Institute for Systems Research
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Item Analysis of a complex activator-inhibitor equation(1999) Justh, Eric W.; Krishnaprasad, Perinkulam S.; ISR; CDCSSBasic properties of solutions and a Lyapunov functionalare presented for a complex activator-inhibitor equation witha cubic nonlinearity.Potential applications include control of coupled-oscillator arrays(for quasi-optical power combining and phased-array antennas),and control of MEMS actuator arrays (for micro-positioning small items).(This work to appear in Proc. 1999 American Control Conference.)
Item Control Problems of Hydrodynamic Type(1998) Krishnaprasad, Perinkulam S.; Manikonda, Vikram; ISR; CDCSSIt has been known for some time that the classical work of Kirchhoff, Love,and Birkhoff on rigid bodies in incompressible, irrotational flows provideseffective models for treating control problems for underwater vehicles.This has also led to a better appreciation of the dynamics of suchsystems. In this paper, we develop results based on geometric mechanics andcenter manifold theory to solve controllability and stabilization questionsfor a class of under-actuated left invariant mechanical systems on Liegroups that include approximate models of underwater vehicles and surfacevehicles. We also provide numerical evidence to capture the globalproperties of certain interesting feedback laws.(This work appears as an invited paper in the Proc. IFAC Sympo. on NonlinearControl Systems Design (NOLCOS'98), (1998), 1:139-144)
Item A Lyapunov Functional for the Cubic Nonlinearity Activator-Inhibitor Model Equation(1998) Justh, Eric W.; Krishnaprasad, Perinkulam S.; ISR; CDCSSThe cubic nonlinearity activator-inhibitor model equation is a simpleexample of a pattern-forming system for which strong mathematical resultscan be obtained. Basic properties of solutions and the derivation ofa Lyapunov functional for the cubic nonlinearity model are presented.Potential applications include control of large MEMS actuator arrays.(In Proc. IEEE Conf. Decision and Control, December 16-18, 1998)Item Convergence Analysis and Analog Circuit Applications for a Class of Networks of Nonlinear Coupled Oscillators(1996) Justh, Eric W.; Krishnaprasad, Perinkulam S.; Kub, Francis J.; ISRThe physical motivation and rigorous proof of convergence for a particular network of nonlinear coupled oscillators are reviewed. Next, the network and convergence proof are generalized in several ways, to make the network more applicable to actual engineering problems. It is argued that such coupled oscillator circuits are more natural to implement in analog hardware than other types of dynamical equations because the signal levels tend to remain at sufficiently large values that effects of offsets and mismatch are minimized. Examples of how analog implementations of these networks are able to address actual control problems are given. The first example shows how a pair of coupled oscillators can be used to compensate for the feedback path phase shift in a complex LMS loop, and has potential application for analog adaptive antenna arrays or linear predictor circuits. The second example shows how a single oscillator circuit with feedback could be used for continuous wavelet transform applications. Finally, analog CMOS implementation of the coupled oscillator dynamics is briefly discussed.Item Adaptive Friction Compensation for Bi-Directional Low-Velocity Position Tracking(1992) Leonard, Naomi E.; Krishnaprasad, Perinkulam S.; ISRThis paper presents a comparative investigation of friction- compensating control strategies designed to improve low-velocity position tracking performance in the presence of velocity reversals for servomechanisms. The methods considered include adaptive control and estimation-based control. Additionally, the various controller designs incorporate different friction models ranging from classical friction and Stribeck friction to the less popular Dahl friction model. This investigation of friction models is motivated by the fact that there is little consensus in the literature on how best to model friction for dynamic friction compensation. the control strategies are compared in an extensive test program involving sinusoidal position trajectory tracking experiments on a direct-drive dc motor. We focus attention on comparative experimental results of friction compensation especially with repeated velocity reversals. The results show that the adaptive experiments also yield insight into the appropriateness of the different friction models under the tested operating conditions. In particular, the Dahl model, typically ignored in the literature proves to be significant for the firction-compensating control problem with repeated zero- velocity crossings.Item Dissipation Induced Instabilities(1992) Bloch, Anthony M.