Root Locus Research Articles

Stability control of high-speed magnetic levitation turbomolecular pumps with shock-excited disturbance

The disturbance suppression of magnetic levitation turbomolecular pumps is a critical problem in industrial applications. This work addresses the stability control of high-speed magnetic levitation turbomolecular pumps with shock-excited disturbance. A disturbance suppression method based on improved linear extended state observer is proposed to attenuate the impact of external low-frequency disturbing force on a magnetic levitation turbomolecular pump. Firstly, a linear extended state observer of an active magnetic bearing is obtained by analyzing the rotor dynamics model. Then, the detailed descriptions of external disturbance suppression method using linear extended state observers and adaptive notch filters are discussed for the system. The boundary condition of the parameters of the improved linear extended state observer is determined. The root loci of the closed-loop system with improved linear extended state observers is also investigated. Finally, simulation and experimental results on a magnetic levitation turbomolecular pump show the applicability of the proposed method. The results show that the proposed method can attenuate the rotor vibration displacement caused by impact by 46.9%.

Attitude Control and Parameter Optimization: A Study on Hubble Space Telescope

Abstract In this work, we build a satellite attitude Proportional-Integral-Derivative (PID) controlled system by using the Hubble Space Telescope (HST) parameters as a reference and tune its controller parameters using various tuning methods. First, we give the equations for the motion of a satellite. We elaborate the control structure as controller, actuator, dynamics, and kinematics subsystems and construct an external disturbance model. We use a reaction wheel assembly used in the HST with the same configuration as the actuator. We evaluate the performance of the linearization by comparing it with the nonlinear model output. By working on the linearized model, we tune the PID controller parameters using two different methods: “Model-Based Root Locus Tuning” and “Genetic Algorithm Based Tuning”. First, we obtain the controller parameters by manipulating the poles on the root locus plot of the linearized system. In addition, we use genetic algorithms to find the optimized controller values of the system. Finally, we compare the performances of the two methods based on their cost function values and find that the Genetic Algorithm-based tuned parameters are more fruitful in terms of the cost function value than the parameters obtained by the Root Locus-based tuning. However, it is found that the Root Locus-based tuning performs better in disturbance rejection.

An Adaptive Virtual Impedance Method for Grid-Connected Current Quality Improvement of a Single-Phase Virtual Synchronous Generator under Distorted Grid Voltage.

The proportion of distributed generation systems in power grids is increasing, leading to the gradual emergence of weak grid characteristics. Moreover, using voltage-sourced grid-connected inverters can enhance the stability of a weak grid. However, due to the presence of background harmonics in weak grids, the grid voltage can cause significant distortions in the grid-connected current, which adversely affects the quality of the grid-connected current. This paper begins by briefly introducing the principle of the virtual synchronous generator (VSG). Then, the output current of the voltage source inverter is analyzed to elucidate the mechanism of harmonic current generation. Considering the distortion in the grid-connected current of the voltage source grid-connected inverter caused by background harmonics in the grid voltage, a harmonic current suppression strategy based on an adaptive virtual harmonic resistor is proposed. The proposed strategy employs a signal separation module based on multiple second-order generalized integrators connected through a cross-feedback network. This module effectively separates the fundamental and harmonic currents from the grid-connected current, extracts the amplitudes of the fundamental and harmonic currents through coordinate transformation, and adaptively adjusts the virtual harmonic resistance magnitude through the negative feedback control of the harmonic content (the ratio of the harmonic current amplitude to the fundamental current amplitude). These measures are used to enhance the quality of the grid-connected current. Additionally, the stability of the system is analyzed using the root locus of the open-loop transfer function. Finally, the effectiveness of the proposed method is validated through a combination of MATLAB/Simulink simulations and experimental results.

Comparative Study of Furuta Pendulum Based on LQR and PID Control

Furuta pendulum is an underdriven and unstable controlled object with one input and two outputs, and has an open-loop zero point at the origin of the complex plane. In this paper, a nonlinear dynamic model of the Furuta pendulum is established by the Eulerian-Lagrangian method, and a series-connected PID control strategy is designed by using the root locus method and the frequency domain analysis method based on the local linearization model at the steady-state operating point. Besides, an LQR control strategy is designed based on the discrete state-space model. In addition, a comparative analysis of the PID control strategy and the LQR control strategy is conducted, and it is found that the LQR control strategy can achieve stable control of the outer loop in the case of unstable inner loop. Finally, the effectiveness of the two control schemes is verified by real-time control experiments.

