IN Part-I paper [1], the proposed control protocol under. Strategic Topology Switching for Security Part II: Detection & Switching Topologies

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1 1 Strategic Topology Switching for Security Part II: Detection & Switching Topologies Yanbing Mao, Emrah Akyol, and Ziang Zhang arxiv: v1 cs.ma 3 Nov 17 Abstract This two-part paper considers strategic topology switching for the second-order multi-agent system under attack. In Part II, we propose a strategy on switching topologies to reveal zero-dynamics attack. We first study the detectability of zero-dynamics attack for the second-order multi-agent system under switching topology, which has requirements on the switching times and switching topologies. Based on the strategy on switching times proposed in Part I and the strategy on switching topologies proposed in Part II, a decentralized strategic topology-switching algorithm is derived. The primary advantages of the algorithm are: 1) in achieving consensus in the absence of attacks, control protocol does not need velocity measurements and the algorithm has no constraint on the magnitude of coupling strength; ) in revealing zero-dynamics attack, the algorithm has no constraint on the size of misbehaving-agent set; 3) in revealing zero-dynamics attack, if the Xor graph generated by every two-consecutive topologies has distinct eigenvalues, only one output is enough for the algorithm. Simulation examples are provided to verify the effectiveness of the strategic topologyswitching algorithm. Index Terms Multi-agent system, second-order consensus, strategic topology switching, zero-dynamics attack. I. INTRODUCTION IN Part-I paper 1, the proposed control protocol under switching topology is governed by only relative positions of agents, which is different from the well-studied control protocols that need both relative position measurements and velocity measurements 6. The second-order multi-agent system under the proposed control protocol are described by ẋ i = v i (1a) n v i = γ a σ ji (x j x i ), i = 1,, n (1b) j=1 where x i R is the position state, v i R is the velocity state, γ > is the coupling strength, σ :, ) S {1,,..., s}, s N, is the topology-switching signal, i.e., σ = p k S for t t k, t k+1 ) means the p th topology is activated over the time interval t k, t k+1 ), and a p k ij is the element of the coupling matrix that describes the activated p th topology of undirected communication network. The final objective of this two-part paper is the strategic topology-switching algorithm for the second-order multi-agent system under attack. The algorithm is based on two strategies that are the strategy on switching times and the strategy on Y. Mao, E. Akyol and Z. Zhang are with the Department of Electrical and Computer Engineering, Binghamton University SUNY, Binghamton, NY, 139 USA, ( ymao3@binghamton.edu; eakyol@binghamton.edu; zhangzia@binghamton.edu). Table I CONDITIONS ON DETECTABLE ATTACK Reference Conditions Dynamics 8 connectivity is not small than K + 1 Discrete 9 K is smaller than connectivity Discrete 1 size of linking (K M) is smaller than K Continuous switching topologies. The strategy on switching times studied in Part-I paper 1 enables the strategic topology-switching algorithm the ability of reaching the second-order consensus in the absence of attacks. In the following, we present the precise definition of the second-order consensus in this context. Definition 1: The second-order consensus in the multiagent system (1) is achieved, if and only if the following holds lim x i x j =, t lim v i v j =, i, j = 1,, n. t (a) (b) for any initial condition. The strategy on switching topologies studied in this Part-II paper enables the strategic topology-switching algorithm the ability of revealing zero-dynamics attack without constraint on the misbehaving-agent set. A. Related Work For the cyber-physical systems, several incidents, such as Stuxnet malware attack and Maroochy Shire Council Sewage control incident, has raised the attention to security challenge 7. One of the fundamental problems of security is the attack detection However, there exists an attack called stealthy attack, where the attacker s goal is to make the user accept false aggregation results, which are significantly different from the true results determined by the measured values, while not being detected by the user 16. The undetectable property has raised attention to the detectability of stealthy attack 8 1. For the large scaled networked system, due to the lack of centralized measurements, the multiagent systems are prone to attack 1. The obtained results of detectability of zero-dynamics attack in recent years are summarized in Table I, which shows that the detectability obtained therein has constraints on the connectivity of network topology and the size of misbehaving-agent set. To reveal zerodynamics attack, the strategic topology-switching algorithm studied in this two-part paper has no such constraints, i.e., the situation that all agents are misbehaving is allowed.

