Showing papers on "Describing function published in 2022"

Accurate Loop Gain Modeling of Digitally Controlled Buck Converters

Abstract: For the digitally controlled buck converters, the nonlinearity and time periodicity, caused by the pulsewidth modulator (PWM) and sample and hold, make the accurate frequency-domain analysis intractable. In this article, based on the harmonic transfer function (HTF) approach, a precise small-signal continuous-time modeling for the digitally controlled Buck converter operating in continuous-conduction mode (CCM) under constant-frequency voltage-mode control is presented. The sideband components on the closed-loop control are embedded in the model. Thus, this model is accurate within the full frequency domain region, which breaks the limit of Nyquist frequency. Furthermore, by overcoming the barrier of infinite series introduced by the sideband effects, the analytical loop gain expression is derived, which contributes to accurate stability assessment and reduction of computation burden. In addition, the proposed exact small-signal model has explained the reasons why different information injection points lead to different measured loop gains. Simulations and experimental results are conducted to verify the effectiveness of the proposed method.

Accurate Loop Gain Modeling of Digitally Controlled Buck Converters

Abstract: For the digitally controlled buck converters, the nonlinearity and time periodicity, caused by the pulsewidth modulator (PWM) and sample and hold, make the accurate frequency-domain analysis intractable. In this article, based on the harmonic transfer function (HTF) approach, a precise small-signal continuous-time modeling for the digitally controlled Buck converter operating in continuous-conduction mode (CCM) under constant-frequency voltage-mode control is presented. The sideband components on the closed-loop control are embedded in the model. Thus, this model is accurate within the full frequency domain region, which breaks the limit of Nyquist frequency. Furthermore, by overcoming the barrier of infinite series introduced by the sideband effects, the analytical loop gain expression is derived, which contributes to accurate stability assessment and reduction of computation burden. In addition, the proposed exact small-signal model has explained the reasons why different information injection points lead to different measured loop gains. Simulations and experimental results are conducted to verify the effectiveness of the proposed method.

Impact of Controller Saturation on Instability Behavior of Grid-Connected Inverters

Abstract: For the power-electronics-based power system, a kind of stable operation with constant-amplitude oscillation was often witnessed in the reported instability incidents. In some cases, the root is proved to be negative damping by the theoretical analysis based on the linear model and the impedance-based stability theory. However, the negative damped resonances should diverge. This leaves a void why the analysis result is not in accord with the field measurement. This article aims to provide a potential answer for the disaccord, i.e., the controller saturation which is commonly ignored in the modeling process due to its nonlinearity. The controller saturation is linearized by the describing function and the improved linear model is developed. By applying the generalized Nyquist stability criterion, the impact of the controller saturation on the instability behavior of grid-connected inverters is identified, which reveals the underlying mechanism of the constant-amplitude oscillation. The theoretical expectations are finally validated by the experimental results.

Enhanced Small-Signal Modeling for Charge-Controlled Resonant Converters

Abstract: Charge control for resonant converters introduces an inner feedback loop that improves the system dynamic characteristics. However, small-signal modeling is not as straightforward with charge-controlled resonant converters as it is with current mode hard-switching topologies, due to the nature of resonant behavior. Conventional small-signal models for charge-controlled resonant converters are developed based on the converter input and output energy balance. This simplified approach overlooks the dynamics of the magnetizing inductor current and generates errors in small-signal frequency response. To improve the analytical model and enable high-bandwidth design, this article proposes a new methodology for modeling charge-controlled resonant converters. The energy stored in the resonant tank is analyzed using the theory of extended describing function, which accounts for the effect of the magnetizing current. This methodology applies to a wide range of charge control variants, such as bang-bang charge control and hybrid-hysteretic control. To demonstrate the modeling procedure, this article considers a high-order, five resonant-component half-bridge CLLC resonant converter as a case study. The proposed analytical model is applied to a 1 kW, 400-V/3.3-A power supply prototype for simulation and experimental validation. The proposed model successfully predicts the frequency response of the resonant converter across frequency and load conditions, providing rapid frequency-domain evaluation in the design process.

