Describing Function Research Articles

Stability Analysis Of Two-Position Control System In Interacting Tanks Using Describing Function Method

In a water storage unit consisting of 3 interacting tanks, in its operation there will be problems if manual control is used, because if manual control is not optimal, then when filling water in the three tanks it will cause water overflow, which results in a lot of water being wasted. Two-position control, or "on-off," is a control system that is suitable for use in regulating the water level in the storage unit tank. In a two-position control system, if the frequency of water up and down (Limit Cycle) is large, it will cause an increase in the number of "switching" and "on-off" operations of the switch and solenoid valve per minute, so that the system becomes unstable and will shorten the life of the component. The Describing Function (DF) method is an analysis method that can determine the Limit Cycle frequency based on the width of the water up and down range (hysteresis) of the two-position control system, with the aim of determining the hysteresis parameters that cause the system to be stable.

Describing Function and Event Based Analysis of a General Class of Piecewise Feedback Control Systems

The analysis of nonlinear systems can be greatly improved using computer software. In this work the Sridhar nonlinearity is described, and a new parameterization is given that allows to systematically obtain a large number of the classical piecewise nonlinearities as particular cases. The calculation of limit cycles using the describing function (DF) approach for the case of autonomous nonlinear control systems and an event-based simulation method are presented and compared. Finally, a user-friendly, graphically oriented interactive tool based on Sysquake is presented. In this way the students can easily assimilate some concepts of introductory courses in nonlinear control such as the behavior of piecewise linear systems, stable and unstable limit cycles, event-based simulation and the DF method.

Aeroelastic investigation on an all-movable horizontal tail with free-play nonlinearity

Free-play-induced nonlinear dynamic behavior has been one of the most important topics of aeroelastic research in recent decades. In this paper, the describing function (DF) method is developed to investigate the complex dynamic response of a popular all-movable horizontal tail with free-play. Piecewise expressions for the time history and phase portrait of limit cycle oscillation (LCO) are derived by the developed DF method, which is conducive to understand the mechanism of free-play-induced LCO. Another advantage of the developed DF method is the ability to predict the high-order harmonics, which cannot be realized by the classic DF method. A three-dimensional (3D) all-movable horizontal tail model with torsional free-play was designed and manufactured to implement wind tunnel tests via various initial parameters. A good agreement was found between the numerical and experimental results, which can demonstrate the effectiveness of the proposed method. The influence of the initial parameters of the all-movable horizontal tail on the LCO characteristics is analyzed by both numerical calculations and wind tunnel tests. The method and results in this paper can provide a significant reference for the design of all-movable horizontal tail versus the free-play-induced LCO.

Effects of Nonlinear MPPT Control and PV Array on Stability Analysis for a Utility-Scale PV Energy System

Various studies have focused on the stability analysis of PV energy systems. However, the majority of these methods rely on a linearized model around an equilibrium point, which cannot exactly reveal the influence of the nonlinear power controller and PV array on the PV energy system at low frequencies. To conquer this gap, this article contributes to providing intuitive insight into stability analysis and establishing a complete system model including the PV panel, boost converter, DC line, central inverter and AC grid. The describing function (DF) of the nonlinear discontinuous element in PV and the transfer function of the linear elements in the system are derived to determine the system stability. To support the stability hypothesis based on the DF-based stability analysis method, a comparative study in contrast to existing stability analysis methods is provided. The impedance-based method may not be accurate when the division of the ‘source-load’ subsystem is different. The proposed DF method can offer further concluding remarks on MPPT control, the number of PV panels, DC/DC converter, etc. Based on a hardware-in-the-loop (HIL) operation, the proposed model is explicitly validated under different constraints, and the influence of control and circuit parameters on system stability are illustrated.