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; Ratiu, Tudor S.; ISRThe main goal of this paper is to prove that if the energy- momentum (or energy-Casimir) method predicts formal instability of a relative equilibrium in a Hamiltonian system with symmetry, then with the addition of dissipation, the relative equilibrium becomes spectrally and hence linearly and nonlinearly unstable. The energy-momentum method assumes that one is in the context of a mechanical system with a given symmetry group. Our result assumes that the dissipation chosen does not destroy the conservation law associated with the given symmetry group -- thus, we consider internal dissipation. Our result also includes the special case of systems with no symmetry and ordinary equilibria. Our result is proved by combining the techniques of Chetaev, who proved instability theorems using a special Chetaev- Lyapunov function, those of Hahn, which enable one to strengthen the Chetaev results from Lyapunov instability to spectral instability. Our main achievement is to strengthen these results to the context of the block diagonalization version of the energy momentum method given by Lewis. Marsden, Posbergh, and Simo. However, we also give the eigenvalue movement formulae of Krein, MacKay and others both in general and adapted to the context of the normal form of the linearized equations given by the block diagoanl form as provided by the energy-momentum method. A number of specific examples, such as the rigid body with internal rotors, are provided to illustrate the results.Item On the Geometry and Dynamics of Floating Four-Bar Linkages(1992) Yang, R.; Krishnaprasad, Perinkulam S.; ISRIn this paper, we investigate the kinematics and dynamics of floating, planar four-bar linkages. The geometry of configuration space is analyzed through the classical theory of mechanisms due to Grashof. The techniques of symplectic and Poisson reduction are used to understand the dynamics of the system. Bifurcations of relative equilibria for linkages admitting symmetric shapes are studied using the techniques of singularity theory. The problem of reconstruction of the full dynamics and its relation to geometric phases is discussed through some examples. This research reveals that a coupled mechanical system with kinematic loops possesses richer and more complicated dynamical aspects in comparison with systems which have the same number of degrees of freedom, but no kinematic loops.Item Comparative Study of Friction-Compensating Control Strategies for Servomechanisms(1991) Leonard, Naomi E.; Krishnaprasad, Perinkulam S.; ISRThis paper describes a comparative investigation of friction- compensating control strategies designed to improve low-velocity position tracking performance for servomechanisms. Several control methods are considered including adaptive control and estimation-based control. Additionally, the various controller designs incorporate different friction models ranging from classical friction and Stribeck friction to the less popular Dahl friction model. This investigation of friction models is motivated by the fact that there is little consensus in the literature on how best to model friction for dynamic friction compensation. The control strategies are compared in an extensive test program involving position tracking experiments on a direct- drive de motor. This effort addresses the current lack of comparative experimental results on friction compensation. The results show that the adaptive and estimation-based controllers outperform more traditional linear controllers. The experiments also yield insight into the appropriateness of the different friction models under the tested operating conditions. In particular, the Dahl model is observed to provide a reliable representation of friction behavior near zero velocity.Item Gyroscopic Control and Stabilization(1991) Wang, L.S.; Krishnaprasad, Perinkulam S.; ISRIn this paper, we consider the geometry of gyroscopic systems with symmetry, starting from an intrinsic Lagrangian viewpoint. We note that natural mechanical systems with exogenous forces can be transformed into gyroscopic systems, when the forces are determined by a suitable class of feedback laws. To assess the stability of relative equilibria in the resultant feedback systems, we extend the energy-momentum block-diagonalization theorem of Simo, Lewis, Posbergh, and Marsden to gyroscopic systems with symmetry. We illustrate the main ideas by a key example of two coupled rigid bodies with internal rotors. The energy-momentum method yields computationally tractable stability criteria in this and other examples.Item Steady Rigid-Body Motions in a Central Gravitational Field(1991) Wang, L.S.; Maddocks, J.H.; Krishnaprasad, Perinkulam S.; ISRIn recent work, the exact dynamic equations for the motion of a finite rigid body in a central gravitational field were shown to be of Hamiltonian form with a noncanonical structure. In this paper, the notion of relative equilibrium is introduced based upon this exact model. In relative equilibrium, the orbit of the center of mass of the rigid body is a circle, but the center of attraction may or may not lie at the center of the orbit. This feature is used to classify great-circle and non-great-circle orbits. The existence of non-great-circle relative equilibria for the exact model is proved from various variational principles. While the orbital offset of the non-great-circle solutions is necessarily small, a numerical study reveals that there can be significant changes in orientation away from the classic Lagrange relative equilibria, which are solutions of an approximate model.