Bilinear Transformation Research Articles

Tracking Control of Nanopositioning Stages Using Parallel Resonant Controllers for High-Speed Nonraster Sequential Scanning

The resonant controller (RC), as a promising candidate for high-speed nonraster nanopositioning applications, can track the sinusoidal reference with zero steady-state error. This article presents a controller composed of several RCs in parallel for tracking nonraster sequential scanning trajectories. The selection for each RC in the parallel array is based on two considerations: one is the spectrum of the reference signal and the other is the harmonics caused by the nonlinearities of the nanopositioning stage. The performance of RC is highly dependent on the accurate placement of the resonant poles, but unfortunately, many existing digital implementation methods could cause a deviation of the resonant poles from their initial locations. To address this problem, a modified Tustin (MTus) method is proposed in this article to implement the controller with better accuracy. Furthermore, the fractional-order (FO) calculus is introduced to improve the transient performance of the RCs. To validate the proposed methods, a comprehensive examination of several types of the nonraster sequential scanning trajectories with a wide frequency range has been carried out on a nanopositioning stage. The results have been compared with other methods, showing that the tracking errors are reduced significantly under the controller implemented by the MTus method especially in high-frequency conditions and that the application of the FO calculus reduces the settling time of the controller by more than 30% in most cases. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The demand for high-speed atomic force microscope (AFM) increases rapidly. However, the commonly used raster trajectory limits the achievable scan speed of the AFM. An effective way to improve the scanning and imaging speed of the AFM is the application of the sequential nonraster scanning methods. The trajectories of sequential nonraster scanning patterns mainly composed of few sinusoid signals with different frequencies. Therefore, the resonant controller (RC) is introduced in this article as it is capable to track the sinusoidal reference with zero steady-state error. Several RCs are selected first based on the spectrum analysis of the reference and the consideration of the harmonics caused by the system nonlinearities, and then, they are connected in parallel to form the controller for precise tracking of the reference. In order to realize the digital implementation of the designed controller, a modified Tustin discretization method is proposed, which ensures the accurate resonant pole placement of the RC and thus maintains the tracking performance of the RC. In addition, the fractional-order (FO) calculus is introduced to speed up the convergence of the designed controller while preserving the tracking accuracy, and this parallel-structure FO RC (PSFORC) design can be implemented to other systems that require high-speed and high-accuracy tracking of the periodic signals.

Model predictive control for regular linear systems

The present work extends known finite-dimensional constrained optimal control realizations to the realm of well-posed regular linear infinite-dimensional systems modeled by partial differential equations. The structure-preserving Cayley–Tustin transformation is utilized to approximate the continuous-time system by a discrete-time model representation without using any spatial discretization or model reduction. The discrete-time model is utilized in the design of model predictive controller accounting for optimality, stabilization, and input and output/state constraints in an explicit way. The proposed model predictive controller is dual-mode in the sense that predictive controller steers the state to a set where exponentially stabilizing unconstrained feedback can be utilized without violating the constraints. The construction of the model predictive controller leads to a finite-dimensional constrained quadratic optimization problem easily solvable by standard numerical methods. Two representative examples of partial differential equations are considered.

Modeling the dynamic self-discharge effects of supercapacitors using a controlled current source based ladder equivalent circuit

Abstract Modeling the self-discharge effects can be very useful for the energy awareness of supercapacitors. In this paper, the conventional charge redistribution, the residual charge redistribution and the leakage of charge during self-discharge of a supercapacitor are investigated by a set of experimental results. A novel controlled current source based ladder equivalent circuit model (ECM) is proposed for depicting the dynamic self-discharge effects of the supercapacitor. The originality of the proposed model is that the controlled current source can reflect the diffusion phenomenon of the supercapacitor residual charge during rest phases. Based on the experimental data, the ladder ECM parameters are identified by the least square and bilinear transformation methods. Furthermore, the controlled current source is designed according to the identified parameters and the estimated voltage error of the ladder ECM. Validation is done based on experimentally observed results and simulation results. The comparative results show that the percent relative error of the proposed model is within 0.8%. In addition, the energy awareness and the estimated energy efficiency of the supercapacitor would also be improved by using the proposed model.

