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A Graph-Theoretic Classification for the Controllability of the Laplacian

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A Graph-Theoretic Classification for the Controllability of the Laplacian 53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA A graph-theoretic classification for the controllability of the Laplacian leader-follower dynamics Cesar O. Aguilar1 and Bahman Gharesifard2 Abstract— In this paper, we revisit the controllability problem control vector, and finally, conditionally controllable graphs for the Laplacian based leader-follower dynamics with the are controllable for a strict subset of the control vectors. The aim of addressing some fundamental gaps within the existing particular focus in this paper is on the case where the control literature. We introduce a notion of graph controllability classes for Laplacian based leader-follower control systems, namely, set is the binary vectors. The motivation for this choice is to the classes of essentially controllable, completely uncontrollable, capture the situation where the control signal is broadcasted and conditionally controllable graphs. In addition to the topol- by the leader nodes. Moreover, it includes as a special case ogy of the underlying graph, our controllability classes rely on the scenario considered in [2]. We prove that essentially the richness of the set of control vectors. The particular focus controllable graphs are necessarily asymmetric. Notably, in this paper is on the case where this set is chosen as the set of binary vectors, which captures the case when the control we show that the example of the 6-graph provided in the signal is broadcasted by the leader nodes. We first prove that literature [2] to demonstrate that asymmetry is not necessary the class of essentially controllable graphs is a strict subset of for uncontrollability, is in fact essentially controllable; thus the class of asymmetric graphs. We provide a non-trivial class this is not an appropriate example of such phenomenon. of completely uncontrollable asymmetric graphs, namely the We next focus on the class of completely uncontrollable class of large block graphs of Steiner triple systems. Several constructive examples demonstrate our results. graphs. We provide an explicit class of graphs, namely, the class of block graphs of Steiner triple systems, that I. INTRODUCTION are asymmetric yet completely uncontrollable. This result In recent years, there has been a surge of activity within indicates that characterizing graph uncontrollability via graph the control theory community to understand how the network symmetries targets a narrow class. Given that having a structure of multi-agent systems affects the fundamental repeated Laplacian eigenvalue results in complete uncontrol- properties of controllability and stabilizability. With regards lability, we prove that completely uncontrollable graphs on to controllability, the growing body of literature has fo- four and five vertices are completely uncontrollable if and cused on the Laplacian based consensus dynamics, see for only if they have a repeated eigenvalue. Numerical evidence instance [1], [2], [3], [4], [5], [6], and references therein. suggests that the same result holds for graphs of orders six Specifically, starting with a Laplacian consensus algorithm, and seven. However, we provide explicit examples of graphs a subset of the agents are classified as leaders and act as on eight and nine vertices that are completely uncontrollable control agents that can change the dynamics of the network. and yet have simple eigenvalues. Throughout the paper, The remaining agents, called the followers, are indirectly several examples demonstrate the results. controlled by the leaders via the connectivity of the network. II. PRELIMINARIES Most of the effort in the current literature has focused In this section, we establish notation and basic notions on obtaining graph theoretic conditions under which such from graph theory, and state a result on the controllability of systems are uncontrollable. For example, in [2] it is shown linear systems when the system matrix is diagonalizable. that if leader nodes are chosen that preserve a non-trivial graph symmetry then the resulting system is uncontrollable. A. Graph theory In this paper, we introduce graph controllability classes Our notation from graph theory is standard, see for in- for the Laplacian leader-follower dynamics over undirected stantce [7], [8]. By a graph we mean a pair G = (V, E) graphs, namely, essentially controllable, completely uncon- consisting of a finite vertex set V and an edge set E ⊆ trollable, and conditionally controllable graphs. In addition [V]2 := {{v, w} | v, w ∈ V}. The order of the graph G to the topology of the underlying graph, our