Tyler H. Summers

Coordinated Standoff Tracking of Moving Targets: Control Laws and Information Architectures

In this paper, we present work on control of autonomous vehicle formations in the context of the coordinated stando tracking problem . The objective is to use a team of unmanned aircraft to fly a circular orbit around a moving target with prescribed inter-vehicle angular spacing using only local information. We use the recently introduced Lyapunov guidance vector field approach to achieve the desired circular trajectory. The contributions of this paper involve both single vehicle path planning and multiple vehicle coordination. For single vehicle path planning, we complete a proof of heading convergence using feedback, which has thus far not been fully addressed in the literature, and also oer a novel approach for heading convergence that does not require continuous feedback in the ideal case (no wind, stationary target), taking advantage of an analytical solution to the guidance field. Further, we use a variable airspeed controller to maintain the circular trajectory despite unknown wind and unknown constant velocity target motion. Adaptive estimates of the unknown wind and target motion are introduced to ensure stability to the circular trajectory. A novel feature of our results is rigorous satisfaction of vehicle specific kinematic constraints on heading rates and airspeed variations. For multiple vehicle coordination, we again use a variable airspeed controller to achieve the prescribed angular spacing. In an eort towards a unified framework for control of autonomous vehicle formations, we make a connection with some recent work that addresses information architecture in vehicle formations using graph theory. Specifically, we utilize two types of information architectures, symmetric and asymmetric, and implement decentralized control laws. The information architectures are scalable in the sense that the number of required communication/sensing links increases linearly with the number of vehicles. The control laws are decentralized in the sense that they use only local information.

On Submodularity and Controllability in Complex Dynamical Networks

Controllability and observability have long been recognized as fundamental structural properties of dynamical systems, but have recently seen renewed interest in the context of large, complex networks of dynamical systems. A basic problem is sensor and actuator placement: choose a subset from a finite set of possible placements to optimize some real-valued controllability and observability metrics of the network. Surprisingly little is known about the structure of such combinatorial optimization problems. In this paper, we show that several important classes of metrics based on the controllability and observability Gramians have a strong structural property that allows for either efficient global optimization or an approximation guarantee by using a simple greedy heuristic for their maximization. In particular, the mapping from possible placements to several scalar functions of the associated Gramian is either a modular or submodular set function. The results are illustrated on randomly generated systems and on a problem of power-electronic actuator placement in a model of the European power grid.

Control of Minimally Persistent Leader-Remote-Follower and Coleader Formations in the Plane

This paper solves an n -agent formation shape control problem in the plane. The objective is to design decentralized control laws so that the agents cooperatively restore a prescribed formation shape in the presence of small perturbations from the prescribed shape. We consider two classes of directed, cyclic information architectures associated with so-called minimally persistent formations: leader-remote-follower and coleader. In our framework the formation shape is maintained by controlling certain interagent distances. Only one agent is responsible for maintaining each distance. We propose a decentralized control law where each agent executes its control using only the relative position measurements of agents to which it must maintain its distance. The resulting nonlinear closed-loop system has a manifold of equilibria, which implies that the linearized system is nonhyperbolic. We apply center manifold theory to show local exponential stability of the desired formation shape. The result circumvents the non-compactness of the equilibrium manifold. Choosing stabilizing gains is possible if a certain submatrix of the rigidity matrix has all leading principal minors nonzero, and we show that this condition holds for all minimally persistent leader-remote-follower and coleader formations with generic agent positions. Simulations are provided.

Optimal Sensor and Actuator Placement in Complex Dynamical Networks

Abstract Controllability and observability have long been recognized as fundamental structural properties of dynamical systems, but have recently seen renewed interest in the context of large, complex networks of dynamical systems. A basic problem is sensor and actuator placement: choose a subset from a finite set of possible placements to optimize some real-valued controllability and observability metrics of the network. Surprisingly little is known about the structure of such combinatorial optimization problems. In this paper, we show that an important class of metrics based on the controllability and observability Gramians has a strong structural property that allows efficient global optimization: the mapping from possible placements to the trace of the associated Gramian is a modular set function. We illustrate the results via placement of power electronic actuators in a model of the European power grid.

