CONTENTS 1 A MOTIVATING EXAMPLE: FLOCKING Lecture: linear algebra Sonia Mart´ınez April 12, 2018 Abstract The treatment corresponds to selected parts from Chapter 1 in [1] and from Chapter 2 in [2]. Contents 1 A motivating example: flocking 1 2 Matrix concepts 2 2.1 Basic notation ............................................ 2 2.2 Discrete-time linear systems and (semi-)convergent matrices ................... 3 2.3 Special matrix sets .......................................... 3 2.4 Eigenvalues, singular values, and induced norms .......................... 5 2.5 Spectral radius and general criterion for convergent matrices .................. 5 3 Birkho↵–von Neumann theorem 6 4 Perron-Frobenius theory 7 4.1 Positive matrices ........................................... 8 4.2 Nonnegative matrices ........................................ 9 4.3 Application to dynamical systems .................................. 11 1 A motivating example: flocking Consider n agents all moving in the plane with the same speed but with di↵erent headings. Each agent’s heading is updated using a simple local rule based on the average of its own heading plus the headings of its neighbors. For our purposes here, let us assume that the interaction topology is fixed, i.e., the set of neighbors of each agent remains the same throughout the evolution. The heading of agent i 1,...,n ,written✓i, evolves in discrete time in accordance with a model of the form 2{ } 1 ✓i(t + 1) = ✓i(t)+ ✓j(t) . (1) 1+di 0 1 j X2Ni @ A 1 MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 2 MATRIX CONCEPTS Here, i denotes the set of neighbors of agent i, and di is the cardinality of this set. Headings are represented as realN numbers in the interval [0, 2⇡). If we think of this model as an algorithm, then certainly this algorithm is distributed, in the sense that each agent only needs to interact with its neighbors in the graph to execute it. Exercise 1.1 Ponder a bit about the advantages and disadvantages of interpreting a heading as a number in [0, 2⇡). • As long as the interaction graph is connected, it turns out that the algorithm (1) will make all agents agree on the same heading and “flock” in one direction. Even more, even if the interaction topology is changing between timesteps (say, at time t, agents i and j are neighbors, but they are not at time t + 1), one can show that, as long as it remains connected, the same flocking behavior emerges. Even more, for dynamic interaction topologies, they might not be connected ever, but as long as they are not “too disconnected” (and we will make this very loose statement fully precise in the future), flocking will be achieved. This example will serve us as a guide in the introduction of several important concepts in this lecture. It is instructive to write the heading update equations (1) in compact form, ✓(t + 1) = F✓(t), (2) where F is the matrix with elements 1 Fii = , 1+di 1 1+d j is a neighbor of i, Fij = i (0 j is not a neighbor of i. Several properties of F are worth noticing. First, the matrix is square and nonnegative (i.e., all its elements are greater than or equal to zero). Second, the rows of F all sum up to one. These properties play a critical role in explaining the convergence properties of the algorithm (1). 2 Matrix concepts 2.1 Basic notation We let C, R, N, and Z denote the set of complex, real, natural and integer numbers, respectively; also R 0 ≥ and Z 0 are the set of non-negative real numbers and non-negative integer numbers. For a<b, two real numbers,≥ we let [a, b]= x R a x b , ]a, b] or (a, b]= x R a<x b , { 2 | } { 2 | } [a, b[ or [a, b)= x R a x<b , ]a, b[ or (a, b)= x R a<x<b , { 2 | } { 2 | } n m n m We let R ⇥ and C ⇥ denote the set of n m real and complex matrices. Given a real matrix A and a T ⇥ complex matrix U,weletA> (or A ) and U ⇤ denote the transpose of A and the conjugate transpose matrix of U,respectively.WeletIn denote the n n identity matrix, the vector whose entries are all ones (resp. zeros) n n ⇥ n is 1n R (resp. 0n R ). For a matrix A, λ C is an eigenvalue and v C is its corresponding (right) 2 2 2n 2 eigenvector if Av = λv. A left eigenvector w C of an eigenvalue λ satisfying w>A = λw>. 