Performance Analysis of Speed-Sensorless Induction Motor Drive Using Discrete Current-Error Based MRAS Estimators

In the literature on sensorless control of induction motors, many algorithms have been presented for rotor flux and speed estimation. However, all these algorithms have been developed in the continuous–time domain. The digital realization of the control systems, requires the implementation of those estimation methods in a discrete–time domain. The main goal of this article is comparison of the impact of different numerical integration methods, used in analogue emulation under the digital implementation of the control systems, to the operation of classical Model Reference Adaptive System; CC-based on two current models (MRASCC) speed estimator and its three modified versions developed for the extension of the estimator stability region. In this paper the generalized mathematical model of MRASCC estimator is proposed, which takes into account all known methods for the extension of the stability region of classical speed estimator of this type. After the short discussion of the discretization methods used for the microprocessor implementation of control algorithms the impact of different numerical integration methods on the stable operation range of the classical and modified MRASCC estimators is analyzed and validated in simulation and experimental tests. It is proved that Modified Euler discretization method is much more accurate than forward and backward Euler methods and gives almost as accurate results as Tustin method, however is much less complicated in practical realization.

A Novel Method for Implementing Harmonic Compensation with Multi-Inverters in a Micro-Grid

This paper presents a new multi-inverter-based compensation scheme for the harmonic and reactive influences in a micro-grid, which not only realizes the integration of renewable energy, but also combines the functions of improving power quality at the point of common coupling (PCC) simultaneously. The proposed method uses the remaining power energy in the inverter to implement local harmonic compensation, which achieves the flexible customization of power quality and economics. Specifically, an accurate and stable modulated hopping Discrete Fourier Transform (mHDFT) algorithm with merits of computation of a sparse DFT result is applied to measure the harmonic current. In addition, a proportional resonant (PR) controller selected to track the command current due to its infinite gain for a specific point ensures there is no steady-state error (ess). Besides, delay compensation and discretization by pre-swapping bilinear transformation are performed to improve the stability of the current loop and enable the controller for digital implementation respectively. The simulation results based on Matlab/Simulink are given to verify the performance of the proposed method.

The analysis of conditions for preservation of gain-frequency and phase-frequency characteristics optimality under analog and digital filters transformation

Prototype filters have wide usage for the design of filters with required quality indexes (QI) of gain-frequency response (GFR). The designed filter is obtained from a prototype filter b means of frequency transformation, which preserves these QI. But most of employed frequency transformations result in variations of QI of phase-frequency response (PFR). In this paper we proposed to use prototype filters that are Pareto-optimal for QI of GFR and PFR. Transfer functions of these filters may be found by means of heuristic optimization algorithms. This method will be efficient if the frequency transformation preserves the optimality of filters. It was shown that frequency transformation has this feature if it preserves the result of QI comparison (more or less) for filters with equal orders. Compliance of this criterion was checked for bilinear transformation of analog low pass filters (LPF) into digital LPF and for Konstantinidis transformation of digital LPF into other digital LPF. The analysis showed that Pareto-optimality for QI of GFR and PFR is preserved if the delay-frequency characteristic of the filter has a minimum at zero frequency and has a maximum at the upper boundary of the pass band. These conditions are complied for LPF with sufficiently small unevenness of GFR in the pass band and sufficiently fast decline of GFR at higher frequencies. Examples confirming these conclusions are given.

