Root Locus Diagram Research Articles

Closed‐form optimal solution of two‐degree‐of‐freedom system with Inerter based on equal modal damping with potential application in non‐structural elevator for seismic control

AbstractTargeting the great demand for adding non‐structural elevators to old residential buildings, this article proposes an updated configuration of tuned mass damper inerter (U‐TMDI) applied in external non‐structural elevators. An analog 2‐DOF system is established to describe the residential building controlled by an inerter elevator. Then, three closed‐form optimal solutions for designing the U‐TMDI are derived via the fixed‐point method, equal modal damping criterion, and the infinity damping assumption. Subsequently, these optimal solutions are compared and discussed involving the expressions of tuning frequency ratio and the optimal damping parameters, root locus diagram and supplemental damping ratios, transfer function, and robustness to frequency variation, respectively. Finally, a residential building example is adopted to validate the feasibility of the proposed retrofitting strategy and the closed‐form optimal design solutions. It is demonstrated that the optimal tuning frequency ratios derived by the fixed‐point method and equal modal damping criterion are different for U‐TMDI due to the influence of elevator stiffness ratio , while its degradation forms for tuned mass damper (TMD) are identical, recognizing the importance of the elevator stiffness. Moreover, the proposed retrofitting strategy of using an inerter elevator can significantly mitigate the main structure displacement by about 18%∼23% for both far‐field and near‐fault earthquakes.

Two-port feedback analysis on Miller-compensated amplifiers

In this paper, various Miller-compensated amplifiers are analyzed by using the two-port feedback analysis together with the root-locus diagram. The proposed analysis solves problems of Miller theorem/approximation that fail to predict a pole-splitting and that require an impractical assumption that an initial lower frequency pole before connecting a Miller capacitor in a two-stage amplifier should be associated with the input of the amplifier. Since the proposed analysis sheds light on how the closed-loop poles originate from the open-loop poles in the s-plane, it allows the association of the closed-loop poles with the circuit components and thus provides a design insight for frequency compensation. The circuits analyzed are two-stage Miller-compensated amplifiers with and without a current buffer and a three-stage nested Miller-compensated amplifier.

Mapping Relationships Between Airship Original Design and Stability Modes

In this paper, we study the influence of an original airship design on its stability modes. Based on the verified model of the YEZ-2A airship, the Sobol sensitivities are calculated to filter the design parameters. The airship hull length, the slenderness ratio of the airship hull, the exponent of the tail, the fin installation position on the hull, and the fin half-span are found to be of large influences on all stability modes, whereas the stability modes are comparatively insensitive to the exponent of the bow and the fin relative thickness. The empirical formulas for mapping relationships are fitted to the experimental data and are verified by both sensitivities and root locus diagrams. The empirical formulas representing the mapping relationships between the original design parameters and the stability mode parameters of the airship can be used as references to simplify and estimate the stability and performance in the preliminary design of the airship.

A Novel Flight Dynamics Modeling Using Robust Support Vector Regression against Adversarial Attacks

<div>An accurate Unmanned Aerial System (UAS) Flight Dynamics Model (FDM) allows us to design its efficient controller in early development phases and to increase safety while reducing costs. Flight tests are normally conducted for a pre-established number of flight conditions, and then mathematical methods are used to obtain the FDM for the entire flight envelope. For our UAS-S4 Ehecatl, 216 local FDMs corresponding to different flight conditions were utilized to create its Local Linear Scheduled Flight Dynamics Model (LLS-FDM). The initial flight envelope data containing 216 local FDMs was further augmented using interpolation and extrapolation methodologies, thus increasing the number of trimmed local FDMs of up to 3,642. Relying on this augmented dataset, the Support Vector Machine (SVM) methodology was used as a benchmarking regression algorithm due to its excellent performance when training samples could not be separated linearly. The trained Support Vector Regression (SVR) predicted the FDM for the entire flight envelope. Although the SVR-FDM showed excellent performance, it remained vulnerable to adversarial attacks. Hence, we modified it using an adversarial retraining defense algorithm by transforming it into a Robust SVR-FDM. For validation studies, the quality of predicted UAS-S4 FDM was evaluated based on the Root Locus diagram. The closeness of predicted eigenvalues to the original eigenvalues confirmed the high accuracy of the UAS-S4 SVR-FDM. The SVR prediction accuracy was evaluated at 216 flight conditions, for different numbers of neighbors, and a variety of kernel functions were also considered. In addition, the regression performance was analyzed based on the step response of state variables in the closed-loop control architecture. The SVR-FDM provided the shortest rise time and settling time, but it failed when adversarial attacks were imposed on the SVR. The Robust-SVR-FDM step response properties showed that it could provide more accurate results than the LLS-FDM approach while protecting the controller from adversarial attacks.</div>