2 B. Motivation of Part-II Paper Recent experiment of stealthy false-data injection attacks on networked control system 17 showed the changes in the system dynamics could be used to reveal stealthy attack. To have changes in the system dynamics to reveal zero-dynamics attack, Teixeira et al. 11 considered the method of modifying input matrix or modifying output matrix. However, in more realistic situations, such as the defender or the system operator has no knowledge of the attack-beginning time, or the attacks happen infinitely over infinite time, to reveal zero-dynamics attack the system dynamics has to be time-varying, i.e., the system dynamics changes infinitely over infinite time. However, before using the time-varying dynamics to reveal zerodynamics attack, the question that whether the time-varying dynamics can destroy the system stability in the absence of attacks must be investigated. If the time-varying dynamics can destroy the stability of original system, the changes in system dynamics would be an attack, such as the topology attack 18, 19. For the dynamical networks, recent studies have highlighted the important role played by the network topology 3. For example, Menck et al. find that in the numerical simulations of artificially generated power grids, tree-like connection schemes, so-called dead ends and dead trees, can strongly diminish the stability; Schultz et al. 3 show that how the addition of links can change the synchronization properties of the network. These motivate us to consider the method of topology switching such that the multi-agent system can have changes in its system dynamics to reveal zero-dynamics attack. The strategy on switching times proposed in Part-I paper 1 shows that if the dwell time of switching topologies satisfies certain condition, the second-order consensus can be achieved, i.e., following the strategy on switching times, the time-varying dynamics of multi-agent system, which is caused by topology switching at switching time, does not destroy the agents ability of reaching consensus. Based on the work in Part-I paper 1, this Part-II paper focuses on the strategy on switching topologies that addresses the problem of switching to what topologies to reveal zero-dynamics attack. C. Contribution of Part-II Paper The contribution of this paper is threefold, which can be summarized as follows. Based on the detectability of attack, we propose a strategy on switching topologies which can reveal zero-dynamics without constraint on the size of misbehaving-agent set. Under the strategy setting that the Laplacian matrix of Xor graph generated by every two-consecutive topologies has distinct eigenvalues, only one output is enough to reveal zero-dynamics attack. Based on the strategy on switching times and the strategy on switching topologies, a strategic topologyswitching algorithm is proposed. To reveal zero-dynamics attack, the algorithm has no constraint on the size of misbehaving-agent set. To achieve the second-order consensus, the algorithm has no constraint on the magnitude of coupling strength and the control protocol does not need velocity measurements. Based on the strategy on switching times and the strategy on switching topologies, an attack-detection algorithm working under strategic topology-switching algorithm is derived. The remainder of this paper is organized as follows: Section II presents the preliminaries and problem formulation; in Section III, the detectability of zero-dynamics attack is studied; Section IV presents the strategic topology-switching algorithm. Numerical examples are given in Section V; finally, Section VI concludes the two-part paper. II. PRELIMINARIES AND PROBLEM FORMULATION A. Preliminaries 1) Notation: For a set V, V denotes the cardinality (i.e., size) of the set. In addition, for a set K V, V\K denotes the complement set of K with respect to V. R n and R m n denote the set of n-dimensional real vectors and the set of m n- dimensional real matrices, respectively. Let C denote the set of complex number. N represents the set of the natural numbers and N = N {}. Let I and n n be the identity matrix with compatible dimension and n-dimensional zero matrix, respectively. 1 n R n and n R n denote the vector with all ones and the vector with all zeros, respectively. The superscript stands for matrix transpose. ) Graph Theory: The interaction among n agents is modeled by an undirected graph G = (V, E), where V = {ϑ 1, ϑ,, ϑ n } is the set of vertices that represent n agents and E V V is the set of edges of the graph G. An undirected edge in G is denoted by a ij = (ϑ i, ϑ j ) E, where a ij = a ji = 1 if agents i and j interact with each other, and a ij = a ji = otherwise. Assume that there are no self-loops, i.e., for any ϑ i V, a ii / E. A path is a sequence of connected edges in a graph. A graph is a connected graph if there is a path between every pair of vertices. Lemma 1: 4 If the undirected graph G is connected, then its Laplacian L R n n has a simple zero eigenvalue (with eigenvector 1 n ) and all its other eigenvalues are positive and real. Lemma : 5 The Laplacian of a path graph P n has the eigenvalues as ( ) (k 1) π λ k = cos, k = 1,, n. (3) n B. Problem Formulation For simplicity, let K = {ϑ 1, ϑ, } V = {ϑ 1,, ϑ n } denote the set of misbehaving agents, and M = {ϑ 1, ϑ, } K denote the set of outputs. The multiagent system (1) with its output under attack can be described by ẋ i = v i, (4a) n v i = γ a σ ij (x j x i ) + g i u i (t κ), (4b) i=1 y i = x i + g i u i (t κ), i M (4c)