Model order reduction of interval systems using an arithmetic operation

ABSTRACTThe paper presents an extension of the differentiation method for model order reduction of large-scale interval systems. This is an alternative approach to the existing differentiation method of interval systems. The proposed method has been applied for both continuous and discrete-time interval systems. The reduction of discrete-time interval systems is achieved by using simple linear transformation and bilinear transformation , where . The proposed method always generates stable reduced-order models, and also it retains the zeroth-order interval time moment. Four numerical examples exemplify the accuracy of the method and computational simplicity. Furthermore, the difficulties associated with the extension of Routh-based approximations to interval systems for obtaining stable reduced-order models are discussed. The stability of interval systems is verified by using Kharitonov's theorem.

A traction controller of a two-wheel mobile robot with feedforward compensator

A simple traction controller was derived from a model following control (MFC) using bilinear transformation. The controller was simplified by adjusting some dynamic parameters. The experiments were conducted to evaluate the validity of the controller by comparing with the classical PD controller. In the experiments, while the traction controller can prevent slippage, tracking control performance needed to be improved. The feedforward compensation was added to reduce the traction error. The experimentation using the simple controller with feedforward shows that the position error has been reduced while maintaining the traction force between floor and wheels.

New discrete-time fractional derivatives based on the bilinear transformation: Definitions and properties

In this paper we introduce new discrete-time derivative concepts based on the bilinear (Tustin) transformation. From the new formulation, we obtain derivatives that exhibit a high degree of similarity with the continuous-time Grünwald-Letnikov derivatives. Their properties are described highlighting one important feature, namely that such derivatives have always long memory.

Improved design of noise transfer functions with monotonic passband response for Delta‐Sigma modulators

A method for the design of a noise transfer function (NTF) is presented in which the stopband of a continuous-time high-pass filter with monotonic passband characteristics is minimised by optimally placing an arbitrary number of imaginary-axis zeros. Towards this end, the Papoulis, Halpern, and least-squares monotonic polynomials have been used in the high-pass filter design. The NTF is derived by applying the bilinear transformation on a continuous-time high-pass filter that minimises the in-band energy and satisfies all the realisability constraints of a Delta-Sigma modulator (DSM). By way of examples, it is shown that the signal-to-quantisation noise ratio achieved by the NTF designed by the proposed method is higher than that obtained by the NTF designed using a recently published method. The signal-to-noise ratio of the NTF designed with the proposed method is higher for a wide range of signal amplitude.

A step edge detector based on bilinear transformation

Nowadays, Canny edge detector is considered to be one of the best edge detection approaches for the images with step form. Various overgeneralized versions of these edge detectors have been offered up to now, e.g. Saryazdi edge detector. This paper proposes a new discrete version of edge detection which is obtained from Shen-Castan and Saryazdi filters by using bilinear transformation. Different experimentations are conducted to decide the suitable parameters of the proposed edge detector and to examine its validity. To evaluate the strength of the proposed model, the results are compared to Canny, Sobel, Prewitt, LOG and Saryazdi methods. Finally, by calculation of mean square error (MSE) and peak signal-to-noise ratio (PSNR), the value of PSNR is always equal to or greater than the PSNR value of suggested methods. Moreover, by calculation of Baddeley’s error metric (BEM) on ten test images from the Berkeley Segmentation DataSet (BSDS), we show that the proposed method outperforms the other methods. Therefore, visual and quantitative comparison shows the efficiency and strength of proposed method.

Implementation of Control Algorithms in Small Embedded Systems

This work describes how a control algorithm can be implemented in a small (8-bit) microcontroller for the main purpose of merging embedded systems and control theory in electrical engineering undergraduate classes. Two different methods for discretizing the control expression are compared: Euler transformation and bilinear transformation. The sampling rate’s impact on the algorithm is discussed and theoretical results are verified by an application to a temperature control system in a heating plant. Four control algorithms are compared: PID and PI algorithms discretized with Euler and bilinear transformation, respectively. It is shown that for the heating plant used in this work, a bilinear PI algorithm implemented in a small 8-bit microcontroller outperforms a commercial controller from Panasonic. It is also demonstrated that all the derived algorithms can be implemented using integer calculations only, obviating the need for expensive and time-consuming floating-point calculations. This work bridges the gap between control theory equations and the implementation of control systems in small embedded systems with no inherent floating-point processing power.

