Control of Large Actuator Arrays Using Pattern-Forming Systems

Pattern-forming systems are used to model many diverse phenomena from biology,chemistry and physics. These systems of differential equations havethe property that as a bifurcation (or control) parameter passes through acritical value, a stable spatially uniform equilibrium state gives way to astable pattern state, which may have spatial variation, time variation, orboth. There is a large body of experimental and mathematical work on pattern-forming systems.However, these ideas have not yet been adequately exploited inengineering, particularly in the control of smart systems; i.e.,feedback systems having large numbers of actuators and sensors. With dramatic recent improvements in micro-actuator and micro-sensortechnology, there is a need for control schemes betterthan the conventional approach of reading out all of the sensor informationto a computer, performing all the necessary computations in a centralizedfashion, and then sending out commands to each individual actuator.Potential applications for large arrays of micro-actuators includeadaptive optics (in particular, micromirror arrays), suppressingturbulence and vortices in fluid boundary-layers, micro-positioning smallparts, and manipulating small quantities of chemical reactants. The main theoretical result presented is a Lyapunov functional for thecubic nonlinearity activator-inhibitor model pattern-forming system.Analogous Lyapunov functionals then follow for certain generalizations ofthe basic cubic nonlinearity model. One such generalization is a complex activator-inhibitor equation which, under suitable hypotheses,models the amplitude and phase evolution in the continuum limitof a network of coupled van der Pol oscillators, coupled to a network of resonant circuits, with an external oscillating input. Potentialapplications for such coupled van der Pol oscillator networks includequasi-optical power combining and phased-array antennas. In addition to the Lyapunov functional, a Lyapunov function for the truncated modal dynamics is derived, and the Lyapunov functional isalso used to analyze the stability of certain equilibria. Basic existence, uniqueness, regularity, and dissipativity properties ofsolutions are also verified, engineering realizations of the dynamicsare discussed, and finally, some of the potential applications areexplored

Analysis of a high-resolution optical wave-front control system

We 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.

A Simple Control Law for UAV Formation Flying

This 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

Analysis of a complex activator-inhibitor equation

Basic 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.

A Lyapunov Functional for the Cubic Nonlinearity Activator-Inhibitor Model Equation

The 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

Opto-Electronic Zernike Filter for High-Resolution Wavefront Analysis using a Phase-Only Liquid Crystal Spatial Light Modulator

An opto-electronic technique for high-resolution wave-front phase imagingis presented and demonstrated experimentally. The technique is analogousto the conventional Zernike phase-contrast approach, but uses modernspatial light modulator technology to increase robustness and improveperformance. Because they provide direct measurements of wave-frontphase (rather than wave-front slope measurements, as in Shack-Hartmannsensors), robust phase-contrast sensors have potential applications inhigh-speed, high-resolution adaptive optic systems. Advantages of theopto-electronic approach over alternative advanced phase-contrasttechniques (such as a related phase-contrast sensor which uses a liquid-crystal light valve exhibiting a Kerr-type optical response toperform Fourier filtering) are discussed. The SLM used for theexperimental results is a 128x128-element pixilated phase-only liquidcrystal spatial light modulator from Boulder Nonlinear Systems, Inc. SPIE Proc., High-Resolution Wavefront Control: Methods,Devices, and Applications II, Vol. 4124, pp. 138-147, 2000.</Center

Delay Induced Instabilities in Self-Propelling Swarms

We consider a general model of self-propelling particles interacting through a pairwise attractive force in the presence of noise and communication time delay. Previous work by Erdmann, et al. [Phys. Rev. E {\bf 71}, 051904 (2205)] has shown that a large enough noise intensity will cause a translating swarm of individuals to transition to a rotating swarm with a stationary center of mass. We show that with the addition of a time delay, the model possesses a transition that depends on the size of the coupling amplitude. This transition is independent of the initial swarm state (traveling or rotating) and is characterized by the alignment of all of the individuals along with a swarm oscillation. By considering the mean field equations without noise, we show that the time delay induced transition is associated with a Hopf bifurcation. The analytical result yields good agreement with numerical computations of the value of the coupling parameter at the Hopf point.Comment: 4 pages, 5 figures Final revision to appear in PRE Rapid Communication

Convergence Analysis and Analog Circuit Applications for a Class of Networks of Nonlinear Coupled Oscillators

The 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

Advanced Phase-Contrast Techniques for Wavefront Sensing and Adaptive Optics

High-resolution phase-contrast wavefront sensors based on optically addressed phase spatial light modulators and micro-mirror/LC arrays areintroduced. Wavefront sensor efficiency is analyzed for atmosphericturbulence-induced phase distortions described by the Kolmogorov and Andrews models. A nonlinear Zernike filter wavefront sensor based on anoptically addressed liquid crystal phase spatial light modulator isexperimentally demonstrated. The results demonstrate high-resolutionvisualization of dynamically changing phase distortions within the sensortime response of 10 msec.SPIE Proc., High-Resolution Wavefront Control: Methods,Devices, and Applications II, Vol. 4124, pp. 98-109, 2000

Adaptive Wavefront Control using a Nonlinear Zernike Filter

A conventional Zernike filter measures wavefront phase by superimposing the aberrated input beam with a phase-shifted version of its zero-orderspectral component. The Fourier-domain phase-shifting is performed by afixed phase-shifting dot on a glass slide in the focal plane of a Fourier-transforming lens. Using an optically-controlled phase spatiallight modulator (SLM) instead of the fixed phase-shifting dot, we havesimulated and experimentally demonstrated a nonlinear Zernike filterrobust to wavefront tilt misalignments. In the experiments, a liquid-crystal light valve (LCLV) was used as the phase SLM. The terminology "nonlinear" Zernike filter refers to the nonlinear filteringoperation that takes place in the Fourier domain due to the phase changefor field spectral components being proportional to the spectral componentintensities. Because the Zernike filter output intensity is directlyrelated to input wavefront phase, a parallel, distributed feedback systemcan replace the wavefront reconstruction calculations normally requiredin adaptive-optic phase correction systems. Applications include high-resolution phase distortion suppression for atmospheric turbulence,optical phase microscopy, and compensation of aberrations in optical system components. A factor of eight improvement in Strehl ratio wasobtained experimentally, and simulation results suggest that even betterperformance could be obtained by replacing the LCLV with a more sophisticated optically-controlled phase SLM.SPIE Proc., High-Resolution Wavefront Control:Methods, Devices, and Applications II, Vol. 4124, pp. 198-200, 2000.</Center