graph control- is the cardinality of its vertex set V. The neighbors of v ∈V lability classes rely on the richness of the set of control is the set N := {w ∈V|{v, w} ∈ E} and the degree vectors. Essentially controllable graphs are controllable for v of v, denoted d , is cardinality of N , that is, d := |N |. any choice of non-trivial control vectors, completely un- v v v v A graph G is connected if there is a path between any pair controllable graphs are uncontrollable for any choice of the of vertices, that is, given a pair of vertices v and w there 1Cesar O. Aguilar is with the Department of Mathematics, is a sequence of distinct vertices (v0, v1,...,vk) such that California State University, Bakersfield, CA 93311, USA {vi 1, vi}∈E and v0 = v and vk = w. − [email protected] Henceforth, without loss of generality, we let V = 2Bahman Gharesifard is with the Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada {1,...,n}, where n is the order of G. The adjacency matrix [email protected] of G is the n × n matrix A defined as Aij =1 if {i, j}∈E 978-1-4673-6088-3/14/$31.00 ©2014 IEEE 619 and Aij = 0 otherwise, where Aij denotes the entry of A The following result characterizes the controllability of in the ith row and jth column. single-input linear systems (F, b) when F is diagonalizable. We denote by D the degree matrix of G, i.e., the diagonal Proposition 2.1: (Controllability and eigenvalue multi- n n matrix whose ith diagonal entry is di. The Laplacian matrix plicity): Let F ∈ R × and suppose that F is diagonalizable. of G is given by The following hold: L = D − A. (i) For any open set B⊂ Rn, the pair (F, b) is uncontrol- lable for every b ∈ B if and only if F has a repeated L The Laplacian matrix is symmetric and positive semi- eigenvalue. L definite, and thus the eigenvalues of can be ordered (ii) Suppose that F has distinct eigenvalues and let U 1 T λ1 ≤ λ2 ≤···≤ λn. The ones vector n := [1 1 ··· 1] be a matrix whose columns are linearly independent L is an eigenvector of with eigenvalue λ1 = 0, and if G eigenvectors of F. If b ∈ Rn then the dimension of L is connected then λ1 = 0 is a simple eigenvalue of . We hF; bi is equal to the number of nonzero components assume throughout that G is connected so that 0 < λ . 1 2 of v = U− b. In particular, (F, b) is controllable if A mapping ϕ : V→V is an automorphism of G if it is and only if no component of v is zero. a bijection and {i, j}∈E implies that {ϕ(i), ϕ(j)}∈E. The proof of (i) follows from the properties of the determi- An automorphism ϕ of G induces a linear transformation on nant and for the proof of (ii) see for instance [1]. n R , denoted by Pϕ or just P when ϕ is understood, whose matrix representation in the standard basis is a permutation III. PROBLEM STATEMENT AND PRELIMINARY RESULTS matrix, i.e., as a linear mapping ϕ acts as a permutation Let G = (V, E) be a graph with vertex set V = e e Rn on the standard basis { 1,..., n} of . It is well known {1, 2,...,n}. The Laplacian dynamics on G is the linear and straightforward to show that ϕ is an automorphism of system G if and only if PA = AP. Moreover, an automorphism x˙ (t)= −Lx(t), P preserves degree of vertices and therefore di = dϕ(i) for x Rn R every i ∈ {1, 2,...,n}, in other words PD = DP, and where ∈ and t ∈ . Suppose that a subset of consequently PL = LP. the vertices V ⊂V are actuated by a single control u : R A graph is called k-regular if all its vertices have degree [0, ∞) → and consider the resulting single-input linear b T n k ∈ N. A k-regular graph G = (V, E) is called strongly control system.e Explicitly, let = [b1 b2 ··· bn] ∈{0, 1} regular, denoted by SGR(n,k,λ,µ), if there exists λ, µ ∈ N be the binary vector such that V = Vb := {i ∈ V | bi =1}, such that and consider the single-input linear control system e i) |Nv ∩ Nu| = λ, for every v ∈V and every u ∈ Nv; x˙ (t)= −Lx(t)+ bu(t). (1) ii) |Nv ∩ Nu| = µ, for every v ∈V and every u∈ / Nv. The vertices Vb are seen as control or leader nodes and It is known that strongly regular graphs have exactly three influence the remaining follower nodes V\Vb through the eigenvalues [9]. A subclass of strongly regular graphs, the control signal u(·) and the connectivity of the network. The so-called block graph of a Steiner triple system, plays an motivation behind the set of binary control vectors is that it important role in one of our main results. captures the scenario of when an external agent connected to Definition 2.1: (Steiner triple systems): A (t,k,n)- the nodes Vb is unable to distinguish between its followers. Steiner triple system of order n, denoted by STS(t,k,n), Hence, all the followers receive the same control input from is a set S of n elements together with a set of k-element the leader, i.e., the control signal is broadcasted.
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