Robust Consensus for Multi-Agent Systems Communicating over Stochastic Uncertain Networks \thanksreffootnoteinfo
In this paper, we study the robust consensus problem for a set of discrete-time linear agents to coordinate over an uncertain communication network, which is to achieve consensus against the transmission errors and noises resulted from the information exchange between the agents. We model the network by means of communication links subject to multiplicative stochastic uncertainties, which are susceptible to describing packet dropout, random delay, and fading phenomena. Different communication topologies, such as undirected graphs and leader-follower graphs, are considered. We derive sufficient conditions for robust consensus in the mean square sense. This results unveil intrinsic constraints on consensus attainment imposed by the network synchronizability, the unstable agent dynamics, and the channel uncertainty variances. Consensus protocols are designed based on the state information transmitted over the uncertain channels, by solving a modified algebraic Riccati equation.
China]Zhongkui Li, UK]Jie Chen
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China
Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong
Key words: Cooperative control, consensus, uncertainty, communication channel, robustness.
Over the last decade, consensus and other cooperative control problems of multi-agent systems have received compelling attention from the control community; see the recent works [1, 2, 3, 4] and the references therein. Roughly speaking, a consensus problem of multi-agent systems is composed of three essential components, both independent and inter-related: the agent dynamics, the network communicating among the agents, and the consensus protocol required to achieve consensus . The network topology is without any question the central mechanism that enables multiple agents to cooperate via information exchange between the agents and over the network. In the literature, a directed or undirected graph, commonly represented by a known, often constant matrix, is adopted to characterize the information flow among the agents in a highly idealized manner. In practice, however, the information exchanges among agents must be executed over a communication network consisting of various communication links, which cannot be immune to transmission errors, channel noises, and system constraints. A realistic multi-agent model must therefore account for the network communication errors and noises, and for it to be viable, the theory of cooperative control of multi-agent systems must address the robustness against the noises and uncertainties in the communication channels.
Communication errors may result from a variety of channel constraints and transmission imperfections, such as data rate limit, quantization precision, time delay, packet dropout, and channel capacity, etc. These issues pose novel challenges to feedback design and have been a focal subject of the recent research on networked control systems [5, 6, 7]. By now it is widely known that for a system controlled over a communication channel, the channel specification, be it data rate, capacity, quantization density, and signal-to-noise ratio, must in one way or another satisfy an intrinsic bound in order to ensure feedback stabilization [8, 9, 10]. Unsurprisingly, these advances have made their way into the study of multi-agent systems. Consensus problems using quantized information exchanges are considered in, e.g., [11, 12]. Conditions are derived in [13, 14] to achieve mean square consensus in the presence of communication or measurement noises. Efforts are made in [15, 16] to address the effect of data loss or packet dropout on consensus. Generally, in the presence of such communication errors, one finds the design of a robust consensus protocol ever more difficult and indeed a highly nontrivial task, due to the interplay between the agent dynamics, the communication graph, and the communication channels.
In this paper we study the robust consensus problem for general discrete-time multi-agent systems with stochastic uncertain communication channels. Motivated by the random quantizers in  and the fading channel model in , we model each communication link between the agents as a noisy communication channel subject to a multiplicative stochastic uncertainty. A graph network so formulated provides a more realistic model of the communication network consisting of erasure channels, which are particularly susceptible to describing such network losses as packet dropout and random delay. Indeed, the uncertain channel model herein includes the random quantizers (as pointed out in ), the stochastic i.i.d. packet loss (as demonstrated in ), and random relative-state-dependent measurement noises (as shown in ) as special cases. Our task is twofold. First, we design robust consensus protocols so that the agents can achieve consensus despite the presence of the channel uncertainties. Second and perhaps more importantly, we investigate the fundamental limits of the uncertainties under which consensus remains attainable. We address these issues for two different communication topologies, including undirected graphs and leader-follower graphs. The following passages summarize our main contributions.
