Decentralized pole assignment for interconnected systems

Given a general proper interconnected system, this paper aims to design a LTI decentralized controller to place the modes of the closed-loop system at pre-determined locations. To this end, it is first assumed that the structural graph of the system is strongly connected. Then, it is shown applying generic static local controllers to any number of subsystems will not introduce new decentralized fixed modes (DFM) in the resultant system, although it has fewer inputoutput stations compared to the original system. This means that if there are some subsystems whose control costs are highly dependent on the complexity of the control law, then generic static controllers can be applied to such subsystems, without changing the characteristics of the system in terms of the fixed modes. As a direct application of this result, in the case when the system has no DFMs, one can apply generic static controllers to all but one subsystem, and the resultant system will be controllable and observable through that subsystem. Now, a simple observer-based local controller corresponding to this subsystem can be designed to displace the modes of the entire system arbitrarily. Similar results can also be attained for a system whose structural graph is not strongly connected. It is worth mentioning that similar concepts are deployed in the literature for the special case of strictly proper systems, but as noted in the relevant papers, extension of the results to general proper systems is not trivial. This demonstrates the significance of the present work

Dictionary Matching with One Gap

The dictionary matching with gaps problem is to preprocess a dictionary $D$ of $d$ gapped patterns $P_1,\ldots,P_d$ over alphabet $\Sigma$, where each gapped pattern $P_i$ is a sequence of subpatterns separated by bounded sequences of don't cares. Then, given a query text $T$ of length $n$ over alphabet $\Sigma$, the goal is to output all locations in $T$ in which a pattern $P_i\in D$, $1\leq i\leq d$, ends. There is a renewed current interest in the gapped matching problem stemming from cyber security. In this paper we solve the problem where all patterns in the dictionary have one gap with at least $\alpha$ and at most $\beta$ don't cares, where $\alpha$ and $\beta$ are given parameters. Specifically, we show that the dictionary matching with a single gap problem can be solved in either $O(d\log d + |D|)$ time and $O(d\log^{\varepsilon} d + |D|)$ space, and query time $O(n(\beta -\alpha )\log\log d \log ^2 \min \{ d, \log |D| \} + occ)$, where $occ$ is the number of patterns found, or preprocessing time and space: $O(d^2 + |D|)$, and query time $O(n(\beta -\alpha ) + occ)$, where $occ$ is the number of patterns found. As far as we know, this is the best solution for this setting of the problem, where many overlaps may exist in the dictionary.Comment: A preliminary version was published at CPM 201

Pole Assignment With Improved Control Performance by Means of Periodic Feedback

This technical note is concerned with the pole placement of continuous-time linear time-invariant (LTI) systems by means of LQ suboptimal periodic feedback. It is well-known that there exist infinitely many generalized sampled-data hold functions (GSHF) for any controllable LTI system to place the modes of its discrete-time equivalent model at prescribed locations. Among all such GSHFs, this technical note aims to find the one which also minimizes a given LQ performance index. To this end, the GSHF being sought is written as the sum of a particular GSHF and a homogeneous one. The particular GSHF can be readily obtained using the conventional pole-placement techniques. The homogeneous GSHF, on the other hand, is expressed as a linear combination of a finite number of functions such as polynomials, sinusoidals, etc. The problem of finding the optimal coefficients of this linear combination is then formulated as a linear matrix inequality (LMI) optimization. The procedure is illustrated by a numerical example