Reliable supervisory control for general architecture of decentralized discrete event systems

In this paper, we investigate the reliable decentralized supervisory control of discrete event systems (DESs) under the general architecture, where the decision for controllable events is a combination of the conjunctive and disjunctive fusion rules. By reliable control, we mean that the performance of closed-loop systems will not be degraded even in the face of possible failures of some local supervisors. The main contributions are twofold. First, a necessary and sufficient condition for the existence of a k-reliable decentralized supervisor under the general architecture is presented after introducing notions of -controllability and k-reliable -coobservability. Second, a polynomial-time algorithm to verify the reliable -coobservability of a specification is proposed.     

Regularization parameter estimation for large-scale Tikhonov regularization using a priori information

Solutions of numerically ill-posed least squares problems for A∈R^m×n by Tikhonov regularization are considered. For D∈R^p×n, the Tikhonov regularized least squares functional is given by where matrix W is a weighting matrix and is given. Given a priori estimates on the covariance structure of errors in the measurement data , the weighting matrix may be taken as which is the inverse covariance matrix of the mean 0 normally distributed measurement errors in . If in addition is an estimate of the mean value of , and σ is a suitable statistically-chosen value, J evaluated at its minimizer approximately follows a χ² distribution with degrees of freedom. Using the generalized singular value decomposition of the matrix pair , σ can then be found such that the resulting J follows this χ² distribution. But the use of an algorithm which explicitly relies on the direct solution of the problem obtained using the generalized singular value decomposition is not practical for large-scale problems. Instead an approach using the Golub-Kahan iterative bidiagonalization of the regularized problem is presented. The original algorithm is extended for cases in which is not available, but instead a set of measurement data provides an estimate of the mean value of . The sensitivity of the Newton algorithm to the number of steps used in the Golub-Kahan iterative bidiagonalization, and the relation between the size of the projected subproblem and σ are discussed. Experiments presented contrast the efficiency and robustness with other standard methods for finding the regularization parameter for a set of test problems and for the restoration of a relatively large real seismic signal. An application for image deblurring also validates the approach for large-scale problems. It is concluded that the presented approach is robust for both small and large-scale discretely ill-posed least squares problems.     

Narrowing power vs efficiency in synchronous set agreement: Relationship, algorithms and lower bound

The k-set agreement problem is a generalization of the uniform consensus problem: each process proposes a value, and each non-faulty process has to decide a value such that a decided value is a proposed value, and at most k different values are decided. It has been shown that any algorithm that solves the k-set agreement problem in synchronous systems that can suffer up to t crash failures requires rounds in the worst case. It has also been shown that it is possible to design early deciding algorithms where no process decides and halts after rounds, where f is the number of actual crashes in a run (0≤f≤t).This paper explores a new direction to solve the k-set agreement problem in a synchronous system. It considers that the system is enriched with base objects (denoted has [m,?]_SA objects) that allow solving the ?-set agreement problem in a set of m processes (m<n). The paper makes several contributions. It first proposes a synchronous k-set agreement algorithm that benefits from such underlying base objects. This algorithm requires rounds, more precisely, rounds, where . The paper then shows that this bound, that involves all the parameters that characterize both the problem (k) and its environment (t, m and ?), is a lower bound. The proof of this lower bound sheds additional light on the deep connection between synchronous efficiency and asynchronous computability. Finally, the paper extends its investigation to the early deciding case. It presents a k-set agreement algorithm that directs the processes to decide and stop by round . These bounds generalize the bounds previously established for solving the k-set agreement problem in pure synchronous systems.     

Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities

The conjecture that periodically switched stability implies absolute asymptotic stability of random infinite products of a finite set of square matrices, has recently been disproved under the guise of the finiteness conjecture. In this paper, we show that this conjecture holds in terms of Markovian probabilities. More specifically, let S_k∈C^n×n,1≤k≤K, be arbitrarily given K matrices and , where n,K≥2. Then we study the exponential stability of the following discrete-time switched dynamics S: where can be an arbitrary switching sequence.For a probability row-vector and an irreducible Markov transition matrix with , we denote by the Markovian probability on corresponding to . By using symbolic dynamics and ergodic-theoretic approaches, we show that, if S possesses the periodically switched stability then, (i) it is exponentially stable -almost surely; (ii) the set of stable switching sequences has the same Hausdorff dimension as . Thus, the periodically switched stability of a discrete-time linear switched dynamics implies that the system is exponentially stable for “almost” all switching sequences.     

