2050

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 11, NOVEMBER 2014

Studies on Resilient Control Through Multiagent Consensus Networks Subject to Disturbances Deyuan Meng, Member, IEEE, and Kevin L. Moore, Senior Member, IEEE

Abstract—Resiliency is one of the most critical objectives found in complex industrial applications today and designing control systems to provide resiliency is an open problem. This paper proposes resilient control design guidelines for industrial systems that can be modeled as networked multiagent consensus systems subject to disturbances or noise. We give a general analysis of multiagent consensus networks in the presence of different disturbances from the input-to-output stability point of view. Using a nonsingular linear transformation, some necessary and sufficient results are established for disturbed multiagent consensus networks by taking advantage of the input-to-state stability theory, based on which the disturbance rejection performance is analyzed in three cases separated by the spaces of disturbances and state disagreements between agents. It is shown that the linear matrix inequality technique can be adopted to determine the optimal disturbance rejection indexes for all the three cases. In addition, two illustrative numerical examples are given to demonstrate the derived consensus results for different types of directed graphs and subject to different classes of disturbances. Index Terms—Directed graphs, disturbances, linear matrix inequality (LMI), multiagent consensus networks, resilient control.

I. Introduction ESILIENCEhas recently come to be considered one of the most critical abilities of a practical control system. A resilient control system is one that can maintain or recover an accepted level of operational normalcy in the face of unexpected disturbances or threats [1]. Recently, it has also been noted that many complex plants can be modeled through the use of networked multiagent systems, especially in those cases where the systems can be viewed as a consensus network. In this case, it is possible to develop strong results for both analysis and design by analyzing the network’s consensus performance when subjected to disturbances or noise [2]–[7]. In this paper, we take the view that networked multiagent systems

R

Manuscript received March 13, 2013; revised September 30, 2013; accepted January 9, 2014. Date of publication February 10, 2014; date of current version October 13, 2014. This work was supported in part by the National Basic Research Program of China 973 Program under Grant 2012CB821200, Grant 2012CB821201, in part by the NSFC under Grant 61104011, Grant 61134005, Grant 61221061, and Grant 61327807, in part by the MOE under Grant 20111102120031, and in part by the Beijing Natural Science Foundation under Grant 4122046. This paper was recommended by Associate Editor D. Scheidt. D. Meng is with the Seventh Research Division and the Department of Systems and Control, Beihang University (BUAA), Beijing 100191, China (e-mail: [email protected]). K. L. Moore is with the Department of Electrical Engineering and Computer Science, Colorado School of Mines, Golden, CO 80401 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2301555

can provide reasonable description for many practical complex plants operating through interaction with each other and that a good consensus performance of such networks can reflect the practical system resilience in the presence of disturbances (see [2] and [3] for power networks subject to cyber attacks). For our purposes, a multiagent system refers to a group of integrating agents that cooperate mutually in a certain network topology to perform coordination tasks for the group. Typically such a topology is delineated by a weighted interaction matrix between agents or nodes in the system, where these weights might be dynamic, but are typically static gains. As claimed and demonstrated in [8]–[14], consensus is one of the fundamental problems in the coordination control of multiagent systems. A consensus protocol aims to eliminate disagreement between the states of the agents. It is worth noting that consensus control algorithms are generally implemented in a distributed manner, using nearest neighbor or local information between the networked agents (given the harder-to-achieve accessibility of global information). Usually, such an information application rule used to design consensus algorithms explicitly takes into account the local nature of information exchange mechanisms in multiagent networks, which is generated because of the limitation of communication or sensing ability of each agent. It can also be the case that physical dynamics of a large-scale interconnected system define limit interactions between the agents. In either case (communication/sensing abilities or the physics of the particular system) the interaction topology leads to a distributed multiagent consensus network whose performance in the face of disturbances will differ significantly from that of centralized or lumped systems. Our approach is to model the resilient control problem as the problem of disturbance or noise attenuation in consensus networks. To motivate our approach we first note that a number of physically relevant systems exist that admit system dynamics that can be modeled as a consensus network. One such physical system is the power system [2], [3], [15]. Another example is the model of thermal processes in a building [16]. A third model is a set of autonomous vehicles maneuvering under a consensus control algorithm [12]. In the first two examples, the interactions in the system dynamics lead to a consensus network. In the third, it is the control algorithm itself that produces the consensus network. In all three cases, however, if we call X the global state variable, the resulting global system dynamics take the form ˙ = −L(t)X + U + V X

c 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267  See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

MENG AND MOORE: STUDIES ON RESILIENT CONTROL THROUGH MULTIAGENT CONSENSUS NETWORKS

where U is the vector of inputs to the system, V is the vector of noise and disturbances driving the system, and L(t) is the possibly time-varying system Laplacian, defined by the interconnection topology between the individual components in the system. Second, we note that in much of the literature related to resilient control a key consideration is the cyber-environment associated with the system and the affect of cyber attacks in such systems. Indeed, most complex industrial systems are implemented via a distributed control system (DCS). In modern DCS, controllers are almost exclusively implemented using computers. Thus, computer-to-computer communications are digital communications, either through a dedicated link (wire) or over a network. In addition, it is increasingly common to find sensors made up of a transducer combined with an analog-to-digital converter, some type of digital processing on an embedded processor, and a communication mechanism. Similar configurations are found in modern actuators and their drives. Fig. 1 depicts these features and highlights those parts of the system that are vulnerable to cyber attack. As depicted, cyber attacks can affect signals in communication streams as well as functionality in code executed on a computer. Further, these vulnerabilities can exist in both wired and wireless communication channels, though the mechanisms by which they occur may be different in the two cases. To relate the impact of digital communications to the resilience of the DCS, we need to identify ways signals can be corrupted. While component failure for digital systems can be handled in the same ways as for physical components (e.g., as changes in dynamics), it can be very complicated to articulate a detailed model of the possible failure mechanisms that can occur in the signal chains in Fig. 1. One approach is to consider the well-being of the cyber-related devices (sensors/actuators) in the system and incorporate this well-being by defining a modified form of the plant equations for those parts of the process that are impacted by cyber-related devices [17]. Other approaches in the literature follow more conventional fault tolerant control strategies that focus on comparing signals and estimates derived from those signals to what has been determined a priori to be ideal or nominal. Yet other approaches focus on encryption or encoding of signals, which can protect against some malicious attacks. An example of this kind of approach is the use of so-called wave variables and the power junction idea found in [18], which combines these ideas with assumptions of passivity to ensure stability of DCS. For our purposes, we model cyber-related faults in a DCS as causing either 1) a parametric or structural change in a controller or plant component or 2) an interruption or corruption of a signal (e.g., noise). Fig. 2 demonstrates the failure mechanisms we assume. The dependence on time and on a time-varying parameter vector for the controller and plant subsystems (c(t) and p(t), respectively) captures the possibility of component failure or degradation in those systems. The addition of noise n(t) or disturbances d(t) can model the possibility of signal corruption or interruption through cyber attack. Thus, Fig. 2 can be used to represent generically a complex, interconnected system with a distributed controller, where the plant can be modeled as a consensus network. For this setting, the design

Fig. 1.

2051

Points of cyber attack in a distributed control system.