Robust Model Predictive Current Control of PMSM Based on Nonlinear Extended State Observer

The control performance of traditional model predictive control (MPC) usually deteriorates dramatically with the permanent magnet synchronous motor (PMSM) parameter changes in operation, leading to current harmonics and torque ripple. This article proposes a robust model predictive current control (MPCC) method based on a nonlinear extended state observer (NESO) to resolve this problem. The NESO is used to observe the current disturbance part to avoid the influence of motor parameters on the current model. Because this method is not sensitive to changes in motor parameters, the system is robust and has a low current harmonic. Second, the ESO designed in this article based on the nonlinear function has a lower observation error. In addition, this article uses the fixed coefficient method to simplify the design process. A more accurate parameter design method is proposed based on the bounded analysis of observation error. At the same time, the root locus diagram proves that NESO contributes to the stability of the current loop. Finally, hardware and simulation experiments proved that the robust MPCC based on NESO has lower harmonic content and better dynamic performance.

Boundary Analysis of Control Systems using Schwarz Lemma

In this paper, Schwarz lemma at the boundary is considered for analysis of transfer functions used in control systems. Two theorems are presented with their proofs by performing boundary analysis of the derivative of positive real functions evaluated at the origin. Considering that the transfer function,H(s) , is an analytic function defined on the right half of the s- plane, inequalities for |H(0)| are obtained by assuming that is also analytic at the boundary point s=0 on the imaginary axis with H(0)=0. Finally, the sharpness of these inequalities is proved. As result of the sharpness analysis, different extremal functions corresponding to different transfer functions are obtained. The related block diagrams and root-locus curves are also presented for considered transfer functions. According to the root-locus diagrams, marginally stable transfer functions are obtained as the natural results of the theorems proposed in the study.

Power Decoupling Control of MMC and Small-Signal Stability Analysis of AC/DC Distribution Network With Renewable Energy

AC/DC hybrid distribution network can realize the high penetration utilization of renewable energy and modular multilevel converter (MMC) is the key equipment to connect AC grid and DC grid. In this paper, the direct modulation based on the virtual resistor and the reference value of dc-bus voltage is adopted, and instantaneous-value model of three-phase half-bridge MMC is derived by introducing differential-mode and common-mode component representation. Then, the decoupling control of AC active power and reactive power is designed based on the linear active disturbance rejection control theory (LADRC). The total disturbances including the coupling term and capacitor voltage fluctuation can be estimated by extended state observer (ESO), and then cancelled exactly. Based on the simplified average-value model of MMC, the small signal stability analysis of three-terminal AC/DC distribution network with 26-level MMCs corresponding to the Tangjiawan-Jishan1-Jishan2 demonstration project is carried out by root locus and bode diagram method, which gives the guidance for the choice of the main circuit component and controller parameters, and shows that the virtual resistor greatly reduces the resonant peak of AC/DC distribution network. Simulation and experiment results verify the effectiveness of the power decoupling control strategy of MMC and the AC/DC distribution network can realize complex multi-directional power flow.

