Transverse Feedback Linearization with Partial Information for Single-Input Systems

“First Published in SIAM Journal on Control and Optimization in 2014, published by the Society for Industrial and Applied Mathematics (SIAM)” and the copyright notice as stated in the article itself (e.g., “Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.”')This paper is motivated by the problem of asymptotically stabilizing invariant sets in the state space of control systems by means of output feedback. The sets considered are smooth embedded in submanifolds and the class of system is nonlinear, finite-dimensional, autonomous, deterministic, single-input and control-affine. Given an invariant set and a control system with fixed output, necessary and sufficient conditions are presented for feedback equivalence to a normal form that facilities the design of output feedback controllers that stabilize the set using existing design techniques.This work was supported by supported by the National Science and Engineering Research Council (NSERC) of Canad

Input to State Stability of Bipedal Walking Robots: Application to DURUS

Bipedal robots are a prime example of systems which exhibit highly nonlinear dynamics, underactuation, and undergo complex dissipative impacts. This paper discusses methods used to overcome a wide variety of uncertainties, with the end result being stable bipedal walking. The principal contribution of this paper is to establish sufficiency conditions for yielding input to state stable (ISS) hybrid periodic orbits, i.e., stable walking gaits under model-based and phase-based uncertainties. In particular, it will be shown formally that exponential input to state stabilization (e-ISS) of the continuous dynamics, and hybrid invariance conditions are enough to realize stable walking in the 23-DOF bipedal robot DURUS. This main result will be supported through successful and sustained walking of the bipedal robot DURUS in a laboratory environment.Comment: 16 pages, 10 figure

Feedback and Partial Feedback Linearization of Nonlinear Systems: A Tribute to the Elders

Arthur Krener and Roger Brockett pioneered the feedback linearization problem for control systems, that is, the transforming of a nonlinear control system into linear dynamics via change of coordinates and feedback. While the former gave necessary and sufficient conditions to linearize a system under change of coordinates only, the latter introduced the concept of feedback and solved the problem for a particular case. Their work was soon extended in the earlier eighties by Jakubczyk and Responder, and Hunt and Su who gave the conditions for a control system to be linearizable by change of coordinates and feedback (full rank and involutivity of the associated distributions). It turned out that those conditions are very restrictive; however, it was showed later that systems that fail to be linearizable can still be transformed into two interconnected subsystems: one linear and the other nonlinear. This fact is known as partial feedback linearization. For input-output systems with well-defined relative degree, coordinates can be found by differentiating the outputs. For systems without outputs, necessary and sufficient geometric conditions for partial linearization have been obtained in terms of the Lie algebra of the system; however, both results of linearization and partial feedback linearization lack practicability. Until recently, none has provided a way to actually compute the linearizing coordinates and feedback. In this paper, we propose an algorithm allowing to find the linearizing coordinates and feedback if the system is linearizable, and in the contrary, to decompose a system (without outputs) while achieving the largest linear subsystem. Those algorithms are built upon successive applications of the Frobenius theorem. Examples are provided to illustrate

Spatial correlations in hexagons generated via a Kerr nonlinearity

We consider the hexagonal pattern forming in the cross-section of an optical beam produced by a Kerr cavity, and we study the quantum correlations characterizing this structure. By using arguments related to the symmetry broken by the pattern formation, we identify a complete scenario of six-mode entanglement. Five independent phase quadratures combinations, connecting the hexagonal modes, are shown to exhibit sub-shot-noise fluctuations. By means of a non-linear quantum calculation technique, quantum correlations among the mode photon numbers are demonstrated and calculated.Comment: ReVTeX file, 20 pages, 7 eps figure

Maneuvering Control of a Spacecraft with Propellant Sloshing

Propellant slosh has been a problem studied in spacecraft designs since the early days of large, liquid-fuel rockets. The conventional design solution involves physical structures inside the fuel tanks that limit propellant motion. Although effective, baffles and bladders add to spacecraft mass and structural complexity. In this research, the sloshing fuel mass is treated as an unactuated degree of freedom within a rigid body. Specifically, the propellant is modeled as a pendulum mass anchored at the center of a spherical tank. After obtaining the coupled equations of motion, several linear controllers are developed to achieve planar spacecraft pitch-maneuvers while suppressing the slosh mode. The performance of these linear controllers will be compared to that of a nonlinear controller developed using Lyapunov’s Second Method. It is shown that the linear controllers are ill-equipped to achieve the desired spacecraft attitude and transverse velocity simultaneously, especially during aggressive pitch-maneuvers; while the Lyapunov controller is superior in this regard