Showing papers on "Describing function published in 2019"

High-Bandwidth Secondary Voltage and Frequency Control of VSC-Based AC Microgrid

Abstract: This paper proposes a novel secondary control strategy for the power-electronic-based ac microgrid. This approach restores the voltage and frequency deviations by utilizing only local variables with very high bandwidth. This is realized with a finite control set model predictive control technique that is adopted in the inner level of the primary control of voltage source converters. In the outer level of the primary control, droop control and virtual impedance loops are exploited to adjust the power sharing among different distributed generation (DGs). As inner control level operates with a very high bandwidth, need for filtering of the calculated active and reactive powers in the outer level of the primary control is insignificant. Therefore, the secondary control can be operated with a far superior bandwidth compared to the case when the conventional cascaded linear control is used. Merits of the proposed approach are investigated analytically with the help of the describing function methodology that allows the quasi-linear approximation of the inner control level. Finally, simulation and experimental results are presented.

Review of Small-Signal Modeling Methods Including Frequency-Coupling Dynamics of Power Converters

Abstract: Over the past years, the linearized modeling techniques for power converters have been continuously developed to capture the small-signal dynamics beyond half the switching frequency. This paper reviews and compares the small-signal modeling approaches based on a buck converter with voltage-mode control. The study includes the small-signal averaged modeling approach, the describing function method, and the harmonic state-space modeling approach, in order to be able to better select the correct method when modeling and analyzing a power electronic circuit as well as a power-electronic-based power system. The model comparison points out that the describing-function-based models do improve the modeling accuracy beyond the half-switching frequency of the converter, yet they fail to predict the frequency-coupling interactions (e.g., beat frequency oscillations) among multiple converters, and instead, harmonic state-space models in the multiple-input multiple-output form are required.

“Constant in Gain Lead in Phase” Element– Application in Precision Motion Control

Abstract: This paper presents a novel “Constant in gain Lead in phase” (CgLp) element using nonlinear reset technique. PID is the industrial workhorse even to this day in high-tech precision positioning applications. However, Bode's gain phase relationship and waterbed effect fundamentally limit performance of PID and other linear controllers. This paper presents CgLp as a controlled nonlinear element which can be introduced within the framework of PID allowing for wide applicability and overcoming linear control limitations. Design of CgLp with generalized first-order reset element and generalized second-order reset element (introduced in this paper) is presented using describing function analysis. A more detailed analysis of reset elements in frequency domain compared to existing literature is first carried out for this purpose. Finally, CgLp is integrated with PID and tested on one of the DOFs of a planar precision positioning stage. Performance improvement is shown in terms of tracking, steady-state precision, and bandwidth.

Describing Function Method Based Power Oscillation Analysis of LCL -Filtered Single-Stage PV Generators Connected to Weak Grid

Abstract: For the LCL -Filtered single-stage photovoltaic (PV) generators, the low-frequency power oscillation ( LCL -Filtered single-stage PV generators. First, the complete model of the PV generators is established including the perturbation and observation (P&O) based PV power loop and dc voltage loop, which are rarely considered by the existing stability analysis methods. Especially, the P&O based PV power loop is nonlinear and discontinuous, and the conventional small-signal modeling is inapplicable. Hence, its influence is completely ignored in the existing stability analysis methods. Second, considering the nonlinear discontinuous link, the describing function method is adopted to analyze the whole system stability based on the established complete model. In this way, the accuracy and completeness of the stability analysis are enhanced. Furthermore, for the critically stable state, both the oscillation amplitude and frequency can be calculated accurately. At the same time, different influence factors are analyzed quantitatively including the operation points, grid strength, and the interaction of multiple control loops. It is first revealed that the grid impedance has different influence on the system high-frequency and low-frequency stability. Also, the proposed analysis method and the conventional method are compared, which presents the advantages of the proposed analysis method. Finally, all the theoretical analyses are verified by the real-time hardware-in-loop tests.

