Search Results (137)

In response to the growing complexity and uncertainty in real-world decision-making, this study introduces a novel framework based on the polytopic fuzzy tensor (PFT) model, which unifies the geometric structure of polytopes with the representational power of fuzzy tensors. The PFT framework is specifically designed to handle high-dimensional, imprecise, and ambiguous information commonly encountered in multi-criteria group decision-making scenarios. To support this framework, we define a suite of algebraic operations, aggregation mechanisms, and theoretical properties tailored to the PFT environment, with comprehensive mathematical formulations and illustrative validations. The effectiveness of the proposed method is demonstrated through a real-world application involving the evaluation of six pancreatic cancer treatment strategies. These alternatives are assessed against five key criteria: quality of life, side effects, treatment accessibility, cost, and duration. Our results reveal that the PFT-based approach outperforms traditional fuzzy decision-making techniques by delivering more consistent, interpretable, and reliable outcomes under uncertainty. Moreover, comparative analysis confirms the model’s superior ability to handle multidimensional expert evaluations and integrate conflicting information. This research contributes a significant advancement in the field of fuzzy decision science by offering a flexible, theoretically sound, and practically applicable tool for complex decision problems. Future work will focus on improving computational performance, adapting the model for real-time data, and exploring broader interdisciplinary applications. Full article

Nonlinear robotic systems often operate under widely varying conditions that challenge traditional linear control approaches. The Linear Parameter-Varying (LPV) paradigm overcomes these limitations and offers a unifying framework by representing the system’s time-varying dynamics as a convex blend of linear models. This enables both controller and observer synthesis through convex optimization, while considering nonlinearities and time-dependent behavior. This paper presents a linear matrix inequality (LMI)-based methodology for simultaneous stabilization control and state estimation in robotic application within the LPV/polytopic setting. Parallel to controller design, the full-state estimation challenge posed by limited sensors in robotics is addressed. An LPV observer architecture, based on the Luemberger observer, is proposed. The simultaneous observer/controller gains synthesis is then reduced to an LMI feasibility problem. The efficacy of our approach is then demonstrated and illustrated through simulations. Full article

We give a variational characterization of the Born rule. For each measurement context, we project a quantum state ρ onto the corresponding abelian algebra by minimizing Umegaki relative entropy; Petz’s Pythagorean identity makes the dephased state the unique local minimizer, so the Born weights $p_C\left(i\right)=\mathrm{Tr}\left(\rho P_i\right)$ arise as a consequence, not an assumption. Globally, we measure contextuality by the minimum classical Kullback–Leibler distance from the bundle $\left\{p_C\left(\rho \right)\right\}$ to the noncontextual polytope, yielding a convex objective $\Phi \left(\rho \right)$. Thus, $\Phi \left(\rho \right)=0$ exactly when a sheaf-theoretic global section exists (noncontextuality), and $\Phi \left(\rho \right)>0$ otherwise; the closest noncontextual model is the classical I-projection of the Born bundle. Assuming finite dimension, full-rank states, and rank-1 projective contexts, the construction is unique and non-circular; it extends to degenerate PVMs and POVMs (via Naimark dilation) without change to the statements. Conceptually, the work unifies information-geometric projection, the presheaf view of contextuality, and categorical classical structure into a single optimization principle. Compared with Gleason-type, decision-theoretic, or envariance approaches, our scope is narrower but more explicit about contextuality and the relational, context-dependent status of quantum probabilities. Full article

We present a graphical framework to represent entanglement in three-qubit states. The geometry associated with each entanglement class and type is analyzed, revealing distinct structural features. We explore the connection between this geometric perspective and the tangle, deriving bounds that depend on the entanglement class. Based on these insights, we conjecture a purely geometric expression for both the tangle and Cayley’s hyperdeterminant for non-generic states. As an application, we analyze the energy eigenstates of physical Hamiltonians, identifying the sufficient conditions for genuine tripartite entanglement to be robust under symmetry-breaking perturbations and level repulsion effects. Full article