A new look at the Equivalent Linearization Technique to predict LCO in aeroelastic systems with discrete nonlinearities

Control surfaces nonlinearities can lead to limit cycle oscillations (LCO). Several methods have been proposed to predict LCO, such as Harmonic Balance-based methods (HB). Describing function (DF) is the HB with a single harmonic motion assumed, and the approach can be combined with a classic eigenvalue stability analysis via the Equivalent Linearization Technique (ELT) to predict LCO. On the other hand, High-Order Harmonic Balance (HOHB) methods consider higher number of harmonics, but they lead to a more complex nonlinear algebraic system of equations. This paper introduces a new look at the ELT combining both eigenvalue analysis and describing functions. Two applications are considered. The first one is a new DF written in a matrix form for the ELT to consider both first and third harmonics in LCOs due to freeplay. The second application is an iterative procedure to combine the eigenvalue analysis with a describing function to predict LCOs in systems with both freeplay and friction nonlinearities. Numerical simulations are performed for the aeroelastic typical section airfoil. The results show that the new DF improves the classic DF by providing a more accurate prediction of LCOs that accounts for third harmonic. Also, the iterative-ELT is shown as an excellent predictor of LCO for systems with both freeplay and friction.

Frequency-Domain Properties of the Hybrid Integrator-Gain System and Its Application as a Nonlinear Lag Filter

This brief investigates frequency-domain properties of the hybrid integrator-gain system (HIGS) by analytically deriving its higher-order sinusoidal input describing functions (HOSIDFs). Furthermore, a novel application of this nonlinear control element is proposed, where it is utilized as a nonlinear lag filter. Hereby, the closed-loop stability can be guaranteed based on a non-parametric plant model—while ensuring robustness against plant uncertainties—by utilizing a circle criterion-like argument. Control design is initially based on a describing function (DF) analysis of the HIGS. This yields magnitude characteristics similar to those of a linear lag filter, while having the freedom to tune the induced phase lag. However, the excitation of higher-order harmonics by the HIGS is, in that case, neglected. Based on the derived HOSIDFs, it is shown that the validity of this assumption depends on the properties of the plant and the controller, as well as the spectral properties of the external inputs. Finally, the potential of the nonlinear lag filter is investigated by means of simulations and experiments on a high-precision industrial motion stage, showing performance improvements compared with a linear lag filter.

Band-Passing Nonlinearity in Reset Elements

This article addresses nonlinearity in reset elements and its effects. Reset elements are known for having less phase lag based on describing function (DF) analysis compared to their linear counterparts; however, they are nonlinear elements and produce higher-order harmonics. This article investigates the steady-state higher-order harmonics for reset elements with one resetting state and proposes an architecture and a method of design that allows for band-passing the nonlinearity and its effects, namely, higher-order harmonics and phase advantage. The nonlinearity of reset elements is not entirely useful for all frequencies, for example, they are useful for reducing phase lag at crossover frequency regions; however, higher-order harmonics can compromise tracking and disturbance rejection performance at lower frequencies. Using the proposed “phase shaping” method, one can selectively suppress the nonlinearity of a single-state reset element in a desired range of frequencies and allow the nonlinearity to provide its phase benefit in a different desired range of frequencies. This can be especially useful for the reset elements in the framework of the “constant in gain, lead in phase” (CgLp) filter, which is a newly introduced nonlinear filter, bound to circumvent the well-known linear control limitation—the waterbed effect.

Describing Function Approach with PID Controller to Reduce Nonlinear Action

The nonlinear effect in the control system is so important and it may have a hard or soft effect on the electrical, mechanical, biological, and many other systems. This paper analyzes the describing function (DF) which is the transfer function of the nonlinear (NL) control systems of many NL elements found such as saturation, and backlash. The effect of the NL on the third-order delayed system is considered. The PID controller is considered the heart of the control system and continuously finds the error between input and output, and formulates the desired signal for the actuator to control the plant. Experimental tanning of PID controller with the saturation NL as a case study with buffer Operation Amplifier (Op-Amp) to maintain the gain and phase shift. In addition, a low pass filter (LPF) is used in the feedback to minimize and attenuate the effect of the NL in the closed-loop control system. The Fourier series is used to analyze the DF. The results show the effectiveness of the proposed algorithm.

