Bilinear Transformation Research Articles

Selective weighted least square and piecewise bilinear transformation for accurate satellite DSM generation

One of the main products of multi-view stereo (MVS) high-resolution satellite (HRS) images in photogrammetry and remote sensing is digital surface model (DSM). Producing DSMs from MVS HRS images still faces serious challenges due to various reasons such as complexity of imaging geometry and exterior orientation model in HRS, as well as large dimensions and various geometric and illumination variations. The main motivation for conducting this research is to provide a novel and efficient method that enhances the accuracy and completeness of extracting DSM from HRS images compared to existing recent methods. The proposed method called Sat-DSM, consists of five main stages. Initially, a very dense set of tie-points is extracted from the images using a tile-based matching method, phase congruency-based feature detectors and descriptors, and a local geometric consistency correspondence method. Then, the process of Rational Polynomial Coefficients (RPC) block adjustment is performed to compensate the RPC bias errors. After that, a dense matching process is performed to generate 3D point clouds for each pair of input HRS images using a new geometric transformation called PWB (pricewise bilinear) and an accurate area-based matching method called SWLSM (selective weighted least square matching). The key innovations of this research include the introduction of SWLSM and PWB methods for an accurate dense matching process. The PWB is a novel and simple piecewise geometric transformation model based on superpixel over-segmentation that has been proposed for accurate registration of each pair of HRS images. The SWLSM matching method is based on phase congruency measure and a selection strategy to improve the well-known LSM (least square matching) performance. After dense matching process, the final stage is spatial intersection to generate 3D point clouds, followed by elevation interpolation to produce DSM. To evaluate the Sat-DSM method, 12 sets of MVS-HRS data from IRS-P5, ZY3-1, ZY3-2, and Worldview-3 sensors were selected from areas with different landscapes such as urban, mountainous, and agricultural areas The results indicate the superiority of the proposed Sat-DSM method over four other methods CATALYST, SGM (Semi-global matching), SS-DSM (structural similarity based DSM extraction), and Sat-MVSF in terms of completeness, RMSE, and MEE. The demo code is available at https://www.researchgate.net/publication/377721674_SatDSM.

Superposition and Interaction Dynamics of Complexitons, Breathers, and Rogue Waves in a Landau–Ginzburg–Higgs Model for Drift Cyclotron Waves in Superconductors

This article implements the Hirota bilinear (HB) transformation technique to the Landau–Ginzburg–Higgs (LGH) model to explore the nonlinear evolution behavior of the equation, which describes drift cyclotron waves in superconductivity. Utilizing the Cole–Hopf transform, the HB equation is derived, and symbolic manipulation combined with various auxiliary functions (AFs) are employed to uncover a diverse set of analytical solutions. The study reveals novel results, including multi-wave complexitons, breather waves, rogue waves, periodic lump solutions, and their interaction phenomena. Additionally, a range of traveling wave solutions, such as dark, bright, periodic waves, and kink soliton solutions, are developed using an efficient expansion technique. The nonlinear dynamics of these solutions are illustrated through 3D and contour maps, accompanied by detailed explanations of their physical characteristics.

Adaptive control of an amplified piezoelectric-driven micropositioning stage based on notch filter structure inspiration and neural network online tuning

Abstract Piezoelectric-driven flexure micro/nanopositioning stages often exhibit a low-damping resonance mode, which can easily excite mechanical resonance during high-speed movement, and significantly impact the control system's stability, control bandwidth, and trajectory tracking accuracy. To mitigate the reliance on the precise modeling of stage dynamics inherent in current resonant controllers, an adaptive control method based on a back propagation (BP) neural network was designed to suppress resonance in real-time. First, a piezoelectric-driven flexure micro/nanopositioning stage system was constructed. Next, a feedback controller similar to a notch filter was designed, with bilinear transformation applied based on the system's inherent parameters to determine the initial values. Finally, the designed adaptive control method was tested through trajectory tracking experiments using a triangular wave signal. The experimental results showed that, when tracking the triangular wave signal, the maximum tracking error was reduced by 72.66% compared to proportional-integral (PI) control alone and by 68.66% compared to proportional integral control combined with a traditional notch filter. The tracking results demonstrate a significant improvement in the stage's stability and tracking accuracy.

