Lecture 1 and 2: Introduction and Graph theory basics. Spring EE 194, Networked estimation and control (Prof. Khan) January 23, 2012
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1 Lecture 1 and 2: Introduction and Graph theory basics Spring EE 194, Networked estimation and control (Prof. Khan) January 23, 2012
2 Spring 2012: EE Networked estimation and control Schedule Monday and Wednesday, 4:30pm - 5:45pm, Halligan 108 Instructor Prof. Usman Khan 135 Halligan khan@ece.tufts.edu Office hours: M-W, 6:00pm 7:00pm (or by ) Web: Helpful references Modern Graph Theory, B. Bollobás, Springer, New York, NY, Algebraic Graph Theory, C. Godsil and G. F. Royle, Springer, New York, NY, Matrix Computations, G. H. Golub and C. F. Van Loan, Johns Hopkins Press, Mathematical Statistics, P. J. Bickel and K. A. Docksum, Prentice Hall, Statistical Signal Processing, L. Scahrf, Prentice Hall, A Course in Robust Control Theory: A Convex Approach, G. E. Dullerud and F. Paganini, Springer, Feedback Control of Dynamic Systems, G. F. Franklin, J. D. Powell and A. Emami-Naieni, Prentice Hall, Assignments Class presentations Based on a series of readings comprising of recent research papers related to the course Homework assignments Up to five problem sets distributed throughout the course Final project There will be a course project in place of the final exam. The project goals and nature of study has to be approved by the instructor. A team of up to 2 members is required. Deliverable: A 4-page research paper submitted by Apr. 30, Course Structure Class participation and presentations: 30% HW assignments: 30% Final project: 40% Active class participation and engagement in creative discussions is absolutely required.
3 EE Networked Estimation and Control Course Schedule Assigned Due Misc. Jan-23 (Mon) Lecture 01 No office Hours Jan-25 (Wed) Lecture 02 Jan-30 (Mon) Lecture 03 Feb-01 (Wed) Lecture 04 Feb-06 (Mon) Lecture 05 Feb-08 (Wed) Lecture 06 Feb-13 (Mon) Lecture 07 Feb-15 (Wed) Lecture 08 Feb-20 (Mon) Feb-22 (Wed) No classes (President's Day) Lecture 09 Feb-23 (Thu) Lecture 10 Substitute Lec Feb-27 (Mon) Lecture 11 Feb-29 (Wed) Lecture 12 Mar-05 (Mon) Lecture 13 Mar-07 (Wed) Lecture 14 Mar-12 (Mon) Lecture 15 Mar-14 (Wed) Lecture 16 Mar-19 (Mon) Mar-21 (Wed) Mar-26 (Mon) No classes (Spring Recess) No classes (Spring Recess) Lecture 17 Mar-28 (Wed) Lecture 18 Apr-02 (Mon) Lecture 19 Apr-04 (Wed) Lecture 20 Apr-09 (Mon) Lecture 21 Apr-11 (Wed) Lecture 22 Apr-16 (Mon) Apr-18 (Wed) No classes (Patriot's Day) Lecture 23 Apr-23 (Mon) Lecture 24 Apr-25 (Wed) Lecture 25 Apr-30 (Mon) Lecture 26 Prof. Usman Khan Spring 2012 ECE Dept., Tufts University
4 4 I. INTRODUCTION Estimation theory Control theory Distributed algorithms Dynamical system DT-LTI or DT-LTV x k+1 = A k x k + v k y k = C k x k + r k. Given the statistics of v k and r k, what is the optimal estimate, x k k, of x k, when we only observe y k? Dynamical system DT-LTI or DT-LTV x k+1 = A k x k + Bu k y k = C k x k + r k. Given the optimal estimate, x k k, of x k, what is the optimal control input (or actuation), u k = K k x k k, when we only observe y k? Suppose the observations (system measurements) are given as y i k = C i kx k + r i k, where i is the ith sensor. How do we modify optimal estimation (control) when yk i s are not available at one location but the sensors interact over a network? Agents (sensors) interacting over a network can be modeled as nodes interacting over a graph. Graph theory deals with: Formal models of such interactions. Information flow within such interactions. Example 1: Suppose we have N agents, i.e., i = 1,..., N. Each agent possesses a quantity denoted by y i. We are interested in computing the following functions of y i s, y = max i y i, y = min i y i, ỹ = 1 N N y i, when there is no central location where all y i s are available. Each agent i interacts with a subset of other agents over a network. A typical distributed algorithm is i=1 z i k+1 = j N i w ij z j k, zi 0 = y i, i, where N i is the set of neighbors of agent i. Where does the above converge (lim k z i k ) and under what conditions on w ij and agent connectivity? How fast is the convergence? How does the convergence speed change as a function of the agent connectivity? Graph theory provides a means to design and analyze such computations.
