Equivalent Transformation Research Articles

Composite learning prescribed performance control of strict-feedback nonlinear systems with mismatched parametric uncertainties

A composite learning prescribed performance control (CLPPC) approach is presented for strict-feedback nonlinear systems with mismatched parametric uncertainties. A finite-time performance function based on polynomial is introduced to predefined a restriction region with respect to the tracking error. By introducing an error transformation function, the tracking error restriction problem of the original system is transformed into the stability problem of an equivalent transformation system. To guarantee the convergence of unknown parameters, online recording data and instantaneous are used to generate a prediction error, which is used together with filtered tracking errors to update parameter estimations under an interval excitation condition. All signals are proven to be ultimately uniformly stable under the proposed CLPPC strategy. Furthermore, the tracking error is limited within the predefined region after the predefined time. Simulation results verify the effectiveness of the proposed approach.

A mediational theory of equivalence relations and transformation of function.

In this article, we describe a mediational theory of emergent or derived relations resulting from matching-to-sample procedures that produce equivalence and transformation of function. According to a mediational theory, behaviors that occur at the time of reinforcement mediate subsequent behavioral relations referred to as "derived" or "emergent." Such relations have been documented for decades in studies using mostly matching-to-sample procedures with humans and nonhumans. In both verbal human and nonhuman participants, the mediating behaviors consist of differential responding to the sample stimulus. In humans, such behaviors are mostly, but not necessarily, verbal; in nonhumans they include a variety of sample-specific responses, sometimes called "coding." The proposed mediational theory, based only on the four-term contingency and the basic principles of operant learning, makes specific predictions and explains results from a broad range of experiments. There are at least three important implications of a mediational theory. First, if by "derived" or "emergent" one means untrained or unreinforced, then derived relations may not exist. Second, if there are no derived relations, then theories of such relations may not be necessary. Third, a mediational theory of relational responding has potentially important implications for clinical practice.

Optimizing integrated berth allocation and quay crane assignment: A distributionally robust approach

In this research, we have formulated a Two-Stage Distributionally Robust Optimization (TDRO) model within the context of a mean–variance ambiguity set, specifically designed to address the challenges in the Integrated Berth Allocation and Quay Crane Assignment Problem (BACAP). A key consideration in this study is the inherent uncertainty associated with ships’ arrival times. During the initial stage, we derive a baseline schedule governing berth allocation and quay crane assignment. Anticipating potential disruptions arising from uncertain arrival delays, the second stage is meticulously formulated to determine the worst-case expectation of adjustment costs within the mean–variance ambiguity set. Subsequently, we undertake an equivalent transformation, converting the general TDRO model into a Two-Stage Robust Second-Order Cone Programming (TRO-SOCP) model. This transformation facilitates the application of the Column and Constraint Generation (C&CG) algorithm, ensuring the derivation of an exact solution. To address the computational intricacies associated with second-order cone programming, we propose two enhancement strategies for upper and lower bounds, aimed at expediting the solution process. Additionally, to contend with large-scale instances, we introduce a refinement and approximation method, transforming the TDRO model into a Mixed-Integer Programming (MIP) model. Furthermore, extensive numerical experiments are executed on both synthetic and real-life instances to validate the superior performance of our model and algorithms. In terms of the total cost, the TDRO model demonstrates superior performance compared with Two-Stage Stochastic Programming (TSP) and Two-Stage Robust Optimization (TRO) models.

A one‐dimensional phenomenological constitutive model of shape memory alloys considering the cyclic degradation of two‐way memory effect

AbstractThis study introduces a one‐dimensional phenomenological constitutive model designed to describe two‐way shape memory effect (TWSME) and its associated cyclic degradation. The model utilizes the logistic function to formulate the phase transformation equation, incorporates the expansion of key parameters into their respective fatigue functions to characterize the fatigue phenomenon, and integrates a phase transformation rate regulation function into the differential form of the phase transformation equation. This integration facilitates the control over the entire phase transformation process and the simulation of incomplete transformations. The model is distinguished by its comprehensive functionality, simple form, ease of calculation, and the clear and direct influence of key parameters. Furthermore, it offers a degree of flexibility because each function within the framework is replaceable. The simulation of the TWSME strain and recovery stress hysteresis loop deformation has been successfully conducted, enabling the description of internal hysteresis loops caused by incomplete transformation. The validity of the model is corroborated by comparing it with existing experimental results.