Fig. 2. Modeling uncertainty via cyber attack and component failure in a distributed control system as noise, disturbance, and parameter variation.

problem is to determine the controller that provides system resilience to noise and disturbances (corresponding to cyber attack) or system failure (resulting in changes in system dynamics). In this paper, we consider resilient control from this framework, but restrict our attention to the attenuation of noise and disturbances in consensus networks, assuming plant and controller constants do not change. Recently, the tools of robust H∞ control have been applied to analyze the consensus performance of distributed multiagent networks in the face of disturbances [19]–[25]. An advantage of the H∞ consensus analysis is that the well known Lyapunov stability theory (like bounded real lemmas [26]) can be generalized to networked multiagent systems by appropriately choosing controlled output functions for all the agents, together with the disturbance rejection index determined simultaneously. However, the H∞ analysis requires the disturbances belonging to the L2 space or, in other words, is only appropriate for the finite-energy disturbances. Conversely, there are malicious attacks that constantly threaten the resilient control systems (see [5]–[7] for the Gaussian white noise). In addition, there have been practically considered other classes of disturbances, e.g., bounded disturbances belonging to the L1 space addressed in [27]–[31] by applying the tools based on, respectively, adaptive neural networks [27], [28], input-tostate stability (ISS) [29], and Lyapunov stability [30], [31]. For more results presented to address the consensus performance of multiagent networks with disturbances, see [32] for exponentially fast convergence analysis, [33] for disturbance observer-based controller design, and [34] for H2 performance synthesis. Also, there have been reported many studies that aim to suppress measurement noise existing in the transmission channels of multiagent consensus networks [35]–[38]. Motivated by the aforementioned analyses and observations,

2052

this paper considers resilient control from the viewpoint of networked multiagent consensus systems subject to disturbances. Two different classes of disturbances are considered, which is to examine the system resilience in the face of two essentially different threats or attacks in practice. A general framework is proposed to study the disturbance attenuation performance of networked multiagent consensus systems that are associated with directed graphs for these two classes of disturbances. To this end, we first present developments to demonstrate that the disturbance rejection issue for multiagent consensus networks can be equivalently treated as an inputto-output stability (IOS) problem. However, it turns out that such an IOS problem is related to a system that is not completely observable. Thus, we use a nonsingular linear transformation to decompose the system and then employ the decomposed system to develop general necessary and sufficient consensus results by taking advantage of the ISS theory. Moreover, we handle the disturbance rejection problem in three cases that are separated by the spaces of two different classes of disturbances and state disagreements of agents. For each case, we propose a performance index to evaluate the system resilience and to devise guidelines to compute its optimal or suboptimal values through solving a class of linear matrix inequality (LMI) conditions. Our developed consensus results for disturbed multiagent networks are very general and widely applicable, especially in comparison with the H∞ consensus results in [19]–[23]. This applicability is demonstrated by considering two illustrative numerical examples for multiagent consensus networks under different classes of directed graphs and in the face of different classes of disturbances. The remainder of the paper is organized as follows. We give the problem statement by considering consensus of multiagent networks in the face of disturbances in Section II. The problem analysis is implemented from the IOS point of view in Section III, together with making a problem transformation and giving preliminary consensus lemmas. In Section IV, the performance is analyzed for the multiagent consensus networks subject to two different classes of disturbances, for which two illustrative numerical examples are also given. The conclusions are made in Section V. In the Appendix, the detailed proofs are provided for all the lemmas proposed in the Sections III and IV. Before proceeding to Section II, we end this introduction by presenting some notations and preliminaries related to graph theory. Notations: In = {1, 2, · · · , n}, 1n = [1, 1, · · · , 1]T ∈ Rn , I and 0 denote the identity matrix and the null matrix with required dimensions, respectively, and a bigstar () used in symmetric block matrices denotes a term that is induced by the symmetry. Given vectors or matrices A and B, A denotes the Euclidean (or spectral) norm of a vector (or matrix) A, A ≥ 0 denotes a nonnegative vector or matrix whose elements are nonnegative, A  0 (respectively, A  0) denotes a positive (respectively, semipositive) definite matrix, and A  B (respectively, A  B) if and only if A − B∞  0 (respectively, 1/2 A−B  0). For x(t) ∈ Rn , let x(t)2 = 0 x(t)2 dt and x(t)∞ = supt≥0 x(t). We say that x(t) ∈ L2 (respectively, x(t) ∈ L∞ ) if and only if x(t)2 < ∞ (respectively,

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 11, NOVEMBER 2014

x(t)∞ < ∞). Preliminaries in the graph theory: Let G denote an nth order directed graph with a vertex set V(G) and an edge set E(G), where V(G) = {vi : i ∈ In } and E(G) ⊆ {(vi , vj ) : vi , vj ∈ V(G)}. For an edge (vi , vj ) in E(G), it means that there exists information flowing from vj to vi , and vj is a neighbor of vi . The index set of the neighbors of vi is denoted by Ni = {j : (vi , vj ) ∈ E(G)}. A path in the directed graph G is a finite sequence vi1 , vi2 , · · · , vij of vertices such that (vil , vil+1 ) ∈ E(G) for l = 1, 2, · · · , j−1. If there exists a special vertex that can be connected to all other vertices through paths, then G is said to have a spanning tree, and this special vertex is called the root vertex. A nonnegative weighted adjacency matrix A = [aij ] ≥ 0 associated with the directed graph G is defined to model the information exchange between any two agents, where aij > 0 ⇔ (vi , vj ) ∈ E(G) and aij = 0 otherwise. Here, the weighted directed graph is denoted by G(A), and aii = 0 is assumed for i ∈ In . The Laplacian matrix of the directed graph G(A) is defined as L=  − A, where n  = diag{11 , 22 , · · · , nn } and ii = j=1,j =i aij for all i ∈ In .

II. Problem Statement Let us consider networked systems with a group of n agents (or dynamic systems representing interconnected components) that are labeled 1 through n. Without loss of generality, every agent is regarded as a vertex of the directed graph G(A), where the agents share a common state space R. The ith agent is considered to have the following dynamics: x˙ i (t) =



ail [xl (t) − xi (t)] + ωi (t)

for all i ∈ In

(1)

l∈Ni

where xi (t) is the state, and ωi (t) is the disturbance that reflects unexpected threats or attacks faced by the agent vi . As carried out in [35]–[38], the disturbance ωi (t) can also be extended to take into account the measurement noise in the communication channels of multiagent consensus networks. Two classes of disturbances are addressed: 1) ωi (t) ∈ L2 and 2) ωi (t) ∈ L∞ . In practice, ωi (t) ∈ L2 can represent the sudden high-energy disturbances. Since this class of disturbances have only an instantaneous duration, they have finite energy and will vanish with the increase of time even though their energy may be quite huge. A typical example for ωi (t) ∈ L2 is the impulse signal. In contrast to this, there are also a class of disturbances that happen continually, such as the square wave signal and the sine or cosine signal. Since this class of disturbances occur all the time, they may result in infinity energy with the increase of time, but their magnitude at each time is finite, mainly because they only possess finite energy over an instantaneous duration. Consequently, ωi (t) ∈ L∞ can give appropriate descriptions of continually happening disturbances. The objective of this paper is to consider the resilient control system through examining the consensus performance of (1) in the face of disturbances ωi (t) ∈ L2 or ωi (t) ∈ L∞ . For the multiagent networks given by (1), we say that all agents accomplish consensus if their states are guaranteed to satisfy

MENG AND MOORE: STUDIES ON RESILIENT CONTROL THROUGH MULTIAGENT CONSENSUS NETWORKS

[9] lim [xi (t) − xl (t)] = 0

t→∞

for all i, l ∈ In .

(2)

For consensus of networked multiagent systems, there are many promising results [5]–[25], [27]–[38]. It follows from these existing consensus results that certain conditions on the network topology of (1) are required to achieve (2), even when (1) is in the absence of disturbances. For the sake of our following analysis, such a consensus result is summarized in the next proposition (see [9] for details). Proposition 1: Consider the networked multiagent system (1) in the absence of disturbances (i.e., ωi (t) ≡ 0 for t ≥ 0 and i ∈ In ). Then the consensus objective (2) holds if and only if G(A) has a spanning tree. Proposition 1 shows the basic consensus result of networked multiagent systems. However, it is easy for us to demonstrate that the existence of disturbances may invalidate the consensus result of Proposition 1. This necessarily requires (1) to possess certain system resilience when it is in the face of disturbances. Motivated by this observation, we consider how to evaluate the resilience for the networked multiagent consensus system (1) and (2) in the face of different classes of disturbances through enabling it to achieve good disturbance rejection performance. It is worth pointing out that the results to be established in the following sections can be extended to the vector state case via introducing the Kronecker product. Such extensions, however, are omitted here for simplicity. In the sequel, the time variable t will be omitted when no confusions can arise.

