Determining the best set of nodes to receive pinning controls in networked systems is a complex combinatorial problem. In this section, we propose an effective algebraic technique to identify one such node to receive pinning control without resorting to an exhaustive search algorithm or heuristic methods based on various centrality measures, such as degree, betweenness, closeness, and the eigenratio [180, 168].
Optimal Pinning Problem: The optimal pinning control problem is concerned with the selection of the best control site(s). That is, given a network of n oscillators, find the node j to pin such that the control effort is minimized, i.e., for a fixed coupling strength σ , the feedback gain K is minimized; or for a fixed feedback gain K, the coupling strength is minimized.
The problem at hand is the stabilization of the origin through pinning control, which can be treated as pole assignment in control theory [181]. And as we have shown in Section 6.4, the critical
coupling is inversely proportional to the dominant eigenvalue of the matrix ¯Gn. Suppose that the
eigenvalues of ¯Gnare ordered as λ1< λ26 · · · 6 λn(for a connected graph), then in the absence of feedback, i.e., K = 0, the smallest eigenvalue λ1= 0 given by (6.18) (for a chain or a ring network) is independent of the coupling strength σ , while λ2, · · · , λn are proportional to σ . Therefore, in order to stabilize a network of coupled (SL) oscillators, we need to move the dominant eigenvalue
λ1 away from zero to a value greater than the constant α in the SL oscillator model by applying
the feedback gain K. As a result, the problem of finding the optimal control site is reduced to identifying the node j at which the feedback gain K will be the most effective (in the sense that for a given fixed coupling and gain (σ , K), the feedback at the optimal control site will have the
most stabilizing impact on the networked system than at any other location) in placing λ1 at a
desired location in the complex plane. This problem can be tackled using the notion of geometric measure of modal controllability [182], which allows one to quantify the influence of each input on a particular eigenvalue. However, extra care must be taken when this method is applied to systems with repeated eigenvalues [183], and it requires the computation of eigenvalues at least n times, corresponding to each possible control site in the network, which is not very practical when dealing with large complex networks. A comprehensive description of this approach as it applies to networks of oscillators is given in Appendix D.1.
To circumvent the computational burden of the modal controllability method, we propose a method- ology based on Lyapunov’s direct method [181, 184], where a Lyapunov function V (x) is es- tablished to determine stability of a dynamical system. We would refer to this approach as the
Theorem 3: Consider the time-invariant linear autonomous system
˙
x= Ax, (6.34)
with the origin as an equilibrium point, where x ∈ Rn and A ∈ Rn×n. The system described by
(6.34) has the origin as a globally asymptotically stable equilibrium point if a Lyapunov function V(x) can be found such that (i) V (x) > 0 for all x 6= 0 and V (0) = 0, (ii) ˙V(x) < 0 for all x 6= 0, and (iii) V (x) → ∞ as kxk → ∞ [184].
A canonical Lyapunov function is V (x) = x0Px, with P 0 a real symmetric, positive definite (PD)
matrix. Taking the time derivative of the Lyapunov function and substituting for the dynamics in (6.34) lead to ˙V(x) = x0[A0P+ PA + ˙P]x. If the system matrix A is Hurwitz, then ˙V must be negative
definite (ND), and for any positive definite (PD) matrix Q ∈ Rn×n, there exists a matrix P(t) such
that the matrix differential equation,
A0P+ PA + ˙P= −Q (6.35)
is satisfied. Furthermore, for a given constant matrix A, a constant matrix P will suffice for (6.35),
which implies ˙P= 0 and then (6.35) becomes
A0P+ PA = −Q. (6.36)
The main idea of our method is to consider a control system described by the Laplacian dynamics associated with a network of n nodes, given by
˙
where L ∈ Rn×nis the graph’s Laplacian matrix of the given network, B ∈ Rn×mis an input matrix,
and u = u(x) = −Kx ∈ Rm is a feedback control law. It is well known that the linear quadratic
regulator (LQR) control of the form u = −Kx = −B0Px, where P is the solution to the algebraic
Riccati equation (ARE) as in (6.36) with A = −σ L, is a stabilizing feedback control, namely,
the closed loop matrix Acl = A − BB0P is Hurwitz [181]. Observe that if we substitute A by Acl
in the Lyapunov equation (6.36) we would obtain an ARE and the computed P matrix will be PD. However, solving the ARE associated with a large network system would be computationally expensive. In the following, we would then derive simpler solutions.
To apply the Lyapunov formalism, we will shift the real part of the eigenvalues of A and consider the Lyapunov equation [A −1nI]0P+ P[A −1nI] = −Q, which gives
P= −1
2[A − 1 nI]
−1Q 0, (6.38)
where Q is taken as the identity matrix I. Note that the Laplacian dynamics in (6.37) is stable in the sense of Lyapunov [181], in other words, it is marginally stable. Hence, we can relax the condition for P and allow it to be positive semidefinite (PSD) without destabilizing the system. Note that by
doing so, the Lyapunov function V (x) = x0Px no longer satisfies condition (i) in Theorem 3 and
V(x) > 0, ∀x 6= 0. Nonetheless, we obtain the same information about the relative importance of
each control site as if (6.38) was used. By considering this relaxed condition and taking advantage of the symmetry of A, P, and Q, we obtain a simple solution to the Lyapunov equation (6.36),
P= −1
2A
aim thus far is not to compute the stabilizing gain, but rather to find some measure that will allow for the determination of the best control site. We refer to this measure as the control centrality. Once the optimal control site is known, then the appropriate stabilizing gain K can be computed taking into consideration the dynamics of the oscillators in the network.
Let us now define the vector of control centrality measures, vc= (1/d1, · · · , 1/dn)0, as the reciprocal of the diagonal entries of the matrix P, where d = diag(P) = (d1, . . . , dn)0, and the control centrality index ic(with ic= 1, · · · , n) as the index of the ithentry of vc, respectively. To compute the control centralities, we consider that n feedback controls are applied, and thus the input matrix B = I and the gain matrix K = P.
It is known that modes (eigenvalues of the A matrix) that are less controllable from a given input require higher gains to stabilize them [185]. Therefore, by computing the gain as proposed, the most influential node in the network will require the least amount of feedback gain. This is the intuition behind the proposed method. Therefore, pinned nodes that have less control (or influence) over the dominant eigenvalue of the network will have low control centrality values, while pinned nodes that have more control over the dominant eigenvalue of the network will have higher values of control centrality. Hence, the node with the highest control centrality is the optimal control site, and its index indicates the location in the network.
Remark 1: The concept of control centrality has been introduced before to quantify the ability of each node to control a directed network [186], however, in this thesis we use it as a measure to quantify the ability of each node to stabilize a network through pinning.