Linear Fractional Transformation Research Articles

An improved solution approach for solving multi-objective linear and linear fractional transportation problem

This work shows a transportation problem (TP) that can be thought of as the sum of a linear function and a linear fractional function. Here, the linear function shows the total transportation cost, and the linear fractional function shows the proportion of damaged objects and the transport profit. We aim to set up a transportation plan that reduces the total transportation cost and the proportion of total damage to the object to the total profit. Here we are discussing three types of objective functions as: linear plus linear fractional, linearfractional, and linear separately. We formulated a TP using the three objective functions for the solution. We solve all three objective functions for different weights with the proposed method and get a compromise solution.

Approbation of the State of Track Circuit Data Processing Method in Intelligent Systems of Control and Diagnosis

The article presents the results of experimental researches of the method of processing data on the state of a track circuit, based on the use of mathematical apparatus of conformal mappings and fractional-linear transformations. The correlation of the results of calculations with experimental data is 93%, which proves the possibility of practical application of the developed methods.

Large-Signal Stability Guarantees for Cycle-by-Cycle Controlled DC–DC Converters

Stability guarantees are critical for cycle-by-cycle controlled dc-dc converters in high performance applications including microprocessors and LiDAR. Traditional stability analysis on cycle-by-cycle dc-dc converters is incomplete because the inductor current ramps are considered fixed; however, inductor ramps are dependent on the output voltage in large-signal transients, which results in a previously neglected feedback path that often creates instability. We present a new modeling approach together with large-signal stability theory based on a linear fractional transformation of the feedback system. This analysis reveals analytical stability criteria that are straightforward to ensure in practice; the criteria bound sufficient conditions for two practical time constants that are design parameters familiar to power electronics engineers: <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L/R$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$RC$ </tex-math></inline-formula> time constants of the dc-dc converter. These time constants determine the amount of coupling between the current ramp and the output voltage. Specifically, we perform the analysis, simulation, and hardware verification on a buck converter, but the theory and modeling methods apply to other hard-switching power converters.

Mathematical Aspects of &amp;lt;i&amp;gt;SU&amp;lt;/i&amp;gt; (2) and &amp;lt;i&amp;gt;SO&amp;lt;/i&amp;gt;(3,R) Derived from Two-Mode Realization in Coordinate-Invariant Form

Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.

Fractional Transformation-Based Intelligent H-Infinity Controller of a Direct Current Servo Motor

Direct current (DC) servo motors are central to many complex systems, such as electrical, electro-mechanical, and electro-hydraulic frameworks. In practice, these systems can have nonlinear characteristics and parameter variations. Accurate model representation and position tracking of DC motors are the main issues in many real systems, such as twin rotors, aircraft, airships, and robot manipulators. The precise position tracking of these systems has already been achieved using conventional H-infinity (H∞) controllers. However, the order and structure become more intricate when employing complex weights to shape the closed-loop system, which limits the current proposals. To overcome the above-mentioned limitations, in this article, we provide a precise angular position tracking of a DC servo motor utilizing an intelligent, robust linear controller based on a fixed-structure linear fractional transformation. The conventional H∞ controllers are based on the minimization of an unstructured linear fractional transformation objective function that leads to a complex design of these controllers. The main advantage of the proposed intelligent H∞ synthesis is the fixed and simple structure that increases its practical implementation. The methodology is formulated in the MATLAB software for the robust design of the proposed synthesis based on an intelligent fixed-structure H∞ optimization. Simulation results are compared with conventional H∞ and proportional-integral-derivative controllers. The results are also validated experimentally.

A Robust Controller of a Reactor Electromicrobial System Based on a Structured Fractional Transformation for Renewable Energy

The focus on renewable energy is increasing globally to lessen reliance on conventional sources and fossil fuels. For renewable energy systems to work at their best and produce the desired results, precise feedback control is required. Microbial electrochemical cells (MEC) are a relatively new technology for renewable energy. In this study, we design and implement a model-based robust controller for a continuous MEC reactor. We compare its performance with those of traditional methods involving a proportional integral derivative (PID), H-infinity (H∞) controller and PID controller tuned by intelligent genetic algorithms. Recently, a dynamic model of a MEC continuous reactor was proposed, which describes the complex dynamics of MEC through a set of nonlinear differential equations. Until now, no model-based control approaches for MEC have been proposed. For optimal and robust output control of a continuous-reactor MEC system, we linearize the model to state a linear time-invariant (LTI) state-space representation at the nominal operating point. The LTI model is used to design four different types of controllers. The designed controllers and systems are simulated, and their performances are evaluated and compared for various operating conditions. Our findings show that a structured linear fractional transformation (LFT)-based H∞ control approach is much better than the other approaches against various performance parameters. The study provides numerous possibilities for control applications of continuous MEC reactor processes.