III. Consensus Problem Analysis A. Analysis of Multiagent Consensus Objective It is clear that the consensus objective (2) is a fundamentally different problem from the state stability (i.e., limt→∞ xi (t) = 0 for all i ∈ In ) of the networked multiagent system (1). It will be found that (2) reflects essentially an output stability problem of (1). Given any j ∈ In , let us denote  xi − x j , i < j yi = (3) for all i ∈ In−1 . xi+1 − xj ,i ≥ j With (3), it can be easily validated that the consensus objective (2) is essentially the same as lim yi (t) = 0

t→∞

for all i ∈ In−1 .

(4)

T T In addition,  let x = [x1 , x2,T· · · , xn ] , ω = [ω1 , ω2 , · · · , ωn ] , and y = y1 , y2 , · · · , yn−1 . Then, we can easily rewrite (1) and (3) in a compact form of  x˙ = −Lx + ω (5) y = Qx

where L is the Laplacian matrix of G(A), and Q is an (n − 1) × n matrix given by ⎧ j=1 ⎪ ⎨ [−1n−1 , I], Q = [E1 , · · · , Ej−1 , −1n−1 , Ej , · · · , En−1 ],1 < j < n ⎪ ⎩ j = n. [I, −1n−1 ]

2053

Here, (and in what follows), Ei is the ith column of the (n − 1)th order identity matrix for i ∈ In−1 . It can be easily seen that the networked multiagent consensus system is represented by (5), where (4) [therefore, (2)] shows essentially an output stability problem for this system. Moreover, we can study the consensus performance of (5) with respect to disturbances by considering the transfer function matrix Tyω (s) = Q(sI + L)−1 : ω → y. In addition, it can be verified that Q is a fullrow rank matrix (i.e., rankQ = n−1). Notice also that Q1n = n−1 i=1 Ei −1n−1 = 0 holds. Thus, the null space of Q is spanned by 1n . This, together with L1n = 0 (see [9] for the properties

of Laplacian T

T

matrices), leads to QT , LT QT , · · · , Ln−1 QT 1n = 0. Then, we can obtain

T T n − 1 = rankQ ≤ rank QT , LT QT , · · · , Ln−1 QT < n



T T which yields rank QT , LT QT , · · · , Ln−1 QT = n−1. Using the standard linear system theory, we know that the multiagent consensus system (5) is not completely observable, which has n−1 observable state variables. Consequently, Tyω (s) discloses essentially the relationship between the disturbances and n − 1 state variables that are both controllable and observable in (5). We will use the regular system decomposition to clearly reveal this fact, in order to better study the performance of networked multiagent consensus systems subject to disturbances. Toward this end, let ⎧ ⎨ [0, I]T , j = 1 n×(n−1) [E1 , · · ·, Ej−1 , 0, Ej , · · ·, En−1 ]T , 1 < j < n P ∈R = ⎩ [I, 0]T , j = n.  j 0, the consensus objective (2) holds with Tyω 2−2 < γ if and only if there exists a symmetric matrix W = W T ∈ R(n−1)×(n−1) satisfying the following Riccati equation: WQLP + P T LT QT W − I − γ −2 WQQT W = 0.

(13)

Proof: Necessity: By Lemma 2, we know that if (2) holds with ω ∈ L2 , G(A) has a spanning tree. Then, it follows from Lemma 1 that −QLP is Hurwitz stable. By noticing this result and then applying the bounded   real lemma [26] to the system (11), we can obtain that Tyω 2−2 < γ is guaranteed only if the Riccati equation (13) holds. Sufficiency: If the Riccati equation (13) is feasible, we know W  0 [26], and consequently I +γ −2 WQQT W  0. By again using (13), we can validate that −QLP is a Hurwitz stable matrix. This, together with Lemmas 1 and 2, can guarantee the consensus objective (2). Furthermore, the application of   the bounded real lemma [26] leads to Tyω 2−2 < γ. Remark 3: With the use of the bounded real lemma [26], we deduce a necessary and sufficient condition (13) to achieve   the consensus of disturbed multiagent networks with Tyω  < γ. Using the Schur’s complement formula, we can 2−2 validate that (13) holds if and only if there exists a positive definite matrix 0 ≺ X ∈ R(n−1)×(n−1) such that ⎡ ⎤ XQLP + P T LT QT X XQ I ⎣ () γI 0 ⎦  0. (14) () () γI Note that (14) is an LMI with respect to X and γ. Consequently, we can employ the LMI solver “feasp” from the MATLAB LMI control toolbox to demonstrate the feasibility of (14). We can also optimize the performance index γ by computing (14) with the LMI solver “mincx”. By the above discussion, we know that the system resilience for multiagent consensus networks in the face of disturbances belonging to L2 can be quantitatively evaluated by considering   Tyω  whose optimal upper bound can be computed using 2−2 the condition (13) or (14). For the two conditions, we propose the following result to provide them a necessary guarantee. Corollary 1: A necessary condition for the Riccati equation (13) or the LMI (14) is that G(A) has a spanning tree. Proof: From the proof of Theorem 1, we can deduce that the satisfaction of (13) needs the Hurwitz stability of −QLP. This, together with the result of Lemma 1 and the equivalence between (13) and (14), can give the result of this corollary. Remark 4: In the existing literature of H∞ consensus results [19]–[23], the Lyapunov analysis approach has been developed such that the H∞ method can be employed to derive the robust consensus for multiagent networks subject to many other practical uncertainties such as communication delays and switching topologies. It can be validated that we can also carry out the H∞ analysis to achieve such robust consensus results together with guaranteeing a certain disturbance

2056

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 11, NOVEMBER 2014

rejection index by our presented problem transformation approach, especially in comparison with the approach given in [19]–[23]. Moreover, these robust H∞ consensus results for multiagent networks can be viewed as a generalization of Theorem 1 by just taking into account, e.g., communication delays and switching topologies. B. Case 2: ω ∈ L2 and y ∈ L∞ We proceed to investigate the consensus performance for disturbed multiagent networks by evaluating y ∈ L∞ in response to ω ∈ L2 . This analysis can help to better examine the system resilience. Also, we have from Lemma 2 that y ∈ L∞ can be achieved in the presence of ω ∈ L2 . By this observation, we study the effects of disturbances on networked multiagent consensus systems by considering an induced transfer function matrix norm for Tyω (s) as   Tyω  = sup y∞ . ∞−2 ω2 ≤1

    Clearly, we know  y∞ ≤ Tyω ∞−2 ω2 . This yields that we can apply Tyω ∞−2 as an alternative to quantitatively measure the system resilience of multiagent consensus networks in the presence of disturbances satisfying ω ∈ L2 . In this case, we present the following result forthe consensus performance evaluation through considering Tyω ∞−2 . Theorem 2: Consider the networked multiagent system (1), and let ωi ∈ L2 for all i ∈ In . Then, the consensus objective 1/2 (2) holds with Tyω ∞−2 = W2 if and only if there exists a positive definite matrix 0 ≺ W ∈ R(n−1)×(n−1) satisfying the following Lyapunov equation: QLPW + WP T LT QT − QQT = 0.

