Experimenting with Hybrid Control

There is a growing realization among educators andemployers that students of automatic control should be encouraged tothink of the subject in broader terms. The systems approach shouldembrace communication requirements, signal processing, data logging,etc. all the way up to and including the level of complexity suggestedby the phrase "enterprise control." Designing a controlexperiment that is illustrative and instructional in this broadersense presents a number of challenges beyond those discussedabove. The systems under consideration must be very flexible. Ofcourse the hardware must continue to be reliable and relatively easyto understand at an intuitive level. They should also reflect thecomplexity of purpose and the possibility of multi-modal operationthat one expects to find in complex systems. With these qualities inmind, we have assembled and extensively exercised an experimentalhybrid control system for use in an instructional/research laboratoryat Harvard. Our goal with this paper is to describe for others thestructure of the system and to present a sample of the experimentsthat were facilitated by it.An important feature of the facility we describe is that it uses severaltypes of sensing modalities including position sensing, tactile sensingand more conventional vision sensing. It can interact with objectsof different complexity and is subject to communication constraints arising in a completelynatural and generic way. In constructing it we have used off-the-shelfcomponents wherever possible and made choices with an eye towardflexibility and reliability.The research and scientific content in this material has been submitted to the IEEE Control Systems Magazine.</Center

Sub-Riemannian Geometry and Time Optimal Control of Three Spin Systems: Quantum Gates and Coherence Transfer

Many coherence transfer experiments in Nuclear Magnetic Resonance Spectroscopy, involving network of coupled spins, use temporary spin-decoupling to produce desired effective Hamiltonians. In this paper, we show that significant time can be saved in producing an effective Hamiltonian, if spin-decoupling is avoided. We provide time optimal pulse sequences for producing an important class of effective Hamiltonians in three spin networks. These effective Hamiltonians are useful for coherence transfer experiments and implementation of quantum logic gates in NMR quantum computing. It is demonstrated that computing these time optimal pulse sequences can be reduced to geometric problems that involve computing sub-Riemannian geodesics on Homogeneous spaces

Time Optimal Control in Spin Systems

In this paper, we study the design of pulse sequences for NMR spectroscopy as a problem of time optimal control of the unitary propagator. Radio frequency pulses are used in coherent spectroscopy to implement a unitary transfer of state. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. Moreover, the general proofs are no more difficult than the specific problems of current interest. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group, what is the minimum time required to steer the system from some initial point to a specified final point? In NMR spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two spin system.Comment: 20 Pages, 3 figure

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In this paper, we study the problem of determining a mathematical description of the surface defined by the shape of a membrane based on an image of it and present an algorithm for reconstructing the surface when the membrane is deformed by unknown external elements. The given data are the projection on an image plane of markings on the surface of the membrane, the undeformed configuration of the membrane, and a model for the membrane mechanics. The method of reconstruction is based on the principle that the shape assumed by the membrane will minimize the elastic energy stored in the membrane subject to the constraints implied by the measurements. Energy minimization leads to a set of nonlinear partial differential equations. An approximate solution is found using linearization. The initial motivation, and our first application of these ideas, comes from tactile sensing. Experimental results affirm that this approach can b

On explicit steady-state solutions of Fokker-Planck equations for a class of nonlinear feedback systems

We study the question of existence of steady-state probability distributions for systems perturbed by white noise. We describe a class of nonlinear feedback systems for which an explicit formula for the steady-state probability density can be found. These systems include what has been called monotemperaturic systems in earlier work. We also establish relationships between the steady-state probability densities and Liapunov functions for the corresponding deterministic systems. 1 Introduction The study of linear systems excited by white noise of constant intensity is greatly facilitated by the fact that one has an explicit formula for the solution of the Fokker-Planck equation which describes the evolution of the probability density. For nonlinear systems the situation is quite different. Namely, not only are the transient solutions difficult to find, but even the steady-state solutions are hard to characterize. In this paper we show that for a certain class of nonlinear systems the s..