Reza Olfati-Saber

Consensus problems in networks of agents with switching topology and time-delays

In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.

Distributed cooperative control of multiple vehicle formations using structural potential functions

In this paper, we propose a framework for formation stabilization of multiple autonomous vehicles in a distributed fashion. Each vehicle is assumed to have simple dynamics, i.e. a double-integrator, with a directed (or an undirected) information flow over the formation graph of the vehicles. Our goal is to find a distributed control law (with an efficient computational cost) for each vehicle that makes use of limited information regarding the state of other vehicles. Here, the key idea in formation stabilization is the use of natural potential functions obtained from structural constraints of a desired formation in a way that leads to a collision-free, distributed, and bounded state feedback law for each vehicle.

Graph rigidity and distributed formation stabilization of multi-vehicle systems

We provide a graph theoretical framework that allows us to formally define formations of multiple vehicles and the issues arising in uniqueness of graph realizations and its connection to stability of formations. The notion of graph rigidity is crucial in identifying the shape variables of a formation and an appropriate potential function associated with the formation. This allows formulation of meaningful optimization or nonlinear control problems for formation stabilization/tacking, in addition to formal representation of split, rejoin, and reconfiguration maneuvers for multi-vehicle formations. We introduce an algebra that consists of performing some basic operations on graphs which allow creation of larger rigid-by-construction graphs by combining smaller rigid subgraphs. This is particularly useful in performing and representing rejoin/split maneuvers of multiple formations in a distributed fashion.

Ultrafast consensus in small-world networks

In this paper, we demonstrate a phase transition phenomenon in algebraic connectivity of small-world networks. Algebraic connectivity of a graph is the second smallest eigenvalue of its Laplacian matrix and a measure of speed of solving consensus problems in networks. We demonstrate that it is possible to dramatically increase the algebraic connectivity of a regular complex network by 1000 times or more without adding new links or nodes to the network. This implies that a consensus problem can be solved incredibly fast on certain small-world networks giving rise to a network design algorithm for ultra fast information networks. Our study relies on a procedure called random rewiring due to Watts & Strogatz (Nature, 1998). Extensive numerical results are provided to support our claims and conjectures. We prove that the mean of the bulk Laplacian spectrum of a complex network remains invariant under random rewiring. The same property only asymptotically holds for scale-free networks. A relationship between increasing the algebraic connectivity of complex networks and robustness to link and node failures is also shown. This is an alternative approach to the use of percolation theory for analysis of network robustness. We also show some connections between our conjectures and certain open problems in the theory of random matrices.

Approximate distributed Kalman filtering in sensor networks with quantifiable performance

We analyze the performance of an approximate distributed Kalman filter proposed in recent work on distributed coordination. This approach to distributed estimation is novel in that it admits a systematic analysis of its performance as various network quantities such as connection density, topology, and bandwidth are varied. Our main contribution is a frequency-domain characterization of the distributed estimators steady-state performance; this is quantified in terms of a special matrix associated with the connection topology called the graph Laplacian, and also the rate of message exchange between immediate neighbors in the communication network.

Collision avoidance for multiple agent systems

Techniques using gyroscopic forces and scalar potentials are used to create swarming behaviors for multiple agent systems. The methods result in collision avoidance between the agents as well as with obstacles.

Global configuration stabilization for the VTOL aircraft with strong input coupling

Trajectory tracking and configuration stabilization for the vertical takeoff and landing (VTOL) aircraft has been so far considered in the literature only in the presence of a slight (or zero) input coupling (i.e., for a small /spl epsi/). In this paper, our main contribution is to address global configuration stabilization for the VTOL aircraft with a strong input coupling using a smooth static state feedback. In addition, the differentially flat outputs for the VTOL aircraft are automatically obtained as a by-product of applying a decoupling change of coordinates.

Normal forms for underactuated mechanical systems with symmetry

We introduce cascade normal forms for underactuated mechanical systems that are convenient for control design. These normal forms include three classes of cascade systems, namely, nonlinear systems in strict feedback form, feedforward form, and nontriangular quadratic form (to be defined). In each case, the transformation to cascade systems is provided in closed-form. We apply our results to the Acrobot, the rotating pendulum, and the cart-pole system.

Distributed structural stabilization and tracking for formations of dynamic multi-agents

We provide a theoretical framework that consists of graph theoretical and Lyapunov-based approaches to stability analysis and distributed control of multi-agent formations. This framework relays on the notion of graph rigidity as a means of identifying the shape variables of a formation. Using this approach, we can formally define formations of multiple vehicles and three types of stabilization/tracking problems for dynamic multi-agent systems. We show how these three problems can be addressed mutually independent of each other for a formation of two agents. Then, we introduce a procedure called dynamic node augmentation that allows construction of a larger formation with more agents that can be rendered structurally stable in a distributed manner from some initial formation that is structurally stable. We provide two examples of formations that can be controlled using this approach, namely, the V-formation and the diamond formation.

Global stabilization of a flat underactuated system: the inertia wheel pendulum

Inertial Wheel Pendulum (IWP) is a planar pendulum with a revolving wheel (that has a uniform mass distribution) at the end. The pendulum is unactuated and the wheel is actuated. Our main result is to address global asymptotic stabilization of the inertia wheel pendulum around its up-right position. Simulation results are provided for parameters taken from a real-life model of the IWP.