A Describing Function-Based Stability Analysis Method for Cascaded DC-DC Converters

Abstract: The parameters of DC-DC converters in a cascaded converters system are often set up separately, which may cause instability or even breakdown of the whole system. The traditional Middlebrook impedance stability criterion and Nyquist criterion are based on the small-signal model, where the nonlinear links are ignored or linearized, which, however leads to inaccurate modeling, and the difficulty for analytically obtaining stability range. In this paper, the nonlinear links, i.e. pulse width modulation links, of cascaded DC-DC converters system are first modeled by the describing function (DF); then, a novel stability analysis method for the cascaded DC-DC converters system is proposed by combining the impedance analysis method, Nyquist criterion and DF to accurately determine the stability range of the cascaded converters system. Since the DF method is only used into single converter so far for stability analysis, a source/load converter equivalent method is proposed in this paper to apply DF method into the cascaded converters. To this end, the two-stage cascaded boost converters systems with and without considering the parasitic parameters are taken as examples to conduct the simulation and experiment for verifying the correctness and the accuracy of the proposed DF-based stability analysis method. This work provides an analytical method to determine not only the stable and unstable ranges but also the critical stability range of the cascaded system, as well as provides a reference to quantificationally design the control parameters.

Sustained Oscillation Analysis of VSC Considering High-Order Oscillation Components

Abstract: Oscillations of grid-connected voltage source converter (VSC) usually reaches sustained state because of the limiters in controller. Describing function has been adopted to analyze the amplitude and frequency of the sustained oscillations, but high-order oscillation components caused by the limiter are ignored in previous literatures. This letter proposes an analysis method for sustained oscillations which considers high-order oscillation components. This method could obtain a more precise oscillation frequency and the amplitude of 1st, 2nd, and 3rd order components. The proposed method is verified by case studies and simulations in PSCAD/EMTDC.

Application of Nonlinear Correction Method for Attitude Control and Landing Oscillations Prevention

Abstract: The purpose of this work is to demonstrate the application of the nonlinear correction method to prevent attitude oscillations of flying vehicles. This paper investigates the complications of manual control tasks in aeronautics caused by employing the integral component in the control loop. If the actuators of aircraft controlling surfaces reach their rate limits, the aircraft oscillations can appear for both manually and automatically piloted cases. In this paper, the oscillations alleviation problem is solved using the nonlinear correction method. A general methodology for pilot-induced oscillations prevention is described briefly by the example of an aircraft pitch control system. The obtained pilot model parameters are used for altitude control system examination. The nonlinear systems analysis is carried out by calculating the generalized sensitivity function for the control and disturbance input signal. In the second part, the application of the nonlinear correction method to prevent aircraft and satellite oscillations caused by integrator windup. A comparative analysis of the nonlinear correction and the customary anti-windup feedback is carried out. The results obtained demonstrate that a sequential nonlinear corrective device can be efficiently used to suppress pilot-induced and integrator windup oscillations.

Analysis of Nonlinear Characteristics for Forced Oscillation Affected by Quadratic Nonlinearity

Abstract: Forced oscillation is challenging power system stability. Most previous research was based on linear models, so analysis of nonlinear characteristics of forced oscillation in power system is missing. Thus, this letter aims to theoretically reveal the effect of the quadratic nonlinearity on forced oscillation. A multi-scale technique is employed, by which the amplitude-frequency function of the steady response and the approximated first-order analytical expression of the dynamic response are derived for nonlinear forced oscillation. Interesting phenomena of frequency deviation and jumping caused by nonlinearity are found and explained. Above expressions and phenomena are verified by numerical simulation. Although revealing all characteristics of nonlinear forced oscillation in power system is difficult, the analysis results in this letter could offer insights.

Occasional coupling enhances amplitude death in delay-coupled oscillators.

