2016 American Control Conference (ACC) Boston Marriott Copley Place July 6-8, 2016. Boston, MA, USA The Observability Radius of Network Systems
Gianluca Bianchin, Paolo Frasca, Andrea Gasparri, and Fabio Pasqualetti
Abstract— This paper introduces the observability radius of be unreliable and in fact fail to recognize unobservable network systems, which measures the robustness of a network systems: instead, our notion of observability, that is, the to perturbations of the edges. We consider linear networks, size of the smallest perturbation preventing observability where the dynamics are described by a weighted adjacency matrix, and dedicated sensors are positioned at a subset of or, equivalently, the distance to the nearest unobservable nodes. We allow for perturbations of certain edge weights, realization, can be measured more reliably [10]. Among with the objective of preventing observability of some modes our contributions, we highlight connections between the of the network dynamics. Our work considers perturbations robustness of a network and its structure, and we propose with a desired sparsity structure, thus extending the classic an algorithmic procedure to construct optimal perturbations. literature on the controllability and observability radius of linear systems. We propose an optimization framework to Our work finds applicability, for instance, in network control determine a perturbation with smallest Frobenius norm that problems where the network weights are time varying, in renders a desired mode unobservable from a given set of security applications where an attacker gains control of some sensor nodes. We derive optimality conditions and a heuristic network edges, and in network science for the classification optimization algorithm, which we validate through an example. of network edges and the design of robust complex networks. Related work Our notion of robustness is inspired by classic works on the observability radius of dynamical systems [11], I.INTRODUCTION [12], [13], which is defined as the norm of the smallest per- Network systems are broadly used to model engineering, turbation yielding unobservability. In particular, for a linear social, and natural systems. An important property of such system with matrices (A, C), the radius of observability is systems is their robustness to different contingencies, includ- ∆A ing failure of components affecting the flow of information, µ(A, C) = min , ∆A,∆C ∆C external disturbances altering individual node dynamics, and 2 variations in the network topology and weights. It remains s.t. (A + ∆A,C + ∆C ) is unobservable. an outstanding problem to quantify how different topological features enable robustness, and to engineer complex network As a known result [12], the observability radius satisfies systems that ensure functionality and operability in the face sI A µ(A, C) = min σn − , of arbitrary and perhaps malicious perturbations. s C Observability of a network system guarantees the ability where or if complex perturbations are allowed. to reconstruct the state of individual nodes from sparse s R s C The optimal∈ perturbations∈ and are typically full measurements. While observability is a binary notion [1], ∆A ∆C matrices and, to the best of our knowledge, all existing results the degree of observability (or controllability) of a network and procedures are not applicable to the case where the can be quantified in different ways, including the energy perturbations must satisfy a desired sparsity constraint (e.g., associated with the measurements [2], [3], the novelty of see [14]). This scenario is in fact the relevant one for network the output signal [4], the number of necessary sensor nodes systems, where the nonzero entries of the network matrices [5], [6], [7], and the robustness to removal of interconnection and correspond to existing network edges, and it would edges [8]. A graded notion of observability is preferable, as A C be undesirable or unrealistic for a perturbation to modify the it allows us to compare different networks, select optimal interaction of disconnected nodes. A notable exception is [8], sensor nodes, and identify features favoring observability. where, however, the discussion is limited to edge removal. In this work we measure the robustness of a network We depart from the literature by requiring