Structure Identifiability of an NDS With LFT Parametrized Subsystems

Requirements on subsystems have been made clear in this paper for a linear time invariant (LTI) networked dynamic system (NDS), under which subsystem interconnections can be estimated from external output measurements. In this NDS, subsystems may have distinctive dynamics, and subsystem interconnections are arbitrary. It is assumed that system matrices of each subsystem depend on its (pseudo) first principle parameters (FPPs) through a linear fractional transformation (LFT). It has been proven that if in each subsystem, the transfer function matrix (TFM) from its internal inputs to its external outputs is of full normal column rank (FNCR), while the TFM from its external inputs to its internal outputs is of full normal row rank (FNRR), then the structure of the NDS is identifiable. Moreover, under some particular situations like there are no direct information transmission from an internal input to an internal output in each subsystem, a necessary and sufficient condition is established for NDS structure identifiability. A matrix valued polynomial (MVP) rank based equivalent condition is further derived, which depends affinely on subsystem (pseudo) FPPs and can be independently verified for each subsystem. From this condition, some necessary conditions are obtained for both subsystem dynamics and its (pseudo) FPPs, using the Kronecker canonical form (KCF) of a matrix pencil.

Robust Gain-Scheduling Path Following Control of Autonomous Vehicles Considering Stochastic Network-Induced Delay

This paper concerns the robust gain-scheduling control issue for autonomous path following systems with stochastic network-induced delay. Firstly, to effectively approximate the highly nonlinear tire dynamics, the linear fractional transformation formulations are employed to describe the tire cornering stiffness with a norm-bounded uncertainty. Secondly, by taking the data dropout and delay encountered in signal computation and transmission into account, a more generalized lumped delay form is proposed to unify the time-varying data dropout and network-induced delay. Moreover, a Markovian process is presented to describe the lumped delay as a stochastic distribution. Thirdly, to address the issue of varying vehicle velocity, a linear parameter varying model is established to capture vehicle lateral behaviors. Based on the stochastic stability theory, a new robust gain-scheduling path following control method is proposed for the autonomous vehicles. Finally, the experimental study is presented to bridge the gap between the theoretical and practical investigations on path following control of autonomous vehicles, and results validate the superior performance of the proposed method compared with existing works.

Some Results on Invariant Measures for 1-dimensional Maps

For many fibred systems the existence of an invariant measure can be proved but considerably less is known about the shape of the density. In this note various examples of invariant densities are discussed: Piecewise fractional linear maps with four branches and maps which are associated to continued fractions with increasing digits. There are ergodic maps with a non-integrable density which do not have an indifferent fixed point and maps such that the set of points which miss the digit $k=1$ has positive Lebesgue measure.

Sensor fault diagnosis for uncertain dissimilar redundant actuation system of more electric aircraft via bond graph and improved principal component analysis

This paper develops a sensor condition monitoring method integrating the model-based bond graph (BG) technique and data-driven principal component analysis (PCA) for the dissimilar redundant actuation system of more electric aircraft with uncertain parameters. The uncertain dissimilar redundant actuation system is modeled by BG in linear fractional transformation form. After that, the analytical redundancy relations containing the nominal part and the uncertain part can be derived, based on which the adaptive thresholds and the fault signature matrix (FSM) can be obtained for robust fault detection and fault isolation. To improve the fault isolation performance under the multiple faults condition, a new fault isolation method integrating FSM and improved PCA (IPCA) is developed, where the possible fault set generated from the FSM is further refined by the IPCA module with an improved reconstruction algorithm and cyclic PCA monitoring model to achieve a more efficient fault isolation result. The effectiveness of the proposed approach is validated by simulation investigations.

A Monge normal form for the rolling distribution

Using a parametrisation of s l 2 sl_2 given by the second prolongation of the group action of unimodular fractional linear transformations as presented in an article of Clarkson and Olver, we find a Monge normal form for the distribution associated to the rolling of two hyperboloid surfaces over each other without slipping or twisting.

Advances in fine line-of-sight control for large space flexible structures

The increased need in pointing performance for Earth observation and science Space missions together with the use of lighter and flexible structures directly come with the need of a robust pointing performance budget from the very beginning of the mission design. An extensive understanding of the system physics and its uncertainties is then necessary in order to push control design to the limits of performance and constrains the choice of the set of sensors and actuators. A multi-body framework, the Two Input Two Output Ports approach, is used to build all the elementary flexible bodies and mechanisms involved in a fine pointing mission. This framework allows the authors to easily include all system dynamics with an analytical dependency on varying and uncertain mechanical parameters in a unique Linear Fractional Transformation (LFT) model. This approach opens the doors to modern robust control techniques that robustly guarantee the expected fine pointing requirements. In particular, a novel control architecture is proposed to reduce the microvibrations induced both by reaction wheel imbalances and Solar Array Drive Mechanism driving signal, by letting them work during the imaging phase. Thanks to a set of accelerometers placed at the isolated base of the payload and in correspondence of the mirrors with the largest size in a Space telescope (typically the primary and secondary ones), it is possible to estimate the line-of-sight error at the payload level by hybridizing them with the low-frequency measurements of the camera. While a classical Fast Steering Mirror in front of the camera can compensate for a large amount of microvibration, an innovative architecture with a set of six Proof-Mass Actuators installed at the payload isolator level can further improve the pointing performance. In particular, it is shown how the proposed architecture is able to robustly guarantee an absolute performance error of 10 arcsec in face of system parametric uncertainties at low frequency (≈ 1rad/s) with a progressive reduction of the jitter down to 40 marcsec for higher frequencies where micro-vibration sources act.