(15)

To prove Theorem 2, we need two useful lemmas which are introduced as follows. Lemma 3: For a positive definite matrix 0 ≺  ∈ Rm×m and a vector η ∈ Rm , if there exists a positive scalar θ > 0 such that ηT −1 η < θ 2 holds, then η < θ1/2 . Lemma 4: There exists a positive definite matrix 0 ≺ W ∈ R(n−1)×(n−1) satisfying the Lyapunov equation (15) if and only if there exists a positive definite matrix 0 ≺ X ∈ R(n−1)×(n−1) satisfying the following Lyapunov inequality: QLPX + XP T LT QT − QQT  0.

(16)

Moreover, we premultiply   and postmultiply  T the above matrix inequality with yT , ωT and yT , ωT , respectively, and consider the system description of (11) to obtain 0 < 2yT X−1 QLPy − 2yT X−1 Qω + ωT ω = −2yT X−1 y˙ + ωT ω d  T −1  y X y + ωT ω =− dt which leads to  t yT X−1 y < ω(τ)2 dτ ≤ ω22 .

With Lemma 3, it follows from (18) that y < ω2 X1/2 , and thus we can deduce sup y y∞ t≥0 = ≤ X1/2 ω2 ω2   which, together with the definition of Tyω ∞−2 and the fact of (17), yields   1/2 Tyω  ≤ W2 . (19) ∞−2 In addition, if we select a particular disturbance signal as  T T T −1/2 λT QT e−P L Q (T −t) vT ,0 ≤ t ≤ T ωT = 0, t>T where λT is the maximum eigenvalue of the matrix XT , vT is its corresponding eigenvector with modulus equal to one (i.e., vT  = 1), and XT has the form of  T T T T XT = e−QLPt QQT e−P L Q t dt. 0

It can be easily validated that ωT 2 = 1 for T > 0, XT  = λT , and limT →∞ XT = W. Then, considering the system (11), we can obtain from the standard linear system theory that  T   T T T −1/2 yT (T ) = e−QLP(T −t) Q λT QT e−P L Q (T −t) vT dt = λT

(17)

Proof of Theorem 2: Necessity: From Lemmas 1 and 2, it can be obtained that −QLP is Hurwitz stable if Tthe consensus T objective (2) holds. Note that QQT = n−1 i=1 Ei Ei +1n−1 1n−1 = T I + 1n−1 1n−1  0 is a positive definite matrix. Hence, it can be easily deduced that the Lyapunov equation (15) has a unique positive definite solution expressed by  ∞ T T T e−QLPt QQT e−P L Q t dt  0. W= 0

Sufficiency: If (15) has a positive definite solution W  0, we can use the Lyapunov theory to derive that −QLP is

(18)

0

0 −1/2

Moreover, it follows that: W = inf {X : X  0 subject to (16)} .

Hurwitz stable. Note that ωi ∈ L2 for all i ∈ In , i.e., ω ∈ L2 holds. Then, it follows from Lemmas 1 and 2 that the consensus objective (2) can be achieved. By noting the result of Lemma 4 and then applying the Schur’s complement formula to (16), we have  −1  X QLP + P T LT QT X−1 −X−1 Q  0. () I

=

XT vT

1/2 λT vT

which leads to yT (T )2 = λT . As a consequence, we can deduce that limT →∞ yT (T )2 = limT →∞ λT = W. This clearly implies that   1/2 Tyω  ≥ W2 . (20) ∞−2   1/2 By combining (19) with (20), we can see Tyω ∞−2 = W2 . The proof is complete. Remark  From Theorem 2, we know that one can exactly  5: obtain Tyω ∞−2 through solving the Lyapunov equation  (15). This is obviously different from the computation of Tyω 2−2 ,

MENG AND MOORE: STUDIES ON RESILIENT CONTROL THROUGH MULTIAGENT CONSENSUS NETWORKS

2057

TABLE I Performance of MultiAgent Consensus Networks for ω ∈ L2

Fig. 3. Two types of six-agent directed graphs with spanning trees. (a) Tree-type graph Ga . (b) Ring-like graph Gb .

which is achieved by instead determining it with an appropriate upper bound. It can be verified that the Lyapunov equation (15) is only a necessary condition of the Riccati equation (13).This   means thatwe need  a more relaxed condition when Tyω ∞−2   instead of Tyω 2−2 is used to evaluate the system resilience. Moreover, it is easy to see from Theorem 2 and Lemma 4 that   Tyω  can be computed as follows: ∞−2    Tyω  = inf X1/2 : X  0 subject to (16) ∞−2 where the inequality (16) can be rewritten in an LMI form as   QLPX + XP T LT QT Q  0. (21) () I   This implies that Tyω ∞−2 can also be derived by computing the LMI conditions. With the above development, we know that three equivalent conditions,  i.e., (15), (16), and (21), are provided to accomplish Tyω ∞−2 , to quantitatively assess the system resilience for multiagent consensus networks in the face of disturbances belonging to L2 . For the three conditions, we can also establish a network topology condition associated with G(A) to provide them a necessary and sufficient guarantee. Corollary 2: The following four conditions are equivalent: 1) there exists W  0 satisfying the Lyapunov equation (15); 2) there exists X  0 satisfying the Lyapunov inequality (16); 3) there exists X  0 satisfying the LMI (21); 4) G(A) has a spanning tree. Proof: To derive this corollary, we only need to prove the equivalence between 1) and 4). From the proof of Theorem 2, we can see that (15) holds if and only if −QLP is Hurwitz stable. Then this proof can be developed from Lemma 1, which is not detailed here. By comparing  also observe that  Corollaries 1 and 2, we  can the use of Tyω ∞−2 , in contrast to Tyω 2−2 , may provide us a more easily accessible way to investigate the system resilience of disturbed multiagent consensus networks. Next, we present an illustrative numerical example to demonstrate the consensus performance of networked multiagent systems in the presence of disturbances satisfying ω ∈ L2 . Example 1: We consider a class of consensus networks with six agents. As shown in Fig. 3, the two six-agent networks have spanning trees. The following two cases will be discussed for these networks by adopting different edge weights. EW1) All the edges of both graphs in Fig. 3 share the same weight equal to 10.

EW2) All the edges of both graphs in Fig. 3 share the same weight equal to 0.05. By noting the discussions of Remarks 3 and   5,we know that we can determine two norms Tyω 2−2 and Tyω ∞−2 through computing LMIs. Therefore, we apply the LMI solver “mincx” from the LMI control toolbox of the MATLAB to determine their least upper bounds, which are achieved by optimizing γ subject to LMI constraint (14) and X1/2 subject to LMI constraint (21), respectively. Without loss of generality, we will consider y defined in (3) by adopting j = n (also in Example 2 given in Section IV-C).     The least upper bounds of Tyω 2−2 and Tyω ∞−2 computed under the directed graphs Ga and Gb with the edge weights of EW1) and EW2) are depicted in Table I. It can be obviously seen that we have performance   different indices when applying Tyω 2−2 and Tyω ∞−2 to evaluate the system resilience for multiagent consensus networks subject to ω ∈ L2 . To be more specific, we can see the following facts. 1) Different   performance  evaluation results are achieved by Tyω 2−2 and Tyω ∞−2 even though they are performed for the same multiagent consensus network. 2) The topology of multiagent consensus networks   impacts Tyω  or their performance index, no matter whether 2−2   Tyω  is considered and which edge weight of EW1) ∞−2 or EW2) is adopted. 3) The edge weight has effects on the performanceindex  of Tyω  multiagent consensus networks, regardless of 2−2   or Tyω ∞−2 is applied to the resilience evaluation under what type of directed graph Ga or Gb . Clearly, these facts imply that the resilience of disturbed multiagent consensus systems can be improved by designing appropriately their networks including the topologies and adjacency weights associated with them. In particular,  it can  be clearly observed from TableI that the evaluation of Tyω ∞−2 is much smaller than that of Tyω 2−2 when we consider Ga and Gb under the edge weightof EW2).  It implies that the system performanceevaluated by Tyω ∞−2  has advantages over that evaluated by Tyω 2−2 for networked multiagent consensus systems subject to disturbances ω ∈ L2 . To make this point more obvious, we perform simulations with 10  ω= cos(2πt), cos(2.5πt), cos(3πt), − cos(2πt), 20t + 1 T − cos(2.5πt), − cos(3πt) . It can be validated that ω ∈ L2 . To clearly depict the simulation results, we propose the following definitions of a vector-valued function ξ(t) ∈ Rn :  t 1/2 ξ2,[0,t] = ξ(τ)2 dτ , ξ∞,[0,t] = sup ξ(τ) 0

