Set Theoretic Framework Research Articles

Set membership state estimation for discrete-time linear systems with binary sensor measurements

This paper addresses the problem of state estimation for discrete-time linear systems, based on measurements provided by an output binary sensor. The problem is formulated and solved in a set theoretic framework. Two algorithms are devised for recursively computing outer approximations of the set of state vectors compatible with the information provided by the binary sensor. This allows one to obtain a nominal state estimate and to characterize the associated uncertainty. The procedures can be tuned to suitably trade off the quality of set approximations and the required computational load. An input design technique based on the computed feasible state sets, which is aimed at promoting uncertainty reduction, is provided. The case of time-varying sensor threshold is also considered and a strategy for selecting online the value of the threshold is formulated. All the proposed methods are validated in simulations on two numerical examples.

Active input design for simultaneous fault estimation and fault-tolerant control of LPV systems

This paper considers the problem of integrating fault estimation (FE) and fault-tolerant control (FTC) for linear parameter varying (LPV) systems with inexact scheduling variables from a new perspective. Unlike the conventional passive FE approaches that maintain the performance of FE by only adjusting parametric matrices, the proposed method guarantees the precision and optimality of FE by actively designing optimal inputs at each time instant under the set-theoretic framework, which is named as active fault estimation (AFE). Meanwhile, by constructing a quadratic Lyapunov function, the stability of augmented estimation-error dynamics can be guaranteed based on H∞ setting by solving a group of matrix inequalities. In addition, the optimal inputs are also designed to implement the objective of integrated AFE and FTC for LPV systems, which leads to a balance between the goal of AFE and the reference output tracking performance. Although the original optimization problem computing the optimal inputs is not convex, it is proved that this problem can be equivalently transformed to a family of convex quadratic programming problems with linear constraints. At the end, a physical vehicle model is used to verify the effectiveness of our proposed integrated AFE and FTC method.

A novel geometry optimization strategy to online active fault diagnosis of LPV systems

This paper proposes a novel geometry optimization strategy for the online robust active fault diagnosis (AFD) of discrete-time linear parameter varying (LPV) systems under set-theoretic framework. By establishing a bank of zonotopic set-value observers to match healthy/faulty system modes, the key of the optimization strategy comes down to the design of the optimal system inputs and gain matrices simultaneously. The criterion on the design of optimal inputs is characterized by fully utilizing the geometry property of zonotopes, which formulates a non-convex fractional programming problem able to be solved under 0–1 mixed integer quadratic programming framework based on a series of transformations. While the optimal gain matrices can be obtained analytically based on a so-called zonotopic kalman filtering process. The proposed method can improve the sensitivity of AFD as much as possible while ensuring the stability of the designed observer. At the end, two physical models are used to verify the effectiveness of our proposed method.

Superiorized Adaptive Projected Subgradient Method With Application to MIMO Detection

In this paper, we show that the adaptive projected subgradient method (APSM) is bounded perturbation resilient. To illustrate a potential application of this result, we propose a set-theoretic framework for MIMO detection, and we devise algorithms based on a superiorized APSM. Various low-complexity MIMO detection algorithms achieve excellent performance on i.i.d. Gaussian channels, but they typically incur high performance loss if realistic channel models (e.g., correlated channels) are considered. Compared to existing low-complexity iterative detectors such as individually optimal large-MIMO approximate message passing (IO-LAMA), the proposed algorithms can achieve considerably lower symbol error ratios over correlated channels. At the same time, the proposed methods do not require matrix inverses, and their complexity is similar to IO-LAMA.

A Reassessment of Cantorian Abstraction based on the varepsilon -operator

Cantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s varepsilon -operator. Moreover, it is argued that the BK system and the varepsilon -operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first the axiomatic setting of BK and the definition of cardinal numbers by means of the varepsilon -operator. Then, after presenting Cantor’s abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo–von Neumann and Frege–Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege’s objections to Cantor’s proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the varepsilon -operator in the BK definition of cardinal numbers.

Z-number-based AQI in rough set theoretic framework for interpretation of air quality for different thresholds of PM2.5 and PM10.