A pseudo-random network mobile automaton with linear growth

Based on the mobile automaton model, an algorithm is introduced that grows planar, tri-valent graphs by exhibiting a peculiar, twofold dynamics. In a first phase, graph growth appears to be pseudo-random and O(n) then it settles to a very regular behavior and rate. A pseudo-random mobile automaton is already known; the new automaton provides now a finite, but surprisingly long, pseudo-random, linear growth process. Applications of mobile automata to fundamental physics and quantum gravity have been recently suggested.     

On generalized middle-level problem

Let be the subgraph of the hypercube Q_n induced by levels between k and n-k, where n?2k+1 is odd. The well-known middle-level conjecture asserts that is Hamiltonian for all k?1. We study this problem in for fixed k. It is known that and are Hamiltonian for all odd n?3. In this paper we prove that also is Hamiltonian for all odd n?5, and we conjecture that is Hamiltonian for every k?0 and every odd n?2k+1.     

Pentagons: A weakly relational abstract domain for the efficient validation of array accesses

We introduce Pentagons (), a weakly relational numerical abstract domain useful for the validation of array accesses in byte-code and intermediate languages (IL). This abstract domain captures properties of the form of . It is more precise than the well known Interval domain, but it is less precise than the Octagon domain.The goal of is to be a lightweight numerical domain useful for adaptive static analysis, where is used to quickly prove the safety of most array accesses, restricting the use of more precise (but also more expensive) domains to only a small fraction of the code.We implemented the abstract domain in , a generic abstract interpreter for.NET assemblies. Using it, we were able to validate 83% of array accesses in the core runtime library in a little bit more than 3 minutes.     

Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons

Motivated by routing issues in ad hoc networks, we present polylogarithmic-time distributed algorithms for two problems. Given a network, we first show how to compute connected and weakly connected dominating sets whose size is at most times the optimum, Δ being the maximum degree of the input network. This is best-possible if and if the processors are required to run in polynomial-time. We then show how to construct dominating sets that have the above properties, as well as the “low stretch” property that any two adjacent nodes in the network have their dominators at a distance of at most in the output network. (Given a dominating set S, a dominator of a vertex u is any v∈S such that the distance between u and v is at most one.) We also show our time bounds to be essentially optimal.     

Probabilistic integer sorting

We introduce a probabilistic sequential algorithm for stable sorting n uniformly distributed keys in an arbitrary range. The algorithm runs in linear time and sorts all but a very small fraction of the input sequences; the best previously known bound was . An EREW PRAM extension of this sequential algorithm sorts in O((n/p+lgp)lgn/lg(n/p+lgn)) time using p?n processors under the same probabilistic conditions. For a CRCW PRAM we improve upon the probabilistic bound of obtained by Rajasekaran and Sen to derive a bound. Additionally, we present experimental results for the sequential algorithm that establish the practicality of our method.     

A compact data structure for representing a dynamic multiset

We develop a data structure for maintaining a dynamic multiset that uses bits and O(1) words, in addition to the space required by the n elements stored, supports searches in worst-case time and updates in amortized time. Compared to earlier data structures, we improve the space requirements from O(n) bits to bits, but the running time of updates is amortized, not worst-case.     

Invariance properties in the root sensitivity of time-delay systems with double imaginary roots

If is an eigenvalue of a time-delay system for the delay τ₀ then is also an eigenvalue for the delays τ_k?τ₀+k2π/ω, for any k∈Z. We investigate the sensitivity, periodicity and invariance properties of the root for the case that is a double eigenvalue for some τ_k. It turns out that under natural conditions (the condition that the root exhibits the completely regular splitting property if the delay is perturbed), the presence of a double imaginary root for some delay τ₀ implies that is a simple root for the other delays τ_k, k≠0. Moreover, we show how to characterize the root locus around . The entire local root locus picture can be completely determined from the square root splitting of the double root. We separate the general picture into two cases depending on the sign of a single scalar constant; the imaginary part of the first coefficient in the square root expansion of the double eigenvalue.     

Learning of event-recording automata

In regular inference, a regular language is inferred from answers to a finite set of membership queries, each of which asks whether the language contains a certain word. One of the most well-known regular inference algorithms is the L^∗ algorithm due to Dana Angluin. However, there are almost no extensions of these algorithms to the setting of timed systems. We extend Angluin’s algorithm for on-line learning of regular languages to the setting of timed systems. Since timed automata can freely use an arbitrary number of clocks, we restrict our attention to systems that can be described by deterministic event-recording automata (DERAs). We present three algorithms, , and , for inference of DERAs. In and , we further restrict event-recording automata to be event-deterministic in the sense that each state has at most one outgoing transition per action; learning such an automaton becomes significantly more tractable. The algorithm builds on , by attempts to construct a smaller (in number of locations) automaton. Finally, is a learning algorithm for a full class of deterministic event-recording automata, which infers a so called simple DERA, which is similar in spirit to the region graph.     