An Effective Scheme for Modeling and Compensating Differential Age Errors in Real-Time Kinematic Positioning

In many real-time kinematic (RTK) positioning applications, reference observations are transmitted over wireless links that can experience frequent interruptions or substantial delays. This results in large differential ages between base and rover observations, which, in turn, leads to a deterioration in positioning performance. To bridge the significant age difference, in this work, we propose a simple and effective scheme for modeling and compensating for such errors. Firstly, the overall differential age error was modeled using truncated Taylor expansion. Then, a time-differenced carrier phase (TDCP)-based observation model was established to estimate the errors with the Kalman framework. Since estimating the receiver’s clock error is unnecessary, equivalent transformation and sequential filtering technology were adopted to significantly reduce the computational complexity. Furthermore, a predictor performance monitor was introduced to mitigate the integrity risks that may occur due to model mismatches. The effectiveness of this scheme was validated by static and dynamic field experiments. The static experiment results showed that when the differential age was 60 s, the GPS and BDS satellites’ overall root mean square error (RMSE) with the asynchronous RTK (ARTK) prediction method was 2.8 and 5.5 times that of the proposed method, respectively. Moreover, when the differential age was 120 s, these values were 3.3 and 5.4 times that of the proposed method, respectively. The field experiment results showed that when the differential age was 60 s, the integer ambiguity fixed rate and false fixed rate of the ARTK method were 0.90 and 1.63 times that of the proposed method, respectively. Finally, at a 120 s differential age, these values were 0.78 and 4.78 times that of the proposed, respectively.

Modelling of negative equivalent magnetic reluctance structure and its application in weak-coupling wireless power transmission

In weak-coupling wireless power transmission, increasing operating frequency, and incorporating metamaterials, resonance structures or ferrite cores have been explored as effective solutions to enhance power efficiency. However, these solutions present significant challenges that need to be addressed. The increased operating frequency boosts ferrite core losses when it exceeds the working frequency range of the material. Existing metamaterial-based solutions present challenges in terms of requiring additional space for slab installation, resulting in increased overall size. In addition, limitations are faced in using Snell’s law for explaining the effects of metamaterial-based solutions outside the transmission path, where the magnetic field can not be reflected or refracted. To address these issues, in this work, the concept of a negative equivalent magnetic reluctance structure is proposed and the metamaterial theory is extended with the proposed magnetic reluctance modelling method. Especially, the negative equivalent magnetic reluctance structure is effectively employed in the weak-coupling wireless power transfer system. The proposed negative equivalent magnetic reluctance structure is verified by the stacked negative equivalent magnetic reluctance structure-based transformer experiments and two-coil mutual inductance experiments. Besides, the transmission gain, power experiments and loss analysis experiments verify the effectiveness of the proposed structure in the weak-coupling wireless power transfer system.

Study on rigid-flexible coupling modeling of planetary gear systems: Incorporation of the pass effect and random excitations

A rigid-flexible coupling model for the planetary gear system is developed, in which the flexibility of the ring gear, input shaft, and carrier are considered. Subsequently, rotational modal projection is proposed to simulate the meshing of the flexible ring gear at any angle of planet gear revolution, and the coordinate transformation equations are presented to establish the coupling between rigid body and rotating flexible body. These two approaches incorporate the pass effect into the coupling model. Additionally, a method employing pre-emphasized noise to simulate random excitations is introduced. The model is validated by comparing computed modal frequencies and dynamic responses with those from ANSYS software and experiment, respectively. Results indicate that the proposed method accurately simulates the modulation signal observed in the experiment. Closely matched resonances excited by random excitations are apparent in experimental and simulated acceleration spectra, highlighting the validity of the proposed method and aiding in the determination of operational system modes. In addition, analyses of parameter influences are conducted, which further reveal the dynamic characteristics of the system.

Lattice Boltzmann method for the linear complementarity problem arising from American option pricing

In this paper, a lattice Boltzmann method is proposed for solving the linear complementarity problem (LCP) arising in single and multi-asset American put option pricing. The LCP for American option is a variable coefficient parabolic model defined on an unbounded domain. Initially, using the far field estimate method and the penalty method respectively, the LCP could be reformulated into a nonlinear parabolic partial differential equation on a bounded domain. To construct a unified lattice Boltzmann model for the option pricing problems, the above transformation equations are rewritten into an equivalent divergence form. Then, through the incorporation of an amending function into the evolution equation, which assists in recovering the source term and eliminating the error term, the lattice Boltzmann model with spatial second-order accuracy is constructed. Finally, the present model is validated using numerical simulations, and the numerical results agree well with the option values obtained by existing methods, which indicates that the present lattice Boltzmann model is efficient for solving the American put option pricing problem.