The increasing penetration of distributed photovoltaic (PV) and energy storage (ES) systems in power grids, while advancing the transition to clean energy and enhancing grid flexibility, poses resource dispersion, uncertainty, and scheduling challenges. Consequently, it becomes crucial to manage and optimize these resources. In this paper, we innovatively propose a distribution network day-ahead optimal scheduling model that takes into account distributed resource aggregation and uncertainty. Firstly, distributed PV aggregation (PVA) is performed using the Minkowski summation method, and distributed ES aggregation (ESA) is performed using the polytope inner approximation method. Then, in order to deal with the uncertainty, the supply–demand balance of flexibility is modeled using kernel density estimation (KDE). Finally, the aggregation model and the flexibility supply–demand balance model are incorporated into the distribution network day-ahead optimization. The simulation study of the IEEE 69-node distribution system shows that the aggregate feasible region (AFR) is improved by about 90% and the active flexibility is improved by about 10% compared to box inner approximate aggregation methods, demonstrating their effectiveness in managing operational uncertainties and optimizing the utilization of distributed energy resources in day-ahead scheduling. Full article

Tropical Principal Component Analysis (PCA) is an analogue of the classical PCA in the setting of tropical geometry, and applied it to visualize a set of gene trees over a space of phylogenetic trees, which is a union of lower-dimensional polyhedral cones in an Euclidean space with dimension $m(m-1)/2$, where m is the number of leaves. In this paper, we introduce a projected gradient descent method to estimate the tropical principal polytope over the space of phylogenetic trees, and we apply it to an Apicomplexa dataset. With computational experiments against Markov Chain Monte Carlo (MCMC) samplers, we show that our projected gradient descent method yields a lower sum of tropical distances between observations and their projections onto the estimated best-fit tropical polytope, compared with the MCMC-based approach. Full article

The paper presents the design of iterative learning control for a class of mobile robots. This control strategy allows driving the considered system, which executes the same control task in trials, to the predefined reference within the consecutive iterations by improving the control signal gradually. The control problem being stated concerns a mobile robot, and hence, its kinematic model is presented. The considered model is nonlinear as it is related to the robot orientation angle. Thus, the linearization strategy is introduced by dividing the range of possible orientation angles to four quarters and then deriving a linear parameter-varying system. As a distinct research topic, the feasible/optimal number selection of polytope vertices of each LPV submodel are considered. Next, for the resulting bank of models, the switched iterative control scheme is transformed into closed-loop differential linear repetitive processes. Subsequently, based on the fact that ensuring the so-called stability along the trial is equivalent to the convergence of the original model output to the predefined reference, an appropriate stabilization condition is applied in order to compute the feedback controller gains. The overall effectiveness and performance of the proposed methodology are evaluated through comprehensive simulation examples. Full article

Beginning with a symmetrical multiple-choice individual as the foundation, I develop a sociophysics model of decision-making. By simplifying the range of choices, the framework incorporates the complex Polytopic fuzzy model to capture nuanced dynamics. This approach enables a deeper analysis of decision-making processes within social systems. Decision-making problems commonly involve uncertainty and complexity, posing considerable challenges for organizations and individuals. Due to their structure and variable parameters, the Einstein t-norm (ETN) and t-conorm (ETCN) offer more elasticity than the algebraic t-norm (ATN) and t-conorm (ATCN). This flexibility makes them commonly effective and valuable in fuzzy multi-attribute decision-making (MADM) problems, where nuanced valuations are critical. Their application enhances the ability to model and analyze vagueness and uncertain information, eventually leading to more informed decision outcomes. The complex Polytopic fuzzy set (CPFS) improves the Polytopic fuzzy set (PFS) and complex fuzzy set (CPFS), allowing for a more precise valuation of attributes in complex (MADM) problems. This study aims to propose a MADM scheme using the ETN and ETCN within the framework of a complex Polytopic fuzzy environment. It begins by presenting the Einstein product and sum operations for complex Polytopic fuzzy numbers (CPFNs) and explores their necessary properties. This method enhances the accuracy and applicability of DM processes in ambiguous environments. Subsequently, three complex Polytopic fuzzy operators with known weighted vectors are developed: the complex Polytopic fuzzy Einstein weighted averaging (CPFEWA) operator, complex Polytopic fuzzy Einstein ordered weighted averaging (CPFEOWA) operator, complex Polytopic fuzzy Einstein hybrid averaging (CPFEHA) operator. Moreover, some substantial properties of the operators are studied. Finally, a method based on novel operators is planned, and a numerical example is provided to prove the practicality and effectiveness of the new proposed methods. Full article