Stability Analysis of PV Generators With Consideration of P&O-Based Power Control

Abstract: Photovoltaic (PV) generators have continuously increased in recent years, whose power is usually controlled through the perturbation and observation (P&O) method. In essence, the P&O method is nonlinear and discontinuous. Hence, the conventional small-signal stability analysis is not suitable anymore when the influence of the P&O-based power control is considered. Focusing on this problem, this paper adopts the nonlinear describing function (DF) method to conduct the accurate stability analysis of PV generators with consideration of P&O-based power control. The detailed procedures about the DF method are introduced, and then the related influence factors like perturbation size, filters, and so on are analyzed quantitatively. Furthermore, the comparison with the conventional stability analysis methods is made, which suggests that the DF method can effectively enhance the accuracy of the stability analysis. All the conclusions are verified by the real-time hardware-in-loop tests.

Complex order control for improved loop-shaping in precision positioning

Abstract: This paper presents a complex order filter developed and subsequently integrated into a PID-based controller design. The nonlinear filter is designed with reset elements to have describing function based frequency response similar to that of a linear (practically non-implementable) complex order filter. This allows for a design which has a negative gain slope and a corresponding positive phase slope as desired from a loopshaping controller-design perspective. This approach enables improvement in precision tracking without compromising the bandwidth or stability requirements. The proposed designs are tested on a planar precision positioning stage and performance compared with PID and other state-of-the-art reset based controllers to showcase the advantages of this filter.

A Dual-Edge Pulsewidth Modulator for Fast Dynamic Response DC–DC Converters

Abstract: Voltage regulator modules are dc–dc converters that power modern microprocessors. They must exhibit a fast dynamic response, in order to achieve satisfactory regulation performance in spite of the rapid load current variations. The pulsewidth modulator is one of the elements that determine the converter transient response. This letter introduces a dual-edge modulator that outperforms conventional modulation schemes in terms of both small- and large-signal dynamic performances: first, its small-signal transfer function exhibits a phase boost action in the high-frequency range, enabling the design of robust, wide-bandwidth control loops. Furthermore, it introduces a reduced expected delay in response to a large-signal perturbation. The modulator transfer function is derived in this letter using the describing function approach, and it is validated by comparing its predictions to SIMPLIS simulations and to the measured loop transfer function of a synchronous buck converter, designed in a 130 nm CMOS technology.

Nonlinear Robust Approaches to Study Stability and Postcritical Behavior of an Aeroelastic Plant

Abstract: Two approaches to tackle the nonlinear robust stability problem of an aerospace system are compared. The first employs a combination of the describing function method and $\mu $ analysis, while the second makes use of integral quadratic constraints (IQCs). The model analyzed consists of an open-loop wing’s airfoil subject to freeplay and linear time-invariant parametric uncertainties. The key steps entailed by the application of the two methodologies and their main features are critically discussed. Emphasis is put on the available insight on the nonlinear postcritical behavior known as limit cycle oscillation. It is proposed a strategy to apply IQCs, typically used to find absolute stability certificates, in this scenario, based on a restricted sector bound condition for the nonlinearity. Another contribution of this paper is to understand how the conservatism usually associated with the IQCs multipliers selection can be overcome by using information coming from the first approach. Nonlinear time-domain simulations showcase the prowess of these approaches in estimating qualitative trends and quantitative response’s features.

Damage detection in nonlinear systems using an improved describing function approach with limited instrumentation

Abstract: The describing function approach is a powerful tool for characterizing nonlinear dynamical systems in the frequency domain. In this paper, we extend the describing function approach to detect and localize the damage in initially healthy nonlinear systems with limited measurements. The requirement of complete FRF of the underlying linear system by describing function approach is overcome by using a newly developed nonparametric principal component analysis-based model. Numerical simulation studies have been carried out by considering a cantilever beam with multiple local nonlinear attachments to demonstrate the localization process of the improved describing function approach with limited instrumentation. Parametric estimation of a shear building model is considered as a second numerical example to demonstrate the capability of the proposed approach in identifying the different types of nonlinearities and as well as combined types of nonlinearities (i.e. more than one type of nonlinearity). These combined nonlinearities can exist either in the same or different spatial locations. Experimental investigations have also been presented in this paper to complement the numerical investigations to demonstrate the practical applicability.