Improved Power Sharing and Energy Management Platform in Microgrid Considering Stochastic Dynamic Behavior of the Electric Vehicles

In this study, an improved model of optimal power sharing and energy management as well as total harmonic distortion (THD) control in AC microgrids in order to control the voltage and frequency parameters by considering the random behavior of electric vehicles (EVs) is presented. The proposed model is developed based on the primary and secondary controller. The primary controller is planned with the purpose of optimal power-sharing, reduction of feeder's impedance mismatch and also THD reduction. The primary controller is modeled with the hybrid Event-Triggered approach and virtual impedance method. The secondary controller has been modeled using the particle swarm optimization (PSO) algorithm with the aim of reducing the power distribution error and optimizing the virtual impedance parameters, as well as considering the participation of EVs in the stochastic power flow equations. In addition to comparing the performance of the proposed model with conventional methods, it has also been compared with other meta-heuristic methods. The results of this have been verified in two software environments (MATLAB/Simulink) and experimental setup (dSPACE model 1202) in different scenarios and also the stability of the proposed model based on the Nyquist stability criterion, the Root Locus method and small signal analysis has been presented.

Virtual Synchronous Generator (VSG) Control Strategy Based on Improved Damping and Angular Frequency Deviation Feedforward

The output active power of a grid-connected inverter controlled by a traditional virtual synchronous generator (VSG) has the problems of oscillation and steady-state errors. A VSG control strategy based on improved damping and angular frequency deviation feedforward is proposed. This strategy reduces the steady-state error of active power by adding a transient damping link to a traditional VSG damping feedback channel. At the same time, the angular frequency deviation feedforward compensation is used to improve the response speed of the VSG to the active power instruction and reduce the active power overshoot in the dynamic process. First, the VSG active power closed-loop small-signal model is established. The effects of inertia and damping on the dynamic and steady-state performance of the VSG are analyzed by the root locus method. The effect of the proposed control strategy on the system is analyzed by using a closed-loop zero-pole diagram. This strategy improves the precision of active power control the dynamic performance of the system effectively. Finally, the effectiveness and superiority of the proposed control strategy are verified by Matlab/Simulink simulation and semi-physical simulation platform RT-LAB.

Automatic Excitation Control Using PI Method Based on Analysis Root Locus for Synchronous Generator

The demand for electrical energy is increasing (at 2–3 percent per year). It is necessary to utilize the energy available in nature optimally and stably to fulfill electricity needs. Synchronous generators have an essential role in fulfilling electricity needs. Maintaining and improving the operational stability of the synchronous generator is fundamental to the safe and economical operation of an electric power system. The use of control methods to increase control excitation is recognized as an economical and effective way. Previous research shows there are still areas for improvement in the excitation control method. Thus, this research focuses on optimizing the automatic excitation system using the PI control method with stability analysis using the root locus. The effect of PI control on the results of the characteristics response has Tr is 1.9 s, Ts is 6.6 s, Mp is 38.95 percent, and Ess is 11.02 percent. While the generator excitation current will increase in line with the increase in load current, and the resulting output voltage will be stable with errors that occur when compared to the nominal voltage is 10.51 percent because the sensor reading is affected by a frequency that is not following the operating frequency.DOI: 10.17977/um024v8i12023p036

Asymptotic mean square stability of Predictor-Corrector methods for stochastic delay ordinary and partial differential equations

In this paper, we focus on the asymptotic mean square stability of stochastic delay ordinary and partial differential equations. By virtue of root locus technique, the sufficient and necessary conditions of asymptotic mean square stability for Predictor-Corrector methods are obtained, which are suitable for solving above two types of problems. Furthermore, by regulating the value of parameter θ in drift term of the schemes, we could investigate a series of schemes to explore different size of stable regions, this amounts to provide a road to obtain the optimal stable regions. Several theorems prove that the Predictor-Corrector schemes could inherit the asymptotic mean square stability for above two original problems. Numerical experiments verify the stability theorems, precision and the parallelism.

An Unconditionally Stable Integration Method for Structural Nonlinear Dynamic Problems

This paper presents an unconditionally stable integration method, which introduces a linearly implicit algorithm featuring an explicit displacement expression. The technique that is being considered integrates one Newton iteration into the mean acceleration method. The stability of the proposed algorithm in solving equations of motion containing nonlinear restoring force and nonlinear damping force is analyzed using the root locus method. The objective of this investigation was to assess the accuracy and consistency of the proposed approach in contrast to the Chang method and the CR method. This is achieved by analyzing the dynamic response of three distinct structures: a three-layer shear structure model outfitted with viscous dampers, a three-layer shear structure model featuring metal dampers, and an eight-story planar frame structure. Empirical evidence indicates that the algorithm in question exhibits a notable degree of precision and robustness when applied to nonlinear dynamic problem-solving.

