Describing Function Method Research Articles

Fractional-order lead networks to avoid limit cycle in control loops with dead zone and plant servo system

Abstract The fractional-order controllers (FOCs) have recently had a significant impact on control applications. However, they still need further research for feedback systems with hard nonlinearities, such as dead zones. The above compelling evidence motivates the design of a new robust FOC to avoid limit cycles caused by dead zones in the control loops. The proposed FOC consists of the cascade of two shifted in frequency, fractional-order lead networks. They provide high-value and sufficiently flat phase leads in sufficiently large frequency intervals. In this way, the linear part of the control loop can be easily shaped to achieve avoidance of limit cycles. The article applies classical concepts, such as the Nyquist plot and describing function method, to derive guidelines for designing the free parameters of the FOC. Moreover, a realization algorithm and a parameter setting procedure make the new FOC easily implementable in engineering practice.

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.

Force-Limited Control of Wave Energy Converters using a Describing Function Linearization

Actuator saturation is a common nonlinearity. In wave energy conversion, force saturation conveniently limits drivetrain size and cost with minimal impact on energy generation. However, such nonlinear dynamics typically demand numerical simulation, which increases computational cost and diminishes intuition. This paper instead uses describing functions to approximate a force saturation nonlinearity as a linear impedance mismatch. In the frequency domain, the impact of controller impedance mismatch (such as force limit, finite bandwidth, or parameter error) on electrical power production is shown analytically and graphically for a generic nondimensionalized single degree of freedom wave energy converter in regular waves. Results are visualized with Smith charts. Notably, systems with a specific ratio of reactive to real mechanical impedance are least sensitive to force limits, a criteria which conflicts with resonance and bandwidth considerations. The describing function method shows promise to enable future studies such as large-scale design optimization and co-design.

Describing function of the nonlinear dynamics of viscous unsteady lift response for a pitching airfoil

In recent years, there has been a growing interest in low-Reynolds-number, unsteady flight applications, leading to renewed scrutiny of the Kutta condition. As an alternative, various methods have been proposed, including the combination of potential flow with the triple-deck boundary layer theory to introduce a viscous correction for Theodorsen's unsteady lift. In this research article, we present a dynamical system approach for the total circulatory unsteady lift generation of a pitching airfoil. The system's input is the pitching angle, and the output is the total circulatory lift. By employing the triple-deck boundary layer theory instead of the Kutta condition, a new nonlinearity in the system emerges, necessitating further investigation to understand its impact on the unsteady lift model. To achieve this, we utilize the describing function method to represent the frequency response of the total circulatory lift. Our analysis focuses on a pitching flat plate about the mid-chord point. The results demonstrate that there is an additional phase lag due to viscous effects, which increase as the reduced frequency increases, the Reynolds number decreases, and/or the pitching amplitude increases.

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.

Stability analysis of the Chua’s circuit with generic odd nonlinearity

This paper analyzes the stability of the Chua’s circuit with generic odd nonlinearity using two traditional analytical tools of the control theory. The first tool is based on the linear approximation of a nonlinear system for the local stability analysis around equilibrium points or manifolds using the Jacobian matrix eigenvalues, which are mapped in the complex plane using the root locus method. The second tool is an extension of linear techniques based on frequency response known as describing function method, which allows analyze effects of nonlinearities in dynamical systems and predict several nonlinear phenomena with reasonable accuracy. These two analytical tools are jointly applied to the stability analysis of an example of this class of nonlinear systems to identify and map its dynamical behavior in parameter space. Numerical investigations based on computational simulations corroborate the theoretical predictions obtained in this stability analysis.

On the Applicability of Describing Function Method

In the present paper, the problem of describing function method is justified by reducing to the procedure to search for sufficient conditions in the special class of system solvability of operator equations. These conditions are derived under the framework of the geometrical functional analysis methods. The applicability of the proposed technique is demonstrated on the study of particular systems from automatic regulation control theory.

Nonlinear impedance matching control for a submerged wave energy converter

AbstractThe impedance matching control, also known as approximate complex conjugate control (ACC), is one of the main strategies for improving the capture of energy by point absorber wave energy converters. Such a strategy shapes the mechanical impedance related to the floater dynamics via the control law. Since the traditional ACC is given by a linear control law, this work proposes a generalization denoted as nonlinear complex conjugate control (NCC) that considers the presence of nonlinear viscous damping in addition to the usual linear damping and stiffness. The energy maximization conditions for the proposed NCC are derived in the frequency domain through the describing function method. These conditions show that the ACC is a special case of the NCC when the total damping on the floater is approximated as a linear function of its velocity. From numerical simulations of a point absorber wave energy converters with nonlinear damping, which is based on the Archimedes wave swing prototype, it is shown that the NCC provides greater energy conversion than the ACC, as well as a robust performance in the presence of variations of the damping coefficient and the excitation force peak frequency.