Multiple solitons, multiple lump solutions, and lump wave with solitons for a novel (2+1)-dimensional nonlinear partial differential equation

The primary aim of this paper is to explore exact solutions to a novel (2+1)-dimensional water wave equation that models oceanic wave phenomena. We begin by applying the Hirota bilinear transformation method to derive multi-soliton solutions, including 3-soliton and 4-soliton solutions. Then, utilizing the bilinear form of the equation and the long-wave limit method, we identify multiple lump solutions and interaction solutions between lumps and solitons. These include 1-lump, 2-lump, and 3-lump solutions, as well as interactions between a 1-lump and a 1-soliton, and between a 1-lump and 2-solitons. The physical dynamics of these solutions are visually represented, offering insight into the corresponding oceanic wave phenomena.

Numerical Solution of Linear Second-Kind Convolution Volterra Integral Equations Using the First-Order Recursive Filters Method

A new numerical method for solving Volterra linear convolution integral equations (CVIEs) of the second kind is presented in this work. This new approach uses first-order infinite impulse response digital filters method (IIRFM). Three convolutive kernels were analyzed, the unit kernel and two singular kernels: the logarithmic and generalized Abel kernels. The IIRFM is based on the combined use of the Laplace transformation, a first-order decomposition, and a bilinear transformation. This approach often leads to simple analytical expressions of the approximate solutions, enabling efficient numerical calculation, even using single-precision floating-point numbers. When compared with the method of homotopic perturbations with Laplace transformation (HPM-L), the IIRFM approach does not present, in linear cases, the convergence difficulties inherent to iterative approaches. Unlike most solution methods based on the Laplace transform, the IIRFM has the dual advantage of not requiring the calculation of the Laplace transform of the source function, and of not requiring the systematic calculation of inverse Laplace transforms.

Robust digital voltage mode control of buck converter with enhanced tracking performance

The digital voltage controller of power converters is used because it has many advantages over its analog counterparts. Thus, in this paper, a controller of digital voltage mode is designed and analyzed for a non-isolated DC-DC buck converter in continuous conduction mode (CCM). First, an analog controller is designed using a Bode plot to achieve the desired stability margins in the frequency domain. Next, the analog controller is discretized using bilinear transformation with the pre-warping method. The characteristics of the discretized compensator can be maintained close to the corresponding analog compensator as long as a proper sampling time is selected. The digital control scheme is simulated with a nonlinear DC-DC buck converter model in MATLAB/Simulink to investigate the controller performance. The simulation results demonstrated the validation of the proposed digital voltage-mode control design approach. The developed digital controller eliminates DC error by tracking the reference voltage, achieving a fast transient response, maintaining specified stability margins, and effectively rejecting large disturbances.

Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations

Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations

A transfer function based on bilinear transformation for narrow-band seismic record.

For a narrow-brand seismograph with a flat response range limited, it cannot precisely record the signal of a ground motion and output the records with the low-frequency components cut down. A transfer function is usually used to spread the spectrum of the narrow-brand seismic records. However, the accuracy of the commonly used transfer function is not high. The authors derive a new transfer function based on the Laplace transform, bilinear transform, and Nyquist sampling theory to solve this problem. And then, the derived transfer function is used to correct the narrow-band velocity records from the Hi-net. The corrected velocity records are compared with the velocities integrated from the synchronously recorded broad-band acceleration at the same station with Hi-net. Meanwhile, the corrected records are compared with those corrected by the Nakata transfer function. The results show that the calculation accuracy of the derived transfer function is higher than the Nakata transfer function. However, when the signal-to-noise ratio is below 24, its accuracy diminishes, and it is unable to recover signals within the 0.05-0.78Hz frequency band.

