Institute for Systems Research

A new approach, namely the set-valued RtR controller with ellipsoidapproximation, is proposed to estimate the process model from acompletely different point of view. Because the set-valued RtRcontroller identifies the process model in the feasible parameter setwhich is insensitive to noises, the controller is robust to theenvironment noises.Ellipsoid approximation can significantly reduce the computation load for the set-valued method.

In this paper, the Modified Optimal Volume Ellipsoid (MOVE) algorithm is used toestimate the process model in each run. Designof the corresponding controller and parameter selection of the controller are introduced.Simulation results showed that the controller is robust toenvironment noises and model errors.

(This work to appear in Proc. 1999 American Control Conference.)

Previous and concurrent research in the modeling of magnetostrictive actuators and the related area of electrostrictive actuators have produced models that are of low order and reproduce their quasi-static response reasonably well. But the main interest in using these and other smart actuators is at a high frequency -- for producing large displacements with mechanical rectification, producing sonar signals etc. The well known limitation of smart actuators that are based on electro-magneto-thermo-elastic behaviors of smart materials is the complex, input-rate dependent, hysteretic behavior of the latter.

The model proposed in this dissertation is a bulk model and describes the behaviour of a magnetostrictive actuator by a system with 4 states. We develop this model using phenomenological arguments following the work done by Jiles and Atherton for describing bulk ferromagnetic hysteresis. The model accounts for magnetic hysteresis; eddy current effects; magneto-elastic effects; inertial effects; and mechanical damping. We show rigorously that the system with the intial state at the origin has a periodic orbit as its $Omega$ limit set. For the bulk ferromagnetic hysteresis model - a simplification of the magnetostrictive model, we show that all trajectories starting within a certain set approach this limit set.

It is envisioned that the model will help application engineers to do simulation studies of structures with magnetostrictive actuators. Towards this end, an algorithm is proposed to identify the various parameters in the model.

In control applications, one may require the actuator to follow a certain trajectory. The complex rate dependent behaviour of the actuator makes the design of a suitable control law a challenging one. As our system of equations do not model transient effects, they do not model the minor-loop closure property common to ferromagnetic materials. Therefore, the design of control laws making explicit use of the model (without modifications) is not possible.

A major reason to use model free approaches to control design is that magnetostrictive actuators seem to have slight variations in their behavior with time. Therefore, we tried to use a direct adaptive control methodology that uses features of our model. The system is now looked at as a relative degree two linear system with set-valued input nonlinearity. Extensions of Eugene Ryan's work on universal tracking for a relative degree one linear system and Morse's work on stablization for relative degree two linear systems were sought.

Experimental verification of our method confirmed our intuition about the model structure. Though the tracking results were not very satisfactory due to the presence of sensor noise, the experimental results, nevertheless validate our modeling effort.

In this paper, participation factors are revisited, resulting in new definitions. The aim ofthese definitions is to achieve a conceptual framework that doesn't hinge on anyparticular choice of initial condition. The initial condition is modeled as an uncertainquantity, which can be viewed either in a set-valued or a probabilistic setting.If the initial condition uncertainty obeys a symmetry condition, the new definitionsare found to reduce to the original definition of participation factors.

Bothpitchfork bifurcation and transcritical bifurcation are addressed.The results obtained forpitchfork bifurcations apply to general nonlinear models smoothin the state and the control. For transcritical bifurcations,the results require the system to be affine in the control.

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.