Dynamics and Control of Higher-order Nonholonomic Systems

A theoretical framework is established for the control of higher-order nonholonomic systems, defined as systems that satisfy higher-order nonintegrable constraints. A model for such systems is developed in terms of differential-algebraic equations defined on a higher-order tangent bundle. A number of control-theoretic properties such as nonintegrability, controllability, and stabilizability are presented. Higher-order nonholonomic systems are shown to be strongly accessible and, under certain conditions, small time locally controllable at any equilibrium. There are important examples of higher-order nonholonomic systems that are asymptotically stabilizable via smooth feedback, including space vehicles with multiple slosh modes and Prismatic-Prismatic-Revolute (PPR) robots moving open liquid containers, as well as an interesting class of systems that do not admit asymptotically stabilizing continuous static or dynamic state feedback. Specific assumptions are introduced to define this class, which includes important examples of robotic systems. A discontinuous nonlinear feedback control algorithm is developed to steer any initial state to the equilibrium at the origin. The applicability of the theoretical development is illustrated through two examples: control of a planar PPR robot manipulator subject to a jerk constraint and control of a point mass moving on a constant torsion curve in a three dimensional space

FEEDBACK CONTROL OF QUANTUM SYSTEMS AND ENTANGLEMENT

Ph.DDOCTOR OF PHILOSOPH

Magnetic Stabilization of Nadir-Pointing Small Satellites

Since magnetic control systems are relatively lightweight, require low power and are inexpensive, they are attractive for small, inexpensive satellites in low Earth orbits. In this thesis we present averaging-based feedback control laws that achieve three-axis stabilized nadir-pointing attitude. Two types of nonlinear feedback control laws are proposed: full-state feedback and passivity-based feedback. Full-state feedback uses the attitude and angular velocity measurements to regulate the spacecrafts dynamics. Passivity-based feedback uses the attitude measurement and doesn’t require the rate sensors. The control laws are tested using two magnetic field models: the tilted dipole model and the International Geomagnetic Reference Field (IGRF) model. Computer simulations are included to illustrate the effectiveness of the proposed control laws

Nonlinear control of underactuated mechanical systems with application to robotics and aerospace vehicles

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (leaves 308-316).This thesis is devoted to nonlinear control, reduction, and classification of underactuated mechanical systems. Underactuated systems are mechanical control systems with fewer controls than the number of configuration variables. Control of underactuated systems is currently an active field of research due to their broad applications in Robotics, Aerospace Vehicles, and Marine Vehicles. The examples of underactuated systems include flexible-link robots, nobile robots, walking robots, robots on mobile platforms, cars, locomotive systems, snake-type and swimming robots, acrobatic robots, aircraft, spacecraft, helicopters, satellites, surface vessels, and underwater vehicles. Based on recent surveys, control of general underactuated systems is a major open problem. Almost all real-life mechanical systems possess kinetic symmetry properties, i.e. their kinetic energy does not depend on a subset of configuration variables called external variables. In this work, I exploit such symmetry properties as a means of reducing the complexity of control design for underactuated systems. As a result, reduction and nonlinear control of high-order underactuated systems with kinetic symmetry is the main focus of this thesis. By "reduction", we mean a procedure to reduce control design for the original underactuated system to control of a lowerorder nonlinear or mechanical system. One way to achieve such a reduction is by transforming an underactuated system to a cascade nonlinear system with structural properties. If all underactuated systems in a class can be transformed into a specific class of nonlinear systems, we refer to the transformed systems as the "normal form" of the corresponding class of underactuated systems. Our main contribution is to find explicit change of coordinates and control that transform several classes of underactuated systems, which appear in robotics and aerospace applications, into cascade nonlinear systems with structural properties that are convenient for control design purposes. The obtained cascade normal forms are three classes of nonlinear systems, namely, systems in strict feedback form, feedforward form, and nontriangular linear-quadratic form. The names of these three classes are due to the particular lower-triangular, upper-triangular, and nontriangular structure in which the state variables appear in the dynamics of the corresponding nonlinear systems. The triangular normal forms of underactuated systems can be controlled using existing backstepping and feedforwarding procedures. However, control of the nontriangular normal forms is a major open problem. We address this problem for important classes of nontriangular systems of interest by introducing a new stabilization method based on the solutions of fixed-point equations as stabilizing nonlinear state feedback laws. This controller is obtained via a simple recursive method that is convenient for implementation. For special classes of nontriangular nonlinear systems, such fixed-point equations can be solved explicitly ...by Reza Olfati-Saber.Ph.D