Kolkata has a reputation for being one of the world’s most polluted cities, particularly in the post-monsoon months of October, November, and December. Diwali, a Hindu festival, coincides with these months where a large number of firecrackers are set off followed by high emissions of air pollutants. As a result, the air quality index (AQI) deteriorates to “very poor” (301 ≤ AQI ≤ 400) and “poor” (201 ≤ AQI ≤ 300) categories. This situation stays for several days to a month. The present study aims to identify the thresholds for PM2.5 and PM10 that cause the AQI of Kolkata to deteriorate to “very poor” and “poor.” For this purpose, we have used a rough set theory-based condition-decision support system to predict the aforementioned categories of AQI. We have developed a Z-number-based novel quantification measure of semantic information of AQI to assess the reliability of the outcomes, as generated from the condition-decision-based decision rules, during post-monsoon season. The result reveals the best possible forecast of AQI with linguistic summarization of the reliability or confidence for different threshold ranges of PM10 and PM2.5. Inverse-decision rules based on rough set theory are utilized to justify and validate the forecasts. The explainability of the condition-decision support system is demonstrated/visualized using a flow graph that maps rough-rule-based different decision paths between input and output with strength, certainty, and coverage. The investigation resulted in an advanced intelligent environmental decision support system (IEDSS) for air-quality prediction.

Stabilization for constrained switched positive linear systems via polyhedral copositive Lyapunov functions

The constrained stabilization problem of switched positive linear systems (SPLS) with bounded inputs and states is investigated via the set-theoretic framework of polyhedral copositive Lyapunov functions (PCLFs). It is shown that the existence of a common PCLF is proved to be necessary and sufficient for the stabilizability of an SPLS. As a primary contribution of this paper, we propose a PCLF-based approach for stabilization with a larger estimate of the domain of attraction for the constrained SPLSs. The analysis problems are converted into optimization problems whose constraints become linear matrix inequalities when a few variables are fixed. Finally, a turbofan engine model is employed to demonstrate the potential and effectiveness of the theoretical conclusions.

Introduction: Causation, inferences, and solution types in configurational comparative methods

This special issue addresses questions of causality and validity of different solution types in configurational comparative methods (CCMs). First, what main parameters characterize the debate about correct causal interpretation of solution types? Second, to what extent has this debate been linked to a theory of causation? The special issue contribution by Mahoney and Acosta bases qualitative comparative analysis (QCA) within a regularity theory of causation integrating type-level inferences and counterfactual cases. Swiatczak clarifies how the different algorithms underlying QCA and Coincidence Analysis (CNA) produce non-identical models. Baumgartner defines and benchmarks QCA solution types against the search target of minimal robust sufficiency. Alamos-Concha et al. identify the conservative solution as most appropriate for a multimethod design combining a counterfactual causal understanding at the cross-case level with an in-depth mechanistic explanation at the within-case level. Finally, Mahoney and Owen develop a general set-theoretic framework for the study of necessity and sufficiency in quantitative research using a counterfactual understanding of causality. Our introduction reviews the state of the art, identifies current limitations and open questions regarding the theoretical basis for causal interpretation of QCA solutions.

Set theoretical variants of the teaching–learning-based optimization algorithm for optimal design of truss structures with multiple frequency constraints

In this paper, a set theoretical framework is proposed for the population-based metaheuristic algorithms. Using the proposed framework, two new set theoretical variants of the teaching–learning-based optimization (TLBO) algorithm are developed. These algorithms are named as OST-TLBO and STMP-TLBO, which are acronyms for the “ordered set theoretical teaching–learning-based optimization” and “set theoretical multi-phase teaching–learning-based optimization”, respectively. The present framework can be applied to other population-based metaheuristic algorithms. In order to verify the stability and robustness of the presented algorithms, some optimization problems are examined. These problems include four truss optimization problems with multiple natural frequency constraints. Comparing the results obtained from the proposed algorithms with those of the standard version of the TLBO algorithm shows that the proposed set theoretical framework improves the performance of the standard TLBO in terms of robustness and convergence characteristics.

Stabilization of constrained switched systems via multiple Lyapunov R-functions

This paper investigates the stabilization problem of switched linear systems subject to state constraints, input saturation and external disturbance. Within the set-theoretic framework of R-function, the multiple Lyapunov R-functions (MLRFs) are presented, and merge both the multiple polyhedral and the multiple quadratic Lyapunov functions. The advantages of the MLRFs are that the external level sets are designed in accordance to the state constraints and inner level sets arbitrary can be made as close as possible to the smooth one. Based on the MLRFs method, stabilization conditions for the constrained switched system under a gradient-based switching law are derived. The union of the invariant truncated ellipsoidal sets is provided as an alternative approximation to the domain of attraction of the constrained switched system. Each truncated ellipsoidal set is the intersection of a polyhedral set and an ellipsoidal set, which come from the polyhedral Lyapunov function and the quadratic Lyapunov function, respectively. The sub-optimality bounds are investigated by a policy iteration. Two examples are simulated to show the benefits of the proposed strategy.

