Optimal control of multiscale systems using reduced-order models

We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy "first reduce, then optimize"--rather than "first optimize, then reduce"--is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the "first reduce, then control" strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the "first reduce, then optmize" approach and discuss possible pitfalls

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed

Control strategies and motion planning for nanopositioning applications with multi-axis magnetic-levitation instruments

This dissertation is the first attempt to demonstrate the use of magnetic-levitation (maglev) positioners for commercial applications requiring nanopositioning. The key objectives of this research were to devise the control strategies and motion planning to overcome the inherent technical challenges of the maglev systems, and test them on the developed maglev systems to demonstrate their capabilities as the next-generation nanopositioners. Two maglev positioners based on novel actuation schemes and capable of generating all the six-axis motions with a single levitated platen were used in this research. These light-weight single-moving platens have very simple and compact structures, which give them an edge over most of the prevailing nanopositioning technologies and allow them to be used as a cluster tool for a variety of applications. The six-axis motion is generated using minimum number of actuators and sensors. The two positioners operate with a repeatable position resolution of better than 3 nm at the control bandwidth of 110 Hz. In particular, the Y-stage has extended travel range of 5 mm ÃÂ 5 mm. They can carry a payload of as much as 0.3 kg and retain the regulated position under abruptly and continuously varying load conditions. This research comprised analytical design and development, followed by experimental verification and validation. Preliminary analysis and testing included open-loop stabilization and rigorous set-point change and load-change testing to demonstrate the precision-positioning and load-carrying capabilities of the maglev positioners. Decentralized single-input-single-output (SISO) proportional-integral-derivative (PID) control was designed for this analysis. The effect of actuator nonlinearities were reduced through actuator characterization and nonlinear feedback linearization to allow consistent performance over the large travel range. Closed-loop system identification and order-reduction algorithm were developed in order to analyze and model the plant behavior accurately, and to reduce the effect of unmodeled plant dynamics and inaccuracies in the assembly. Coupling among the axes and subsequent undesired motions and crosstalk of disturbances was reduced by employing multivariable optimal linear-quadratic regulator (LQR). Finally, application-specific nanoscale path planning strategies and multiscale control were devised to meet the specified conflicting time-domain performance specifications. All the developed methodologies and algorithms were implemented, individually as well as collectively, for experimental verification. Some of these applications included nanoscale lithography, patterning, fabrication, manipulation, and scanning. With the developed control strategies and motion planning techniques, the two maglev positioners are ready to be used for the targeted applications