Abstract: This paper aims to study amplitude death in time delay coupled oscillators using the occasional coupling scheme that implies intermittent interaction among the oscillators. An enhancement of amplitude death regions (i.e., an increment of the width of the amplitude death regions along the control parameter axis) can be possible using the occasional coupling in a pair of delay-coupled oscillators. Our study starts with coupled limit cycle oscillators (Stuart-Landau) and coupled chaotic oscillators (Rössler). We further examine coupled horizontal Rijke tubes, a prototypical model of thermoacoustic systems. Oscillatory states are highly detrimental to thermoacoustic systems such as combustors. Consequently, a state of amplitude death is always preferred. We employ the on-off coupling (i.e., a square wave function), as an occasional coupling scheme, to these coupled oscillators. On monotonically varying the coupling strength (as a control parameter), we observe an enhancement of amplitude death regions using the occasional coupling scheme compared to the continuous coupling scheme. In order to study the contribution of the occasional coupling scheme, we perform a detailed linear stability analysis and analytically explain this enhancement of the amplitude death region for coupled limit cycle oscillators. We also adopt the frequency ratio of the oscillators and the time delay between the oscillators as the control parameters. Intriguingly, we obtain a similar enhancement of the amplitude death regions using the frequency ratio and time delay as the control parameters in the presence of the occasional coupling. Finally, we use a half-wave rectified sinusoidal wave function (motivated by practical reality) to introduce the occasional coupling in time delay coupled oscillators and get similar results.

Frequency-Response of Non-Singular Terminal Sliding Mode Control With Actuators

Abstract: Non-singular terminal sliding mode control (NTSMC) is widely popular due to its various industrial applications. In many practical circuit systems, the control input is actually given to an actuator rather than directly to the plant. Therefore, it is important to study the effect of actuators on NTSMC performance. In this brief, frequency-domain characteristics of NTSMC are analyzed using the describing function method. It is shown that in the presence of actuator dynamics, the system exhibits a self-sustained periodic oscillation known as chattering. The relationship between the fundamental frequency, the amplitude of this periodic motion and the NTSMC parameters is explored. The effectiveness of the theoretical analysis is verified via a practical example based on an industrial emulator system.

An interactive application to analyse the existence of limit cycles using the describing function

Abstract: This work describes the development of an interactive tool to illustrate the analysis of Lure systems through the use of the describing function. This is a classic tool that is usually introduced in the introductory courses of non-linear control systems present in the majority of master's degrees in control engineering.

Identifying A(s) and β(s) in Single-Loop Feedback Circuits Using the Intermediate Transfer Function Approach

Abstract: It is common practice to model the input–output behavior of a single-loop feedback circuit using the two parameters, A and β. Such an approach was first proposed by Black to explain the advantages and disadvantages of negative feedback. Extensive theories of system behavior (e.g., stability, impedance control) have since been developed by mathematicians and/or control engineers centered around these two parameters. Circuit engineers rely on these insights to optimize the dynamic behavior of their circuits. Unfortunately, no method exists for uniquely identifying A or β in terms of the components of the circuit. Rather, indirect methods, such as the injection method of Middlebrook or the break-the-loop approach proposed by Rosenstark, compute the return ratio RR of the feedback loop and inferred the parameters A and β. While one often assumes that the zeros of (1 + RR) are equal to the zeros of (1 + A × β), i.e., the closed-loop poles are equivalent, this is not true in general. It is the objective of this paper to present an exact method to uniquely identify each feedback parameter, A or β, in terms of the circuit components. Further, this paper will identify the circuit conditions for which the product of A × β leads to the correct closed-loop poles.

Analysis of Limit Cycle Oscillations in Dual-rate Haptic Rendering: Effect of Dual-rate Sampling

Abstract: The onset of instability during virtual or remote haptic interaction is most often associated with the existence of limit cycles that are perceptible to a human operator’s touch and even hearing. Time discretization due to sampling, position quantization due to ubiquitous use of optical encoders, and the device Coulomb friction are the main factors that contribute to the occurrence of limit cycles in haptic systems. The frequency and magnitude of such limit cycles are important as they limit the control gains that characterize the dynamics of the rendered environments. In earlier works, the effect of discrete sampling and position quantization on the behavior of limit cycles exhibited in uniform-rate haptics controllers has been studied but a similar analysis for multi-rate haptics controllers has not been reported yet. Towards this end, the present work explores how sampling of the position data at two distinct rates - dual-rate sampling - for implementation of stiff virtual walls affect the limit cycle behavior in a particular class of multi-rate haptics controllers. Specifically, we intend to evaluate the dependence of limit cycle frequency and magnitude on the dual-rate sampling rates. We utilize the non-linear control theory tool of describing function analysis along with numerical simulations to establish the same.

