There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We consider the problem of cooperation among a collection of vehicles performing a shared task using intervehicle communication to coordinate their actions. Tools from algebraic graph theory prove useful in modeling the communication network and relating its topology to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that decomposes formation stability into two components: Stability of this is achieved information flow for the given graph and stability of an individual vehicle for the given controller. The information flow can thus be rendered highly robust to changes in the graph, enabling tight formation control despite limitations in intervehicle communication capability.

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Information flow and cooperative control of vehicle formations

In this note, we revisit the problem of scheduling stabilizing control tasks on embedded processors. We start from the paradigm that a real-time scheduler could be regarded as a feedback controller that decides which task is executed at any given instant. This controller has for objective guaranteeing that (control unrelated) software tasks meet their deadlines and that stabilizing control tasks asymptotically stabilize the plant. We investigate a simple event-triggered scheduler based on this feedback paradigm and show how it leads to guaranteed performance thus relaxing the more traditional periodic execution requirements.

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Event-Triggered Real-Time Scheduling of Stabilizing Control Tasks

This paper describes decentralized control laws for the coordination of multiple vehicles performing spatially distributed tasks. The control laws are based on a gradient descent scheme applied to a class of decentralized utility functions that encode optimal coverage and sensing policies. These utility functions are studied in geographical optimization problems and they arise naturally in vector quantization and in sensor allocation tasks. The approach exploits the computational geometry of spatial structures such as Voronoi diagrams.

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Coverage control for mobile sensing networks

This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.

Symbolic models of control systems have recently been used to synthesize controllers enforcing specifications given by temporal logics, regular languages, or automata. These specification mechanisms can be regarded as qualitative since they divide the set of trajectories into bad trajectories (those that should be eliminated by control) and good trajectories (those that need not be eliminated). In many situations, however, a quantitative specification, where each trajectory is assigned a cost, is more appropriate. As a first step towards the synthesis of controllers enforcing qualitative and quantitative specifications we investigate in this paper the use of symbolic models for timeoptimal controller synthesis. Our results show that it is possible to obtain upper and lower bounds for the time to reach a desired target by an algorithmic analysis of the symbolic model. Moreover, we can also algorithmically synthesize a feedback controller enforcing the upper bound. All the algorithms have been implemented using Binary Decision Diagrams and are illustrated by some examples.

Approximate time-optimal control via approximate alternating simulations

Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of approximating nonlinear differential equations with linear ones is not new, we propose a new approximation scheme that captures both local as well as global properties. This is achieved via a hierarchy of approximations, where the Nth degree of the hierarchy is a linear differential equation obtained by globally approximating the Nth Lie derivatives of the trajectories. 
We show how the proposed approximation scheme has good approximating capabilities both with theoretical results and empirical observations. In particular, we show that our approximation has convergence range at least as large as a Taylor approximation while, at the same time, being able to account for asymptotic stability (a nonlocal behavior). We also compare the proposed approach with recent and classical work in the literature.

Non-local Linearization of Nonlinear Differential Equations via Polyflows

We present a method to decompose synthesis of controllers for safety specifications into smaller controller synthesis problems. The method applies to systems that we call decomposable, which means that their transition relations can be expressed as the meet of some number of transition relations over disjoint inputs. The method presented here is based on assume-guarantee reasoning and is shown to be correct and complete: a controller enforces the safety specification if and only if it can be obtained by this method.

Decomposing controller synthesis for safety specifications

This work is motivated by privacy concerns as a result of the growing rate of information exchange among components of complex cyber-physical systems, agents in a network, or actuators/sensors of a process. We propose a deterministic notion of privacy for a dynamical system, and completely characterize it for linear time-invariant dynamics. The proposed notion relies on a "plausible deniability" principle, which implies that a curious party will always be in doubt about the actual value of private variables of the system. In case privacy is guaranteed, we propose analytical metrics to assess the degree of privacy or privacy margin of the system. The size of the latter depends on the amount and structure of the information on the system which can be accessed by a curious party. We study the proposed notions and metrics for a class of distributed averaging algorithms.

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Plausible deniability as a notion of privacy

Mechanical control systems are a very important class of nonlinear control systems. They possess a rich mathematical structure which can be extremely important for the solution of various control problems. We expand the applicability of design methodologies developed for mechanical control systems by locally rendering nonlinear control systems, mechanical by a proper choice of feedback. In particular, we characterize control systems which can be transformed to Hamiltonian control systems by a local feedback transformation.

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Paulo Tabuada

Papers

Approximate time-optimal control via approximate alternating simulations

Non-local Linearization of Nonlinear Differential Equations via Polyflows

Decomposing controller synthesis for safety specifications

Plausible deniability as a notion of privacy

From nonlinear to Hamiltonian via feedback