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.

/pdf/coverage-control-for-mobile-sensing-networks-3gs75qafnm.pdf

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.

Computing controlled invariant sets for hybrid systems with applications to model-predictive control

Feedback control laws have been traditionally treated as periodic tasks when implemented on digital platforms. However, the growing complexity of systems calls for efficient implementations of control tasks that reduce resource utilization while keeping desired levels of performance. In this paper we drop the periodicity assumption in favour of self-triggered strategies for the execution of control laws. Such strategies determine the next execution time based on the current state of the plant. Under the self-triggered policy, the inter-execution times scale in a predictable manner: a scaling of the state of the plant entails a scaling in the inter-execution times. This property allows us to derive a simple formula for the next execution time guaranteeing performance.We illustrate the proposed techniques on the control of a jet engine compressor.

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Space-time scaling laws for self-triggered control

This paper is concerned with a detailed comparison of two different abstraction techniques for the construction of finite state symbolic models for controller synthesis of hybrid systems. Namely, we compare quotient based abstractions (QBA) with different realizations of strongest (asynchronous) l-complete approximations (SAlCA). Even though the idea behind their construction is very similar, we show that they are generally incomparable both in terms of behavioral inclusion and similarity relations. We therefore derive necessary and sufficient conditions for QBA to coincide with particular realizations of SAlCA. Depending on the original system, either QBA or SAlCA can be a tighter abstraction.

/pdf/comparing-asynchronous-l-complete-approximations-and-45o27z4k1x.pdf

Comparing asynchronous l-complete approximations and quotient based abstractions

Control Lyapunov functions (CLFs) and control barrier functions (CBFs) have been used to develop provably safe controllers by means of quadratic programs (QPs), guaranteeing safety in the form of trajectory invariance with respect to a given set. In this manuscript, we show that this framework can introduce equilibrium points (particularly at the boundary of the unsafe set) other than the minimum of the Lyapunov function into the closed-loop system. We derive explicit conditions under which these undesired equilibria (which can even appear in the simple case of linear systems with just one convex unsafe set) are asymptotically stable. To address this issue, we propose an extension to the QP-based controller unifying CLFs and CBFs that explicitly avoids undesirable equilibria on the boundary of the safe set. The solution is illustrated in the design of a collision-free controller.

Control Barrier Function based Quadratic Programs Introduce Undesirable Asymptotically Stable Equilibria

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. 11A full version of this “work, containing all the proofs, can be found in [1]

Paulo Tabuada

Papers

Computing controlled invariant sets for hybrid systems with applications to model-predictive control

Space-time scaling laws for self-triggered control

Comparing asynchronous l-complete approximations and quotient based abstractions

Control Barrier Function based Quadratic Programs Introduce Undesirable Asymptotically Stable Equilibria

Non-local Linearization of Nonlinear Differential Equations via Polyflows