Bifurcation Behavior Analysis and Stability Region Discrimination for Series-Parallel Architecture Electric Energy Router

Electric energy routers (EERs) can effectively deal with the energy management issues caused by the access of multi-sources and multi-loads. Different from the energy transmission form in the existing series architecture EER (SA-EER) for low-voltage distribution networks, a new series-parallel architecture EER (SPA-EER) has the advantages of high-power transmission capability and reactive power flexible operation. However, if controller parameters change beyond their critical stability boundaries, SPA-EER will produce slow- and fast-scale bifurcation behaviors, which are manifested as low- and high-frequency oscillations of port voltages or port currents in varying degrees. To avoid the system instability caused by controller parameter changes, how to discriminate the stability regions of controller parameters for SPA-EER is a key research content in this paper. To this end, the stroboscopic mapping discrete model of SPA-EER system is first established, and system bifurcation behaviors are analyzed based on bifurcation theory. Furthermore, the critical stability boundaries of controller parameters are discriminated by using Jacobian matrix, root locus and bifurcation diagram. After that, a time delay feedback control (TDFC) is employed to suppress system bifurcation behaviors. Finally, the correctness of bifurcation analysis and the effectiveness of TDFC are verified by a hardware-in-the-loop (HIL) experimental platform.

LOCUS OF CONTROL THEORY IN TREATING TOURIST BEHAVIOR: THE THEORY ROOTS AND RESEARCH DIRECTION IN DESTINATION BRANDING FIELD

We aim to shed light on this issue by reviewing the roots and development of the locus of control theory. Moreover, we will introduce how we can use this development, in theory, to provide a new research direction in the tourism service field. A theory-based review was conducted to investigate the locus of control theory roots and its potential implications in the tourism industry using the Australian Business Deans Council (ABDC) list to explore the current literature. We followed the PRISMA methodology to collect the data from the Scopus database as well as Google Scholar and ResearchGate. The study found that the locus of control theory has its roots in social psychology and has been developed over the years to explain individual differences in behavior and decision-making. In the tourism service field, we found that understanding the locus of control can help service providers tailor their services to meet the needs and expectations of different types of tourists. This will contribute to attribution literature in psychological aspects and tourism literature with a deep understanding of how tourists behave and interpret differently.

Theoretical Upper and Lower Limits for Normalized Bandwidth of Digital Phase-Locked Loop in GNSS Receivers.

Determining the loop noise bandwidth and the coherent integration time is essential and important for the design of a reliable digital phase-locked loop (DPLL) in global navigation satellite system (GNSS) receivers. In general, designers set such parameters approximately by utilizing the well-known fact that the DPLL is stable if the normalized bandwidth, which is the product of the integration time and the noise bandwidth, is much less than one. However, actual limit points are not fixed at exactly one, and they vary with the loop filter order and implementation method. Furthermore, a lower limit on the normalized bandwidth may exist. This paper presents theoretical upper and lower limits for the normalized bandwidth of DPLL in GNSS receivers. The upper limit was obtained by examining the stability of DPLL with a special emphasis on the digital integration methods. The stability was investigated in terms of z-plane root loci with and without the consideration of the computational delay, which is a delay induced by the calculation of the discriminator and the loop filter. The lower limit was analyzed using the DPLL measurement error composed of the thermal noise, oscillator phase noise, and dynamic stress error. By utilizing the carrier-to-noise density ratio threshold which indicates the crossing point between the measurement error and the corresponding threshold, the lower limit of the normalized bandwidth is obtained.

Linearly Implicit Algorithm with Embedded Newton Iteration of Velocity and its Application in Nonlinear Dynamic Analysis of Structures

With the application of energy dissipation and damping techniques such as viscous dampers and viscoelastic dampers in seismic engineering, dynamic equilibrium equations of overall structures involve nonlinear damping force. Therefore, numerical integration methods with explicit velocity expressions are more favorable. Based on the average acceleration method, a novel linearly implicit algorithm with explicit expression of velocity is proposed in which once Newton iteration is embedded. The stability of the proposed algorithm in solving equations of motion containing nonlinear restoring force and nonlinear damping force is analyzed using the root locus method. The nonlinear stability of the proposed algorithm is also examined using a single-degree-of-freedom shear-type structure. The dynamic response analysis of a three-storey shear structure model equipped with viscous dampers, a three-storey shear structure equipped with metal dampers, and an eight-storey planar frame structure, are carried to investigate the accuracy and stability of the proposed method with respect to the Chang method and the CR method. The results show that the proposed algorithm exhibits higher accuracy and finer stability for nonlinear dynamic problems.