Comparison of Tustin Transform and Backward Difference in Digital PID Controller Using Limited Pole Placement Method

Comparison of Tustin Transform and Backward Difference in Digital PID Controller Using Limited Pole Placement Method

Mixed lump and soliton solutions for a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation

Under investigation is a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation which can be used to describe nonlinear wave propagation in dissipative media. Via the bilinear transformation method, the mixed lump and soliton solutions are obtained for the equation. The asymptotic behavior of the mixed solutions are analyzed. Furthermore, the fusion and fission behaviors of the lump and soliton are observed for the first time. The lump and soliton can merge into a whole soliton over time, or, on the contrary, the soliton may differentiate into a lump and a new soliton. During the processes, the amplitude of the lump will greatly vary, while the amplitude of the soliton will change slightly.

Modeling and Evaluation of Stator and Rotor Faults for Induction Motors

The modeling of stator and rotor faults is the basis of the development of online monitoring techniques. To obtain reliable stator and rotor fault models, this paper focuses on dynamic modeling of the stator and rotor faults in real-time, which adopts a multiple-coupled-circuit method by using a winding function approach for inductance calculation. Firstly, the model of the induction machine with a healthy cage is introduced, where a rotor mesh that consists of a few rotor loops and an end ring loop is considered. Then, the stator inter-turn fault model is presented by adding an extra branch with short circuit resistance on the fault part of a stator phase winding. The broken rotor bar fault is then detailed by merging and removing the broken-bar-related loops. Finally, the discrete models under healthy and faulty conditions are developed by using the Tustin transformation for digital implementation. Moreover, the stator and rotor mutual inductances are derived as a function of the rotor position according to the turn and winding functions distribution. Simulations and experiments are performed on a 2.2-kW/380-V/50-Hz three-phase and four-pole induction motor to show the performance of the stator and rotor faults, where the saturation effect is considered in simulations by exploiting the measurements of a no load test. The simulation results are in close agreement with the experimental results. Furthermore, magnitudes of the characteristic frequencies of 2f1 in torque and (1 ± 2s)f1 in current are analyzed to evaluate the stator and rotor fault severity. Both indicate that the stator fault severity is related to the short circuit resistance. Further, the number of shorted turns and the number of continuous broken bars determines the rotor fault severity.

Adaptive fractional‐order control of power system frequency in the presence of wind turbine

Inertial and droop control are common methods for improving grid frequency control. Using the first-order derivative of the frequency in the inertial control loop only investigates the rate of frequency variation, while other useful information is not taken into consideration. Accordingly, in this study, the application of a fractional-order derivative (between 1 and 2) in the inertial control loop is proposed to have more useful frequency-based information. To achieve this goal, firstly continued fraction approximations and secondly, the discretization is adopted by using the Tustin Method with Prewarping. Inertia loop and droop loop gain have a significant impact on the operation of the wind turbines and the power system frequency control. Therefore, in this paper, a novel method for the adaptive adjusting of droop and inertia control loops gains in DFIG is proposed. To utilize more frequency error information, fractional-order adaptive law is proposed to update the loop gains and the inertia control loop. In order to have a robust system, which is permanently stable for every fractional order (FO), is used parameter space method. Simulation results demonstrate the proper performance of the FO-based adaptive approach to increase the frequency nadir (FN) and decrease the frequency variations.