In this paper, we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integer combination of given, relatively prime, (strictly) positive integers (known as the Frobenius number). The main contribution of this paper are observations regarding a previously known upper bound on the Frobenius number where, in particular, we observe that a previously presented argument features a subtle error, which alters the value of the upper bound. Despite this, we demonstrate that the subtle error does not impact upon on the validity of the upper bound, although it does impact on the upper bounds tightness. Notably, we formally state the corrected result and additionally compare the relative tightness of the corrected upper bound with the original. In particular, we show that the updated bound is tighter in all but only a relatively “small” number of cases using both formal techniques and via Monte Carlo simulation techniques. Full article

In this paper, we present a method to optimize the attainable moment set (AMS) to increase the control authority for electrical vertical take-off and landing vehicles (eVTOLs). As opposed to 3D AMSs for conventional airplanes, the hover control of eVTOLs requires vertical thrust produced by the powered lift system in addition to three moments. The limits of the moments and vertical thrust are coupled due to input saturation, and, as a result, the concept of the traditional AMS is extended to the 4D generalized moment set to account for this coupling effect. Given a required moment set (RMS) derived from system requirements, the optimization is formulated as a 4D convex polytope coverage problem, i.e., the AMS coverage over the RMS, such that the system’s available control authority is maximized to fulfill the prescribed requirements. The optimization accounts for not only nominal flight, but also for one critical engine inoperative situation. To test the method, it is applied to an eVTOL with eight rotors to optimize for the rotors’ orientation with respect to the body axis. The results indicate highly improved coverage of the RMS for both failure-free and one-engine-inoperative situations. Closed-loop simulation tests are performed for both optimal and non-optimal configurations to further validate the results. Full article

This paper presents a robust sliding mode fault-tolerant control (FTC) strategy for a class of linear parameter variant (LPV) systems with variable time-delays and uncertainties. First fault estimation (FE) is conducted using a robust sliding mode observer, synthesized to simultaneously estimate the states and actuator faults of LPV polytopic delayed systems. Second, a sliding mode FTC is developed, ensuring all states of the closed-loop system converge to the origin. This paper presents an integrated sliding mode FTC strategy to achieve optimal robustness between the observer and controller models. The integrated design approach offers several advantages over traditional separated FTC methods. Our novel approach is based on incorporating adaptive law into the design of the Lyapunov–Krasovskii functional to improve both robustness and performance. This is achieved by combining the concept of sliding mode control (SMC) with the Lyapunov–Krasovskii function under the $H_{\infty }$ criteria, which plays a key role in guaranteeing the stability of this class of system. The effectiveness of the proposed method is demonstrated through a diesel engine example, which highlights the validity and benefits of the integrated and separated FTC strategy for uncertain nonlinear systems with time delays and the sliding mode control. Full article

This paper addresses the problem of computing the optimal stacking of two monotone polytopes P and Q in ${\mathbb{R}}^d$. A monotone polytope in ${\mathbb{R}}^d$ is defined as a polytope whose intersection with any line parallel to the last coordinate axis $x_d$ is connected, and the stacking of P and Q is defined as a translation of Q, such that “Q touches P from above”. To evaluate the stack, we use three different scoring criteria: (1) the height of the stack, (2) the maximum pointwise distance along the $x_d$-axis, and (3) the volume between P and Q. We propose exact algorithms to compute the optimal stacking for each scoring criterion. Full article

This paper presents a trajectory-tracking controller of an inverted pendulum system on a self-balancing differential drive platform. First, the system modeling is described by considering approximations of the swing angles. Subsequently, a discrete convex representation of the system via the nonlinear sector technique is obtained, which considers the nonlinearities associated with the nonholonomic constraint. The design of a discrete parallel distributed compensation controller is achieved through an alternative method due to the presence of uncontrollable points that avoid finding a solution for the entire polytope. Finally, simulations and experimental results using a prototype illustrate the effectiveness of the proposal. Full article

In this paper, we investigate a novel T-S fuzzy parameter varying system with Markov jump, in which parameters depend not only on a Markov chain but also on linear parameter varying elements that take values in convex polytopic sets. Stable conditions and the gain-scheduling controller design method for this system are obtained. Applying Lyapunov function depending on the operation mode and full block S-procedure lemma, we obtain stochastic stabilization conditions. We find that this novel system has two distinct advantages. On the one hand, it inherits the advantages of traditional T-S fuzzy systems in handling nonlinear objects under the frame of T-S fuzzy systems; on the other, it obtains the advantages of dealing with time-varying characteristics from the point of linear parameter varying (LPV) systems. Finally, the theory results are illustrated via numerical simulation. Full article