5 5 II. GRAPH THEORY A graph, G, is defined to be a collection of two sets: (i) a vertex-set, V = {1,..., N}, that is a collection of nodes (vertices); and an edge-set, E V V, that is a collection of edges. The edge-set, E, is defined as a set of ordered pairs (i, j) with i, j V such that j is connected to i to be interpreted as j can send information to i. Formally, and a graph is denoted ny G = (V, E). E = {(i, j) j i}, (1) A graph is said to be undirected if (i, j) E (j, i) E for all i and j. A graph that does not satisfy this property is called a directed graph or a digraph. Unless otherwise stated, we deal explicitly with undirected graphs in the following. The neighborhood of a node i is defined as N i = {j (i, j) E}. (2) Sometimes, the above is called the open neighborhood of a node, whereas the closed neighborhood is defined as N i {i}. The degree of a node i is defined as the number of nodes that can send information to node i, i.e., N i. A. Graph theory and Linear algebra Analysis of graphs is typically carried out via matrix theory. For this purpose, we define matrices that can define a graph (as opposed to the set notation earlier). The adjacency matrix, A = {a ij }, of a graph is defined as a ij = { 1, j i, 0, otw. Sometimes it is assumed that (i, i) E. With this assumption, the adjacency matrix has all 1 s on the main diagonal and is called extended adjacency. Remark 1: The adjacency matrix of an undirected graph is symmetric. The incidence matrix, C = c ij, of a graph is defined as an N M matrix (where M is the total number of edges) such that for the mth edge (i, j) E, the mth column of C has a 1 at the ith location, a 1 at the jth location, and zeros everywhere else. Note that if (i, j) is counted then (j, i) is not counted. We fix any one arbitrary orientation, i.e., either incoming or outgoing. (3)
6 6 The degree matrix, D, is defined as a diagonal matrix that has N i as the ith element on the main diagonal. The following definitions of a graph Laplacian, L = {l ij }, are equivalent: (i) L = D A. (4) N i, j = i, (ii) l ij = 1, i j, j i, (5) 0, otw. (iii) L = CC T. (6) Remark 2: The Laplacian, L, is symmetric and positive-semidefinite. Proof: Obvious from definition (iii). The eigenvalues of L are denoted by λ 1, λ 2,..., λ N ; the following conventional is typically employed, 0 = λ 1 λ 2... λ N. Remark 3: The Laplacian, L, is singular (rank-deficient), i.e., it has at least one 0 eigenvalue. Proof: Row-sum is 0. A path between node i 1 V and node i K+1 V of length K is defined as a sequence of edges (i 1, i 2 ), (i 2, i 3 ),... (i K, i K+1 ) in E for any distinct i 2,..., i K. An undirected graph is said to be connected if there exists a path from each i V to each j V. A graph is said to be fully-connected or an all-to-all networkif (i, j) E, for all i and j. If a graph is not connected then it can be partitioned into connected components.
