Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases

Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, ${\int_t^{t+T}\alpha(s){\rm d}s \geq \mu}Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, . In particular, when α(t) becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback u = Kx, with K only depending on (A, B) and possibly on μ, T, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when A is neutrally stable and when the system is the double integrator. Notation  A continuous function is of class , if it is strictly increasing and is of class if it is continuous, non-increasing and tends to zero as its argument tends to infinity. A function is said to be a class -function if, for any t ≥ 0, and for any s ≥ 0. We use |·| for the Euclidean norm of vectors and the induced L ₂-norm of matrices.     

Fuzzy logic models in a category of fuzzy relations

We investigate interpretations of formulas ψ in a first order fuzzy logic in models which are based on objects of a category SetR(Ω) which consists of Ω-sets, i.e. sets with similarity relations with values in a complete MV-algebra Ω and with morphisms defined as special fuzzy relations between Ω-sets. The interpretations are then morphisms in a category SetR(Ω) from some Ω-set to the object . We define homomorphisms between models in a category SetR(Ω) and we prove that if is a (special) homomorphism of models in a category SetR(Ω) then there is a relation between interpretations of a formula ψ in models . Supported by MSM6198898701, grant 201/07/0191 of GAČR and grant 1M0572.     

Processing fuzzy set membership functionals as vectors

A framework is presented for processing fuzzy sets for which the universe of discourse X = {x} is a separable Hilbert Space, which, in particular, may be a Euclidian Space. In a given application, X would constitute a feature space. The membership functions of sets in such X are then “membership functionals”, that is, mappings from a vector space to the real line. This paper considers the class Φ of fuzzy sets A, the membership functionals μ _A of which belong to a Reproducing Kernel Hilbert Space (RKHS) F(X) of bounded analytic functionals on X, and satisfy the constraint . These functionals can be expanded in abstract power series in x, commonly known as Volterra functional series in x. Because of the one-to-one relationship between the fuzzy sets A and their respective μ _A , one can process the sets A as objects using their μ _A as intermediaries. Thus the structure of the uncertainty present in the fuzzy sets can be processed in a vector space without descending to the level of processing of vectors in the feature space as usually done in the literature in the field. Also, the framework allows one to integrate human and machine judgments in the definition of fuzzy sets; and to use concepts analogous to probabilistic concepts in assigning membership values to combinations of fuzzy sets. Some analytical and interpretive consequences of this approach are presented and discussed. A result of particular interest is the best approximation of a membership functional μ _A in F(X) based on interpolation on a training set {(v ⁱ , u ⁱ ),i = 1, . . . , q} and under the positivity constraint. The optimal analytical solution comes out in the form of an Optimal Interpolative Neural Network (OINN) proposed by the author in 1990 for best approximation of pattern classification systems in a F(X) space setting. An example is briefly described of an application of this approach to the diagnosis of Alzheimer’s disease.     

Strong non-standard completeness for fuzzy logics

In this paper we are going to introduce the notion of strong non-standard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic is strong non-standard complete means that, for any countable theory Γ over and any formula φ such that , there exists an evaluation e of -formulas into a -algebra such that the universe of is a non-Archimedean extension of the real unit interval [0,1], e is a model for Γ, but e(φ) < 1. Then we will apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinite-valued events are complete with respect to classes of models defined starting from non-standard measures, that is measures taking value in .     

Computing Population Variance and Entropy under Interval Uncertainty: Linear-Time Algorithms

In statistical analysis of measurement results it is often necessary to compute the range of the population variance when we only know the intervals of possible values of the x _i . While can be computed efficiently, the problem of computing is, in general, NP-hard. In our previous paper “Population Variance under Interval Uncertainty: A New Algorithm” (Reliable Computing 12 (4) (2006), pp. 273–280) we showed that in a practically important case we can use constraints techniques to compute in time O(n · log(n)). In this paper we provide new algorithms that compute (in all cases) and (for the above case) in linear time O(n). Similar linear-time algorithms are described for computing the range of the entropy when we only know the intervals of possible values of probabilities p _i . In general, a statistical characteristic ƒ can be more complex so that even computing ƒ can take much longer than linear time. For such ƒ, the question is how to compute the range in as few calls to ƒ as possible. We show that for convex symmetric functions ƒ, we can compute in n calls to ƒ.     