Distributed State Observer for Systems with Multiple Sensors under Time-Delay Information Exchange.

The issues of state estimations based on distributed observers for linear time-invariant (LTI) systems with multiple sensors are discussed in this paper. We deal with the scenario when the information exchange has known time delays, and aim at designing a distributed observer for each subsystem such that each distributed observer can estimate the system state asymptotically by rejecting the time delay. To begin with, by rewriting the target system in a connecting form, a subsystem which is affected by the time-delay states of other nodes is established. And then, for this subsystem, a distributed observer with time delay is constructed. Moreover, an equivalent state transformation is made for the observer error dynamic system based on the observable canonic decomposition theorem. Further, in order to ensure that the distributed observer error dynamic system is asymptotically stable even if there exists a time delay, a linear matrix inequality (LMI) which is relative to the Laplace matrix is elaborately set up, and a special Lyapunov function candidate based on the LMI is considered. Next, based on the Lyapunov function and Lyapunov stability theory, we prove that the error dynamic system of the distributed observer is asymptotically stable, and the observer gain is determined by a feasible solution of the LMI. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.

Disturbance-Observer-Based Adaptive Prescribed Performance Formation Tracking Control for Multiple Underactuated Surface Vehicles

This study proposes a new disturbance-observer-based adaptive distributed formation control scheme for multiple underactuated surface vehicles (USVs) subject to unknown synthesized disturbances under prescribed performance constraints. A modified sliding mode differentiator (MSMD) is applied as a nonlinear disturbance observer to estimate unknown synthesized disturbances, which contain unknown environmental disturbances and system modelling uncertainties, thus enhancing the robustness of the system. Based on this, we impose the time-varying performance constraints on the position tracking error between the neighboring USVs. A novel differentiable error transformation equation is embedded in the prescribed performance control, and an adaptive prescribed performance controller is constructed by employing the backstepping method to ensure that the position tracking error remains within the prescribed transient and steady performance, and each USV realizes collision-free formation motion. Furthermore, a novel second-order nonlinear differentiator is introduced to extract the derivative information of the virtual control law. Finally, the numerical simulation results verify the effectiveness of the proposed control scheme.

Development of a non-standard system for simplifying boolean functions

The object of this study is models of low-power digital logic circuits. The problem being solved is the effectiveness of the technique for simplifying Boolean functions to obtain optimal structures of logic circuits. A new theorem of a non-standard system of simplification of Boolean functions has been formulated, according to which in order to obtain a minimal function it will suffice to perform all non-redundant operations of simple and/or super-gluing of variables, which ultimately provides a minimal function in the main basis without using an implicant table. Thus, the problem of simplifying Boolean functions to the simplest normal equivalent is solved in one step. The interpretation of the result is that the properties of 2-(n, b)-design combinatorial systems make it possible to reproduce the definition of logical operations of super-gluing variables, to represent logical operations in a different way, and vice versa. This, in turn, ensures the establishment of the locations of equivalent transformations on the binary structure of the truth table and the implication of a systematic procedure for simplifying Boolean functions by an analytical method. Special feature of the results is that unambiguous identification of the locations of equivalent transformations is possible even when different intervals of the Boolean space containing the 2-(n, b)-design systems have common modules. It has been experimentally confirmed that the non-standard system improves the efficiency of simplifying Boolean functions, including partially defined ones, by 200–300 % compared to analogs. In terms of application, a non-standard system for simplifying Boolean functions will ensure the transfer of innovations to material production: from conducting fundamental research, expanding the capabilities of digital component design technology to organizing serial or mass production of novelties

The variable coefficient Boiti–Leon–Pempinelli and similar systems: The Lie symmetry approach

We consider the variable coefficient Boiti–Leon–Pempinelli and two similar systems. We present the enhanced Lie group classification. This task is achieved with the aid of the equivalence transformations. We construct a number of similarity reductions which are used to derive exact solutions for the three ‐dimensional systems. The reductions are obtained using both one‐ and two‐dimensional subalgebras. In a specific case, each system admits a hidden symmetry.

Dynamic gain–based neural network backstepping control with its applications to a quad-rotor hover

In this paper, a novel neural network (NN) backstepping control method is proposed for a class of uncertain nonlinear systems with unknown control direction functions. To solve the control problems caused by unknown control direction functions, a method called “virtual control coefficient” is presented using equivalence transformations. However, to improve the flexibility of the control method, a function named “dynamic gain” is introduced to design the feedback gains of the NN backstepping controllers. Moreover, instead of using σ-modification, gradient descent algorithm is applied to train the weights of NNs such that the unknown nonlinearities of the system can be well approximated by NNs. With the help of Lyapunov stability criterion, it can be proved that the system tracking error is semi-global uniformly ultimately bounded. Finally, the effectiveness of the proposed control method is verified by the experimental results on a quad-rotor hover platform.