Small Signal Analysis of Active Clamp Flyback Converters in Transition Mode and Burst Mode

Abstract: Active clamp flyback (ACF) converters in transition mode improves the full load efficiency of high-frequency AC/DC and DC/DC power supplies with zero-voltage switching (ZVS). To minimize the impact of the circulating energy for ZVS, burst mode control is used to benefit ACF light load efficiency. However, due to the resonance nature of ACF switching current waveforms, it is difficult to characterize the small-signal properties analytically. In this paper, two new small signal models are developed, and analytical design guides are investigated. An average modeling technique for the transition mode ACF is developed which eliminates the complexity of describing the second-order resonance waveforms, and the model enables a higher system bandwidth design. A high-frequency model for burst mode is derived with the describing function method, which identifies the loop compensation impact and differentiates the stability criteria from the conventional ripple-based (V2) control. Enlightened from the models, a novel nonlinear ramp compensation method and a parallel damping technique are proposed to improve the loop stability and optimize the dynamic response. Finally, SIMPLIS simulations verify the model accuracy, and the fast transient response with the compensation methods are demonstrated on a high-density 45W GaN ACF notebook adapter.

Robust Stability and Nonlinear Loop-Shaping Design for Hybrid Integrator-Gain-Based Control Systems

Abstract: In this paper, the use of quasi-linear tools for the closed-loop design and analysis of Hybrid Integrator-Gain Systems (HIGS) is considered. A nonlinear motion control design procedure is proposed in which quasi-linear loop-shaping methods, based on describing functions, are combined with rigorous conditions for closed-loop stability. The latter are established by means of multiple piecewise quadratic Lyapunov functions. Admissible functions are found by solving a set of numerically tractable linear matrix inequalities (LMIs). The potential of the robust design method is illustrated by simulation results of a two-mass-spring-damper system.

Investigation on the small signal characteristic based on the LLC hybrid hysteretic charge control

Abstract: In this paper, an analytical small-signal model applied for hybrid hysteretic charge (HHC) control has been proposed and analyzed with the advantages over direct frequency control (DFC). Based on the approach of extended describing function method and average concept, for the first time, the systematical analytical open loop transfer functions from control to output, input to output, output impedance and the closed transfer functions of the overall loop, audio susceptibility and output impedance are proposed and verified through simulation. Additionally, some important physical insights have been extracted, analyzed and verified. Finally, the experiments on a design example of 12 VDC&12 A output power are conducted and verified. It shows that the calculations match well with the results from both the simulation and experiment, which reveals the proposed analytical transfer functions are very useful for the practical power design to achieve good prediction result.

A Robust Relay Feedback Structure for Processes Under Disturbances: Analysis and Applications

Abstract: In this paper, a robust relay feedback structure is used to provide a stable oscillation under large static disturbances or drift. This relay structure is composed of a block to remove static disturbance or drift followed by a relay. The robust relay structure is analyzed using the describing function method. Furthermore, based on Poincare map, the conditions for the existence and local stability of the limit cycle are obtained. In order to estimate multiple frequency points of the process, a step change and periodic square waves with frequencies lower than the ultimate frequency are applied at the process input. Thus, these frequencies information are used to identification of first-order plus time delay models over a wider frequency region. For these models, a phase adjustment is proposed from the frequencies information of the additional signal. From simulated and experimental studies, it is verified that the robust relay structure provides a symmetrical oscillation at the process output, independent of the disturbance in the process input and iterations between the loops.

Accurate Small-Signal Model for LLC Resonant Converters

Abstract: Resonant converters are the preferred topologies of a wide range of applications. The major benefits, especially for LLC converters, are zero-voltage switching (ZVS), low turn-off current of primary-side switches, zero-current switching of rectifier, and circuit simplicity. Despite the popularity, no simple and accurate small-signal model is available. In most applications, a feedback loop is incorporated into the control system to provide a regulated output. Therefore, small-signal models for those converters are critical for an optimal controller design. Based on describing function method, this paper proposes an accurate small-signal modeling for LLC converters. The study on spectrum shows the non-linear portion in LLC converters consisting of voltage-controlled oscillator (VCO), inverter, resonant tank, and rectifier has to be modeled as a whole. The describing functions of the non-linear block are derived. The model is accurate beyond switching frequency, which establishes a solid base for further development of a simplified equivalent circuit model.

