problemset2_Soln - The University of Texas at Austin...

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The University of Texas at Austin Department of Electrical and Computer Engineering EE362K: Introduction to Automatic Control—Fall 2009 Solutions for Problem Set Two C. Caramanis Due: Monday, September 21, 2009. This problem set is intended to get us started thinking about linear algebra, and also to continue giving us some practice with Matlab. 1. The first column of A 1 is a multiple of the second column: (1 , - 2) * 2 = ( - 2 , 4). Therefore, any vector of the form (2 x, - x ) will be in the null space of A 1 . Generically, the range of any matrix is the span of the columns. Since the columns of A 1 are linearly dependent, the range is just one-dimensional, and equal to Span(1 , - 2). For A 2 , you can check that any vector of the form: (2 x,x, - 5 x/ 2) is in the null space. Moreover, there are two linearly independent columns, which means that the pan of the columns is two dimensional, and that therefore the dimension of the range is 2. Since the dimension of the null space plus the dimension of the range equals n , or in this case 3, we know that indeed the null space is the one-dimensional subspace: Span(2 , 1 , - 5 / 2). The range will be the entire ( x,y )-plane, or, Span { (1 , 0 , 0) , (0 , 1 , 0) } . 2. As mentioned in the book, the consensus algorithm is realizable by the update law: x [ k + 1] = x [ k ] - γ ( D - A ) x [ k ] where A is the adjacency matrix of the graph and D is a diagonal matrix with entries cor- responding to the number of neighbors of each node. The constant γ describes the rate at which the estimate of the average is updated based on information from neighboring nodes. (a) A and D for K 10 graph are given by: A = 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 ,D = 9 0 0 0 0 0 0 0 0 0
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problemset2_Soln - The University of Texas at Austin...

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