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.

In this paper, we present a novel solution to the minimum attention control problem. In minimum attention control, the objective is to minimise the ‘attention’ that a control task requires, given certain performance requirements. Here, we interpret ‘attention’ as the inverse of the time elapsed between two consecutive executions of a control task. Instrumental for the solution will be a novel extension of the notion of a control Lyapunov function. By focussing on linear plants, by allowing for only a finite number of possible intervals between two subsequent executions of the control task and by taking the extended control Lyapunov function to be ∞-norm based, we can formulate the minimum attention control problem as a linear program, which can be solved efficiently online. Furthermore, we provide a technique to construct suitable ∞-norm-based (extended) control Lyapunov functions for our purposes. Finally, we illustrate the theory using a numerical example, showing that minimum attention control can outperform an alternative implementation-aware control law available in the literature.

On the minimum attention control problem for linear systems: A linear programming approach

In this paper we revisit the problem of computing controlled invariant sets for controllable discrete-time linear systems. We propose a novel algorithm that does not rely on iterative computations. Instead, controlled invariant sets are computed in two moves: 1) we lift the problem to a higher dimensional space where a controlled invariant set is computed in closed-form; 2) we project the resulting set back to the original domain to obtain the desired controlled invariant set. One of the advantages of the proposed method is the ability to handle larger systems.

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Computing controlled invariant sets in two moves

It is well known that event-triggered and self-triggered controllers implemented on dedicated platforms can provide the same performance as the traditional periodic controllers, while consuming considerably less bandwidth. However, since the majority of controllers are implemented by software tasks on shared platforms, on one hand, it might no longer be possible to grant access to the event-triggered controller upon request. On the other hand, due to the seemingly irregular requests from self-triggered controllers, other applications, while in reality schedulable, may be declared unschedulable, if not carefully analyzed. The schedulability and response-time analysis in the presence of self-triggered controllers is still an open problem and the topic of this paper.

Self-triggered controllers and hard real-time guarantees

Networked Embedded Sensing and Control

In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a general sufficient condition for a residual network to have the power of universal approximation by asking the activation function, or one of its derivatives, to satisfy a quadratic differential equation. Many activation functions used in practice satisfy this assumption, exactly or approximately, and we show this property to be sufficient for an adequately deep neural network with $n+1$ neurons per layer to approximate arbitrarily well, on a compact set and with respect to the supremum norm, any continuous function from $\mathbb{R}^n$ to $\mathbb{R}^n$. We further show this result to hold for very simple architectures for which the weights only need to assume two values. The first key technical contribution consists of relating the universal approximation problem to controllability of an ensemble of control systems corresponding to a residual network and to leverage classical Lie algebraic techniques to characterize controllability. The second technical contribution is to identify monotonicity as the bridge between controllability of finite ensembles and uniform approximability on compact sets.

Paulo Tabuada

Papers

On the minimum attention control problem for linear systems: A linear programming approach

Computing controlled invariant sets in two moves

Self-triggered controllers and hard real-time guarantees

Networked Embedded Sensing and Control

Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theory