0≤τ≤t

2058

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 11, NOVEMBER 2014

Fig. 4. Under the directed graph Ga with the edge weight of EW2), the consensus performance of multiagent networks with respect to ω ∈ L2 . (a) State disagreements between agents. (b) Process of the Eulidean norm y. (c) Process of y2,[0,t] and ω2,[0,t] . (d) Process of y∞,[0,t] and ω2,[0,t] .

where limt→∞ ξ2,[0,t] = ξ2 and limt→∞ ξ∞,[0,t] = ξ∞ . Figs. 4 and 5 plot the simulation results obtained under the directed graphs Ga and Gb , respectively, where the edge weight of EW2) is used. By Figs. 4 and 5, we have the following facts. 1) From Fig. 4(a) and (b) and Fig. 5(a) and (b), it is obvious that all agents achieve the consensus objective (2) in the presence of ω ∈ L2 , regardless of what type of directed graph Ga or Gb is under consideration. 2) From Fig. 4(c) and (d)  and Fig. 5(c) and (d),  it can clearly be seen that Tyω ∞−2 < 1 < Tyω 2−2 . This  implies that the performance evaluation through Tyω  provides a better description of the system ∞−2 resilience for disturbed multiagent consensus networks   in comparison with that through Tyω 2−2 from the disturbance rejection point of view, no matter what type of directed graph Ga or Gb is considered. Clearly, the simulation test results of Figs. 4 and 5 coincide with the computed results of Table I. This can also demonstrate that the proposed results with respect to the case where ω ∈ L2 and y ∈ L∞ bring a good alternative way to consider consensus performance of networked multiagent systems in the presence of disturbances satisfying ω ∈ L2 , especially when comparing with the existing H∞ consensus results [19]–[23].

C. Case 3: ω ∈ L∞ and y ∈ L∞ Next, we consider the consensus performance of networked multiagent systems in the face of disturbances that fulfill ω ∈ L∞ . In practice, there are many disturbances belonging to L∞ , but not belonging to L2 , especially those continually happening disturbances. Moreover, we know from the result 1) of Lemma 2 that y ∈ L∞ can be derived in response to ω ∈ L∞ . Noting this fact, we define an induced transfer function matrix norm of Tyω (s) as   Tyω  = sup y∞ . ∞−∞ ω∞ ≤1

  With this definition, we have y∞ ≤ Tyω ∞−∞ ω∞ .  Clearly, Tyω ∞−∞ can be applied to quantitatively measure the system resilience of multiagent consensus networks with the bounded disturbances through examining y∞ in response to ω∞ .  In this case, we present the following theorem for evaluating Tyω  by any prescribed performance index. ∞−∞ Theorem 3: Consider the networked multiagent system (1) and let ωi ∈ L∞ for all i ∈ In . For  a prescribed scalar γ > 0, the objective (12) holds with Tyω ∞−∞ < γ if there exist two scalars μ > 0 and ψ > 0 and a positive definite matrix 0 ≺

MENG AND MOORE: STUDIES ON RESILIENT CONTROL THROUGH MULTIAGENT CONSENSUS NETWORKS

2059

Fig. 5. Under the directed graph Gb with the edge weight of EW2), the consensus performance of multiagent networks with respect to ω ∈ L2 . (a) State disagreements between agents. (b) Process of the Eulidean norm y. (c) Process of y2,[0,t] and ω2,[0,t] . (d) Process of y∞,[0,t] and ω2,[0,t] .

W ∈ R(n−1)×(n−1) satisfying the following matrix inequalities:   WQLP + P T LT QT W − ψW −WQ 0 (22) () μI ⎡ ⎤ ψW 0 I ⎣ () (γ − μ)I 0 ⎦  0. (23) () () γI Proof: From (22), we can apply the Schur’s complement formula to obtain WQLP + P T LT QT W − ψW  0. Due to ψW  0, the use of the Lyapunov stability theory yields that −QLP is Hurwitz stable. Hence, it follows from Lemmas 1 and 2 that  (12)  can be achieved. To derive the performance index Tyω ∞−∞ < γ, let us define an auxiliary Lyapunov-like function V = yT Wy. By considering the system description of (11), we premultiply and the matrix   postmultiply T inequality (22) with yT , ωT and yT , ωT , respectively, to obtain 0 < 2yT WQLPy − 2yT WQω − ψyT Wy + μωT ω = −2yT W y˙ − ψV + μωT ω = −V˙ − ψV + μωT ω.

(24)

By the Schur’s complement formula, (23) holds if and only if there exists a certain ∈ (0, γ) such that   0 ψW − (γ − )−1 I 0 () (γ − μ)I which, by premultiplying and this matrix    postmultiplying T inequality with yT , ωT and yT , ωT , respectively, leads to ψV − (γ − )−1 yT y + (γ − μ)ωT ω > 0.

(25)

By combining (24) and (25), we have (γ − )V˙ + yT y < γ(γ − )ωT ω ≤ γ(γ − ) ω2∞

for t ≥ 0

which can further guarantee (γ − )V˙ + y2∞ ≤ γ(γ − ) ω2∞ . Since V ≥ 0, we can derive from the above inequality that y2∞ t ≤ (γ − )V + y2∞ t  t   = (γ − )V˙ (τ) + y2∞ dτ 0 t ≤ γ(γ − ) ω2∞ dτ 0

= γ(γ − ) ω2∞ t

for t ≥ 0

2060

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 11, NOVEMBER 2014

√ which clearly implies  y∞ ≤ γ(γ − ) ω∞ . Consequently, we have Tyω ∞−∞ < γ. The proof is complete. We know from Theorem 3 that (12), rather than the consensus objective (2), holds in general when networked multiagent systems are subject to bounded disturbances ω ∈ L∞ . By noting Lemma 2, we can further obtain that (2) holds if the bounded disturbances also vanish with the increase of time. Motivated by this observation, we can summarize the following corollary. Corollary 3: Consider the networked multiagent system (1), and let ωi ∈ L∞ and limt→∞ ωi = 0 for all i ∈ In . Then, for a prescribed   scalar γ > 0, the consensus objective (2) holds with Tyω ∞−∞ < γ if there exist two scalars μ > 0 and ψ > 0 and a positive definite matrix 0 ≺ W ∈ R(n−1)×(n−1) satisfying the matrix inequalities (22) and (23). Proof: A consequence of Lemma 2 and Theorem 3. Remark 6: With the Schur’s complement formula, we know that the matrix inequality (23) holds if and only if   0 ψW − γ −1 I 0 () (γ − μ)I which, together with the satisfaction of (22), leads to   WQLP + P T LT QT W − γ −1 I −WQ  0. () γI

(26)

