SUMMARYIn this article, a synthesis method for linear systems subject to uncertain time-varying parameters is proposed. Starting with the given values for the nominal parameters, the first objective is to find a constant state feedback such that the controlled system is stabilized and given dynamic specifications, such as a minimal decay rate, are met. In a second step, the constant feedback is then locally optimized such that the uncertain system parameters are allowed to vary around their nominal point as much as possible, whereas stabilization and dynamic specifications still hold. In addition, the procedure is extended toward observer-based state feedback. Finally, this approach is applied to a synchrotron example, where the particle beam in longitudinal direction is to be stabilized and the coherent synchrotron frequency as well as the damping rate are uncertain.
The bunched particle beam in a synchrotron can perform various longitudinal oscillation modes of which the dipole mode occurs most frequently. Although naturally damped by Landau damping, these oscillations can become unstable if driven accordingly. In any case Landau damping is accompanied by filamentation of the bunch which leads to rms emittance blow up and thus reduces the beam quality. Therefore a beam-phase feedback is used to damp dipole oscillations. At GSI Helmholtzzentrum für Schwerionenforschung GmbH the feedback is designed as an FIR filter [1]. However, the feedback performance may be improved using a matched filter instead of the current filter setting as is demonstrated in this work by comparing the different filter designs.
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