7 7 Lec 2: Wednesday, Jan. 25, 2012 A diagonally-dominant matrix, A, is such that a ii j i a ij, i. A strictly diagonallydominant, A, is such that a ii > j i a ij, i. Lemma 1 (Gershgorin circle theorem): Let A = {a ij } C N N. Let D i be the closed disc centered at a ii with radius j i a ij. Then every eigenvalue of A lies in i D i. Corollary 1: A symmetric diagonally-dominant matrix with non-negative diagonals is PSD. Proof: Follows from Gershgorin circle theorem. Corollary 2: A Laplacian matrix is PSD. Proof: Laplacian matrices are symmetric diagonally-dominant with non-negative elements on the main diagonal. Lemma 2: Let G be connected and let λ 1 λ 2... λ N be the Laplacian eigenvalues. Then λ 2 > 0. Proof: Let u = [u 1, u 2,..., u N ] T be an eigenvector of L with eigenvalue 0. Since Lu = 0 and u T Lu = u T CC T u, we have C T u = 0. Now C T u u i u j = 0, (i, j) E. (7) This implies that u i = u j for all (i, j) E. As the graph is connected, we have u i = u j for all i, j V and the only normalized eigenvector that satisfies Lu = 0 is u = 1 [1, 1,..., 1] T. (8) N }{{} N elements Hence, the there is only one 0 eigenvalue and λ 2 > 0 since L is PSD. Lemma 3: The number of connected components equals to the multiplicity of 0 eigenvalues in its Laplacian. Proof: A disconnected graph is a union of some number of connected components. Each of such component is a connected graph on its own and has exactly one 0 eigenvalue. Example 2: Consider a network with N nodes and no edges. There are N connected components (each node). From the above lemma, the Laplacian should have N 0 eigenvalues. Can be verified as the Laplacian in this case is a 0 matrix. An irreducible matrix is such that it cannot be transformed into a block-diagonal matrix with any row-column permutation. Remark 4: A matrix is irreducible if and only if its associated graph is strongly-connected. A symmetric matrix is irreducible if and only if its associated graph is connected.
8 8 A primitive matrix is such that it is non-negative, square, and its pth power (p > 0) has all positive elements. Remark 5: A primitive matrix is irreducible. Proof: Can be proved by contradiction. The following statements can be proved. (i) A graph is connected if and only if its Laplacian is irreducible. (ii) For a fully-connected graph, λ 2 =,..., λ N = N. The algebraic connectivity of the graph is defined as the second-smallest eigenvalue of its Laplacian, i.e., λ 2. For connected graphs, this measures the strength of connectivity. Remark 6: In a connected graph, adding an edge does not decrease λ 2. B. Types of graphs A k-regular graph is such that each node is connected to exactly k other nodes. A nearest neighbor graph is such that each node is connected to all the nodes within a certain communication radius. An m-circulant graph is such that each node is connected to m forward and m backward neighbors. Remark 7: The adjacency and Laplacian matrices of a circulant graph are circulant matrices. The eigenvalues and eigenvectors of a circulant matrix are known in closed-form. The above graphs are referred to as structured graphs. Typically such graphs are highly clustered (how many of my neighbors are neighbors of each other) but have a large average shortest path. Can also be related to graph diameter (largest shortest path). A random graph with 1 p 0 is such that every two nodes, i and j, are connected with a probability p. Random graphs have smaller average shortest path but suffer from weak clustering. Example 3: The above is one of the Erdös-Renýi graph generating model. The other is to randomly pick (with uniform probability) one graph out of all possible graphs with N nodes and K edges. Consider the following graph generation: Take a structured graph and a positive number 0 p 1. For each edge in the graph, rewire it to a randomly chosen (uniform probability) node with the probability p. Watts-Strogatz model: When p is small and the starting graph is circulant, the resulting graph is shown to exhibit the small-world principle, i.e., small average shortest path and large clustering. Example 4: Transportation network, electric power grids, network of brain neurons, social networks, six degrees of separation. The sending experiment. The author collaboration
9 9 network, the famous Erdös number (mine is is 5 from three paths, may be 4-cannot prove), mean is 4.65: Khan, U. Moura, J. Püschel, M. Beth, T. Mullin, R. Erdös, P.)
Lecture 18 and 19. and a graph is denoted by G = (V, E).
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