1,722,024 research outputs found
A Novel Non-Orthogonal Joint Diagonalization Cost Function for ICA
Get PDFWe present a new scale-invariant cost function for non-orthogonal joint-diagonalization of a set of symmetric matrices with application to Independent Component Analysis (ICA). We derive two gradient minimization schemes to minimize this cost function. We also consider their performance in the context of an ICA algorithm based on non-orthogonal joint diagonalization
The Berry-Hannay Phase of the Equal-Sided Spring-Jointed Four-Bar Mechanism
Get PDFIn this work we apply the moving systems approach developed by Marsden, Montgomery, and Ratiu to a free-floating, equal-sided, spring-jointed, four-bar mechanism that is being slowly rotated about its central axis and derive a formula for the induced geometric phase
Degenerate Gradient Flows: A Comparison Study of Convergence Rate Estimates
Get PDFDegenerate gradient flows arise in the context of adaptive control of linear systems when the usual gradient algorithm is used for the parameter update law. It is well known that in general parameter convergence is not guaranteed without further assumptions. The standard approach utilizes the notion of a persistently exciting input and different authors have derived different convergence rate estimates. In a recent paper Brockett re-examined this issue and developed a rate estimate using a property of symmetric matrices related to the condition number. In this paper we compare two well-known convergence rate estimates from the persistently exciting point of view with Brockett's estimate through a semianalytical numerical study. We establish a common footing by relating the assumptions of each theorem to the parameters specified under the persistently exciting condition. Our analysis shows that for all parameter values Anderson's result yields a tighter bound than the other two estimates. In each case the magnitude of the difference depends on the time it takes for the uniform observability condition to hold in the persistently exciting assumption. The shorter the time is, the larger the difference is
The Berry-Hannay Phase of the Equal-Sided, Spring-Jointed, Four-Bar Mechanism: A Detailed Story
Get PDFIn this work we apply the moving systems approach developed by Marsden, Montgomery, and Ratiu to a free-floating, equal-sided, spring-jointed, four-bar mechanism that is being slowly rotated about its central axis and derive a formula for the induced geometric phase. We investigate the phase for a few specific systems using both analytic analysis and simulation
Control of Small Formations Using Shape Coordinates
Get PDFFormations that contain a small number of robots are modeled as controlled Lagrangian systems on Jacobi shape space. This allows a block-structured control of position, orientation and shape of the formation. Feedback control laws are derived using control Lyapunov functions. The controlled dynamics converges to the invariant set where desired shape is achieved. Controllers are implemented in a layered fashion via the extended motion description language(MDLe) system. Group MDLe plans are constructed to allow structured controller design for formations
The Hannay-Berry Phase of the Vibrating Ring Gyroscop
Get PDFIn an analysis published in 1890 G.H. Bryan investigated the retrograde precession of the nodal points in a vibrating, rotating shell and wrote down a formula relating the rate of precession to the rate of rotation. This effect has been utilized in the design of various vibratory gyroscopes including modern MEMS-based devices. Existing analyses model these systems with a pair of harmonic oscillators coupled through the Coriolis force (the normal mode method). In this work we utilize the theory of moving systems developed by Marsden, Montgomery, and Ratiu to show that the nodal precession can be understood as a geometric phase with respect to the Cartan-Hannay-Berry connection. This approach allows us to explicitly characterize the simplifications of the linearizing assumptions common to previous analyses. Our results match those of Bryan for small amplitude vibrations of the ring. We use the inherently nonlinear nature of the moving systems approach to calculate a (small) correction to the rate of precession of the nodes
Computing Balanced Realizations for Nonlinear Systems
Get PDFThis paper addresses the problem of computability pertaining to the Scherpen(1994) theory and procedure for balancing of nonlinear systems. In contrastto Moore's (1981) balancing method for linear systems, the Scherpen procedurefor nonlinear balancing is not immediately amenable to computationalimplementation. For example, the controllability energy function correspondsto the value function for a nonlinear optimal control problem. Also, theMorse-Palais lemma guarantees the existence of a local coordinatetransformation under which the controllability energy function takes acanonical quadratic form, but provides no constructive procedure for obtainingit. Thus, tools have not yet appeared for computing balanced realizations fornonlinear systems, and the procedure has not yet been applied as a tool formodel reduction.First, we consider the problem of computing the controllability energyfunction without numerically solving the family of optimal control problems,or the associated Hamilton-Jacobi-Bellman equation, implied in its definition.Stochastically excited systems play a major role in our methodology. Wepresent a stochastic method for computing an estimate of the controllabilityfunction, and show that in certain situations the method provides an exactsolution. The procedure is tested on applications via Monte-Carlo experiments.Then, we address the problem of numerically determining a Morse transformationfor a function with non-degenerate critical point at 0. We develop analgorithm for computing the desired nonlinear transformation and estimatingthe neighborhood on which the transformed controllability function isquadratic. In the literature, examples of applied nonlinear balancing have been limited topseudo-balancing of 2-dimensional gradient systems and noting that in the caseof linear systems the energy functions approach reduces to the usual setting ofgramians. We apply our approach to numerically derive, for the first time,balanced representations of nonlinear state-space models. In particular, wepresent applications to a forced damped pendulum system and a forced dampeddouble pendulum system.The research and scientific content in this material has been published in theProceedings of the 14th International Symposium on Mathematical Theory of Networks and Systems, Perpignan, France, June 19-23, 2000.</Center
Formation Dynamics under a Class of Control Laws
Get PDFA system of two earth satellites is analyzed as a controlled mechanical system. The orbit of an earth satellite can be represented by a point inthe vector space of ordered pairs of angular momentum and Laplace vectors. Control laws are obtained by introducing a Lyapunov function on this space. Formations of two satellites are achieved asymptotically by the controlled dynamics
- …