Again using the Schur’s complement formula, we can demonstrate the equivalence between (14) and (26). This fact together with Theorem 1 can guarantee that the matrix inequalities  (22) and (23) provide a sufficient condition to achieve Tyω  < γ. Similarly, we can further show from (26) 2−2 that the satisfaction of (26) makes the LMI (21) hold. We can thus apply Theorem 2 and Corollary 2 to develop that (22) and (23) achieve the Lyapunov equation  are  sufficient to 1/2 (15) with Tyω ∞−2 = W2 . Clearly, we can observe that the conditions (22) and (23) can be applicable even when the networked multiagent consensus systems are subject to disturbances satisfying ω ∈ L2 . Actually, this observation matches the reality. That is mainly because if ω ∈ L2 holds, then we usually have ω ∈ L∞ . Nevertheless, the converse is usually not true. Remark 7: From the discussion of Remark 6, we can derive that the matrix inequalities (22) and (23) provide a very general condition to guarantee the consensus performance of disturbed multiagent networks. But, this also makes (22) and (23) hard to meet. In particular, it is worth noting that both (22) and (23) are not described in the form of LMIs. There exists a coupled term ψW which is nonlinear with respect to the variables ψ > 0 and W  0 in (22) and (23). An alternative way to solve them with the LMI control toolbox of MATLAB is to first predetermine the scalar ψ appropriately. If ψ > 0 is known, it can be easily validated that (22) and (23) become LMIs with respect to γ > 0, μ > 0 and W  0. Therefore, we can solve the performance index γ by computing LMIs, to gain better disturbance rejection results for networked multiagent consensus systems. In addition to the above developments on the two conditions (22) and (23), we give the following corollary to provide a

TABLE II Performance of MultiAgent Consensus Networks for ω ∈ L∞

necessary guarantee from the network topology perspective. Corollary 4: A necessary condition for the matrix inequalities (22) and (23) is that G(A) has a spanning tree. Proof: By the proof for Theorem 3, we can easily observe that the Hurwitz stability of −QLP is necessarily required to ensure the conditions (22) and (23). Thus, we can develop this corollary by combining the result of Lemma 1. Next, we again consider the multiagent consensus networks used in Example 1 to illustrate the system resilience in the face of disturbances satisfying ω ∈ L∞ . Example 2: Let us consider disturbances such that  10 ω= cos(2πt), cos(2.5πt), cos(3πt), − cos(2πt) 1/2 20t + 1 T − cos(2.5πt), − cos(3πt) . It can be validated that ω ∈ L∞ with limt→∞ ω = 0, but ω ∈ L2 . Consequently, the development from two previous subsections cannot be applied. We will use the development for this subsection to discuss the consensus performance of networked multiagent systems subject to such bounded disturbances. As pointed out in Remark 7, the matrix inequalities (22) and (23) are not in the form of LMIs, and we should predetermine ψ, in order to apply the LMI control toolbox from the MATLAB to solve Table II gives the computed least  them.  upper bound of Tyω ∞−∞ , which is obtained by using the LMI solver “mincx” under the predetermined ψ [i.e., to choose certain ψ and then optimize γ subject to the LMI constraints (22) and (23)]. Also, we have applied the LMI solver “feasp” to verify the feasibility of (22) and (23) for the predetermined ψ and its corresponding performance index γ, and found that (22) and (23) are feasible, no matter what type of directed graph Ga or Gb is considered and which edge weight of EW1) or EW2) is used. Moreover, we can see from Table II that the network topologies and their associated edge have  weights  effects on the performance index given for Tyω ∞−∞ . This implies that the we can improve system resilience in the face of bounded disturbances by appropriately designing the multiagent consensus networks including their topologies and associated adjacency weights. In addition, we carry out simulation tests under the directed graphs Ga and Gb , respectively, with the edge weight of EW1). Figs. 6 and 7 depict the corresponding simulation results. It is obvious from the two figures that the consensus objective (2) can be accomplished for the agents in the presence of ω ∈ L∞ with limt→∞ ω = 0, regardless of what type of directed graph Ga or Gb is considered. This, as well as Table II, illustrates the proposed consensus results for networked multiagent systems in the face of disturbances that satisfy ω ∈ L∞ but ω ∈ L2 .

MENG AND MOORE: STUDIES ON RESILIENT CONTROL THROUGH MULTIAGENT CONSENSUS NETWORKS

Fig. 6. Under the directed graph Ga with the edge weight of EW1), the consensus performance of multiagent networks with respect to ω ∈ L∞ . (a) State disagreement between agents. (b) Process of the Eulidean norm y.

Remark 8: From the discussion made for three cases in this section as well as Examples 1 and 2, we can see that Theorems 2 and 3 can provide more general consensus results than the H∞ consensus result derived in Theorem 1. To our knowledge, there have been presented no such results in the literature with respect to the disturbance rejection issues of networked multiagent consensus systems. Thus, our proposed results can bring new insights into the consensus analysis for networked multiagent systems in the face of disturbances, through which new ways to resilient control can be provided. In addition, it can be seen from Tables I and II that it is necessary to appropriately design the multiagent networks in order to make them achieve good resilience in the presence of disturbances through gaining good performance index for disturbance rejection in enabling the networked agents to reach consensus. V. Conclusion In this paper, we have studied the problem of resilient control from the perspective of disturbance attenuation in multiagent consensus networks. We have adopted a general analysis approach to consider the disturbance rejection problems for multiagent consensus networks from the IOS point

2061

Fig. 7. Under the directed graph Gb with the edge weight of EW1), the consensus performance of multiagent networks with respect to ω ∈ L∞ . (a) State disagreements between agents. (b) Process of the Eulidean norm y.

of view. This provides a good alternative method to develop ISS-based approaches to establishing the consensus results for networked multiagent systems in the presence of disturbances. Furthermore, this method allowed consideration of system performances in consensus networks subject to different classes of disturbances, specifically disturbances belonging to the spaces ω ∈ L2 and ω ∈ L∞ . It has been shown that the system resilience of disturbed multiagent consensus networks can be measured by investigating the performance indexes of disturbance rejection, which studies three of induced  classes    Tyω  , Tyω  transfer function matrix norms, i.e., , 2−2 ∞−2   and Tyω ∞−∞ . In each case, the performance index can be quantitatively computed through solving LMI conditions. This has demonstrated via two illustrative numerical examples that demonstrate the proposed consensus results of multiagent networks subject to disturbances ω ∈ L2 and ω ∈ L∞ , respectively. We finally note that the results developed in this paper can be extended to take into account switching topologies and communication delays in the consensus network, a topic of our future research.

2062

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 11, NOVEMBER 2014

Appendix Proof of Lemma 1: Notice that L is the Laplacian matrix of G(A). By following [9], we can easily deduce that L has exactly one zero eigenvalue and all the other eigenvalues of L are in the open right half plane if and only if G(A) has a spanning tree. Moreover, we have     RL1n RLP 0 RLP −1 ˜ ˜ P LP = = . QL1n QLP 0 QLP Since P˜ −1 LP˜ has the same eigenvalues with L, we can easily develop from this fact that all the eigenvalues of QLP are in the open right half plane (that is, −QLP is Hurwitz stable) if and only if G(A) has a spanning tree. Hence, this completes the proof of Lemma 1. Proof of Lemma 2 1) For the system (11), we can apply the standard linear system theory to develop that y ∈ L∞ holds for ω ∈ L∞ if and only if  ∞ ! ! !hij (t)! dt ≤ β < ∞ 0

  where e−QLPt Q  hij (t) ∈ R(n−1)×n . As the system (11) is both controllable and observable, we can further derive that this integral condition holds if and only if −QLP is a Hurwitz stable matrix. Then, from Lemma 1, it follows that y ∈ L∞ holds for ω ∈ L∞ if and only if G(A) has an spanning tree.  By noting y = Qx, we have 2 1/2 y = x − x  which implies that xi − i j i=1 xj  ≤ y holds for all i ∈ In . This, together with y ∈ L∞ , leads to     xi − xl  =  xi − xj − xl − xj  ≤ xi −xj +xl −xj  ≤ 2 y for all i, l ∈ In (27) and, therefore, (12) can be achieved if y ∈ L∞ . On the contrary, if (12) holds, then we can obtain y2 =

n 

xi − xj 2

i=1,i =j

≤ (n − 1)