Fuzzy sliding mode control design based on stability margins

Abstract: Fuzzy Sliding Mode Control (FSMC) algorithms have been widely studied and implemented to combine the robustness of the classical sliding mode control with lower chattering levels, but industrial implementation requires a robustness measurement such as the stability margins. Due to the significance of stability margins in control systems, the computation of the phase Margin (PM) and gain margin (GM) in FSMC has already been proposed with the self-sustained oscillations considered as an unstable behavior. However, a procedure to obtain the desired PM and GM from the FSMC design itself has not been developed yet. Hence, in this study, a method to design FSMC algorithms with desired PM and GM is presented. The describing function (DF) and harmonic balance (HB) techniques were used to identify the limit cycles; then, a step-by-step process to tune the FSMC parameters was provided. Examples and simulations are presented to validate the proposed method.

Analysis of primitive genetic interactions for the design of a genetic signal differentiator.

Abstract: We study the dynamic and static input-output behavior of several primitive genetic interactions and their effect on the performance of a genetic signal differentiator. In a simplified design, several requirements for the linearity and time-scales of processes like transcription, translation and competitive promoter binding were introduced. By experimentally probing simple genetic constructs in a cell-free experimental environment and fitting semi-mechanistic models to these data, we show that some of these requirements can be verified, while others are only met with reservations in certain operational regimes. Analyzing the linearized model of the resulting genetic network, we conclude that it approximates a differentiator with relative degree one. Taking also the discovered nonlinearities into account and using a describing function approach, we further determine the particular frequency and amplitude ranges where the genetic differentiator can be expected to behave as such.

Controlling self-excited vibration using acceleration feedback with time-delay

Abstract: This paper investigates the control of vibration of a self-excited system, namely Rayleigh Oscillator, by using Acceleration Feedback Control method. In this control scheme, acceleration of the vibrating system is fed back to a second-order compensator and the control force is produced by amplifying the signal obtained from the compensator. Linear and non-linear stability analyses are performed. Linear stability analysis is used to obtain the stability regions and the optimal system parameters. Non-linear analysis is performed using Describing Function method. Acceleration feedback control is found to be effective in controlling the self-excited vibration. The presence of time-delay is also studied in this paper. It is observed that the presence of uncertain time-delay in the feedback loop can be detrimental. The optimal system (optimized for no delay case) can result in instability of the static equilibrium leading to finite amplitude oscillation. However, one can improve the situation by increasing the loop-gain. In order to circumvent this problem, it is proposed that a pre-determined time-delay may be introduced in the feedback circuit and the control parameters are re-optimized considering this time-delay. As a result, the system equilibrium can be stabilized even in the presence of time-delay. The results of theoretical analysis are validated with the simulation results performed in MATLAB Simulink.

Pattern Analysis in Networks of Diffusively Coupled Lur’e Systems

Abstract: In this paper, a method for pattern analysis in networks of diffusively coupled nonlinear systems of Lur'e form is presented. We consider a class of nonlinear systems which are globally asymptotically stable in isolation. Interconnecting such systems into a network via diffusive coupling can result in persistent oscillatory behavior, which may lead to pattern formation in the coupled systems. Some of these patterns may coexist and can even all be locally stable, i.e. the network dynamics can be multistable. Multistability makes the application of common analysis methods, such as the direct Lyapunov method, highly involved. We develop a numerically efficient method in order to analyze the oscillatory behavior occurring in such networks. We focus on networks of Lur'e systems in which the oscillations appear via a Hopf bifurcation with the (diffusively) coupling strength as a bifurcation parameter and therefore display sinusoidal-like behavior in the neighborhood of the bifurcation point. Using the describing function method, we replace nonlinearities with their linear approximations. Then we analyze the system of linear equations by means of the multivariable harmonic balance method. We show that the multivariable harmonic balance method is able to accurately predict patterns that appear in such a network, even if multiple patterns coexist.