We formulate and solve the robust consensus problem in the mean square sense. Here we say that a group of agents achieve mean square consensus if the states of the agents converge asymptotically to a common state under the mean square criterion. We derive sufficient conditions guaranteeing the mean square consensus, under which accordingly, a robust consensus protocol is designed based on the relative state information; the latter is achieved by solving a modified algebraic Riccati equation. This result sheds important light into the intrinsic limit to the channel uncertainty variance that a multi-agent system can tolerate to achieve consensus with the given graph topology and the feedback protocol, which is seen to depend on the graph synchronizability, the topological entropy of the agents, and the uncertainty variance. Interestingly, the condition recovers the conventional results on consensus of discrete-time linear multi-agent systems with ideal channels in, e.g., [19, 20, 21], which is both necessary and sufficient in the noise-free case.
A further expansion of these results concerns the case where uncertainties are present in the control input channels of the agents. In this case, each agent communicates with its neighbors through the same communication channel perturbed by a multiplicative uncertainty. A sufficient condition is obtained similarly to guarantee the mean square consensus.
Previous works pertinent to this paper include [22, 23, 13, 24, 25], where robust consensus over uncertain communication channels is studied. It should be noted, nonetheless, that only single-integrator agents are considered in these works, while the present paper addresses general high-order linear agent dynamics. This difference in the agent dynamics is by no means simple. Indeed, even with single-integrator agents, the robust consensus conditions (see, e.g., ) appear rather complex, lest the conditions for general linear agents. Equally difficult is the task of designing feedback gain matrices of the consensus protocols. This latter issue is moot with the aforementioned case of single-integrator agents, as therein only scalar feedback gains need to be designed. More generally, due to the need to incorporate the graph structure, the analysis and design of robust consensus against the multiplicative stochastic uncertainties presents in general a highly nontrivial task, whose difficulty goes considerably beyond that found in networked control problems [5, 6, 7]. Finally, it is worth pointing out that robustness issues will also arise from the uncertainties in agent dynamics. This, however, is not the concern of the current paper; there is a bulk of research devoted to the study of heterogeneous agents, see, e.g.,  and references therein.
The mathematic preliminaries used in this paper are summarized in Section 2. The uncertain communication channel model is presented and the robust consensus problem is formulated in Section 3. The robust consensus problem in the presence of stochastic channel uncertainties is studied in Section 4. The uncertain input channel model is discussed in Section 5. Numerical simulation results are presented for illustration in Section 6. Finally, Section 7 concludes our paper.
2 Mathematical Preliminaries
The notations used in this paper are standard. denotes the set of real matrices. The symbol denotes a column vector with all entries equal to 1. denotes the 2-norm of a vector and denotes the induced 2-norm of a real matrix . The matrix inequality means and are symmetric matrices and is positive definite. represents the Kronecker product of matrices and . The expectation operator is denoted by .
2.2 Algebraic Graph Theory
A directed graph is defined by , where is the set of nodes and denotes the set of edges. For an edge , node is called the parent node (i.e., the head), is the child node (i.e., the tail), and is a neighbor of . A graph with the property that implies that for any , i.e., each edge is an unordered pair of nodes, is said to be undirected. A path from node to node is a sequence of ordered edges in the form of , . A directed graph contains a directed spanning tree if there exists a node called the root which has directed paths to all other nodes in the graph. For a directed graph , its adjacency matrix, denoted by , is defined such that , is a positive value if and otherwise, where denotes the weight for the edge . For an undirected graph, assume that each of its edges has a head and a tail. The incidence matrix associated with the undirected graph, denoted by , is defined as if is the tail of , if is the head of , and otherwise. The Laplacian matrix associated with a graph is defined as and , .
Lemma 1 ()
For an undirected graph, , where is an diagonal matrix, with positive on the diagonal.
Lemma 2 ()
Zero is an eigenvalue of with as a right eigenvector and all nonzero eigenvalues have positive real parts. Besides, zero is a simple eigenvalue of if and only if has a directed spanning tree.