Linear Active Disturbance Rejection Control Strategy with Known Disturbance Compensation for Voltage-Controlled Inverter

When the linear active disturbance rejection control (LADRC) is applied for the voltage-controlled inverter, the discrete period and the measurement noise limits the observer bandwidth, which affects the anti-disturbance performance of the system. This results in a poor ability to deal with the output voltage fluctuation under the load switch. In this paper, a novel LADRC strategy based on the known disturbance compensation is proposed for the voltage-controlled inverters. Firstly, the original LADRC scheme is designed. The dynamic performance and robustness of the system are analyzed by a root locus diagram, and the anti-disturbance ability is studied through amplitude-frequency characteristics. Then the partial model information and the load current are treated as the known disturbance and introduced to the linear extended state observer (LESO) to improve observation accuracy. The difference in anti-disturbance performance with the original scheme is compared and the stability of the LESO and LADRC is analyzed. Finally, the effectiveness of the proposed scheme is verified by the simulation and experimental results.

Bifurcation analysis of cascaded H-bridge converter controlled by proportional resonant

This paper is concerned with the nonlinear dynamics of Cascaded H-Bridge Converter under PR (Proportional Resonant) control. First, a stroboscopic mapping model of PWM (Pulse Width Modulation) converter is established and the corresponding dynamics, including bifurcations and chaos is investigated analytically. Taking the switching frequency, controller parameters as the bifurcation parameter, some different dynamic behaviors are identified over a wide range of some specified parameter. Moreover, the bifurcation condition is further verified based on the root locus diagram and Lyapunov exponential spectrum. Second, aiming at deriving the stability regions of PWM converter, the variable substitution approach is utilized to transform the non-autonomous converter model into an autonomous one. Then, the PR (Proportional Resonant) controller is applied to control the grid-side current, which can suppress the chaos and bifurcation in the current loop by increasing the proportional gain. Finally, the experiment results are presented to illustrate the validity of the control scheme.

Stability system design of flying body elevator control

This paper, suggest the solution of a dynamic non-homogenous longitudinal pitch-mode flying body equation to analysis motion stability by determining the transfer function of a pitch mode control system from feedback control system diagram. It used a flying actuator design system to estimate optimum system stability and control using the time response method and root locus diagram for different gain values (K) and Gyro sensitivity values (GR) to obtain the best stability, peak value and time response. The results shows that the best stability and control behavior are achieved when K=1.4 and GR=1.2.

Transceiver pseudolite carrier frequency self-alignment closed-loop system

Due to instability of a normal pseudolite, it is vital to ensure that the signal transmitted by a pseudolite is stable. In this paper, a carrier frequency self-alignment closed-loop system is designed for enhancing the stability of a transceiver pseudolite. Carrier frequency feedback is used to compensate for interference from external noise, so that it can make the signal more stable. This system is able to be stable via the related root locus and bode diagram analysis. PID compensation is used to enhance transient-state and static-state performances of carrier frequency self-alignment closed-loop system. According to MATLAB Simulink simulation, carrier frequency self-alignment is achievable and it can suppress external noise interference, which makes the signal more stable.

Root-Locus Analysis of Delayed First and Second Order Systems

For finite dimensional linear system the root-locus method is well established however for the case of delayed systems the method has some problems due to the transcendental term involved. This work intends to illustrate the problems that arises when a root-locus diagram is performed as well as to develop a Matlab function that provides the root-locus diagram for delayed low order systems. In this way, some comments about the problems that should be tackled to obtain a generalization of the computational method for delayed systems with real m poles and n zeros

A Nonlinear Disturbance Observer Based Virtual Negative Inductor Stabilizing Strategy for DC Microgrid with Constant Power Loads

For the dc microgrid system with constant power loads (CPLs), the dc bus voltage can easily cause high-frequency oscillation owing to the complicated impedance interactions. The large line inductance and the CPL-side capacitance will form an undamped LC circuit on the dc bus, which, together with the CPL, will make the system fall into the negative-damping region, thus causing the system instability. To address this problem, a virtual negative inductor (VNI) is built on the source side converter in this paper, which can effectively counteract the large line inductance, thus alleviating the instability problem. Moreover, a nonlinear disturbance observer (NDO) is proposed for estimating the converter output current, which relieves the strong dependence of the proposed VNI strategy on the output current measurement. And the proposed strategy is implemented in a totally decentralized manner, thus alleviating the single-point-failure problem in the central controller. For assuring the optimal parameter value for the proposed stabilizing strategy, a system root-locus diagram based parameter designing approach is adopted. And comparative Nyquist diagram based stability analyses are taken for studying the robustness of the proposed strategy to the system perturbations. Finally, detailed real-time simulations are conducted for validating the effectiveness of the proposed stabilizing strategy.