Geometrical Interpretation of Space Contraction in Two-dimensional Lorentz Transformation

This paper points out that the transformation equations for the spatial and temporal coordinates between two frames of reference in the existing generally accepted version of the Lorentz transformation are deficient, since transformation equations are based on one dimensional motion between inertial frames. Therefore, all possible space-time coordinate transformation equations between moving and stationary frames by prolonging Lorentz transformation in a two-dimensional plane are thoroughly proposed in this article. If denotes the relative velocity between stationary frame (x, y) and moving frame (x', y'), then the transformation equations along X and Y-axis under two-dimensional Lorentz are given, respectively, by the formulas (see full text) . In this work we conclude that length and breadth of a rectangle appears to be shortened to the observer when there is the relative velocity between the rectangle and observer along both X and Y-axis. In particular, we present a concise and carefully reasoned account of a new aspect of Lorentz transformation which decently allows for the determination of space-time coordinates transformation equations in two dimensions of space.

Unifying flavors of fault tolerance with the ZX calculus

There are several models of quantum computation which exhibit shared fundamental fault-tolerance properties. This article makes commonalities explicit by presenting these different models in a unifying framework based on the ZX calculus. We focus on models of topological fault tolerance – specifically surface codes – including circuit-based, measurement-based and fusion-based quantum computation, as well as the recently introduced model of Floquet codes. We find that all of these models can be viewed as different flavors of the same underlying stabilizer fault-tolerance structure, and sustain this through a set of local equivalence transformations which allow mapping between flavors. We anticipate that this unifying perspective will pave the way to transferring progress among the different views of stabilizer fault-tolerance and help researchers familiar with one model easily understand others.

Reduced Outer Space Algorithm for Globally Computing Affine Sum-of-Ratios Problems

This paper investigates the affine sum-of-ratios problem (SRP) which has numerous applications in many fields of economy and engineering. For globally computing the affine SRP, based on equivalent transformation and new affine relaxation problem, a reduced outer space branch-and-bound algorithm is designed. The designed algorithm has been proven to converge to a global optimal solution of the problem (SRP) eventually. Meanwhile, the maximum iteration time for the algorithm in the worst case is estimated by analyzing its computational complexity. Finally, numerical experimental results indicate that the presented algorithm is robust and effective.

How the Space-Time as A Field of Mass Reconciles Quantum and Classical Mechanics

The invariance of the speed of light in RFs in relative motion between them is a clear indication of the nature of space-time. Lorenz's transformation equations were obtained by analysing the propagation of a single ray of light in a single direction and therefore are valid only for that single case. A set of general transformation equations can be obtained by analysing the propagation of an omnidirectional flash of light emitted by a source in motion with respect to a stationary observer. According to these equations, motion generates a spatial component normal to the velocity v and to the coordinates (x, y, z) of the source’s RF, displacing it into an “imaginary” fourth dimension for a value v/c, thus reducing its "density" by a ratio. The density of a RF plays a major role in the perception of physical reality. In fact, two observers in motion between them, watching the same phenomenon, would measure different spaces and times, same velocities, but different acceleration and therefore different forces acting on the same object. The transformation equations show that motion modifies only the RF, i.e the space-time, of objects, but as we cannot separate them from their space, they too are modified by a component transverse to the motion, and their “density” is also reduced. If the object is an electric charge, the transverse component of its field is identified with the magnetic field, which induces attractive or repulsive (or nil) forces only on other charges in motion, depending on the angle between their velocities. The same happens for the motion of a mass, with the fundamental difference that a rotating mass, in addition to a gravito-magnetic field, produces a transverse field which lines of flux are parallel to the axis of rotation, and therefore it propagates indefinitely, without attenuation, inside a cylinder with the same diameter of the mass. This field has a critical importance if we assume that the space-time is a field of the mass, coinciding with its gravitational field. The space-time of the Earth-bound observer would be made by the sum of the gravitational fields of the surrounding atomic masses, plus that of the transverse fields generated by the motion of those same masses; the main part of it would be produced by a sort of “ether” formed by “quanta” of space-time generated by the rotation around themselves of the atomic components of the whole universe. This would imply that the space-time at the atomic level is extremely "denser" than that at macroscopic level, therefore the Earth-bound observer would measure forces interacting amongst the atomic masses extremely higher than those calculated with Newton’s law. He will then be forced to postulate the existence of a mysterious agent, the electric charge, capable of developing those forces. With this assumption, instead, the atomic world would be populated only by masses subject, at their level, to gravitational and gravito-magnetic forces. It will become similar to the macroscopic world.