3 3 where γ > is the coupling strength; σ is the topology switching signal of the system under attack; g i is agent i s attack signal; u i (t κ) is the unit step function: u i (t κ) = 1 if t κ and i K, and u i (t κ) = otherwise; κ is the attack-beginning time. The agent under attack is usually named the misbehaving agent. Definition : 9 Consider multi-agent system (4), an agent i is misbehaving if there exists a time t such that g i u i (t κ). The multi-agent system (4) can be rewritten in the form of switched system under attack: {ż = Aσ z + Bg (5) y = C M z + D M g where z = x 1,, x n, v 1,, v n R n and A σ = n n I, (6) γl σ n n B B = K K V\K n n, (7) V\K K V\K V\K B K = e ϑ1,, e ϑ K R K K, (8) C M = e χ1 e χ M M (n M ), (9) D M = M n e χ1 e χ M M (n M ) (1) n g = g, (11) V\K g = g 1 u i (t κ),, g K u K (t κ), (1) with e ϑi R K and e χi R M are the i th vectors of the canonical basis. For the multi-agent system (1), consider the constant velocity v = 1 n v i = 1 n v i (t ). (13) n n i=1 Based on v, define fluctuations: x i = x i 1 n ṽ i = v i v, i=1 n x i () vt, i=1 (14a) (14b) where v is given by (13). It follows from (14a) and (14b) that 1 n x =, (15) 1 n ṽ =. (16) Using the defined fluctuations (14a) and (14b), with the same outputs in (5) that are in the absence of attacks, the multi-agent system (1) can be written equivalently as { z = A σ z (17) ỹ = C M z where z = x 1,, x n, ṽ 1,, ṽ n R n, A σ and C M are given by (6) and (9), respectively. Now we make the following assumptions on the attacker and defender. Assumption 1: Assume that attacker: 1) has the knowledge of topology switching sequences, including the switching times and the activated topologies at switching times; ) can modify the initial conditions arbitrarily if the attackbeginning time is the initial time; 3) can attack the second-order multi-agent system infinitely over infinite time. Assumption : Assume defender has no knowledge of the attack-beginning time and the misbehaving agents. To end this section, we present the following lemmas that can be used to derive the strategic topology-switching algorithm in the following section. Lemma 3: 1 Consider the following system: x = γl t x (τ)dτ + ṽ(), (18) where γ >, L R n n is the Laplacian matrix of a connected undirected graph; x () R n and ṽ () R n satisfy (15) and (16), respectively. The individual position solutions x i, i = 1,, n, are x i = n l= q li q l ( x () cos( γλ l t) + ṽ() γλl sin( ) γλ l t), t (19) where λ l (λ 1 = ), l =,, n, are the none-zero eigenvalues of L, q l = q l1,, q ln R n is the eigenvector associated with the eigenvalue λ l of L. Let σ = r S for t t k, t k+1 ), k N, the t dynamics in (17) can be rewritten as x = γl r t k x (τ)dτ + ṽ(t k ), t t k, t k+1 ). Therefore, Lemma 3 implies that the multi-agent agent system (1) under each fixed topology has a period P such that {ṽi = ṽ i (t + P), i = 1,, n x i = x i (t + P), t t k, t k+1 ), k N. Lemma 4: 1 Consider the function F = ϖ x x () + ṽ ṽ, (1) with ṽ = x R n. Along the system (18), if the Laplacian matrix L has distinct eigenvalues and ϖ satisfies < ϖ γλ i (L), i =,, n, () where λ i (L) denotes the i th eigenvalue of L and λ 1 (L) =. Then the following situation would never happen: F φ, t. (3) Lemma 5: 1 Consider the second-order multi-agent system (17). For the given period P satisfying (), scalars 1 > β >, α > and L N. If the dwell time τ satisfies (β 1 L L 1) α ξ < τ min τ = ˆτ max + mp, m N, (4)

4 4 with ξ < α, (5) < ˆτ max < ln β α, (6) < ˆτ max + m P ( ) β 1 L L 1 α ξ, (7) ξ = max {1 γλ i (L r ), 1 + γλ i (L r )}, (8) r S,i=1,,n where γ is the coupling strength and λ i (L r ) is the i th eigenvalue of the Laplacian matrix L r. Then the second-order consensus can be achieved by Definition 1. Remark 1: Lemma 5 is a strategy on switching times that when the multi-agent system (5) should have changes, which are caused by topology switching, in its dynamics to reveal zero-dynamics attack 17, such that the changes on the system dynamics do not destroy the system stability in the absence of attacks. Remark : In achieving the second-order consensus, the strategy on switching times Lemma 5 has no constraint on the magnitude of coupling strength γ >. The obtained solution in Lemma 3 shows the coupling strength can affect the required period P, which works with Lemma 5 implies that the coupling strength can control the dwell time of switching topologies, thus can affect the convergence speed of consensus. The strategic topology-switching algorithm and the attackdetection algorithm that are to be proposed in the following sections are illustrated by Figure 1, where the clock icon along communication link denotes the logic of link is driven by time, i.e., the topology-switching signal is time-dependent. on the detection of zero-dynamics attack by strategy setting on switching topologies. Before proceeding on, we present the formal definitions of undetectable attack and zero-dynamics attack of continuoustime multi-agent systems. Definition 3: (Undetectable Attack 1) For the multiagent systems (5) and (17). The attack signal g in (5) is undetectable if j M : y j = ỹ j, t. Definition 4: (Zero-Dynamics Attack 1) Consider the following two systems {ṗ = Ap + Bg (9) ȳ = Cp + Dg { q = Aq (3) ỹ = Cq where p, q R n, ȳ, ỹ R m with m n, g R o with o n, A R n n, B R n o, C R m n and D R m o. The attack signal g = g (κ) e λ(t κ), t κ with κ, is a zero-dynamics attack if g(κ) o and λ C satisfy λi A B p (κ) q (κ) n =. (31) C D g (κ) m The state and output of system (9) satisfy ȳ = ỹ, t (3) p = q + (p (κ) q (κ)) e λ(t κ), t κ. (33) Remark 3: The detailed proof of the results (3) and (33) can be found in 6. The resulted state (33) shows that through choosing the parameter λ, the attacker can achieve its attack objective: Re (λ) > : unstable system; Re (λ) =, Im (λ) : oscillating; Re (λ) < : modifying the steady-state value. The resulted output (3) means the undetectable property of zero-dynamics attack. Figure 1. Strategically Topology-Switching for the Second-Order Multi-Agent System: two communication links control four topologies. III. DETECTABILITY OF ZERO-DYNAMICS ATTACK This section will show the advantages of strategic topology switching in revealing zero-dynamics attack. A. Zero-Dynamics Attack Zero-dynamics attack is one class of stealthy attacks whose one important property is undetectable. This section will focus B. Strategy on Switching Topologies To better present the strategy on the switching topologies, we introduce the definition of components in a graph. Definition 5: 7 The components of a graph G are its maximal connected subgraphs; a subgraph is a trivial component if it has no edges; otherwise, it is a nontrivial component. Definition 6: The Xor graph G xor is a graph generated by two graphs G 1 and G that have same number of vertices: a xor ij = a 1 ij xor a ij. Based on the components of Xor graph, we can rewrite the agent set as V = C 1 C Cd where d is the number of the components of Xor graph and C q, q = 1,, d, is the set of all agents in the q th component. Obviously, C p Cq = if p q. Take the Xor graph generated in Figure as an example, the Xor graph has two nontrivial components as C 1 = {ϑ 1, ϑ, ϑ 3, ϑ 4 }, C = {ϑ 5, ϑ 6 }, and two trivial components as C 3 = {ϑ 7 }, C 4 = {ϑ 8 }.

