Institute for Systems Research Technical Reports
Permanent URI for this collectionhttp://hdl.handle.net/1903/4376
This archive contains a collection of reports generated by the faculty and students of the Institute for Systems Research (ISR), a permanent, interdisciplinary research unit in the A. James Clark School of Engineering at the University of Maryland. ISR-based projects are conducted through partnerships with industry and government, bringing together faculty and students from multiple academic departments and colleges across the university.
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Item The Hannay-Berry Phase of the Vibrating Ring Gyroscop(2004) Andersson, Sean B.; Krishnaprasad, Perinkulam S.; Krishnaprasad, Perinkulam S.; ISR; CDCSSIn 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.Item Control of Small Formations Using Shape Coordinates(2003) Zhang, Fumin; Goldgeier, Michael; Krishnaprasad, Perinkulam S.; Krishnaprasad, Perinkulam S.; ISR; CDCSSFormations 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.Item Control of Hysteresis in Smart Actuators with Application to Micro-Positioning(2003) Tan, Xiaobo; Baras, John S.; Krishnaprasad, Perinkulam S.; ISR; CDCSSHysteresis in smart material actuators makes the effective use of these actuators quite challenging. The Preisach operator has been widely used to model smart material hysteresis. Motivated by positioning applications of smart actuators, this paper addresses the value inversion problem for a class of discretized Preisach operators, i.e., to find an optimal input trajectory given a desired output value. This problem is solved through optimal state transition of a finite state machine (FSM) that corresponds to the discretized Preisach operator. A state-space reduction scheme for the FSM is developed, which significantly saves the memory and the computation time. As an example, micro-positioning control of a magnetostrictive actuator is investigated. Experimental results are presented to demonstrate the effectiveness of the proposed approach.Item Analysis of a high-resolution optical wave-front control system(2002) Justh, Eric W.; Krishnaprasad, Perinkulam S.; ISR; CDCSSWe consider the formulation and analysis of a problem of automaticcontrol: correcting for the distortion induced in an optical wave frontdue to propagation through a turbulent atmosphere. It has recentlybeen demonstrated that high-resolution optical wave-front distortionsuppression can be achieved using feedback systems based on high-resolution spatial light modulators and phase-contrast techniques.We examine the modeling and analysis of such systems for the purposeof refining their design. The approach taken here might also beapplicable to other problems involving feedback controlof physical fields, particularly if the field sensing is performedoptically. (In Proc. Conf. on Information Sciences and Systems, Vol. 2, pp. 718-723, 2001.)Item Degenerate Gradient Flows: A Comparison Study of Convergence Rate Estimates(2002) Andersson, Sean B.; Krishnaprasad, Perinkulam S.; Krishnaprasad, Perinkulam S.; ISRDegenerate 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.Item Quotient Signal Decomposition and Order Estimation(2002) Napoletani, D.; Berenstein, Carlos A.; Krishnaprasad, Perinkulam S.; ISRIn this paper we propose a method for blind signal decomposition that does not require the independence or stationarity of the sources. This method, that we consider a simple instance of non-linear projection pursuit, is based on the possibility of recovering the areas in the time-frequency where the original signals are isolated or almost isolated with the use of suitable quotients of linear combinations of the spectrograms of the mixtures.We then threshold such quotients according to the value of their imaginary part to prove that the method is theoretically sound under mild assumptions on the mixing matrix and the sources. We study one basic algorithm based on this method. Moreover we propose a practical measure of separation for the sources in a given time frequency representation.
The algorithm has the important feature of estimating the number of sources with two measurements, it then requires n-2 additional measurements to provide a reconstruction of n sources. Experimental results show that the method works even when severalshifted version of the same source are mixed.
Item A Dynamic Model for Magnetostrictive Hysteresis(2002) Tan, Xiaobo; Baras, John S.; Krishnaprasad, Perinkulam S.; Baras, John S.; Krishnaprasad, Perinkulam S.; ISR; CDCSSThe rate-dependent hysteresis present in thin magnetostrictive actuators can be captured by a dynamic model, consisting of a Preisach operator coupled to an ordinary differential equation in an unusual way. The model presents interesting problems in analysis and computation due to its special structure. In this paper we first transform the model into a more amenable form and gain insight into the model by introducing a new hysteretic operator. Then we investigate some system-theoretic properties of the model: stability of equilibria, input-output stability, reachability and controllability. Existence of periodic solutions under periodic forcing is also established. Finally numerical integration schemes for the model are discussed.Item A Simple Control Law for UAV Formation Flying(2002) Justh, Eric W.; Krishnaprasad, Perinkulam S.; ISRThis paper presents a Lie group setting for the problem of control of formations, as a natural outcome of the analysis of a planar two-vehicle formation control law.The vehicle trajectories are described using planar Frenet-Serret equations of motion, which capture the evolution of both the vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The set of all possible (relative) equilibria for arbitrary G-invariant curvature controls is described (where G = SE(2) is a symmetry group for the control law). A generalization of the control law for n vehicles is presented, and the corresponding (relative) equilibria are characterized. Work is on-going to discover stability and convergence results for the n-vehicle problem.
The practical motivation for this work is the problem of formation control for meter-scale UAVs; therefore, an implementation approach consistent with UAV payload constraints is also discussed.
Item Coordinated Orbit Transfer for Satellite Clusters(2002) Zhang, Fumin; Krishnaprasad, Perinkulam S.; ISRWe propose a control law which allows a satellite formation to achieveorbit transfer. During the transfer, the formationcan be either maintained or modified to a desired one.Based on the orbit transfer control law proposed by Chang, Chichka and Marsden forsingle satellite,we add coupling terms to the summation of Lyapunov functions forsingle satellites. These terms are functionsof the difference between the mean anomalies (or perigee passing times) of formation members. The asymptotic stability of the desired formationin desired orbits is proved.Item Cayley Transforms in Micromagnetics(2001) Krishnaprasad, Perinkulam S.; Tan, Xiaobo; ISR; CDCSSMethods of numerical integration of ordinary differential equations exploiting the Cayley transform arise in a variety of contexts, ranging from the classical mid-point rule to symplectic and (almost) Poisson integrators, to numerical methods on Lie Groups. In earlier work, the first author investigated the interplay between the Cayley transform and the Jacobi identity in establishing certain error formulas for the mid-point rule (with applications to coupled rigid bodies). In this paper, we use the Cayley transform to lift the Landau-Lifshitz-Gilbert equation of micromagnetics to the Lie algebra of the group of currents (on a compact magnetic body) with values in the 3-dimensional rotation group. This follows an idea of Arieh Iserles and, we use the lift to numerically integrate the Landau-Lifshitz-Gilbert equation conserving automatically the norm of the magnetization everywhere.