Shattering Thresholds for Random Systems of Sets, Words, and Permutations

This paper considers a problem that relates to the theories of covering arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability thresholds. Specifically, we want to find the number of subsets of [n]:={1,2,....,n} we need to randomly select, in a certain probability space, so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to words, we ask for the number of n-letter words on a q-letter alphabet that are needed to shatter all t-subwords of the q^n words of length n. Finally, we explore the number of random permutations of [n] needed to shatter (specializing to t=3), all length 3 permutation patterns in specified positions. We uncover a very sharp zero-one probability threshold for the emergence of such shattering; Talagrand's isoperimetric inequality in product spaces is used as a key tool.Comment: 25 page

On suboptimal control design for hybrid automata using predictive control techniques

In this paper we propose an on-line design technique for the target control problem, when the system is modelled by hybrid automata. First, we compute off-line the shortest path, which has the minimum discrete cost, from an initial state to the given target set. Next, we derive a controller which successfully drives the system from the initial state to the target set while minimizing a cost function. The model predictive control (MPC) technique is used when the current state is not within a guard set, otherwise the mixed-integer predictive control (MIPC) technique is employed. An on-line, semi-explicit control algorithm is derived by combining the two techniques. Finally, as an application of the proposed control procedure, the high-speed and energy-saving control problem of the CPU processing isconsidered