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 Simplified Three-Order Small-Signal Model for Capacitive Power Transfer System Using Series Compensation

Capacitive power transfer is promising for highefficiency wireless charging. A small-signal model is required to evaluate the system stability under different control approaches. This paper focuses on the modeling and simplification of the highorder system using series compensation. Based on the extended describing function method, this paper proposes a series of model transformation and approximation to effectively reduce the circuit order of the resonant tank. The number of resonant components is reduced from ten to two. Combining with the inverter and rectifier model, a third-order equivalent circuit model is firstly derived for the whole system with clear physical meaning and explicit transfer function. The experiment results show the model is accurate up to 1/5 of the switching frequency.

Describing function method with pointwise balancing in time domain for periodic responses of non-smooth self-excited system

Describing function method with pointwise balancing in time domain for periodic responses of non-smooth self-excited system

Describing function method with pointwise balancing in two-dimensional regularized time domain for quasi-periodic responses

This paper proposes a new semi-analytical approach, called as describing function method with pointwise balancing (DFPB), to solve quasi-periodic (QP) responses of nonlinear dynamical systems especially those with non-smooth nonlinearities. This method is to describe the QP motion by generalized Fourier series with multiple regularized time scales. By discretizing the solution in the regularized time domain, the original equation is transformed into over-determined algebraic equations governing Fourier series coefficients via pointwise balancing. The coefficients are determined by solving the balance equations via the Newton–Raphson iteration algorithm. In addition, the Tikhonov regularization is utilized to enhance the convergence of the iteration scheme, so that the least squares solution of the over-determined equations can be obtained efficiently. On the one hand, the pointwise balancing avoids repeated transformations of non-smooth functions between time- to frequency-domain. On the other hand, the regularized time scales make it convenient to discretize the system equations regardless of unknown fundamental frequencies. With these procedures, the DFPB is well-designed to solve non-smooth systems for QP response with frequencies to be determined. In the numerical examples, the effectiveness and accuracy are elucidated in detail through both smooth and non-smooth models. The DFPB results are in excellent agreement with those attained by the Runge–Kutta method. In addition, it is worthy of pointing out the DFPB can track both stable and unstable QP responses.

Closed-Loop Frequency Analysis of Reset Control Systems

This article introduces a closed-loop frequency analysis tool for reset control systems. To begin with sufficient conditions for the existence of the steady-state response for a closed-loop system with a reset element and driven by periodic references are provided. It is then shown that, under specific conditions, such a steady-state response for periodic inputs is periodic with the same period as the input. Furthermore, a framework to obtain the steady-state response and to define a notion of closed-loop frequency response, including high order harmonics, is presented. Finally, pseudosensitivities for reset control systems are defined. These simplify the analysis of this class of systems and allow a direct software implementation of the analysis tool. This methods gives deeper insight into the performance of the system than that achieved with the describing function method.

Nonlinear Aeroelastic Characteristics of a Supersonic All-Movable Fin with Single or Multiple Freeplay Nonlinearities

Establishing the aeroelastic characteristic of all-movable fins with freeplay nonlinearities is one of the most common problems in the design of supersonic flight vehicles. In this context, this study provided novel points of view on the nonlinear aeroelastic characteristics of an all-movable fin with freeplay nonlinearities in its root. The unsteady aerodynamic model that was employed uses the second-order piston theory considering thickness effects. For a system with multiple freeplay nonlinearities, a discrete scanning method based on the describing function method was established to solve the limit-cycle oscillations (LCOs) and avoid the loss of solutions. Combining this with the time-domain integration method, the influences of the support stiffness at the root of the fin and the freeplay size ratio of the bending and torsional degrees of freedom on the dynamical response of the system were analyzed. The results demonstrate that systems with a single freeplay nonlinearity exhibit two completely different types of LCO, while systems with multiple freeplay nonlinearities exhibit complex dynamical behaviors such as LCO and quasi-periodic and chaotic motions. The path of a quasi-periodic torus breaking into chaos was observed. Furthermore, a harmonic initial condition for the time-domain integration is proposed; this can be used for a quick check of the frequency-domain calculation results.

A cascade dead-zone extended state observer for a class of systems with measurement noise

<abstract><p>For high frequency noise, a new $ 2n $-th order cascade extended state observer with dynamic dead-zone structure is proposed in this paper. Dead zone dynamic consists of two parts. One is to "trim" the effect of noise by cutting off the part that falls in the dead zone. The other part pushes the dead zone amplitude to converge to 0 as soon as possible to ensure the convergence of the estimation error. Moreover, in the cascade structure, the high-gain parameter grows only to a second power, thus avoiding excessive amplification of the measurement noise and solving numerical implementation problems. The design procedure ensures that the extended state observer is input-to-state stable. Numerical simulations show the improvement in terms of total disturbance estimation and noise attenuation. The frequency-domain analysis of the proposed ESO using the describing function method investigates the effect of the dead zone nonlinear parameter on the performance of a closed-loop system.</p></abstract>