Precise Mathematical Model of PSCM Class-D Amplifiers

Abstract: A mathematical analysis of four-level (three-phase) phase-shifted carrier modulation class-D amplifiers with analog feedback is proposed. The nonlinear difference equations are extracted and solved using the perturbation theorem. In this way, a closed-form function describes the low-frequency part of the output for an arbitrary input signal. The results are generalized for an arbitrary number of levels by analogy, well matched with the previous model of two-, three-, and the proposed four-level ones. The analytical results confirm that, apart from switching frequency, the multilevel class-D amplifiers converge to linear amplifiers as the number of output levels increases. An analytical equation expresses the harmonic distortion of practically interesting five-level amplifiers with bridge-tied load. The model is verified by system-level simulations and circuit-level implementation. In this way, a prototype fully differential five-level circuit is implemented on the printed circuit board to confirm the model.

Bernstein polynomials based compensator design for coupled tank system with saturation nonlinearity

Abstract: This study deals with the design of Bernstein polynomials based compensator in order to attenuate the degrading effect of saturation nonlinearities in process control applications. The proposed compensator is structured in terms of Bernstein polynomials and a frequency dependent cost function is utilized in order to calculate the optimal coefficients for the proposed compensator. In order to illustrate the efficiency of proposed compensator, a well-known benchmark process control problem, the coupled tank system, is taken into consideration in this study. The success of the proposed compensator is illustrated via frequency based harmonic plots and time domain plots of saturated control signal.

Effective Controller Design of Non-Ideal Sheppard-Taylor DC-DC Converter

Abstract: This paper presents a detail dynamic modeling, small-signal analysis, controller design and its stability assessment of non-ideal Sheppard-Taylor DC-DC converters. This class of DC-DC converter provides non-pulsing input and output current via two inductors at the input and output sides with improved current ripple. This paper provides the detail small-signal analysis that can be used as a systematic guideline for designing a robust control system for this class of DC-DC converters, which is a missing piece of the puzzle within the literature. The state space model and the canonical model of the system are also provided. Furthermore, the open loop line-to-output transfer function and control-to-output (open loop and closed loop) for non-ideal Sheppard-Taylor DC-DC converter are derived and used for developing a PID-based compensator to validate the derived models. The provided analysis is supported with simulation studies to validate the theoretical derivations.

Sliding mode control for DC-DC buck and boost converters

Abstract: Analysis of a SM DC-DC buck and boost converters dynamics using the LPRS methodology is proposed. Some design aspects are considered too. The buck converter model can be directly transformed to a RFS, which allows one to directly apply the LPRS method for analysis of the SM converter dynamics and control design. The boost converter model was transformed from a switching model to a RFS too. However, the plant dynamics after this transformation are nonlinear. And there two different approaches to analysis: linearization of the nonlinear dynamics and the use of the formulation of the LPRS for a nonlinear plant, with subsequent derivation of the LPRS function. Both options are explored in this book chapter, and the analysis of the boost converter was carried for both the linear and the nonlinear dynamic models. The functionality of the LPRS method, which features exact determination of the frequency and amplitude of chattering and possibility of analysis of external signal propagation, was fully utilized in the presented development. Chattering (ripple) in inductor current and output voltage, as well as, the effect of source voltage fluctuations, which may come from rectification of AC voltage, were analyzed through the concepts of the LPRS method. It was possible to identify the reason and explain the effects of the source voltage fluctuations, which are a combination of AM and input signal propagation effects. The proposed analysis is supported by simulations and experimental results. The presented methodology makes it a superior tool of analysis of SM DC-DC buck and boost converters.

Detection of Nonlinear Behavior in Voltage Source Converter Control in Wind Farms Based on Higher-Order Spectral Analysis

