In this paper, a new control scheme called repetitive control is proposed, in which the controlled variables follow periodic reference commands. A high accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period L into the closed-loop system. Sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small gain theorem and the stability theorem for time-lag systems. Synthesis algorithms are presented both by the state-space approach and the factorization approach. In the former approach, the technique of the Kalman filter and perfect regulation is utilized, while coprime factorization over the matrix ring of proper stable rational functions and the solution of the Hankel norm approximation are used in the latter one.

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We observe the formation in a single-photon transition of two core holes, each at a different carbon atom of the C2H2 molecule. At a photon energy of 770.5 eV, the probability of this 2-site core double ionization amounts to 1.6 ± 0.4% of the 1-site core double ionization. A simple theoretical model based on the knockout mechanism gives reasonable agreement with experiment. Spectroscopy and Auger decays of the associated double core hole states are also investigated.

The formation of hollow molecules (with a completely empty K shell in one constituent atom) through single-photon core double ionization has been demonstrated using a sensitive magnetic bottle experimental technique combined with synchrotron radiation. Detailed properties are presented such as the spectroscopy, formation, and decay dynamics of the N(2)(2+) K(-2) main and satellite states and the strong chemical shifts of double K holes on an oxygen atom in CO, CO2, and O2 molecules.

In this article, a new control scheme named repetitive control is proposed for a class of nonlinear systems described by x(t) = Ax(t) + Bu(t) + n(x(t)) and y(t) = Cx(t), in which the controlled variables follow periodic reference commands. The stability condition is derived by applying the passivity theorem. We show how to apply the repetitive control scheme to the trajectory control of a manipulator. A simple repetitive control scheme is developed for the trajectory control of a manipulator by using nonlinear compensation and feedbacks of position and velocity signals. Experimental results for a three link manipulator verify that the proposed repetitive control reduces the tracking error to a very low level.