Convergence analysis of the time-varying discrete-time Altafini model on signed network

This paper mainly studies the convergence of time-varying discrete-time Altafini model. Firstly, we convert the bipartite consensus of the Altafini model with n nodes into consensus of the DeGroot model with 2 n nodes by adopting lifting approach. Secondly, based on structural balance theory and node relabelling theory, the consensus of enlarged DeGroot model can be further equivalently converted into the consensus of one simple DeGroot model with n nodes. Thirdly, by using the equivalent transformation and the ergodic coefficient theory, we change the sufficient condition from ‘repeatedly jointly strongly connected’ graphs to ‘repeatedly jointly scrambling’ graphs, and verify that the new condition also can lead to the bipartite consensus of the Altafini model. This novel criterion allows for determining the bipartite consensus of the Altafini model without the condition of ‘repeatedly jointly strongly connected’ graphs. Next, we provide a concise proof for the case of opinions converging to 0 with the condition of ‘repeatedly jointly strongly connected’ by the ergodic coefficient theory. Finally, we develop a reasonable coevolution model to illustrate the effectiveness and strong applicability of our main theoretical results.

Correlation and consistency between two detection methods for serum 25 hydroxyvitamin D levels in human venous blood and capillary blood.

The study assessed the correlation and concordance of 25-hydroxyvitamin D [25(OH)D] levels in capillary and venous plasma collected simultaneously after vitamin D3 supplementation in 42 healthy adults. They were randomly divided into three groups by random number table method. Group A took 1,000 IU vitamin D3 daily, group B took 10,000 IU vitamin D3 every 10 days, and group C took 30,000 IU vitamin D3 every 30 days until the end of the 12th month. Venous blood serum 25(OH)D level was detected by chemiluminescence immunoassay (CLIA) and mass spectrometry (LC-MS) at day 1, day 14, day 28, month 6, and month 12 respectively, the capillary blood serum 25(OH)D level was detected by chemiluminescence immunoassay (CLIA) at the same time. Pearson correlation analysis and linear regression analysis were employed to investigate the relationship and transformation equation between the findings of the two samples and the results obtained from different detection methods within the same sample. The Bland-Altman method, Kappa analysis, and receiver operating characteristic (ROC) curve were utilized for assessing consistency, sensitivity, and specificity. The three groups all reached a stable peak at 6 months, and the average levels of the three groups were 49.21, 42.50 and 43.025 nmol/L, respectively. The average levels of group A were higher than those of group B and group C (P < 0.001). The mean values of serum 25(OH)D measured by LC-MS and CLIA in 42 healthy adults were 45.32 nmol/L and 49.88 nmol/L, respectively, and the mean values of 25(OH)D measured by LC-MS in capillary blood were 52.03 nmol/L, and the difference was statistically significant (P < 0.001). Pearson correlation analysis showed that the linear fitting formula of scatter data was as follows: venous 25(OH)D concentration (nmol/L) = 1.105 * capillary 25(OH)D concentration -7.532 nmol/L, R2 = 0.625. Good agreement was observed between venous and corrected capillary 25(OH)D levels in clinical diagnosis (Kappa value 0.75). The adjusted serum 25(OH)D in capillary blood had a high clinical predictive value. The agreement between the two methods is good when the measured 25(OH)D level is higher. Standardized capillary blood chemiluminescence method can be used for 25(OH)D detection.

A novel single-loop estimation method for predictive failure probability-based global sensitivity analysis

The existing predictive failure probability (PFP)-based global sensitivity analysis methods may get stuck in practical application due to the large amount of calculation or insufficient precision. Therefore, a novel single-loop estimation method is proposed in this paper to effectively estimate PFP-based global sensitivity measures. Firstly, by introducing standard normal auxiliary variables and equivalent probability transformation, the overall uncertainty of each input variable is decomposed into its inherent uncertainty and distribution parameter uncertainty. Then, Bayes theorem-based formulations are derived for three PFP-based global sensitivity measures, where only the unconditional PFP and the conditional marginal and joint distributions of standard normal auxiliary variables and distribution parameters need to be known. These items can be evaluated simultaneously using only one set of functional calls. Finally, adaptive Kriging model is embedded into the proposed method in place of the time-demanding performance function to further enhance computational efficiency. The effectiveness of the proposed methods is validated through four benchmark reliability problems asnd one engineering problem.