Generalized fuzzy bi-ideals of semigroups

After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of (∈, ∈ ∨q)-fuzzy subgroups introduced by Bhakat is one among them. In this paper, using the relations between fuzzy points and fuzzy sets, the concept of a fuzzy bi-ideal with thresholds is introduced and some interesting properties are investigated. The acceptable nontrivial concepts obtained in this manner are the (∈, ∈ ∨q)-fuzzy bi-ideals and -fuzzy bi-ideals, which are extension of the concept of a fuzzy bi-ideal in semigroup.     

Fast and exact projected convolution for non-equidistant grids

Usually, the fast evaluation of a convolution integral requires that the functions f, g are discretised on an equidistant grid in order to apply the fast Fourier transform. Here we discuss the efficient performance of the convolution in locally refined grids. More precisely, the convolution result is projected into some given locally refined grid (Galerkin approximation). Under certain conditions, the overall costs are still where N is the sum of the dimensions of the subspaces containing f, g and the resulting function.     

A Novel Class of Symmetric and Nonsymmetric Periodizing Variable Transformations for Numerical Integration

Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the integral via a variable transformation that maps [0,1] to itself, and then approximates the resulting transformed integral by the trapezoidal rule. In this work, we propose a new class of symmetric and nonsymmetric variable transformations which we denote , where r and s are positive scalars assigned by the user. A simple representative of this class is . We show that, in case , or but has algebraic (endpoint) singularities at x = 0 and/or x = 1, the trapezoidal rule on the transformed integral produces exceptionally high accuracies for special values of r and s. In particular, when and we employ , the error in the approximation is (i) O(h ^r ) for arbitrary r and (ii) O(h ^2r ) if r is a positive odd integer at least 3, h being the integration step. We illustrate the use of these transformations and the accompanying theory with numerical examples.      

Topology on the set of R 0 semantics for R 0 algebras

This paper introduces the concepts of R ₀ valuation, R ₀ semantic, countable R ₀ category , R ₀ fuzzy topological category , etc. It is established in a natural way that the fuzzy topology δ and its cut topology on the set Ω _M consisting of all R ₀ valuations of an R ₀ algebra M, and some properties of fuzzy topology δ and its cut topology are investigated carefully. Moreover, the representation theorem for R ₀ algebras by means of fuzzy topology is given, that is to say the category is equivalent to the category . By studying the relation between valuations and filters, the Loomis–Sikorski theorem for R ₀ algebras is obtained. As an application, K-compactness of the R ₀ logic is discussed.     

A simple population protocol for fast robust approximate majority

We describe and analyze a 3-state one-way population protocol to compute approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of x’s, y’s and blanks that contains at least one non-blank, the goal is for the agents to reach consensus on one of the values x or y. Additionally, the value chosen should be the majority non-blank initial value, provided it exceeds the minority by a sufficient margin. We prove that with high probability n agents reach consensus in O(n log n) interactions and the value chosen is the majority provided that its initial margin is at least . This protocol has the additional property of tolerating Byzantine behavior in of the agents, making it the first known population protocol that tolerates Byzantine agents.     

On the computability power and the robustness of set agreement-oriented failure detector classes

Solving agreement problems deterministically, such as consensus and k-set agreement, in asynchronous distributed systems prone to an unbounded number of process failures has been shown to be impossible. To circumvent this impossibility, unreliable failure detectors for the crash failure model have been widely studied. These are oracles that provide information on failures. The exact nature of such information is defined by a set of abstract properties that a particular class of failure detectors satisfy. The weakest class of such failure detectors that allow to solve consensus is Ω. This paper considers failure detector classes from the literature that solve k-set agreement in the crash failure model, and studies their relative power. It shows that the family of failure detector classes (1 ≤ x ≤ n), and (0 ≤ y ≤ n), can be “added” to provide a failure detector of the class Ω ^z (1 ≤ z ≤ n, a generalization of Ω). It also characterizes the power of such an “addition”, namely, , can construct Ω ^z iff y + z > t, and can construct Ω ^z iff x + z > t + 1, where t is the maximum number of processes that can crash in a run. As an example, the paper shows that, while allows solving 2-set agreement (but not consensus) and allows solving t-set agreement (but not (t − 1)-set agreement), a system with failure detectors of both classes can solve consensus for any value of t. More generally, the paper studies the failure detector classes , and Ω ^z , and shows which reductions among these classes are possible and which are not. The paper also presents a message-passing Ω ^k -based k-set agreement protocol and shows that Ω ^k is not enough to solve (k − 1)-set agreement. In that sense, it can be seen as a step toward the characterization of the weakest failure detector class that allows solving the k-set agreement problem. An extended abstract of this paper has appeared in the proceedings of PODC 2006 [20]. This work has been supported partially by a grant from LAFMI (Franco-Mexican Lab in Computer Science), the European Network of Excellence ReSIST and PAPIIT-UNAM.     