"

#2

and some δ, λ > 0. For any t0 ≥ 0, we can obtain the solution to (11) as  t y = e−QLP(t−t0 ) y(t0 )+ e−QLP(t−τ) Qω(τ)dτ for all t ≥ t0 t0

which can be estimated by y ≤ δe−λ(t−t0 ) y(t0 ) +

≤ (n − 1)2 < ∞ √ which guarantees y∞ ≤ n − 1, i.e., y ∈ L∞ . Hence, we can easily conclude that the equivalent statement of 1) holds. 2) Necessity: By Proposition 1, we know that accomplishing the consensus objective (2) for the networked multiagent system (1) requires that G(A) has a spanning tree, even in the particular case where ω ≡ 0. This observation clearly implies that the necessity of our statement 2) holds. Sufficiency: If G(A) has a spanning tree, then we can derive from Lemma 1 that −QLP is Hurwitz stable. Hence, we have e−QLP(t−τ)  ≤ δe−λ(t−τ) for t ≥ τ ≥ 0

t

δe−λ(t−τ) Qω(τ)dτ

t0

δQ ≤ δe y(t0 )+ sup ω(τ) for all t≥t0 . λ t0 ≤τ≤t (28) For any given > 0, we can easily gain from limt→∞ ω = 0 that there is T1 > 0 such that ω ≤ λ /2δQ holds for all t ≥ T1 . Note that limt→∞ δe−λ(t−t0 ) y(t0 ) = 0. There is T2 > 0 such that δe−λ(t−t0 ) y(t0 ) < /2 for all t ≥ T2 . If we take t0 ≥ T1 , then we can easily deduce from (28) that y ≤ for all t ≥ T = max{t0 , T2 } −λ(t−t0 )

which implies limt→∞ y = 0. Note that this convergence result is equivalent to the consensus objective (2). The sufficiency is complete. Consequently, the statement 2) is proved. 3) Note that if ω ∈ L2 , then we have limt→∞ ω = 0. Hence, the proof of the statement 3) can be developed by following the lines used in the proof of the statement 2). 4) The necessity of this statement can be easily obtained by following the similar steps as in deriving that of the statement 2), which is not detailed here. Under the condition that G(A) has a spanning tree, Lemma 1 ensures the Hurwitz stability of −QLP, which implies e−QLP(t−τ)  ≤ δe−λ(t−τ) for t ≥ τ ≥ 0 and some δ, λ > 0. Since limt→∞ ω = 0 exponentially fast, there are some α, σ > 0 (σ = λ) such that ω ≤ αe−σt holds for all t ≥ 0. Let λmin = min{λ, σ} and λmax = max{λ, σ}. Then we have λmax > λmin > 0. Note that the solution to (11) can be given as  t e−QLP(t−τ) Qω(τ)dτ y = e−QLPt y(0) + 0  t −QLPt =e y(0) + e−QLPτ Qω(t − τ)dτ for all t ≥ 0. 0

max xi − xl 

∀i,l∈In



(29)

Using the facts of (29), we can always obtain  t y ≤ δe−λt y(0) + δe−λmin (t−τ) Qαe−λmax τ dτ 0

δαQ ≤ δe y(0) + e−λmin t λ − λ max min # " δαQ e−λmin t ≤ δy(0) + λmax − λmin −λt

for all t ≥ 0

(30) which implies limt→∞ y = 0 exponentially fast. By combining (30) with (27), we can conclude that the consensus objective (2) is achieved exponentially fast. The proof of the statement 4) is complete.  Proof of Lemma 3: This lemma naturally holds for η = 0. Next, we will complete this proof by considering η = 0. By

MENG AND MOORE: STUDIES ON RESILIENT CONTROL THROUGH MULTIAGENT CONSENSUS NETWORKS

applying the Schur’s complement formula, ηT −1 η < θ 2 is equivalent to  2  θ ηT 0 ()  which, again by using the Schur’s complement formula, leads to  − θ −2 ηηT  0. This, together with   I, implies     η2  − θ −2 η2 = ηT I − θ −2 ηηT η   ≥ ηT  − θ −2 ηηT η >0 which guarantees η < θ1/2 . The proof is complete. Proof of Lemma 4: From the Lyapunov stability theory, it follows that each of (15) and (16) holds if and only if the matrix −QLP is Hurwitz stable. This implies the equivalence between (15) and (16). In addition, we can deduce from (15) and (16) that QLP(X − W) + (X − W)P T LT QT  0.

(31)

Based on the Hurwitz stability of −QLP, we can obtain from (31) that X − W  0, i.e., X  W. This guarantees W < X. Moreover, for any given > 0 and the solution W to (15), let  ∞

T T T   ∞ X=W+ e−QLPt e−P L Q t dt   T T T 0  e−QLPt e−P L Q t dt    0

and then it is easy to verify that X − W ≤ and QLP(X − W) + (X − W)P T LT QT

T   =  ∞ −QLPt −P T LT QT t  QQ  0.   e e dt   0

Thus, we can see that (17) holds. The proof is complete. References [1] C. G. Rieger, D. I. Gertman, and M. A. McQueen, “Resilient control systems: Next generation design research,” in Proc. 2nd Conf. Human Syst. Interactions, May 2009, pp. 632–636. [2] A. Teixeira, H. Sandberg, and K. H. Johansson, “Networked control systems under cyber attacks with applications to power networks,” in Proc. Amer. Control Conf., Jun./Jul. 2010, pp. 3690–3696. [3] C. M. Colson, M. H. Nehrir, and R. W. Gunderson, “Distributed multiagent microgrids: A decentralized approach to resilient power system self-healing,” in Proc. 4th Int. Symp. Resilient Control Syst., Aug. 2011, pp. 83–88. [4] C. G. Rieger, Q. Zhu, and T. Basar, “Agent-based cyber control strategy design for resilient control systems: Concepts, architecture and methodologies,” in Proc. 5th Int. Symp. Resilient Control Syst., Aug. 2012, pp. 40–47. [5] S. Biswas, Q. Dong, and L. Bai, “Consensus control for linear systems in the presence of environmental and channel noise,” in Proc. 4th Int. Symp. Resilient Control Syst., Aug. 2011, pp. 126–130. [6] S. Biswas, F. Ferrese, Q. Dong, and L. Bai, “Resilient consensus control for linear systems in a noisy environment,” in Proc. Amer. Control Conf., Jun. 2012, pp. 5862–5867. [7] F. Ferrese, S. Biswas, Q. Dong, and L. Bai, “Resiliency of linear system consensus in the presence of channel noise,” in Proc. 5th Int. Symp. Resilient Control Syst., Aug. 2012, pp. 137–142. [8] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 998–1001, Jun. 2003. [9] W. Ren and R. W. Beard, “Consensus seeking in multi-agent systems under dynamically changing interaction topologies,” IEEE Trans. Autom. Control, vol. 50, no. 5, pp. 655–661, May 2005.