3 Stochastically Perturbed Channels and Problem Formulation
Consider a network consisting of discrete-time linear agents. The dynamics of the -th agent are described by
where is the state vector and is the control input vector of the -th agent, respectively, and the following assumption holds.
Assumption 1 The pair is stabilizable.
The information flow among the agents is depicted by a graph , which can be either directed or undirected. Agent can obtain information from agent , if is an edge. The graph merely depicts the information flow topology among the agents. In practice, the agents cooperate via information exchange over a network of communication channels, which in general are subject to transmission errors resulted from communication noises and channel constraints. Motivated by the random quantizers in  and the fading channel model in , in this paper we model each communication channel as an ideal transmission system subject to a multiplicative stochastic uncertainty. Specifically, each agent obtains the relative state information from its neighbors through uncertain channels, which are ideal channels with a unity transfer function perturbed by an stochastic uncertainty , as shown in Fig. 1.
The control input of each agent, i.e., the consensus protocol, is designed as
where is a positive scaling factor, is the feedback gain matrix, denotes the -th entry of the non-weighted (i.e., ) adjacency matrix of the graph , and denotes the stochastic uncertainty associated with the communication channel between agents and .
We assume that the stochastic uncertainties satisfy the following assumption.
Assumption 2 , in (3) are white noise processes, independent from each other and for each , with and .
Remark 1 The stochastically perturbed communication channel in this paper is fairly general and can be used to describe several communication losses in digital networks, such as quantization , signal distortion , packet drop , and multiplicative (relative-state-dependent) communication noises . Specifically, the binary i.i.d. packet loss and Rice fading channel  can be described by a structured bounded variance uncertainty in the form of (3). It is also shown in  that if is uniformly distributed over a certain interval for every , and , then the uncertainty can be viewed as a random quantization error.
The protocol (3) solves the robust consensus problem in the mean square sense, i.e., the mean square consensus problem, in the presence of the uncertain channel model described as above, if , . Our objective in this paper is two-fold. We first determine conditions such that the agents in (3) will achieve consensus under the protocol (3). This in turn leads to intrinstic limits relating the channel uncertainty specification to the network synchronizability and the agents’ unstable dynamics. Next, we design the feedback gain matrix in (3) such that the states of the agents reach agreement in the mean square sense.
4 Consensus over Uncertain Communication Channels
In this section, we consider the robust consensus problem with the stochastically perturbed channel model for different network topologies.
4.1 Undirected Graphs
In this subsection, we consider the case where the communication graph is undirected, which satisfies the following assumption.
Assumption 3 The graph is undirected and connected.
where , with , , and , and is the symmetric Laplacian matrix associate with . Evidently,
Let , where . It is easy to see that is a simple eigenvalue of with as the eigenvector and 1 is the other eigenvalue with multiplicity . Then, it follows that if and only if . Hereafter, we will refer to as the consensus error. In light of the facts that and , we can deduce from (4.1) that satisfies the following dynamics:
Before moving forwards, we introduce the following modified algebraic Riccati equation (MARE) :
where , , and . The lemma given below concerns the existence of a solution to the MARE (4.1).
Lemma 3 ()
Assume that be stablizable and . Then, the MARE (4.1) has a unique positive-definite solution , if , where denotes the Mahler measure of , which is the absolute value of the product of the unstable eigenvalues of .
The following theorem presents a sufficient condition to solve the mean square consensus problem.