2 For a square matrix A,wewriteA>0, resp. A 0, if A is symmetric positive definite, resp. symmetric positive semidefinite. Recall that a symmetric matrix≥ is positive definite (resp. positive semidefinite) if all of n its eigenvalues are positive (resp. nonnegative.) For a real matrix A,weletkernel(A)= x R Ax = 0n { 2 | } and image(A)= y Rn Ax = y, for some x Rn , rank(A) = dim(image(A)) denote the kernel and rank of A, respectively.{ Given2 | a vector v, we let diag(2 v)} denote the square matrix whose diagonal elements are equal to the component v and whose o↵-diagonal elements are zero. 2 MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 2.2 Discrete-time linear systems and (semi-)convergent matrices 2 MATRIX CONCEPTS 2.2 Discrete-time linear systems and (semi-)convergent matrices Definition 2.1 (Discrete-time linear system) A square matrix A defines a discrete-time linear system by x(k + 1) = Ax(k),x(0) = x0, (3) k or, equivalently by x(k)=A x0, where the sequence x(k) k Z 0 is called the solution, trajectory or evolution of the system. { } 2 ≥ Sometimes it is convenient to adopt the shorthand x+ = f(x) to denote the system x(k + 1) = f(x(k)), for a given map f : Rn Rn. We are interested in understanding when a solution from an arbitrary initial condition has an asymptotic! limit as time diverges and to what value the solution converges. We formally de ne this property as follows. n n Definition 2.2 (Convergent and semi-convergent matrices) AmatrixA R ⇥ is 2 ` (i) semi-convergent if lim` + A exists; and ! 1 ` (ii) convergent if it is semi-convergent and lim` + A =0. ! 1 • k It is immediate to see that, if A is semi-convergent with limiting matrix A =limk + A ,thenlimk + x(k)= 1 ! 1 ! 1 A x0. 1 Remark 2.3 (Modal decomposition for symmetric matrices) Before treating the general analysis method, we present a self-contained and instructive case of symmetric matrices. Recall that a symmetric matrix A has real eigenvalues λ1 λ2 λn and corresponding orthonormal (i.e., orthogonal and unit-length) ≥ ≥···≥ n eigenvectors v1,...,vn. Because the eigenvectors are an orthonormal basis for R , we can write the modal th decomposition x(k)=y (k)v + + y (k)v ,wherethei normal mode is defined by y (k)=v>x(k). 1 1 ··· n n i i We then left-multiply the two equalities (3)byvi> and exploit Avi = λivi to obtain yi(k + 1) = λiyi(k), k yi(0) = vi>x0,whichimpliesyi(k)=λi (vi>x0). In short, the evolution of the linear system is k k x(k)=λ (v>x )v + + λ (v>x )v . 1 1 0 1 ··· n n 0 n Therefore, each evolution starting from an arbitrary initial condition satisfies (i) limk x(k)=0n if and only if λi < 1 for all i 1,...,n and !1 | | 2{ } (ii) limk x(k)=(v1>x0)v1 + +(vm> x0)vm if and only if λ1 = = λm = 1 and λi < 1 for all i !1m +1,...,n ··· ··· | | 2{ } 2.3 Special matrix sets The provided examples of MAS are defined via special matrices, whose properties are key to establish whether they are convergent or not. In the following we recall a few definitions. n n A matrix A R ⇥ with entries aij, i, j 1,...,n ,is 2 2{ } T (i) Orthogonal if AA = In, and is special orthogonal if it is orthogonal with det(A) = +1. The set of orthogonal matrices is a group.1 1AsetG with a binary operation, denoted by G G (a, b) a?b G,isagroup if: (i) a?(b?c)=(a?b) ?c for all a, b, c G (associativity property); (ii) there exists e⇥ G3such that7! a?e2= e?a= a for all a G (existence of an identity element);2 and (iii) there exists a 1 G such that a?a2 1 = a 1 ?a= e for all a G (existence of2 inverse elements). − 2 − − 2 3 MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 2.3 Special matrix sets 2 MATRIX CONCEPTS (ii) Nonnegative (resp., positive) if all its entries are nonnegative (resp., positive). n (iii) Row-stochastic (or stochastic for brevity) if it is nonnegative and j=1 aij = 1, for all i 1,...,n ; in other words, A is row-stochastic if 2{ } P A1n = 1n. (iv) Column-stochastic if it is nonnegative and n a = 1, for all j 1,...,n . i=1 ij 2{ } (v) Doubly stochastic if A is row-stochastic andP column-stochastic. (vi) Normal if AT A = AAT . (vii) A permutation matrix if A has precisely one entry equal to one in each row, one entry equal to one in each column, and all other entries equal to zero.