We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the control of vehicular platoons, flow control, microelectromechanical systems (MEMS), smart structures, and systems described by partial differential equations with constant coefficients and distributed controls and measurements. For fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator (LQR), 2 and , optimal controllers can be obtained by solving a parameterized family of standard finite-dimensional problems. We show that optimal controllers have an inherent degree of decentralization, and this provides a practical distributed controller architecture. We also prove a general result that applies to partially distributed control and a variety of performance criteria, stating that optimal controllers inherit the spatial invariance structure of the plant. Connections of this work to that on systems over rings, and systems with dynamical symmetries are discussed.

We study the (perfect Bayesian) equilibrium of a model of learning over a general social network. Each individual receives a signal about the underlying state of the world, observes the past actions of a stochastically-generated neighborhood of individuals, and chooses one of two possible actions. The stochastic process generating the neighborhoods defines the network topology (social network). The special case where each individual observes all past actions has been widely studied in the literature. We characterize pure-strategy equilibria for arbitrary stochastic and deterministic social networks and characterize the conditions under which there will be asymptotic learning-that is, the conditions under which, as the social network becomes large, individuals converge (in probability) to taking the right action. We show that when private beliefs are unbounded (meaning that the implied likelihood ratios are unbounded), there will be asymptotic learning as long as there is some minimal amount of "expansion in observations". Our main theorem shows that when the probability that each individual observes some other individual from the recent past converges to one as the social network becomes large, unbounded private beliefs are sufficient to ensure asymptotic learning. This theorem therefore establishes that, with unbounded private beliefs, there will be asymptotic learning in almost all reasonable social networks. We also show that for most network topologies, when private beliefs are bounded, there will not be asymptotic learning. In addition, in contrast to the special case where all past actions are observed, asymptotic learning is possible even with bounded beliefs in certain stochastic network topologies.Keywords: information aggregation, learning, social networks, herding, information cascades.JEL Classification: C72, D83. * We thank Lones Smith and Peter Sorensen for useful comments and suggestions. We gratefully acknowledge financial support from the AFOSR and the NSF.

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The operation of an autonomous vehicle in an unknown, dynamic environment i s a v ery complex problem, especially when the vehicle is required to use its full maneuvering capabilities, and to react in real time to changes in the operational environment.A new class of algorithms, based on the construction of probabilistic roadmaps, has been recently introduced, and proven to provide a very fast and e cient s c heme for motion planning for robots with many degrees of freedom, while maintaining completeness guarantees in a probabilistic sense. In this paper we will present an extension of the probabilistic roadmap approach, which is able to deal e ectively with the system dynamics, in an environment c haracterized by m o ving obstacles. This is accomplished through a L y apunov function based approach to the construction of the roadmap.The proposed algorithm can be directly applied to a very general class of dynamical systems, including traditional state space systems, as well as hybrid systems systems including both discrete and continuous dynamics. Simulation examples, involving a small autonomous helicopter, will be presented and discussed.

The paper proposes a framework for modeling and analysis of the dynamics of supply, demand, and clearing prices in power system with real-time retail pricing and information asymmetry. Real-time retail pricing is characterized by passing on the real-time wholesale electricity prices to the end consumers, and is shown to create a closed-loop feedback system between the physical layer and the market layer of the power system. In the absence of a carefully designed control law, such direct feedback between the two layers could increase volatility and lower the system's robustness to uncertainty in demand and generation. A new notion of generalized price-elasticity is introduced, and it is shown that price volatility can be characterized in terms of the system's maximal relative price elasticity, defined as the maximal ratio of the generalized price-elasticity of consumers to that of the producers. As this ratio increases, the system becomes more volatile, and eventually, unstable. As new demand response technologies and distributed storage increase the price-elasticity of demand, the architecture under examination is likely to lead to increased volatility and possibly instability. This highlights the need for assessing architecture systematically and in advance, in order to optimally strike the trade-offs between volatility, economic efficiency, and system reliability.

The operation of an autonomous vehicle in an unknown, dynamic environment is a very complex problem, especially when the vehicle is required to use its full maneuvering capabilities, and to react in real time to changes in they operational environment. A possible approach to reduce the computational complexity of the motion planning problem for a nonlinear, high dimensional system, is based on a quantization of the system dynamics, leading to a control architecture based on a hybrid automaton, the states of which represent feasible trajectory primitives for the vehicle. This paper focuses on the feasibility of this approach, in the presence of disturbances and uncertainties in the plant and/or in the environment: the structure of a Robust Hybrid Automaton is defined and its properties are analyzed. In particular, we address the issues of well-posedness, consistency and reachability. For the case of autonomous vehicles, we provide sufficient conditions to guarantee reachability of the automaton.

We study the (perfect Bayesian) equilibrium of a model of learning over a general social network. Each individual receives a signal about the underlying state of the world, observes the past actions of a stochastically-generated neighborhood of individuals, and chooses one of two possible actions. The stochastic process generating the neighborhoods defines the network topology (social network). The special case where each individual observes all past actions has been widely studied in the literature. We characterize pure-strategy equilibria for arbitrary stochastic and deterministic social networks and characterize the conditions under which there will be asymptotic learning-that is, the conditions under which, as the social network becomes large, individuals converge (in probability) to taking the right action. We show that when private beliefs are unbounded (meaning that the implied likelihood ratios are unbounded), there will be asymptotic learning as long as there is some minimal amount of "expansion in observations". Our main theorem shows that when the probability that each individual observes some other individual from the recent past converges to one as the social network becomes large, unbounded private beliefs are sufficient to ensure asymptotic learning. This theorem therefore establishes that, with unbounded private beliefs, there will be asymptotic learning in almost all reasonable social networks. We also show that for most network topologies, when private beliefs are bounded, there will not be asymptotic learning. In addition, in contrast to the special case where all past actions are observed, asymptotic learning is possible even with bounded beliefs in certain stochastic network topologies.Keywords: information aggregation, learning, social networks, herding, information cascades.JEL Classification: C72, D83. * We thank Lones Smith and Peter Sorensen for useful comments and suggestions. We gratefully acknowledge financial support from the AFOSR and the NSF.

In this paper we present a tracking controller for a class of underactuated mechanical systems, based on a backstepping procedure. This class includes an approximation of small helicopter dynamics. The need to avoid artificial singularities due to the attitude representation is the main driver behind the control design presented in this paper: t o achieve this goal, we will operate directly in the configuration manifold of the vehicle. The control design provides asymptotic tracking for an approximate model of small helicopters, and bounded tracking when more complete models are considered. Simulation examples, including both point stabilization and aggressive maneuver tracking, are presented and discussed. IntroductionIn the recent past, the design and implementation of control algorithms for autonomous helicopters has been the object of a relevant body of research, due to an identified need for maneuverable autonomous aerial vehicles, for both military and civil applications. While slower and less fuel-efficient than airplanes, helicopters are capable of vertical take off and landing, hover, and in general are more maneuverable in tight spaces than airplanes. As a consequence, helicopters are one of the best platforms for operations in urban or otherwise cluttered environments. However, in many respects the dynamics of a helicopter are more complicated than the dynamics of a fixed wing aircraft: a helicopter is inherently unstable at hover, and the flight characteristics change dramatically over the entire flight envelope.