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Ellipsoidal Calculus for Estimation and Control
01 Sep 1996-
TL;DR: In this article, the authors give an account of an ellipsoidal calculus and ellipssoidal techniques that allow presentation of the set-valued solutions to these problems in terms of approximating ellipsseidal-valued functions.
Abstract: This text gives an account of an ellipsoidal calculus and ellipsoidal techniques that allows presentation of the set-valued solutions to these problems in terms of approximating ellipsoidal-valued functions. Such an approach leads to effective computation schemes, an dopens the way to applications and implementations with computer animation, particularly in decision support systems. The problems treated here are those that involve calculation of attainability domains, of control synthesis under bounded controls, state constraints and unknown input disturbances, as well as those of "viability" and of the "bounding approach" to state estimation. The text ranges from a specially developed theory of exact set-valued solutions to the description of ellipsoidal calculus, related ellipsoidal-based methods and examples worked out with computer graphics. the calculus given here may also be interpreted as a generalized technique of the "interval analysis" type with an impact on scientific computation.
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TL;DR: An extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response, via an ellipsoid that gives the possible values of the array for a particular look direction is introduced.
Abstract: This paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response. Sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold. In our method, uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction. We choose weights that minimize the total weighted power output of the array, subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid. The robust weight selection process can be cast as a second-order cone program that can be solved efficiently using Lagrange multiplier techniques. If the ellipsoid reduces to a single point, the method coincides with Capon's method. We describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response. We form separate uncertainty ellipsoids for each component in the signal path (e.g., antenna, electronics) and then determine an aggregate uncertainty ellipsoid from these. We give new results for modeling the element-wise products of ellipsoids. We demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples.
709 citations
01 Jul 2000
TL;DR: This work presents a method to design controllers for safety specifications in hybrid systems, using analysis based on optimal control and game theory for automata and continuous dynamical systems to derive Hamilton-Jacobi equations whose solutions describe the boundaries of reachable sets.
Abstract: We present a method to design controllers for safety specifications in hybrid systems. The hybrid system combines discrete event dynamics with nonlinear continuous dynamics: the discrete event dynamics model linguistic and qualitative information and naturally accommodate mode switching logic, and the continuous dynamics model the physical processes themselves, such as the continuous response of an aircraft to the forces of aileron and throttle. Input variables model both continuous and discrete control and disturbance parameters. We translate safety specifications into restrictions on the system's reachable sets of states. Then, using analysis based on optimal control and game theory for automata and continuous dynamical systems, we derive Hamilton-Jacobi equations whose solutions describe the boundaries of reachable sets. These equations are the heart of our general controller synthesis technique for hybrid systems, in which we calculate feedback control laws for the continuous and discrete variables, which guarantee that the hybrid system remains in the "safe subset" of the reachable set. We discuss issues related to computing solutions to Hamilton-Jacobi equations. Throughout, we demonstrate out techniques on examples of hybrid automata modeling aircraft conflict resolution, autopilot flight mode switching, and vehicle collision avoidance.
571 citations
Cites methods from "Ellipsoidal Calculus for Estimation..."
...A similar idea is to use ellipsoids as inner and outer approximations to the reach set [59], [60]....
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TL;DR: This paper presents a new approach to guaranteed state estimation for non-linear discrete-time systems with a bounded description of noise and parameters with an algorithm to compute a set that contains the states consistent with the measured output and the given Noise and parameters.
555 citations
23 Mar 2000
TL;DR: The proposed techniques, combined with calculation of external and internal approximations for intersections of ellipsoids, provide an approach to reachability problems for hybrid systems.
Abstract: This report describes the calculation of the reach sets and tubes for linear control systems with time-varying coefficients and hard bounds on the controls through tight external and internal ellipsoidal approximations. These approximating tubes touch the reach tubes from outside and inside respectively at every point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes. The proposed approximation scheme induces a very small computational burden compared with other methods of reach set calculation.
In particular such approximations may be expressed through ordinary differential equations with coefficients given in explicit analytical form. This yields exact parametric representation of reach tubes through families of external and internal ellipsoidal tubes. The proposed techniques, combined with calculation of external and internal approximations for intersections of ellipsoids, provide an approach to reachability problems for hybrid systems.
515 citations