Proof Because Assumption 3 holds, it follows from Lemma 2 that zero is a simple eigenvalue of and all the other eigenvalues are positive. Let be a unitary matrix such that . Since the right and left eigenvectors of corresponding to the zero eigenvalue are and , respectively, we select , where . It is not difficult to check that
Let . Then, (4.1) can be rewritten into
In virtue of the definitions of and , we can verify that Therefore, the mean square consensus problem is reduced to checking the mean square stability of . Since and , it follows that
By noting that and in light of (4.1), we have
Under Assumption 2, it is not difficult to derive that the expectation of satisfies
where denotes the -th entry of the matrix and we have also used the fact that . It is not difficult to verify that corresponds to certain weighted Laplacian matrix of the undirected graph . Suppose that there are edges in the graph and denote by the incidence matrix of . In light of Lemma 1, we can verify that
Since the matrices on the right side of (16) are block diagonal, it is evident that , if
where we have used the fact that is convex with respect to . In light of (17) and (18), it follows from (16) that Then, we obtain that satisfies (10), i.e., the system (9) is mean square stable. Therefore, the consensus protocol (3) with chosen as above solves the mean square consensus problem provided that the condition (4) holds.
Remark 2 The scalar in (3) is a design scaling parameter, which is intended to regulate the left-hand side of (4). By simple calculations, we can choose such that left-hand side of (4) is minimized for given , , and . In this case, the condition (4) becomes
On the other hand, If the variances of the channel uncertainties are not available, in this case we can choose and the condition (4) becomes
When the communication channels are noise-free, which is equivalent to for all , the condition (19) or (20) reduces to . The latter is the necessary and sufficient condition for consensus with ideal channels, obtained in, e.g., [19, 20, 21].
Remark 3 Theorem 4 uncovers the intrinsic constraints induced by the channel uncertainties, the communication topology, and the unstable dynamics of the agents. The eigenratio in (19) is commonly used to indicate the synchronizability of dynamical networks [30, 31]. A larger eigenratio implies a better synchronizability. The Mahler measure of , or the topological entropy , plays an important role in networked control systems [6, 32], as it describes the minimum data rate for stabilizing an unstable system. The largest variance can be used to measure the maximum mean square channel capacity for the stochastic communication channels, as discussed in . As demonstrated in the condition (19), the synchronizability factor, the topological entropy of , and the mean square channel capacity impose intrinsic limitations on achieving consensus over stochastic uncertain channels.
Remark 4 For the case where the agents have no eigenvalues outside the unit circle, i.e., , it is worth noting that the condition (19) always holds and the consensus protocol (3) can be constructed as in Theorem 1 and Remark 2 to ensure mean square consensus for any channel uncertainties satisfying Assumption 3 as long as is stabilizable. In this sense, the condition (19) is both necessary and sufficient. Nevertheless, the conservatism of the condition (19) remains unclear for the case of . The condition (20) is generally not necessary. For instance, consider a group of single integrators with a complete graph and assume for simplicity that the variances of the channel uncertainties are all equal, denoted by . In this case, the condition (20) becomes . In light of the frequency-domain mean-square small gain theorem in [18, 33], a necessary and sufficient condition to achieve mean square consensus is that when , when , and when . It is easy to verify that the condition (20) is tight for the special cases with two or three single-integrator agents with complete graphs, and is conservative if .
4.2 Leader-follower Graphs
In this subsection, we consider the case where there exists a leader. Without loss of generality, let the agent in (3) indexed by be the leader and the rest be the followers. The leader receives no information from any follower and only a subset of the followers have access to the leader’s information. It is assumed that the leader’s control input is equal to zero, i.e., , and the control inputs of the followers are described by (3). We assume that the communication graph among the agents satisfies the following assumption.
Assumption 4 The graph contains a directed spanning tree with the leader as the root and the subgraph associated with the followers is undirected.
Because the leader has no neighbors, the Laplacian matrix associated with can be partitioned as , where and . By Lemma 2 and Assumption 4, it is clear that .
where . Let . Then, we rewrite (21) into
where , with , , and . Evidently, , , approach , if in (22) converges to zero. Therefore, the consensus protocol (3) solves the mean square leader-follower consensus problem for the agents in (3), if the system (22) is mean square stable.
where denotes the -th entry of the matrix .