Fault Detection of Landing Gear Retraction/Extension Hydraulic System Based on Bond Graph-Linear Fractional Transformation Technique and Interval Analytic Redundancy Relations

Fault detection in the landing gear retraction/extension hydraulic system is difficult due to uncertainties in component parameters and sensor measurement values. This work lies in the introduction of linear fractional transformation technology and uncertainty analysis theory for the construction of the diagnostic bond graph of the landing gear retraction/extension hydraulic system. Thus, interval analytical redundancy relations can be derived as well as fault signature matrices. By using the fault signature matrix, existing faults can be detected and isolated preliminary. Furthermore, interval analytical redundancy relations can be used to detect system faults in detail. The analysis results of the failure cases of the internal and external leakage of the actuator and landing gear selector valve reversing stuck show that compared to the traditional analytical redundancy relations, this method takes into account the negative factors of uncertainty, so it can effectively reduce missed diagnosis and misdiagnosis; compared to the traditional absolute diagnostic threshold, the interval diagnostic threshold is more accurate and sensitive.

Algebraic methods for generating optimal s-blocks

The paper presents a review of algebraic methods for generating cryptographic S-boxes. In this paper, we consider methods for generating S-boxes based on the use of polynomial transformations (linear, quadratic, cubic), fractional linear transformations, and other special types of algebraic transformations. A number of examples of generating S-boxes using quasigroups and other algebraic structures are also given. The considered methods for generating S-boxes make it possible to obtain new S-boxes that have the necessary cryptographic properties that are the same or even better than those of the S-box built for the Rijndael algorithm.

Smooth dependence of fixed points of Hamiltonians

We consider the Lie semigroup of symplectic Hamiltonians acting on the open convex cone of positive definite matrices via linear fractional transformations. Each member of its interior contracts strictly the invariant Finsler metric, the Thompson metric on the cone, and has a unique positive definite fixed point. We show that the fixed point map is smooth. As applications, we obtain the smooth dependence of the solutions of discrete algebraic Riccati equations and a family of smooth maps from the Siegel upper half-plane over the cone of positive definite matrices into its imaginary part.

Robust diagnosis of PEM electrolysers using LFT bond graph

A model based diagnosis for Proton Exchange Membrane (PEM) water electrolyser has been presented in this paper using Linear Fractional Transformation (LFT) bond graph technique. LFT helps in calculating the adaptive thresholds in order to make the diagnosis robust against parameters and measurement uncertainties, hence avoiding false alarms. A modular approach has been followed that allows the model to be adapted to different configurations of PEM electrolyser. The developed diagnosis has been first tested offline by injecting the faults into behavioural model of the PEM electrolyser and then implemented online for a single cell PEM electrolyser.

LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis

This paper presents the design and experimental validation of grid-based and Linear Fractional Transformation (LFT) approaches for the lateral control of autonomous vehicles. These new methodological approaches are compared together with the classical polytopic approach from the theoretical design to the real implementation on a real automated Renault ZOE vehicle. A solution is proposed to deal with both lane change and lane tracking problems, using a single LPV controller, by adapting the look-ahead distance. Each LPV controller is designed based on LPV/ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_\infty $ </tex-math></inline-formula> concept. Performance comparison includes computational costs, vehicle performance (i.e. lateral tracking error or control effort optimization) and on-board integration complexity. Simulation and experimental results on a private test track are included to support main findings.

A note on the action of the Hecke group H(2) on subsets of the form ℚ∗(n)

Here, we study the action of the groups [Formula: see text] generated by the linear fractional transformations [Formula: see text] and [Formula: see text] is a positive integer, on the subsets [Formula: see text] and [Formula: see text] is a square-free positive integer. We prove that this action has a finite number of orbits if and only if [Formula: see text] or [Formula: see text]. Moreover, we give an upper bound for the number of orbits for [Formula: see text].

Linear combinations of composition operators induced by linear fractional maps on the infinite polydisk

The aim of this paper is to characterize the compactness of linear combinations of composition operators induced by linear fractional maps on the Hardy space of analytic functions over the infinite polydisk.

Schur Functions and Inner Functions on the Bidisc

We study representations of inner functions on the bidisc from a fractional linear transformation point of view. We provide sufficient conditions, in terms of colligation matrices, for the existence of two-variable inner functions. Here the sufficient conditions are not necessary in general, and we prove a weak converse for rational inner functions that admit a one variable factorization. We present a classification of de Branges–Rovnyak kernels on the bidisc (which also works in the setting of the polydisc and the open unit ball of \({\mathbb {C}}^n\), \(n \ge 1\)). We also classify, in terms of Agler kernels, two-variable Schur functions that admit a one variable factorization.