In Section 5.2, we have used graph theory to characterize necessary and sufficient con- dition for structural stabilizability – see Theorem 15. However, such a condition is valid only when the underlying system is captured by an undirected graph, i.e., the state ma- trix A is symmetric. In this section, we aim to extend this result to general structural systems. More specifically, our goal in this section is to solve the following problem:
Problem 11. Given a continuous-time linear time-invariant system x˙ =Ax+Bu, we denote by ( ¯A,B¯) the structural pattern of (A, B). Let G( ¯A,B¯) be the digraph represen- tation of the structural pair. Find necessary and sufficient conditions in G( ¯A,B¯) such that( ¯A,B¯) is structurally stabilizable.
5.5.1 Characterizing General Structural Stabilizability
To solve the above problems, we first recall the definition of structural stabilizability – see Definition 11. As discussed in Lemma 6 earlier this chapter, structural stabilizability is equivalent to structural controllability when the structrual pair ( ¯A,B¯) is irreducible. Therefore, in order to characterize structural stabilizability, we consider permuting the pair ( ¯A,B¯) into the form (2.3) and analyze the graph-unreachable part of the structural system. In other words, according to the definition of stabilizability, we have to find conditions on when there exists a numerical realization ˜A22 such that all its eigenvalues are contained in the open left-half-plane. This motivates us to introduce the following definition.
Definition 16 (Structural Hurwitz-stability). A structured matrix A¯ is called struc- turally Hurwitz-stable if there exists a numerical realization A˜ of A¯such that all eigen- values of A˜ are contained in the open left-half-plane of C.
Next, we present a few sufficient and/or necessary conditions on when a structured matrix is structurally Hurwitz-stable.
Lemma 9. Let A¯ be a structured matrix, andG( ¯A) be the digraph representation ofA.¯ If every vertex xi has a self-loop, thenA¯is structurally Hurwitz-stable.
Proof. If every vertex xi has a self-loop in G( ¯A), then [ ¯A]ii is a ?-entry for alli∈ [n].
In this case, we consider the following assignment on the ?-entries of ¯A:
[ ˜A]ij = −1, ifi=j, 0, ifi6=j. (5.7)
Thus, ˜A =−I is a numerical realization of ¯A and all eigenvalues of ˜A are strictly less than 0.
In the above construction of a numerical realization of ¯A, we have set all off-diagonal
?-entries to zero and all diagonal entries to negative real numbers. Hereafter, we show that it is possible to find a stable numerical realization of ¯Awith all its?-entries are not
equal to 0.
Lemma 10. LetA¯be a structured matrix and all conditions in Lemma 9 holds inG( ¯A). Then there exists a numerical realization of A¯ with all its ?-entries are not equal to 0
such thatA˜ is (strictly) Hurwitz-stable.
Proof. Let dij be the independent parameter of [ ¯A]ij (provided that [ ¯A]ij is a ?-entry.
Since every vertex inG( ¯A) has a self-loop, it is possible to assign values todij such that dii<−Pj6=i|dij|for alli, j∈[n].Subsequently, using Gershgorin’s disk theorem [168],
by setting [ ˜A]ij =dij for all [ ¯A]ij =? and [ ˜A]ij = 0 otherwise, the matrix ˜A is Hurwitz-
stable.
The above lemmas provide sufficient conditions on structural stabilizability. However, such a condition is not necessary. Consider the following example where
¯ A= ? ? ? 0 .
In the digraph representation of ¯A, only the first vertex has a self-loop, violating the assumption in both Lemma 9 and 10. However, consider the following numerical real- ization: ˜ A= −10 3 −3 0 .
It is easy to see that ˜A is Hurwitz-stable since both of its eigenvalues are strictly less than 0. To partly strengthen the above graph-theoretical condition, we next present a lemma that characterizes a necessary condition for structural Hurwitz-stability.
Lemma 11. Let A¯ be a structured matrix, if there exists a (strictly) Hurwitz-stable numerical realization A˜, then G( ¯A) must contain at least one self-loop.