Control technique for the operation of grid-tied converters with high penetration of renewable energy resources

This paper deals with a control technique based on inherent characteristics of synchronous generators (SG) for control of interfaced converters with high penetration of renewable energy resources (RERs) into the power grid, as a new contribution to earlier studies. To present an appropriate assessment of the proposed control technique, under dynamic operating condition, a P–Q curve is extracted and analysed based on the different components and characteristics of the interfaced converter as well as the conventional relationship between the active and reactive power. By combining the swing equation of SG and the power-based dynamic model, a Pm–Q curve is achieved and the effects of the variations of embedded virtual inertia on virtual mechanical power are assessed. Moreover, by using small-signal linearization, the grid frequency stability is investigated based on both virtual inertia and mechanical power variations. In order to assess the power sharing ability of the proposed control technique, two transfer functions are obtained and then, the impacts of variations of virtual mechanical power on the active and reactive power of interfaced converter are evaluated through Nyquist and Root Locus diagrams. Simulation results confirm that the proposed control technique can guarantee the operation of interfaced converters, based on inherent characteristics of SG, to deal with the power grid stability with high penetration of RERs.

PID Controller Tuning Based on Phase Margin (PM) for Wireless Temperature Control

Performance of proportional-integral-derivative (PID) controller which tuning based on phase margin was investigated. A process simulator modified for wireless process control purpose and antennas with modules connected for wireless communication applications. The impact of robust error control code on wireless temperature control experiments is investigated by means of three tuning parameters of the PID controller which are initially determined by using non-parametric methods. The well-tuned control parameters were assessed by means of a MATLAB/SISO tool (Control System Designer tool) for tuning singe input–single output controllers based on root locus diagram. To determine the bias values of the system, an initial steady state was obtained and the system output was monitored at the constant heater capacity (10%) for 300 s. At the end of 5 min, the control key in MATLAB/Simulink block diagram was changed and the PID control algorithm was activated for different set point changes were given to the system at the same time and the effect of the parameters was observed. It was seen that the wireless temperature control successfully carried out desired set values which selected different set points by using the robust error control with well-tuned parameters. Integral of Squared Error (ISE) and Integral of Absolute Error (IAE) values of different set points were calculated for this PID controller.

Analytical phase‐lead controller design using pole placement: a pole‐zero ratio minimisation approach

Owing to their simple structure and considerable improvement to the transient response, phase-lead controllers are broadly used in industrial applications. They are designed using root locus or Bode diagrams to introduce extra phase shift at the crossover frequency. Traditional phase-lead design techniques are either graphical or based on trial-and-error. In design techniques using root locus diagrams, there is no direct control on the values of a given time-domain performance index. To make this criterion satisfactory, the designer has to try different locations of the controller's zero and pole. Based on minimisation of the controller's pole-zero ratio subject to pole placement requirements, a simple analytical design technique is developed in this Letter. The necessary and sufficient conditions to determine optimal controller parameters are provided. To evaluate the performance of the proposed method, it is applied to a single integrating plant with time delay.

Compliance control based on PSO algorithm to improve the feeling during physical human-robot interaction.

Robots play more important roles in daily life and bring us a lot of convenience. But when people work with robots, there remain some significant differences in human–human interactions and human–robot interaction. It is our goal to make robots look even more human-like. We design a controller which can sense the force acting on any point of a robot and ensure the robot can move according to the force. First, a spring–mass–dashpot system was used to describe the physical model, and the second-order system is the kernel of the controller. Then, we can establish the state space equations of the system. In addition, the particle swarm optimization algorithm had been used to obtain the system parameters. In order to test the stability of system, the root-locus diagram had been shown in the paper. Ultimately, some experiments had been carried out on the robotic spinal surgery system, which is developed by our team, and the result shows that the new controller performs better during human–robot interaction.