The complexity of constructing pseudorandom generators from hard functions

We study the complexity of constructing pseudorandom generators (PRGs) from hard functions, focussing on constant-depth circuits. We show that, starting from a function computable in alternating time O(l) with O(1) alternations that is hard on average (i.e. there is a constant such that every circuit of size fails to compute f on at least a 1/poly(l) fraction of inputs) we can construct a computable by DLOGTIME-uniform constant-depth circuits of size polynomial in n. Such a PRG implies under DLOGTIME-uniformity. On the negative side, we prove that starting from a worst-case hard function (i.e. there is a constant such that every circuit of size fails to compute f on some input) for every positive constant there is no black-box construction of a computable by constant-depth circuits of size polynomial in n. We also study worst-case hardness amplification, which is the related problem of producing an average-case hard function starting from a worst-case hard one. In particular, we deduce that there is no blackbox worst-case hardness amplification within the polynomial time hierarchy. These negative results are obtained by showing that polynomialsize constant-depth circuits cannot compute good extractors and listdecodable codes.     

On the complexity of graph self-assembly in accretive systems

We study the complexity of the Accretive Graph Assembly Problem (). An instance of consists of an edge-weighted graph G, a seed vertex in G, and a temperature τ. The goal is to determine if the graph G can be assembled by a sequence of vertex additions starting from the seed vertex. The edge weights model the forces of attraction and repulsion, and determine which vertices can be added to a partially assembled graph at the given temperature. A vertex can be added when the total weight to its already built neighbors in the graph is at least τ. The assembly process is sequential meaning that only one vertex can be added at a time. Our first result is that is NP-complete even on planar graphs with maximum degree 3 when edges have only two different types of weights. This resolves the complexity of in the sense that the problem is poly-time solvable when either the maximum degree is at most 2 or the number of distinct edge weights is one, and is NP-complete otherwise. Our second result is a dichotomy theorem that completely characterizes the complexity of on graphs with maximum degree 3 and two distinct weights: w _p and w _n . We give a simple system of linear constraints on w _p , w _n , and τ that determines whether the problem is NP-complete or is poly-time solvable. In the process of establishing this dichotomy, we give a poly-time algorithm to solve a non-trivial class of Finally, we consider the optimization version of where the goal is to assemble a largest-possible induced subgraph of the given input graph. We show that even on graphs that can be assembled and have maximum degree 3, it is NP-hard to assemble a (1/n ^1-ε)-fraction of the input graph for any here n denotes the number of vertices in G.     

Stability of Linear Difference and Differential Inclusions

Any given n×n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N×N matrix B of spectral radius (B) arbitrarily close to (A). A difference inclusion , where is a compact set of matrices, is asymptotically stable if and only if can be extended to a set of nonnegative matrices B with or . Similar results are derived for differential inclusions.     

Low-latency atomic broadcast in the presence of contention

The Atomic Broadcast algorithm described in this paper can deliver messages in two communication steps, even if multiple processes broadcast at the same time. It tags all broadcast messages with the local real time, and delivers all messages in the order of these timestamps. Both positive and negative statements are used: “m broadcast at time 51” vs. “no messages broadcast between times 31 and 51”. To prevent crashed processes from blocking the system, the -elected leader broadcasts negative statements on behalf of the processes it suspects () to have crashed. A new cheap Generic Broadcast algorithm is used to ensure consistency between conflicting statements. It requires only a majority of correct processes (n > 2f) and, in failure-free runs, delivers all non-conflicting messages in two steps. The main algorithm satisfies several new lower bounds, which are proved in this paper.     

A new bound for the D0L sequence equivalence problem

The D0L sequence equivalence problem consists of deciding, given two morphisms , and a word , whether or not g ⁱ (w) = h ⁱ (w) for all i ≥ 0. We show that in case of smooth and loop-free morphisms, to decide the D0L sequence equivalence problem, it suffices to consider the terms of the sequences with , where n is the cardinality of X.