CHIMAN KWAN

To control the beam in the synchrotron there may be six
different primary feedback loops interacting with the beam
at a given time. Three loops are local to the rf cavity. They
are: high bandwidth cavity phase and amplitude loops
used to minimize the effects due to beam loading and a low
bandwidth cavity tuning loop. The loops global to the ring
accelerating system are: a radial loop to keep the beam on
orbit, a beam phase loop to damp the dipole synchrotron
oscillations, and a synchronization loop to essentially lock
with the succeeding machine. There are various ways in
which these loops may be designed. Designs currently in
use in operating machines are based on classical frequency
domain techniques. To apply modern feedback controllers
and study the interaction of all the feedback loops, a good
mathematical model of the beam is extremely useful. In
this paper we show the derivation of a non-linear tracking
model in terms of differential equations obtained from a
set of time varying finite difference equations. The model
compares well with the results of thin element tracking
codes.

The theory of unifying sliding mode control and classical control for linear SISO systems has been extended to the MIMO linear case. The approach can be viewed as an extension of conventional sliding mode control by including control inputs in the sliding variable definition. Another viewpoint is that when the system is in sliding mode, a classical transfer matrix is realized. This novel approach retains the merits of both sliding and classical controllers on one hand and eliminates their respective limitations on the other. The method is robust and applies to non-minimum phase systems as well as systems with structural uncertainties. No state measurement is required. Chattering can also be alleviated.

The main purpose of this brief paper is to report some preliminary results of a new approach of population control. The frequency-domain model of population system is revisited. Without any simplifications, this model has an infinite-dimensional transfer function with timedelay. Here, a reduced-order model is obtained which is further applied to the stability analysis of population system. Algorithms relating the average fertility rate to the total population have been derived and verified. These algorithms can be applied to population forecast and the construction of population policy. Unlike the previous approaches where functional analysis and nonlinear control theory are being used, all the materials here can be undprrfmd with very limited exposure to classical control and linear systems theory.

Phase-locking the Low Energy Booster to the Mechanism Energy Booster using Trip-plan'' approach is under development. With this scheme it is possible to phase lock the two machines at any time while ramping, even with wide frequency range in the low energy machines. This loop also has the potential to damp the phase oscillations and keep the beam in orbit by using the Low Energy Booster beam signal and a master clock as two moving references. A brief description of the bench test loop set up and the experimental results are shown in this paper to demonstrate the idea. We have investigated the ability of the loop to damp oscillations and also synchronize reference bunches. With the use of special algorithms it looks possible to operate the machine without the radial and beam phase loop. The implementations of such a scheme depends on the computational speed of the processors and the ability of fast Direct Digital Synthesizers to produce the guiding RF signal.