Proof. We proof this lemma by contradiction. Let us suppose thatG( ¯A) does not contain any self-loop. Following this assumption, all diagonal elements of ¯Aare fixed-zeros. Let
˜
˜
A.Since ˜Ais asymmetric, we can write λi=σi+jωi for alli∈[n].In order words, we
denote byσi and ωi the real and imaginary part of the eigenvalue λi.Subsequently, we
have that Tr( ˜A) = n X i λi = n X i=1 σi+jωi = n X i=1 σi<0,
where the last equality is due to the fact that solutions to characteristic polynomials come in pairs. However, since all diagonal elements of ˜A are zero, Tr( ˜A) is equal to 0,
Part II
Networked System Analysis via
Measure Theory
Chapter 6
Bounds on the Spectral Radius of
Digraphs
In the first part of the thesis, we have used tools from structural systems theory and graph theory to characterize properties in symmetrically structured linear systems. In this chapter, we analyze global system properties using, solely, local structural informa- tion. This chapter is organized as follows: In Section 6.1, we introduce certain notions from algebraic graph theory used in our derivations. In Section 6.2, we relate the spec- tral moments of a digraph to subgraph counts and introduce the truncatedK-moment problem from functional analysis, which we then use to upper and lower bound the spec- tral radius using subgraph counts. In Section 6.3, we propose a refine approach to find more accurate bounds on spectral radius by analyzing the skew-symmetric part of the adjacency matrix. We numerically validate the quality of our bounds using randomly generated directed graphs, as well as real networks in Section 6.4.
6.1
Adjacency Matrix and Digraph Isomorphism
In the rest of this chapter, we adopt notations introduced in Subsection 2.1.2. However, we assume that the digraph under consideration is simple. Additionally, a subgraph
Gs is called a bidirected edge if Vs = {i, j} and Es = {(i, j),(j, i)}, where i, j ∈ V. A
subgraph Gs is called a directed triangle if Vs = {i, j, k} and Es = {(i, j),(j, k),(k, i),
wherei, j, k ∈ V.
A digraph G can be represented by an adjacency matrix A ∈Rn×n,whose entries are
defined as [A]ij = 1 if (j, i) ∈ E; [A]ij = 0 otherwise. Particularly, if the graph is
undirected, thenA =A> and all its eigenvalues are real. When the digraph is simple, all the diagonal entries of A are zero. In what follows, we use λ1, . . . , λn to denote
the eigenvalues of A. The eigenvalue spectrum of A is denoted by spec(A) ={λi}ni=1. Moreover, the real part (respectively, imaginary part) ofλiis denoted asσi(respectively, ωi). Without loss of generality, we assume |λ1(A)| ≤ · · · ≤ |λn(A)|. Thespectral radius
ofA is defined as |λn(A)|.Furthermore, we denote byωmax(A) = maxi|ωi|.
Two directed subgraphsGs, Gh ⊆Gare said to beisomorphic[22], denoted byGs'Gh,
if there exist a bijection f :Vs → Vh such that (u, v) ∈ Es if and only if (f(u), f(v))∈ Eh for all u, v ∈ Vs. When Gs and Gh are non-isomorphic, we write Gs 6' Gh. In
particular, whenVs=Vh,the bijectionf is called anautomorphismand the two directed
subgraphsGs and Gh are said to be automorphic, denoted by Gs a
'Gh.Consequently,
an automorphism is an equivalence relation on the set of directed subgraphs of the same order, i.e., it classifies all possible directed subgraphs into equivalent classes. Based on these notions, we define theisomorphic group (respectively, automorphic group) of a directed subgraph Gs ⊆ G by Iso(Gs, G) = {Gh ⊆ G : Gh ' Gs} (respectively,
Auto(Gs, G) ={Gh⊆G:Gh a
'Gs}). Given a directed subgraph Gs⊆G, the count of Gs is defined by
Count(Gs, G) =
|Iso(Gs, G)|
|Auto(Gs, G)| .
Finally, let Ξsbe the set of weakly-connected digraphs of orders.We denote by Ωs⊆Ξs,