The chemostat reactor: A stability analysis and model predictive control

This contribution develops the model predictive control for an unstable chemostat reactor. Initially, we analyze the system’s model — a nonlinear first-order hyperbolic partial integro-differential equation (PIDE) — and carry the model linearization around an unstable operating condition. Employing the structure-preserving Cayley–Tustin transformation, we obtain a discrete-time model representation of the continuous model. Subsequently, we solve the operator Ricatti equations in the discrete-time setting to derive a full state feedback controller that stabilizes the closed-loop and design a Luenberger observer for state reconstruction given the system output measures. Finally, we formulate a dual-mode MPC ensuring constraint satisfaction and optimality, integrating the gain-based unconstrained full-state feedback optimal control obtained from the Ricatti equation. This dual-mode strategy describes an optimization problem where the predictive controller acts only if constraints become active within the control horizon. Simulation studies validate the controller performance, where the MPC only takes action if the constraints are predicted to be active within the control horizon while also guaranteeing closed-loop stabilization under only output feedback. This type of controller can be easily implemented with other control strategies and significantly decreases the computational costs of solving the optimal control problems when compared to other MPC approaches.

Breather, lump, M-shape and other interaction for the Poisson–Nernst–Planck equation in biological membranes

This paper investigates a novel method for exploring soliton behavior in ion transport across biological membranes. This study uses the Hirota bilinear transformation technique together with the Poisson–Nernst–Planck equation. A thorough grasp of ion transport dynamics is crucial in many different scientific fields since biological membranes are important in controlling the movement of ions within cells. By extending the standard equation, the suggested methodology offers a more thorough framework for examining ion transport processes. We examine a variety of ion-acoustic wave structures using the Hirota bilinear transformation technique. The different forms of solitons are obtained including breather waves, lump waves, mixed-type waves, periodic cross-kink waves, M-shaped rational waves, M-shaped rational wave solutions with one kink, and M-shaped rational waves with two kinks. It is evident from these numerous wave shapes that ion transport inside biological membranes is highly relevant, and they provide important insights that may have an impact on various scientific disciplines, medication development, and other areas. This extensive approach helps scholars dig deeper into the complexity of ion transport, illuminating the complicated mechanisms driving this essential biological function. Additionally, to show the physical interpretations of these solutions we construct the 3D and their corresponding contour plots by choosing the different values of constants. So, these solutions give us the better physical behaviors.

Concept of Bilinear Transformation, Jacobian, and Conformal Mapping with Applications

This article explores the concepts of bilinear transformation, Jacobian transformation, and conformal mapping, focusing on their essential properties and presenting key results. The discussion revolves around isogonal transformation, conformal transformation, Jacobian transformation, and bilinear transformation, as well as critical points and fixed points.

Necessary and Sufficient Conditions for Ensuring Stability and Avoiding Sinusoidal Oscillation of Uncertain Interval Systems

In this paper, two types of second-order uncertain interval systems are proposed and discussed. Based on control theory, time-domain approach, and bilinear transformation method, the necessary and sufficient conditions are derived for above two interval systems to ensure that stability can be achieved and that the output signal will not produce sinusoidal oscillations. Finally, several numerical simulations are given to illustrate the feasibility and effectiveness of the obtained results.

Current Harmonics and Unbalance Suppression of Dual Three-Phase PMSM Based on Adaptive Linear Neuron Controller

A strategy of current harmonics and unbalance suppression based on the vector space decomposition control with adaptive linear neuron (ALN) controller for the dual three-phase permanent magnet synchronous machine (DTP-PMSM) drive system is proposed in this paper. According to the conventional vector space decomposition theory, the compensation voltages from the ALN controller in the dq-frame and dqz-frame are employed to suppress the 5th and 7th current harmonics and unbalances. Compared with the VSD control using only the PI controller, the current harmonics and unbalance are sufficiently suppressed in the proposed strategy. In addition, unlike the resonant controller discretized by conventional Tustin transformation, the ALN controller has no resonant pole shift to the desired resonant frequency. Therefore, the harmonic suppression effect of the proposed strategy exhibits better performance than the VSD control with the resonant controller, especially in the high-speed region. Finally, the feasibility and effectiveness of the proposed strategy are validated by experimental results under both steady and transient states.

