Distributed Control Of Networked Lur’e Systems

Large-scale networked systems, i.e. multi-agent networks that consist of a team of single systems(agents) working together in a cooperative fashion are more powerful, more flexible and cheaper than single ones. Such systems can represent many practical ones such as formation spacecraft systems, multiple robot systems and smart grids. In order to make networked systems work in a cooperative fashion, each single system in the network should be controlled in a distributed way to achieve assigned tasks. The key of distributed control is that by only using(local) relative information between adjacent agents in a network, how to drive their collective behaviors to reach desired ones at the network level, e.g. synchronization. Considering interval nonlinearities, for example, saturation and dead zone, spacecraft dynamics can be described by Lur’e systems. Combining algebraic graph theory, absolute stability, linear matrix inequality and robust H∞/optimization control, this thesis focuses on synchronization of a class of nonlinear multi-agent networks, where each node dynamics is described by a Lur’e system. The main contents are as follows:We first consider connected undirected dynamical networks of leaderless diffusively interconnected Lur’e systems with incrementally passive nonlinearities and respectively incrementally sector bounded nonlinearities. It has been shown that the matrix multiplying its transpose, which is spanned by the eigenvectors associated with the nonzero eigenvalues of the Laplacian matrix of an arbitrary connected undirected graph, is exactly the Laplacian matrix associated with a complete graph having the same nodes as that connected undirected graph. Based on this property, sufficient synchronization conditions are obtained for static relative state feedback protocols. The knowledge of the agent dynamics and the entire interconnection topology is required in the computation of suitable protocols. The Kalman-Yakubovich-Popov lemma and dissipativity are employed to discuss the synchronization criteria we obtain. An originality is that self-synchronization of Lur’e dynamical networks is solved in this thesis for the first time. Subsequently, adaptively updated coupling gains are introduced to remove the requirement of the knowledge of the entire interconnection topology which is a kind of global information. By using the redesigned protocols, the Lur’e dynamical networks are synchronized in a fully distributed fashion.Next we study if a network of Lur’e systems on directed graphs can be synchronized.The key properties we use for undirected graphs as well as the embedding of adaptively updated coupling gains are not applicable in the case of directed graphs since they have inherent asymmetry. We investigate the concept of general algebraic connectivity for directed graphs and solve the robust synchronization problems for directed Lur’e networks.It is also focused on self-synchronization here. Therefore, the results obtained in Chapter4 generalize those in Chapter 3 partially. The fully distributed protocol design problem for directed Lur’e networks is left to future research.Because(relative) state information is hard to be derived and not always available,we next move to the dynamic relative measurement output feedback case. Since only relative output measurements are available between neighboring agents, dynamic protocols are required. Here, we do not assume the Lur’e-type nonlinearities to be known exactly and do not use them in the dynamic protocol design. By means of a H∞optimization control technique, a linear dynamic protocol is designed to fulfil the job. The Laplacian eigenvalues emerge in the protocol computation nonlinearly and cannot be removed by using the adaptively updated coupling gains that indeed work in the relative state feedback case. This problem is also left to future research.Finally, we study heterogeneous Lur’e networks. Here the output regulation theory and in particular inter model principle are employed, and we assume that there is an exosystem generating the reference signal that all the agents are required to track, but being connected to only one of the agents. By designing a fully distributed observer, a copy of the reference signal is made at each agent asymptotically. Then the output of each agent can be regulated to the common reference signal locally and thus the output synchronization is achieved. In fact, the distributed cooperative protocols form a homogeneous interconnection network in which synchronization of the estimates for the reference signal happens.In this sense, these nonidentical Lur’e systems are not interconnected directly with each other but through the homogeneous observer network. So, this approach independent of node dynamics is applicable to other heterogeneous networks.