On Quantized And Robust Consensus For Multi-Agent Systems With Directed Network Topologies

Due to the fact that centralized control and global information of networks are notrequired, the distributed coordination of the networked multi-agent systems hasrecently attracted wide interest within the control community. The networkedmulti-agent system has a broad range of applications, since it can not only reduce thecost but also improve the robustness and adaptivity, particularly when the networkworks under adverse environment. As a fundamental problem of the coordination ofnetworked multi-agent systems, distributed consensus algorithms have receivedincreasing attention in various contexts. Consensus algorithms involve appropriatedistributed control laws or protocols that use local neighboring information to lead thestates of the agents to an agreement upon a common value of interest. Thus, thenetwork topology and the information transmitted across the network are twoimportant factors needed to be considered in the analysis and design of distributedconsensus algorithms. In a distributed manner, particularly when networks workunder adverse conditions, packet losses and node failure usually happen. In thesituation that agents have different sense ranges, symmetric exchange protocols andbalanced communications are hard to enforced or even infeasible. Thus, unidirectionaland unbalanced information flow among agents is the primary feature of themulti-agent networks in the real world. However, algebraic graph theory, especiallythe algebraic spectral theory, has not been well developed yet for directed graphs. Asa result, there still exist significant technical challenges when establishingconvergence and robustness properties of consensus algorithms in directed networks.Allowing unidirectional and unbalanced information transmission, rather thanrequiring bidirectional and balanced information exchange, does not only impose lessstringent constraints on network topologies, but also increase robustness to communication failures, and potentially reduce the communication energy and theamount of information flow. Furthermore, in a real world scenario, the agents arespatially distributed and the digital communication channels are subject to bandwidthand energy constraints, which make the information transmitted among agents indigital networks usually quantized prior to being communicated. Therefore, thisdissertation mainly focuses on the analysis and design of distributed quantizedconsensus algorithms, and the obtained results address the distributed consensusproblem by simultaneously taking both network topology and quantized informationcommunication into consideration. Meanwhile, in practical applications, networksusually suffer from the effect of uncertain communication environments and are ofteninterfered by various kinds of noises during the information reception among agents.Hence, we also investigate the robust consensus problem for multi-agent systems withmeasurement noises under general fixed directed topologies.The main contents and contributions of this dissertation are summarized as follows:1. By adopting infinite-level static logarithmical quantization strategy, the thesisfirst studies the weighted average consensus problem with quantizedinformation communication under mild restriction on the interaction topology.By dropping the typical requirement of double stochasticity for the updatematrix in the literature, we study the quantized consensus problem for generaldirected networks with fixed topologies. An upper bound for the quantizationprecision parameter is derived to design a suitable logarithmic quantizer, underwhich the proposed quantized protocol ensures the agents converging to thedesired weighted average consensus. For directed networks, the technicaldifficulty about consensus convergence analysis is that the asymmetric updatematrix can no longer be decomposed as a clean diagonal matrix. Therefore, itis hard to make use of the algebraic spectral theory of undirected graphs tosolve the problem. Actually, the algebraic spectral theory is one of the mainanalysis tools that has been heavily relied on by most existing results. Toovercome this inherent technical difficulty, the consensus convergence analysisis mainly carried out by matrix transformation and Lyapunov stability theory,and the convergence condition is characterized by an easily testable linearmatrix inequality (LMI).2. By adopting finite-level dynamic uniform quantization strategy, we further study the weighted average consensus problem via quantized informationcommunication on general directed unbalanced networks. It is established that,only one bit quantized information transmitted along each connected digitalchannel suffices for achieving weighted average consensus with an exponentialconvergence rate by designing a protocol with a finite-level dynamic uniformquantization scheme, no matter how large the network is, as long as thedirected unbalanced network is strongly connected. Furthermore, ourconvergence analysis can be applied to balanced networks, which leads toaverage consensus. In the existing related work, which heavily reliesheavily on the algebraic spectral theory, average consensus can be achievedat an exponential convergence rate only when each pair of neighboring agentsreciprocally send one bit of quantization information to each other. Motivatedby an intricate interpretation of the left eigenvector associated with themaximal eigenvalue of the weighted update matrix of the digraph, analternative convergence analysis method is developed based on the Lyapunovstability theory. The construction of the generalized quadratic Lyapunovfunction is related to topology properties of the directed network.3. Based on edge-based adaptive finite-level dynamic uniform quantizationstrategy, we also investigate how to design the distributed quantized protocolfor consensus seeking in multi-agent systems with directed switchingtopologies. The quantization information exchange among agents may sufferfrom link failures and recoveries or packet losses. Consequently, the interactiontopology is in fact directed and dynamically switching, and thus theconservation property of state (weighted) average invariance of the network ispreserved, which makes the final consensus value hard to be specified. Withstandard assumptions on the weights rule of the adjacent matrices and networkconnectivity, we propose an effective distributed protocol with an adaptivefinite-level uniform quantized strategy, under which consensus among agentscan be guaranteed without utilizing existing symmetry error-compensationmethod. Consensus convergence analysis mainly relies on non-quadraticLyapunov function method and input-to-output stability theory, and thusavoids the typical common left eigenvector requirement for the existence ofcommon quadratic Lyapunov function. In particular, it is established that, provided the duration of link failure in the directed network is bounded, at eachtime instant each agent may be non-reciprocally sends5-level quantizationinformation to each of its direct neighbors, together with3-level quantizationinformation to itself, which suffices for consensus at an exponentialconvergence rate. Furthermore, the final consensus value still lies in the convexhull of agents’ initial values. The proposed quantized protocol has favorablemerits of requiring little communication protocol overhead and increasingmore robustness to link unreliability, and naturally it fits well into the digitalnetwork framework.4. Finally, we study the robust consensus problem for discrete-time single-integermulti-agent systems with measurement noises under general fixed directedtopologies, and a suitable distributed consensus protocol is proposed. Thetime-varying control gains satisfying the stochastic approximation conditionsare introduced to attenuate noises; therefore the closed-loop multi-agent systemis intrinsically a linear time-varying stochastic difference system. Then by theconstruction of the generalized quadratic Lyapunov function, the mean squareconsensus convergence analysis is developed based on the Lyapunov stabilitytheory. Under the proposed protocol, it is proved that the state of each agentconverges in mean square to a common random variable whose mathematicalexpectation is the weighted average of agents’ initial state values. Meanwhile,the variance of the random variable is bounded. The construction of thegeneralized quadratic Lyapunov function especially does not require the typicalbalanced network topology condition assumed for the existence of quadraticLyapunov function. Thus, the proposed consensus protocol can be applied tomore general networked multi-agent systems, particularly when bidirectionaland/or balanced information exchange between agents is not required.