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.

/pdf/information-flow-and-cooperative-control-of-vehicle-4ws61wioze.pdf

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.

/pdf/event-triggered-real-time-scheduling-of-stabilizing-control-t1gebqsndp.pdf

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.

Can we conclude the stability of an unknown dynamical system from the knowledge of a finite number of snapshots of trajectories? We tackle this black-box problem for switched linear systems. We show that, for any given random set of observations, one can give probabilistic stability guarantees. The probabilistic nature of these guarantees implies a trade-off between their quality and the desired level of confidence. We provide an explicit way of computing the best stability-like guarantee, as a function of both the number of observations and the required level of confidence. Our proof techniques rely on geometrical analysis, chance-constrained optimization, and stability analysis tools for switched systems, including the joint spectral radius.

Data Driven Stability Analysis of Black-box Switched Linear Systems

In this paper we propose a methodology for stabilizing single-input single-output feedback linearizable systems when no system model is known and no prior data is available to identify a model. Conceptually, we have been greatly inspired by the work of Fliess and Join on intelligent PID controllers and the results in this paper provide sufficient conditions under which a modified version of their approach is guaranteed to result in asymptotically stable behavior. One of the key advantages of the proposed results is that, contrary to other approaches to controlling systems without a model (or with a partial model), such as reinforcement learning, there is no need for extensive training nor large amounts of data. Technically, our results draw heavily from the work of Nesic and co-workers on observer and controller design based on approximate models. Along the way we also make connections with other well established results such as high-gain observers and adaptive control. Although we focus on the simple setting of single-input single-output feedback linearizable systems we believe the presented results are already theoretically insightful and practically useful, the last point being substantiated by experimental evidence.

Data-driven Stabilization of SISO Feedback Linearizable Systems.

Control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs. Although these models are very powerful when dealing with physical phenomena, they are less suitable to describe software and hardware interfacing the physical world. This has spurred a recent interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics, where each "symbol" corresponds to an "aggregate" of continuous states in the continuous model. Since these symbolic models are of the same nature of the models used in computer science to describe software and hardware, they provided a unified language to study problems of control in which software and hardware interact with the physical world. In this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent (bisimilar) to symbolic model with a precision that can be chosen a-priori. We also show that for digital controlled systems, in which inputs are piecewise-constant, and under the stronger assumption of incremental input-to-state stability, the symbolic models can be obtained, based on a suitable quantization of the inputs.

https://www-ljk.imag.fr/membres/Antoine.Girard/Publications/cdc2007b.pdf

Symbolic models for nonlinear control systems using approximate bisimulation

Mechanical control systems are an especially interesting and important class of nonlinear control systems. They possess a rich mathematical structure and yet, physical considerations are extremely important for the solution of a large class of control problems. We broaden the applicability of design methodologies developed for mechanical control systems by rendering nonlinear control systems, mechanical by a proper choice of feedback. In particular, we characterize which control systems can be transformed to Hamiltonian control systems by a feedback transformation.

/pdf/from-nonlinear-to-hamiltonian-via-feedback-qxxi0g8oge.pdf

From nonlinear to Hamiltonian via feedback

Control synthesis is slowly transcending its traditional application domain within engineering to find interesting and useful applications in computer science. Synthesis of interfaces, distributed network monitors or reactive programs are some examples that benefit from this design paradigm. In this paper we shed new light on the interplay between the fundamental notion of bisimulation and the control synthesis problem. We first revisit the notion of alternating simulation introduced by Alur and co-workers as it naturally captures important ingredients of the control synthesis problem. We then show that existence of controllers enforcing specifications through bisimulation, alternating simulation or simulation can be characterized by the existence of certain alternating simulations and bisimulations between the specification and the system to be controlled. These results highlight and unify the role of simulations and bisimulations in the control synthesis setting for a wide range of concurrency models. This is achieved by developing our study within the framework of open maps. We illustrate our results on transition systems and timed transition systems.

/pdf/open-maps-alternating-simulations-and-control-synthesis-4ud5g0nkmf.pdf

Paulo Tabuada

Papers

Data Driven Stability Analysis of Black-box Switched Linear Systems

Data-driven Stabilization of SISO Feedback Linearizable Systems.

Symbolic models for nonlinear control systems using approximate bisimulation

From nonlinear to Hamiltonian via feedback

Open Maps, Alternating Simulations and Control Synthesis