ZKF-based optimal robust fault estimation of descriptor LPV systems with measurement error-affected scheduling variables

This paper proposes a robust state and fault estimation (SFE) method for discrete-time descriptor linear-parameter-varying (LPV) systems with inexact scheduling variables. As an important robust method dealing with system uncertainties, the set-membership estimation method is combined with the technique of generalized fault detectability indices and matrix to compute a state and fault tube to contain the real system states and fault signals at each time instant under the assumption that the system uncertainties (i.e., modeling errors, process disturbances, measurement noises and errors of scheduling variables) are bounded to guarantee the robustness of SFE. Theoretically, any trajectory in the tube can be used as a point-wise estimation of the real system states and fault signals. Meanwhile, the optimal parametric matrices both for set-membership estimation and unknown input observer (UIO) are designed by using the zonotopic Kalman filter (ZKF) procedure to guarantee the optimality of SFE under a set-theoretic framework. Furthermore, a collection of stability conditions for the proposed optimal SFE method are established based on the linear matrix inequalities (LMIs). At the end, a practical electric circuit and a vehicle example are used to illustrate the effectiveness of the proposed SFE method.

Indistinguishability and the origins of contextuality in physics.

In this work, we discuss a formal way of dealing with the properties of contextual systems. Our approach is to assume that properties describing the same physical quantity, but belonging to different measurement contexts, are indistinguishable in a strong sense. To construct the formal theoretical structure, we develop a description using quasi-set theory, which is a set-theoretical framework built to describe collections of elements that violate Leibnitz's principle of identity of indiscernibles. This framework allows us to consider a new ontology in order to study the properties of quantum systems. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

A neuroadaptive architecture for model reference control of uncertain dynamical systems with performance guarantees

Neuroadaptive control systems have the capability to approximate unstructured system uncertainties on a given compact set using neural networks. Yet, a challenge in their design is to guarantee the closed-loop system trajectories stay in this set such that the universal function approximation property is satisfied and the overall system stability is achieved. To address this challenge, we present and analyze a new neuroadaptive architecture in this paper for model reference control of uncertain dynamical systems with strict performance guarantees. Specifically, the proposed architecture is predicated on a novel set-theoretic framework and has the capability to keep the closed-loop system trajectories within an a-priori, user-defined compact set without violating the universal function approximation property. A transport aircraft example is also given to complement the presented theoretical results.

On regular but not completely regular spaces

We present how to generate some non-comparable regular but not completely regular spaces. We provide a generalization of Mysior's example by extracting a purely set-theoretic framework that underlies it. This enables us to build some simple counterexamples with the help of the Niemytzki plane, the Sorgenfrey plane or Lusin gaps.

From zero-sum to win-win - Organisational conditions for successful shared value strategy implementation

The number of publications on shared value strategies (SVSs) is continuously increasing. Literature does not answer the question which organisational conditions are necessary and which conjunctions of conditions are sufficient for the successful implementation of SVSs. This research aims at filling the gap by condensing empirical evidence found in the literature on successful SVSs, stakeholder management, corporate social responsibility, and positive social change. The article synthesises the findings of these streams of research into a set-theoretical framework and proposes five complex conditions necessary for successful SVS implementation: shared value-oriented entrepreneurial vision, strategic alignment (STA), shared value-oriented innovation (SVI), networking capabilities (NCs), and impact monitoring. Additional empirical evidence from cases claiming successful implementation of SVS allows the suggestion of several simple conditions constituting the complex conditions. Qualitative comparative analysis can be used for empirically testing the necessity of the suggested conditions and the sufficiency of conjunctions of complex conditions for SVS implementation success. The proposed framework enhances theoretical knowledge concerning successful SVS implementation and is an important step to support management efforts in reorienting their companies from zero-sum strategies to win-win SVSs.