the perturbation based on the size of the smallest perturbation needed to to be real, with a desired sparsity pattern, and confined to prevent observability. Our notion of robustness is motivated the network matrix, that is, . Our approach builds by the fact that network weights are rarely known without ∆C = 0 on the theory of total least squares [15], [16]. With respect uncertainty, and observability is a generic property [9]. For to existing results, our work proposes tailored procedures for these reasons, numerical tests to assess observability may network systems, fundamental bounds, and insights into the This material is based upon work supported in part by NSF robustness of different network topologies. Although our pre- awards #BCS-1430279 and #ECCS-1405330. Gianluca Bianchin and sentation focuses on perturbations preventing observability, Fabio Pasqualetti are with the Mechanical Engineering Department, University of California at Riverside, [email protected], the extension to controllability is straightforward. [email protected]. Andrea Gasparri is with the Department Contributions The contribution of this paper is twofold. of Engineering, Roma Tre University, [email protected]. Paolo Frasca is with the Department of Applied Mathematics, University of First, we define a metric of network robustness that cap- Twente, [email protected]. tures the resilience of a network system to structural per- 978-1-4673-8682-1/$31.00 ©2016 AACC 185 turbations (Section II). Our metric evaluates the distance perturbation to achieve unobservability, and encodes of a network from the set of unobservable networks with the desired sparsity pattern of the perturbation.AH It should the same interconnection structure, and it extends existing be observed that (i) the minimization problem (2) is not works on the controllability and observability radius of linear convex because the variables ∆ and x are multiplied by each systems. Second, we formulate an optimization problem to other in the eigenvector constraint (A + ∆)x = λx, (ii) if determine optimal perturbations (with smallest Frobenius A , then the minimization problem is feasible if and norm) preventing observability. We show that the problem is only∈ AifH there exists a network matrix A + ∆ = A˜ not convex, derive optimality conditions, and show that any satisfying the eigenvalue constraint, and (iii) if = ∈, then AH optimal solution solves a (nonlinear) generalized eigenvalue the perturbation modifies the weights of existingH edgesG only. problem (Section III-A). Based on this analysis, we propose We make the following assumption, which implies that ∆ a numerical procedure based on the power iteration method must be nonzero to satisfy the constraints in (2): to determine (sub)optimal solutions (Section III-B). Finally, (A1) The pair (A, C ) is observable. O we analytically compute optimal perturbations for three di- The minimization problem (2) can be solved by two mensional line networks (Section III-C), and we validate the subsequent steps. First, we fix the eigenvalue λ, and compute effectiveness of our numerical procedure to work in practice. an optimal perturbation that solves the minimization problem for the fixed eigenvalue. This computation is the topic of II.PROBLEM SETUPAND PRELIMINARY RESULTS the next section. Second, we search the complex plane for the optimal eigenvalue yielding the perturbation with Consider a directed graph = ( , ), where = λ minimum cost. We observe that (i) the exhaustive search of 1, . . . , n and areG the vertexV E and edgeV sets, the optimal eigenvalue is an inherent feature of this class respectively.{ } LetEA ⊆= [Va ×] be V the weighted adjacency matrix λ ij of problems, as also highlighted in prior work [13]; (ii) in of , where aij R denotes the weight associated with G ∈ some cases and for certain network topologies the optimal the edge (i, j) , and aij = 0 whenever (i, j) . ∈ E 6∈ E λ can be found analytically; and (iii) in several practical Let ei denote the i-th canonical vector of dimension n. Let applications, the choice of λ is guided by the structure of = o1, . . . , op be the set of sensor nodes, and define theO network{ output} ⊆ matrix V as the problem or it is a given constraint in the optimization. T III.OPTIMALITY CONDITIONSAND ALGORITHMSFOR C = eo1 eop . O ··· THE NETWORK OBSERVABILITY