Modeling and Control of Gate-Controlled Series Capacitor Interfaced With a DFIG-Based Wind Farm

This paper presents application and control of the gate-controlled series capacitor (GCSC) for series compensation and subsynchronous resonance (SSR) damping in doubly-fed induction generator (DFIG)-based wind farms. The GCSC is a new series FACTS device composed of a fixed capacitor in parallel with a pair of antiparallel gate-commuted switches. The study considers a DFIG-based wind farm, which is connected to a series-compensated transmission line whose parameters are derived from the IEEE first benchmark model for computer simulation of the SSR. The small-signal stability analysis of the system is presented, and the eigenvalues of the system are obtained. Using both modal analysis and time-domain simulation, it is shown that the system is potentially unstable due to the SSR mode. Therefore, the wind farm is equipped with a GCSC to solve the instability of the wind farm resulting from the SSR mode, and an SSR damping controller (SSRDC) is designed for this device using residue-based analysis and root locus diagrams. Using residue-based analysis, the optimal input control signal to the SSRDC is identified, which can damp the SSR mode without destabilizing other modes, and using root-locus analysis, the required gain for the SSRDC is determined. MATLAB/Simulink is used as a tool for modeling, design, and time-domain simulations.

Analysis of sub‐synchronous resonance in doubly‐fed induction generator‐based wind farms interfaced with gate – controlled series capacitor

This paper first presents a step-by-step tutorial on modal analysis of a doubly-fed induction generator (DFIG)-based series compensated wind farm. The model of the system includes a wind turbine aerodynamics, a sixth-order order induction generator, a second-order two-mass shaft system, a fourth-order order series compensated transmission line, an eighth-order rotor-side converter (RSC) and grid-side converter (GSC) controllers, and a first-order DC-link model. Then, using modal analysis and time-domain simulations, it is shown that a fixed-series compensated DFIG is highly unstable due to the sub-synchronous resonance (SSR) mode. In order to damp the SSR mode, the wind farm is interfaced with the gate-controlled series capacitor (GCSC) - which is a new series flexible AC transmission system (FACTS) device. A SSR damping controller (SSRDC) is designed for the GCSC using residue-based analysis and root locus diagrams, and an effective input control signal (ICS) to the SSRDC is identified in order to simultaneously increase damping of both the SSR and super-synchronous (SupSR) modes. The IEEE first benchmark model on SSR is adapted with an integrated DFIG-based wind farm to perform studies. Matlab/Simulink is used as a tool for designing process, and PSCAD/EMTDC is used for time-domain simulations.

Tuning Method Aimed at Optimized Settling Time and Overshoot for Synchronous Proportional-Integral Current Control in Electric Machines

Implementation of proportional-integral controllers in synchronous reference frame is a well-established current control solution for electric machines. Nevertheless, their gain selection is still regarded to be poorly reported, particularly in relation to the influence of the computation and modulation delay. To fill this gap, a design procedure to set the maximum gains for an acceptable damped response, with the delay being considered, has been recently proposed. In contrast, this paper presents a simple rule of thumb to achieve nearly the minimum settling time in combination with negligible overshoot for reference changes. This conclusion is theoretically demonstrated by the analysis of root locus diagrams and of overshoot versus settling time trajectories for sweeps of gain values. The design approaches aimed at gain maximization and the one developed here are compared, revealing that the latter provides shorter settling time and much lower overshoot in the command tracking response, while allowing greater stability margins. On the other hand, the proposed tuning method leads to a worse disturbance rejection, but by including an active resistance with enhanced pole/zero cancellation as a second degree of freedom, both design procedures attain comparable and optimized attenuation of disturbances. Matching simulation and experimental results validate the theoretical study.