A Frequency-Weighting Digital Filter in Sound Level Meter Based on Neural Computing Method

Frequency weighting networks are a critical component of a sound level meter (SLM), and their error characteristics directly determine the performances of SLM. For reducing the high-frequency error of the [Formula: see text] frequency-weighting filters with the bilinear transformation method (BTM), a design method for [Formula: see text] frequency-weighting filters based on neural computing method (NCM) is proposed. A detailed algorithm for solving the filter coefficients is provided, and the amplitude-frequency characteristics of the [Formula: see text] frequency-weighting filters with BTM and NCM are compared in detail. The experimental results show that the amplitude-frequency characteristics of the [Formula: see text] frequency-weighting filters in SLM with NCM are significantly better than those of BTM. The filter meets the requirements of the first class SLM defined by IEC61672, which demonstrates the effectiveness of this proposed method.

Serial digital QAM-modem based on complex band-pass IIR filters with LF-prototypes of Bessel

The article is devoted to the development and implementation of algorithms for the single-channel sequential quadrature amplitude modulation and demodulation based on digital complex band-pass IIR filters with LF-prototypes of Bessel. The use of complex IIR filters of Bessel allows to obtain orthogonal carrier signals which are close in shape to symmetric signals with practical non-overlapping spectra. The procedure for calculating the 4th-order digital low-pass Bessel filter using the generalized bilinear transformation method, which is used in the theory of digital signal processing to convert a continuous-time system into a discrete-time system, is considered. The variants of implementing the digital complex bandpass filters with a sequential structure using the complex delay method are considered, which allows to obtain the block diagrams of complex bandpass filters without additional calculations and to realize the restructuring of the central frequency without changing the frequency response shape by changing only two coefficients in complex delays. The procedure for obtaining of the expression of the impulse response of digital complex band-pass IIR filters of Bessel is considered. With the help of circuit modeling, the amplitude-frequency response and impulse response of such filter are determined. If the IIR filters are used in the modulator, the pulse characteristics of which are infinite and asymmetric, then it is necessary to use digital complex FIR filters consistent with the truncated pulse characteristic in the demodulator. But as an exception, the Bessel digital IIR filter has a phase-frequency response similar in shape closely to the linear one and, accordingly, a pulse response close to the symmetric one. In this case, the same recursive complex Bessel filters, similar to the filter in the modulator, can be used as matched filters in the demodulator. Therefore, the serial block diagram of the modem contains the same complex band-pass IIR filters with LF-prototypes of Bessel. The choice of the central frequency of the channel digital complex bandpass IIR filter is carried out by setting two coefficients. The results of circuit modeling of the data transmission system with the quadrature amplitude modulation (QAM-16) in the Microcap environment are presented, confirming the operability of the proposed option.

Approximate moving horizon estimation for switching conservative linear infinite-dimensional systems

This work addresses moving horizon estimation for switching conservative linear infinite-dimensional systems described by partial differential equations (PDE), where the plant and measurement equations are corrupted with bounded disturbances, and the system mode is regarded as an unknown and unpredictable discrete state to be estimated. To address the issues associated with unbounded operators (induced by boundary or point observation and disturbance) and facilitate discrete-time moving horizon estimator design, the Cayley–Tustin transformation is deployed for model time-discretization without any spatial discretization or model reduction while preserving model essential properties. A series of observability concepts along with corresponding properties are proposed and analyzed for the switching linear infinite-dimensional discrete-time systems. A moving horizon estimation algorithm that accounts for state/output and mode estimation and constraint handling is proposed. Based on the proposed observability properties, we prove the stability of the proposed moving horizon estimator. The derived results are demonstrated by an undamped Schrödinger equation with switching group-velocity dispersion.