RADIUS Let xi(t) R denote the state of node i at time t, and let We now consider problem (2) with a fixed choice for λ. Let ∈ n x : N 0 R be the map describing the evolution over time 2 ≥ → ∆∗ be an optimal solution to (2). Then, we refer to ∆∗ F of the network state. The network dynamics are described by as to the observability radius of the network A withk sensork the discrete time-invariant linear system model: nodes , constraint graph , and unobservable eigenvalue λ. O H x(t + 1) = Ax(t), y(t) = C x(t), (1) O A. Optimal network perturbation p where y : N 0 R is the output of the sensor nodes . ≥ → O In this section we manipulate the minimization prob- In this work we characterize structured network perturba- lem (2) to facilitate its solution. Without affecting generality, tions that prevent observability from the sensor nodes. To relabel the network nodes such that the sensor nodes satisfy this aim, let = ( , ) be the constraint graph, and H H define the setH of matricesV E compatible with as = 1, . . . , p , so that C = Ip 0 . (3) H O { } O Accordingly, = M : M R|V|×|V|,Mij = 0 if (i, j) . AH { ∈ 6∈ EH} A11 A12 ∆11 ∆12 Recall from the eigenvector observability test that the net- A = , and ∆ = , (4) A21 A22 ∆21 ∆22 work (1) is observable if and only if there is no right p p p n p n p p eigenvector of A that lies in the kernel of C , that is, where A11 R × , A12 R × − , A21 R − × , O and ∈n p n p. Let ∈ be the∈ unweighted C x = 0 whenever x = 0, Ax = λx, and λ C [21]. A22 R − × − V = [vij] O 6 6 ∈ ∈ We consider and study the following optimization problem: adjacency matrix of , where vij = 1 if (i, j) , and H ∈ EH vij = 0 otherwise. Following the partitioning of A in (4), let 2 min ∆ F, k k V11 V12 s.t. (A + ∆)x = λx, (eigenvalue constraint), V = . V21 V22 x 2 = 1, (eigenvector constraint), (2) We perform the following three simplifying steps. k k C x = 0, (unobservability), (1 – Rewriting the structural constraints) Let B = A + ∆, O 2 Pn Pn 2 and notice that ∆ F = (bij aij) . Then, ∆ , (structural constraint), k k i=1 j=1 − ∈ AH the minimization problem (2) can equivalently be rewritten where the minimization is carried out over the network restating the constraint ∆ , as in the following equality: ∈ AH n n n n n perturbation ∆ R × , the eigenvector x C , and the X X ∈ ∈ 2 2 2 2 1 unobservable eigenvalue . The cost function ∆ F = B A F = (bij aij) v− . λ C ∆ F k k k − k − ij quantifies the total cost in the∈ form of magnitude ofk edgek i=1 j=1 186 2 Notice that ∆ F = whenever ∆ does not satisfy the the diagonal matrix with entries d1, . . . , dn. We now derive structural constraint,k k that∞ is, when v = 0 and b = a . the optimality conditions for the minimization problem (7). ij ij 6 ij (2 – Minimization with real variables) Let λ = λ + iλ , Theorem 3.3: (Optimality conditions) Let x∗ , and x∗ be where i denotes the imaginary unit. Let < = a solution to the minimization problem (7). Then,< = 1 1 x x A¯ N¯ M¯ x∗ Sx Tx y1 x = 2< , and x = 2= , − < = σ , < x = x M¯ A¯ N¯ x∗ Tx Qx y2 < = − − = | {z } | {z∗ } | {z } |{z∗ } denote the real and imaginary parts of the eigenvector x, A˜ x Dx y 1 p 1 p 2 n p 2 n p T (8) with x R , x R , x R − , and x R − . We ¯ ¯ ¯ < ∈ = ∈ < ∈ = ∈ A N M y1 Sy Ty x∗ now prove that the minimization problem (2) can be restated − = σ < , M¯ A¯ N¯ y2 Ty Qy x∗ and solved over real variables only. − − = | {z } |{z∗ } | {z } | {z∗ } Lemma 3.1: (Minimization with real eigenvector con- A˜T y Dy x straint) The constraint (A + ∆)x = λx can equivalently 2n for some σ > 0 and y∗ R with y∗ = 1, and where be written as ∈ k k D = diag(v , . . . , v , v , . . . , v , . . . , v , . . . , v ), (A + ∆ λ I)x = λ x , 1 11 1n 21 2n n1 nn − < < − = = (5) (A + ∆ λ I)x = λ x . D2 = diag(v11, . . . , vn1, v12, . . . , vn2, . . . , v1n, . . . , vnn), < = = < − T T Proof: By considering separately the real and imaginary Sx = (I x∗ ) D1(I x∗ ),Tx = (I x∗ ) D1(I x∗ ), ⊗ < ⊗ < ⊗ < ⊗ = part of the eigenvalue constraint, we have (A+∆)x = λ x+ T T < Qx = (I x∗ ) D1(I x∗ ),Sy = (I y1) D2(I y1), iλ x and (A + ∆)¯x = λ x iλ x¯, where x¯ denotes the ⊗ = ⊗ = ⊗ ⊗ = < − = T T complex conjugate of x. Notice that Ty = (I y1) D2(I y2),Qy = (I y2) D2(I y2). ⊗ ⊗ ⊗ ⊗ (9) (A + ∆)(x +x ¯) = (λ + iλ )x + (λ iλ )¯x, | {z } | < = {z < − = } Proof: We adopt the method of Lagrange multipliers (A+∆)2x 2λ