5 5 Figure. Components of Xor Graph We present the following strategy on switching topology that can reveal zero-dynamics attack without constraint on the size of misbehaving-agent set. Theorem 1: Consider the multi-agent system under attack (5). If the strategy satisfies (r1) (r) after the attack begins at κ, the first topologyswitching signal is time-dependent, i.e, σ(t k ) where k = arg min {t r > κ}, is time-dependent; r N for the Xor graph generated by the two consecutive topologies around the time-dependent topologyswitching time t k defined in (r1): (ra) the Laplacian matrix of each nontrivial component has distinct eigenvalues, (rb) at least one agent in each component is equipped with the monitor output of position, then without constraint on the misbehaving-agent set, the strategy can reveal zero-dynamics attack. Proof of Theorem 1: Consider the requirement (rb) on the set of outputs. To finish the proof we can consider the extreme situation: only one agent in each component has output. Without loss of generality, we can let the first agent in each component has output. Thus, we can define a set of positions of the first agent in each component: O = { 1, 1 + C 1, 1 + C 1 + C,, 1 + } d C h, (34) h=1 where d is the number of components of the Xor graph generated by the two consecutive topologies around the topologyswitching time t k. Therefore, in this situation that only one output in each component is available, we can re-arrange the agents such that the multi-agent systems (5) and (17) can be written as {ż = Aσ z + Bg S 1 : (35) y j = x j + g j u j (t κ), j O { z = A σ z S : (36) ỹ j = x j, j O where O is given by (34). Without loss of generality, we assume the attack-begging time falls into the interval t k 1, t k ), k N. The strategy (r1) means after the attacker attacks at κ, the switching signal σ over two consecutive intervals κ, t k ) t k, δ) is timedependent. Consider the systems (35) and (36). Let z = z, t < κ, so during the time t t, κ), g K and the systems (35) and (36) are the same. Because over the two consecutive intervals κ, t k ) t k, δ), the topology-switching signals σ and σ in (35) and (36) are time-dependent. Hence, we can conclude A σ = A σ, t κ, t k ) t k, δ). Let z = z z with z () = z () z (), y j = y j ỹ j and x j = x j x j. From (35) and (36) we have { z = Aσ z + Bg y j = x j + g j u j (t κ), t κ, t k ) t k, δ), j O. (37) The rest proof can be finished by contradiction. We assume the multi-agent system (35) has zero-dynamics attack. Denote σ = 1 for t κ, t k ). By (31) we can obtain λin n A 1 B ē j ê j z(κ) g(κ) = n where g(κ) is given by (11) with (1), ē j = n ê j =. e j Noting the matrix in B (7), substituting x x z = z z = v ṽ B K = g K (κ) = n = B K K V\K V\K K V\K V\K, j O ej n x = v (38) and, (39a), (39b), (39c) g(κ), (39d) V\K and A 1 given by (6) into (38) yields, for j O, x(κ) λi I n n n n γl 1 λi n n BK v(κ) e j n n e n = j n n (4) Assume the topology switches from topology 1 to topology at instance t k. Let ξ = t k κ. If the multi-agent system (35) has zero-dynamics attack over the time t k, δ), z(κ)e λξ and g(κ)e λξ would be the state-zero direction and attack-zero direction for system (35) over the time interval.