Design of Wide‐Beam Leaky‐Wave Antenna Arrays Based on the Bilinear Transformation of IIR Digital Filters and the Z Transform

Abstract In the pioneering work, the radiation diagrams of leaky wave antenna arrays can achieve attenuation nulls and gains at specific angles by manually placing the zeros and the poles in the Z domain of the corresponding discrete linear time‐invariant (LTI) system. This handcrafted design procedure does not allow radiation diagrams with wide beams since the interaction between poles involved in the wide beam and their corresponding leaky modes cannot be easily handled. To overcome this limitation, this paper describes a novel method for designing radiation diagrams of leaky‐wave antenna arrays based on the theory of IIR discrete filters. The proposed method relies on the design of discrete filters with the prototypes of analog low‐pass filters defined by Butterworth and Chebyshev type I polynomials, whose roots along with the bilinear transformation provide the location of the poles and the zeros of the discrete LTI system and, therefore, the parameters of the leaky‐wave antenna array. Results with different designs and a comparison with other approaches show the utility and effectiveness of this novel method to design wide‐beam leaky‐wave antenna arrays.

On the distribution of unique range sets and its elements over the extended complex plane

In the paper, we discussed the distribution of unique range sets and its elements over the extended complex plane from a different point of view and obtained some new results regarding the structure and position of unique range sets. These new results have immense applications like classifying different subsets of C to be or not to be a unique range set, exploring the fact that every bi-linear transformation preserves unique range sets for meromorphic functions, providing simpler and shorter proofs of existence of some unique range sets, unfolding the fact that zeros or poles of any meromorphic function lie in a unique range set, in particular,identifying the Fundamental Theorem of Algebra to a more specific region and many more applications. We have also posed some open questions to unveil the mysterious arrangement of the elements of unique range sets.

Vector Control Technology of Five‐Phase Induction Motor without Speed Sensor Based on Speed‐Adaptive Full‐Order Observer

The five‐phase induction motor, being the most elementary form of multiphase induction motors, boasts low noise, low torque fluctuations, and high load‐bearing capacity, and is hence suitably employed in domains where high power and reliability are required. In light of the susceptibility of the speed encoder to interference from external environments, this paper proposes a sensorless vector control technique for the five‐phase induction motor based on a speed‐adaptive full‐order observer. The proposed approach first employs the nearest four‐vector space vector pulse width modulation (NFV‐SVPWM) as the modulation technique for the five‐phase inverter, in order to provide a constant switching frequency and low harmonic losses. Subsequently, the expression for the speed‐adaptive full‐order observer is constructed based on the mathematical model of the fundamental space of the five‐phase induction motor, and the expression for the adaptive speed law is deduced using the Lyapunov function. Additionally, the bilinear transformation method is utilized as the discretization technique for the full‐order observer during the process of digital implementation, in order to achieve higher discrete accuracy. Finally, an experimental study involving a 4.5 kW five‐phase induction motor was conducted to validate the accuracy and effectiveness of the proposed method. © 2023 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC.

Two-Level Feature-Fusion Ship Recognition Strategy Combining HOG Features with Dual-Polarized Data in SAR Images

Due to the inherent characteristics of synthetic aperture radar (SAR) imaging, SAR ship features are not obvious and the category distribution is unbalanced, which makes the task of ship recognition in SAR images quite challenging. To address the above problems, a two-level feature-fusion ship recognition strategy combining the histogram of oriented gradients (HOG) features with the dual-polarized data in the SAR images is proposed. The proposed strategy comprehensively utilizes the features extracted by the HOG operator and the shallow and deep features extracted by the Siamese network in the dual-polarized SAR ship images, which can increase the amount of information for the model learning. First, the Siamese network is used to extract the shallow and deep features from the dual-polarized SAR images, and then the HOG feature of the dual-polarized SAR images is also extracted. Furthermore, the bilinear transformation layer is used for fusing the HOG features from dual-polarized SAR images, and the grouping bilinear pooling process is used for fusing the dual-polarized shallow feature and deep feature extracted by the Siamese network, respectively. Finally, the catenation operation is used for fusing the dual-polarized HOG features and dual-polarized shallow feature and deep feature, respectively, which are used for the recognition of the SAR ship targets. Experimental results tested on the OpenSARShip2.0 dataset demonstrate the correctness and effectiveness of the proposed strategy, which can effectively improve the recognition performance of the ship targets by fusing the different level features of the dual-polarized SAR images.