Mathematical Modelling and Fluidic Thrust Vectoring Control of a Delta Wing UAV

Pitch control of an unmanned aerial vehicle (UAV) using fluidic thrust vectoring (FTV) is a relatively novel technique requiring no moving control surfaces, such as elevators. In this paper, the authors’ previous work on the characterization of a static co-flow FTV rig is further validated using the free to pitch dynamic test bench. The deflection of a primary jet after injection of a high-velocity secondary jet was captured using the tuft flow visualization technique, along with the experimental recording of subsequent normal force impinged on the Coanda surface resulting in the pitching moment. The effect of primary and secondary flow velocities on exhaust jet deflection, as well as on the pitch angle of the aircraft, is examined. Aerodynamic gain as well as the inertia of a delta wing UAV test bench are computed through experiments and fed into the equation of motion (e.o.m). The e.o.m developed aided in the design of a model-based PID controller for pitch motion control using the multi-parameter root locus technique. The root locus tuned controller serves as a benchmark controller for performance evaluation of the genetic algorithm (GA) and particle swarm optimization (PSO) tuned controllers. Furthermore, the frequency domain metric of gain and phase margins were also employed to reach a robust control design. Experiments conducted in a simulation environment reveal that PSO-PID results in a better response of the UAV in comparison to the baseline pitch controller.

High-frequency oscillation mechanism analysis of wind farm-side MMC station considering converter transformer stray capacitance

High-frequency oscillation is one of the critical issues threatening the stability of modular multilevel converter (MMC) based high-voltage direct current (HVDC) system. A new 2 kHz high-frequency oscillation has occurred in the wind farm-side MMC station of Rudong wind farm integrated MMC-HVDC project of China during no-load charging process. The mechanism of this high-frequency oscillation is different to previous ones. Apart from the time delay of MMC, the stray capacitance of converter transformer makes a significant contribution to the high-frequency oscillation. This paper investigates the high-frequency oscillation mechanism of wind farm-side MMC station during no-load charging process. First, the harmonic state space (HSS) model of a wind farm-side MMC station is established. In addition to the leakage inductance, the stray capacitance is considered in the converter transformer model. To reveal the significance of the converter transformer stray capacitance in this new high-frequency oscillation, the root loci of the system are calculated with varying stray capacitance. Then, the state variables with high degree of participation in the high-frequency oscillation modes are determined by the participation factor analysis. A reduced-order model suitable for the high-frequency oscillation analysis is further established based on the participation factor analysis results. Next, the high-frequency oscillation mechanism is revealed by impedance analysis. Finally, the effects of system parameters are analyzed by the combination of eigenvalue analysis and impedance analysis. The analysis results are validated by electromagnetic transient simulations.

How Does the First Molar Root Location Affect the Critical Stress Pattern in the Periodontium? A Finite Element Analysis.

The first molar root location plays a pivotal role in neutralization of forces applied to the teeth to prevent injury. This study aimed to assess the effect of maxillary and mandibular first molar root location on biomechanical behavior of the periodontium under vertical and oblique loadings. In this three-dimensional (3D) finite element analysis (FEA), the maxillary and mandibular first molars and their periodontium were modeled. The Young's modulus and the Poisson's ratio for the enamel, dentin, dental pulp, periodontal ligament (PDL), and cortical and cancellous bones were adopted from previous studies. The changes in maximum von Misses stress (MVMS) values of each component were analyzed. The MVMS values were the highest in the enamel followed by dentin, cortical bone, cancellous bone, and PDL. The maxillary and mandibular first molars with different root locations and their periodontium showed different biomechanical behaviors under the applied loads. An interesting finding was that the stress concentration point in the path of load degeneration changed from the cervical third in dentin to the apical third in the cancellous bone, which can greatly help in detection of susceptible areas over time.