6 6 t k, δ). Using the same analysis method to derive (4), we x(κ) λi I n n n n can obtain γl λi n n BK e λξ v(κ) e j n n e n = j n n, j O, which is equivalent to, for j O, λi I n n n n γl λi n n BK e j n n e j From Equations (4) and (41), we have x(κ) v(κ) n = n n. (41) λ x (κ) v (κ) = n, (4) B K + γl 1 x (κ) + λ v(κ) = n, (43) B K + γl x (κ) + λ v(κ) = n, (44) x j (κ) g j (κ) =, j O. (45) Considering the defined components of Xor graph, L 1 L can be written as L 1 L = diag {±L (C 1 ), ±L (C ),, ±L (C d )}, (46) where L (C q ) denote the Laplacian matrix of the q th, q {1,,, d}, component of Xor graph. Obviously, L (C q ) = 1 if C q = 1. Considering (46), from (4) (45) we can observe: (a) Equation (4) is equivalent to λ x (κ) = v (κ). (b) From (43) and (44), γ (L 1 L ) x (κ) = n. (c) If L (C q ), q {1,,, d}, has distinct eigenvalues, it has properties: (i) zero is one of its eigenvalues with multiplicity one; (ii) the eigenvector that corresponds to the eigenvalue zero is 1 Cq, q {1,,, d}. (d) Equation (45) is equivalent to x j (κ) = g j (κ), j O. (e) Combing properties (c) and (d) can yield the solution of Equation γ (L 1 L ) x (κ) = n as x j (κ) = x j+1 (κ) = = x O(ij+1) 1 (κ) = g j (κ), j O, where i j is index of element j in the set O, i.e., O(i j ) = j. Now, based on the above observations (a) (e) we consider the following two different cases. Case One λ = : In this case, if the Laplacian matrix of each component has distinct eigenvalues, from Lemma 1, observation (a), observation (e) and (43) or (44), we have B K = n which is equivalent to g 1 (κ) = = g K (κ) =, so there is no zero-dynamics attack by Definition 4. Case Two λ : In this case, if the Laplacian matrix of each component has distinct eigenvalues, observation (a) and observation (e) implies v j (κ) = ṽ j+1 (κ) = = v O(ij+1) 1 (κ) = λg j (κ), j O. (47) Considering Lemma 1 and observation (e), substituting (47) into (43) or (44) yields the same result as BK = λ x j (κ) 1 n, which is equivalent to g j (κ) = = g O(ij+1) 1 (κ) = λ x j (κ), j O. (48) From (48) and observation (d) we have g j (κ) = λ x j (κ) = x j (κ), j O, which is equivalent to (1 + λ ) x j (κ) =, which also implies x j (κ) =, j O, or λ = ±i. Considering observation (e) and (48), we can conclude g j (κ) = = g O(ij+1) 1 (κ) =, j O, if x j (κ) =, so there is no zero-dynamics attack by Definition 4. If λ = ±i, observation (a) means ±i x (κ) = v (κ), which means the requirements as x(κ) R n and v(κ) R n in Definition 4 are not satisfied. Therefore, the definition of zero-dynamics attack is ruled out. All these analysis results show that there is no zerodynamics attack, which contradicts the assumption. Remark 4: The strategy (r) in Theorem 1 implies that if the Laplacian matrix of each nontrivial component has distinct eigenvalues, then the minimum number of the outputs required to reveal zero-dynamics attack is equivalent to the number of components of Xor graph. Take Figure as an example, two nontrivial components are path graphs; with (k 1)π n < π, k = 1,, n, Lemma implies that the Laplacian matrix of path graph has distinct eigenvalues; hence, the Xor graph in Figure satisfies (ra) in Theorem 1; therefore, we can conclude if the topology set S includes these two graphs, the minimum number of outputs is four. If Xor graph has only one component and its Laplacian matrix has distinct eigenvalues, then only one output is enough to reveal zero-dynamics attack. Corollary 1: Consider the multi-agent system under attack (5). Under the topology-switching strategy: (r1) after the attack begins at κ, the first topologyswitching signal is time-dependent, i.e, σ(t k ), where k = arg min {t r > κ}, is time-dependent; r N (r) the switching topologies around the time-dependent topology-switching time t k, defined in (r1), satisfy that L σ(tk ) L σ(κ), t k > κ, has distinct eigenvalues; without constraint on the misbehaving-agent set, only one output of position is enough to reveal zero-dynamics attack. Proof of Corollary 1: The strategy (r) in Corollary 1 means the Xor graph has only one component that its Laplacian matrix has distinct eigenvalues, then the set O defined by (34) would be O = {1}. Then using the same proof of Theorem 1, this proof can be finished, here it is omitted. IV. STRATEGIC TOPOLOGY SWITCHING A. Finite- Consensus Algorithm To derive a decentralized strategic topology-switching algorithm, we re-use the finite-time consensus network. Lemma 6: 8 (Simplified Version Without External Disturbances) Consider the multi-agent system n ṙ i = α b ij (r j r i ) m n + β n b ij (r j r i ) p q, i = 1,, n j=1 j=1 (49)

