when we want the temperature profile to closely match T des R n . We also wish to satisfy a constraint that bardbl T (0) bardbl should be not be too large. To formalize these requirements, we use the objective (1 / n ) bardbl T ( t des ) T des bardbl and the constraint (1 / n ) bardbl T (0) bardbl T max . The first expression is the RMS temperature deviation, at t = t des , from the desired value, and the second is the RMS temperature deviation from ambient at t = 0. T max is the (given) maximum inital RMS temperature value. (a) Explain how to find T (0) that minimizes the objective while satisfying the constraint. (b) Solve the problem instance with the values of n , c , a , b , t des , T des and T max defined in the file temp_prof_data.m . Plot, on one graph, your T (0), T ( t des ) and T des . Give the RMS temperature error (1 / n ) bardbl T ( t des ) T des bardbl , and the RMS value of initial temperature (1 / n ) bardbl T (0) bardbl . Solution. (a) We can express the temperature dynamics as T = AT , where A is a tridiagonal matrix with A 11 = 1 /c 1 ( a 1 + b 1 ) A ii = 1 /c i ( a i + b i + b i- 1 ) , i = 2 , . . ., n, A i,i- 1 = b i- 1 /c i , i = 2 , . . ., n, A i,i +1 = b i /c i , i = 1 , . . ., n 1 ....
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This note was uploaded on 12/04/2011 for the course EE 263 at Stanford.