Bottleneck flows in unit capacity networks

The bottleneck network flow problem (BNFP) is a generalization of several well-studied bottleneck problems such as the bottleneck transportation problem (BTP), bottleneck assignment problem (BAP), bottleneck path problem (BPP), and so on. The BNFP can easily be solved as a sequence of O(logn) maximum flow problems on almost unit capacity networks. We observe that this algorithm runs in O(min{m3/2,n2/3m}logn) time by showing that the maximum flow problem on an almost unit capacity graph can be solved in O(min{m3/2,n2/3m}) time. We then propose a faster algorithm to solve the unit capacity BNFP in time, an improvement by a factor of at least . For dense graphs, the improvement is by a factor of . On unit capacity simple graphs, we show that BNFP can be solved in time, an improvement by a factor of . As a consequence we have an algorithm for the BTP with unit arc capacities.     

Calculation of the stability ratio of suspensions using Emulsion Stability Simulations

According to Emulsion Stability Simulations (ESS), the flocculation of two non-deformable drops in the primary minimum of the interaction potential, necessarily leads to their coalescence. This property is used here for the evaluation of the stability ratio (W) of solid particles, interacting with the same inter-particle potential as the one of non-deformable droplets. Two different methodologies are used. The first one consists on the repeated evaluation of the coalescence time between two particles. The second one consists on the estimation of the time required for a decrease in the number of aggregates of the dispersion equal to n₀/2 (where n₀ is the initial number of aggregates). The results of the simulations are contrasted with the stability ratio of an anionic latex suspension subject to several ionic strengths (400-1000 mM). The first methodology is far more efficient for the evaluation of W although it misses the development of the aggregates and their growth. Absolute coagulation rates (kf) can also be obtained using one N-particle simulation for the calculation of the fast flocculation rate , and several two-particle simulations for the evaluation of W. This combined procedure is also more efficient than the N-particle evaluation of .     

Embedding paths and cycles in 3-ary n-cubes with faulty nodes and links

The k-ary n-cube, denoted by , is one of the most important interconnection networks for parallel computing. In this paper, we consider the problem of embedding cycles and paths into faulty 3-ary n-cubes. Let F be a set of faulty nodes and/or edges, and n?2. We show that when |F|?2n-2, there exists a cycle of any length from 3 to in . We also prove that when |F|?2n-3, there exists a path of any length from 2n-1 to between two arbitrary nodes in . Since the k-ary n-cube is regular of degree 2n, the fault-tolerant degrees 2n-2 and 2n-3 are optimal.     

Spectral radius minimization for optimal average consensus and output feedback stabilization

In this paper, we consider two problems which can be posed as spectral radius minimization problems. Firstly, we consider the fastest average agreement problem on multi-agent networks adopting a linear information exchange protocol. Mathematically, this problem can be cast as finding an optimal such that x(k+1)=Wx(k), , and W∈S(E). Here, is the value possessed by the agents at the kth time step, is an all-one vector and S(E) is the set of real matrices in with zeros at the same positions specified by a network graph G(V,E), where V is the set of agents and E is the set of communication links between agents. The optimal W is such that the spectral radius is minimized. To this end, we consider two numerical solution schemes: one using the qth-order spectral norm (2-norm) minimization (q-SNM) and the other gradient sampling (GS), inspired by the methods proposed in [Burke, J., Lewis, A., & Overton, M. (2002). Two numerical methods for optimizing matrix stability. Linear Algebra and its Applications, 351-352, 117-145; Xiao, L., & Boyd, S. (2004). Fast linear iterations for distributed averaging. Systems & Control Letters, 53(1), 65-78]. In this context, we theoretically show that when E is symmetric, i.e. no information flow from the ith to the jth agent implies no information flow from the jth to the ith agent, the solution from the 1-SNM method can be chosen to be symmetric and is a local minimum of the function . Numerically, we show that the q-SNM method performs much better than the GS method when E is not symmetric. Secondly, we consider the famous static output feedback stabilization problem, which is considered to be a hard problem (some think NP-hard): for a given linear system (A,B,C), find a stabilizing control gain K such that all the real parts of the eigenvalues of A+BKC are strictly negative. In spite of its computational complexity, we show numerically that q-SNM successfully yields stabilizing controllers for several benchmark problems with little effort.     

Finding frequent items over sliding windows with constant update time

In this paper, we consider the problem of finding ?-approximate frequent items over a sliding window of size N. A recent work by Lee and Ting (2006) [7] solves the problem by giving an algorithm that supports query and update time, and uses space. Their query time and memory usage are essentially optimal, but the update time is not. We give a new algorithm that supports O(1) update time with high probability while maintaining the query time and memory usage as .