Two Problems in Cooperative Control of Complex Multi-agent Systems

Abstract

The cooperative control of multi-agent systems naturally arises from many engineering applications such as cooperative synchronization of multiple robots, unmanned aerial vehicle formations, attitude consensus of rigid spacecraft systems, and cooperative output regulation of multi-agent systems. One of the main challenges posted by the cooperative control of multi-agent systems is that, due to the communication constraints imposed on a multiagent systems, the control of some subsystems of the multi-agent system cannot access the information of the leader system. To overcome this difficulty, a so-called distributed observer for the leader system has been developed to estimate and pass the information of the leader system to various follower subsystems. This distributed observer was first developed for solving the cooperative output regulation problem for linear multi-agent systems and then applied to deal with a variety of other cooperative control problems of multi-agent systems such as the consensus of Euler-Lagrange systems, the synchronization of rigid-body systems, and the connectivity-preserving control of multi-agent systems.

In this thesis, we will further advance the distributed observer approach to some cooperative control problems of the multi-agent systems in two directions. Firstly, we will apply the distributed observer approach to solve the cooperative output regulation problem for linear singular multi-agent systems. Singular systems are a class of dynamic systems whose state variables are governed by both differential and algebraic equations and they include normal systems as special case if the state variables are not constrained by any algebraic equations. Singular systems are natural descriptions of many physical process such as electrical networks, power systems, aerospace systems, and chemical processes. Thus it is interesting to conduct a thorough investigation on this problem. Secondly, the existing distributed observer assumes that the dynamics of the leader system is precisely known. In many practical applications, the dynamics of the leader system may contain some unknown parameters. For example, a sinusoidal signal can be generated by a two dimensional linear leader system. The parameters of the leader system are determined by the frequency of the sinusoidal signal. If the frequency of the sinusoidal signal is unknown, then the system matrix of the leader system is also unknown. To deal with this more practical scenario, we will further consider how to synthesize a distributed observer for a linear leader system whose system matrix contains unknown parameters using adaptive control technique. We call such a distributed observer the adaptive distributed observer for an uncertain leader system. Moreover, we will apply the adaptive distributed observer for an uncertain leader system to several cooperative control problems such as the consensus of Euler-Lagrange systems with an uncertain leader, and the synchronization of rigid-body systems with an uncertain leader, and the cooperative output regulation problem of multi-agent systems with an uncertain leader.

Corresponding to the two advances mentioned above, this thesis consists of two parts which are described as follows.

In Part 1, we study the cooperative output regulation problem of singular linear multiagent systems subject to jointly connected switching graphs by the distributed observer approach. The existing result on the cooperative output regulation problem of singular linear multi-agent systems were obtained under the two restrictive assumptions, namely, the communication network of the multi-agent systems is static and connected, and every follower system satisfies the so-called standard assumption. In this thesis, we will remove both of these assumptions. The main contributions to this part are summarized as follows:

1. By employing the distributed observer for a linear leader system subject to jointly connected networks, we solve the cooperative output regulation problem of singular linear multi-agent subject to jointly connected switching networks. This result is interesting because the jointly connected assumption is the mildest assumption on the communication network and it allows the network to be disconnected at every time instant.

2. By introducing a static output feedback control, we remove the unnecessary standard assumption on the singular system in the existing result. We accomplish our objective by first deriving a reduced-order normal multi-agent systems and obtaining a distributed output measurement feedback control law based on the existing result on the normal multi-agent systems. Then, we show that the composition of the static output feedback control law and the distributed output measurement feedback control law solves the cooperative output regulation problem for the original singular multi-agent systems. This result will significantly enlarge the class of systems whose cooperative output regulation problem is solvable.

In Part 2, we turn to establishing the adaptive distributed observer for an uncertain leader system under various scenarios and applying it to several cooperative control problems. The main contributions to this part are summarized as follows:

1. We first establish an adaptive distributed observer for a linear neutrally stable uncertain leader system under the assumption that the communication network of the follower system is static, connected, and undirected. Under the additional assumption that the leader’s state is persistently exciting, this adaptive distributed observer can also asymptotically learn the unknown parameters of the leader’s system matrix. On the basis of this adaptive distributed observer, we further synthesize an adaptive distributed control law to solve the leader-following consensus problem of multiple Euler-Lagrange systems with an uncertain leader system.

2. In order to further apply the adaptive distributed observer to solve the cooperative output regulation problem for linear heterogeneous multi-agent systems subject to an uncertain leader system, we further establish that the adaptive distributed observer is capable of estimating the unknown parameter vector of the leader system exponentially fast as long as the leader’s signal is persistently exciting. This strengthened result leads to an iterative solution to the cooperative output regulation problem of linear heterogeneous multi-agent systems subject to an uncertain leader system.

3. The assumption that the communication network of the follower system is undirected may not be desirable in some applications. For this reason, we further establish a set of sufficient conditions for guaranteeing the validity of the adaptive distributed observer for an uncertain leader subject to a directed connected graph. These sufficient conditions apply to a few interesting scenarios such as the case where the graph contains no loop, and the case where the graph is balanced. We also apply this result to the leader-following attitude consensus problem of multiple rigid body systems subject to an uncertain leader system.