Robustly Complete Temporal Logic Control Synthesis for Nonlinear Systems

Modern systems such as spacecrafts and autonomous vehicles are complex yet safety-critical, and therefore the control methods that can deal with different dynamics and constraints while being provably correct are sought after. Formal methods are rigorous techniques originally used for developing and verifying finite-state systems with respect to specifications in formal languages. This thesis is concerned with using formal methods in control synthesis for nonlinear systems, which can guarantee the correctness of the resulting control strategies. For nonlinear continuous-state dynamical systems, formal control synthesis relies on finite abstractions of the original system by discretizing the system state space and over approximating system transitions. Without further assumptions, control synthesis is usually not complete in the way that no control strategies can be found even if there exists one. To deal with this problem, this thesis proposes a formal control synthesis approach that is sound and robustly complete in the sense that correct control strategies can be found whenever the specifications can be realized for the system with additional disturbance. Fundamental to the soundness and robust completeness is a fixed-point characterization of the winning set of the system with respect to a given specification, which is the set of initial conditions that can be controlled to satisfy the specification. Regarding discrete-time systems, such characterizations are first presented by using iterative computation of predecessors for basic linear temporal logic (LTL) specifications, including invariance, reachability and reach-and-stay. A more general class of LTL formulas, which can be translated into deterministic B\"uchi automata (DBA), is also considered, and an algorithm guided by the graph structure of the LTL-equivalent DBA is proposed for characterizing the winning set in this situation. It is then shown that the computational complexity of the algorithm can be reduced by using a pre-processing procedure to the graphs of the DBA. Because of the general nonlinearity, exact computation of winning sets is currently almost impossible. In this work, the conditions for set approximations are derived so that control synthesis is robustly complete. To meet such conditions, the proposed approach adopts interval arithmetic and a subdivision scheme in the approximation of predecessors. Under such a scheme, the system state space is adaptively partitioned with respect to both the given dynamics and specification and set approximation can be made arbitrarily precise to satisfy the robust completeness conditions. The proposed method is also shown applicable to sampled-data systems by computing validated solutions over one sampling period based on high-order Taylor expansion. Applications such as converter voltage regulation, parallel parking, and reactive locomotion planning problems are studied to show the effectiveness and efficiency of the proposed approach

Robust hybrid control for autonomous vehicle motion planning

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2001.Includes bibliographical references (p. 141-150).This dissertation focuses on the problem of motion planning for agile autonomous vehicles. In realistic situations, the motion planning problem must be solved in real-time, in a dynamic and uncertain environment. The fulfillment of the mission objectives might also require the exploitation of the full maneuvering capabilities of the vehicle. The main contribution of the dissertation is the development of a new computational and modelling framework (the Maneuver Automaton), and related algorithms, for steering underactuated, nonholonomic mechanical systems. The proposed approach is based on a quantization of the system's dynamics, by which the feasible nominal system trajectories are restricted to the family of curves that can be obtained by the interconnection of suitably defined primitives. This can be seen as a formalization of the concept of "maneuver", allowing for the construction of a framework amenable to mathematical programming. This motion planning framework is applicable to all time-invariant dynamical systems which admit dynamic symmetries and relative equilibria. No other assumptions are made on the dynamics, thus resulting in exact motion planning techniques of general applicability. Building on a relatively expensive off-line computation phase, we provide algorithms viable for real-time applications. A fundamental advantage of this approach is the ability to provide a mathematical foundation for generating a provably stable and consistent hierarchical system, and for developing the tools to analyze the robustness of the system in the presence of uncertainty and/or disturbances.(cont.) In the second part of the dissertation, a randomized algorithm is proposed for real-time motion planning in a dynamic environment. By employing the optimal control solution in a free space developed for the maneuver automaton (or for any other general system), we present a motion planning algorithm with probabilistic convergence and performance guarantees, and hard safety guarantees, even in the face of finite computation times. The proposed methodologies are applicable to a very large class of autonomous vehicles: throughout the dissertation, examples, simulation and experimental results are presented and discussed, involving a variety of mechanical systems, ranging from simple academic examples and laboratory setups, to detailed models of small autonomous helicopters.by Emilio Frazzoli.Ph.D

Control Theory: A Mathematical Perspective on Cyber-Physical Systems

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently the control field is facing new challenges motivated by application domains that involve networks of systems. Examples are interacting robots, networks of autonomous cars or the smart grid. In order to address the new challenges posed by these application disciplines, the special focus of this workshop has been on the currently very active field of Cyber-Physical Systems, which forms the underlying basis for many network control applications. A series of lectures in this workshop was devoted to give an overview on current theoretical developments in Cyber-Physical Systems, emphasizing in particular the mathematical aspects of the field. Special focus was on the dynamics and control of networks of systems, distributed optimization and formation control, fundamentals of nonlinear interconnected systems, as well as open problems in control