On Stability of Time-Invariant Current Controlled Weak-Grid-Tied Inverters Considering Sinusoidal Pulsewidth Modulation Saturation and Parameter Uncertainties

In this article, the stability problem is investigated for a class of time-invariant current controlled <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LCL</i> -type weak-grid-tied inverter (WGTI) system. Considering that the existing stability analysis methods rarely effectively handle the adverse effects of both sinusoidal pulsewidth modulation (SPWM) saturation and plant uncertainty, and the corresponding theoretical results are poor in accuracy and completeness, this article proposes an intuitive stability analysis method in Laplace-domain to ensure the system robustness against SPWM saturation and plant uncertainty. Specifically, the describing function (DF) is employed to characterize the averaged SPWM saturation nonlinear model in Laplace domain. By designing equivalent functions without the poles inside the right-half <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> -plane, the Nyquist criterion required by the DF is equivalently converted into the Hurwitz criterion of closed-loop characteristic polynomial with interval coefficients containing uncertain circuit parameters. Subsequently, by means of the Kharitonov’s theorem, the interval characteristic polynomial is transformed into four Kharitonov polynomials with fixed parameters that are easy to analyze. The state-and-disturbance-observer (SDO)-based current controlled WGTI system is adopted as a case of application study. In order to reduce the conservatism of the Kharitonov’s theorem applied here, a “pointwise” stability analysis on the grid impedance is implemented. Finally, the controller parameters are determined by stability region analysis with different grid impedance points, and the effectiveness of the proposed method is further verified by a series of steady-state and transient experiments.

Detection of subcritical Hopf and fold bifurcations in an aeroelastic system via the Describing Function method

The paper deals with Limit Cycle Oscillations (LCOs) in a Pitch & Plunge model of the wing dynamics, where the stiffness on the plunge displacement is assumed to be an odd fifth-order polynomial. First, it is shown that the model dynamics can be equivalently described via a Lur’e system, i.e., the feedback interconnection between a linear time-invariant subsystem, whose transfer function depends on the wind speed, and a nonlinear memoryless one. Then, the Describing Function (DF) method is used to establish the existence of Predicted Limit Cycles (PLCs) whose amplitude can be analytically expressed in terms of the wind speed, once the intersections of the Nyquist diagram of the linear subsystem with the real axis are obtained. The related bifurcation diagram permits to predict that a stable equilibrium point and two LCOs (one unstable and one stable) coexist for wind speeds smaller than the flutter velocity, at which a subcritical Hopf bifurcation occurs, and larger than the value at which a cyclic fold bifurcation is detected. The closeness of each PLC to the related LCO is measured via the so-called distortion index, which also admits a direct expression in terms of the wind speed. The comparison of the predicted bifurcation diagram with that obtained via numerical simulations confirms the quite good accuracy of the PLCs obtained via the DF method, especially when the distortion index is small.

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

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&#x002F;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.

Analysis of synchronization of multiple identical metronomes on a cart via describing function approach

AbstractBased on the describing function (DF) approach, we study how to obtain analytically the synchronization of n identical metronomes on a cart in this paper. To start with, by using the fact that the torque of each metronome is a discontinuous function of its angle and angular velocity, we obtain the formulae of its DF. Next, we present a unified DF solution of the two types of synchronization for which the amplitude of the first harmonic of the angle of each metronome is identical. One type is called the out‐of‐phase synchronization for which all metronomes do not have an identical phase, and the other type is called the in‐phase synchronization for which all metronomes have an identical phase. We reveal some properties of these two types of synchronization. We validate the filter hypothesis and show that it is indeed critical for guaranteeing the success of using the DF approach to analyze the synchronization modes of the metronomes on a cart. Finally, we report in detail our numerical simulations for two and three identical metronomes using the physical parameters of the metronomes in the literature to validate the obtained analytical results and show certain robustness regarding small parameter differences of the synchronization prediction.

A novel method for estimating three-domain limit cycles in a 3D wing-aileron model with freeplay in aileron deflection

Freeplay-induced limit-cycle oscillations (LCOs) have been a subject of concern for decades This paper proposes a novel method named the dual quasi-harmonic balance (DQHB) method for obtaining quick estimations on LCO behaviors. Two quasi-harmonic motions based on eigenvalues of the underlying and overlying linear systems are constructed so that a three-domain LCO can be interpreted as a dynamic balance between the two motions. Practicable formulations for LCO amplitudes and frequencies and hysteresis are proposed and applied to a 3D wing-aileron model with control-surface freeplay. Wind tunnel tests of flutter and LCOs were carried out with various degrees of freeplay. Numerical results show that the DQHB method produces the same air speeds of LCO onset (31.37 m/s) and critical instability (47.44 m/s) as the describing function (DF) method and time integrations do, which are also verified by test data. The DQHB method also predicts jump and hysteresis phenomena, which are confirmed by test data but not obtained by the DF method or time integrations. Notably the DQHB method can be implemented as fast and convenient as the DF method while provides more physical insights into the mechanism of freeplay-induced three-domain LCOs.