A preordainment approach for design of auxiliary damping controller and SSSC tuning to enhance SSR mode stability in DFIG based windfarm

ABSTRACT Sub-synchronous resonance (SSR) is a growing concern in wind turbine generators-based series capacitive compensated networks at low wind speed and peak compensation levels. It is feasible to increase the stability of SSR mode caused by a transmission line’s passive series compensation by providing active compensation using a flexible ac transmission system controller cooperating with an auxiliary damping controller. This paper presents preordainment and enhancement of SSR mode in DFIG-based series capacitive compensated line by incorporating an additional damping controller into the static synchronous series compensator (SSSC). The auxiliary damping controller modulates the reactive voltage injected by SSSC into the transmission line by considering the appropriate input control signal, reflecting the SSR instability, and giving the maximum damping to the SSR mode. The optimal input control signal is determined via residue analysis, and the design procedure of the auxiliary damping controller is presented using root locus plots. The root locus plots offer a visual representation of the loci for each mode and preordain eigenvalues for new stability, which aids in the design of the auxiliary damping controller. This feature allows to predetermine the SSR mode eigenvalue while before connecting it with SSSC. Accordingly, it provides a controlled dampening to SSR mode such that the desired damping ratio for SSR mode can be achieved without influencing the stability of other modes. The impact of the designed auxiliary damping controller is examined at high compensation levels and low wind speed. The eigenvalue analysis results show that the preordained eigenvalues found using root locus plots are correct and the proposed approach predominantly enhances the stability of SSR mode. The fast Fourier transform (FFT) analysis and transient simulations are used to show the detuning and dampening of SSR mode. The proposed work is executed using MATLAB/Simulink.

Performance analysis of modeling scale on multiband oscillations in grid-connected wind farm

Grid-connected permanent magnet synchronous generator (PMSG) wind farms may be susceptible to oscillation when connected to weak grids. This paper explores the root causes of the oscillations from two perspectives: the model and the oscillation frequency band. First, the small signal model of PMSG wind farm grid-side system is established, and the small disturbance comparison with the PMSGA full-order electromagnetic transient model in MATLAB/Simulink is carried out. Then, we calculatue the eigenvalues of the small signal model and use model analysis techniques such as participation factors calculation and root locus method to identify the critical factors that cause the oscillation modes to be dominant. Finally, time-domain simulation is used to verify the theoretical analysis. Two dominant modes are identified: the subsynchronous oscillation mode and the low-frequency oscillation mode. The subsynchronous oscillation mode is closely related to the direct current (DC) voltage control and the dynamics of DC link. The low-frequency oscillation is significantly weakened with the decrease in grid strength, and it is closely related to the phase-locked loop (PLL) control. The conclusions can provide a reference for tuning the control parameters of PMSG converter.

Stability and bifurcations in scalar differential equations with a general distributed delay

For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, relying on fundamental properties of the characteristic function of the random variable whose probability density function is the delay distribution. Based on the root locus method, the bifurcation curves are determined in the considered parameter plane, also providing the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed in the general setting, and the criticality is analyzed by employing the method of multiple times scales. In contrast with some previously reported results from the literature, our analysis is accomplished in a general context and only then exemplified for particular types of delay distributions (e.g. Dirac, Gamma, uniform and triangular). The theoretical results are showcased in the framework of a simple neural model.

Soil amendments impact root‐associated fungal communities of balsam poplar on a phosphogypsum reclamation site

During reclamation organisms like fungi directly interact at the soil–plant interface and can potentially impact plant growth, health, and overall revegetation success. Therefore, a study was conducted to determine how topsoil amendments (mycorrhizae, biochar, compost, manure) influence root‐associated fungal communities and growth of fast growing Populus balsamifera trees on a phosphogypsum (PG) reclamation site in Alberta, Canada. Tree height, diameter, biomass, and diversity of Ascomycota and Basidiomycota were assessed using Illumina MiSeq sequencing targeting the fungal internal transcribed spacer region 1 (ITS1) and ITS2. Topsoil amendments and root location (topsoil vs. PG within a treatment), greatly affected the root fungal community. Basidiomycota preferentially grew in PG, and had a high treatment specific root fungal community with highest diversity of ectomycorrhizal fungi and basidiomycete yeast in biochar PG and compost PG. Populus balsamifera roots were associated with a high diversity of Ascomycota, found equally in roots in topsoil and PG, with a trend towards higher diversity and abundance in PG. The core genus found across all samples was Fusarium, a potential plant pathogen. All topsoil amendments strongly reduced abundance of Fusarium and other Ascomycota in poplar roots. Biochar and compost reduced number and abundance of potential fungal pathogens and increased number of potential plant growth promoting Basidiomycota, indicating favorable growth conditions for P. balsamifera on a phosphygypsum reclamation site by application of these amendments. This study indicates compost and biochar can be effective soil amendments to support P. balsamifera establishment and growth on a PG reclamation site.