7 7 Figure 3. Xor Graph Has Distinct Eigenvalues where b ij is the element of the coupling matrix that describes topology of an undirected connected communication network and its corresponding Laplacian matrix is denoted as L A, α >, β >, the odd numbers m >, n > p > and q > that satisfy m > n and p < q. Its global finite-time consensus can be achieved, i.e., r i 1 n r i () =, t T, i = 1,, n. (5) n i=1 Further, the setting time T is bounded by T < 1 λ (L A ) (n m n n α ) n m n + 1 β q. (51) q p Remark 5: Adjust parameters α > and β > in the finite-time consensus network (49) such that ) 1 (n m n n n λ (L A ) α m n + 1 β q < τ min, (5) q p where τ min satisfy the left-hand of (4). Therefore, the setting time T in (51) satisfies T < τ min. B. Strategic Topology-Switching Algorithm We make the following assumption on the topology set and the output set for the strategic topology-switching algorithm. Assumption 3: The topology set S includes at least two topologies: a) at least one topology that its Laplacian matrix has distinct eigenvalues; b) at least two topologies: b1) the Laplacian matrix of each nontrivial component of their Xor graph has distinct eigenvalues, and b) each component of their Xor graph is equipped with at least one output of position. Based on Lemma 4, Lemma 5 and Theorem 1, through employing the finite-time consensus network (49), the derived decentralized strategic topology-switching algorithm that can reveal zero-dynamics attack is described by Algorithm 1. Algorithm 1: Strategic Topology-Switching Algorithm Input: Topology set S satisfying Assumption 3, individual functions F i = ϖ x i + ṽ i with ϖ satisfying (), initial time t k =, initial topology G σ(tk ), initial topology-switching time t k+1 = t k + τ with τ generated by Lemma 5, loop-stopping criteria δ. 1 while F (t k 1 ) > δ do Input individuals F i (t k ) and F i (t k ) to agent i in the finite-time consensus network (49) at time t k ; 3 Output F (t k ) (1) and F (t k ) and from the finite-time consensus network (49) to the agents in (4) at time t k + τ min ; 4 Run until the time t k+1 = t k + τ; 5 if F (t k ) = then 6 Switch the topology of network (4b) to σ(t k+1 ) that satisfies: σ(t k+1 ) σ(t k ), L σ(tk+1 ) has distinct eigenvalues, 7 else Laplace matrix of each component of G xor = G σ(tk+1 ) xor G σ(tk ) has distinct eigenvalues; 8 Switch the topology of network (4b) to σ(t k+1 ) that satisfies: σ(t k+1 ) σ(t k ), 9 end 1 end Laplace matrix of each component of G xor = G σ(tk+1 ) xor G σ(tk ) has distinct eigenvalues; 11 Update the topology-switching time: t k 1 t k ; 1 Update the topology-switching time: t k t k+1. Theorem : Consider the multi-agent system under attack (5). If the topology-switching signal is generated by Algorithm 1, then the following properties hold. (i) (ii) (iii) In the absence of attacks, if the loop-stopping criteria δ = (in Line 1 of Algorithm 1), the agents can achieve the second-order consensus by Definition 1. In the absence of attacks, if the loop-stopping criteria δ > (in Line 1 of Algorithm 1), by finitely topology switching the agents can achieve the second-order consensus under admissible consensus error δ, i.e., F (t k) δ with < k < and F given by (1). Without constraint on the misbehaving-agent set, Algorithm 1 is able to reveal zero-dynamics attack. Proof of Theorem : Considering Lemma 5 and Lemma 4, the proofs of property (i) and property (ii) directly follows the proof of Theorem in Part-I paper 1, here it is omitted. Proof of property (iii): Line 4, Line 11 and Line 1 of Algorithm 1 mean all the topology-switching times, i.e., t k, k N, are time-dependent. So, the requirement (r1) in Theorem 1 is satisfied. Line 6 and Line 8 of Algorithm 1 implies the Laplacian matrix of each component of Xor graph, which is generated by every two-consecutive topologies, has distinct eigenvalues. Note the required topology set S in Input

8 8 of Algorithm 1 follows Assumption 3. Thus the requirement (r) in Theorem 1 is satisfied. Therefore, by Theorem 1 we can conclude the property (iii) under Algorithm 1, which completes the proof. Remark 6: In the extreme situation that only one output of position is available, by Corollary 1 the requirements on switching topology in Line 6 and Line 8 of Algorithm 1 would be changed as L σ(tk ) has distinct eigenvalues and L σ(tk+1 ) L σ(tk ) also has distinct eigenvalues. Because Lemma implies that the Laplacian matrix of a path graph has distinct eigenvalues, it is not difficult to choose such two topologies. Figure 3 provides one example on how to choose such two consecutive topologies: one path graph and one ring graph. C. Attack Detection Before proceeding on the attack-detection algorithm, consider the following multi-agent system under switching topology: ė x = e y, ė y = γl σ e x + η Λe x, (53a) (53b) where e x R n, e y R n, L σ is a Laplacian matrix of a connected undirected graph and Λ = diag{ 1 }{{} O(1),,, 1 }{{} O(),,, 1 }{{} O(d),,, } R n n, (54) with O given by (34). Now, we present the following lemmas that can be used to derive an attack-detection algorithm under strategic topology switching. Lemma 7: For the system ë x = ( η Λ γl ) e x + e y (), t (55) where e x R n, L R n n is the Laplacian matrix of a connected undirected graph and Λ is given by (54). If A = η Λ γl <, (56) the system solutions, e xi, i = 1,, n, are solved as e xi = n q li q l l=1 ( ) e x () cos( λl t) + ey() sin( λl t), t λl (57) where λ l, l = 1,, n, are the eigenvalues of the matrix A defined in (56) and q l = q l1,, q ln R n is the eigenvector associated with the eigenvalue λ l of A. Proof of Lemma 7: The proof is as the same as the proof of Lemma 3 in Part-I paper 1, here it is omitted. Let σ = r S for t t k, t k+1 ), k N, the dynamics (53) can be rewritten as ė x = γl r t k t e x (τ)dτ + e y (t k ), t t k, t k+1 ). Therefore, Lemma 7 implies that for the multi-agent agent system (53) under each fixed topology, there exist a period P such that { ( exi = e xi ( t + P), i = 1,, n, (58) e yi = e yi t + P), t tk, t k+1 ), k N. Remark 7: Lemma 7 shows that either the coupling strength γ > or the control gain η can control the period P that satisfies (58). Lemma 8: Consider the second-order multi-agent system under switching topology (53) with A r = η Λ γlr <, r S. (59) where Λ is given by (54). For the given common period P of P satisfying () and P satisfying (58), scalars 1 > β >, α > and L N. If the dwell time τ satisfies ϕ max < τ min τ = ˆτ max + m P, m N, (6) with { } ϕ max = (β 1 L L 1) max α ξ, L, α ϱ (61) α > max {ξ, ϱ}, (6) < ˆτ max < ln β α, (63) < ˆτ max + m P ϕ max (64) ξ = max i (L r ), 1 + γλ i (L r )}, r S,i=1,,n (65) ϱ = max i (A r ), 1 + λ i (A r )}, r S,i=1,,n (66) where λ i (L r ) and λ i (A r ) are the i th eigenvalues of the Laplacian matrix L r and the matrix A r, respectively. Then the system is globally uniformly asymptotically stable. Proof of Lemma 8: Noting (6), the left-hand of (6) implies the (β 1 L 1) L α ξ < τ min. Because P is a common period of P and P. Hence, we can conclude if Lemma 8 holds, Lemma 5 also holds. Then the rest proof is as the same as that of Lemma 5 in Part-I paper 1, here it is omitted. Inspired by the attack-detection algorithm proposed in 1, which is based on Luenberger observer. Using Luenberger observer under switching topology, we present the detection algorithm that has the minimum number of outputs for the zero-dynamics attack. Theorem 3: For the multi-agent system (35), consider the attack-detection filter ẋ = v, ( v = γl σ + η Λ ) x η Λy, Λr = Λx Λy, (67a) (67b) (67c) where x() = x(), v() = v() and Λ is given by (54). If (59) holds, the dwell time τ in Algorithm 1 is generated by Lemma 8, and the topology-switching signals in both the filter (67) and the multi-agent system (35) are the same and generated by Algorithm 1, then i) Λr d, t, if and only if g n, t, ii) in the absence of attacks, the filter errors e x = x x and e v = v v are globally uniformly asymptotically stable,

9 9 iii) in the absence of attacks, if the loop-stopping criteria δ = (in Line 1 of Algorithm 1), the agents can achieve the second-order consensus by Definition 1, iv) in the absence of attacks, if the loop-stopping criteria δ > (in Line 1 of Algorithm 1), by finitely topology switching the agents can achieve the second-order consensus under admissible consensus error δ, i.e., F (t k) δ with < k < and F given by (1). Proof of Theorem 3: (Proof of i)) Define the errors e x = x x, e v = v ṽ and e = e x, e y. From (35) and (67) we have where Ă σ = B = { ė = Ă σ e + Bg r j = e xj g j, j O n n I γl σ + η Λ n n (68), (69) n n BK + η Λ K V\K V\K K V\K V\K, (7) with Λ given by (54) and B K is given by (39b). Replace A σ and B in the proof of Theorem 1 by Ă σ (69) and B (7), respectively. Then using the same analysis method to obtain (41) we have e λi I n n x (κ) n n γl η Λ λi n n BK + η Λ e y (κ) e j n n e n j n = n, j O. (71) The third equation of (71) is equivalent to η Λe x (κ) = η Λg (κ) with g (κ) given by (39c) with (39d), from which (71) can be rewritten equivalently as λi I n n n n γl λi n n BK e j n n e j e x (κ) e y (κ) n = n n, j O. Then using the same proof of Theorem 1, we can conclude that there is no zero-dynamics attack. (Proof of ii)) In the absence of attacks, the dynamics (68) is equivalent to the dynamics (53). Hence, by Lemma 8 we can conclude ii) in Theorem 3. (Proof of iii)) Since, Lemma 8 implies Lemma 5, by Lemma 5 we can conclude iii) in Theorem 3 The Proof of iv) is as the same as the proof of (ii) in Theorem, here it is omitted. V. SIMULATION The simulations on a second-order multi-agent system with n = 4 agents will be presented to demonstrate the effectiveness of the proposed strategic topology-switching algorithm. Table II CANDIDATE TOPOLOGIES Index σ a σ 1 a σ 13 a σ 14 a σ 3 a σ 4 a σ In the simulation setting, the initial position and velocity conditions are chosen as x() = v() = 1,, 3, 4. To convincingly verify the effectiveness of Algorithm 1 in revealing zero-dynamics attack, we consider the extremely bad situation: all the agent are misbehaving agents, i.e., K = {ϑ 1, ϑ, ϑ 3, ϑ 4 }, the connectivity of each candidate topology is significantly low: one for path graph and two for ring graph, only one output of position is available, let M = {ϑ 1 }. Set the coupling strength in the multi-agent system (4) and the control gain in the filter (67) as γ = and η = 8. The candidate topologies are given in Table II. Because both the topology 3 and topology 1 are path graphs, the eigenvalues of the Laplacian matrices of topologies 1 and are enough to calculate the common period. The required eigenvalues of Laplacian matrices are solved as λ 1 (L 1 ), λ (L 1 ), λ 3 (L 1 ), λ 4 (L 1 ) =,.6,, 3.4, (7) λ 1 (L ), λ (L ), λ 3 (L ), λ 4 (L ) =,,, 4, (73) λ 1 (A 1 ), λ (A 1 ), λ 3 (A 1 ), λ 4 (A 1 ) =.5, 1, 1.6, 7.9, (74) λ 1 (A ), λ (A ), λ 3 (A ), λ 4 (A ) =.5, 1.3, 3.9, 1.1. (75) Using the solved eigenvalues (7) (75) and the state solutions (19) and (57), following (6) and (64), we can choose the dwell time τ = P +.1 = For