Abstract: In recent years, the sub-synchronous oscillation (SSO) accidents caused by wind power have received extensive attention. A method is needed to distinguish if nonlinear behavior exists in the recorded equal-amplitude accident waveforms, so that different methods can be adopted to analyze the mechanism of the oscillation. The theory of higher-order statistics (HOS) has become a powerful tool for detection of nonlinear behavior (DNB) in production quality control since 1960s. However, HOS analysis has been applied in mechanical condition monitoring and fault diagnosis, even after being introduced into the power system and wind farms. This paper focuses on the voltage source converter (VSC) control systems in wind farms and tries to detect the nonlinear behavior caused by the bilateral or unilateral saturation hard limits based on HOS analysis. First, the traditional describing function is extended to obtain more frequency domain information, and hereby the harmonic characteristics of bilateral and the unilateral saturation hard limit are studied. Then the bispectrum and trispectrum are introduced as HOS, which are extended into bicoherence and tricoherence spectrums to eliminate the effects from linear parts in the VSC control system. The effectiveness of DNB and classification based on HOS is strictly proved and its detailed calculation and estimation process is illustrated. Finally, the proposed method is demonstrated and further discussed through simulation results.

Aeroelastic Characteristics of Fin-Actuator System Based on High Order Harmonic Balance Method and Pseudo-arclength Continuation

Abstract: Although the time-domain aeroelastic analysis of the fin-actuator system is accurate, it is very time-consuming. Some work in the past used the describing function method to calculate the dynamic stiffness of the actuator, and then obtained the aeroelastic stability of the system in the frequency domain. This greatly shortened the time, but there was a loss in accuracy. In order to improve the accuracy and speed up the calculation to a certain extent, this paper uses high order harmonics to describe the response of the system. The frictional hysteresis loop is difficult to obtain a closed-form solution in the frequency domain. In this paper, a truncated Taylor series expansion is used to smooth the LuGre friction model so that the harmonic balance method can be used. The pseudo-arclength continuation method is used to solve the problem, and the bifurcation diagram of the limit cycle of the system is obtained. The results show that the method used in this paper has achieved a balance between time and accuracy in calculating the aeroelastic stability of the fin-actuator system.

An Easily Used Phenomenological Magnetization Model and Its Empirical Expressions Based on Jiles–Atherton Parameters

Abstract: In this paper, a simple magnetization model convenient for engineering applications is presented based on the expressions of the first-order LTI system model. Considering the trade-off between the nonlinearity of anhysteretic magnetization and the hysteresis width, the proposed model employs two different equations with different magnetic field amplitudes. Furthermore, the proposed model utilizes the first-order LTI system model with a low magnetic field amplitude and a simple nonlinear function, based on the amplitude–frequency function, with a high magnetic field amplitude. Two important characteristic parameters for engineering applications, namely, amplitude and the equivalent phase lag, were exacted and analyzed to validate the computation precision of the proposed model. Then, the model was verified through comparisons to the validated Jiles–Atherton model. For easy use, similar to a physics-based model instead of a fitting method, empirical expressions for the model parameters were given, and applicable ranges of these equations were determined using the parameters of the Jiles–Atherton model. Finally, an example of the magnetization model applied to an on/off type device was computed to further verify the effectiveness of the proposed model with quite a simple expression.

Frequency‐Domain (Transfer Function) Approach

Abstract: This chapter presents a review of the classical feedback control approach that is based on the concepts of Laplace transform and transfer functions (TFs). The TF model can only represent the system to certain level. There are three phenomena that are not included in this way of modeling such as high frequency dynamics, nonlinear dynamics, and time-varying dynamics. These types of uncertainty are unstructured uncertainties. The chapter explains the fundamental concepts pertaining to this approach including the definitions of stability, poles and zeros, and TFs. It reviews the main analysis and design tools such as root-locus method and Bode diagrams. The chapter also explains the stability margins in this domain such as the phase margin, gain margin, and delay margin. It introduces the popular proportional-integrating-derivative controller and explains its properties.

Results on approximate controllability for a second-order semilinear nonlocal control system with monotonic nonlinearity

Abstract: In this paper, we propose the approximate controllability analysis for a second semilinear system under nonlocal conditions using monotone nonlinearity as a tool. Monotonicity is a significant feature in many communications applications that use digital-to-analog converter circuits. While non-monotonicity prevents such applications from working, nonlinearity makes them work. Therefore, it is of great interest to study a problem while assuming the nonlinear function is monotonic. Nonlocal conditions have a finer effect on the solution and are additional specifications for the physical measurements beyond the conventional ones. In order to determine the controllability of the considered system, we formulate the control function. In addition to this, we have broadened the scope of the results for the impulsive system. At last, we discuss two examples.