| The subject of this thesis is the control of singularly perturbed Markov and semi-Markov jump linear systems, as well as a particular control problem in accelerator physics known as the Bunch-Train Cavity Interaction (BTCI). It is shown that the BTCI is in fact an example of a semi-Markov jump linear system.; We begin by developing a new state-space linear model for the BTCI. Practically speaking, this model is more general than its frequency-domain predecessors and is well suited for state-space control design.; The delta operator is next introduced, which, save for some anomalous cases, unifies both continuous-time and discrete-time system theory. Next, a composite control method (of fast and slow modes) for singularly-perturbed delta-operator formulated discrete-time systems is derived. Then, we present a means for obtaining the upper-bound on the stability-invariant singular perturbation parameter. Tools for stability analysis, including a bound on the singular perturbation parameter for jump-switched systems are also developed.; The remainder of the thesis is devoted to stochastically jumping linear systems, specifically those whose regime sojourn times are exponentially or non-exponentially distributed. The two cases are respectively referred to as Markov and semi-Markov jump linear systems. In the case of Markov jump linear systems we again unify existing singular perturbation results for continuous time, extending them via the delta operator to the discrete-time case. Then, a new near-optimal composite control for singularly perturbed delta-operator formulated jump linear systems is proposed. We also apply our results on switched systems to introduce an almost sure bound for the singular perturbation parameter for the Markov-jump-linear-system case.; Finally, we develop a new algorithm for semi-Markov jump linear system control design. The algorithm is based on the stochastic stopping time idea used often in martingale theory. This algorithm is numerically tractable and convergent under mild conditions, and retrieves a control law numerically identical to the closed-form Markov case when the system under consideration is in fact Markov. Extensions to singularly-perturbed semi-Markov jump linear systems are also considered. |