simplicity, let the attack-beginning time is the initial time, i.e., κ = t =, which means the attacker can modify the initial conditions arbitrarily. A. Zero-Dynamics Attack Design First, we considered the topology set S = {, 3 } where the representations of and 3 are given in Table II. Obviously, the set S = {, 3 } does not satisfy the strategy (r) in Theorem 1. Thus, the attacker can easily design a zerodynamics attack such that Algorithm 1, with only one output of position, cannot reveal it. Let the attacker s goal is to attack the multi-agent system under Algorithm 1 to be unstable, while not being detected. To choose this goal, the attacker should first choose a positive λ. Following the zero-dynamics attack design method (31) in Definition 4, one of its zero-dynamics attack strategies is easily designed as introduce attack signal: g = e t, e t, 4e t, 4e t, modify the initial conditions: x () = v () =, 3, 5, 6. The trajectories of the attack-detection signal r 1 deigned by Theorem 3 and the state x 1 are shown in Figure 4,

10 1 which shows the attacker s goal of attacking the multi-agent system to be unstable while not being detected by the defender is achieved. Thus, using only one output, the designed zerodynamics attack is not revealed under the topology set S = {, 3 } attack-detection signal r 1 state x Figure 4. State x 1 : multi-agent system under attack is unstable; Attack- Detection Signal r 1 : the attack is not detected. B. Reveal Zero-Dynamics Attack In the extremely bad situation, the existing results 8 1, 14 for the multi-agent systems under fixed topology failure to work. This is manly because the misbehaving-agents set K = 1, and the connectivity of topology 1 (and topology ) is 1 (and ), and the output M = 1. All these violate the conditions on the connectivity of communication network, the size of misbehaving-agent set and the size of output set, which are summarized in Table I. Now we turn to the topology set S = {1, } to real the attack. As illustrated by Figure 3, the Xor graph generated by topologies 1 and has only one component that is a path graph. By Lemma, the Laplacian matrix of the Xor graph has distinct eigenvalues. Thus, by Corollary 1 we can conclude that using only one output, the strategic topology-switching algorithm Algorithm 1 is able to reveal the designed zerodynamics attack. The trajectory of the attack-detection signal is shown in Figure 5, which illustrates that with all the agents being misbehaving, using only one output of position, Algorithm 1 succeeds in revealing zero-dynamics attack. C. Filter in The Absence of Attacks Now using the same dwell time, we show the effectiveness of strategic topology switching for the filter (67) in estimating the states of the multi-agent system (1), i.e., the system (4) is in the absence of attacks. Input the attacker modified initial conditions to the filter (67), i.e., x () = v () =, 3, 5, 6. The trajectories of filter errors are shown in Figure 6, which illustrates that using the strategic-topology switching algorithm, the filter errors are indeed globally uniformly asymptotically stable. This simulation shows Algorithm 1 works successfully Attack-Detection Signal r Non-Zero Figure 5. Attack-Detection Signal r 1 : using only output of position, the designed zero-dynamics attack is revealed. for the filter to track the real multi-agent system in the absence of attacks. VI. CONCLUSION This two-part paper studies strategic topology switching for the second-order multi-agent system under attack. Based on the strategy on switching times studied in Part-I paper and the strategy on switching topologies studied in Part-II paper, we propose a strategic topology-switching algorithm. Two obvious merits of strategic topology-switching algorithm can be summarized as In achieving the second-order consensus in the absence of attacks, the velocity measurements are not needed for the control protocol and the algorithm has no constraint on the magnitude of coupling strength. In revealing zero-dynamics attack, the algorithm has no constraint on the set of misbehaving agents. Through strategy setting on switching topologies, only one monitor of position is enough to reveal zero-dynamics attack. The algorithm allows the system operator or the defender having no knowledge of the attack-beginning time, and allows the attacks happening infinitely over infinite time. This two-part paper studies the detection of a single attack strategy, that is zero-dynamics attack. This paper provides an insight that strategic topology switching can be a promising method for the detection of mixed attack strategies, such as zero-dynamics attack working with DoS attack, zero-dynamics attack working with topology attack, topology attack working with delay attack, etc. How to reveal such mixed attacks by strategic topology switching would be our future research work. REFERENCES 1 Y. Mao, E. Akyol, and Z. Zhang. Strategic topology switching for security Part I: consensus & switching times. submitted to IEEE Transactions on Control of Network Systems. W. Yu, G. Chen, and M. Cao. Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems. Automatica, 46(6): , 1.

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