2063

[10] W. Yu, G. Chen, M. Cao, and J. Kurths, “Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 881–891, Jun. 2010. [11] Z. Meng, W. Ren, Y. Cao, and Z. You, “Leaderless and leader-following consensus with communication and input delays under a directed network topology,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 1, pp. 75–88, Feb. 2011. [12] J. Qin, W. X. Zheng, and H.-J. Gao, “Coordination of multiple agents with double-integrator dynamics under generalized interaction topologies,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 1, pp. 44–57, Feb. 2012. [13] Y. Sun and J. Huang, “Cooperative output regulation with application to multi-agent consensus under switching network,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 3, pp. 864–875, Jun. 2012. [14] Z. Wu, H. Fang, and Y. She, “Weighted average prediction for improving consensus performance of second-order delayed multi-agent systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 5, pp. 1501–1508, Oct. 2012. [15] K.-K. Oh, F. Lashhab, K. L. Moore, T. L. Vincent, and H.-S. Ahn, “Consensus of positive real systems cascaded with a single integrator,” Int. J. Robust Nonlinear Control, published online, DOI: 10.1002/rnc.3093. [16] K. L. Moore, T. L. Vincent, F. Lashhab, and C. Liu, “Dynamic consensus networks with application to the analysis of building thermal processes,” in Proc. 18th IFAC World Congr., Aug./Sep. 2011, pp. 3078–3083. [17] Q. Zhu and T. Basar, “Towards a unifying security framework for cyberphysical systems,” in Proc. Workshop Found. Dependable Secure CyberPhys. Syst., Apr. 2011, pp. 47–50. [18] N. Kottenstette, G. Karsai, and J. Sztipanovits, “A passivity-based framework for resilient cyber physical systems,” in Proc. 2nd Int. Symp. Resilient Control Syst., Aug. 2009, pp. 43–50. [19] P. Lin, Y. Jia, and L. Li, “Distributed robust H∞ consensus control in directed networks of agents with time-delay,” Syst. Control Lett., vol. 57, no. 8, pp. 643–653, Aug. 2008. [20] P. Lin and Y. Jia, “Robust H∞ consensus analysis of a class of second-order multi-agent systems with uncertainty,” IET Control Theory Applicat., vol. 4, no. 3, pp. 487–498, Mar. 2010. [21] Y. Liu and Y. Jia, “H∞ consensus control of multi-agent systems with switching topology: A dynamic output feedback protocol,” Int. J. Control, vol. 83, no. 3, pp. 527–537, Mar. 2010. [22] Y. Liu and Y. Jia, “Consensus problem of high-order multi-agent systems with external disturbances: An H∞ analysis approach,” Int. J. Robust Nonlinear Control, vol. 20, no. 14, pp. 1579–1593, Sep. 2010. [23] L. Mo and Y. Jia, “H∞ consensus control of a class of high-order multiagent systems,” IET Control Theory Applicat., vol. 5, no. 1, pp. 247–253, Jan. 2011. [24] Z. Li, Z. Duan, G. Chen, and L. Huang, “Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 1, pp. 213–224, Jan. 2010. [25] Y. Sun and L. Wang, “H∞ consensus of second-order multi-agent systems with asymmetric delays,” Syst. Control Lett., vol. 61, no. 8, pp. 857–862, Aug. 2012. [26] M. Green and D. J. N. Limebeer, Linear Robust Control. Englewood Cliffs, NJ, USA: Prentice Hall, 1995. [27] Z.-G. Hou, L. Cheng, and M. Tan, “Decentralized robust adaptive control for the multiagent system consensus problem using neural networks,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 3, pp. 636–647, Jun. 2009. [28] H. Zhang and F. L. Lewis, “Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics,” Automatica, vol. 48, no. 7, pp. 1432–1439, Jul. 2012. [29] G. Shi, Y. Hong, and K. H. Johansson, “Connectivity and set tracking of multi-agent systems guided by multiple moving leaders,” IEEE Trans. Autom. Control, vol. 57, no. 3, pp. 663–676, Mar. 2012. [30] S. Li, H. Du, and X. Lin, “Finite-time consensus algorithm for multiagent systems with double-integrator dynamics,” Automatica, vol. 47, no. 8, pp. 1706–1712, Aug. 2011. [31] G. Hu, “Robust consensus tracking of a class of second-order multiagent dynamic systems,” Syst. Control Lett., vol. 61, no. 1, pp. 134–142, Jan. 2012. [32] L. Schenato and F. Fiorentin, “Average TimeSynch: A consensusbased protocol for clock synchronization in wireless sensor networks,” Automatica, vol. 47, no. 9, pp. 1878–1886, Sep. 2011. [33] H. Yang, Z. Zhang, and S. Zhang, “Consensus of second-order multiagent systems with exogenous disturbances,” Int. J. Robust Nonlinear Control, vol. 21, no. 9, pp. 945–956, Jun. 2011.

2064

[34] D. Zelazo, S. Schulerb, and F. Allg¨ower, “Performance and design of cycles in consensus networks,” Syst. Control Lett., vol. 62, no. 1, pp. 85–96, Jan. 2013. [35] D. B. Kingston, W. Ren, and R. W. Beard, “Consensus algorithms are input-to-state stable,” in Proc. Amer. Control Conf., Jun. 2005, pp. 1686–1690. [36] C. Ma, T. Li, and J.-F. Zhang, “Consensus control for leader-following multi-agent systems with measurement noises,” J. Syst. Sci. Complex., vol. 23, no. 1, pp. 35–49, Jan. 2010. [37] A. Garulli and A. Giannitrapani, “Analysis of consensus protocols with bounded measurement errors,” Syst. Control Lett., vol. 60, no. 1, pp. 44–52, Jan. 2011. [38] S. Liu, L. Xie, and H. Zhang, “Distributed consensus for multi-agent systems with delays and noises in transmission channels,” Automatica, vol. 47, no. 5, pp. 920–934, May 2011.

Deyuan Meng (M’12) received the B.S. degree in mathematics and applied mathematics from the Ocean University of China, Qingdao, China, in June 2005, and the Ph.D. degree in control theory and control engineering from Beihang University, Beijing, China, in July 2010. He is currently with the Seventh Research Division and the Department of Systems and Control, Beihang University. From 2012 to 2013, he was a Visiting Scholar with the Department of Electrical Engineering and Computer Science, Colorado School of Mines, Golden, CO, USA. His current research interests include iterative learning control, and distributed control of multiagent systems.

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 11, NOVEMBER 2014

Kevin L. Moore (M’80-SM’97) received the B.S. and M.S. degrees in electrical engineering from Louisiana State University, Baton Rouge, LA, USA, and the University of Southern California, Los Angeles, CA, USA, respectively, and the Ph.D. degree in electrical engineering, with an emphasis on control theory, from Texas A&M University, College Station, TX, USA, in 1989. He is the Dean of the College of Engineering and Computational Sciences, Colorado School of Mines, Golden, CO, USA. He was an Assistant and Associate Professor with Idaho State University, Pocatello, ID, USA, from 1989 to 1998; an Associate and a Full Professor of electrical and computer engineering with Utah State University, Logan, UT, USA, where he was the Director of the Center for Self-Organizing and Intelligent Systems, directing multidisciplinary research teams of students and professionals developing a variety of autonomous robots for government and commercial applications, from 1998 to 2004; a Senior Scientist with Applied Physics Laboratory, Johns Hopkins University, Baltimore, MD, USA, during a one-year research stay, where he worked in the area of unattended air vehicles, cooperative control, and autonomous systems from 2004 to 2005; and a Full Professor of Engineering with the Colorado School of Mines from 2005 until now, where he was a Director of the Center for Robotics, Automation, and Distributed Intelligence and the G.A. Dobelman Distinguished Professor from 2005 to 2011. He was also engaged in the industry for three years pre-Ph.D as a member of the Technical Staff with Hughes Aircraft Company, Glendale, CA, USA. His current research interests include iterative learning control, autonomous systems and robotics, and applications of control to industrial and mechatronic systems, including the cooperative control of networked systems. He is the author of the research monograph Iterative Learning Control for Deterministic Systems, the co-author of the book Sensing, Modeling, and Control of Gas Metal Arc Welding, and the co-author of the research monograph Iterative Learning Control: Robustness and Monotonic Convergence for Interval Systems. He is a licensed Professional Engineer, involved in several professional societies and editorial activities, and is interested in engineering education pedagogy, particularly capstone senior design. Dr. Moore is an ABET Program Evaluator, a member of the IEEE Control System Society Technical Committee on Intelligent Control, and serves on several editorial boards.