Stability analysis and controller synthesis for a class of piecewise smooth systems

This thesis deals with the analysis and synthesis of piecewise smooth (PWS) systems. In general, PWS systems are nonsmooth systems, which means their vector fields are discontinuous functions of the state vector. Dynamic behavior of nonsmooth systems is richer than smooth systems. For example, there are phenomena such as sliding modes that occur only in nonsmooth systems. In this thesis, a Lyapunov stability theorem is proved to provide the theoretical framework for the stability analysis of PWS systems. Piecewise affine (PWA) and piecewise polynomial (PWP) systems are then introduced as important subclasses of PWS systems. The objective of this thesis is to propose efficient computational controller synthesis methods for PWA and PWP systems. Three synthesis methods are presented in this thesis. The first method extends linear controllers for uncertain nonlinear systems to PWA controllers. The result is a PWA controller that maintains the performance of the linear controller while extending its region of convergence. However, the synthesis problem for the first method is formulated as a set of bilinear matrix inequalities (BMIs), which are not easy to solve. Two controller synthesis methods are then presented to formulate PWA and PWP controller synthesis as convex problems, which are numerically tractable. Finally, to address practical implementation issues, a time-delay approach to stability analysis of sampled-data PWA systems is presented. The proposed method calculates the maximum sampling time for a sampled-data PWA system consisting of a continuous-time plant and a discrete-time emulation of a continuous-time PWA state feedback controller

Dynamics and control of electromagnetic satellite formations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2007.Includes bibliographical references (p. 197-203).Satellite formation flying is an enabling technology for many space missions, especially for space-based telescopes. Usually there is a tight formation-keeping requirement that may need constant expenditure of fuel or at least fuel is required for formation reconfiguration. Electromagnetic Formation Flying (EMFF) is a novel concept that uses superconducting electromagnetic coils to provide forces and torques between different satellites in a formation which enables the control of all the relative degrees of freedom. With EMFF, the life-span of the mission becomes independent of the fuel available on board. Also the contamination of optics or sensitive formation instruments, due to thruster plumes, is avoided. This comes at the cost of coupled and nonlinear dynamics of the formation and makes the control problem a challenging one. In this thesis, the dynamics for a general N-satellite electromagnetic formation will be derived for both deep space missions and Low Earth Orbit (LEO) formations. Nonlinear control laws using adaptive techniques will be derived for general formations in LEO. Angular momentum management in LEO is a problem for EMFF due to interaction of the magnetic dipoles with the Earth's magnetic field. A solution of this problem for general Electromagnetic (EM) formations will be presented in the form of a dipole polarity switching control law. For EMFF, the formation reconfiguration problem is a nonlinear and constrained optimal time control problem as fuel cost for EMFF is zero. Two different methods of trajectory generation, namely feedback motion planning using the Artificial Potential Function Method (APFM) and optimal trajectory generation using the Legendre Pseudospectral method, will be derived for general EM Formations.(cont.) The results of these methods are compared for random EM Formations. This comparison shows that the artificial potential function method is a promising technique for solving the real-time motion planning problem of nonlinear and constrained systems, such as EMFF, with low computational cost. Specifically it is the purpose of this thesis to show that a fully-actuated N-satellite EM formation can be stabilized and controlled under fairly general assumptions, therefore showing the viability of this novel approach for satellite formation flying from a dynamics and controls perspective.by Umair Ahsun.Ph.D