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 of Large Actuator Arrays Using Pattern-Forming Systems(1998) Justh, Eric W.; Krishnaprasad, P.S.; ISR; CDCSSPattern-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.
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 Convergence Analysis of a Class of Networks of Nonlinear Coupled Oscillators(1994) Justh, Eric W.; Krishnaprasad, P.S.; ISRA network of nonlinear coupled oscillators is presented, and a convergence proof is given along with physical motivation. Next, the network architecture is generalized by allowing interconnections between oscillators to be controlled in an adaptive fashion, and convergence of the generalized network is proved. An example network is presented to illustrate the utility of such networks and to show why the problem of undesired stable equilibria must be addressed. Two alternative approaches are then presented which overcome the problem of undesired stable equilibria appearing in the network dynamics. Finally, an analog VLSI approach to implementation of such networks is presented, and tradeoffs among power dissipation, bandwidth, and network size are discussed.