question archive Consider the same sensor network as in Problem 3 of HW #3
Consider the same sensor network as in Problem 3 of HW #3
Subject:Electrical EngineeringPrice: Bought3
Consider the same sensor network as in Problem 3 of HW #3. A quadratic function (potential function) can be defined as f(x) = _(ij) (xi - x;)2 with (i, j) ranges over all pairs of sensors that are connected by a communication link (each such pair counts once in the summation). (a) The potential function f(x) can be written as f(x) = x Px for some proper symmetric matrix (called the graph Laplacian matrix) P E R4X4. Explain why P is positive semidefinite. (b) Show that f(x) = 0 if and only if xi, i = 1, 2, 3, 4, are in consensus (i.e., identical). What does this say about the null space of P? (c) Suppose a[k] evolves according to the LTI dynamics x[k + 1] = Ax[k] as in Problem 3 of HW #3. Show that the potential function is non-increasing: f(x[k + 1]) < f(x[k]). Note that this is equivalent to checking if a certain matrix defined using A and P is negative semidefinite. (d) Show that the potential function is strictly decreasing: f(x[k + 1]) < f(x[k]), whenever x[k] is not in consensus, i.e., f(x[k]) * 0. Note that this is equivalent to checking if a certain matrix defined using A and P is negative semidefinite with its null space being exactly the 1-D consensus subspace (null space of P).
