Search Results (134)

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

The grand antiprism $\mathbf{A}$ is an outlier among the uniform 4-polytopes, since it is not obtainable from Wythoff’s construction. Its symmetry group $G\left(\mathbf{A}\right)$ has been incorrectly described as $\left[\right[10,2^{+},10\left]\right]$ or even as an ‘ionic diminished Coxeter group’. In fact, $G\left(\mathbf{A}\right)$ is another group of order 400, namely the group $\pm [D_{10}\times D_{10}]·2$, in the notation of Conway and Smith. We explain all this and so correct a persistent error in the literature. This fresh look at the beautiful geometry of the polytope $\mathbf{A}$ is also a fine opportunity to introduce the reader to the elegance of Wythoff’s construction and to the less familiar use of quaternions to classify the finite 4-dimensional isometry groups. Full article

When a ground vehicle runs at high speeds, even a slight excess in the wheel steering angle can immediately cause the vehicle to slide sideways and lose control. In this study, we propose an active safety control system designed to address emergency situations where the driver applies excessive steering input and the vehicle speed varies significantly during control. The system combines the direct yaw moment (DYM) method with a steering saturation scheme that prevents excessive driver steering input from adversely influencing the front-wheel steering. Consequently, the control system allows the DYM to focus more on other stabilization tasks and maintain tire/road friction within its workable linear range. The implementation relies on a reference steering angle and a reference vehicle state, derived from a linear vehicle model considering tire/road friction limitations. When the driver’s steering angle and the system state deviate from these reference values, the control system intervenes by applying both the steering saturation scheme and DYM method. This ensures the front-wheel steering angle and system state remain close to the reference values. The control strategy is developed using the polytopic Linear Parameter Varying (LPV) technique and Linear Matrix Inequality (LMI) to account for the changes in vehicle speed. It is further enhanced with an input saturation technique based on a high-gain approach, which improves control utilization and system response during emergency situations. The advantages of the proposed control strategy are demonstrated through simulation results. Full article

For a gas undergoing a compression process, it is more appropriate to think of either isentropic or polytropic efficiency as process-defining parameters indicating that a given end state of compression has been achieved, rather than a measure of effectiveness of conversion of one form of energy into another. The polytropic efficiency, as defined in ASME PTC-10 standard for compressor field trials and acceptance tests, actually involves the comparison of two distinct compression processes, neither of which are actually connected to the performance of the compressors producing them. Consequently, it is not rational to compare the ASME PTC-10 polytropic efficiency of a compressor designed to compress a gas predominantly adiabatically with that for a compressor designed to compress a gas predominantly isothermally. A framework correcting this situation is set out and is illustrated with several numerical examples. Suggestions for maintaining backward compatibility with ASME PTC-10 are also put forward. Full article