Internationalization Behavior of Small and Medium- Sized Enterprises from Emerging Markets - Implications for Sustainability

This study examines the internationalization behavior of small and medium-sized enterprises from Paraguay exploring their internationalization activities using the Uppsala model as the theoretical framework. The research methodology is a comparative multiple-case study research design. The sample consists of 28 typical Paraguayan firms from different industries. The findings of this study show how and why the the firms of the case study select new foreign markets as well as market entry and market growth modes, then compares the results with the set theoretical framework, and offers explanations for the ones that differ. The study provides a guideline for internationalization to international managers and offers insights to the literature on the Uppsala model.

Internationalisation behaviour of small and medium-sized enterprises from emerging markets: implications for sustainability

This study examines the internationalisation behaviour of small and medium-sized enterprises from Paraguay exploring their internationalisation activities using the Uppsala model as the theoretical framework. The research methodology is a comparative multiple-case study research design. The sample consists of 28 typical Paraguayan firms from different industries. The findings of this study show how and why the firms of the case study select new foreign markets as well as market entry and market growth modes, then compares the results with the set theoretical framework and offers explanations for the ones that differ. The study provides a guideline for internationalisation to international managers shows the implications of internationalisation on sustainability and offers insights to the literature on the Uppsala model.

Set-theoretic methodology using fuzzy sets in rule extraction and validation - consistency and coverage revisited

The use of set-theoretic concepts of consistency and coverage and their fuzzified versions needs to be accompanied with an extension of the general (non-fuzzy) methodology. In the fuzzy context, the same piece of data can provide partial evidence in favour of the existence of a given relationship, and at the same time contribute to its disproof, which constitutes a significant interpretation and methodological issue. We point out the possible problems of the use of these measures fuzzified in direct analogy to their non-fuzzy counterparts in the investigation of relationships and causality. We propose two alternative fuzzifications of these measures to be used in the set-theoretic framework and fuzzy set qualitative comparative analysis (fsQCA). We also introduce novel degree-of-support and degree-of-disproof measures to handle simultaneous subset relations. In this way, simultaneous partial support for- and disproof of a given rule by the given set of data can be analyzed in detail. The suggested measures can enhance the insights needed in theory building and be widely used in research using fuzzified set-theoretic methods for the assessment of the plausibility of the rules. We compare the newly suggested consistency and coverage measures with standard ones and discuss their properties in practical examples.

“Invariants” in Koffka’s Theory of Constancies in Vision: Highlighting Their Logical Structure and Lasting Value

Summary By introducing the concept of “invariants”, Koffka (1935) endowed perceptual psychology with a flexible theoretical tool, which is suitable for representing vision situations in which a definite part of the stimulus pattern is relevant but not sufficient to determine a corresponding part of the perceived scene. He characterised his “invariance principle” as a principle conclusively breaking free from the “old constancy hypothesis”, which rigidly surmised point-to-point relations between stimulus and perceptual properties. In this paper, we explain the basic terms and assumptions implicit in Koffka’s concept, by representing them in a set-theoretic framework. Then, we highlight various aspects and implications of the concept in terms of answers to six separate questions: forms of invariants, heuristic paths to them, what is invariant in an invariant, roots of conditional indeterminacy, variability vs. indeterminacy, and overcoming of the indeterminacy. Lastly, we illustrate the lasting value and theoretical power of the concept, by showing that Koffka’s insights relating to it do occur in modern perceptual psychology and by highlighting its role in a model of perceptual transparency.

A granular computing framework for approximate reasoning in situation awareness

We present our results on the adoption of a set-theoretic framework for granular computing to situation awareness. The proposed framework guarantees a high degree of flexibility in the process of creation of granules and granular structures allowing to satisfy the wide variety of requirements for perception and comprehension of situations where some elements must be perceived per similarity, others per spatial proximity, some must be fused to improve their comprehension, and so on. A second value is the support for approximate reasoning in situation awareness. A granular structure in particular represents a snapshot of a situation, and is a building block for the development of tools and techniques to reason on situation in order to reduce situation awareness errors and accelerate the process of decision-making. To this purpose, we show a technique to support operators in the analysis of conformity between a recognized situation and an expected one. A third value is the fact that we can support operators in having rapid and indicative measures of how two situations, e.g. a recognized and a projected, may differ. A preliminary evaluation instantiating our approach with self-organizing maps is reported and discussed. The results are encouraging with respect to the capability of improving perception and comprehension of a situation, reducing comprehension errors and supporting projection of situations.