Without loss of generality, assume that the followers indexed by , have access to the leader. Further, suppose that there are edges in the graph . Thus, there exist edges in the subgraph among the followers. Let where the last items correspond to the channels in . Define , where denotes the incidence matrix associated with and is formed by the first columns of .
We first show the following claim.
Claim 1 The matrix , defined as in (25), is equal to .
Let and . Denote by the weighted Laplacian matrix corresponding to the undirected subgraph , which, in virtue of Lemma 1, is equal to . Then,
By the definition of , it is not difficult to verify that the right hand of the last equation in (26) is equal to .
In light of Claim 1, it follows from (25) that
Following similar steps in the proof of Theorem 4, it is not difficult to show that if the condition (23) holds, then (28) and also (24) hold, implying that (22) is mean square stable, i.e., the consensus protocol (3) solves the mean square leader-follower consensus problem. The details are omitted here for conciseness.
Remark 5 For the leader-follower graphs considered in this subsection, communications among the followers are bidirectional, while those between the followers and the leader are directional. Therefore, it is reasonable to define the maximum mean square channel capacity in this case by the maximum of the sum of the variances of the stochastic uncertainties on every bidirectional channel and the variances of the uncertainties on the directional channels. This is different from the undirected graph case in the previous subsection.
5 Consensus via Uncertain Input Channels
In this section, we consider the case where the agents in (3) are perturbed by uncertainties in their control input channels. The stablization problem of a single LTI system with uncertain input channels has been considered in [32, 34]. The uncertain input channel model is illustrated in Fig. 2, where denotes the stochastic uncertainty in the input channel of agent and denotes the signal available to agent , which will go through the uncertain input channel for controller design. The stochastic uncertainties satisfy the following assumption.
Assumption 5 , , in (29) are independent zero-mean white noise processes with variances .
In this section, for simplicity we only consider the case where the communication graph is undirected and satisfies Assumption 3. The consensus protocol is given as follows:
where , , and are defined as in (3).
This uncertain input channel model is related to the uncertain communication channel in the previous sections. By comparing (29) to (3), we can observe that the uncertain input channel model can be regarded in the sense that each agent communicates with its neighbors through the same communication channel perturbed by a multiplicative uncertainty . The uncertain channels are different for different agents, even though the agents may have bidirectional communication links. This differs from the uncertain communication channel model in the previous sections.
where and is the Laplacian matrix of .
The following theorem ensures that (29) solves the mean square consensus problem.
Proof Let the consensus error be defined as in (4.1). We can verify that in this case satisfies
Since and both and have zero row and column sums, we can choose , with , be the unitary matrix such that and , where . Let , where . Since we can derive that satisfies
The rest of the derivations can be completed by following steps in the proof of Theorems 4, by further noting that
where we have used the facts that and . The details are omitted here for brevity.
6 Numerical Simulations
Consider a group of six discrete-time double integrators, i.e., the matrices in (3) are given by , The communication graph is an undirected cycle, where the communication channels are perturbed by stochastic uncertainties satisfying Assumption 2. The smallest and largest nonzero eigenvalues of the Laplacian matrix are 1 and 4, respectively. For illustration, let the variances of the uncertainties of the six channels be 1.5. Solving the MARE (3) with and gives . Therefore, it follows from Theorem 4 that the feedback gain matrix of (3) is obtained as . The initial states of the agents are chosen to be , , , , , and . The average of 1000 sample trajectories of the relative states , , under (3) with and designed as above, are depicted in Fig. 3, which is an approximation of the expectation of and from which it can be observed that mean square consensus is indeed achieved.
In this paper we have studied a robust consensus problem for a network of general discrete-time linear agents coordinating through uncertain communication channels. We model the communication channel as an ideal transmission system subject to a multiplicative stochastic perturbation. We have presented sufficient robust consensus conditions, which exhibit intrinstic limitations imposed by the synchronizability factor, the topological entropy of the agents, and the mean square channel capacity. Under this limit, consensus protocols can be designed by solving an MARE.
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