Memristor Circuits for Simulating Neuron Spiking and Burst Phenomena.

Since the introduction of memristors, it has been widely recognized that they can be successfully employed as synapses in neuromorphic circuits. This paper focuses on showing that memristor circuits can be also used for mimicking some features of the dynamics exhibited by neurons in response to an external stimulus. The proposed approach relies on exploiting multistability of memristor circuits, i.e., the coexistence of infinitely many attractors, and employing a suitable pulse-programmed input for switching among the different attractors. Specifically, it is first shown that a circuit composed of a resistor, an inductor, a capacitor and an ideal charge-controlled memristor displays infinitely many stable equilibrium points and limit cycles, each one pertaining to a planar invariant manifold. Moreover, each limit cycle is approximated via a first-order periodic approximation analytically obtained via the Describing Function (DF) method, a well-known technique in the Harmonic Balance (HB) context. Then, it is shown that the memristor charge is capable to mimic some simplified models of the neuron response when an external independent pulse-programmed current source is introduced in the circuit. The memristor charge behavior is generated via the concatenation of convergent and oscillatory behaviors which are obtained by switching between equilibrium points and limit cycles via a properly designed pulse timing of the current source. The design procedure takes also into account some relationships between the pulse features and the circuit parameters which are derived exploiting the analytic approximation of the limit cycles obtained via the DF method.

Tuning procedure for event-based PI controllers under regular quantization with hysteresis

In this paper a tuning procedure is proposed for event based PI controllers with Regular Quantization with Hysteresis (RQH) sampling law. The RQH is a generalization of Symmetric Send on Delta (SSOD) strategy which decreases the robustness requirements to avoid limit cycle oscillations and reduce the number of events needed for control, improving the overall performance of PI controllers in networked control systems. The tuning procedure takes into account not only classical robustness margins but also takes advantage of some specific robustness measures to avoid limit cycle oscillations induced by the sampler. As the robustness analysis depends on the Describing Function (DF) method, a study evaluating the effect of high order harmonics is provided, showing the validity of the tuning procedure. Some examples are included in which the usefulness of the tuning procedure is shown.

Loop-shaping for reset control systems: A higher-order sinusoidal-input describing functions approach

The ever-growing demands on speed and precision from the precision motion industry have pushed control requirements to reach the limitations of linear control theory. Nonlinear controllers like reset provide a viable alternative since they can be easily integrated into the existing linear controller structure and designed using industry-preferred loop-shaping techniques. However, currently, loop-shaping is achieved using the describing function (DF) and performance analysed using linear control sensitivity functions not applicable for reset control systems, resulting in a significant deviation between expected and practical results. This major bottleneck to the wider adaptation of reset control is overcome in this paper with two important contributions. First, an extension of frequency-domain tools for reset controllers in the form of higher-order sinusoidal-input describing functions (HOSIDFs) is presented, providing greater insight into their behaviour. Second, a novel method that uses the DF and HOSIDFs of the open-loop reset control system for the estimation of the closed-loop sensitivity functions is proposed, establishing for the first time — the relation between open-loop and closed-loop behaviour of reset control systems in the frequency domain. The accuracy of the proposed solution is verified in both simulation and practice on a precision positioning stage and these results are further analysed to obtain insights into the tuning considerations for reset controllers.

Inclusion of Current Limiter Nonlinearity in the Characteristic Analysis of Sustained Subsynchronous Oscillations in Grid-Connected PMSGs