Signal corrector and decoupling estimations for UAV control

AbstractFor a class of uncertain systems with large-error sensing, the low-order stable signal corrector and observer are presented for signal correction and uncertainty estimation according to completely decoupling estimation. The model-free signal corrector can reject the bounded stochastic disturbance/error in global position sensing, and system uncertainty can be estimated by the observer, even the existence of large disturbance in position sensing. Furthermore, a general form of signal corrector is given. The describing function method is used to analyse the robustness of the corrector in frequency domain, and the parameter selection rules are presented. The merits of the signal corrector includes its model free, gain-bounded stable structure, sufficient rejection of bounded stochastic disturbance/error in sensing and ease of parameters’ regulation. The corrector and observer are applied to a UAV navigation and control for large disturbance/error corrections in position/attitude angle and the uncertainties estimation in the UAV flight dynamics. The control laws are designed according to the correction-estimation results. Finally, experiments demonstrate the effectiveness of the proposed method.

Application of describing function method and active disturbance rejection in LFC systems with nonlinear characteristics

In load frequency control (LFC) systems, the presence of governor dead zones and generation rate constraint is a common problem which can affect the dynamic response of the system and even make it unstable. To improve the stability of load frequency control systems with governor dead zone or generation rate constraint (GRC). In this paper, an error compensation strategy is proposed in conjunction with the linear active disturbance rejection control (LADRC) technique, and then the reliability of the system is analyzed using the descriptive function method to obtain a range of values for the compensation coefficients. It is found that the error compensation strategy proposed in this paper can well recover and improve the control performance of the system, at the same time, it is feasible and effective to obtain a range of values for the compensation factor by means of the descriptive function method.

Nonlinear sliding mode controller using disturbance observer for permanent magnet synchronous motors under disturbance

We propose a nonlinear sliding mode controller (NSMC) using disturbance observer (DOB) for permanent magnet synchronous motors (PMSMs) under disturbance. A nonlinear gain, which is a function of a sliding surface, is incorporated to improve the performance of the sliding mode controller (SMC). The chattering in the proposed controller is mathematically proven to be lower than that in a conventional sliding mode controller (CSMC). In the NSMC, the sliding surface exponentially converges to zero when the sliding surface is far from zero, and then reaches zero within a finite time, similar to the behavior of the CSMC near zero. The exponential convergence property ensures stability without disturbance information. The chattering performance under the unmodeled dynamics of the PMSM is mathematically analyzed using the describing function method. A DOB is used to compensate for disturbances, such as parameter uncertainties and load torque. The closed-loop stability under the perturbation term, which is caused by the disturbance estimation error, is guaranteed without disturbance information.

A receptance based method for the calculation of nonlinear normal modes of large ordered structures with distributed localized nonlinearities

In recent years, the concept of nonlinear normal modes (NNMs) has gained interest for interpreting a broad range of nonlinear dynamic phenomena. Present study introduces a simple and efficient computational framework to compute NNMs of large order nonlinear structures overcoming difficulties associated with commonly used time-integration and discretization methods. In this paper, a Receptance Based Nonlinear Normal Mode Calculation Method (RBNM) is developed to determine the NNMs of large ordered realistic finite element models in frequency domain. Describing Function Method (DFM), which makes it possible to write the nonlinear internal forcing vector as a nonlinear displacement dependent stiffness matrix multiplied by displacement vector, is used to model the internal nonlinear forcing vector in frequency domain. A unique matrix manipulation based on dynamic stiffness and receptance concepts is used, as a result of which the number of governing nonlinear equations becomes independent from the total number of degrees of freedom of the model and it only depends on the number of degrees of freedom associated with the nonlinear elements. This brings a significant computational advantage over the classical reduced order modeling techniques. A simple 2-degree of freedom (DOF) system, the results of which are available in the literature, is used to verify the proposed method in determining NNMs. A 20-DOF lumped parameter system model and later a complex finite element model of a large structure are used to demonstrate the applicability of RBNM in determining NNMs of larger systems with several different types of nonlinear elements distributed among the degrees of freedom.

Time Invariant Static Nonlinearities of the Dynamical Systems

Abstract Dynamical technical systems are famous for large scale of time invariant nonlinearities applied inside. Some kinds of nonlinearities describe physical properties of devices used in the technical system. Some special kind of time invariant static nonlinearities are for ensure stability limiting and truncating signals inside the dynamical systems. Moreover, nonlinearity is a property of materials used both in static or dynamical induction machines. One of the widely spread and applied method to handle static nonlinearities is dynamical technical systems is the describing function method (DFM). The purpose of the author is to introduce and apply this technique to evaluate stability conditions of the automatic flight control system of the unmanned aerial vehicles (UAVs).