Comprehensive Study on Numerical Aspects of Modified Lorentz Model-Based Dispersive FDTD Formulations

Finite-difference time domain (FDTD) has been widely used to analyze electromagnetic wave interaction with dispersive media. It is of great necessity to incorporate a dispersion model into FDTD formulation for electromagnetic wave analysis of dispersive media. Recently, it was reported that the modified Lorentz model can cover Debye, Drude, Lorentz, critical point, and quadratic complex rational function models. In this work, it is illustrated that the modified Lorentz model can also cover the complex-conjugate pole-residue model which is one of the most popular dispersion models. Modified Lorentz-based dispersive FDTD has not been thoroughly studied, especially for numerical aspects. In this work, we investigate auxiliary differential equation (ADE)-FDTD formulations for the modified Lorentz model based on electric flux density (D), current (J), or polarization (P). We perform a comprehensive study on memory requirement, the number of arithmetic operations, numerical stability, and numerical permittivity for the above three ADE-FDTD formulations. In addition, the bilinear transformation (BT) is incorporated into modified Lorentz-based FDTD formulations and it will be shown that the utilization of the BT can lead to better performance in terms of numerical stability and numerical accuracy. Numerical examples are used to demonstrate our work.

Computationally Efficient CN-PML for EM Simulations

Since nearly all studies concerning the approximate Crank–Nicolson perfectly matched layer (CN-PML) are limited to 2-D cases, a computationally efficient implementation that can be used to truncate 3-D finite-difference time-domain (FDTD) lattices is presented in this article. More precisely, it is based on the CN direct-splitting (DS) scheme and the bilinear transform (BT) method. This article can fully exploit the unconditional stability of the standard CN-FDTD method and can be free from the Courant–Friedrich–Lewy (CFL) limit; hence, it is especially suitable for situations where space discretization step is much smaller than 1/10th or 1/20th of the smallest wavelength of interest. Aiming at further reducing the requirement of the computer resources, this new implementation can be reformulated in more simple forms if proper auxiliary variables are introduced. It therefore shows a higher iteration speed than other published unconditionally stable PMLs as fewer numbers of arithmetic operations are involved. Finally, three numerical examples, including scatting, transmission, and radiation, are also provided to validate its running time, unconditional stability, and absorption characteristic.

Improved smoothness priors using bilinear transform

Smoothness priors is a well-known and most commonly used method in the analysis of stochastic processes making it very useful in the field of stochastic signal processing. It is particularly suited for smoothing the noisy data and detrending the time-series signals. The method is based on an optimization problem where the n-th order derivative of the signal enters as a constraint. When the method is designed in discrete time domain, the backward difference rule is used to perform differential-to-difference conversion. Moreover, the solution depends on a smoothness trade-off parameter. An efficient algorithm for the trade-off parameter selection remains an important and challenging issue. In this paper, first, we propose a closed-form expression for the trade-off parameter. The closed-form expression resulted from a frequency domain interpretation of the smoothness priors procedure. The trade-off parameter determines the amount of frequency components that the procedure allows to pass. We show that the trade-off parameter is related to the arbitrary choice of cutoff frequency. Second, we introduce a new way to the design and implementation of smoothness priors using bilinear transformation method. Frequency analysis and experiments on both synthetic and real world signals with different levels of noise demonstrate that bilinear transform is indeed more effective for smoothness priors implementation when compared with the traditional ones, i.e., the backward difference rule.

NEW CLASS OF DIGITAL MALMQUIST-TYPE ORTHOGONAL FILTERS BASED ON THE GENERALIZED INNER PRODUCT; APPLICATION TO THE MODELING DPCM SYSTEM

A new class of cascade digital orthogonal filters of the Malmquist type based on bilinear transformation for mapping poles to zeroes and vice versa is presented in this paper. In a way, it is a generalization of the majority of the classical orthogonal filters and some newly designed filters as well. These filters are orthogonal with respect to the generalized inner product which is actually a generalization of the classical inner product. Outputs of these filters are obtained by using polynomials orthogonal with respect to the new inner product. The main quality of these filters is that they are parametric adaptive. The filter with six sections is practically realized in the Laboratory for Modeling, Simulation and Control Systems. Performances of the designed filter are proved on modeling and identification of the system for differential pulse code modulation. Real response and response from the proposed filter are compared with regard to the chosen criteria function. Also, a comparative analysis of the proposed filter with some existing filters is performed.