Circuit Topology and Small Signal Modeling of Variable Duty Cycle Controlled Three-Level LLC Converter

Abstract: With a view to regulate the output voltage with fixed frequency, without using any additional component or complex modulation technique, a topology of a variable duty controlled three-level LLC converter is proposed and an equivalent small signal model for Electric Vehicle (EV) charger applications is deduced in this paper. The steady state equations of each operating region are derived in time domain. Based on an Extended Describing Function (EDF) approach, a small signal equivalent circuit is modeled which includes both frequency and duty controlled terms. The equivalent circuit is further simplified to derive a transfer function of duty control to output voltage. The transfer function is verified through simulation software. Analyzing the transfer function, a voltage controller is designed and implemented with a PI compensator. The simulation results of the proposed control schemes are illustrated and discussed. The topology is compared to a conventional frequency control topology and the merits of the proposed topology are presented.

Unmodeled dynamics induced harmonic analysis method for Buck converter sliding mode control system

Abstract: The invention belongs to the technical field of Buck converter sliding mode control systems, and particularly relates to an unmodeled dynamics induced harmonic analysis method for a Buck converter sliding mode control system. The method specifically comprises the following steps: (1) modeling of an unmodeled dynamics Buck converter sliding mode control system for a sensor; (2) Buck converter sliding mode control; and (3) analysis of the amplitude-frequency characteristic of output voltage harmonics of a Buck converter based on a describing function method. A novel output voltage harmonic amplitude-frequency characteristic quantitative analysis method based on the describing function method is provided for the Buck converter sliding mode control system, and a novel harmonic induction mechanism under the co-existence of switching nonlinearity of a sliding mode controller and unmodeled dynamics of the sensor is provided.

Discrete Time Modeling of Wireless Power Transfer System Using LCC Compensation Topology

Abstract: In recent years, wireless power transfer (WPT)technology has developed rapidly. However, conventional modeling methods for the dynamics of WPT suffer from the order increase problem and complicated derivations and expressions. In this paper, a discrete time modeling method is used for wireless power transfer systems which use magnetic resonant coupling. The proposed method aims on the dynamics of the overall WPT system, including the nonlinear inverter and rectifier. The orders of the developed models are lower than that of the generalized state space averaging method and the extended describing functions technique. Control to output voltage small-signal behaviors predicted by the discrete time model are verified by circuit simulation results.

A New Anti-Windup Compensator Based on Quantitative Feedback Theory for an Uncertain Linear System with Input Saturation

Abstract: This paper devotes to the robust stability problem for an uncertain linear time invariant (LTI) feedback system with actuator saturation nonlinearity. Based on a three degree of freedom (DOF) non-interfering control structure, the robust stability is enforced with the describing function (DF) approach for an uncertain LTI system to avoid the limit cycle. A new type of anti-windup (AW) compensator is designed using the quantitative feedback theory (QFT) graphical method, which results in a simple design procedure and low-order AW control system. One of the most significant benefits of the proposed method is free of the non-convexity (intractable) drawback of the linear matrix inequality (LMI)-based approach. The analysis conducted on the benchmark problem clearly reveals that the proposed QFT-based anti-windup design is able to handle both saturation and uncertainty in a very effective manner.

Limit Cycle Analysis for Drive Systems with Backlash Nonlinearity using an Eigenvalue Method

Abstract: This paper proposes an analysis of limit cycles for drive systems with backlash nonlinearity using an eigenvalue method. The analysis of limit cycles by means of eigenvalue analysis has the advantage that several nonlinearities can be treated simultaneously, e.g. simultaneous occurrence of backlash and friction. Here, a speed-controlled drive system is investigated. At first, a nonlinear state space model for the drive is set up in combination with the control algorithm. Afterwards, the resulting limit cycles of the controlled system are examined by means of the harmonic balance, which specifies the describing function of the nonlinearity for the eigenvalue analysis. Finally, the analytically calculated limit cycle is experimentally validated using a prototype dual-mass system, with backlash.