The interactions between permanent magnet synchronous generators (PMSGs) and weak grids may cause negatively damped or growing subsynchronous oscillation (SSO). The growing-SSO would evolve into a sustained-SSO when it is predominantly governed by the nonlinearities in the PMSG's converter control, such as current limiter. Most of the existing frequency-domain modeling and stability analysis methods are in the small-signal sense and thus unable to consider the impact of such nonlinearities. Besides, these methods can hardly give vital information to characterize the sustained-SSO, for instance, the magnitude. This paper presents a detailed characteristic analysis of the sustained-SSO while accounting for the effect of current limiter nonlinearity. First, a single-input-single-output (SISO) frequency-domain model of the PMSG is developed. This SISO model comprises the transfer functions of outer-/inner-loop controls and phase-locked loop (PLL). The nonlinear part, i.e., current limiter, is incorporated into the model using the describing function (DF) method. Then, the equivalent transformation on the system's closed-loop characteristic equation is used to separate the SISO model into independent linear and nonlinear parts. This separation enables the use of the DF-based Nyquist criterion for evaluating the system stability and determining oscillation attributes, such as the magnitude and frequency. Both the analytic results and time-domain simulations reveal that the current limiter has a significant impact on the characteristics of the sustained-SSO.

Stability analysis of dynamic interval type‐2 fuzzy control systems in frequency domain

AbstractThe present paper aims to analyze the stability of dynamic Interval Type‐2 fuzzy control systems (IT2 FCSs) with adjustable parameters by investigating the existence of limit cycles based on the describing function (DF) method. To simplify the stability analysis, the following structured procedure is described. First, to avoid complex computation, a simple architecture of the IT2 FCS using two embedded Type‐1 fuzzy control systems (T1 FCSs) is proposed. Then, the DF of IT2 FCS is obtained based on the DFs of embedded T1FCSs. Subsequently, integrating the stability equation and parameter plane approaches provides a solution to identify the limit cycle and the asymptotically stability regions. Moreover, particle swarm optimization (PSO) technique is utilized to minimize the limit cycle region. Other focus of this paper is on the stability margins analysis for robust design. For this purpose, a gain‐phase margin tester is applied to determine the minimum Gain Margin (GMmin) and Phase Margin (PMmin) when limit cycles can arise. Finally, two simulation examples are given to verify the effectiveness of the proposed approach.

Analysis and design of an asynchronous pulse‐width modulation technique for switch mode power supply

This paper presents a new mathematical approach to design an asynchronous pulse width modulation (APWM) for the switched mode power supply (SMPS). Unlike the conventional APWM that utilises a Hysteretic comparator based Asynchronous Pulse Width Modulation (HAPWM) to provide the self-oscillating property, the proposed APWM is based on a Binary comparator based Asynchronous Pulse Width Modulation (BAPWM) and a delay cell. This way, compared to the HAPWM, the mathematical analysis of the modulator is significantly simplified. This was achieved by replacing the describing function (DF) -based graphical method with a ?linearised mathematical model. The introduced mathematical approach is extended to study the behaviour of the higher order self-oscillating modulators in terms of the harmonic distortion. To confirm the effectiveness of the analytical derivations, the BAPWMs are employed in a classic DC-DC buck converter and the system-level simulation results are provided. It is concluded that there is an acceptable degree of similarity between the simulation results and the deployed analytical calculations for different orders of the modulators. Finally, to validate these analytical and simulation results and also to compare the BAPWM performance with its HAPWM counterpart, both modulators are implemented using off-the-shelf components and their measurement results are presented.

Validity of continuous tuning rules in event-based PI controllers using symmetric send-on-delta sampling: An experimental approach

One of the difficulties encountered in the tuning of event-based proportional-integral (PI) controllers using symmetric send-on-delta sampling (SSOD) is the possible appearance of a stable limit cycle, especially when tuning rules designed for continuous control loops are applied. This oscillation is a consequence of the intersection in the Nyquist map of the open loop transfer function (process + controller) with the negative reciprocal of the describing function (DF) of the SSOD sampler. As the DF-based approach is an approximated method and therefore it introduces errors in the locations of the intersections in the Nyquist map, a set of oscillation-free PI controller parameters regions generated by an experimental approach are presented in this paper. The relevant aspect of these regions is that any parameters’ couple located inside them implies a controller that can be applied for the control of first order or integrating processes without generating stable limit cycles. As these regions constitute a resource to check whether a continuous tuning rule can be safely applied in an event-based control loop, a study of the validity of some PI tuning rules is provided in the paper. Also, the regions provide a framework for the design of tuning rules specifically formulated for event-based PI controllers and an example of that is presented in the paper. Finally, it is analyzed whether the classical sensitivity functions can be used as a tool to check and design tuning rules for SSOD-PI controllers instead of the experimental regions.