Controlling Four Agent Formations

Abstract This paper considers formation shape control of a team of four point agents, for the most part in the plane. Control laws based on specified interagent distances are used. For a complete graph, specification of all interagent distances determines the formation shape uniquely. Krick, Broucke and Francis showed that for a now almost standard control law, there may exist equilibrium formation shapes with incorrect interagent distances. This paper studies such equilibria, shows that in some cases they are necessarily unstable.

Distributed model predictive consensus via the Alternating Direction Method of Multipliers

We propose a distributed optimization method for solving a distributed model predictive consensus problem. The goal is to design a distributed controller for a network of dynamical systems to optimize a coupled objective function while respecting state and input constraints. The distributed optimization method is an augmented Lagrangian method called the Alternating Direction Method of Multipliers (ADMM), which was introduced in the 1970s but has seen a recent resurgence in the context of dramatic increases in computing power and the development of widely available distributed computing platforms. The method is applied to position and velocity consensus in a network of double integrators. We find that a few tens of ADMM iterations yield closed-loop performance near what is achieved by solving the optimization problem centrally. Furthermore, the use of recent code generation techniques for solving local subproblems yields fast overall computation times.

Submodularity of energy related controllability metrics

The quantification of controllability and observability has recently received new interest in the context of large, complex networks of dynamical systems. A fundamental but computationally difficult problem is the placement or selection of actuators and sensors that optimize real-valued controllability and observability metrics of the network. We show that several classes of energy related metrics associated with the controllability Gramian in linear dynamical systems have a strong structural property, called submodularity, which allows for an approximation guarantee by using a simple greedy heuristic for their maximization. The results are illustrated for randomly generated systems and placement of power electronic actuators in a model of the European power grid.

Topology design for optimal network coherence

We consider a network topology design problem in which an initial undirected graph underlying the network is given and the objective is to select a set of edges to add to the graph to optimize the coherence of the resulting network. We show that network coherence is a submodular function of the network topology. As a consequence, a simple greedy algorithm is guaranteed to produce near optimal edge set selections. We also show that fast rank one updates of the Laplacian pseudoinverse using generalizations of the Sherman-Morrison formula and an accelerated variant of the greedy algorithm can speed up the algorithm by several orders of magnitude in practice. These allow our algorithms to scale to network sizes far beyond those that can be handled by convex relaxation heuristics.

Computational aspects of distributed optimization in model predictive control

This paper presents a systematic computational study on the performance of distributed optimization in model predictive control (MPC). We consider networks of dynamically coupled systems, which are subject to input and state constraints. The resulting MPC problem is structured according to the systems dynamics, which makes the problem suitable for distributed optimization. The influence of fundamental aspects of distributed dynamic systems on the performance of two particular distributed optimization methods is systematically analyzed. The methods considered are dual decomposition based on fast gradient updates (DDFG) and the alternating direction method of multipliers (ADMM), while the aspects analyzed are coupling strength, stability, initial state, coupling topology and network size. The methods are found to be sensitive to coupling strength and stability, but relatively insensitive to initial state and topology. Moreover, they scale well with the number of subsystems in the network.

Approximate dynamic programming via sum of squares programming

We describe an approximate dynamic programming method for stochastic control problems on infinite state and input spaces. The optimal value function is approximated by a linear combination of basis functions with coefficients as decision variables. By relaxing the Bellman equation to an inequality, one obtains a linear program in the basis coefficients with an infinite set of constraints. We show that a recently introduced method, which obtains convex quadratic value function approximations, can be extended to higher order polynomial approximations via sum of squares programming techniques. An approximate value function can then be computed offline by solving a semidefinite program, without having to sample the infinite constraint. The policy is evaluated online by solving a polynomial optimization problem, which also turns out to be convex in some cases. We experimentally validate the method on an autonomous helicopter testbed using a 10-dimensional helicopter model.